.

Advances in

HEAT TRANSFER

Edited by

James P. Hartnett Energy Resources Center University of Illinois Chicago, Illinois

Volume 20

Thomas F. Irvine, Jr. Department of Mechanical Engineering State University of New York at Stony Brook Stony Brook, New York

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PRIN- r lD IN Tl iE LINITlil) STATES 0 1 . AMIiRICA

YO 91 91 93 Y x 7 h 5 4 3 2 I

ADVANCES IN HEAT TRANSFER, VOLUME 20

Nucleate Boiling Heat Transfer and Its Augmentation

KANEYASU NISHIKAWA AND YASUNOBU FUJITA

Department of Mechanical Engineering Kyushu University Fukuoka, Japan

I. Introduction

The advent of high-power density systems such as the nuclear reactor, rocket, and spacecraft gave a great impetus to research in the boiling heat transfer. Although a great amount of effort has been made so far to clarify the boiling phenomena and to correlate the experimental data of the heat transfer in nucleate boiling, the results have not been satisfactory because of the complexity and irreproducibility of the phenomena. This is caused by the fact that the surface conditions (i.e., the surface roughness, the deposition of foreign materials, or the absorption of gas on the surface) become inherent factors that influence bubble generation. This complexity in the bubble generation is a basic characteristic of boiling and may also become the fundamental cause of the irreproducibility of boiling phenomena. Although the outline of boiling characteristic curves is known for various combinations of liquid and surface conditions, there is still considerable disagreement in the exact shape and location of the curves, and especially in the details concerning boiling mechanisms. Since the high heat flux densities realized in the nucleate boiling are interpreted as the result of the bubble generation, which induces the strong agitation of the liquid near the heating surface, many correlations have been formulated on the basis of the bubble agitation model.

Recently, considerable attention has been paid to using the organic fluids as the working fluids of the power cycle, which makes use of the heat sources at moderate temperature levels. This trend reflects the current situation of a deficient energy supply worldwide. However, the lack of the data for their thermal properties seems to prevent the application of

1 Copyright 0 1990 hy Academic Press. Inc

All rights of reproduction in any form reserved

2 KANEYASU NISHIKAWA AND YASUNOBA FUJITA

nondimensional correlations over the wide ranges of temperature and pressure. From this point of view, the simplified formulas, which do not include thermal properties, are required for the industrial application of nucleate boiling.

The heat-transfer coefficient in nucleate boiling depends strongly on the conditions of the heating surface. This makes it difficult to correlate the heat-transfer coefficient only in terms of liquid properties. The surface condition is the inevitable factor in correlating the heat transfer, while such factors as shape, size, and configuration of the heating surface have been treated as secondary factors and are usually omitted from the correlations. However, there seems to be considerable differences between the horizontal and the vertical surfaces in the behaviors of boiling bubbles and also in the liquid motion and void distribution near the heating surface. Conse- quently, it also becomes important to investigate in detail the effect of the surface configuration not only for the industrial application of its results but for the clarification of the mechanism in nucleate boiling.

On the other hand, the bubble behavior of nucleate boiling in a confined space is quite different from that in conventional unconfined pool boiling. It becomes an important problem to clarify the characteristics of the nucleate boiling in a confined space or in a liquid film in connection with the cooling of the nuclear reactors and various electronic devices.

To realize the utmost use of the thermal energy, it is necessary to maximize the effective temperature difference of the power cycle operating between the heat source and sink. Therefore the development of a high- performance heat exchanger is an urgent problem. The nucleate boiling heat transfer is one of the most important modes of heat exchange occurring in many constituent devices of thermal plant. Thus the augmentation of nucleate boiling heat transfer is sure to go a long way toward the efficient use of the thermal energy.

In this chapter, a correlation method of nucleat boiling heat transfer is shown by considering the factors that affect the heat-transfer processes. Then the effect of surface configuration on the nucleate boiling and the nucleate boiling heat transfer in a narrow space and in a liquid film are stated. Finally the potential measures for augmentation of nucleate boiling heat transfer and some results on them are mentioned.

11. Nondimensional Correlating Equation of Nucleate Boiling Heat Transfer

A. ANALOGY BETWEEN NUCLEATE BOILING AND FREE CONVECTION In nucleate boiling, two different driving forces are conceivable as the

cause of convection current. One is the buoyancy force W, due to the

NUCLEATE BOILING HEAT TRANSFER 3

change in liquid density, which is usually encountered in the problem of single-phase free convection, and the other is the stirring force Wb of rising bubbles, which originates from the apparent change in fluid density due to the containment of bubbles in it. Both driving forces act independently on the heat transfer in nucleate boiling and their sum gives the effective driving force W . The driving force W and the corresponding Grashof number Gr are expressed as follows:

w = wt+ w,

where

Gr = (g13/vtCt5,)W

and 6, is the thickness of thermal boundary layer; p the coefficient of thermal expansion of liquid; T, T, , and T, the local temperature, the bulk temperature, and the temperature of heating surface, respectively; N / A the bubble population density; do and f the diameter of a bubble leaving the heating surface and the frequency of bubble generation, respectively; x the product of do and f (i.e., x = d , f ) ; y the normal distance from the heating surface; V ( y ) and V, the bubble volume rising at a point y and leaving the heating surface, respectively; U ( y ) and Urn the rising velocity of a bubble at a pointy and its averaged value, respectively; g the gravitational acceleration, 1 the representative length of the heating surface; vL the kinematic viscosity of liquid; He the effective stirring length of bubbles [l] (i.e., the distance from the heating surface to the point beyond which the effect of the liquid stirring by generated bubbles disappears); and F ( y ) a function expressing the variation of the liquid stirring effect of rising bubbles due to their position from the heating surface, whose limiting values should be F(0) = 1 and F(He) = 0 as illustrated in Fig. 1.


FIG. 1. Schematic representation of a bubble agitation model for nucleate pool boiling.

Since the stirring action of bubbles may become dominant after initiation of nucleate boiling, W, in Eq. (1) is neglected in comparison with W, [ 2 ] . Thus, the first term of the right-hand side of Eq. (2) is neglected compared to the second term in nucleate boiling.

It is assumed that the same type of correlation used in the free convection of the single-phase fluid is applicable in nucleate boiling provided that the bubble Grashof number defined by the second term of the right-hand side of Eq. (2) is adopted in place of Gr in

Nu = K(Gr - Pr)m (5)

where Nu and Pr are the Nusselt and the Prandtl numbers, respectively. There exists the empirical relation between the heat-transfer coefficient

a and the thickness of thermal boundary layer 6, in nucleate boiling [3], which runs as aSi$=constant. This relation can be reduced to the nondimensional form:

NuS(6,)/1 = B, NU al/hL (6)

where hL is the thermal conductivity of liquid, and constants s and B take different values depending on whether the flow in the boundary layer is laminar or turbulent.

From Eqs. ( 2 ) , ( 5 ) , and (6),

N u = K* ( Pr-dd,-l A" 2Xy: 2 g - (7)

where

k m / ( l - sm) (8)

(9) ( Cb 4 Ct B ) K* K l / ( ' - S m )

A correlation for the heat transfer in nucleate boiling is derived as follows on the basis of Eq. (7), which includes the bubble population density.


B. FORMULATION OF ELEMENTARY PROCESSES I N NUCLEATE BOILING

In order to apply Eq. (7) to the nucleate boiling of liquids with different physical properties, it is necessary to know the values of H e , d o , x, U, , and N / A .

Assuming that the heat flux from the heating surface is ultimately carried away to vapor space as the latent heat of bubbles, the following relation holds:

where L is the latent heat of evaporation, pv the density of vapor, q the heat flux, and d , the diameter of a bubble arriving at the free liquid surface.

From the recognition that the difference in nucleation characteristics of heating surface must appear as the difference in bubble population density, heat flux q should be expressed in terms of the bubble population density N / A as well as the degree of wall superheat AT. This relationship among them is sometimes called the two-parameter expression of heat flux in nucleate boiling [3, 41:

AT = C,(N/A)"qY (11) where C, is a constant that depends on the physical properties of liquid. Numerical values of x , y, and C, change with the condition whether the boundary layer is laminar or turbulent. Figure 2a shows an example of the experimental results [5] at low heat flux on the roughened horizontal surfaces with artificial grooves, whose average depth h , is selected as the measure of roughness. The data scattering due to every roughness in plotting of q versus AT has seemingly disappeared in Fig. 2b where the data are correlated after the two-parameter expression.

From Eqs. (7), (lo), and (ll), the following relationships are derived:

(dU/4J3 = M(N/A)"I" (12)

N / A = [(6/.rrLpv)(qR-"/Md:f )I1/('+*) (13)

u = (3k - 1)/(1 - y) (14)

2 = ( k + x + y - 1)/(1- y ) (15)

M G (CqCk)1/(l-Y) (6/.?rLpv)(d3,f)(k+Y-')/('-Y) (16)

c k =A,K*[Pr (q /uV; ) (He / l ) lk (17)

where

6 KANEYASU

4

-3 Y 2 v 2 2

tl

NISHIKAWA AND YASUNOBA FUJITA

a

lo3 1 I I I I l l 1 I 3 10

A T (K)

b 15

10

c - Y

G 3 102 3 103 3 10' 3

NIA (rn-*)

FIG. 2. (a) Relationship between the heat-transfer coefficient and the temperature difference between heating surface and liquid in nucleate boiling from roughened surface with grooves. (b) Relationship among heat flux, bubble population density, and temperature difference between heating surface and liquid (i.e., two-parameter expression for heat flux in nucleate boiling).

and C, is a constant depending on the physical properties of liquid, while M is considered a constant independent of the physical properties.

In nucleate boiling, do and Urn may be given by the following correlations:

where u and pL are the surface tension and the density of liquid, respectively. Numerical constants a and b are given empirically as a = 1.04 by Fritz [6] and as b = 1.18 by Peebles-Garber [7].


If the distilled water boiling on a clean and smooth surface at atmospheric pressure is chosen as the standard, and its relevant values are distinguished by adding a subscript “s”, then the bubble of an arbitrary liquid carries away I(, times as much as energy of the distilled water at its departure from the heating surface. Hence,

+ = (d0/d0,)3( p v / p v , ) ( L / L s ) (20) If the frequency of bubble generation varies with the following relation

according to Jakob’s study [8],

f = f s / * (21) then the product of the departure diameter of a bubble and the frequency of bubble generation, x , is written as

x = do f = P/(d%VL)

P (do , fs )d~spvsLs ~sd?J‘vsLs (23)

(22)

where a constant xs was found to take a value of 0.111 m/sec in the previous study [3].

The effective stirring length of bubbles He seems to change with the kind of liquid and it is empirically related to the bubble Reynolds number Reb as expressed by the following equation

Hel l = T*/Reb (24)

Reb U,,,do/v, (25) where T* is a proportional constant. The value of T* = 1100 was obtained from the Nishikawa’s experiment made on various liquids [9] as shown in Figs. 3a and 3b.

By substitution of Eqs. (13), (18), (19), (22), and (24) into Eq. (7), a correlating equation for the heat transfer in nucleate boiling is finally derived:

(26) y = K*@X-k/(l+z)

where

and cL is the specific heat of liquid.

8

3.0

3.6

34

3.2

3 0 n

cu Y

E 2.8

Y 2.6

U

2 v

2.4

2.2

2 .o

1.8

1.6

1.4

KANEYASU NISHIKAWA AND YASUNOBA FUJITA

a

T r"

FIG. 3. (a) Relationship between the heat-transfer coefficient and the liquid level in nucleate boiling. For distilled water, (V) I = 35 mm and q = 29 kW/mz; (X) I = 50 mm and q = 20 kW/m2; (A) I = 70 mm and q = 14.5 kW/mz. For 30% sugared water, (0) 1 = 50 mm and q = 20.0 kW/mz. (b) Relationship between the effective stirring length of bubbles H, and the bubble Reynolds number Re, in nucleate boiling.

2


C. EVALUATION OF CONSTANTS

Constants and exponents in the correlating equation (26) are estimated

The value of C, is obtained by assuming the temperature profile in the as follows.

boundary layer by

( T - T,)/(Tw - T,) = (1 - r1)2 (30)

Since it is difficult to evaluate accurately the value of c b , the change in volume and velocity of a rising bubble departing from the surface is estimated from the photographic records of Jakob and Fritz [lo], which run as follows:

V(Y)/VO = 1 + 2(Y/W2

U(Y)/Um = (2/3)[1 + (Y/K)I

F(Y) = 1 - (Y/He)

(32)

(33)

(34)

By using the following linear function of F(y) [ l l ] ,

then Cb can be evaluated as

As for the constants and exponents in the two-parameter expression [Eq. ( l l ) ] , Nishikawa's results [3] are applicable when the flow in the boundary layer is laminar:

= -1 6, y = 9 , c, = 0.0412 K(m2/W)2/3(l/m2)'/6 (36)

x = - 4 , y = ' 5, C, = 0.0987 K(m2/W)3/5(l/m2)'/5 (37)

For turbulent flow, the result given by Zuber [12] may be used:

Although the values of x and y do not change with the liquid, C, is the value relevant to the distilled water at atmospheric pressure. In order to evaluate M , these values are substituted into Eq. (16).

In Eq. (6), representing the relationship between the heat-transfer coefficient and the thickness of thermal boundary layer, the exponent and constant are determined from the experimental results shown in Fig. 4:

laminar flow region:

s = 1, B = 3.22 (38)


FIG. 4. bound a ry

.- 6 t l I

Relationship between the heat-transfer coefficient and the thickness layer in nucleate boiling.

of thermal

turbulent flow region:

s = 3, B = 0.292 (39) Lippert-Dougall [ 131 used the thickness of the equivalent conduction layer as S, , while Yamagata et al. [3] adopted the conventional thickness of thermal boundary layer. Both data confirm the validity of the relationship expressed by Eq. (6). The above values of constant B and exponent s are evaluated by using the conventional thickness of the thermal boundary layer.

The constants and exponents evaluated from the procedure examined in this section are summarized in Table I, where the subscripts “1” and “t” refer to the laminar and turbulent flows, respectively. Thus, the correlation for the heat transfer becomes as follows:



TABLE I VALUES OF CONSTANTS AND EXPONENTS IN EQ. (26)

Laminar Turbulent

K 0.56 0.13 m 4 f S 1 f

k = m/(l - sm) f 3 X -3 -4 Y 4 3 u = (3k - 1)/(1 - y ) 0 4 z = ( k + x + y - 1)/(1 - y ) -4 -3 n = (1 + z)/(l + 2 - k ) K' 0.328 0.105 4 14.08 23.89 M 805 10540

B 3.22 0.292

3 5

[ m - ' = M (- f)1 [ m - 3 / 2 ( - ~ , ) 1

P(in watts) 1 .0 1.0


Y = 3.58X4/', 2~ [ (1 /Mt2P)(cLp2,g /AL~~LpV)1l /*ql (41)

D. GENERALIZED CORRELATION FOR HEAT TRANSFER

In the correlation derived previously by Nishikawa and Yamagata [l], the following two empirically determined values were commonly used without distinction of the laminar and turbulent flows:

P = 1 . 9 7 6 W, M = 9 0 0 m-l (42)

The measured value of do = 0.00363 m by Yamagata et al. [3] was used to evaluate the value of P in Eq. (42). Since this value loses consistency with Eq. (18), the calculated value from Eq. (18) is used for evaluation of P in Table I . Values M I and M , in Eqs. (40) and (41) for the laminar and turbulent flow regions, respectively, differ dimensionally as well as numerically, and they are separately calculated from Eq. (16) for both regions.

To correlate the experimental data according to Eq. (40) or (41), it is necessary to make sure which region the data points fall in because the definition of X is different between the laminar and turbulent flow regions. Since this discrimination is difficult in practice, it is convenient to use the


numerical values of Eq. (42) as P and M, regardless of the laminar and turbulent flow regions. Then the correlation for the heat transfer in nucleate boiling can be put in the following unified form, in which the same coordinates are used in correlating the data for both regions:


Y = 6.24X2l3 (43)

(44)

(45)

turbulent flow region: y = 0.661-2/5~4/5

where x= [(1/M2P)(CLp2Lg/hLaLp")l 1/2 q 12/3

and the transition Nusselt number yt from the laminar to the turbulent flow region is given by

yt = 4.71 X 10512 (1 in meters) (46) Since the units of MI and M, are m-l and m-3/2, respectively, and the unit of M in Eq. (42) is m-', the value of 0.66 in Eq. (44) becomes a dimensional constant with the unit of m2/5 because of using X instead of X .

E. NUCLEATION FACTOR Since the correlations (43) and (44) are derived only for the clean and

smooth surface, a modification is necessary when they are applied to a heating surface having different nucleation ability from the clean and smooth surface. The difference in the nucleation ability of a surface comes out phenomenally as a difference in the bubble population density and it causes a change in the proportional constant C of the boiling characteristic curve,

q = C A T " (47)

where the following relation is obtained for the exponent n from the comparison of Eqs. (26) and (47):

n = (1 + z)/(l + z - k ) (48)

Hence the nucleation factor fs may be defined by

LY = Co( fLAT)"-'

q = Cof a-' AT"

(49)

(50)


where C, is a proportional constant for the clean and smooth surface (fs = l ) , and its value changes with liquid. Thus, fs becomes a function of the nucleation ability of the surface alone. From Eq. (49)

a = C p ( f s 4 ) ( n - l ) / n = C,!/n(fsq)W(l + z ) (51)

( A T ~ / A T ~ - ’ ) (52) fL = ( a / a s ) n / ( n - l ) =

Consequently,

where subscript “s” refers to the values for the heating surface of fL = 1. Furthermore, the nucleation factor fs is related to the bubble population

density ( N / A ) [14]:

fs = [ (N/A) / (N/A)s ] -”n / (n - l J (53)

TABLE I1

DATA IN FIG. 5 AND VALUES OF NUCLEATION FACTOR f?

Representative length of

Heating heating surface Key Liquid surface (m) fi Observer

IZ Water HW D = 6.096 X 0.59 Addoms CII Water HT D=6.94X 1.00 8 Ethanol HT D = 6.94 x 1 0 - ~ Borishanskii [XI Ethanol HT D = 4 . 9 9 ~ @ Water HP R = 4.763 x lo-’ 1.56 x Benzene HP R = 4.763 x lo-’ 1.30 Cichelli-Bonilla €3 n-Heptane HP R = 4.763 x lo-’ 1.40 N Water HT D =3.81 X lo-’ 3.38 83 Methanol HT D = 3.81 x lo-’ 3.95} Cryder-Finalborgo Q Carbon tetrachloride HT D = 3.81 X lo-‘ 3.82 D Water HP R = 2.54 x lo-’ Q Water HP R = 2.54 x lo-’ ::::) Gaertner A Water HP R=2.54X lo-’ 0.70 (D Water HP R = 5.0 X lo-’ 1.00 Jakob-Linke + Water HP R = 2.54 x lo-‘ 1.50 Kurihara 0 Water HP R = 7.0 X lo-* 0 Water HP R = 5.0 x lo-’ ::::) Nishikawa 0 Ethanol HP R = 5.0 X lo-’ 1.27 Q Water HP R = 1.886 X lo-’ 1.70 Raben ef al.

a HW, HT, and HP are horizontal wire, tube, and plate, respectively; D and R are diameter and radius, respectively.


Exponent -xn / (n - 1) becomes 4 for both the laminar and the turbulent flow regions. Therefore the nucleation factor fc has a physical meaning of the fourth root of the ratio of the bubble population density on the surface in question to that on the clean and smooth surface.

The correlation, which accounts for the nucleation ability of heating surface, is obtained by using f cX instead of X in Eqs. (43) and (44).


Y = 6.24(fcX)2'3, Y 5 Yt (54)

Y = 0.661-2'5( fcX)"/', Y 2 yt (55)


The available experimental results [9, 15-22] up to the high heat flux at atmospheric pressure are plotted in Fig. 5 by the relation between Y and f sX . Estimated values of nucleation factor are listed in Table 11. It is clear from Fig. 5 that the experimental data are well predicted by the correlations (54) and (55).

104

lo3

102

10'

100 lo-' 100 10' 102 lo3 1oL

frx FIG. 5. Correlation of experimental data in nucleate boiling at atmospheric pressure in the

Y-fiX coordinates.


FIG. 6. Relationship between the surface factor C,, and the nucleation factor fs in nucleate boiling from heating surfaces with various surface conditions.

Rohsenow [23] proposed the following correlation, in which the well- known surface factor C,, is included to account for the condition of heating surface:

where pL is the viscosity of liquid. The exponent s has the constant value of 0.33, while the exponent r ranges from 0.8 to 2.0 and is usually taken as 1.7. Figure 6 shows the relation between Csf andfg evaluated from the data for various combinations of heating surfaces and boiling liquids by many investigators [3, 9, 10, 15-22, 23-34]. There exists a close relationship between C,, andfg, and both factors represent well the nucleation ability of heating surface. Nucleation factor fg is just like the emissivity in radiation heat transfer and it is important to find out the unified rule on this factor. For that purpose the data for various surface conditions should be accumulated in future.

F. PRESSURE FACTOR

As seen in Fig. 5, the correlations (54) and (55) reproduce well the heat-transfer coefficient measured for various liquids at atmospheric pressure. But the data for pressures more or less than the atmospheric pressure shift in parallel with the lines predicted from the correlations, making the pressure a parameter on the Y-Xcoordinates. One possible reason for this shift of the data may be due to the empirical relations that the individual


elementary process of nucleate boiling has introduced into the correlations, but their validity was not examined over a wide range of pressure because of the scarcity of available data. On this account, for example, the factor M is treated as a constant but it may become a function of pressure because it is closely related to the growth rate of a bubble. In order to revise those inadequacies on the effect of pressure, a pressure factorf, was introduced into the correlations, andfpXwas used in place of X. Then the correlating equations are expressed as

Y = 6.24(f,fpX)2/3, Y 5 yt (57)

Y = 0.661-2/5(fsfpX)4/5, Y 2 yt (58)

The pressure factor should be evaluated from the data over a wide range of pressure in experiments, while keeping the heating surface under the same condition. Taking special care of this point, Fujita and Nishikawa [35] measured the heat-transfer coefficient in nucleate boiling of various liquids on a specified heating surface over a wide range of pressure. For the specified liquid boiling on the specified heating surface, the correlations (57) and (58) can be reduced to the following form by combining all the independent terms of pressure into one constant C* and by denoting the exponent with (n - l) /n:

y = C*(fpX)(n-l)/n (59)

Let the pressure factor and the value of Y at the pressure in question be denoJed byf, and YO,,, respectively, and those at the reference pressure by fpa and Yo,po, respectively, then the following equations are obtained corresponding to Xo (i.e., the arbitrarily fixed value of X ) :

Yo,p = C*(f,Xo)(n-l)/n (60) yo,po = C*(f,"XO)

yo,p/yo,p, = ( fp/fpJ

(61)

(62)

(n- l ) /n

By eliminating C* and Xo from these equations ( n - l ) / n

A plot of Yo,p/Yo,po versus the reduced pressure p/pc is shown in Fig. 7, where the data are from the present authors and others [20-22, 36, 371, and pc denotes the critical pressure and the reference pressure po is selected as pc/lOO for convenience. The reduced data points for different liquids lie on a single curve over the entire range of pressure. If the variation of Yo,p/Yo,p, withplp, is represented by the curve shown in Fig. 7, the following expression of fp , with c as a numerical constant, is obtained by considering the Eq. (61):


FIG. 7. Variation of Y(,.p/Yo,po versus reduced pressure p / p c in nucleate boiling.

A final expression of the pressure factor is determined as follows from the condition that the factor is unity at atmospheric pressure p s (i.e., fp' = 1)

fp = ( m 0 ~ 7 { ~ 1 + 3(p/pc)31/[1 + 3(ps/pc)3~> (64)

For liquids with the critical pressure higher than 1 MPa, Eq. (64) is approximated by

f p = (P /Ps )" .7 [1 + 3(P/PJ31 (65)

Furthermore, at the pressure of less thanpJ10 for these liquids, Eq. (65) is further simplified as

f p = ( P / P s ) 0 . 7 (66)

The heat-transfer coefficient in nucleate boiling is predicted by the final correlations (57), (58), and (64), which account for the effect of the surface condition by the nucleation factor fs and the effect of pressure by the pressure factor fp . Figure 8 compares the predicted curves and the experimental data of the present authors [35] on the Y-fsfpX plane. Values of the nucleation factor are given in Fig. 8. It was found that the available data measured by some other investigators [21, 22, 36, 371 are also


FIG. 8. Comparison of the proposed correlations, Eqs. (57) and (58) , with experimental data in the Y-f,f,Xcoordinates.

predicted by the proposed correlations with an accuracy of less than +30% [35], and such an agreement confirms the validity of the proposed correlations.

111. Simplified Formula for Heat Transfer in Nucleate Boiling Based on Thermodynamic Similarity of Thermal Properties

Since the nucleate boiling is a complicated phenomenon accompanying a phase change from liquid to vapor, the heat transfer is affected by many physical properties of liquid and vapor as well as the condition of the heating surface as seen in the nondimensional correlations in Section 11. A nondimensional correlation is very useful in predicting the heat-transfer coefficient for various liquids over a wide range of pressure, if only the physical properties are available. For many liquids, however, the systematic information on physical properties is lacking or available for only a limited range of pressures and temperatures. Thermal properties of freons, for example, are not well known especially at the higher temperature


range. This is because freons have been used so far as the working fluids in a refrigeration cycle. Recently, special attention has been paid to their use as the working fluids of power cycles that utilize heat sources at moderate temperature level or as the cooling medium of electronic equipments. In connection with those applications, it becomes important to clarify the heat transfer in nucleate boiling of freons. From this point of view, lacking physical properties at higher temperature for most freons, a simplified formula that enable the designer to predict heat-transfer coefficient without use of physical properties is very desirable. Such formula is derived as follows on the basis of the thermodynamic similarity of thermal properties. Of course the formula is useful for other liquids whose properties are not well known or complied.

A. CORRELATION DERIVED FROM THERMODYNAMIC SIMILARITY

The nucleate boiling is usually affected by the physical properties of liquid and vapor under the saturated condition. Due to the theory of thermodynamic similarity [38, 391 an arbitrary physical property 5 on the saturation curve can be expressed in terms of the critical constants and the reduced pressure as follows:

5 = f i b 7 P c 9 T c 3 Ro) * fZ(P/PC) (67)

where m is the molecular weight, pc and T, the critical pressure and temperature, respectively, Ro the universal gas constant, and fl and f2 the functions taking different forms for every property.

As far as the heat transfer from the heating surface with an fixed condition is considered, the heat-transfer coefficient a depends on the heat flux q and the physical properties. Hence, it can be expressed as the products of exponential functions of these quantities:

cr = Cfmm'T~p~R~F(p/p,)qY (68)

The exponent of heat flux y is known to be (n - l ) /n from Eq. (51). The other exponents in this equation are determined from the dimensional analysis:

ml = -1/2n, m2 = (1 - 2n)/2n, m3 = l / n , rn4 = 1/2n

Substitution of these exponents into Eq. (68) yields a = C f R ~ / ( 2 n ) r n - 1 / ( 2 n ) T ( 1 - 2 n ) / ( 2 n ) 1 l n

C P c F(P/Pc)@ - lYn


where, as given in Table I, n = 3 for the low heat flux region and n = 5 for the high flux region. Let a subscript “L” refer to the low heat flux region and “H” to the high heat flux region, Eq. (69) becomes as follows:

low heat flux region:

aL = CL(p:/’/m’/’ T:l6) FL (p/pC)q2/’ (70)

a~ = C~(p~’5/mL’10T~’10)F~(p/pc)q4/5 (71)

high heat flux region:

where CL and CH are the constants that are to be determined empirically from the experimental data.

B. EFFECT OF PRESSURE

In Section I1 it was confirmed that the proposed correlations (57) and (58) reproduce well the available experimental data for various liquids over a wide range of pressure. To express the pressure dependence of the heat-transfer coefficient predicted by those correlations, the effect of the surface conditions must be removed from the correlations. Since the heat- transfer coefficient is proportional to the 2 / 3 power of the heat flux in the low heat flux region and to the 4/5 power in the high heat flux region, this removal is done by making the ratio of the quantity, a/q2I3 or a/q4Is at the pressure in question, p , to (a/q2’3)p, or ( c ~ / q ~ / ~ ) ~ , ~ at the reference pressure p o . This ratio was calculated for various liquids and plotted against the reduced pressure in Figs. 9a and 9b where po is selected as pJ100. The data points gather to form a single curve on both figures. Thus the functions of FL(p /pc ) and F H ( p / p c ) are determined irrespectively of liquid as follows:


FH(p/pc) = + 2 ( p / p ~ ) 2 + 8(p/p~)81 (75)

These relationship were proved to be also applicable to freons [40].

NUCLEATE BOILING HEAT TRANSFER

lister

E t h a n o l

o e n z e n c

r e t h a n e

E t h a n e

P r o p a n e

n - B u t a n e

n - P e n t a n e

21

1 o3 102 10' 1 oo P / P c

I I 1 / I l l / I 1 I I I I I I I I I I I I

I I

P / P,

FIG. 9. (a) Effect of pressure on nucleate boiling heat transfer. (Comparison of Eq. (54) for the low heat flux region with data from various sources.) (b) Effect of pressure on nucleate boiling heat transfer. (Comparison of Eq. (55) for the high heat flux region with data from various sources.)


C. SIMPLIFIED FORMULA FOR HEAT TRANSFER

Since the effect of the conditions of heating surface is excluded in deriving the formulas (70) and (71), the nucleation factor & mentioned in Section I1 is reintroduced to give the final form of the simplified formulas of heat transfer in nucleate boiling: low heat flux region:

(76) aL = 1140(pE/3/m'/6T,5/6)FL(p/pc)fi 2/3 q 2/3

2/3 2/3 = &LFL(P/PC)~~ 4


(77) (yH = 492(pE/5/m'/'0T9/'0 4/5 4/5

c )FH(p/pc)fL 4

where the units of a are watts per square meter per degrees Kelvin q in watts per square meter, p and pc in megapascals, T and Tc in degrees Kelvin, and m in kilograms per kilomole. The numerical constants were determined as CL = 1140 and CH = 492 to correlate the boiling data of various freons over a wide range of pressure [40].

TABLE 111 VALUES OF hL, hH, AND q, FOR VARIOUS FREONS IN EQS. (76), (77), AND (78)

P c Tc (It m

Liquid (kg/kmol) (MPa) (K) [y(y3] [y(yS] (kW/mz)

R 11 R 12 R 13 R13B1 R 14 R 21 R 22 R 23 R 112 R 113 R 114 R 115 RC3 18 R 502

137.37 120.91 104.46 148.91 88.01

102.92 86.47 70.01

203.84 187.38 170.92 154.47 200.03 111.63

4.409 4.125 3.870 3.984 3.745 5.166 4.988 4.832 3.444 3.414 3.263 3.120 2.783 4.076

471.15 384.95 302.05 340.15 227.49 451.65 369.35 299.05 551.15 487.25 418.85 353.15 388.45 355.35

4.87 5.76 7.07 6.10 9.12 5.58 6.72 8.21 3.69 4.13 4.69 5.41 4.61 6.22

1.59 1.91 2.37 2.07 3.09 1.75 2.12 2.61 1.27 1.42 1.63 1.90 1.66 2.06

44.6 40.2 36.0 32.9 32.9 59.2 56.4 54.6 31.0 30.1 27.9 25.8 21.2 39.8


The heat flux at the transition point qt between the low and high heat flux regions is derived by equating L Y ~ to a H :

1140 ''I2 p c T r / 2 F L ( p / p c ) ''/* 1 " = ( = ) m1/2 [ F H ( p / p C ) l 5

= OtFt(P/PJ (l/fL) (78) For convenience of the practical use of the simplified formulas, numerical values of iiL, (YH and Ot for various freons are given in Table 111. The calculated heat-transfer coefficients by the formulas (76) and (77) agree well with the experimental data by many investigators [41-471.

IV. Effect of Surface Configuration on Nucleate Boiling

In nucleate boiling, the experimental data for heat transfer have usually been correlated according to the hypothesis that the nucleate boiling is a local phenomenon and the bubble behavior near the heating surface has a dominant role in heat transfer. Furthermore, the heat-transfer coefficient in nucleate boiling depends markedly on the condition of the heating surface. Consequently, such factors as shape, size, and configuration of the heating surface have been treated as secondary, and their effects have been neglected from the detailed consideration. As for the configuration of the heating surface, however, there seem to exist considerable differences in boiling behaviors between the horizontal and vertical surfaces, such as the bubble generation, growth and departure of bubbles, the motion of bubbles and incidental liquid relative to the surface, and the void fraction near the heating surface. In turn these differences may lead to significant differences in heat transfer for these two configurations. Consequently a detailed investigation on the effect of the surface configuration becomes important not only for its industrial implication but also for the understanding of nucleate boiling mechanism in various situations. This section is devoted to the effect of surface configuration on nucleate boiling heat transfer.

A. BOILING CURVES AND BUBBLE BEHAVIOR

Nucleate boiling from an inclined surface was experimented on for saturated water at atmospheric pressure by the use of an apparatus shown in Fig. 10. The heating surface was a rectangular copper plate 175 mm long and 42 mm wide and its inclination angle w from the horizontal plane was changeable without interrupting the boiling. The obtained boiling curves


0 h e a t i n g su r face

@ i n n e r b o i l i n g vesse l

0 o u t e r vessel

(?, condenser

@ thermocouples

CS condenser

a thermocouple

@ a u x i l i a r y h e a t e r

@ a u x i l i a r y h e a t e r

@I heat ing su r face assembly

@ s u p p o r t i n g tube

@g lass window

@ h e a t i n g suv face

@ copper b lock

@ Nichrome h e a t e r

0 power leads

@ glass wool

@ thermocouple leads

@ gu ide p l a t e s

FIG. 10. Experimental apparatus and details of a heating surface assembly used for study- ing the effect of surface inclination on boiling heat transfer.

are shown in Fig. l l a . An interesting feature is as follows. At low heat flux, the boiling curves shift upwards with an increase of inclination angle from facing upwards to facing downwards, which are consistent with the results by other investigators [48-541. An increase of heat flux, however, results in the merge of these curves into a common boiling curve with a slight deviation toward the opposite direction to that observed at low heat flux. This change in the effect of the surface inclination on the heat transfer is supposedly caused by the change in bubble behavior and probably due to the change in the heat-transfer mechanism.

To discuss the effect of surface inclination, the nucleate boiling region is subdivided into three regions for convenience. However, it is impossible to specify the exact boundaries between these regions at the present time.


FIG. 11. Effect of surface inclination on boiling heat transfer (experimental data)

The first one is the region at the low heat flux of less than 7 x lo4 W/m2 where the effect of inclination appears significantly. The second is the region at the middle heat flux from 7 x lo4 to 17 x lo4 W/m2 where the effect of inclination gradually decreases. The third is the region at the high heat flux of more than about 17 X lo4 W/m2 where the effect of inclination almost disappears. Bubble behaviors along the heating surface at various inclination angles were recorded from the side of the surface, and typical results for the three regions are shown in Fig. 12. Features of the bubble behavior in each region are summarized as follows.

In the low heat flux region, an increase of the inclination angle reduces the population density of nucleation sites on the heating surface while, on the contrary, it increases the bubble diameter. The decrease in nucleation sites is confirmed by the observation of the scale deposition at active nucleation sites on the surface after the boiling of water containing dissolved nickel salts. At an inclination angle of O", isolated bubbles generate almost periodically from the uniformly distributed nucleation sites. These characteristics hold true up to an inclination angle of 120". At an inclination angle of more than 150", however, a bubble grows quickly at the nucleation site immediately after its generation and then the enlarged bubble rises up in an elongated form along the heating surface. As a result


--

GI= 900

w = 1200

&J= 150'

W = 175 '

E- . --

a

t Y

--

b C

FIG. 12. Photographs of boiling behavior along the surfaces with various inclination angles: (a) low heat flux region (q = 4 . 0 x lo4 W/m2), (b) middle heat flux region. (q = 1.2 x lo5 W/m2), and (c) high heat flux region (q = 3.7 x lo5 W/m2).


the isolated bubbles disappear gradually with an increase of inclination angle. As the heat flux is raised the frequency of bubble generation increases and an interval between the generation of enlarged bubbles becomes more periodic.

In the middle heat flux region, even at an inclination angle of less than 120" some coalesced bubbles emerge locally on the heating surface although most of the surface is covered with isolated bubbles. At an inclination angle larger than 150", clusters of small bubbles can be seen between the enlarged bubbles. Thus small bubbles coexist with large bubbles in this region.

In the high heat flux region, bubble generation is so vigorous that coalesced bubbles prevail all over the heating surface at any inclination angle. Especially at an inclination angle larger than 150", large elongated bubbles generate continuously and cover almost the whole heating surface. The surface of the elongated bubble pulsates irregularly, which means the vigorous evaporation from the base of the bubble.

B. EFFECT OF SURFACE INCLINATION

The heat-transfer coefficient (Y is plotted against the inclination angle w for selected heat fluxes in Fig. l l b . As seen from the boiling curves in Fig. l l a , and also evident from Fig l l b , with increasing inclination angles up to 175", the heat-transfer coefficient increases at low heat flux, while it is almost constant, independent of the inclination angle, at high heat flux [55] . If an inclination angle is further increased from 175", the heat-transfer coefficient might drop suddenly to a certain minimum value because the bubble movement along the heating surface may be completely impeded. This characteristic has been observed also in Refs. [49,53,54].

The effect of surface inclination on heat transfer is not constant over the whole region of nucleate boiling. This indicates the difference in the heat-transfer mechanisms between the low and high heat flux regions. From visual observations of boiling behavior, the heat-transfer mechanisms at low and high heat fluxes are inferred as follows in connection with the effect of surface configuration.

In the low heat flux region, at an inclination angle of less than 120", heat transfer is mainly controlled by the stirring action of isolated bubbles. Therefore, the heat-transfer coefficient becomes higher with an increase in the nucleation sites as long as isolated bubbles prevail over the heating surface. At an inclination angle of more than 150", on the contrary, heat transfer seems to be controlled by the following two mechanisms because of the emergence of large elongated bubbles and their movement along the heating surface. One is the sensible-heat transport by the compulsory


removal of a superheated liquid layer from the surface at the time when an elongated bubble sweeps the point in question on the surface. The other is the latent-heat transport by the evaporation of a liquid film under an elongated bubble at the time when the point in question is covered with the bubble. If these two mechanisms are dominant, the heat-transfer coefficient may be not affected by either the number of nucleation sites on the surface or the nucleation characteristics of the surface.

In the high heat flux region, the flow conditions in the vicinity of the heating surface become important for heat transfer and the generation and movement of large elongated bubbles do not play an important role. Since the nucleation in the thin liquid film under a coalesced bubble controls the flow conditions in the vicinity of the surface, the nucleation characteristics of the surface seem to affect the heat transfer in this high heat flux region independent of the inclination angle.

To confirm the above consideration, the effects of surface inclination on heat transfer are compared between two surfaces with different nucleation characteristic (i.e., a smooth surface finished with No. 0/10 emery paper and a rough surface finished with No. 0 emery paper). The measured boiling curves for these two surfaces are shown in Fig. 13. At an inclination angle of less than 120", the rough surface gives a higher heat-transfer coefficient than the smooth surface, as generally observed in pool boilhg. At an inclination angle of more than 150", however, this is only true in the high heat flux region and there is no difference in heat transfer in the low heat flux region, as would be expected. These results seem to support the previously mentioned idea concerning the mechanisms in nucleate boiling from the inclined surface.

C. HEAT-TRANSFER MODEL FOR THE SURFACE FACING DOWNWARD

In the low heat flux region, the heat transfer from the surface facing downward is assumed to occur by two mechanisms of sensible-heat transport and latent-heat transport. In the subsequent analysis a rising elongated bubble is assumed to carry away the superheated liquid layer in front of it. The sensible heat removed in this manner is evaluated based on the transient-heat conduction from the heating surface to the liquid during the liquid period t, , that is, the time interval during which the heating surface contacts with the bulk liquid. After the liquid period elapses, a succeeding bubble rises up to the point in question on the surface. While this bubble passes over there a thin liquid film is formed under the bubble and it is evaporated by the heat flux from the surface. The latent heat removed in this manner is also evaluated by the transient-heat conduction through the thin liquid film during the vapor period, t, , that is, the time interval during which the heating surface is covered with a rising bubble.

NUCLEATE BOILING HEAT TRANSFER

106-

- N E

2 lo5 -

10'~

b

29

I 1 1 , , , , - 3

- w= 120' - :SMOOTH SURFACE - 0 : ROUGH SURFACE -

- - - '1; - 8

Q r j 10 30

a AT ( K )

C AT (K) d AT ( K )

FIG. 13. Effect of surface roughness on heat transfer in nucleate boiling from the surfaces with various inclination angles.

1. Sensible-Heat Transport

After the superheated liquid layer has been completely swept away by the preceding bubble, the bulk liquid of saturation temperature T,,, comes in contact with the heating surface at constant temperature T,. As shown in Fig. 14a, the boundary and initial conditions for the one- dimensional heat conduction from the surface to an infinite bulk liquid are

30 KANEYASU NISHIKAWA AND YASUNOBA KANEYASU NISHIKAWA AND YASUNOBA

T

BULK L IQUID

X

FUJITA

a

T T

(i) 6 < 6' (ii) 6 > 6'

b FIG. 14. Temperature profiles near the heating surface facing downwards during (a) liquid

period and (b) vapor period.

as follows:

x = O , T = T w

x = a, T = Tsat

t = 0 , T = T w a t x = O a n d T = T s a , a t x > O


where x is the distance measured from the surface into the bulk liquid. Then the time-averaged heat flux q1 during the liquid period tI and the corresponding heat-transfer coefficient aI are obtained as

= %/AT (80)

where A L and aL are the thermal conductivity and the thermal diffusivity of liquid, respectively, and AT the temperature difference between the heating surface and the bulk liquid. The transient temperature profile during the liquid period is illustrated in Fig. 14a.

2. Latent-Heat Transport

After a rising bubble sweeps away the bulk liquid in contact with the surface, the surface is covered with this bubble. The thin liquid film is assumed to remain under the bubble and to have the same temperature profile as that at the end of the liquid period. This temperature profile becomes the initial condition for transient conduction during the vapor period, which is approximated in a linear profile as shown in Fig. 14b. Then the boundary and initial conditions for the vapor period are as follows:

Case 1. When the liquid film is thinner than the thermal layer at the end of the liquid period:

x = Q , T = T ,

x = 6, T = T,,,

t = 0 , T = T, - ATx/JraLtl

Case 2. When the liquid film is thicker than the thermal layer at the end of the liquid period:

x = Q , T = T ,

x = 6 , T = T,,,

t = Q , T = T , - A T x / G for x < 6 '

T = Tsa, for 6 ' < x < 6 '

where 6 is the thickness of liquid film and 6' the thickness of the superheated liquid layer defined in Fig. 14b. The time-averaged heat flux qv during the vapor period and the corresponding heat-transfer coefficient a,

32 KANEYASU NISHIKAWAA AND YASUNOBA FUJITA

are obtained for both cases:

Case 1.

q v = 1 t" ~of'(-*L$g x = o ) d t

I i I I I l l i 1 i I I T

GI

0 120' A 150'

FIG. 15. Void fraction measured at 0.5 mm above the surfaces with various inclination angles.


The values of periods tl and t , and liquid film thickness S under an elongated bubble are to be evaluated in order to calculate heat fluxes q1 and qv due to sensible-heat transport and latent-heat transport.

3 . Liquid and Vapor Periods

Periods during which the heating surface is either in contact with the bulk liquid or covered with an elongated bubble were measured at 0.5 mm above the center of the heating surface using an electric prabe, which detects vapor and liquid phases. Assuming that the bubble and liquid slugs move at the same velocity in the vicinity of the heating surface, then following time ratio of the measured vapor period to the measured total period gives a void fraction $:

* = 2 t , / p tl + z t,) (83) The void fraction measured in this way is shown in Fig. 15. It becomes higher with increasing inclination angle and heat flux.

The values of measured periods tl and t , are distributed so widely that the data are treated statistically to find their distribution characteristics. The averaged periods and 7, determined like this are shown in Fig. 16 [55]. The rising frequency of a bubbles is given by an inverse of the liquid and vapor period. It decreases with increasing inclination angle and becomes extremely high in the high heat flux region, which corresponds to the occurrence of vigorous boiling.

4. Liquid Film Thickness

Since it is difficult to measure the liquid film thickness under a boiling bubble rising along the heating surface, measurements were done for the liquid film formed between the nonheating surface and an air bubble rising along it.

A pair of electrodes of 1 mm in diameter is embedded in an acrylic plate at a distance of 4 mm apart. The electric resistance between them was measured when the injected air bubble was passing over them [56]. Then


b

v 120' 150'

FIG. 16. Average liquid (a) and vapor (b) periods for bubble generation in boiling from the surfaces with various inclination angles.

TABLE IV

LIQUID FILM THICKNESS FORMED BENEATH AN AIR BUBBLE RISING ALONG THE HEATING SURFACE"

~~~~ ~

Thickness of liquid film (pm) Inclination angle

w(deg.) Maximum Average Minimum

120 473 401 273 150 178 145 112 165 101 78 53 175 64 44 24

Experimental conditions: electric conductivity of water, 6200-49,600 n cm; volume of air bubble, 2, 4, and 8 cm3.


the film thickness was determined from a calibration curve between the electric resistance and the film thickness. Table IV gives the measured results, on which bubble size and frequency had no appreciable effect.

5 . Results of Analysis

Equations (79) to (82) are evaluated by substitution of measured data on the liquid period t l , the vapor period t,, the film thickness 6, and the temperature difference AT. Figures 17a and 17b show the heat-transfer coefficients aI and a, during the liquid and vapor periods. As evident from

FIG. 17. Results of calculation based on a proposed model for boiling heat transfer from the surfaces with various inclination angles. (a) Heat-transfer coefficient E l , (b) mV, (c) relative contribution of latent-heat transport, and (d) calculated time-averaged heat flux qc .


Fig. 17, an inclination angle has a large effect on a”, while aI is rather insensitive to the variation of an inclination angle. Thus the observed effect of surface configuration on heat transfer is concluded to be mainly due to the latent-heat transport from the surface to the rising elongated bubbles.

Since the heating surface is alternately in contact with the bulk liquid and the rising bubble, the calculated time-averaged heat flux qc is expressed in terms of by q l , qv, and 1(1 as follows:

The second term on the right-hand side represents the heat flux due to the latent-heat transport, and this contribution becomes more predominant at larger inclination angles.

Good agreement between the calculated heat flux qc and the measured one qM in Fig. 17d supports the validity of the present theoretical approach on the mechanism of heat transfer from the heating surface facing downwards.

V. Nucleate Boiling in Narrow Spaces

It is often the case in industrial applications that boiling space is restricted to the same size as or less than that of a bubble. But systematic study on the effect of the boiling-space dimension on boiling heat transfer is scarce and there is no unified interpretation on this issue. This section discusses the saturated boiling heat transfer in narrow confined space.

A. BOILING BEHAVIOR IN NARROW SPACES

As the result of observing the boiling behavior within a vertical narrow annulus, which consists of a cyclindrical copper heater and a concentric unheated glass tube, two boiling regions are distinguished by the difference in their bubble behavior [57]. One is an isolated bubble region where many small spherical bubbles generate on the heating surface, depart, and rise from the heating surface. The other is a coalesced bubble region where large coalesced bubbles generate to fill up the narrow space, then depart and rise regularly at a low frequency. The heater size used in this experiment, at atmospheric pressure, was 80 mm in diameter and 304 mm in height, and the unheated glass tube was changed to make the clearance of boiling space from 1 to 20 mm. For the pressurized experiment another heater of 50 mm in diameter and 152 mm high was used and the clearance was changed from 0.6 to 2.0 mm.

When the distilled water, at atmospheric pressure, was boiled at the clearance of 3 mm or more, many small spherical bubbles generated from


FIG. 18. Boiling behavior in narrow spaces (at atmospheric pressure within an isolated bubble region). (a) Clearance is 5.04 mm, liquid is distilled water, q = 47500 W/m2, a = 7090 W/m’ K, A T = 6.70 K. (b) Clearance is 5.04 mm, liquid is 15 ppm sodium oleate aqueous solution, q = 47500 W/m’, a = 7940 W/m2 K, AT = 5.99 K.

the heating surface. Their diameters were 3 to 3.5 mm on an average at the center of the heating surface. Figure 18 shows the typical behaviors of bubble generation in the isolated bubble region.

When the clearance was reduced to less than 3 mm, isolated bubbles were no longer observed. The space filled with large coalesced bubbles, as seen in Fig. 19, and the behavior was quite different from that of the isolated bubble region, shown by Fig. 18. At the clearance of 0.97 mm, for example, one of these coalesced bubbles extended instantaneously to a height of -30 mm and momentarily covered the whole heating surface. These coalesced bubbles regularly emerge one after another in the clearance at a low frequency. When a coalesced bubble is generated, a thin liquid film is observed on the heating surface covered by the bubble. Magnified photographs of the surface occupied by a coalesced bubble are shown in Fig. 20. Figure 20a shows the behavior just after the bubble generation. Liquid at the upper part of a vapor bubble is displaced upwards by the expanding bubble and the vapor-liquid interface forms a shape just like a crater. At the lower part, the interface is smooth because the buoyancy force is balanced with the expansion force. A thin liquid film can also be observed on the surface covered by the bubble. Figure 20b shows

FIG. 19. Boiling behavior in narrow spaces (at atmospheric pressure, within a coalesced bubble region). (a) Clearance is 0.97 mm, liquid is distilled water, q = 2950 W/m2, a = 2750 W/mZ K, AT= 1.08 K. (b) Clearance is 0.97 mm, liquid is 15 ppm sodium oleate aqueous solution, q = 2900 W/mZ, a = 2640 W/m2 K, AT = 1.10 K. (c) Clearance is 1.64 mm, liquid is distilled water, q = 5310 W/mZ, a = 2820 W/mZ K , AT = 1.89 K.

FIG. 20. Photographs of thin film on the heating surface under a bubble for boiling in narrow space. Clearance is 0.97 mm, the liquid is distilled water, at atmospheric pressure, q = 9300 W/mZ, a = 6160 W/mZ K, AT = 1.51 K. (a) Behavior just after bubble generation, (b) same, after a short time has elapsed, and (c) after more time has elapsed. B, liquid; III], vapor bubble (heating surface covered by thin liquid film); 0 , vapor bubble (dried out surface).


the situation after a short time has elapsed from the generation of a bubble. Some portion of the thin film has evaporated and the dried area on the surface is distinguished by a change in color to a white tone. Figure 20c shows the behavior after a longer time has elapsed. The dried-up area becomes wet again due to the rising liquid. The shape and extent of this dried area change with time and their reproducibility is poor.

In either case, when the clearance is extremely small, when the clearance is small but heat flux is comparatively high, or when the system pressure is low, the phenomena become quite different from that just described. Figure 21 shows the photograph of this situation. Namely, a coalesced bubble is generated in the boiling space first. Then, as the time proceeds the portion occupied by vapor bubbles gradually becomes enlarged and, as a result, the ring-shaped liquid band between the successive coalesced bubble becomes narrower. And finally this liquid band collapses into the considerable number of small droplets. Then the droplets adhere to both insides of the annular space and there start to evaporate. At last they are out of sight due to complete evaporation. The region where this dryout phenomena take place on the heating surface in this manner is presently called the liquid deficient region.

FIG. 21. Photographs of the heating surface at -;he liquid deficient region for boiling in narrow space. Clearance is 0.57 mm, the liquid is distilled water, at atmospheric pressure, 9 = 92200 W/m2, a = 4260 W/m2 K, AT =21.6 K.


When the clearance is kept at 0.57 mm and the system pressure is raised to 0.2 MPa and further up to 0.4 MPa, coalesced bubbles are also generated steadily at a low frequency and a thin liquid film can be observed on the surface covered by the coalesced bubbles. A further increase of pressure, up to 1.10 MPa, while keeping heat flux the same, leads to the generation of isolated bubbles in place of coalesced bubbles.

B. HEAT TRANSFER CHARACTERISTICS IN NARROW SPACES

1. Isolated Bubble Region

the isolated bubble region, whose relation is expressed as Figure 22 shows the heat-transfer coefficient a against the heat flux q in

a a 9213 (85)

This proportionality is the same as that at low heat flux in the unconfined pool boiling mentioned in Section 11. But the heat-transfer coefficient in a narrower clearance becomes higher. The effect of the clearance A R of boiling space on the heat transfer is represented as

m q 2 / 3 A R - 0 . 1 3 (86)

l o 4 3

FIG. 22. Relationship among heat-transfer coefficient, heat flux, and clearance of boiling space (distilled water, atmospheric pressure, isolated bubble region).

2

1 0'

h

Y

% s tl

1 0 2 8

FIG. 23. Relationship among heat-transfer coefficient, heat flux, and clearance (in mm) of boiling space (distilled water, atmospheric pressure, coalesced bubble region).


2. Coalesced Bubble Region

Although the bubble behavior in this region is quite different from that in the isolated bubble region, the same relationship as Eq. (85) holds true, as clearly seen in Fig. 23. But the heat-transfer coefficient in this region is remarkably higher than that in the isolated bubble region. For a clearance of 0.97mm, for example, it becomes about 3.8 times as large as the heat-transfer coefficient in the unconfined pool boiling. In Fig. 23 the data for the aqueous solutions of sodium oleate and saponin [57] are also plotted. The separation of the a versus q curves, due to a difference in surface tension, cannot be observed in the coalesced bubble region. Although the 15-ppm aqueous solution of sodium oleate has a surface tension that is 14% smaller than that of distilled water, there is no appreciable difference in their heat-transfer coefficients. No effect of surface tension is one of major characteristics in the coalesced bubble region. This

-. 15ppm sodium oleole solution

dist i l led water

'8 \

@\

-

--Coalesced. bubble Iso la ted bubble region region

103 I I 1 I I I 1 1 1 1 1 1 I I 1 I l t t l 1 10

A R (mm) 100

FIG. 24. Variation of heat-transfer coefficient due to change in clearance of boiling space (atmospheric pressure, constant heat flux). Data for a vertical copper cylinder with 80-mm diameter and 304-mm height are shown for, distilled water (0) and 15 ppm sodium oleate aqueous solution (0) (from Ishibashi and Nishikawa [57]). Data for a vertical copper tube with 19.6-mm diameter and 479-mm height and also for a vertical steel tube 21.5-mm diameter and 479-mm height are shown for distilled water (0) (from Chernobyl'skii and Tananaiko [58]).


fact implies that the bubbles are squeezed into a narrow space and, as a result, their behavior is controlled by the space dimension and the heat flux independent of the surface tension.

In the coalesced bubble region, there is a marked increase in the heat- transfer coefficient in comparison to the isolated bubble region. This difference becomes more emphasized if the heat fluxes in both regions are compared at the same temperature difference. Since Eq. (85) holds true for both region, the relationship between heat flux q and temperature difference AT is commonly expressed as

q a AT3 (87) In this equation the proportional constant in the coalesced bubble region realized at the clearance of 0.97 mm becomes 32 times as large as that in pool boiling.

The effect of clearance on the heat transfer is similar to that in the isolated bubble region but much large in the coalesced bubble region:

(y q 2 / 3 ~ ~ - 2 / 3 (88)

Dependency of the heat-transfer coefficient on the clearance is comprehensively shown in Fig. 24; plotting ct versus AR for two different heat fluxes. The data of Ishibashi and Nishikawa [57] agree well with those of Chernobyl'skii and Tananaiko [%I.

In the coalesced bubble region, the frequency of bubble emission is considered the controlling factor for the heat transfer. Figure 25 shows the emission frequency of a coalesced bubble against the heat flux q. As is clear from the figure, is unaffected by the surface tension and it increases linearly with heat flux:

N(1, cc 4 (89)

N ( l ) a qA R - 2 / 3 (90)

The emission frequency is a function of the clearance as well as heat flux:

Figure 26 shows the relationship between N(l) and q for ethanol in the narrow space with clearance of 0.97 mm. The same dependency of Nc1) on q , as given by Eq. (89), is also found for ethanol, provided that the heat flux is less than a certain level. For heat flux beyond this level, however, the emission frequency is reduced far below that estimated by Eq. (89), and the heat-transfer coefficient depends on heat flux to a smaller degree compared with that estimated by Eq. (85):

a qO.'2 (91) Since the phenomena in this situation can easily shift into the burnout,

this region governed by Eq. (91) is named the preburnout region. The

8 I I I I I I I ) I I I I I I I I I

n t

& - 1 r

2-

0 2

Liquid

Distilled water 0 0.97

a 1.64

0- 1.91

3 ppn sodium oleate solution (aq-) A 0.97 & 1.64

fb 1.91

15 ppm sodium oleate solution (aq.) 0 0.97

0. 191

I 4OOppm s a w solution (aq.) 0.97

a 1.64

2 1 o4 1 o5

q Wm') FIG. 25. Relationship among coalesced bubble emission frequency, heat flux, and clearance of boiling space.


h - I

in u

6

1

0.2 2 1 o4 8

FIG. 26. Relationship between coalesced bubble emission frequency and heat flux for boiling in narrow space. Clearance is 0.97 mm. (0) Distilled water, (A), 3 ppm sodium oleate solution (aq.), (0), 15 ppm sodium oleate solution (aq.), (U), 400 ppm saponin solution (as.), (x) , ethanol. All liquids are at stp.

heat-transfer coefficient in the preburnout region was found to be correlated in terms of the emission frequency of coalesced bubbles in the same manner as that for the coalesced bubble region. This means that the emission frequency of the coalesced bubble also controls the mechanism of the boiling heat transfer in the preburnout region.

Figure 27 shows the heat-transfer coefficient measured at pressures 0.1, 0.2, 0.4, and 1.10 MPa in the narrow space with a clearance of 0.57 mm. The data at 0.1 to 0.4 MPa are in the coalesced bubble region and their pressure dependency is expressed by

ff ,P-o.353 (92) At the highest pressure of l.lOMPa, contrary to Eq. (92), a is larger

than that at 0.4MPa. This is because the coalesced bubble region


L

2 1 o5

FIG. 27. Relationship among heat-transfer coefficient, heat flux, and pressure for boiling in narrow space (clearance is 0.57 mm for distilled water).

disappears at the pressure between 0.4 to 1.10 MPa and only the isolated bubble region prevails at 1.10 MPa. Consequently the relationship of a ~ p ' . ~ , similar to that in pool boiling, holds at these higher pressures. The reverse effect of pressure on heat transfer, as expressed by Eq. (92), is an interesting feature in the coalesced bubble region.

The frequency of a coalesced bubble is finally expressed in terms of heat flux, clearance, and pressure as

iyl) o: 4(P AR)-1'3 (93)

Figure 28 shows' the qualitative dependence of the heat-transfer coefficient on pressure. As clearly seen from Fig. 28, the effect of pressure is different in all three regions. In the coalesced bubble region, the heat- transfer coefficient decreases with increasing pressure, as indicated by Eq. (92). If the pressure exceeds a certain value, however, the coalesced bubbles disappear and isolated bubbles emerge. Then the heat-transfer coefficient asymptotically approaches the curve of a ~ p ' . ~ . If the pressure


Isolated bubble -:;;;;Un, 4 1 bubble Coalerced region izytt - region - - - - region

\Transient region

--c -< ./ . a 0~ p-0.353

. . . . . . . a 0~ pO.4

0 0

109 P

FIG. 28. Relationship between the heat-transfer coefficient and pressure at constant clearance of boiling space and at constant heat flux.

is reduced across another certain value, on the contrary, the liquid deficient region takes place for the coalesced bubble region, and the heat- transfer coefficient falls rapidly for further reduction of pressure.

To distinguish the three regions more distinctively, the boiling-region diagram is drawn for water in Fig. 29, where the pressure and the clearance

Isolated bubble region

0 Coolesced bubble rqion

I Isolated bubble region

0 L ~ q u i d deficient rq ion

fr Boundory between coalesced bubble and isolated bubble r q i o n tested

1 x

I 1 1 1 1 1 1 1 1 ( 1 , ) I , , , I 1h-I 1 1 I l l l l l l l l l , l r , l l l r d 0 4 0 6 0 8 I 2 3 4 6 8 1 0 20 30 4 0 60 80 100 200

01

0 4 0 6 0 8 I 2

region

lmlnlr.( hiihhlr

fr Boundory between coalesced bubble and isolated bubble

A R (mm)

FIG. 29. Boiling region diagram for the boiling of distilled water in narrow space (See Table W.

TABLE V

HEAT FLUX IN RELATION TO DIFFERENT CLEARANCES AND PRESSURES"'~

2870- 2675- 2840- 4490- 2790- 2790- 4690- 7675- 2440- 92575 57800 89120 75620 55240 57570 76580 58030 66520

0.1

PrpcciirP Clearance (mm)

10930- 11400- 10820- 14190- 58150 65360 65830 57100

4390- 6520- 4745- o.2 74850 89400 76060

89030 I u.4

I 1 4450-

4590- 70790

88810 78290 11 130-

92840 73040 1.1

"Heat flux in units of watts per square meters. bPortion above the solid line for the coalesced bubble region; below the solid line, the isolated bubble region.


are taken as coordinates. The boundary between the isolated and the coalesced bubble regions is represented by an inclined line with a slope of 45". In the transient region between them, isolated bubbles on the lower half of the heating surface coexist with coalesced bubbles on the upper half.

Using the following nondimensional characteristic numbers for the coalesced bubble region,

aAR aL YL PL PrL=-, - NU=- , FOE AL N ( l ) AR2' aL Pv

all experimental data are correlated with a single line in Fig. 30, which is expressed as

Nu = 200 Fo-2/3 PrL2l3 ( p L / p V ) - ' l 2 (94) where a is the heat-transfer coefficient, AR the clearance of a space (i.e., the gap between two concentric cyclinders), AL the thermal conductivity of liquid, aL the thermal diffusivity of liquid, uL the kinematic viscosity of liquid, N(l) the emission frequency of coalesced bubble, Nu the Nusselt number, Fo the Fourier number, PrL the Prandtl number of liquid, and pL and pv the density of liquid and vapor, respectively. The data by Cher- nobyl'skii and Tananaiko [58] are also well correlated by Eq. (94).

The emission frequency of coalesced bubble N ( l ) included in Eq. (94) was measured for water, sodium oleate aqueous solution, saponin aqueous solution, and ethanol. The data are correlated in terms of heat flux, clearance, and physical properties, as shown in Fig. 31, and a following correlation is obtained:

N(l) = 1.174 X lOP9q AR-3/2Prk627 (pL/pv)1.0s5 (95) where N(l) is per hour, q in watts per square meter, AR in meters, and pL and pv in kilograms per cubic meter, and the empirical constant is 1.174 X m3.'/W h. This equation is applicable to only the coalesced bubble region in saturated boiling.

C. THEORETICAL CONSIDERATIONS

The heat-transfer characteristics in the coalesced bubble region are analyzed here on the basis of a simple model.

An one-dimensional model is assumed, as illustrated in Fig. 32, where a heating surface with constant temperature T2 faces an adiabatic surface while keeping a clearance of AR. The space between them is filled with the liquid of saturation temperature 7', at the time o f t = 0. This liquid is heated


by conduction from the heating surface and then momentarily displaced by a coalesced bubble after time t has elapsed. Thereafter the liquid at temperature T, refills the space without any delay of time. These processes are repeated periodically.

It is further assumed that heat transfer occurs only when the space is filled with the liquid. This assumption is reasonable because the thermal conductivity of liquid is much larger than that of vapor and the actual time during which the liquid stays in the space is long compared to that of the bubble. Furthermore, it is assumed that the heat is transferred by heat conduction to the liquid. Since the convective heat transfer due to liquid circulation along the heating surface and the latent heat transport due to the bubble formation cannot be ignored, these effects are accounted for in the model by assuming the equivalent heat source in the liquid.

When an equivalent heat source Q exists in the narrow space, the differential equation for heat conduction is written as

' , t > 0 , A R > x > O (96) dT d2T - _ -aL,+- at ax CLPL

m \ N L n

I 1 1 I I , , , I I 1 I , 1 1 1 4 1 2 2 3 4 6 10 20 30 40 60 I00 200

Fo-1 = N(~)AR'/~L

FIG. 30. Generalized relationship of heat-transfer coefficient in the coalesced bubble region by NU-Fo-' for boiling in narrow space.


where uL, cL and pL are the thermal diffusivity, the specific heat, and the density of liquid, respectively. The initial and boundary conditions are, respectively,

T ( 0 , x ) = T1 (97)

T ( t , 0) = T2 (98)

and

(5) = o x = A R

Pressure Heating W a ) surface

Liquid Cl-ce

(nun)

0 0.97

C, 1.91

Q 2.36

0 2.70

A 0.97 1.64

& 1.91

4 2.36 0 0.97

@ 1.91

Q 2.36

9 2.70

0 0.97

IJ- 1.91

X 0.97 () 0.57 0 1.105

f) 1.575 0 2.05

@ 0.57 @ 0.57

1.105 Q 1.575

Q 1.26 + 2.14

cj 1.64

c, 1.64

d 1.64

Distilled water

3 ppm sodium oleate solution (aq.)

15 ppn sodium oleate solution (aq.)

400 ppm saponin solution (aq.)

Ethonol Distilled water

Distilled water

Distilled water

Distilled water

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.4 0.2 0.2

0.1

Venial copper cylinder. 60 nun diameter 30 nun

height

Vertical ~ p p u cylinder. 50 nun diameter 152 nun

height (Cr-plated)

Vercical stccl tube: 21.9 mm diameter 479 mm height

(99)

FIG. 30. (continued)

C

.d

3

0

c

-0

m

c B 6 b

d cz


x=o x=AR

Hea t ing w a l l

T2 c o n s t a n t temp.

T ( t , x ) llasswal T(0,x) =

X

FIG. 32. Model of unsteady heat conduction to liquid for the coalesced bubble region of boiling in narrow space.

The solution is expressed by

X xi Q A R ~ F O

2 T = + T 2 + 2 (-1)"exp

A L n=O

x { -- 'tLR2 ( (2n + 1)T exp[ ("" 1)li)2F~]

where,

A R - x QLt

AR ' A R ~ X = - Fo=-

The average heat flux q transferred from the surface to the liquid during the time interval t is evaluated by the following integration:

Since the heat-transfer coefficient is defined by a = q / ( T 2 - T I ) , the


Nusselt number becomes

=-( l -a 1 8 " 1 Fo n=O (2n + q2

+ exp[ - ( (2n Fo]])

The emission frequency of coalesced bubble iV(l) measured at the lower end of the vertical heating surface is used for evaluating the Fourier number in the correlating equation [Eq. (94)]. As stated before, the frequency at every measuring point is proportional to the heat flux, but it becomes higher at higher positions on the surface. Therefore, if Fo is defined with use of the frequency either at the lower end, at the center, or at the upper end of the heating surface, the empirical correlations of the

-[ I 0 k 8

FIG. 33. Comparison between theoretical equation based on assuming an equivalent heat source in liquid and empirical equation based on bubble emission frequency N(T, for boiling of water in narrow space at atmospheric pressure.


heat transfer to water, at atmospheric pressure, are represented as

Nu = 3.45 F o ( B ) - ~ / ~ (103a)

Nu = 1.39F0(C)-~’~ (103b)

Nu = 1.10 Fo(T)-*l3 (103c)

where B, C, and T refer to the lower end, the center, and the upper end of the heating surface, respectively. When Q is taken as 30% of the total heat transferred from the heating surface, and the emission frequency measured at the upper end of the heating surface is used, the empirical equation (103c) agrees with the theoretical equation (102) as shown in Fig. 33, which confirms the validity of the proposed theoretical analysis.

VI. Nucleate Boiling at Low Liquid Levels

Nucleate boiling at low liquid level, including the boiling in a liquid film, is often encountered in various engineering problems, such as the cooling of a high-temperature steel plate. Furthermore, the boiling in the thin liquid film becomes important when considering the mechanisms of burnout in two-phase flow. The experiment under these confinements of the boiling space is realized by lowering the liquid level, which is different from the boiling in the space restricted by the solid wall as stated in Section V. Also in this case the moving space of the generating or rising bubbles is restricted, and the boiling behavior is quite different from that of unconfined pool boiling. Furthermore, the heat-transfer coefficient is supposed to vary with the liquid level because the intensity of the stirring action of generated bubbles changes vertically. This section discusses the effect of the liquid level on boiling heat transfer and also reviews the mechanism of heat transfer in a thin liquid film.

A. HEAT-TRANSFER CHARACTERISTICS AT Low LIQUID LEVELS

Both the boiling behavior and the mechanisms of heat transfer markedly change with the kind of liquid and the liquid level. According to the experiment in which a horizontal circular heating surface was immersed in distilled water at various liquid level under atmospheric pressure and the heat-transfer characteristics were measured [59], at the liquid level over 30 mm the boiling behavior is similar to that in pool boiling. Namely the boiling water rises upward in the central portion of the heating surface and at the free surface it changes the direction outerward. Then the water flows


down along the wall of the boiling vessel and it turns again on the horizontal heating surface towards the center. Thus bulk circulation is maintained. At the liquid level below 30 mm, however, the boiling water can not keep regular circulation and generating bubbles rise straight upward above the free liquid surface, resulting in local convection. A similar change in the boiling behavior with the liquid level is observed also for other liquids. The nucleation sites increase with decreasing liquid level. At a high liquid level, the bubble diameter remains almost constant between 2 and 3 mm, independent of the position of the nucleation site on the heating surface. At a low liquid level, on the other hand, it varied from 3 to 7 mm depending on the position of the nucleation site.

As a result of the retardation of the liquid circulation by lowering the liquid level, the coalescence of bubbles occurs at the free liquid surface to form large vapor domes of hemisphere. At the liquid level of less than 3 mm, the bottom of the vapor domes contacts with the heating surface and the liquid on the heating surface is in intensive motion corresponding to the growth and collapse of the vapor domes. The heating surface is instantaneously dried out at the bottom of the dome during the growth period. Thus the heat transfer at a low liquid level should be seriously affected by the behavior of vapor domes.

Figure 34 shows an example of the change in the heat-transfer coefficient with the liquid level. The heat-transfer coefficient a becomes almost constant at the liquid levels H from 30 to 10 mm. And it increases markedly with the lowering of the liquid level to less than 5 mm. At the liquid level lower than 1 mm, the boiling, under steady condition, cannot successfully be realized for almost all liquids tested.

Physical properties and variables characterizing the heat-transfer process in a low liquid level are found in the experiments performed by varying the liquid level and heat flux for different liquids. They are combined by the usual dimensional analysis to yield the following set of nondimensional groups:

where a is the heat-transfer coefficient, q the heat flux, L the latent heat of evaporation, cL the specific heat of liquid, A L the thermal conductivity of liquid, pL the viscosity of liquid, u the surface tension of liquid, p L the density of liquid, p v the density of vapor, W the buoyancy due to the density difference between liquid and vapor, and H the liquid level.

The correlating equations were obtained as follows by the regression analysis of the measured data with use of the previously mentioned


12

10

9

- a

€ 7

ld

N

6 v 2 U

5

4

3

2

1 1 2 3 4 5 10 15 20 25 30

ti (I d FIG. 34. Effect of liquid level on the heat-transfer coefficient in nucleate boiling.

variables:

where

PLgo,25

L a Pv 0.75 1.25


FIG brass,

10

3 0.1 1 '0 100

PL 90.25 c L UL

pvl 2 5 X 2 = ~ H ( T ) ' ' ~ L o o 7 5

L

35. Correlation of the boiling heat-transfer coefficient at low liquid level Run No. I. 9 = 26 - 65 kW/m2. Run No. 11: copper, 9 = 19 - 606 kW/m2 Run No 111-1. brass,

q = 13 heating surface was a circular plate.

40 kW/m2. Run No. 111-2: copperiq = 15 - 242 kW/m2. Symbol 4indicates that the

Figure 35 compares correlations (104) and (105) with the experimental data for distilled water and ethanol at the liquid level of less than 10 mm on the horizontal circular heating surfaces of 140 and 41 mm in diameter [59]. The curve representing the data changes its slope at Xz = 2, becoming gentle for Xz < 2. This corresponds to the fact that the effect of the liquid level on heat transfer becomes relatively larger at either the lower heat flux or at the lower level, as seen in Fig. 34.

B. HEAT-TRANSFER CHARACTERISTICS IN LIQUID FILM

To clarify the mechanism of boiling in a liquid film, experiments were performed for distilled water at atmospheric pressure by using the horizontal circular and rectangular flat surfaces [60]. Size and material of heating


surfaces are 41-mm stainless steel (St), 41-mm brass (Br), and 51-mm copper (Cu) (all of which refer to circular plates); and, for rectangular plates, 30 X 50-mm Cu and 15 X 100-mm German silver (GS). Heating surface was roughened by emery paper into three grades [60]. Figures 36 and 37 show a few examples representing how boiling behavior changes with the liquid level. The pound sign in the figures indicates the grade number of emery paper used for polishing the heating surfaces. Three different liquid levels from 3 to 1 mm were tested to obtain the following conclusions.

(1) The temperature of the heating surface rapidly falls when the dried area appears on the heating surface at the bottom of the vapor dome. In the case of the heating surface of the left-hand sample [lo00 # (l)] in

9 Y 9 8 a

H (mm) H (mm) H (mm)

rrI 10

c 9

k g v " , I ",7

FIG. 36. Relationship between boiling behavior and surface temperature for boiling in liquid film. H , water level; h,,, surface roughness; AT temperature difference between heating surface and saturated liquid. Symbol 0 indicates that a rectangular plate was used as the heating surface.

FIG. 37. Relationship between boiling behavior and surface temperature for boiling in liquid film. H , water level; h,,, surface roughness: AT, temperature difference between heating surface and saturated liquid. Symbol 4 indicates that a circular plate was used as the heating surface.


Fig. 37, there is no dried area observed on the heating surface or temperature drop even when the liquid level is reduced to the minimum of 1 mm. But for the heating surface of the next sample [1000#(2)], the dried portion and the corresponding temperature drop are observed already at the liquid level of 2 mm. Similar correspondence is obtained for the other heating surfaces tested.

(2) Appearance of dried area at the bottom of vapor domes and corresponding drop of the surface temperature depend on the material and roughness of the heating surface, the liquid level, and the heat flux. The highest liquid level at which the dried portion appears at first on the surface at the bottom of vapor dome is for copper, followed by brass, German silver, and stainless steel. The drop of heating surface temperature for further reduction of the liquid level after the first appearance of dried area is also largest for copper and it becomes smaller in the same order as mentioned for the liquid level. But the heating surface temperature at the same liquid level and heat flux is highest for stainless steel and it becomes lower in the reverse order.

(3). A smoother heating surface of the same material leads to a higher surface temperature, and the appearance of dried area at higher liquid level is more likely. Temperature drop with decreasing liquid level is larger for the smoother surface. Since the reverse is true for a rougher heating surface, the difference in the heating surface temperature becomes smaller between the smooth and rough surfaces at the lower liquid level. (4) At a liquid level greater than that for the first appearance of dried

area at the bottom of domes, nucleation sites are observed on the surface, and their number is larger for a rougher surface, as in the case of pool boiling.

( 5 ) At higher heat fluxes, the liquid level becomes lower for the appearance of the dried area, and the drop of surface temperature after that is not as large as at lower heat fluxes.

Figure 38 represents an example of the change in heat-transfer coefficient caused by lowering the liquid level at different heat flux and surface roughness. The liquid level was lowered from 3 to 1 mm step by step keeping the conditions steady. Further lowering was done at a rate of -0.5 to 1 mm/min up to the minimum level at which the dryout phenomenon occurs accompanying temperature excursion, and the heat-transfer coefficient was also measured during this unsteady drying progress of the heating surface. It was found, from the unsteady experiments, that the drying at the bottom of vapor dome starts earlier than an occurrence of physical dryout phenomena and the heat-transfer coefficient reaches the maximum before the dryout. The ratio of this maximum heat-transfer


0 20 40 60 80 100

Time (s) -+- 3 2 1

H (rnm) FIG. 38. Change in the heat-transfer coefficient in nucleate boiling when the liquid level is

reduced from 1 mm at a constant rate of 1 to 0.5 mm/min, keeping heat flux and surface roughness constant.

coefficient to that at the liquid level of 1 mm is almost constant for the heating surface of the same material independent of the heat flux or the surface roughness. This result is useful for the estimation of the heat- transfer coefficient at the liquid level less than 1 mm.

It is not clear why the surface temperature drops with the drying of the surface under the vapor dome at a low liquid level. Further study is needed to answer this question.

When a vapor dome is formed on the free liquid surface, the liquid level at the bottom of the dome is reduced in comparison with the liquid level


outside of the dome. This displacement of liquid is caused by the excess pressure of the vapor dome, which balances the surface tension force acting on both sides of the dome wall. The amount of liquid level reduction at the bottom of the dome AH is given by

A H = J 8 a l ( D p L g ) (108) where D is the diameter of the dome, u the surface tension, pL the density of liquid, and g the gravitational acceleration. The liquid level equal to AH [in Eq. (108)] is called the critical liquid level H , , corresponding to the vapor dome of diameter D. For the 20-mm-diameter vapor dome formed on distilled water film at the saturation temperature under atmospheric pressure, a critical liquid level is estimated at 2.5 mm from Eq. (108), and this reduction of film thickness agrees with the observed value. The notable decrease in the heating surface temperature at low liquid level seems to occur at the liquid film of less than the critical value.

At the liquid level less than the critical level, generation of vapor domes, growth, and collapse are repeated in very complicated and multiple man- ners. Since the generation and collapse of vapor domes occur in a very short time frame, most time is devoted to the growth period. During the early stage of the growth period, a dome grows at a very high rate, mainly due to the evaporation at the bottom of the dome and the coalescence with other growing domes. After the time when the central portion of the dome starts drying out (i.e., at a latter stage of the growth period), the growth of the dome becomes retarded because coalescence is no longer taking place, and vapor is being supplied from only the partly wetted area at the bottom of the dome. Judging from this, the latter stage covers about 70% of the total growth period, a governing mechanism for boiling heat transfer at a very low liquid level (lower than the critical level), may be attributable to evaporation from a thin liquid film at the bottom of a dome during the latter stage of the growth period.

VII. Augmentation of Nucleate Boiling Heat Transfer

The heat transfer in nucleate boiling is superior to that in other modes of convective heat transfer. It also becomes one of the most useful means in the process of heat exchange. Recently, the augmentation of nucleate boiling heat transfer has drawn considerable attention in connection with the effective use of thermal energy at the low- or moderate-temperature level. The current research on this subject suggests the existence of many effective methods for the augmentation of the nucleate boiling. There are two passive methods applied to the heating surface to promote the heat


transfer in nucleate boiling. The first one treats the heating surface so as to reduce the wettability of the surface with the boiling liquid. The second fabricates reentrant cavities on the heating surface. There is, however, some limitation as to the application of the first method [61]. For example, this method does not work for liquids with small surface tension. In this section, the augmentation of boiling by using the surfaces with reentrant cavities will be discussed because it is effective, in principle, for any combination of liquids and surfaces.

A. STABILITY OF VAPOR NUCLEUS

1. Conical Cavity

In most correlations of nucleate boiling heat transfer, the surface tension of liquid exerts a negative effect. That is, the larger surface tension results in a lower heat-transfer coefficient for a given heat flux. When the surface tension is large, three characteristics can be seen: (1) the diameter of the bubble enlarges at the instance of its departure from the heating surface, (2) the surface does not become wet easily, and (3) the vapor trapped in the cavity on the surface is prone to stay there, forming a stabilized site of nucleation.

At the boundary of the three phases formed by a liquid droplet put on a solid surface, the contact angle 8 between the solid surface and the vapor- liquid interface is determined by the following equation from the balance of interfacial forces:

cos 8 = (usv - ~ . L ) / ~ L v (109)

where usv, usL, uLv are the interfacial tensions at the solid-vapor, the solid-liquid, and the vapor-liquid interfaces, respectively. Equation (109) indicates that the contact angle 8 becomes larger for liquids with larger surface tension vLV, and as a result the solid surface has a smaller chance to get wet by the liquid. This is the situation stated by the second characteristic. The first characteristic can be confirmed easily because the Laplace constant includes the square root of uLv . Then there follows an examina- tion of the third characteristic, which is related to the stability of the small amount of vapor trapped in a conical cavity on the surface as shown in Fig. 39. The vapor-liquid interface in the conical cavity with the apex angle 24 has the following radius of curvature R:

sin 4 sin [ 6 - 4 * (n/2)]

R = X

(a) +: 8 < n / 2 + 4 (b) - : 8 > ~ / 2 + 4


a l i q u i d

Pm

b liquid

Pm

FIG. 39. Stability of the vapor trapped in a conical cavity: (a) easily wetted with the liquid and (b) poorly wetted with the liquid.

where x is the length of the generatrix from the apex to the interface. The coefficient of x in Eq. (1 10) is positive for both ranges of 8. The difference between the vapor pressure pv and the liquid pressure p.. is:

(a) + : 8 < ~ / 2 + + (b) - : 8 > ~ / 2 + +

If, for whatever reason, the vapor reduces its volume, as in the case of Fig. 39a, that is, if the length of x starts to decrease, then the radius of R also decreases. The vapor pressure then increases according to Eq. (lll), and the corresponding saturation temperature becomes higher. If its temperature is lower than the saturation temperature, the vapor will condense releasing the latent heat. The resulting loss of the mass of vapor will cause a reduction of its volume to some extent. This process will then go repeatedly unless a new thermodynamic equilibrium is established due to the heating by liberated latent heat. The meniscus will descend a downward slope as shown in the lower half of Fig. 40.

In the case of Fig. 39b, on the other hand, there is little possibility that the contraction of the vapor volume will occur because the curvature of the interface will lower the pressure of the vapor and the corresponding saturation temperature. In these circumstances, the temperature of the vapor will rise to the saturation temperature sooner or later, and, therefore, the vapor may be too stable for any imposed disturbances to change


E a

I 3 a

L Y

FIG. 40. Variation of vapor pressure when the vapor nucleus shown in Fig. 39 shrinks [(a) and (b) same as in Fig. 391.


it. The meniscus has to go uphill to reduce the volume of the trapped vapor as shown in the upper half of Fig. 40.

If the surface tensions are large at the vapor-liquid interface uLv and the solid-liquid interface usL , and/or the surface tension at the solid-vapor interface usv is small, the contact angle then becomes large. Then there is considerable chance that the inequality (b) holds and the vapor trapped in a conical cavity becomes stable. This is what is mentioned by the third characteristic. The poor wettability, namely, the large contact angle at the cavity wall is effective for the augmentation of the heat transfer in nucleate boiling by the present method. Surface treatments that would reduce the wettability by the boiling liquid at only the cavity wall should be applied. The poor wettability of the rest of heating surface is by no means effective for the augmentation, as seen by the first and second characteristic.

2. Reentrant Cavity

If every conceivable way to reduce wettability is unsuccessful, the first method discussed in the previous subsection cannot be applied to the augmentation of the heat transfer in nucleate boiling. This is the case when freons or liquefied gases, such as liquefied nitrogen, are boiled. There seem to be no alternative except for the method of using the surface with the so-called reentrant cavities for these kinds of liquids. This second method is essentially effective for every combination of liquid and surface.

Figure 41 depicts the highly idealized reentrant cavity. The cavity with a conical ceiling is connected to the outer liquid space by the cylindrical mouth. The vertex of the cone is in the liquid side. When the meniscus is at the conical portion of the cavity, the radius of the curvature of the vapor- liquid interface is calculated as

(a) -: 8 < ~ / 2 - $ (b) + : 0 > 7 ~ / 2 - 4

where x is the distance of the interface from the vertex of cone along the generatrix. The coefficient of x in Eq. (112) is positive in both cases. The pressure difference between the vapor trapped in the reentrant cavity pv and that in the liquid p.. is

(a) + : 8 < ~ / 2 - 4 (b) - : ~ > ~ / 2 - 4


a liquid

PCU

b li q u i d

Pm

FIG. 41. Stability of the vapor trapped in a reentrant cavity [(a) and (b) same as in Fig. 391.

The corner joining the cylindrical and the conical portions becomes a singular point of the pressure variation with the location of meniscus. The pressure at which the vapor is in equilibrium with the liquid is considered in this context. While the meniscus is at the cylindrical portion, pv is higher or lower than pm by 12aLv cos 6 /xo sin C#JI depending on whether the contact angle is smaller or larger than ~ / 2 . Here xo is the value of x at the corner and xo sin C#J is the radius of the cylindrical mouth. When the meniscus is at the conical portion, the pressure is given by Eqs. (112) and (113).

Figure 42 shows diagrammatically this pressure variation. If the contact angle 6 is small enough to satisfy inequality (a), the vapor in the cavity is unstable because of the well at the corner. Namely, the vapor depicted in Fig. 42a shrinks spontaneously until a new equilibrium is established. When the contact angle 6 is fairly large, on the other hand, inequality (b) holds. Then there is little possibility of the vapor turning around the lower end of the cylindrical mouth of the cavity. There is a barrier at the corner. The situation shown in Fig. 42b can barely occur. The vapor trapped in either the conical cavity (discussed in the previous section) or the reentrant cavity is stable, provided that the meniscus has its center of the curvature radius in the liquid side [61]. The most important feature of the reentrant cavity, however, is the lower threshold of the contact


a

FIG. 42. Variation of vapor pressure when the vapor nucleus shown in Fig. 41 shrinks. [(a) and (b) same as in Fig. 39.1


angle for the stability of the trapped vapor. The relevant contact angle is smaller by 24, (i.e., the apex angle of the cone) than that of the conical cavity. Furthermore inequality (b) in Eqs. (112) and (113) is always fulfilled when the apex angle is r, that is, the ceiling is horizontal. This means that the vapor trapped in the reentrant cavity is stable irrespective of the contact angle.

B . HEAT-TRANSFER CHARACTERISTICS BY PREPARED SURFACES

Nishikawa et al. [62] experimented with boiling through a horizontal copper tube with a diameter of 18 mm. To make a thin porous layer on the surface of the tube, a copper or bronze powder of spherical particles was sintered onto the outer surface of this tube, whose diameter was ranged from 100 to 1000 pm and thickness from 0 (bare tube) to 5 mm. The

3 1 0 0 10' 3

A T (K)

FIG. 43. Some typical examples of the augmentation of nucleate boiling heat transfer by the surfaces with sintered porous layer. The liquid used is saturated R113 at atmospheric pressure. Copper (Co) and bronze (Br) are used as the sintered particle material. The number following the element symbol indicates the diameter of the sintered particle, in micrometers, and the number following the dash indicates the thickness of the sintered layer, in millimeters.


measured porosity of the sintered layer ranged from 0.38 to 0.71. By selecting the outer surface of the copper tube (i.e., the substrate surface) as the reference, the heat-transfer area and the surface temperature are defined.

The test fluids are R11, R113, and benzene at the saturation temperature under atmospheric pressure. Figure 43 shows the typical boiling curves of the sintered surfaces exhibiting a rather favorable performance. The liquid used is R113 saturated at atmospheric pressure. The curve on the right- hand side of Fig. 43 was obtained for a horizontal plain copper cylinder polished by emery paper #1000. As clearly seen in the figure, the heat- transfer coefficient of these sintered surfaces for a specified low heat flux is greater than that of the smooth plain surface by about 10 or more times, while the performance deteriorates at high heat flux. A rough threshold for the heat flux may be around 0.1 MW/m2.

The optimum geometry of the sintered layer has been searched for experimentally. Figure 44 gives some idea about this concept by demonstrating how the heat-transfer coefficient changes with the thickness of

6 / d 0 4 8 12

/ I h

1 2 3 4

6 (mm)

FIG. 44. Effect of the thickness of sintered layer of copper particles on nucleate boiling heat transfer. The liquid used is R113 saturated at atmospheric pressure. The diameter of the particles is 250 p n .


10 .. 0

sintered layer at different heat fluxes. The liquid is again R113 and the diameter of the powder particles is kept constant. The optimum thickness of the sintered layer appears to become thinner as the heat flux increases. The optimum thickness has been estimated for several combinations of material and diameter of particles and liquids. The results are summarized in Fig. 45. The ordinate is the ratio of the estimated optimum thickness to the diameter of a particle, although the diameter of particle is unchange- able on each curve. It is inferred that the optimum thickness decreases slightly as the heat flux increases. The bronze layer of 100-pm particles in R113 (shown by abbreviation Br 100 R113) behaves quite differently from others, and the optimum ratio is about 10. The physical meaning for the existence of the optimum ratio is not certain.

The mechanism of nucleate boiling heat transfer from the porous layer has been described only vaguely in the reports by many investigators. Although it might be too early to correlate experimental data, Nishikawa and Ito [63] tentatively applied the regression analysis to their data and

- Br 100R113 -

I 0" 8 - -

W Br 500R113

6 .

4 -

2 -

0

-

-

Br 500 R11 -

I I I I I I I I I I I I I I

l 4 1 2 t

FIG. 45. Optimum thickness of the sintered layer for the augmentation of boiling heat transfer. The liquid is saturated at atmospheric pressure. Bronze (Br) and copper (Co) are used as the material of the sintered particles. The number following Br or Co is the diameter of the sintered particles in micrometers, and the following abbreviation is the liquid used- R113 or R11.


obtained the following empirical correlation:

x ( A ~ ~ / A J ~ . ~ ~ ~ ( P J P V ) ~ . ~ ' (1 14)

(115) A, = EAL + (1 - &)Ap

where 6 is the thickness of sintered layer, E the porosity of sintered layer, L the latent heat of evaporation, A the thermal conductivity, p the viscosity, p the density, (J the surface tension at the vapor-liquid interface, and subscripts L, V, p, and m denote liquid, vapor, particle, and apparent value for the sintered layer filled with the boiling liquid, respectively.

The left-hand side of Eq. (114) represents the Nusselt number. The first and the third nondimensional variables of the right-hand side of Eq. (114),

I I I 1

10) 4

( r K J " " 8 4 ($0.560 (4 d 7.59' (*) - 0 . 7 0 8 ($T67

E: L uv

FIG. 46. Correlation of nucleate boiling heat transfer from the surfaces with sintered porous layer. The liquids used are saturated R113, R11, and benzene at atmospheric pressure. The materials of sintered particle are bronze and copper. The diameter of sintered particle range from d = 100-1000 pm. The thickness of sintered layer S = 1-5 mm.


respectively, represent the effect of the surface tension and the Reynolds number of vapor flow if it would fill up voids of sintered layer. The proposed correlation is compared with the data obtained from the experiment in Fig. 46, where the ordinate is the left-hand side of Eq. (114) and the abscissa the right-hand side. Experimental data have been obtained for three liquids, two different types of material used for the sintered layer,

1 I 1

10-1 100 10’ 4

A T (K)

FIG. 47. Augmentation of nucleate boiling heat transfer by Thermoexcel, in a saturated liquid at nearly atmospheric pressure. The abbreviations can be defined as WSP: water, smooth, plane; WEP: water, Thermoexcel, plane; 11SC: RII, smooth, cylinder; 11EC: R11, Thermoexcel, cylinder; 12SC: R12, smooth cylinder; 12EC: R12, Thermoexcel, cylinder; 22SC: R22, smooth, cylinder; 22EC: R22, Thermoexcel, cylinder; 113SC: R113, smooth, cylinder; 113EC: R113, Thermoexcel, cylinder; NzSP: nitrogen, smooth, plane; NzEP: nitrogen, Thermoexcel, plane; HeSP: helium-4, smooth, plane; and HeEP: helium-4, Thermo- excel, plane.


particle diameter ranging from 100 to 1000 pm, and sintered layer thickness ranging from 1 to 0.5 mm. Most data at atmospheric pressure seem to be correlated with an error of less than 30% by the proposed correlation.

Hitachi Cable [64] has developed a tube of unique surface for nucleate boiling with the commercial name of Thermoexcel. The catalogue describes the product as follows.

It has tunnels circumferentially under the outer surface skin, with many openings to the outside. The liquid in the tunnels is heated rapidly and changes to the vapor which leaves through openings as bubbles. A part of the vapor remains always in tunnels and therefore the boiling occurs continuously. The quantity which removes as the vapor phase is compensated with the liquid sucked into tunnels from adjacent openings.

The performance of Thermoexcel in the nucleate boiling [65] for various liquids is shown in Fig. 47. The reduction of the degree of surface superheat at the specified heat flux is sometimes by a factor of 1/10, and the promotion is more pronounced at low heat flux, as it was the case in the Nishikawa's result shown in Fig. 42. Nakayama et al. [66] have analyzed the fluid flow in and around the openings of Thermoexcel and traced the bubble history to predict the heat-transfer characteristics. Their model is as encouraging as they claim.

2 l o o 10' 4

A T ( K )

FIG. 48. Augmentation of nucleate boiling heat transfer by UC High Flux in a saturated liquid at atmospheric pressure.


Union Carbide, Linde Division [67-691, has developed a novel boiling surface consisting of a porous metallic matrix, which is bonded on a metallic substrate. The surface layer is about 0.01 to 0.02 in. thick with porosity from 0.5 to 0.65 and contains a number of cavities or pores, which function as sites for generation of vapor bubbles. The surface is commercially referred to as UC High Flux. Substantial enhancements of heat transfer have been proven in a long-term laboratory test as well as in field or prototype tests. The liquids tested include freons, cryogens, ammonia- water solutions, light hydrocarbons, glycol-water solutions, and seawater. Figure 48 shows the comparison between the performance of flat heaters, with smooth and porous surfaces, facing upwards in various liquids, including propylene, ethanol, R11, and water. The data clearly reveal the marked improvements of heat transfer over a smooth surface. Further- more the deterioration of the performance at high heat flux is not recognized in their data.

A comparison of the sintered surface tested by Nishikawa et al. with Thermoexcel and UC High Flux is made in Figs. 49a and 49 b [70], where

5

a

10'

- N E

3 \

v

U

10'

2

I I I l l

I S a t u r a t e d R113 I a t atmospl

2 l o o 10' 5

A T (K)

FIG. 49. Comparison of boiling heat-transfer performance in variously prepared surfaces in (a) saturated R113 and (b) saturated water, both at atmospheric pressure.


Fig. 49. (Continued)

the data are obtained, respectively, for R113 and water at atmospheric pressure. The wettability of the copper surface differs extremely between R113 and water. As seen from these figures, the sintered surface by Nishikawa et al. shows the highest performance for freons, while Thermo- excel is more effective in the case where the substrate is poorly wettable for the boiling liquid.

VIII. Concluding Remarks

This chapter has been prepared with the intention of illustrating the heat-transfer characteristics of nucleate boiling in pool, and give some ideas about the augmentation of boiling heat transfer. Nucleate boiling is a complicated phenomenon accompanied with a phase change from liquid to vapor. Consequently, many physical properties and the characteristics of heating surface must be taken into account when correlating heat transfer. The dependence of the heat-transfer coefficient on heat flux and pressure


has almost been clarified. Alternatively, little is known about the quantita- tive effect of surface conditions although considerable experimental data exists on this issue. As the nucleate boiling heat transfer is greatly influenced by surface conditions, the unified rule of the nucleation factor for various surface conditions, which is similar to the emissivity encountered in the heat transfer by thermal radiation, needs to be researched in the future.

In the present situation where the effect of surface conditions on the nucleate boiling is not clear, such factors as shape, size, and configuration of heating surface and the size of boiling space have been treated as secondary factors and detailed considerations have not been made for their effects. However, the effect of surface orientation or the effect of space confinement on the boiling heat transfer is increasingly important as related to the heat removal from the nuclear reactor or the cooling of electronic devices. The authors look forward to the further advancement of studies on these subjects.

The augmentation of nucleate boiling heat transfer has become of major interest in connection with the efficient use of thermal energy. The effectiveness of reentrant cavities on the prepared surface has been widely recognized. Fields of further study would include (1) the determination of the optimum shape and size of reentrant cavities; (2) the establishment of heat-transfer correlation for prepared surfaces; (3) the increase in the resistivity of specific surfaces against the fouling by long-term operation and against the contamination of boiling liquid; and (4) the overall assessment of prepared surfaces, including manufacturing cost, maintenance in the operation, and so on.

NOMENCLATURE

A a

D d do

du

Fo f fP

fr

C

Gr g H

area of heating surface thermal diffusivity specific heat diameter of vapor dome diameter of sintered particle diameter of a bubble just leaving

the heating surface diameter of a bubble just arriving

at the free liquid surface Fourier number frequency of bubble formation pressure factor nucleation factor Grashof number gravitational acceleration liquid level above the heating

surface

critical liquid level effective stirring length of

latent heat of evaporation representative length of heating

molecular weight bubble emission frequency bubble population density Nusselt number pressure critical pressure atmospheric pressure Prandtl number equivalent heat source assumed in

heat flux of heating surface

bubbles

surface

the liquid


R radius of curvature of the vapor-liquid interface in the cavity

Ro universal gas constant Re, bubble Reynolds number A R r

T temperature T, critical temperature

clearance of a boiling space radius of cylindrical mouth in the

reentrant cavity

T, temperature of heating surface AT degree of wall superheat t time U Urn V volume of a bubble X nondimensional variable defined

Y nondimensional variable defined by

rising velocity of a bubble average rising velocity of a bubble

by Eq. (45)

Eq. (27)

Greek Symbols

a coefficient of heat transfer p coefficient of thermal expansion 6, thickness of thermal boundary

6 thickness of liquid film, layer

or thickness of sintered layer

contact angle (i.e., the angle & porosity 0

between the solid surface and the vapor-liquid interface)

A thermal conductivity

L liquid Subscripts

V

viscosity kinematic viscosity density surface tension half apex angle of cone

void fraction in vapor-liquid mixture

inclination angle of heating surface measured from the horizontal plane

do f

vapor

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20. V. M. Borishanskii, G. I. Bobrovich, and F. P. Minchenko, Heat transfer from a tube to water and to ethanol in nucleate boiling. In “Problems of Heat Transfer and Hydraulics of Two-Phase Media” (S. S . Kutateladze, ed.), pp. 85-106. Pergamon, Oxford, 1969.

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31. S. T. Hsu and F. W. Schmidt, Measured variations in local surface temperature in pool boiling of water. J . Heat Transfer 83, 254 (1961).

32. D. B. Kirby and J . W. Westwater, Bubble and vapor behavior on a heated horizontal plate during pool boiling near burnout. Chem. Eng. Prog., Symp. Ser. 61, 238 (1965).


33. E. D. Piret and H. S. Isbin, Natural-circulation evaporation two-phase heat transfer. Chem. Eng. Prog. 50, 305 (1954).

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36. D. A. Huber and J. C. Hoehne, Pool boiling of benzene, diphenyl, and benzene- diphenyl mixture under pressure. J . Heat Transfer 85, 215 (1963).

37. C. Sciance, Pool boiling heat transfer to liquefied hydrocarbon gases. Ph. D. Thesis, Univ. of Oklahoma, Norman, 1966.

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44. R. B. Mesler and J. T. Banchero, Effect of superatmospheric pressures on nucleate boiling of organic liquids. AIChE J . 4, 102 (1958).

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49. P. M. Githinji and R. H. Sabersky, Some effects of orientation of the heating surface in nucleate boiling. J . Hear Transfer 85, 379 (1963).

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51. C. R. Class, J . R. Dehaan, M. Piccone, and R. B. Cost, Boiling heat transfer to liquid hydrogen from flat surfaces. A h . Cryog. Eng. 5,254 (1959).

52. J . W. Littles and H. A. Walls, Nucleate pool boiling of Freon 113 at reduced gravity levels. A S M E Pap. 70-HT-17 (1970).

53. L. T. Chen, Heat transfer to pool boiling Freon from inclined heating plate. Lett. Hear Mass Transfer 5, 111 (1978).

54. I . P. Vishnev, I. A. Filatov, Y. G . Vinokur, V. V. Gorokhov, and G. G. Svalov, Study of heat transfer in boiling of helium on surfaces with various orientations. Heat Trans- fer-Sov. Res. 8, 104 (1976).

55. K. Nishikawa, Y. Fujita, S. Uchida, and H. Ohta, Effect of surface configuration on boiling heat transfer. Inf. J . Heat Mass Transfer 27, 1559 (1984).

56. Y. Iida, Research on the estimation of flow pattern in the two-phase flow of vapor and liquid. Trans. JSME 45, 895 (1979).

57. E. Ishibashi and K. Nishikawa, Saturated boiling heat transfer in narrow spaces. Int. J . Heat Mass Transfer 12, 863 (1969).


58. I. I. Chernobyl’skii and I. M. Tananaiko, Heat exchange during boiling of liquids in narrow annular tubes. Sov. Phys.-Tech. Phys. (Engl. Transl.) 1, 1244 (1956).

59. K. Nishikawa, H. Kusuda, K. Yamasaki, and K. Tanaka, Nucleate boiling at low liquid levels. Bull. JSME 10, 328 (1967).

60. H. Kusuda and K. Nishikawa, A study on nucleate boiling in liquid film. Mem. Fac. Eng. Kyushu Univ. 27, 155 (1967).

61. K. Nishikawa and T. Ito, Augmentation of nucleate boiling heat transfer by prepared surfaces. In “Heat Transfer in Energy Problems,” p. 119. Hemisphere, Washington, D.C., 1983.

62. K. Nishikawa, T. Ito, and K. Tanaka, Augmented heat transfer by nucleate boiling at prepared surfaces. Proc. ASME-JSME Therm. Eng. Conf. 1, 386 (1983).

63. K. Nishikawa and T. Ito, Augmentation performance of boiling heat transfer. In “Re- search on Effective Use of Thermal Energy,” Vol. 1, SPEY 1 , p. 39. Minist. Educ. Sci. Cult., Tokyo.

64. Sales catalogue of Hitachi Cable, Cat. No. E A 501 (1987). 65. N. Arai, T. Fukushima, A. Arai, T. Nakajima, K. Fujie, and Y. Nakayama, Heat

transfer tubes enhancing boiling and condensation in heat exchangers of a refrigerating machine. ASHRAE Trans. 83, 58 (1977).

66. W. Nakayama, T. Daikoku, H. Kuwahara, and T. Nakajima, Dynamic model of enhanced boiling heat transfer on porous surface. J . Heat Transfer 102, 445 (1980).

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ADVANCES IN HEAT TRANSFER. VOLUME 20

Two-Phase Slug Flow

YEHUDA TAITEL AND DVORA BARNEA

Faculty of Engineering, Department of Fluid Mechanics and Heat Transfer, Tel-Aviv University, Ramat-Aviv 69978, Israel

I. Introduction

Gas-liquid flow in conduits may take on a variety of configurations related to the spatial distribution of the two phases in the pipe, termed as flow patterns (Mandhane et al . , 1974; Taitel and Dukler, 1976; Taitel et a l . , 1980; Barnea, 1987; Weisman et al . , 1979). One of the most complex flow pattern with unsteady characteristics is the intermittent or slug flow. Gas-liquid intermittent flow exists in the whole range of pipe inclination and over a wide range of gas and liquid flow rates. In vertical slug flow most of the gas is located in large bullet-shaped bubbles, which occupy most of the pipe cross section. These bubbles are usually called Taylor bubbles. The Taylor bubbles are separated by liquid slugs containing usually small bubbles. The liquid confined between the bubble and the pipe wall flows around the Taylor bubble in a thin falling film. In horizontal and inclined flow, slugs of liquid that fill the whole cross section of the pipe are separated by a stratified zone with an elongated gas bubble in the upper part of the pipe and the liquid film at the bottom. The intermittent pattern is sometimes subdivided into slug and elongated bubble flow patterns. When the flow is calm and the liquid slug is almost free of gas bubbles the pattern is termed as elongated bubble flow. For high flow rates, when the liquid is aerated with gas bubbles, the flow is designated as slug flow. In spite of the distinction between slug and elongated bubble flows, the term slugflow is still often used for the general intermittent flow.

Slug flow is a highly complex type of flow with an unsteady nature, thus the prediction of pressure drop, heat, and mass transfer for such flow is a difficult task. Obviously an exact solution of the continuum equations is out of the question at this time. Therefore a variety of approximate methods

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84 YEHUDA TAITEL AND DVORA BARNEA

has been developed for calculating the slug hydrodynamic parameters. The former methods simply used correlations of experimental results. More recently there is a tendency to formulate approximate models that are capable of simulating the flow behavior sufficiently accurate so that the calculation of the pressure drop as well as other flow parameters can be performed with a relatively high degree of confidence and generality. Such models were introduced by Dukler and Hubbard (1975) and Nicholson et al. (1978) for horizontal flow; Fernandes et al. (1983), Sylvester (1987), and Orell and Rembrand (1986) for vertical flow; and Bonnecaze et al. (1971) for the inclined flow.

All of the aforementioned models deal with steady slug flow, which is an orderly flow with relatively short slugs (less than 1000) and a constant average flow rate of liquid and gas over the time period of a slug cycle.

There are, however, more complex types of slugs that are of a typical transient nature and are abnormally long. The most common are the terrain-induced slugging, where slugs are generated in a pipeline when the heavy liquid is accumulated in the lower section of a pipe that follows a hilly terrain. The state of the art of transient slugging is not yet well developed and perhaps only the simple case of a system that contains a single riser and a single pipeline was adequately studied (Schmidt et al., 1980; Taitel, 1986; Taitel et al., 1989).

In this Chapter we will treat first the steady slug flow. Various options of modeling the hydrodynamic parameters and pressure drop will be introduced using a unified approach that is applicable for the vertical, horizontal, as well as the inclined cases. Transient phenomena in slug flow will also be reviewed with the detailed example of the case of severe slugging in a pipeline-riser system. This system is of major practical importance for the offshore oil and gas industry. It is also the one that has been treated with sufficient success.

Heat transfer in slug flow is also of major importance for practical applications. The topic of heat transfer during evaporation and condensation is considered in the specialized literature. The treatment of the two- phase flow is usually considered there as a two-phase mixture. Very few studies have been performed that treat the liquid and the gas in slug flow as different entities and that address themselves to questions such as the temperature profiles in the liquid and the gas, the local fluctuation of the wall temperature, the difference between the upper and lower parts of the wall temperature in slug flow, and the heat-transfer coefficient in slug flow with dependence on peripherial and axial positions. Since we feel that the amount of work done on this subject is not yet at the point where a meaningful, coherent summary can be written, this review is limited to the hydrodynamic aspects of slug flow only. Note, however, that the hydrodynamics of slug flow is the basis for any detailed heat-transfer analysis.

TWO-PHASE SLUG FLOW 85

11. Steady-State Slug-Flow Modeling

Models for predicting the steady slug flow were usually treated separately for the horizontal case and for the vertical case. Some papers also consider the inclined case.

For the horizontal case, the most detailed models were presented by Dukler and Hubbard (1975) and Nicholson et af. (1978). For vertical upwards flow, the most detailed work has been done by Fernandes et af. (1983), Sylvester (1987), and Orell and Rembrand (1986). The inclined case was considered by Bonnecaze et al. (1971).

In this presentation, we are not going to describe previous work in any consecutive manner. We will rather present our own approach to the modeling of slug flow that we feel is the best combination of engineering accuracy and ease of calculation. Previous work will be reviewed (in a critical way when appropriate) as we move along in the development of modeling the different mechanisms that take place in slug flow. We will try to present an approach that is as general as possible and can handle vertical, horizontal, and inclined slug flows in an unified fashion.

Slug flow is a very complex fluid-mechanics problem. The purpose of modeling slug flow is to be fairly close to the true physical process that is taking place, but it is also important for a model to be simple enough so that practical solutions for the slug-flow parameters can be obtained with reasonable effort and that the modeling could be used in routine engineering calculations.

The schematic geometry of slug flow is shown in Figure 1. The slug body is subdivided into two main sections: the liquid slug zone of length f, and the film zone of length I f . Although the liquid slug zone can be aerated by dispersed bubbles, it forms a competent bridging and gas cannot penetrate through the slug zone. The liquid holdup within the liquid slug zone is designated as R,. Once the slug is incapable of forming a competent bridging, the slugs are then termed protoslugs (Andritsos and Hanratty, 1987) or wavy annular (Barnea et al., 1980), and this is the beginning of transition to annular flow. The average liquid velocity in the liquid slug body is designated as uL. The average axial velocity of the dispersed bubbles in the liquid slug is termed u b . Note that uL and ub are not necessarily the same, even though for horizontal flow, both velocities are considered equal.

The film zone consists of a liquid film and an elongated gas bubble. For horizontal and inclined pipes, the bubble is in the upper part of the pipe. In vertical and off-vertical pipes, a complete symmetry is assumed, the bubble is in the center of the pipe and a thin film flows around it adjacent to the pipe wall. In this case the large bubble is termed Taylor bubble, and the film zone is usually termed the Taylor bubble zone. The translational


C

FIG. 1 . Slug-flow geometry.

velocity of the elongated bubble ut is the velocity at which the elongated bubble propagates downstream. If one moves at a velocity u, the slug picture is seen as frozen in space. The liquid velocity in the film is designated as uf and that of the gas uG. Note that unlike the liquid slug region, which is considered axially homogeneous, the liquid and gas velocities in the film zone vary along the pipe due to the variation of the film thickness, h,(z) or 6(z) (for the symmetrical-vertical case), behind the liquid slug.

A. MASS BALANCES

Mass balances presented here consider both the liquid and gas as incompressible. For long pipelines, where the density is not constant, we can still consider it as locally constant for the purpose of calculating steady slug flow *

A liquid mass balance over a slug unit can be performed in two ways. One way is to integrate the fluid flow rate at a fixed cross section over the time of the passage of a slug unit. The second one is by considering the


volume of fluid in a slug unit. Both methods obviously yield the same results.

Using the first approach for the liquid mass balance yields

WL =L tU (uLAR,pLtS + ~o'fufARfpLdr) (1)

where W L is the input liquid flow rate; uL is the liquid average velocity in the liquid slug; uf is the liquid velocity in the film; tu , t, , and tf are the times for the passage of the slug unit, the liquid slug, and the film zone, respectively. Since t , = ls/ut and tf = l f / u , , Eq. (1) takes the form

The second way of formulating the mass balance yields the following relation,

The term in the parenthesis is the mass of the liquid in a slug unit. A slug unit is propagating in the pipe at a velocity u, and the time for a slug unit to pass through a fixed point in the pipe is tu = lu /u t . However, at this time, part of the liquid in the film moves upstream (backward) relative to the gas-liquid interface and is captured by the following slug. This amount of picked up liquid X is given by the expression

X = (ut - U L ) P L A R ~ = (ut - Uf)PLARf (4)

Using Eq. (4) for X , one can show that both Eqs. (1) and (2) are indeed the same.

Equations (2) and (4) can be combined to yield

Exactly equivalent mass balances can be performed on the gas. How- ever, it is more convenient to use a mass balance on one species only and a mass balance on the mixture. A very simple continuity balance on both liquid and gas states that for contant densities, the volumetric flow rate through any cross section is constant. Applying this balance on a cross section in the liquid slug zone yields

us = ULS + uGS = uLR, + ub(Ys (6)

where us is the mixture velocity within the liquid slug.


B. AVERAGE VOID FRACTION

The average void fraction of a slug unit is defined as

a , = (a& + j--clfdx)/f, (7)

Using the mass balance [Eq. (5 ) ] to eliminate the integral term in Eq. (7) yields

a, = (-uLs + ULRs + uta,)/ut (8)

Using Eq. (6), the liquid flow rate uLs can be replaced by uGS to yield

= ( U G S - Ubas + U&s)/ut (9)

As pointed out before (Barnea, 1989), Eqs. (8) and (9) present indeed a very interesting result. It shows that the average void fraction of a slug unit depends only on the liquid and gas flow rates ( u G S , uLs), the dispersed bubble velocity ub, the translational velocity ut , and the void fraction within the liquid slug as, and it is independent of the bubble shape, the bubble length, the liquid slug length, and film thickness. This is a very important and convenient result since it shows that the gravitational pressure drop can be calculated independent of the detailed slug structure. For the simple case when the liquid slug is not aerated, Rs = 1 and Eq. (9) reduces to the simple result a, = uGS/ut.

C. HYDRODYNAMICS OF THE LIQUID FILM

The length of the liquid film lf , its shape hf(z), the velocity profile along the liquid film uf(z), and, especially, the film thickness and its velocity just before pickup hfe and ufe , respectively, are important parameters for calculating the pressure drop and heat and mass transfer in slug flow.

The shape of the liquid film is a very complex structure, especially near the tail of the liquid slug. It is a three-dimensional problem, with free surface, of turbulent flow and obviously an exact solution is out of the question at this time. A reasonable approximation is to use the one- dimensional approach of the channel flow theory. This method has been used by Dukler and Hubbard (1975) and Nicholson er af. (1978).

In order to find solutions for the film velocity uf and the film holdup Rf as a function of position from the rear of the slug, z , we will consider momentum balances on the film zone. Refering to Figure 1, the momentum equations for the liquid film and the gas above it relative to a coordi-


nate system moving with a velocity u, are

d U f d P TfSf TjSj dhf pLuf- = - - + - - - + pLg sin /3 - pLgcos p- az d z Af Af d z (10)

where vf = u, - uf and vG = u, - uG. Note that although these equati0n.s are written for the relative velocities vf and v G , the shear stresses are given in terms of the real velocities as follows (see Fig. 1 for the definition of the positive direction for the shear stresses):

? =fi[PGIuG - ufI(uG - uf)/2] (14)

where ff , fG , and fi are the 'friction factors between the liquid and the wall, the gas and the wall, and the gas-liquid interface, respectively; uf and uG are considered positive in the downstream ( x ) direction.

For smooth pipes, the Blasius correlation can be used,

ff = cf(Dhuf/ (15)

where Dh = 4Af/Sf. A similar expression can be used for the gas with the exception that the gas hydraulic diameter is taken as Dh = 4AG/(SG + Si) (Taitel and Dukler, 1976). For laminar flow Cf = 16 and n = -1 , while for turbulent flow Cf = 0.046 and n = -0.2.

For rough pipes the roughness of the pipe should be taken into account. An example for such an expression is the convenient explicit formula (Hall, 1957):

f = 0.001375{1+[2 x 104(&/Dh) + (106/Re)]'/3} (16)

Obviously many other correlations can be used. More problematic is the determination of the interfacial friction fac-

tor fi. For the case of low liquid and gas velocities, the smooth surface friction factor can be used. When the interface is wavy, the wavy structure determines the value of the average friction factor. Unfortunately, due to the complexity of the wavy structure, the interfacial friction factor cannot be predicted accurately and one has to use some crude correlations and assumptions. The nature of the interface (smooth or wavy) can be determined on the basis of flow pattern maps considering the film zone as stratified flow with the appropriate flow rates of liquid and gas.


For wavy stratified flow in the horizontal and inclined cases, a constant value o f 5 = 0.014 was suggested (Cohen and Hanratty, 1968; Shoham and Taitel, 1984). For the vertical case Wallis (1969) correlation for cocurrent annular flow

fi = O.OOS[l + 300(8/D)] (17) can be used, though the flow in the film zone is usually countercurrent. It may be noted that Wallis et al. (1978, 1979) suggested modified correlations for countercurrent flow. However, those correlations are applicable near the flooding point and this is usually not the case for normal slug flow.

As can be seen, the information regarding the interfacial shear is very limited, primarily for inclined pipes. Fortunately the accuracy of the interfacial friction is generally not important since in most cases the interfacial shear in the film zone is negligible.

Eliminating the pressure gradient from Eq. (10) and (11) yields

Using Eq. (4), the relative velocities vf and vG are given by

Vf = (u, - Uf) = (u, - UL)Rs/Rf (19)

VG (ut - uG) = (Ut - u b ) a y , / a f (20)

Likewise

Since Rf as well as af are functions of hf or 8, substituting these values in Eq. (18) yields a differential equation for hf (or 8) as a function of z :

(21) where for the case of stratified film flow


The differential equation [Eq. (21)] is solved numerically for hf (2) and the corresponding u f (z) is found using the mass balance [Eq. (19)]. The integration is performed until the mass balance of Eq. (5) is satisfied, yielding the length of the liquid film lf , as well as the holdup Rfe , and the velocity ufe at the end of the liquid film just before pickup.

For large z, the limiting value of hf, is the equilibrium liquid level hE , which is obtained when dhf/dz = 0, namely, the numerator of Eq. (21) equals zero.

Note that for aerated liquid slugs, the gas velocity in the elongated bubbles usually exceeds that of the dispersed bubbles in the liquid bridge. In addition the liquid film seems to be essentially free of small bubbles. The physical picture that is consistent with this description is that the dispersed bubbles in the liquid slug coalesce at the nose of the elongated bubble, while gas bubbles are reentrained from the back of the bubble into the liquid slug. Thus the liquid holdup in the front of the liquid film Rfi equals the value of R, and ufi equal uL ; h, is the liquid level corresponding to R, . Thus the integration of Eq. (21) starts normally with hf = hfi = h, at z = 0 and hf decreases (dhf/dz < 0) from h, towards the limit of hE .

However, under certain conditions, dhf/dz may be positive. It occurs whenever the critical liquid level h, is less than h, , where h, is the level that equates the denominator to zero. In this case, the liquid level reduces ‘‘instantaneously’’ to the critical level, and the integration of hf starts with hfi = h, at z = 0. This procedure is similar to the analysis of liquid drainage from a reservoir to a super critical channel flow (Henderson, 1966). We may further note that in the event that h, or h, are less than the equilibrium level h E , then hE is reached immediately. For the vertical case, the denominator is never zero and a critical film thickness does not exist.

Equation (21) is the most detailed form of the one-dimensional channel flow approach. This approach with several degrees of simplification has been used by various investigators. Dukler and Hubbard (1975) and Nicholson et aE. (1978) assumed that the pressure drop in the film zone is negligible. Under this assumption, the liquid is treated as an uncoupled free surface channel flow, and Eq. (18) takes the form

Equation (23) is still a differential equation that has to be integrated numerically, and the neglect of the pressure drop along the gas bubble, although usually justified, may be incorrect for very long film zones in which the contribution of the pressure drop in the gas zone is not negligible.

Further simplifications have been proposed in order to avoid the numerical integration. The most common approach is to consider the liquid film


as having a constant thickness in equilibrium. This equilibrium level is indeed the solution of hE . For the case of the vertical pipe this was, in fact, the only approach taken (Fernandes et al . , 1983; Sylvester, 1987; Orell and Rembrand, 1986). In this case, the solution for h (or S for vertical symmetrical flow) should satisfy

In the aforementioned works, however, the pressure drop in the gas zone was also neglected. This neglect is usually justified only for relatively short bubbles.

In summary, several approaches with various degrees of simplicity have been presented for the hydrodynamics of the liquid film. We focus our attention on three cases:

Case 1. This is the most general formulation for the one-dimensional channel flow approximation. It is given by Eq. (18) or (21).

Case 2. The liquid film is treated as a free surface channel flow

Case 3. An uniform film is assumed along the bubble zone [Eq. (24)]. [Eq. (2311.

D. PRESSURE DROP

Since the slug is not a homogeneous structure, the local axial pressure drop is not constant. For practical purposes, we need the average pressure drop over a slug unit, namely, APJl,,.

The pressure drop for a slug unit can be calculated using a global force balance along a slug unit between cuts A-A and B-B (Fig. 1). The momentum fluxes in and out are identical and the pressure drop across this control volume is

+ T G ~ G dz A

APu = pug sin Plu + - A

where pu is the average density of the slug unit:

pu = aupG + (1 - a u ) p L (26) The first term on the right-hand side of Eq. (25) is the gravitational contribution to the pressure drop whereas the second and third terms are the frictional term in the slug and in the film zones.

A second method, which is frequently used for calculating the pressure drop, is to neglect the pressure drop in the film region and to calculate the pressure drop only for the liquid slug zone. In the slug zone a control


volume between the plane cuts A-A and C-C is used. The resulting pressure drop along a slug unit APu in this case is

r, rrD A APu = p,g sin fi ls + - 1, + APmix

where p, is the average density of the liquid slug body, namely,

P s = %PG + RSPL (28) The first term on the right-hand side of Eq. (27) is the gravitational term of the liquid slug; the second term is the pressure loss due to friction, and the third term is the pressure losses in the near-wake region behind the long bubble. Dukler and Hubbard (1975), Nicholson et al. (1978), and Stanislav et al. (1986) proposed that this pressure drop is associated only with the acceleration of the slow moving liquid in the film to the liquid velocity within the liquid slug, namely,

APmix = APacc = ~ ,RsA(ut - UL)(UL - Ufe) (29) However, a careful mass balance between cuts A-A and C-C indicates that the contribution to APmiX is not only due to the acceleration pressure drop, but also due to the change in the liquid level between the film zone and the liquid slug zone (Taitel and Barnea, 1989), namely,

A A p m i , = p L g c o S f i ~ ~ ~ f e ( h f e - y ) b d y -pLgcos f i~~~" (h f i -Y)bdy

+ p~R,A(ut - uL)(ufi - ufe) (30) As discussed in the previous section, hfi is usually h, , and ufi is uL , in which case the last term on the right-hand side of Eq. (30) is equal to the acceleration pressure drop as given by Eq. (29). However, when h, < h,, hfi equals h,, ufi = u,, and the acceleration term in Eq. (30) is different from that in Eq. (29). The integration in Eq. (30) can be carried out and written explicitly

(31) Inspection of Eq. (30) shows that APmix is always less than AP,,, . The use of APmix = AP,,, may cause a minor error for small-diameter pipes but it can lead to a serious error when the pipes are of a large diameter.


Two methods have been presented for the pressure-drop calculation. In the first method, a global force balance is used [Eq. (25)], while the second method is based on the momentum balance only on the liquid slug zone, neglecting the pressure drop in the film zone [Eq. (27)]. It will now be shown that the two methods are identical provided that in the first method we also assume that the pressure along the film zone is essentially constant.

The integrated form of the momentum balance given by Eq. (23) is

hfe ah, xl0" R f dz - pLg cos p Rf - dz 6, dz

(32) The left-hand side of Eq. (32) is exactly the acceleration term APacc.

Integrating by parts, one can show that

lohfAf dhf = lohf (hf - y)b dy (33)

Equation (32) then takes the form

+ PLg cos p lohf' (hfi - y)b dY (34)

Substituting Eq. (34) into Eq. (30) yields another expression for AP,, :

A APmiX = pLg sin p A, dz + qSf dz (35) i:' 1: By substituting APmix of Eq. (35) into Eq. (27), one can see that it is identical to Eq. (25) (for the case where pressure drop in the film zone is zero). Namely, the two methods for pressure drop calculation-(1) a global momentum balance on the whole slug unit [Eq. (25)], and (2) a momentum balance over the liquid slug only [Eq. (27)]-are identical.

For upward inclined and vertical flows the equilibrium level (or film thickness) is frequently reached after a short distance from the liquid slug tail. In this case, one can view the liquid film as composed of two sections. A curved zone and an equilibrium zone where the wall shear stress bal-


ances gravity and the film thickness is uniform. In this case the upper limit of the integrals in Eq. (35) can be the length of the curved zone. This means that the mixing pressure drop APmix equals the sum of the gravitational and the wall shear stress contribution in the liquid film adjacent to the curved nose of the bubble. This observation was first pointed out by Barnea (1989) for the vertical case and Taitel and Barnea (1989) for the inclined case and it has some interesting consequences when the approximation of uniform equilibrium liquid level is used.

When the simplified approach, which considers the liquid film in constant equilibrium thickness, is used (Fernandes et al. , 1983; Sylvester, 1987), the calculation of the pressure drop via Eqs. (25) and (27) is not consistent and yields different results. Equation (25) in this case reads

If (36) T,TD TfSf TGSG AP, = psg sin p l , + -1, + pfg sin plf + -If + -

A A A

where pf = (YfpG + RfpL. Note that in the film zone, gravity is balanced by the shear forces.

Assuming now that the pressure drop in the gas bubble is negligible [this is the same assumption that was used in deriving Eq. (27)], this leaves only the first two terms in Eq. (36). This result is clearly in contradiction to Eq. (27), where APmix is given by the conventional acceleration term [Eq. (29)]. As pointed out by Barnea (1989) and demonstrated by Eq. ( 3 9 , one has to consider a curved nose in order to obtain the same results for the pressure drop by using either Eq. (25) or (27).

Barnea (1989) compared the results of the pressure drop for the constant film thickness model with the more exact solutions. She showed that for the vertical case, when a cylindrical bubble with a flat nose is assumed, the results of the pressure drop, without the acceleration term [in Eq. (27)], is usually closer to the more exact solution than when the acceleration term is used. Note that in the work of Fernandes et al. (1983)) and Sylvester (1987), the acceleration term was considered, although the constant film thickness approximation was used.

E. AUXILIARY RELATIONS

The formulations provided so far are not sufficient yet to obtain a solution. In order to proceed, we should consider the following additional variables: (1) the translational velocity ut , (2) the dispersed bubbles velocity u b , (3) the liquid holdup in the liquid slug zone R,, and (4) the liquid slug length I, or the slug frequency v,. We will term these variables as auxiliary variables. It is convenient for the time being to consider them as


known variables, given in terms of known formulas, and postpone the discussion on determining their values to the end of the chapter. In this manner the solution procedure can be presented in a more general manner. Namely, a solution can be performed for any of these values regardless of the details of their correlations.

The translational velocities are the velocities of the interfaces and this applies, in principle, both to the elongated bubble in the film zone (Taylor bubble in the vertical case) as well as the dispersed bubbles within the liquid slug zone. As mentioned, the translational velocity of the elongated bubble ut is larger than the gas velocity within this bubble uG. The prediction of this translational velocity is not an easy task and, in fact, it is the subject of current research. It is assumed, however, that the translational velocity can be expressed as a linear relation of the slug mixture velocity as

where ud is the drift velocity, namely, the velocity of propagation of a large bubble in stagnant liquid and the factor C is related to the contribution of the mixture velocity. This factor is larger than unity as it is influenced by the liquid velocity profile ahead of the bubble. This expression is very similar to the Zuber and Findlay (1965) distribution parameter, although here it results from an entirely different reason. C and ud are both considered constant for given operation conditions and, in fact, they are usually taken as constants for all flow conditions. At the present time we will assume that both C and ud are known. A special discussion will be given later on the methods used to find these values.

The dispersed bubbles translational velocity can be expressed in a similar manner:

u ~ = B u ~ + u ~ (38)

However, unlike the case of the elongated (Taylor) bubbles, the translational velocity and the gas velocity are the same for the small bubbles. The coefficient B is the distribution parameter and uo is the drift velocity for stagnant liquid. A discussion for the recommended values of B and uo will be presented later.

The liquid holdup within the liquid slug zone R, depends on the flow rates of the liquid and the gas and the pipe inclination. The relation for Rs is obtained either from experimental correlations or by a mechanistic model.

Finally information for the liquid slug length or slug frequency should be given. The liquid slug length and the slug frequency are interrelated variables and it is sufficient if one of these values is given.


As mentioned, we will assume that these four variables are known and proceed towards a solution. Later, a special discussion will follow and the state of the art for determining these variables will be presented.

F. CALCULATION PROCEDURE

The set of equations obtained so far (including the auxiliary relations) allows the calculation of the detailed slug structure, which is the basis for calculating the pressure drop as well as heat and mass transfer. A solution is sought for a given set of operating conditions, namely, liquid and gas flow rates, pipe diameter and inclination, as well as the physical properties of the liquid and the gas. As demonstrated in a previous section, the calculation of the average void fraction a, can be performed immediately using Eq. (8) or (9), which are independent of the detailed slug structure.

Unfortunately the detailed slug structure as well as the pressure drop calculations is not that simple and it requires some numerical efforts. The complexity of the calculations depends largely on the way the shape of the liquid film is calculated. We have distinguished three basic cases as related to the method used for calculating the hydrodynamic of the liquid film (see Section 11,C): (1) accurate film profile, (2) simplified film profile ( T ~ , TG = 0), and (3) constant equilibrium level.

For convenience, we shall start with the simplest case, case 3, where an equilibrium constant liquid level (or constant film thickness) is assumed. This method was used primarily for vertical slug flow (Fernandes et a l . , 1983; Orell and Rembrand, 1986), while for the horizontal case, both Dukler and Hubbard (1975) and Nicholson et al. (1978) considered the shape of the liquid film. In principle there is no reason why the constant equilibrium level was adopted only for the vertical case and not for the horizontal case. A partial justification for it is that in the vertical case the liquid film reaches an equilibrium thickness in a shorter distance than in the horizontal case. Yet this situation depends largely on the flow rates of the liquid and gas.

The solution in this case first requires the calculation of the terminal equilibrium level hE (or tiE). This is done via an implicit solution of Eq. (24) for the film thickness. The calculation sequence can be visualized as a trial-and-error procedure as follows:

1. us is calculated using Eq. (6) (the superficial velocities are assumed to be known).

2. The auxiliary variables u,, ub, R , , and 1 , are determined first; ut is calculated using Eq. (37); u b is calculated using Eq. (38); R, and 1 , are also evaluated by the proper methods (Sections I1,I. and 11,J.).

3. uL is determined using Eq. (6).


4. A first guess for hf is assumed, which allows for the calculation of the geometrical parameters Af , Rf , AG , Sf, SG , and Si . For a stratified film, the following geometrical relations are used:

R f = (1/7~){7~ - c 0 ~ - ' [ 2 ( h f / D ) - 11 + [2 (h f /D) - 1141 - [2(hf/D) - 112} (39)

Sf = D{.n - cos-1[2(hf/D) - 11)

Si = DJ1 - [2(hf/D) - 11' For a symmetrical annular film, simpler geometrical relations can be readily obtained.

5. uf is extracted from Eq. (4). 6. uG can be calculated using a mass balance on a cross section in the

film zone [similar to Eq. (6)]:

U G a f -k ufRf = U s

7. The friction factors ff , fG, and fi are evaluated on the basis of the appropriate Reynolds number. For this purpose there are a few options as discussed earlier. For example, one may use Eq. (16) for ff and fG and a constant value for fi (see Cohen and Hanratty, 1968, for stratified flow) or Eq. (17) for annular flow.

8. The shear stresses Tf, TG , and T~ are calculated using Eqs. (12)-(14). 9. At this point, all the variables in Eq. (24) can be calculated, and one

can check whether the trial film thickness is correct. Obviously, the approach for obtaining the correct solution hf is to use one of the standard methods to ensure fast convergence such as the interval bisection method, the method of false position, Newton-Raphson method, or any other appropriate method.

10. Since slug length I, is considered known, the film length can now be calculated using Eq. (2) for the unknown I,. This results in

(42)

It should also be mentioned that under most conditions T~ and q are very small and can be neglected as was indeed the case in most of the previous reported works. This simplification, however, saves only minor computational efforts and is not recommended for the general case since it can cause a serious error for the case where very long elongated bubbles exist.

Once the film thickness is known, one can proceed and calculate the pressure drop. For this purpose one can use either Eq. (25) or (27). Both, in principle, should give the same results. However, due to the approximation


taken here and the neglect of the curved shape of the bubble nose, Eq. (25) is not consistent with Eq. (27). This point was discussed in detail by Barnea (1989) and Taitel and Barnea (1989). For vertical upward flow, Barnea (1989) showed that in the case of an uniform film thickness, Eq. (25) will somewhat underpredict the pressure drop, whereas Eq. (27) will overpredict the pressure drop, comparing it to the case where the bubble shape is taken into account. However, Eq. (25) is usually much closer to the more exact solution. Thus, when using the constant-film-thickness approach it is recommended to use Eq. (25) rather than Eq. (27), namely, to ignore the acceleration term in the pressure calculation. This is contrary to the way Fernandes et al. (1983) made their calculation (for the vertical case) and is consistent with the Orell and Rembrand (1986) calculations. For the horizontal case, both Dukler and Hubbard (1975) and Nicholson et al. (1978) used Eq. (27). Since they did not neglect the curved nose their method of calculation should have yielded the same result as Eq. (25). However, as pointed out by Taitel and Barnea (1989), they calculated the pressure drop in the mixing zone erroneously, neglecting the integral terms on the right-hand side of Eq. (30), which may result in a serious error for large-diameter pipes.

The term accurate f i lm profile (case 1) is used when the exact Eq. (21) is used to integrate the liquid film level hf(z). The simpliJiedfilrnprofile (case 2) is used when the pressure drop in the film zone is neglected, and hf(z) is obtained via the integration of Eq. (23), instead of Eq. (21). Both cases, however, require numerical integration and their solution is very similar. The solution follows the following steps:

1. The variables u,, u s , R,, u b , and uL are calculated as in case 3. 2. h, is calculated, using Eq. (39) on the basis of the value of R, . 3. The critical level h, is calculated by finding the level at which the

4. The value of hf at z = 0, hf i , is set equal to the lower value of h, or h, . 5. Equation (21) or (23) (for case 2) is integrated numerically to yield

h,(z) . This integration is carried out until the mass balance [Eq. ( 5 ) ] is satisfied and, thus, the film zone length is obtained. Note that Dukler and Hubbard (1975) as well as Nicholson et al. (1978) applied the integration only to the approximate Eq. (23). Also their method of integration was different. Instead of integrating hf(z) (or Rf), they integrated z(Rf).

As mentioned, cases 1 and 2 will usually yield similar results. However, for the case of very long film zones, the pressure drop in the gas cannot be neglected. Since the efforts in the calculation of case 2 are not much easier than in the general case (case l ) , it is recommended to use the exact method (case 1) for all the calculations.

denominator in Eq. (21) equals zero.


6. Since the numerical integration provides the profile of all the variables in the film zone, the pressure drop can be easily obtained using Eq. (25) or (27) (when the pressure drop in the gas zone is ignored). If Eq. (27) is used, the pressure drop in the mixing zone can be calculated by either Eq. (30) or (35).

All in all, the calculation procedure describes a considerable number of options for the user. It is not written in a “ready to use” single format and the results are not presented in the form of a computer program or exact flow chart. It leaves some effort to the user to choose the option appropriate for his specific use and to write his own program. It does, however, contain sufficient details to guide the reader in the use of the options available and present the advantages and drawbacks of the various possibilities.

G. TRANSLATIONAL VELOCITIES OF ELONGATED (TAYLOR) BUBBLES As discussed earlier, Eq. (37) was assumed to apply for the translational

velocities, and the constants C and ud were assumed to be known. We will discuss now the methods by which these variables are determined.

The idea behind the specific form of Eq. (37) is that the translational velocity can be composed as a superposition of the velocity of bubbles in a stagnant liquid (or liquid mixture) ud and the additional contribution of the mixture velocity u s . Also it is assumed that the translational velocity is linearly dependent on the mixture slug velocity. Obviously these assumptions are just an approximation, subject to experimental and theoretical verification.

The motion of elongated bubbles is usually treated separately for the vertical case (Marrucci, 1966; Bendiksen, 1985; Nicklin, 1962; Nicklin et al., 1962; Davies and Taylor, 1949; Dumitrescu, 1943), the horizontal case (Kouba, 1986; Nicholson et al . , 1978; Dukler and Hubbard, 1975), and the more general upward inclined case (Stanislav et al., 1986; Hasan and Kabir, 1986; Zukoski, 1966; Singh and Griffith, 1970; Bendiksen, 1984.) The first attempt, and the most successful one, was the treatment of the vertical case in which reasonable agreement among different researchers exists and both empirical correlations and theoretical approaches seem to be satisfactory. For the horizontal case the situation is less clear. For example, Wallis (1969), Dukler and Hubbard (1975), as well as Bonnecaze et al. (1971) claimed that the drift velocity is zero for the horizontal case since the buoyancy force does not act in the flow direction. Only later Nicholson et al. (1978), Bendiksen (1984), and others showed that a drift velocity exists also for the horizontal case and, in fact, it may even exceed its value in the vertical case (Weber, 1981).

The drift velocity depends on the detailed two-dimensional flow at the front of the bubble. Note that the simple one-dimensional approach based


on channel flow approximation is not suitable for analyzing the immediate region of the bubble nose. Fortunately, approximations that are based on potential flow yield reasonable results. Davies and Taylor (1949) were first to perform such a calculation for the vertical case. Their first approach was quite simple. They consider a potential flow around the nose of the bubble:

q2 = u2 + sin2 8 (44)

where q is the tangential liquid velocity on the surface, 8 is a polar coordinate, and U is the free-stream velocity.

Application of the Bernoulli equation and considering the pressure within the bubble as constant yields that gz = q2/2. Substituting this relation in Eq. (44) yields for small 8:

u = ud = ( 2 / 3 f i ) m = 0.471 m (45)

In Eq. (45) D is the bubble diameter. Modification of this equation, which takes into account the bubble rise in a confined pipe, was also performed by Davies and Taylor (1949). They used a series expansion technique and obtained the same form of Eq. (45) with a constant of 0.328 instead of 0.471. Dumitrescu (1943) performed somewhat more accurate calculations and obtained the constant of 0.35, which thereafter was accepted as the best value that also agrees very well with experimental observations (Nick- lin et al, 1962).

For the horizontal case the situation is less clear. The most interesting fact is that some of the papers do use the drift velocity, whereas others consider the drift velocity as zero on the basis that gravity cannot act in the horizontal direction. The more recent work of Nicholson et al. (1978), Weber (1981), Bendiksen (1984), and Kouba (1986) clearly show that a drift velocity exists also for the horizontal case owing to gravity-induced drift that results from elevation difference in the bubble nose.

Consistent with the approach taken for the vertical case, in the horizontal case also the inviscid theory is applied near the nose region. The drift velocity in horizontal slug flow is the same as the velocity of the penetration of a bubble when liquid is emptied from a horizontal tube (Benjamin, 1968). To predict this velocity the following relations are considered (see Fig. 2): Continuity:

Av, = A 2 ~ 2 (46)

(47)

where A2 is given by

A2 = ( 7 ~ - y + 4 sin 2y)r2

Bernoulli theorem is applied between point (1) and the stagnation point (0). Note that the pressure at the stagnation point, which is the same as


1 2

FIG. 2. Propagation of gas pocket in the draining horizontal pipe.

the pressure in the gas bubble, is taken as the reference pressure:

Pi= -3pv:

Bernoulli theorem between points (0) and (2) along the free surface yields

v; = 2gr( 1 - cos y) (49)

Finally a momentum balance yields

where the integral term in Eq. (50) can be solved explicitly, namely,

As shown in Benjamin (1968), Eqs. (46)-(50) are solved for the liquid level h2 (equivalent to A2 or y) and the liquid velocities v1 and v 2 . The results are

h 2 / D = 0.563 and v1 = u d = 0.542&$ (52)

The result of Eq. (52) is supported experimentally (for bubbles with a negligible effect of surface tension) by Zukoski (1966) and Bendiksen (1984). It is interesting to observe that the drift velocity in the horizontal case is larger than the drift velocity in the vertical case.

For the inclined case, there is no proposed model and one relies primarily on experimental data. The inclined case, as well as the vertical and the horizontal cases, were studied by Zukoski (1966), Singh and Griffith (1970), Bonnecaze et al. (1971), Bendiksen (1984), and Hasan and Kabir (1986). All report a peculiar behavior, that the drift velocity increases as the angle of inclination is declined from the vertical position. The drift velocity then decreases again toward the horizontal position such that the maximum drift velocity occurs at an intermediate angle of inclination


:::I , , , , , ,

OO 45 90

B (degree)

FIG. 3. Dimensionless bubble velocity in stagnant liquid versus inclination angle for different surface tension parameters: (-) predicted; (0) H = 0.064; (0) Z = 0.042; (A) I = 0.01; [Bendiksen, 19841 ( X ) I = 0.001; (+) Z = 0.01, 0.042, 0.064 [Zukoski, 19661 (after Bendiksen, 1984).

around 40" to 60" from the horizontal. Bonnecaze ef al. (1971) were the first to give a qualitative explanation for this peculiar behavior, arguing that the gravitational potential that drives the liquid velocity along the curved surface at the bubble nose increases and then decreases as the angle of inclination changes from the vertical position towards the horizontal position.

Figure 3 shows the results of Bendiksen (1984) and Zukoski (1966) for the change of the dimensionless drift velocity u d / m with the angle of inclination. The upper curve represents the case where surface tension is negligible and, thus, the results for the horizontal and the vertical limits very closely follow the aforementioned theoretical results that were based on potential flow; namely, that u d / m z 0 . 3 5 for the vertical case ( p = 90") and u d / m = 0.54 for the horizontal case. The surface tension effect is given in terms of the surface tension parameter C = 4a/g(p, - pG)D2. Figure 4 shows the experimental data reported by Zukoski (1966). It shows that the effect of surface tension can indeed be substantial, particularly for small-diameter pipes. For a small surface tension parameter C-0.001, the results for the vertical case and the horizontal case are very close to the potential flow theory, namely, u d / m = 0.35 and 0.54, respectively. The drift velocity, however, decreases considerably with an increase in the surface tension parameter (decreasing the pipe diameter) and eventually reaches a zero velocity when C is of the order of unity.


0.0001 0.001 0.01 0.1 1 .o 10

(I

@L-& )gr2 E -

FIG. 4. Variation of normalized velocity with surface tension parameter for @ = O", 45", and 90" (after Zukoski, 1966).

The drift velocity is expected to depend also on the liquid viscosity, or the bubble Reynolds number. However, Zukoski (1966) shows that the dependence of the drift velocity on viscosity is negligible for Reynolds number Re > 300 (Re = udpLD/pL). This is clearly demonstrated by Fig. 5.

Bendiksen proposed, as a practical suggestion, to use the following formula for the drift velocity in the inclined case:

ud=u!cosp+u:sinp (53)

where u i and u: correspond to the drift velocity for the horizontal and the vertical case, respectively.

Hasan and Kabir (1986) proposed the relation:

Ud = u : & j i ( 1 + c0sp)l.Z (54)

which they claim to well correlate experimental data in the range

Next we will consider the additional contribution of the mixture velocity to the elongated bubbles translational velocity, namely, the value of the constant C on the right-hand side of Eq. (37). In developed slug flow, the translational velocity is usually related to the value of the liquid slug

90" > p > 30".


FIG. 5. Bubble velocity versus surface tension parameter for ranges of Reynolds numbers. Flagged symbols from Barr (1926), Dumitrescu (1943), and Goldsmith and Mason (1962) (after Zukoski, 1966).

velocity at the centerline, where the velocity attains its maximum. This is based on the assumption that the propagation velocity of the bubbles is equal to the maximum local liquid velocity in front of the nose tip (Nicklin et al., 1962; Nicklin, 1962; Collins et al . , 1978; Bendiksen, 1984, 1985; Shemer and Barnea, 1987). Although this is a rather simplified approach, it has been found remarkably valid and supported by the more exact approaches (Collins et al . , 1978) and by experimental data. Thus for turbulent flow, C = 1.2, which is the ratio of u , , , / u , ~ ~ ~ for turbulent flow. Nicklin et al. (1962) state that this value is valid for a Reynolds number greater than 8000 but it is also a good approximation for a Reynolds number less than that. For laminar flow the ratio u,,,/u,,,, approaches 2 and indeed there is a strong indication that C increases as the Reynolds number decreases and reaches a value of about 2. A more precise theory shows that C , for laminar flow, equals 2.27 (Taylor, 1961; Collins et al . , 1978) for the case where the surface tension is neglected. Experimental results that were carried out at about 2 = 0.05 show that C is 1.87 (Collins et al . , 1978), 1.94 (Bendiksen, 1985), and 1.8 to 1.95 (Nicklin er al., 1962). Figure 6 is an example of Bendiksen (1985) data on the effect of Reynolds number on the coefficient C. As seen, C has the value of about 1.2 at a high


FIG. 6. A comparison of predicted and measured distribution slip parameters C (0) H = 0.042 [data of Bendiksen (1984)l; (-) predicted with H = 0.042; (-----) predicted with H = 0 (after Bendiksen, 1985).

Reynolds number and it increases as the Reynolds number decreases. The limit of about C = 2 is not shown in this figure.

The exact value of C for the turbulent case, and in particular for the laminar case, is not conclusive. There is still a spread in the experimental data as well as the various theories. Yet, at least to a good engineering approximation, the values of C = 1.2 for turbulent flow and C = 2 for laminar flow is quite a good approximation for the case where the effect of surface tension is small. Furthermore, it is applicable both for the vertical case, the horizontal case, and also the inclined case (Bendiksen, 1984).

It should be stressed that the theoretical consideration just presented is consistent with the assumption that the translational velocity is a linear function of the mixture (slug) velocity as expressed by Eq. (37). Although this is a valid engineering approximation, it is not necessarily the exact representation of the real situation. If one plots experimental data of ut versus us and extrapolates to us = 0, obviously the result for us = 0 is the drift velocity. If the curve u, versus us is not quite a straight line (which is the realistic case), then one has several options. One can draw a straight line that originates at the point us = 0, u, = u d , and draw a best fit over the range form us = 0 up to any desired value. Another option is to fit a straight line over any desired interval where us is larger than zero. In this


case u d is the extrapolation of this straight line to the zero point and it would not be the same as the mentioned drift velocity. Bendiksen (1984) showed that in this case the value of ud is usually less than the one reported here due to the fact that the ut versus us curve bends slightly upwards. However, it seems to us that for any practical application a straight line that originates from the point (0, ud) is adequate for the accuracy needed for engineering calculations.

H. VELOCITIES OF THE DISPERSED BUBBLES IN THE LIQUID SLUG

A rough criterion to distinguish between the elongated (Taylor) bubbles and the dispersed bubbles within the liquid slug is the characteristic value of the pipe diameter. Bubbles with a chord length larger than the pipe diameter D are considered elongated bubbles. Smaller bubbles are usually termed as dispersed bubbles.

As in the case of the translational velocity for elongated bubbles, it is assumed that the velocity of the bubbles in the liquid slug u b is a linear combination of the bubble drift velocity uo and the mixture velocity in the slug zone u s , as it is reflected by the form of Eq. (38). In Eq. (38), B is the distribution parameter (Zuber and Findlay, 1965) and uo is the drift velocity.

For the vertical case the drift velocity is the free-rise velocity of a bubble in the pipeline. This free rise can be evaluated by considering the free-rise velocity of a single bubble in an infinite medium, the free rise of a bubble in a swarm of bubbles (the effect of voids in the liquid slug), and finally the free-rise velocity in a cylindrical pipe. This subject has been discussed extensively (see e.g., Brodkey, 1967; Levich, 1962; Govier and Aziz, 1972; Wallis, 1969).

The free-rise velocity of a single bubble in an infinite media depends largely on the size of the bubble. For very small bubbles the bubbles behave as rigid spheres and the free rise is governed by Stokes law. For larger bubbles, a boundary-layer solution is applicable. As the bubble diameter increases, circulation effects take place and affects the free rise. When the bubbles become larger, their spherical shape is distorted and flattens. This has a drastic effect on slowing down the free rise compared to an equivalent spherical bubble. When the bubble size exceeds some critical value, the rise velocity of the dispersed bubble tends to be constant and independent of the bubble diameter. This critical bubble size is (Brodkey, 1967)


The bubbles in the liquid slug zone are usually larger than dcrit. For a relatively large and deformable bubble, the equation proposed by Har- mathy (1960) for the bubble rise velocity is considered of sufficient accuracy:

(56) 2 1/4 u- = 1 - 5 4 “ d P L - PG) / PLI

Note that Eq. (56) indeed shows that the free-rise velocity is independent of the bubble size.

The free-rise velocity of a bubble within a swarm of bubbles is lower than the free rise of a single bubble. This can be viewed as the decrease of buoyancy that acts on a single bubble in a gas-liquid mixture. This decrease is correlated in the form

uo = u,( 1 - (57)

For relatively large bubbles, Wallis (1969) as well as Govier and Aziz (1972) suggested the use of n = 1.5. Fernandes et al. (1983) used n = 0.5. A value of n = 0 was recommended by Wallis (1969) (after Zuber and Hench, 1962) for the region termed as churn turbulent. The later case is probably most close to the flow of bubbles in the slug region and, thus, the value n = 0 is suggested. It should be noted that no direct information is available on the rise velocity of bubbles within the liquid slug. However, for modeling slug flow, the accuracy of the rise velocity is usually not that important anyway.

Finite size of the pipe also acts to decrease the free-rise velocity. This effect is discussed by Wallis (1969). In general, the effect of the pipe diameter is negligible for d / D < 0.125, and it is suggested to ignore it.

For the case of inclined pipes, we may assume that the drift velocity uo should be multiplied by sin j? (Barnea et al . , 1985).

The value of B depends on the concentration distribution of the bubbles in the liquid slug as demonstrated by the method of Zuber and Findlay (1965). Wallis (1969) points out that B for vertical dispersed flow “usually lies between 1.0 and 1.5 with a most probable value of about 1.2.” For the horizontal case the bubble concentration is definitely not uniform since bubbles tend to concentrate at the top of the pipe. Nevertheless B was taken as unity since it was assumed that the dispersed bubbles have the same average velocity as the liquid in the mixture. Kouba (1986) did measure the distribution parameter in horizontal slug flow and got a value close to 1.2. However, owing to the lack of supporting evidence we would recommend the use of B = 1 for the horizontal case. As can be seen the evaluation of B for horizontal, vertical, as well as the inclined case is still an open question.


I. LIQUID HOLDUP IN THE LIQUID SLUG ZONE

As indicated in Section II,E, the model calculations require as input data the value of the liquid holdup within the liquid slug R, . This value may be obtained experimentally or through a physical model.

Hubbard (1965) measured R, in an air-water horizontal slug system by using an impact Pitot probe system. This technique proved to be very difficult for realization. Considerable scatter was observed and the results obtained showed little consistency.

Experimental values of R,, using a light refined oil-air system in a horizontal pipe, were obtained by Gregory et al. (1978). They used capacitance-type liquid-volume-fraction sensors, which provided a continuous record of the in situ liquid volume fraction. The ranges of flow rates investigated cover virtually the entire region of slug flow that was observed in their flow loop. For air-oil slug flow in horizontal 2.58- and 5.12-cm I.D. pipes, they found a modest diameter effect and suggested a correlation of the following form: 1

1 R, = 1 + ( ~ , / 8 . 6 6 ) ' . ~ ~

where the mixture velocity us has units of meters per second. In spite of the fact that this correlation is limited and does not include the

effect of fluid properties and pipe diameter, it is frequently used because of its simplicity.

Greskovich and Shrier (1971) presented a graphical correlation of R, involving the mixture Froude number (u,/Dg) and the input liquid quality (A). This correlation is based on data collected with air-water in a 1.5-cm I.D. horizontal pipe. Values of R, between 1 and 0.5 were obtained.

Heywood and Richardson (1979) used the yray absorption method in order to determine the average holdup within the liquid slug for an aii- water system in a 4.2-cm horizontal pipeline. The results are presented by a graphical correlation of R, versus uGS with parametric values of uLs. These results are similar to the correlation presented by Gregory et al. (1978).

Schmidt (1977) measured the liquid holdup in the liquid slug in vertical risers by using a capacitance sensor. Void fraction in the liquid slug was correlated with uLs and uGS. The values of the void fraction range from 0.2 to 0.8.

Fernandes (1981) measured R, in a vertical air-water slug flow using a 5-cm I.D. pipe. Based on his experimental results, he suggested that R, be set equal to 0.25. It should be mentioned that Fernandes' data was obtained for a relatively limited uGS-ULS range.


Barnea and Brauner (1985) proposed a method for estimating the gas holdup within the liquid slug, a,. It was suggested that the gas holdup on the transition line from dispersed bubbles is the maximum holdup that the liquid slug can accommodate as fully dispersed bubbles at a given mixture velocity u s . Thus, curves of constant u, within the intermittent region represent the locus where a, or R, is constant and is equal to the holdup of the dispersed bubble pattern at the transition boundary. The transition boundary itself may be obtained by any reliable predictive model or experimentally. Once it is obtained, R, may be determined by the previously mentioned concept. For example, using the theoretical transition boundary from dispersed bubbles for the vertical case, yields (Barnea, 1987),

a, = 1 - R, = 0.058{2[0.4~/( pL - p~)g]”2[(2f,/D)~~]2/5( p ~ / ( + ) ~ / ~ - 0.725)’ (59)

Note that Eq. (59) usually applies also to the inclined and horizontal cases (see Barnea, 1987, for possible exceptions).

The calculated value of R, ranges from 1 to 0.48, where 0.48 is associated with the maximum volumetric packing of the dispersed bubbles in the liquid slug. For the special case of vertical and off-vertical pipes with relatively large diameters (D > 0.05 m for air-water), the maximum value of R, is 0.75 (Barnea and Brauner, 1985).

Barnea and Shemer (1989) used a conductance probe to detect the instantaneous void fraction at the centerline of a vertical 0.05- m I.D. tube in upward air-water slug flow. This information was further processed to obtain the liquid slug holdup and its length. The experimental values of voidage ranges from a,=O.25 on the transition from bubbly flow, to a, = 0.6 near the transition to churn flow, as has been predicted by Barnea and Brauner (1985).

J. SLUG LENGTH AND SLUG FREQUENCY

The slug frequency and the liquid slug length are interconnected properties and are very often alternatively used (Nicholson et al . , 1978). Ex- perimental observations for air-water systems in vertical upward and horizontal slug flows suggest that the stable liquid slug length 1, is relatively insensitive to the gas and liquid flow rates and is fairly constant for a given tube diameter. The stable slug length has been observed to be of about 12-300 for horizontal slugs (Dukler and Hubbard 1975). Nicholson et al. (1978) noted that the variations in the average liquid slug length are much smaller than the corresponding slug unit length and reported an average value of 300. For the vertical case the observed liquid slug length is about 10-200 (Moissis and Griffith, 1962; Moissis, 1963; Akagawa and Saka- guchi, 1966; Fernandes, 1981; Barnea and Shemer, 1989).


Slug frequency has sometimes been considered as an entrance phenomenon, namely, it results from bridging of the liquid at the entrance (Taitel and Dukler, 1977). This is indeed the case in horizontal and slightly inclined flows, near the transition from stratified flow. In this case low- frequency slugs are generated causing relatively long liquid slugs at the entrance which propagates downstream. However, generally short (high frequency) slugs are formed at the entrance of the pipe. These slugs are usually unstable. Shedding of liquid at the rear of the liquid slug seems to be larger for short slugs. As a result, an elongated bubble behind a short slug moves faster and overtakes the bubble ahead of it (Moissis and Griffith, 1962). The bubble and the corresponding liquid slug merge in this process, decreasing the slug frequency. The merging process continues until the liquid slug is long enough to be stable, namely, the trailing bubble is unaffected by the wake of the leading one. This occurs when the velocity profile at the rear of the liquid slug can be considered fully developed (Moissis and Griffith, 1962; Taitel et al., 1980; Barnea and Brauner, 1985; Dukler et al., 1985).

Taitel et al. (1980) and Barnea and Brauner (1985) simulated the mixing process between the film and the slug by a wall jet entering a large reservoir. The process of establishing the stable slug length can be visualized as follows. Referring to Fig. 7, two consequent elongated bubbles are shown. The first is behind a long steady liquid slug. The velocity profiles within this slug are shown as they develop from a mixing wall jet profile to a fully developed pipe flow at the back of the slug. Since the average total mixture velocity at any cross section of the slug is the same and equals u s , it is obvious that the maximum velocity decreases asymptotically towards the value of 1 . 2 4 with distance from the front of the liquid slug. As the bubble velocity is related to the local maximum velocity ahead of it, it is clear that bubble B, which is behind a short liquid slug is faster than bubble A, which is behind a fully developed profile with u,,, = 1 . 2 4 . Thus, bubble B will overtake the leading bubble A. This is the process by which short slugs tend to disappear. This process however is terminated once all the slugs are

SHORT SLUG LONG SLUG

0 0

I ' I 1 I 1 -1c

ut' l .2us - f k

Ut>1.2US

FIG. 7. Velocity profiles in liquid slugs.


long enough so that the velocity profile at the back of the slugs is fully developed. Thus, this process is usually the one that controls the slug length. Taitel et af. (1980) and Barnea and Brauner (1985) suggested that a developed slug length is equal to a distance at which a jet has been absorbed by the liquid. Using this approach, a value of 160 was obtained for the minimum liquid slug length in vertical upflow and a value of 320 was obtained for the case of horizontal flow.

Dukler ef af. (1985), on the other hand, assumed that the liquid at the front of the slug is well mixed with an uniform velocity profile. From this point on, a boundary layer is developed at the pipe wall until a fully developed velocity profile is achieved. They found that the minimum stable slug length 1, is of the order of 200. Although the final results of Dukler et af. are similar to the previous analysis, it does not explain well the merging mechanism. In their approach, the value of the centerline velocity increased with the distance from the liquid slug front, and the maximum velocity, which determines the bubble velocity, is minimal behind short slugs. Namely, elongated bubbles behind short liquid slugs will move slower then those behind longer ones, contrary to experimental observation.

Shemer and Barnea (1987) used the hydrogen-bubble technique to record the velocity profiles behind the elongated bubbles in gas-liquid slug flow. They distinguished between two zones in the development of the velocity profile. The first zone is an annular jet, which terminates at a distance of about 2-30, causing a strongly disturbed velocity profile in the whole cross section of the pipe. At larger distances from the bubble, a gradual decay of the fluctuations occurs until a fully developed profile is obtained. Shemer and Barnea (1987) found that the bubble shape in the wake region closely resembles the liquid velocity profile ahead of it. They, thus, concluded that the propagation velocity of the elongated bubble is related to the maximum instantaneous liquid velocity ahead of it. They found a steep decrease in this maximum velocity in the near-wake region, while a much more gradual decrease is observed at larger distances from the leading bubble until a fully developed velocity profile is observed. In this case the lowest possible value of the instantaneous maximum value is obtained. This distance from the leading bubble determines the minimum length of the stable liquid slugs where all the bubbles have a smooth rounded front shape and propagate with identical velocity. The detected velocity field in the wake of the bubble was utilized by the investigators to estimate the minimum stable slug length. They found that I , is of the order of 200.

Most of the reported data and correlations on slug frequency and slug length are related to the downstream developed slugs and not to the entrance frequency.


Gregory and Scott (1969) measured liquid slug frequencies for the carbon dioxide-water system in an 1.9-cm diameter tube. They correlated their data by the following relation (in SI units):

Greskovich and Shrier (1972) used their own data, as well as Hubbard (1965) data and presented the following dimensional correlation for slug frequency in horizontal pipes,

us =0.0226 [ A (2; - +

where A is the input liquid volumetric quality (A = &us) and the pipe diameter D is given in meters.

Heywood and Richardson (1979) used the ?ray absorption method and determined the probability density function and the power spectral densities of the holdup. From these functions they have estimated the average film and slug holdup, the average slug frequency, and the average liquid slug length. They suggested that the slug frequency data may be summarized by the following relation in SI units:

U, =0.0434 [ A (2; ~ + 5)]1.0*

In all these reports, the slug frequency data exhibit a minimum when plotted versus the mixture velocity us and it is a strong function of the liquid flow rate.

As has been mentioned, slug frequency can replace slug length in the auxiliary relation, and the slug length will be an output of the model calculations. We feel, however, that the input of the slug length as an auxiliary variable is preferred to the slug frequency since the slug length is based on a physical model while the slug frequencies are given primarily by experimental correlations.

K. CONCLUDING REMARKS

Steady-state slug-flow modeling was presented using a general approach that treats the slug hydrodynamics for vertical, horizontal, and inclined pipes in the same fashion. Critical review of previous work is also presented.

Three methods of solution are presented: (1) the exact method that uses the fully one-dimensional channel-flow solution for the liquid film; (2) the same as (1) with the neglect of pressure drop in the film zone; and


(3) considering an uniform equilibrium level for the film in the film zone. These three cases differ with regard to the accuracy of the solution and ease of calculations. Case 3 was used primarily for vertical flow and case 2 for horizontal flows. Method 2 however, can be inaccurate for long film zones. Since method 1 is not more difficult to solve than method 2 it is recommended to use method 1 rather than 2.

A special discussion is devoted to the way pressure drop can be calculated. It is shown that there are basically two methods for calculating the pressure drop, namely, (1) using a global momentum balance on a slug unit, and (2) assuming that the pressure drop in the film zone is negligible. In the latter case, the acceleration-pressure drop should be used. It is also shown that when the uniform equilibrium thickness is considered (case 3), the two methods of calculating the pressure drop are not consistent. Based on some numerical examples it seems that when an uniform film thickness is assumed, the calculation for the pressure drop using method (1) is preferred.

Although the present model is probably the most up to date and consistent model for the calculation of the hydrodynamic parameters of steady slug flow and best suited for practical applications, it is still incomplete and some of the approaches used may be regarded as unsatisfactory. Obviously more research has to be performed for the purpose of bringing the theory closer to reality. We would like, at this time, to point out some of the deficiencies that the reader should be aware of.

The most controvertible treatment is the one related to the bubble shape, especially near the bubble nose. We have used different theories to describe the hydrodynamics of the liquid near the bubble nose and the hydrodynamics of the film further upstream, and then we simply superim- posed their effect. To predict the drift velocity, two-dimensional potential flow analysis was used, which is different for the horizontal and the vertical case and does not include the inclined case. To the drift velocity we add the propagation velocity (Cu,) using the notion that the bubble nose follows the highest local velocity of the velocity profile ahead of it. Then the liquid film was analyzed by using the one-dimensional channel-flow equations. Obviously the behavior of the bubble nose and the body of the film should follow a single formulation. Such a formulation would be too complex and probably impractical for pragmatic calculations. There is, however, a genuine need for detailed calculations and experiments in order to improve the understanding of the slug-flow hydrodynamics as well as to provide accurate analysis with which the simple approaches could be compared.

The friction factors used for the liquid slug and the film zones are also of uncertain accuracy. The flow in the liquid slug is not developed, especially


in the near-wake region. The use of a fully developed friction factor may be inaccurate. We may also note that the interfacial shear stress is virtually unknown. Accurate data on the interfacial shear may be essential for slugs with very long film zones and high gas velocities.

The theories for determining the liquid holdup in the liquid slug zone R, , slug length I , , and slug frequency are also far from being perfect, and considerable work, both experimentally as well as theoretically, should be carried out along these lines.

In spite of the aforementioned deficiencies, of which the reader should be aware, relative simplicity for practical applications is absolutely essential. We do believe that the present model is a reasonable compromise of solid physics and ease of calculations. The scheme presented here is sufficiently flexible to accommodate improvements once more accurate theories or more up-to-date data is available.

111. Severe Slugging

A. TRANSIENT PHENOMENA IN SLUG FLOW

In the previous section steady-state slug flow was considered. In steady slug flow, one is expected to see regular slugs propagating in the pipe. These slugs are of relatively short length (less than 1000) and separated by regular and evenly spaced elongated bubbles. There are many occasions in which the nature of the slug flow is different than the steady-state flow and has a nonsteady behavior.

An example of transient slugging was given by Taitel et al. (1978) when considering unsteady flow of liquid and gas in horizontal pipes. It was shown that one may get temporary slugging of a different nature than that of steady-state slug flow. For example, when one has stratified flow in a pipe and the flow rate of gas is increased slowly, a transition to annular flow will occur. If, however, the increase of the gas flow rate occurs fast, a temporary very long slug will be generated. The occurrence of this phenomenon is easy to explain. In stratified flow the amount of liquid in the pipe is much higher than for the case of annular flow. When suddenly increasing the gas flow rate, there is too much liquid in the pipe for a smooth transition to annular flow. The excess of the liquid is depleted out of the pipe in the form of a very long slug that is being pushed by the gas behind it.

Another example is given by Scott (1987) and Scott et al. (1987). In this work, the existence of very large slugs in the Prudhoe Bay 5-km test


pipeline in Alaska was explained on the basis of transient pressure fluctuations near the stratified-slug transition boundary. The flow that started as stratified wavy established a high liquid content in the pipe. Due to interfacial instability a slug is generated and starts to move downstream. The high content of the liquid film ahead of the slug is consumed by the advancing slug and causes a fast increase in the slug length. At the pipe entrance stratified flow continue to take place. When a slug exits the pipe, an increase in the pressure drop occurs, which is a cause for a sufficient increase of the gas flow rate to trigger another new slug in the stratified flow zone, and so on. The end result are long slugs that grow rapidly near the entrance zone to sizes that are much longer than the slug size in normal steady-state flow.

The most common unsteady slug flow that is very important for practical application is severe slugging or, as it is also called, terrain-induced slugging. In the general case this type of slugging takes place when a pipeline follows a hilly terrain and the liquid tends to accumulate at the lower valleys blocking the gas passage. The gas upstream is compressed while a long liquid slug grows in the valley. Eventually the pressure upstream increases to the point that it overcomes the gravitational head of the liquid and it pushes the liquid in the valley downstream in the form of a long slug.

A general solution to such a problem is quite complex. The simple case, however, that consists of a single pipeline and a riser, is a common occurrence in offshore oil and gas production and has been analyzed quite successfully. The result of this analysis will be detailed here.

B. SEVERE SLUGGING CYCLE

Figures 8-11 show the typical behavior of the severe slugging cycle. The first step is the slug formation (Fig. 8). Liquid entering the pipe accumu- lates at the bottom of the riser, blocks the gas passage and causes the gas in the pipeline to compress. When the liquid height in the riser z reaches the top of the riser z = h, the second step of slug movement into the separator starts (Fig. 9). When the gas that is blocked in the pipeline reaches the bottom of the riser the liquid slug is accelerated to high velocity owing to rapid expansion of the gas in the pipeline. This step is termed blowout (Fig. 10). In the last step, Fig. 11, the remaining liquid in the riser falls back to the bottom of the riser and the process of slug formation starts again.

A model for severe slugging was first presented by Schmidt et al. (1980). The purpose of such a model is to predict the slug length, slug cycle time, and pressure fluctuations. The following analysis is a simplified version based on the Schmidt et al. (1980) analysis.

TWO-PHASE SLUG FLOW

FIG. 8. Slug formation

Po

117

FIG. 9. Slug movement into the separator.


P O

FIG. 10. Blowout.

FALLING FILM - GAS

FIG. 11. Liquid fallback.

Since the severe slugging phenomenon is typical of low flow rates of liquid and gas, the pressure is dominated by gravity and the frictional contribution is neglected. The liquid is considered incompressible and the gas is assumed to behave as an ideal gas. Mass inlet flow rates of the liquid and the gas is assumed to be constant. With reference to Fig. 8, x ( t ) and


z ( t ) can be calculated using the following relations: Conservation of liquid and gas yields

mL = mLi + J1: A U L s P L dt

mG = mGi + 1: AuGSOpGO d f

(63)

(64)

where mLi is the initial value of the liquid in the system and mGi is the initial value of the gas in the pipeline. The determination of these initial values will be discussed later.

The mass of liquid and gas can be written in terms of the values of x and z as

mL = pLA(x + z ) + (1 - a)pLA(I- x) (65)

(LA) is an additional gas volume that may exist between the gas inlet valve and the liquid inlet. Usually L is zero for most practical applications. It is, however, convenient to use L > 0 in experimental facilities (Taitel et al., 1989) to simulate longer pipeline performance when the actual pipeline length 1 is short. Note that mLi and mGi are given by Eqs. (65) and (66) for x = xi and z = z i . Equations (63) and (64) are sufficient to solve for x , z, and Pp as a function of time provided the initial values of x and z (xi and z i ) , and the void fraction in the pipeline a are known. The determination of these initial values will be discussed later.

Substituting mG and mGi from Eq. (66) in Eq. (64) yields for the gas,

1 - + ( z - x sin p ) [ ( I - x ) a + L ]

1 + (zi - xi sin p ) [ ( I - x i ) a + L ] + UGSOPGO dt (67)

Substituting mL and mLi from Eq. (65) into Eq. (63) yields, for the liquid,

z = zi - (Y(X - xi) + uLS dt (68) Id Equations (67) and (68) can be solved now for x ( t ) and z ( t ) , which correspond to the slug formation step (Fig. 8). Once the slug reaches the top of the riser (z = h) the process is continued as shown in step 2 (Fig. 9). Thus, after z = h the solution for x ( t ) is obtained directly from Eq. (67) with z = h.


The initial values of xi and zi depend on the amount of liquid that stays in the riser at the end of the blowout process (Figs. 10 and 11). The blowout process is usually a highly chaotic phenomenon and the prediction of the liquid fallback is difficult. Schmidt et al. (1980) used an experimental correlation to estimate the amount of fallback. Taitel (1986) assumed that the blowout process is in the form of a single fast-moving Taylor bubble. In this case the liquid film left in the riser can be calculated on the basis of a slug-flow model. The result of such calculations showed that the void fraction of such a Taylor bubble a’ is around 10%. Assuming that at the beginning of the fallback the pressure in the pipeline is P, , that the falling liquid blocks the air passage, and that the fallback is very fast, then one can calculate xi, zi , and Pp using the following relations: Hydrostatic pressure:

Pp = P, + pLg(zi - xi sin p) (69) Liquid mass balance requires

ax1 + zi = (1 - a’)h

while the compression of the gas in the pipeline follows the relation

la + L Pp = P,

(1 -Xi). + L

The calculation of the void fraction a in the pipeline can be calculated using a steady-state stratified flow in an inclined pipeline (Taitel and Dukler, 1976). Furthermore, since the flow of the liquid and gas is usually low (this is the a priori condition for severe slugging), the void fraction can be calculated as in an open channel flow. In this case a momentum balance of shear stress and gravity on the liquid phase yields

where (72)

(73)

The friction factor fL can be calculated from the Moody diagram with the appropriate hydraulic diameter. For smooth pipes, for example, the friction factor can be calculated by

where CL = 0.046 and n = 0.2 for turbulent flow, and CL = 16 and n = 1 for laminar flow. The cross-sectional area AL and the wetting periphery SL are given in terms of the equilibrium liquid level hL [see Eqs. (39) and (41)]. Equation (73) can now be solved for the equilibrium level hL. Once hL is


given the void fraction a can be calculated by

= 1 - ( A J A ) (75)

The theory presented here provides sufficient means to calculate some of the major parameters of severe slugging such as the fluctuation of the pressure in the line Pp(t), the length of penetration of the liquid into the pipeline x ( t ) , the slug length that enters the separator, and the cycle time of the process.

C. BOE’S CRITERION FOR SEVERE SLUGGING

The severe slugging pattern is typical of relatively low liquid and gas flow rates. It requires that the flow pattern in the pipeline be stratified. Thus, one condition for the existence of severe slugging is that the flow pattern in the inclined pipeline will be in the stratified flow pattern. For the determination of this condition, one needs to use flow pattern maps or any flow pattern prediction methods (Taitel and Dukler, 1976; Barnea, 1987).

In addition to this condition, the existence of a severe slugging cycle requires that the liquid will penetrate into the pipeline, namely, x > 0 (Boe, 1981). This requirement is usually satisfied for a relatively low gas flow rate. Referring to Fig. 8, the condition for x to stay at zero is when the increase of the pressure owing to the addition of liquid into the riser is balanced by the increase in the pipeline pressure due to the addition of gas. The increase of pressure owing to the addition of liquid is

dPp/dt = PLddZ/dt) = P L W L S (76)

The increase of pressure owing to the addition of gas is

UGSO PGO PpUGS dPp - ki dt V, la + L la + L

RT=- RT=- (77)

Equating the right-hand sides of Eqs. (76) and (77) yield the transition boundary proposed by Boe between the severe slugging pattern and a steady flow in the riser (usually bubbly or slug flow):

Equation (78) can also be derived on the basis of our previous development. Setting x and xi to zero in Eq. (67) leads to the same condition as Eq. (78).

Equation (78) is shown by the boundary A on Fig. 12 for a specific example reported by Taitel et al. (1989). Note that at low liquid flow rates,

122

lot 1 I I I r . t t 1 I I I 1 I I I ' I I 1 ' 1 u - ~ q

STEADY FLOW

/

YEHUDA TAITEL AND DVORA BARNEA

uLs is a monotonic linear function of the gas inlet flow rate uGSO. For high liquid flow rates a approaches 0 and the curve is bent to the left. Note, however, that a here is calculated while neglecting the gas shear [Eq. (72)]. Thus this upper limit is beyond the applicability of the present calculations.

Boe claimed that outside the region bounded by A the flow will be of steady-state nature while inside severe slugging will prevail. This claim, however, will be shown to be not quite accurate. In fact the Boe criterion may be violated and one may get steady-state flow within the region designated by Boe as severe slugging and vice versa, one can get severe slugging in the region designated by the Boe criterion as steady-state flow. The occurrence of such anomalies will be discussed next.

D. STABILITY CRITERION

The stability criterion addresses itself to the blowout step of the severe cycle process. As discussed earlier the blowout process (Fig. 10) was assumed to take place in the form of a spontaneous expansion of the gas in the pipeline. Indeed this is usually the case. The criterion for determining the condition under which a vigorous blowout will occur versus a quasi- equilibrium penetration is termed here the stability criterion (Taitel, 1986).


Assume that the cycle of severe slugging reaches the point at which the liquid slug tail has just entered the riser. Assume a small disturbance y that carries the liquid somewhat higher (see Fig. 10) and that the disturbance is fast enough so that the slow flow rates of liquid and gas are ignored while y changes.

The net force (per unit area) acting on the liquid in the riser is

The first term on the right-hand side is the pipeline pressure driving force. The pressure varies with y as a result of the expansion of the gas in the pipeline. The second term corresponds to the back pressure force applied by the separator pressure and the liquid column of density pL and height (h - y ) . Note that for y = 0 the system is in equilibrium and A F = 0. In Eq. (79) a’ is the gas holdup in the gas cap penetrating the liquid column; a‘ can be estimated on the basis of a slug-flow model. Also, the gas is assumed to expand isothermally following the ideal gas law.

The liquid column will be blown out of the pipe if A F increases with y . Thus, the condition for stability is

d ( A F ) / d y < O at y = O (80) This leads to the criterion for stability

P, [ ( a l + L)/a’] - h

Po PolPLg ->

where Po is the atmospheric, or reference, pressure. Equation (81) is shown in Fig. 12 by boundary B, which divides the

region bounded by the Boe criterion into two subregions. The region below line B is unstable and the blowout process is vigorous. The region above B is characterized by a quasi-equilibrium penetration of the gas into the liquid. Taitel el al. (1989) showed that this penetration can end up either with steady flow in the riser or it can develop into a cyclic operation. The latter is termed quasi-equilibrium severe slugging (to be discussed in the next section).

The stability criterion [Eq. (Sl)] was applied within the region bounded by curve A (Boe criterion). However, this criterion can also be used outside this region where a steady-state flow is assumed to take place (Taitel, 1986). Indeed, it can be shown that an unstable subregion exists outside Boe’s region. In this region a severe slugging process will take place as follows: gas in the pipeline will spontaneously expand into the riser and a blowout will occur, followed by liquid fallback. Thereafter, gas will


continue to penetrate into the riser and bubble through it while the liquid (mixture) level in the riser, 2 , rises towards the top of the riser. At the time the liquid level reaches the top of the riser, a steady state is expected to ensue. However, because of the inherent lack of stability, blowout will reoccur. This gives rise to a cyclic severe slugging process except that the slugs produced into the separator are aerated and shorter than the riser length, unlike the classic severe slugging.

The criterion for the existence of severe slugging under such conditions is obtained using Eq. (81) in which pL is replaced by pL&, where & is the average liquid holdup in the riser under steady-state conditions (the gas density can be ignored).

is obtained by using Eq. (9). Assuming unaerated liquid slugs in the riser, the value of average liquid holdup under steady-state condition is

The value of

- = uGS/[c(uGS + uLS) + ud] (82)

Note that Eq. (82) is also valid for bubbly flow in the riser in which case C and ud are replaced by B and uo.

The gas superficial velocity in the riser, adjusting for the average pressure in the riser, is given by

P,

Equations (82) and (83) yield the liquid holdup 5 in steady state. The stability of this steady state can be evaluated by Eq. (81) (using (a pL). The line of marginal stability is shown in Fig. 12 for the case of slug flow by line C. As can be seen, there is a definite region in which one can obtain unstable steady-state flow outside Boe's region. As a result the flow will be cyclic similar to the severe slugging cycle. We term this cyclic behavior as unstable oscillations.

E. QUASI-EQUILIBRIUM SEVERE SLUGGING The region above line B in Fig. 12, although found to be stable accord-

ing to Eq. (81), may behave in a cyclic fashion termed quasi-equilibrium steady state. In this case it is possible to calculate and predict the behavior of the riser during this process, enabling also to predict whether the system will end up in a steady state or a cyclic operation.

The analysis begins at the point when the riser is full of liquid and gas is just entering the bottom of the riser under equilibrium conditions. We assume that the condition is stable so that no blowout occurs as a result of the penetration of the gas into the riser. Nevertheless, when gas enters the


riser the hydrostatic pressure at the bottom of the riser decreases. This causes an expansion of the gas in the pipeline. As a result the mass flow rate of gas into the riser thG increases. Assuming ideal gas behavior, the instantaneous mass flow rate into the riser can be calculated by

The pressure in the pipeline (and at the bottom of the riser) is the hydrostatic pressure exerted by the weight of the liquid column in the riser (the gas weight is neglected). Designating the local liquid holdup in the pipe as @, one obtains that

pp = ps + @PLgdY (85) i:' The gas that penetrates the bottom of the riser is in the form of either

small bubbles or larger Taylor bubbles. In either case it is assumed that the gas velocity equals the translational velocity, which is given in the form of Eq. (37).

In Eq. (37) us is the mixture velocity in the liquid slug. Note that this mixture velocity is equal to the total superficial velocity, us(us = uLs + uGS). In order to simplify the problem a constant superficial velocity us is assumed. For this purpose we calculate the average gas density as

As can be seen in Eq. (86) the average gas density is calculated based on the local pressure in the riser weighted by the local gas void fraction (1 - a). The local pressure is given by

P(Y) = ps + [@PLgdY (87)

Using Eq. (86), the superficial gas velocity in the riser is

Note that although us is assumed to be constant along the pipe, it is a function of time.

The liquid holdup at the bottom of the riser is given by

@ b = 1 - (uGS/ut) (89)

The local liquid holdup in the riser is determined by simple propagation of the liquid holdup at the bottom of the riser with a velocity u,. Thus, the


local liquid holdup is calculated by

@ ( y ) = @ b On y= jdu ,d t (90)

This mathematical formulation allows one to calculate the variation of the pipeline pressure, gas mass flow rate into the riser as a function of time, and the local instantaneous liquid holdup in the riser @(y, t). Although the formulation is somewhat complex, it is very simple to program using an explicit Lagrangian numerical scheme described next.

At time t = 0 the riser is full of liquid, @ = 1 and r iZG = riZGin. The average density of the gas at this time is the inlet density. The gas superficial velocity is given by Eq. (88) and the translational velocity is calculated by Eq. (37). The riser is subdivided into small segments of length Ah and the time step At is calculated using At = Ah/u , .

(at the bottom of the riser, =@b) is given by Eq. (89); the new pressure is given by Eq. (85); the new average gas density in the riser is given by Eq. (86); and the new gas mass flow rate into the riser is given by Eq. (84). Note that dP/dt in Eq. (84) is approximated numerically by the difference between the new and old pressure divided by At. Once the new ritG is known, the new gas superficial velocity uGS is calculated from Eq. (88) along with the new translational velocity u, from Eq. (37) and the new time step At (At = Ah/u, ) .

At the next time step, @ j + l is set equal to Qj and this takes care of the propagation of the bubbles in the riser;

This analysis can be used provided the penetration of the gas into the riser hG is always positive (which leads finally to a steady-state flow). Boundary D on Fig. 12 is the curve above which the flow will indeed reach a steady state. As seen, a steady state can take place within the Boe region. In the particular example of Fig. 12 boundary D is very close to the upper Boe region. Obviously the region of steady flow within the Boe region can be larger for different operating conditions. Below boundary D ritG is not always positive, resulting in penetration of the liquid into the pipeline. Let x ( t ) be the distance of the liquid interface penetrating into the pipeline. Under hydrostatic equilibrium the pipeline pressure at any time is

After time At,

is calculated as before.

pP = pLg(&t - x sin p) + P, (91)

is the average liquid holdup in the riser. A mass balance on the where gas in the pipeline requires that [see Eq. (67)]

1 - + ( 8 h - x sin p ) [ ( I - x ) a + L]

= [ 2 + T i h ] ( I a + L ) + - &in dt (92)


where i relates to the time when riZG = 0 and penetration of liquid into the pipe starts.

Equation (92) can be solved for x as a function of time. For this purpose the average liquid holdup & should be known as a function of time. The variation of & with time can be calculated as before on the basis of the translational velocity ut from Eq. (37). The calculation of the mixture velocity us is then calculated on the basis of the liquid mass balance to yield

U S = uLS - (Y (dxldt) (93)

At time ti, x = 0 and us = uLs ( I j l G = uGS = 0). For time step At, we then calculate the new @ distribution in the riser and &, the new x , the new us (approximately dx/dt numerically), the new u,, and the new step At. As in the case of severe slugging, x increases to a maximum and than recedes back to zero. When x = 0 the cyclic process is repeated.

This calculation is valid provided no fallback occurs. A condition of fallback is defined when the top of the riser becomes clear of liquid (or liquid mixture) and a visible liquid interface is propagating towards the top of the riser. The condition of fallback is related to the net liquid velocity at the top of the riser. Once the liquid velocity at the top of the riser is less than zero no liquid exits the riser, resulting in fallback of the liquid in the riser. Thus, the point at which fallback occurs is when uL is negative, where uL at the top is given by (simple mass balance)

Once this situation occurs we calculate the liquid height in the riser by z = zi = &h and the calculation proceeds in the exact manner described before for the classic severe slugging. In this calculation x ( t ) as well as z(t) are calculated on the basis of two equations, Eq. (95), which is a mass balance on the gas [similar to Eqs. (92) and (67)],

1 - + ( z - x sin p ) [ ( l - X ) ( Y + L]

A RT P L g I: riZGin dt (95) [(l - Xi). + L ] + -

and Eq. (96), which is a mass balance on the liquid

z = zi - CX(X - xi) +


Equations (95) and (96) are used to calculate x ( t ) and z(t) . Once the slug reaches the top of the riser, then z = h, and x( t ) is calculated by Eq. (95) only. The values of xi and zi are the values of x and z at the time of fallback, namely when uL becomes negative. As in the previous case, once x recedes to zero, the gas penetrates the riser and the cycle is repeated.

F. SUMMARY AND CONCLUSIONS

The severe slugging that consists of one riser and one pipeline is perhaps one of the simplest examples of slug flow under nonsteady conditions. As is evident, even this simple case is not at all trivial and presents quite a number of possible operating conditions. A more detailed discussion and experimental verification of the present theory is given by Taitel et al. (1989) and Vierkandt (1988). A summary of the results is presented using an example of a typical flow map as shown in Fig. 12. This map contains four boundaries: A-the Boe criterion, B-the stability criterion, C-the steady-state stability criterion, and D-the transition to steady flow inside the Boe criterion.

The Boe criterion [Eq. (78)] differentiates quite well between steady and cyclic operations with two exceptions. At high liquid flow rates, a steady flow can also exist within the severe slugging region predicted by the Boe criterion (above boundary D). Also there is a region outside the Boe criterion that is in an unsteady state and leads to unsteady oscillations (between boundaries C and A).

The stability criterion [Eq. (8l)l is applied to the cases of severe slugging (inside the Boe region) where the riser contains only liquid (B), and to the case of steady flow of liquid and gas in the riser (C) (outside the Boe region). The former is an approximate boundary dividing between classical and quasi-equilibrium severe slugging cyclic operations. The latter indicates when steady flow outside the Boe criterion is not possible and one obtains unsteady oscillations.

Unlike boundaries A, B, and C, which are given by simple equations, the condition for boundary D is a more complex one and is obtained during a numerical solution of the quasi-equilibrium case as a dividing line between cases where the gas flow rate into the riser is always positive (hG > 0) and the cases where riZG reaches zero in the cyclic process. Note that in Fig. 12, curve D is very close to the upper boundary of Boe criterion. This is not always the case and, in fact, this boundary can be substantially lower and also higher than the Boe criterion (in which case it is not applicable), depending primarily on the length of the pipeline (1 and L ) (Taitel et al., 1989).

TWO-PHASE SLUG FLOW

NOMENCLATURE

129

A b B C

D d

f F g h 1 L

m n

r P 4 R

Re S t T

U U

V

acc b

d E f fe fi G h i in

C

i

0

pipe cross-sectional area interface width in the pipe constant in Eq. (38) constant in Eq. (37) and in the

friction factor correlation pipe diameter bubble diameter friction factor force acceleration of gravity liquid level and riser height pipe length additional equivalent gas pipeline

length mass constant in the friction factor

pipe radius pressure local absolute velocity liquid holdup, also ideal gas

Reynolds number wetted periphery time absolute temperature velocity in the axial direction free-steam velocity relative velocity, usually relative to

correlation

constant

the translational velocity

V X

X

Y

W z

0

P Y s

0 A

P

P

I

E

V

L7

7

@

volume coordinate in the downstream

direction, also distance of liquid penetration into the pipeline

mass flow rate relative to the translational velocity

coordinate in the perpendicular to the pipe axis direction, also vertical coordinate in the riser

mass flow rate coordinate in the upstream

direction, also liquid height in the riser

void fraction angle of inclination polar angle that defines the interface film thickness roughness polar coordinate liquid volumetric quality, uLs/us viscosity kinematic viscosity, also frequency density surface tension surface tension parameter,

shear stress liquid holdup in the riser

dl[(PL - P&*I

Subscripts and Superscripts

accelaration bubble critical drift equilibrium film film at z = If film at z = 0 gas hydraulic, also horizontal interface, also initial inlet discretization index in the riser

L mix 0

P

S t top

S

U

V 30

liquid related to mixing free rise, also at standard atmospheric conditions pipeline slug, also separator superficial translational top of the riser slug unit vertical for unbounded liquid

Special Symbols

rate - average


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Taitel, Y., Vierkandt, S., Shoham, O., and Brill, J. P. Severe slugging in a pipeline-riser

Taylor, G. I. Deposition of a viscous fluid on the wall of a tube. J . Fluid Mech. 10, 161-165

Vierkandt, S. Severe slugging in a pipeline-riser system, experiments and modeling. M. S.

Wallis, G. B. “One Dimensional Two-Phase Flow.” McGraw-Hill, New York, 1969. Wallis, G. B., Richter, H. J., and Bharathan, D. “Air-Water Countercurrent Annular Flow

in Vertical Tubes.” Rep. EPRI NP-786. Electr. Power Res. Inst. Palo Alto, Calif., 1978. Wallis, G. B., Richter, H. J., and Bharathan, D. “Air-Water Countercurrent Annular Flow

in Vertical Tubes.” Rep. EPRI NP-1165. Electr. Power Res. Inst. Palo Alto, Calif., 1979.

Weber, M: E. Drift in intermittent two-phase flow in horizontal pipes. Can. J. Chem. Eng.

Weisman, J. , Duncan, D., Gibson, J., and Crawford, T. Effect of fluid properties and pipe diameter on two-phase flow pattern in horizontal lines. Int. J . Multiphase Flow 5 ,

Zuber, N., and Findlay, J. A. Average volumetric concentration in two-phase flow systems. J. Heat Transfer 87, 453-468 (1965).

Zuber, N., and Hench, J. “Steady State and Transient Void Fraction of Bubbling Systems and Their Operating Limit. Part I: Steady State Operation,” Rep. 62GL100. Gen. Electr., Schenectady, N. Y., 1962.

Zukoski, E. E. Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes. J. Fluid Mech. 25, 821-837 (1966).

Proc. Int. Conf. Multi-Phase Flow, 3rd, The Hague pp. 55-63 (1987).

slug flow. Physicochem. Hydrodyn. 8, 243-253 (1987).

inclined pipes. AIChE J. 30, 377-385 (1984).

two-phase slug flow in an inclined pipe. J. Eng. Ind. 92, 717-726 (1970).

inclined pipes. Int. J . Multiphase Flow 12, 325-335 (1986).

Resour. Technol. 109, 206-213 (1987).

slug flow. Chem. Eng. Sci. submitted (1989).

and near horizontal gas-liquid flow. AIChE J . 22, 47-55 (1976).

and near horizontal pipes. Int. J . Multiphase Flow 3, 585-596 (1977).

modeling flow pattern transitions. AIChE. J. 24, 920-935 (1978).

upward gas-liquid flow in vertical tubes. AIChE J. 26, 345-354 (1980).

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(1961).

Thesis, University of Tulsa, Tulsa, Oklahoma, 1988.

59, 398-399 (1981).

437-460 (1979).


Unified Regenerator Theory and Reexamination of the Unidirectional

Regenerator Performance

BRANISLAV S. BACLIC

Technicki Fakultet Mihajlo Pupin, University of Novi Sad, 23000 Zrenjanin, Yugoslavia

PETER J. HEGGS

Department of Chemical Engineering, University of Bradford, Bradford, England

I Introduction

The simplest mathematical representation of fixed-bed cyclic thermal regenerators has remained virtually static since the initial publications of Nusselt [l] and Hausen [2], and likewise, the rotary-matrix exchanger has also stayed in the same state since the original work of Coppage and London [3]. In both systems, the sole mechanism of heat transfer between the flowing gases and the regenerator matrix is assumed to be forced convection, and this results in two coupled first-order partial differential equations describing the transfer in each of the two periods of operation. The number of variables in each system was reduced by defining dimensionless groups and normalized temperatures, and it was at this point that separate theories were developed for each of the two regenerator systems-fixed bed and rotary.

A. REGENERATOR PARAMETERS

The fixed-bed system is represented by the dimensionless length A and the dimensionless period II of each period. These were proposed by

133 CoDvriehl 0 1990 bv Academic Press. Inc.

134 BRANISLAV S. BACLIC AND PETER J. HEGGS

Hausen [2] and are defined as

A = aaL/mcf (1)

n = aaP/(l - rp)p,c, (2)

Johnson [4] suggested that the ratio of the dimensionless period to the dimensionless length, termed the utilization factor,

U = n / A = riZcfP/( 1 - rp)p,c,L (3)

should be used as a dimensionless parameter for each regenerator period. The advantage of this factor is that it does not include the heat-transfer coefficient a. A further factor has been introduced [5], the unbalance factor P, which is the ratio of the minimum to maximum utilization factors,

P = Umin/Umax = (n/A)min/(n/A)rnax

= (~Cf%in/(~cfP)max (4)

Hence the performance of a fixed-bed regenerator is a function of four variables, for instance, following Hausen’s approach,

The rotary regenerator is represented by dimensionless parameters similar to those for steady-state ideal 1 : 1 pass shell and tube heat exchangers; the overall number of transfer units, NTU,, and the capacity-rate ratio of the flow streams C * , and two other groups for the transient nature of the system-the capacity ratio Ck and the symmetry factor (CIA)* relating to thermal resistance (or conductance) “balance” of the regenerator. These four parameters are defined as

NTU, = NTUmin[l + (aA)*]-l (6)

C* = Crnin/Cmax (7)

Ck = Cr/Crnin (8)

(9) (@A)* = (aA) on the Cmin side/(aA) on the C,,, side

The parameters in Eqs. (6)-(9) are defined in terms of the rotary regenerator nomenclature as

C the fluid heat-capacity rate = k c f (10) C, the rotor matrix heat-capacity rate = Wc,N (11) NTUmin = a A / C on the Cmin side (12)

UNIFIED REGENERATOR THEORY 135

' h j n

a b

,out

FIG. 1 . Schematics of (a) fixed-bed and (b) rotary regenerators and their respective nomenclature.

Thus the performance of a rotary regenerator is a function of the following four parameters:

fn"TU, 9 c*, Ck , (aA)*I (13)

Figure 1 shows a single fixed-bed regenerator and a rotary regenerator with the respective nomenclature for the two design methods.

B. CYCLIC EQUILIBRIUM AND THERMAL EFFECTIVENESS

A regenerator has reached cyclic equilibrium when the heat transferred to the matrix during the flow of the hot gas stream is equal to the heat released from the matrix during the flow of the cold gas stream. Hence for cyclic equilibrium operation, the regenerator performance is considered in terms of effectiveness E that is, the ratio of the actual heat transferred to the maximum possible enthalpy change per cycle.


For the fixed-bed regenerator:

and for the rotary regenerator:

T in Eq. (14) is time-averaged outlet gas temperature, whereas T in Eq. (15) is the spatially averaged outlet gas temperatures.

C. EQUIVALENCE OF THE Two DESIGN METHODS: ONLY TRUE FOR NUSSELT'S IV MODEL

The two design methods are equivalent as demonstrated by Shah [6]. Coppage and London [3] compared the terminology of the rotary system with that of the fixed-bed case and presented the following equivalence:

The equivalence of the fixed-bed and rotary effectiveness [Eqs. (14) and (15)] becomes apparent if we consider the definition of the periods of a


rotary regenerator [7]. These are as follows

Ph = Afh/(Afh +Afc)N (28)

Pc =Afc/(Afh + Afc)N (29)

h h = kh/Afh (30)

m, = kc /Afc (31)

hCfP/(hCfP),i, = kcf / ( i l c f )min (32)

The mass velocities of the two streams are

Hence it is quite obvious that

and Eqs. (14) and (15) are equivalent and there is a one-to-one correspondence between the two design methods.

D . CLASSIFICATION OF REGENERATOR OPERATION

The operation of a regenerator system may be classified into four main categories depending on the values of the four dimensionless groups. A regenerator is termed symmetric if the dimensionless lengths of each period are identical, otherwise, asymmetric, and is termed balanced if the utilization factors of each period are identical, or otherwise, unbalanced. Table I lists the various combinations of parameters for the four categories in both design methods.

The asymmetric-unbalanced regenerator operation is the most general one, and the others are just subsets. For the symmetric-balanced case, the effectiveness is a function of only two variables-A and II, or NTU, and Cb . For the symmetric-unbalanced and asymmetric-balanced categories, then the effectiveness is a function of three variables-A, Urnin, and U,,, , or NTU, , C*, and Ci; ; and Amin, Amax, and Umin or NTU, , Cb , and (crA)*, respectively.

TABLE I

CLASSIFICATION OF REGENERATOR OPERATION

Classification A-II E-NTU,

Symmetric- balanced C*/(hA)* = 1; C* = 1 Symmetric-unbalanced C*/(hA)* = 1; C' # 1 Asymmetric-balanced A r n i n # A m a x ; P = 1 C'/(hA)* # 1 ; C* = 1 Asymmetric-unbalanced A m m # A m a x ; P # l C*/(hA)* # 1; C* # 1

Amin = Amax ; P = 1 Amin = Amax ; P # 1

138 BRANISLAV S. BACLIC AND PETER J . HEGGS

The two and three-parameter systems have been used to approximate the more general asymmetric-unbalance case and a detailed assessment of these approximations can be found in Mitchell [8] and Shah [6].

E. CLASSICAL REGENERATOR MODEL ASSUMPTIONS

The simplest mathematical representation, which describes the transfer of heat between the flowing gases and the regenerator matrix during both gas flow periods at cyclic equilibrium, is that proposed by Nusselt [l]. This study is based on the same model and, thus, it is imperative to list these assumptions for completeness. The classical idealizations for the regenerator periodic flow theory have been summarized by Shah [6], but here some of them will be revised in a more rigorous way (mainly those following directly from the general forms of governing equations for solid packing-to-gas heat transfer).

The classical model is valid for the following assumptions:

1. The regenerator is thermally insulated from the surroundings. 2. There are no thermal energy sources or sinks within the regenerator

and no phase and/or chemical change occurs. 3. The thermal properties of both fluids and the matrix are constant,

independent of time and position. 4. The velocity and temperature fields of each fluid at the inlet are

uniform over the flow cross section and are constant with time. 5 . The packing voidage scalar field is uniform as well as the surface area

of the matrix. 6. The convective heat-transfer coefficients between the fluids and the

matrix wall are constant throughout the exchanger during both gas flow periods.

7. Radiative heat transfer is negligibly small when compared to the convective transfer of heat between the flowing gases and the regenerator matrix.

8. The wall (packing element) thermal resistance is negligible when compared with the fluid-to-wall convective resistance:

(dlL)[cpl(l - cp)l(Bi/St) -c 1 (33) 9. Heat conduction fluxes in both phases in the flow direction are

negligibly small relative to the fluid-to-packing convective heat transfer:

(d/L)(A~h,,onp/A)(l/St Re Pr) -c 1 (34)

10. Heat conduction fluxes in both phases in the circumferential direction in rotary regenerators are negligibly small relative to both fluid to


matrix cohvective heat transfer:

[LIR,rp12(h,*~,,circ/X)(1/St Re Pr) 1 (35)

[LIR0r,I2(h,*,,,~=/A~~ 7 10%) 1 (36)

and to conductive heat transfer in the flow direction:

11. No mixing of the gases occurs during the switch from one to the other period, and the fluid carry over is negligible relative to the both flow rates (carry over occurs instantaneously).

12. Gas residence (dwell) time in the matrix is negligibly small relative to the flow period.

L/cpvP << 1 (37)

F. OBJECTIVE OF THE DISCOURSE

The aim is to derive the overall performance of a regenerator, which is itself as a “boxed” process, and obtain an exact analytical solution for cyclic equilibrium for the unidirectional case assuming Nusselt’s assumptions [l] with the gas residence time neglected. A new approach will be introduced by defining strong and weak periods, so that hot and cold references become irrelevant. Additionally, the complete cyclic comprising both periods will be described in an unique space and time for cyclic equilibrium conditions, and this is the so-called boxed process.

11. “Boxed” Process for Regenerators

The developments that follow will be presented in the terminology for fixed-bed regenerators, however the results are equally applicable for rotary-matrix regenerators using the equivalence relationships detailed in Section 1,C. Figure 2 schematically shows the heat-exchanger system for transferring heat between two process gas streams; one being at a higher temperature than the other. The box represents the regenerator heat- exchanger system and this could be either rotary or fixed bed. With the rotary system this could be a single-rotary regenerator, because the gas flow is continuous. However, if a fixed-bed system were to be used, then to ensure continuous gas flows, a minimum of two fixed-bed regenerators are required. If the two regenerators are physically identical, then the performance of the overall regenerator heat-exchanger system is equal to that of either of the single fixed-bed regenerators. This same argument holds if more than two fixed-bed regenerators comprise the total system.


c (tic( 10

Tc,ln

I

Regenerator

Exchanger System

- Heat - To,o

A. NEW APPROACH WITH WEAK AND STRONG PERIODS

This new representation will consider weak and strong periods, which are distinguished by the smaller and larger utilization factors. Thus hot and cold become irrelevant attributes. Subscript 1 will refer to the weaker period and subscript 2 to the strong period. The complete regenerator cycle is described in a unique space for both periods, 0 s x s L, and separate time regions, so that the weaker period occupies the subregion, 0 s x s L and 0 s tl s Pl , while the stronger period occupies the remaining region, Osxs L and O s t 2 s P 2 .

A dimensionless system is obtained by defining normalized independent variables and temperatures as

6 = x / L , normalized length (38)

v1 = t / P , , normalized time for weak period (39)

v2 = t / P 2 , normalized time for strong period (40) and

j = 1,2, s l , s2 and i = 1,2, normalized temperatures. (41)


The normalized temperature definition in Eq. (41) means that the gas entering the regenerator in the weaker period is unity (i.e., = 1.0) and the gas entering during the strong period is zero (i.e., = 0.0). The unique normalized space and time regions are bounded as

O ~ ( s 1 and O s q i s l for i = 1, 2 (42)

Hence the weak period starts at ql = 0 and ends at q1 = 1, whereas the strong period commences at q2 = 0 and finishes at q2 = 1.

B. GOVERNING EQUATIONS USING NUSSELT’S ASSUMPTIONS

Nusselt [l] in 1927 considered mathematically five ideal cases, and that the physical properties, heat-transfer coefficients, and flow rates were invariant. His fourth case assumed that the thermal conductivity of the solid is zero parallel to the gas flow and infinitely large normal to the gas flow. This case results in the simplest mathematical representation if the effects of axial dispersion in the gas phase are also negligible and the residence time of the gases within the matrix are neglected. This fourth case is fully represented by the list of assumptions mentioned previously in Section 1,E.

Using these assumptions and performing heat balances first on the gas phase and second over both phases provides the following governing equations. For the weak period, 0 S ( s 1 and 0 =Z q1 s 1:

a e l / g + A1(el - esl) = 0 (43)

(l/ul)(a41/a771) + a e * / x = 0 (44)

These two equations, Eqs. (43) and (44), remain the same for either unidirectional or countercurrent operation. For the strong period, O ~ ( s 1 a n d 0 S q 2 S 1 :

-t ae2/ag + ( ~ ~ / a ) ( e ~ - es2) = 0

(l/ul)(a~,,/a?72) * (l/P)(ae,/an = 0

(45)

(46)

The dimensionless groups and factors in Eqs. (43)-(46) are defined as

Ai = c ~ ~ a L / ( r i z c ~ ) ~ , dimensionless regenerator length, (47)

Ui = ( r iZ~~P)~/ [ ( l - cp)( ~ , c , ) ~ L ] , utilization factor (48)

P = U 1 / U 2 , unbalance factor (49)

a = Al/A2, asymmetry factor (50)


The cyclic equilibrium conditions are identical for both modes of operation and are given by

41(& 771= 0 ) = 6s2(5,772 = 1)

4 2 ( k 772 = 0 ) = f%(& 771 = 1)

(51)

(52)

&(O, vl) = 1 for 0 6 ql 6 1 (53)

The gas inlet condition during the weak period is

for both modes of operation. For the strong period, the inlet condition for the unidirectional operation is

d2(0, q2) = 0 for 0 s v2 6 1 (54)

02(1,r/2)=0 for O s q 2 s 1 (55)

and for the countercurrent operation is

All this is summarized in Fig. 3, where it is obvious that the outlet gas temperatures, t91,0ut and B2,0ut , are the variables of primary interest. If these are known as functions of the pertinent parameters of the system (Al , U,, p, a), then the overall performance of the regenerator at cyclic steady state (equilibrium) can be studied.

The main advantage of introducing these four parameters, A , , U,, p, and CT, is that it is now possible to obtain limiting operating conditions (special cases) that are physically meaningful.

1. U, + 0. Aperiodic, or recuperative, operation, when the matrix temperature distribution is time independent so that it is the same for both periods. The two gas temperature distributions are different but time independent. It must be stressed that this is a hypothetical case for a regenerator due to the assumption 12 in Section I ,E [Eq. (37)], that the gas residence time is much smaller than the period. In fact as the period tends to zero then the effects of the gas residence time must be accommodated in the heat-balance equations [Eqs. (43)-(46)].

The exchanger does not exist because the fluid and solid temperatures are identical everywhere. This case need not be considered.

Long regenerator exchangers when, at some point, there is no transfer potential (driving force) in either period.

Short regenerator exchangers and the exchanger do not exist. This case need not be considered.

Completely unbalanced regenerators where the temperature does not change during the stronger period, and this is true for all values of CT. This case is thermodynamically the best for any given U1 and A1.

2. Ul + 03.

3. A1 + 03.

4. A, + 0.

5 . p + 0.


q2=1 -

el,in = e,(o, ql) = 1 - 1

11 = - Pl

q1=0 -

@SZ(t ! l ) = e s l ( w

STRONG PERIOD (U2 = max{U,U,})

e s 2 ( w = esl(ttl)

WEAK PERIOD (U1= min{Uh,Uc})

t = O t = 1

FIG. 3. Schematic of the boxed process for transferring heat in a regenerator.

6. p + 1. Completely balanced regenerators where the magnitude of the outlet gas temperature variations are identical in each of the two periods.

7. u-+ 0. Completely asymmetric regenerators when there is no transfer potential in the stronger period. This case along with U, + 0 and hl + w yields the highest effectiveness values for any value of p.

Completely symmetric regenerator when the transfer potential is the same in each period.

8. u + 1.


9. (T+ m. Short regenerator exchangers and the exchanger do not exist. This case need not be considered.

It is apparent from the last three (7-9) cases that the asymmetry factor, as defined in Eq. (50) with reference to the weak and strong periods, is bounded by zero and infinity. For ease of presentation of results and to reflect the effect of symmetry between a and l/a all other parameters remaining fixed, then define the asymmetry factor as y where

Y = AminIAmax (56)

y = a for O ~ a s l (59)

y = l / a for I S U G ~ (60) Hence it is only necessary to consider the analytical solution to six of these spatial cases (i.e., 1,3,5-8), and the general case for arbitrary combinations of the values of the four independent parameters-A,, U, , p, and o.

C. OVERALL BALANCE EQUATION FOR THE “BOXED” REGENERATOR

The overall (macro-) energy balance for the boxed regenerator representation in Fig. 3 is obtained by integrating the differential (micro-) balances [i.e., the differential equations, Eqs. (43)-(46)] valid in the corresponding regions of the box, over the domains of the independent variables 5 from 0 to 1, and then with respect to q1 from 0 to 1 gives

2- 1’ [esl(r, 771 = 1) - esl(5,771 = 0)l dS+ f4,out - 1 = 0 (61) Ul 0

where the time-averaged outlet gas temperature (see Fig. 3) is given by

%out = 4(19 771)dVl (62)

The same procedure is carried out on Eq. (46) and results in the expression

where the outlet gas temperature for the strong period has been introduced (see Fig. 3) and this depends on the type of flow. For unidirectional flow,


and for countercurrent flow,

e2 ,out = e2(01772) d7)2 (65) lo1

Addition of Eqs. (61) and (63) eliminates the integral term because of the reversal conditions [Eqs. (51) and (52 ) ] and provides the overall balance for the complete cycle as

e2 ,out = P(1 - e l , o u t ) (66)

Equation (66) is completely equivalent to the enthalpy balance for any adiabatic heat exchanger,

e2,0ut = cR(l - el.out) (67)

and subscripts 1 and 2 correspond to the weaker fluid heat capacity flow rate (Cmin entering the exchanger at = 1) and the stronger fluid heat capacity flow rate (C,,, entering the exchanger at & i n = 0), respectively. It is also well known that for any heat exchanger [9]

e2, out = CRE

e1,out = 1 - E

The expression to the right of the equality sign in Eq. (69) is termed the ineffectiveness of the heat exchanger.

Thus the concept of effectiveness (and ineffectiveness) can be used for regenerators as well as for other types of exchangers, and, in the general case, the effectiveness is a function of four dimensionless parameters, A1, U,, p, (+. However, in Section 1,A it stated that the performance of a regenerator was a function of four variables and following Hausen's approach they were A h , A,, n h , n, . This boxed approach does away with hot and cold, and the normalized independent variables [Eqs. (38)-(40)] and the heat balances [Eqs. (43)-(46)] result in the disappearance of the dimensionless regenerator period Il [Eq. (2)] from the functional relationship. The importance of the utilization factor of a period is now apparent and thus this is the reason that the utilization factor and the unbalance factor p play such an important role in the classification of regenerator operation (see Section 1,D and Table I). It is now appropriate to update Table I with regard to the classification of regenerator operation by Table 11. The classification is now only in terms of the two factors-the asymmetry factor y and the unbalance factor p, where y = Amin/Amax and p = U1/U2 [see Eqs. (56)-(58) and (49), respectively] for the fixed- regenerator nomenclature, and y = C * / ( a A ) * and p = C* [see Eqs. (25)


TABLE I1

CLASSIFICATION OF REGENERATOR OPERATION I N TERMS OF y AND p

Classification Asymmetry factor y Unbalance factor p

Symmetric-balanced Symmetric-unbalanced

= 1 =1

Asymmetric-balanced <1 Asymmetric-unbalanced <1

= 1 <1 = 1 <1

and (27), respectively] for the rotary-regenerator nomenclature. The numerical range of these two factors is bounded between 0 and 1.

It is worthwhile considering the physical meaning of the four dimensionless parameters representing the performance of a regenerator (i.e., Al , Ul , p, u, or y). The dimensionless regenerator length of the weaker period A, [Eq. (l)] is the ratio of the transfer potential of the weaker period to the flowing heat capacity of the weaker period. The utilization factor of the weaker period U, [Eq. (49)] is the ratio of the weaker gas-stream heat- capacity flow per period to the matrix heat capacity. When Ul is equal to unity, then for A1 --* CQ, the effectiveness is equal to unity, because all the matrix enthalpy is transferred to the gas stream or vice versa. The unbalance factor p[Eq. (49)] is the ratio of weaker flowing heat-capacity flow rate to the stronger flow heat-capacity flow rate, and reflects the ratio of the temperature change of the stronger fluid between inlet and outlet to that of the weaker fluid. The asymmetry factor u [Eq. ( S O ) ] or y [Eq. (56)] represents the ratio of the average transfer potential between the stronger and weaker periods divided by the reciprocal of the unbalance factor p. Thus if the regenerator operation is asymmetric-balanced, then the asymmetry factor represents the ratio of the average transfer potential between the stronger and weaker periods. It is this type of operation that allows the designer to use the pseudosymmetric-balanced solution to represent the asymmetric-balanced operation, and the pseudosymmetric dimensionless regenerator length is obtained by the harmonic means as

At this point of time, no simple formula has been obtained for the pseudobalanced parameter U, and p = 1, in terms of Ul and p < 1. How- ever, it is now appropriate to consider the effectiveness of regenerators at cyclic steady state (equilibrium).


D. EFFECTIVENESS OF REGENERATORS AT CYCLIC STEADY STATE (EQUILIBRIUM)

The dimensional form of Eq. (66) is

The absolute value signs in Eq. (71) indicate the irrelevance of hot and cold streams, and each term either side of the equality sign is a measure of the actual heat transfer in each period Qact.

The maximum possible heat transfer in either period Q,,, is

= u m i n l T 1 . i n - Tz. in l= UIIT1,in - T2. in l (72) Hence Umin has the same role as Cmin in heat-exchanger terminology and so does Ul. Thus, a single measure of the regenerator performance is

so that

Hence, to establish the effectiveness of a regenerator as a function of the pertinent parameters, it is necessary to solve the differential equations and obtain expressions for either and 02,,,, are related by the overall balance equations for the various periods, which were developed in Section 2,C in terms of el((, vl), 02((, v2), Osl(& vl), and Os2((, v2) , it is possible to choose any one of the completely equivalent formulae from Table I11 for the evaluation of the effectiveness once-the complete solution is available.

The formulae (a) and (b) in Table I11 relate to the enthalpy changes of the gas streams in each period, and note that in (b) it is necessary to specify whether unidirectional flow [i.e., 02(1, v2)] or countercurrent flow [i.e., 02(0, q2)] is being considered. The average driving forces in each period are represented by formulae (c) and (d), whereas the final formula (e) is the energy change of the complete matrix material during a period.

or 02,,,, . However, since


TABLE I11 REGENERATOR EFFECTIVENESS FORMULAE

(a) From the gas outlet temperature distribution in the weaker period:

~ = 1 - 4 , . . ~ = 1 - 4(1,Tdd111 lo1

(b) From the gas outlet temperature distribution in the stronger period:

(c) From the average driving force (transfer potential) in the weaker period:

E = In1j”1[~l(t3 TI) - esl(t9 Ol)1 d t dTl

(d) From the average driving force (transfer potential) in the stronger period:

(e) From the energy accumulated in the matrix during one period:

The existence of a single measure of the effectiveness of a regenerator, as developed here, alleviates the confusion caused by Razelos [lo] and Shah [6] , who have reported that there are separate effectiveness values for each of the hot and cold periods, and a third value for the overall effectiveness, and substantiates the equivalence relationships between the two design methods, which were developed in Section 1,C.

E. ANOMALY OF THE CONCEPT OF OVERALL HEAT-TRANSFER COEFFICIENT IN REGENERATOR DESIGN

There have been several instances in the development of regenerator theory when the concept of an overall heat-transfer coefficient has been proposed for use in the design method in an analogous manner to that used in steady-state heat-exchanger calculations [ l l , 121. This approach can now be formally analyzed by the methodology developed earlier in this presentation.

In the regenerator system depicted in Fig. 2 it is possible to relate the enthalpy change in each stream to the product of the overall heat-transfer coefficient K , the surface area A , and the average driving force in the


regenerator, as follows using the weak stream:

Qact = (M Cf) lITl , in - Tl .outI = m I T * , a v - ~ 2 , a v I (76) where the subscript "av" refers to averaging over the time and space regions of each period. Thus

This equation is equivalent to Ntu = &/Ad used in steady-state heat- exchanger design, where Ad is some mean dimensionless temperature driving force in the system. In regenerators, this mean driving force is

A e = I' I' el(& 771) d 5 d 7 7 1 - I' I' e2(& 772) 4d772 (78) 0 0 0 0

and this has no clear physical meaning since this difference deals with the mean temperature of the gases in the same space, but in different time intervals. Thus a definition of KA from Eq. (77) leads to the following result:

1 - dl(Al9 771) d771 lo1 (79)

((Ya)lL - I' i' el(& 772) 4d771 - I' I' f32(5, 772) d 5 d 7 7 2

KA --

0 0 0 0

which is not related in any way to the overall heat-transfer coefficient used in steady-state heat-exchanger design.

Hence the regenerator analyst should never attempt to define an overall heat-transfer coefficient for a regenerator system in an analogous fashion to those used in steady-state design.

111. Analytical Solution for the Unidirectional Operation

The symmetric-balanced unidirectional regenerator operation has been studied over rather limited parameter ranges by a number of investigators. Hausen [2] used a numerical solution to cover the range, 0 < A < 50 and II = 0(10)50, for the assumptions quoted in Section II,B and presented his effectiveness values in chart form. These were the first results to confirm that the limiting case of zero periods, n = 0, did not provide the maximum effectiveness for the particular value of the dimensionless length.

For values of A greater than 5 , the effectiveness remains at a value of 0.5. For larger values of II, the effectiveness curves for II = 10,20, 30,40,

150 BRANISLAV S . BACLIC AND PETER J . HEGGS

and 50 exhibited the character of a damped vibration around E = 0.50, with the maximum effectiveness of each curve appearing at n / A = 1. These values were less than the corresponding countercurrent effectiveness values.

Kardas [13] obtained an analytical solution for the case where the thermal conductivity of the matrix material was finite in a direction normal to the transfer surface. He considered a symmetric-balanced arrangement over the range of parameters-0 < A s 10, 0.05 =S Bi < 50-and a frequency parameter M from 0 to 5.0. This frequency factor is the square root of the reciprocal of the Fourier number and thus the equivalent dimensionless period is II = Bi/M2, and its range is 0.002 < - II 6 m. The results are presented in chart form in terms of a local effectiveness $, which he states is a departure from the definition Eq. (14), plotted against M for various Bi values. Six charts are included at fixed values of A. The local effectiveness $, is obtained from the mean driving force in the regenerator over a half cycle by integrating the gas and surface temperature difference. This is equivalent to the effectiveness formulae, (c) and (d) in Table 111, and so, E = $. Unfortunately these results cannot be used for the Nusselt-type representation, because even if we assume that the curves for Bi = 0.05 are identical to Bi = 0.0, it is impossible to read effectiveness values for n > 5 (i.e., when M = 0.1).

Kumar [14] obtained a numerical solution to a model similar to that of Kardas [ 131 for both symmetric-balanced and symmetric-unbalanced unidirectional regenerators. The range of parameters reported in Ref. [5] is as follows: 0 S Bi S 10, 0 6 A / n S 10, and 0 s A S 10 with Ph/Pc = 1, 2, and 3. The effectiveness values were plotted against A / n at fixed values of Bi and PJPC for five values of h(=0.5 , 1, 2, 5 , and 10). This means of presentation illustrated that as the period is increased indefinitely then the effectiveness falls to zero, and if the Bi < 1 for a fixed value of A and the period is progressively decreased, then the effectiveness will reach a maximum value and then decrease to its asymptotic value.

Theoclitus and Eckrich [15] have presented effectiveness results obtained from a numerical solution of the Nusselt-type model in terms of the E - NTU, parameters for all categories of unidirectional regenerator operation (see Table I). The parameter ranges are 1.0 S NTU, s 10.0, 0.2 S C& S 03, 0.5 =s C* S 1.0, and 0.25 S (@A)* S 1.0; and in the fixed- bed regenerator nomenclature, the ranges are 1.25 < hl < 20.0, 2.5 < A2 s 80.0, 0 S II, < 100.0, and 0 S 112 S 400.0. The results are presented in tabular fashion for fixed values of C* and (cuA)*, and each table lists effectiveness values for NTU, = 1.0 (1.0) 10.0 and 12 values of C c starting at 0.2 and ending at infinity. The excellence of this investigation is marred by several errors in the text and in the graphical presentation of the results. Nevertheless the results cover a much wider range of parameters


than any of the other investigations, but the use of the E-NTU, nomenclature obscures the effects of the parameters that are present in the governing differential equations [Eqs. (43)-(46).]

Hence to investigate the effects of the parameters in the governing differential equations over the entire range of possible values, it is intended to develop an analytical solution to Eqs. (43)-(46); the unidirectional regenerator problem.

A. SOLUTION FOR THE GENERAL CASE

In Section I1 it was shown that the regenerator performance and particularly the effectiveness E can be obtained if one is able to establish explicitly the dimensionless temperature fields, O,(& q j ) , where i = 1, 2, s l , s2 and j = 1, 2. This is achieved by applying the Laplace transform method to the governing equations of the unidirectional regenerator [Eqs. (43)-(46) and (51)-(54)] with respect to (+ s, that is,

L&,s{Oi(& vj)) = ei(s, vj), i = 1,2, s l , s2 and j = 1 ,2 (80)

The following equations for the temperature fields in the complex plane are obtained in terms of the cyclic equilibrium conditions:

e l k 771) = [ I + Al&l(% 771)1/(s + A,)

82(s, 772) = Ales2(s, v;)/(Us + 111)

e ~ 2 ( ~ , 772) = e ~ l ( ~ 7 7 1 1 = l ) exp[-u1A1772s/P(Us + A1)]

(81)

(82)

(83)

(84)

esi(s, 771) = (l/s) + [es2(s, 772 = 1) - (l/s)I exp[-Ui~iAis/(s + A,)]

By putting vl = 1 into Eq. (82) and v2 = 1 into Eq. (84) it is possible to solve the resulting two equations to give expressions for esl(s, q1 = 1) and eS2(s, v2 = I) as follows:

1 s 1 - exp[-UIAls/(s +Al)]exp[-UIAls/P(~s +A,)]

1 - exp[ - UIAls/(s + A1)] Osl(s, vl = 1) = -

(86)

With Eqs. (85) and (86) it is now possible to find the explicit relationships for the temperature fields el(& vl), 02(& q2), Osl([, vl), and OS2(5, q2) by applying inverse Laplace formulations to Eqs. (81)-(84).

152 BRANISLAV S . BACLIC AND PETER J . HEGGS

It is important to note that the following manipulation has been used

and the following inverse Laplace transformations have been applied

= Vi,o(a, s> exp[-as/(s + l)] (s + 1)' %..*

wherei= 1,2,3 , . . . , 03,

Equations (90) and (91) are general theorems of scaling the argument and convolution; a and b are any real variables in these expressions.

In Eqs. (88) and (89), the special V functions were introduced by Serov and Korol'kov [16] and the modified Bessel functions can be evaluated by a simple algorithm based on Parl's method of calculating the generalized Marcum Q(& f i ) function [17]. The temperature field for the weaker fluidisOS77,Sl a n d O s ( s 1 i s

x {V1,0[mU1A1+ 771 9 5 - UI - V1,0[(m + 1)UlAl + 771 3 5 - 4 ) du (92)

The first term after the equality sign is the solution for the single-blow


operation and the integrand is a series of convolutions that result from the cyclic equilibrium operation of the unidirectional regenerator.

The temperature field for the strong fluid for 0 d q2 d 1 and 0 d 5 s 1 is

x {V1[mU1Al9 5- U] - v,[(m + 1)ulAlY 5 - u]} du (93) The temperature fields for the regenerator matrix for the weak and

strong periods are x.

eSl(57771) = 1 - Vl(77195) + x {exP[-m~lAl + 1111 - exP[-(" + 1)UIAl m=O

x {V2,0[5- u9 "UlAl+ 1111 - V2,0[5- u9 (" + l)UlAl+ 7)111 du

(94) f o r O d ( S 1 a n d O S q l S 1 , and

m

42(59 772) = E {exP[-"UlAll- exP[-(" + l)UlAlI} m=O

['" + 772)ulAl 7 '1 + i' vl[ (" + 772)4A1 I '3

1

v1 up ~7 o m=O UP U

x {V2,0[5 - u9mU1A11 - V2,0[5 - u9 (" + 1)UlAlI} du (95)

for 0 s 5s 1 and 0 d q2 6 1. The effectiveness of the regenerator is obtained by the use of Eqs. (85)

and (86) and the definition (e) of the effectiveness in Table I11 to give

(1 - exp[-UlAls/P(us + Al)l)(l - exp[-UlA~s/(s + A1)1) Uls2(1 - exp[-UIAls/p(us + A1)] exp[-UIAls/(s + Al)])

(96)

& = 2-1 s+&= 1

or

Finding the inverse transform of either Eq. (96) or (97) is probably the most difficult task in the whole procedure. However it can be dealt with

154 BRANISLAV S . BACLIC AND PETER J. HEGGS

very simple by using Eqs. (87) and (96), so that

-m UIAls -(m + l)UIAls

{ (s + A1) ] - exp[ (s + A1)

Equation (96) is a series of convolutions of Vl functions and is

X {Vl[mUIAl, 1 - u] - Vl[(m + l)UIAl, 1 - u] }du (99) The effectiveness has been evaluated from Eq. (99) by numerical in-

tegration using either Chebyshev integration or the trapezoidal rule. If A, a 10, then the trapezoidal rule is more appropriate. The double precision function UNIDRE only consists of 55 lines of FORTRAN 77, but calls on the subroutine for the evaluation of Vl functions. This function has been used to calculate the effectiveness for any arbitrary combination of the four parameters, Ul , Al , p, and u.

However, due to the rather complicated nature of the equations for the temperature fields [Eqs. (92) to (95)], and, to a lesser extent, that for the effectiveness [Eq. (99)], it is intended to present the formulae for the temperature fields and effectiveness for the special cases highlighted in Section I1,B. These will be for various combinations of the regenerator parameters and will be used to identify the special features of the unidirectional regenerator.

B . APERIODIC OR RECUPERATIVE OPERATION

For Ul + 0 the temperature fields for the matrix are identical for each period, and represented as

The fluid temperature fields are given by the following two expressions:


The regenerator effectiveness is

The expression for the effectiveness [Eq. (103)] is equivalent to the expression commonly used to evaluate the effectiveness of an unidirectional 1 : 1 pass recuperative heat exchanger, that is,

E = [1/(1 + c ~ ) ] { l - exp[-Ntu(1 + CR)]} (104) Comparison of Eqs. (103) and (104) reveals that the unbalance factor P of the regenerator is equivalent to ratio of the minimum to maximum flow heat capacities CR of the recuperator, and the group Al/( l +up) is equivalent to the number of transfer units Ntu of the regenerator. Further manipulation of the equivalent grouping for Ntu gives

so that the grouping is also equivalent to the parameter NTU, proposed by Coppage and London [3] for use in rotary regenerator design theory, and substantiates the equivalence between fixed- and rotary-design theory presented by Heggs [7] [see Section II,C and Eq. (24)].

The effectiveness of the aperiodic unidirectional regenerator, ~ ( 0 , A,, P, u), is shown plotted against the grouping A1/( l + UP) for seven values of the unbalance factor p in Fig. 4. This plot is identical to that for an unidirectional 1 : 1 pass recuperative heat exchanger. Plots of this form are very useful for the designer, but do not give any information regarding the temperature fields in the matrix or the fluids, and the effects at the extremes of the parameter ranges.

Hence it is worthy to note the reduced formulae of Eqs. (100)-(103) depend on the extremes of the parameter ranges, 0 s A, s CQ, 0 s P 6 1, and 0 =s a s 1, and several combinations.

For the symmetric (a = 1) regenerator, the matrix temperature fields are constant along the length of the regenerator and equal to

es, = esz = P/(1 + P)

Ol(5) = P[1+ (1/P) exP(-A,5)1/(1 + P) 62(5) = P[1 - (UP) exP(-A,S)1/(1 + P )

E = 4 0 , A,, P, 1) = (1 - exp(Al))/(l + P )

(106)

(107)

(108)

(109)

while the fluid temperature fields vary as

and the effectiveness formula is


E

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

A1 A1 + ap FIG. 4. Effectiveness for the aperiodic regenerator or unidirectional recuperator.

For the balanced ( p = 1) regenerator all the temperature fields vary along the length:

and the effectiveness is

~ = ~ ( o , A , , l , a ) = - 1-exp - 2 { [2]}

For the symmetric and balanced ( a = p = 1) regenerator, the matrix temperature is constant midway between the two inlet temperatures,

e,, = es2 = 0.5 (1 14) and the weak and strong temperature fields are mirror images of each other


around the matrix temperature

el(%$) = f [ l + exp(-A,t)l (115)

02(0 = - exp(-A1S)1 (116)

E = ( O , A1, 1, l )= i [ l - exp( -Al ) ] (117)

and the effectiveness is

For the long regenerators (A1 + x), the temperature fields become constant when there is no driving force and the value is given by

(118)

(119)

6 = e = e - sl s2 l - f32= P/(1 + P )

E = &(O, x, p, a ) = 1/(1 + p)

The effectiveness is now only dependent on p and is

These are the largest effectiveness values that can be attained in a unidirectional aperiodic (Ul + 0) regenerator and correspond to the limiting values of the curves shown in Fig. 4.

For the short regenerator (Al + 0), the fluid temperatures remain at the extrance values, the effectiveness is zero because in reality the exchanger does not exist. However the matrix temperature are somewhere between the fluid inlet temperatures depending on the values of aand p, that is,

(41 = es2 = aP/(1 + P) (120) For the completely unbalanced regenerator ( p + 0), the fluid tempera-

tures and the matrix temperatures for the stronger period remain constant at the strong fluid inlet temperature, whereas the weaker fluid temperature varies along the regenerator as

el(4 = exp(-A15) (121)

(122)

and the effectiveness is given by

E = ~ ( 0 , A1, 0, a ) = 1 - exp(-Al)

It is interesting to note that for this case, the effectiveness and weaker temperature field formulae are independent of the asymmetry factor a .

For the completely asymmetric regenerator ( a + 0), the matrix temperature fields and that of the stronger fluid are identical and vary along the regenerator as given by

e S l ( 0 = es2(5) = 82(0 = PO - exp[-A,(l + P)51)/(1 + P ) (123)


and the temperature field of the weaker fluid is

el([) = Pi1 + exp[-A1(1+ P)51/P}/(1+ P )

E = 4 0 , A1 , P, 0) = (1 - e x p [ - h ( l + P)1}/(1 + PI

(124)

(125)

The effectiveness is given by

Considering the completely asymmetric regenerator (o. + 0), but now with it being balanced (P + l), the effectiveness formula Eq. (125) becomes

E = ~ ( 0 , A,, 1, 0) = 11 - exp(-2A1)]/2 ( 126) and when A, + m, the limiting value of the effectiveness becomes identical to that of the symmetric and balanced regenerator Eq. (117).

The variation of the effectiveness with respect to A, is obtained from Eq. (103) so that

Hence when A, + 0, the initial slope is simply related to the symmetry and balance conditions of the regenerator.

C. LONG REGENERATOR OR ONE WITH VANISHING TRANSFER POTENTIAL IN BOTH PERIODS

In the previous section for the aperiodic operation (U , + 0), it was sbown that when the regenerator becomes long (A, + m) the temperature fields in the matrix and both fluids are identical [Eq. (118)] and the largest effectiveness values are bounded by Eq. (119). It is imperative at this stage to investigate the periodicity effects, that is, finite values of U, and P, at the upper extreme range of Al , the long regenerator.

The long regenerator can be interpreted as a consequence of either extremely intensive convective heat transfer, a very long matrix, or extremely small fluid mass velocity. It is the first interpretation which is most poignant, because this infers that the transfer of heat between the fluid and matrix is infinitely fast so that the temperatures of the fluid and matrix are identical in time and position. Hence this case is also termed the one with vanishing transfer potential in both periods, that is,

h(5 , 771) - %,(5, rll) + 0

02(5, 1)2) - 4 2 ( 5 , 772) + 0

(128) (129)


The temperature field of the fluid and the matrix in the weaker period is now given by

and the temperature field for the strong period is

where

1, 2 3 0

0, z < o H ( z ) =

is the Heaviside step function. These equations state that the normalized temperatures are either equal to 1 or 0 in the regions governed by the arguments of the Heaviside function. The solution of the problem, thus, degenerates to a form of kinematic waves representing the penetration of the energy into and out of the regenerator at the entrance temperature levels. Hence the shapes of the temperature fields at the cyclic equilibrium conditions are totally dependent on Ul and P.

The regenerator effectiveness also reflects these kinematic waves and is given by

E = 4U1, P , 4

Neither the temperature fields nor the effectiveness are dependent on the symmetry (+ of the regenerator.

A series of figures is now presented to illustrate the temperature fields within the regenerator and the outlet fluid temperatures over a complete


cycle ( T~ + q2) at cyclic equilibrium conditions and various values of balance p for different values of the utilization factor of the weaker period U,.

Figure 5 is for the balanced regenerator (p = 1) and for five values of Ul (CQ, 4 ,1 ,* , f ) . For the limiting case of U, + w, the regenerator is saturated at the inlet temperature levels of the fluid streams for both periods and thus the outlet temperatures are equal to the inlet temperatures. Obviously no heat is transferred between the streams, and, in this case, the effectiveness

a

u, -r(o

E = O

u,=4/3

E =3/4

u,= 1

E= 1

u,=1/2

E = O

U,=1/5

E = l

FIG. 5. Temperature distributions in a long balanced regenerator for U,


of the regenerator will be zero. The progressive decrease of the utilization factor Ul results in the inlet fluid seeing the matrix at the inlet temperature of the other period. The amount of heat transferred from the fluid to the matrix, or vice versa, is dependent on the value of U,. The utilization factor represents the amount of heat to be transferred in the period to the total heat capacity of the matrix. For Ul > 1 more heat passes into the regenerator than can be absorbed by the matrix, and, therefore, at some time during the weaker period the outlet temperature will become equal to the inlet temperature. This may be observed for the case of Ul = 4. For three-quarters of the weak period, the outlet fluid temperature is zero and then the matrix is saturated at the inlet temperature and the outlet temperature changes to the inlet value of unity. The effectiveness in this case is $, that is, the inverse of the value of the utilization factor.

For an utilization factor of unity, the inlet energy is identical to the heat capacity of the matrix and thus the outlet temperature is always at the inlet value of the strong period and vice versa. The effectiveness is now unity. Once the utilization factor falls below unity, the inlet energy is less than the heat capacity of the matrix, and the heat becomes trapped within the matrix. The passage of the heat through the regenerator is totally dependent on the value of the utilization factor and so is the effectiveness. At U, = 4, the effectiveness is zero due to the periodic nature at these conditions. At Ul = 4, the effectiveness is unity.

Figures 6, 7, and 8 show progressively the effect of increasing unbalance on the long regenerator for the same five values of the utilization factor used in the totally balanced case ( p = 1) in Fig. 5 . For the limiting case of U, + m, the effects of unbalance are insignificant because the regenerator is saturated at the inlet temperature levels of the two fluid streams for both periods. However, for finite values of U, , the effects of unbalance can be seen to affect the heat stored in the regenerator. From the definition of p(= U,/U,) [Eq. (49)], the utilization factor of the stronger period is always greater than that for the weaker period and this fact is reflected in the various figures. In the extreme [i.e., totally unbalanced case ( p = 0 ) , Fig. 81, the utilization factor of the strong period is infinite and so the regenerator temperature is zero, (i.e., at the inlet temperature).

For Ul = 4 for all values of the unbalance factor, the effectiveness is equal to $ and similarly for Ul = 1, the effectiveness is equal to unity. For values of U, greater or equal to unity the effectiveness is the reciprocal of the weaker period utilization factor and is independent of the unbalance factor. This result is shown in Fig. 9 for the long regenerator with U, s 1.

For values of U, less than unity, the effectiveness of the long regenerator becomes dependent on the value of the unbalance factor. This is shown in

162 BRANISLAV S. BACLIC

a

b

C

d

#I+-

e

AND PETER J . HEGGS

U,+pP

E = O

U,=l

E = l

u,=1/2

E =l

U, = 1/5

E =1

FIG. 6. Temperature distributions in a long unbalanced (p = 0.5) regenerator for U,.

Fig. 10 for six values of l/U1 (=1.5, 2.0, 2.5, 3.0, 3.5, 4.0). For /3 less than or equal to U, the effectiveness is unity for all values of the utilization factor. For U1 bounded by 1.0 and 0.5, and p greater than U1, the effectiveness progressively falls from unity. For U1 less than 0.5, the effectiveness falls to zero and then increases again for /3 greater than U1.

This very interesting and important phenomenon is better illustrated by plotting the effectiveness against the reciprocal of the utilization factor for various values of the unbalance factor-in Fig. 11 for p = 1.0, 0.8, 0.6,

UNIFIED REGENERATOR THEORY

O l ( ? , , l , )

163

C

/---

d

J$-& e

u,= 1 E =1

E u1=1'2 =l m E = l

FIG. 7. Temperature distributions in a long unbalanced ( p = 0.2) regenerator for U, .

0.4, 0.2, 0.0. It is quite apparent from these plots that the effect of unbalance is to extend the region of the reciprocal of the utilization factor for the weaker period where the effectiveness is unity. The length of the zone of effectiveness equal to unity is only dependent on the value of unbalance factor and is given by (l/P) - 1. This length is periodically repeated depending on the value of l /U l . The positions of the zero effectiveness value, and the two regions where the effectiveness increases from zero to unity or decreases from unity to zero are also dependent on the unbalance factor p.

a

b

, ih /$& &

C

,q& d

e LEI7

E = O uo

U , = l

E = l 0

E = l 1

FIG. 8. Temperature distributions in a long totally unbalanced ( p = 0) regenerator for U, .

2.0 0.6 1

3.0 3:5 410 1 0 1.0 0.5

P

FIG. 9. Effectiveness of the long unidirectional regenerator over the completely unbalance factor range for U, 3 1.0.


0

b 1 .o

C 0.3

0

C 0 . 5

P 1 .o

FIG. 10. Effectiveness of the long unidirectional regenerator for U, < 1.0 at l /UI = (a) 1.5, (b) 2.0, (c) 2.5, (d) 3.0, (e) 3.5, and (f) 4.0.


0

d 0.5

P 1.0

0

e 1.0

P 0 .5

0

f 1 .o

P 0.5

FIG. 10. (Continued)


A

0 2 4 6 8 10 b

0 2 4 6 8 1 0 C

FIG. 11 . Effectiveness of the long unidirectional regenerator plotted against l /U1 for p = (a) 1.0, (b) 0.8, (c) 0.6, (d) 0.4, (e) 0.2, and (f) 0.0.


0 3 6

d

I 1 I I I I * 0 2 4 6 8 10 1 /u, e

1

E

0 1

f FIG. 11. (Continued)


The following equations define all these positions and regions:

E = 0 at l /Ul = i[(l/P) + 11 (134)

(135)

(136)

E increases from zero to unity in the regions

i[(l/P) + 11 s 1/U, s i[(l/P) + 11 + i

E = ( 1 / W - i [ (P + 1>/P1

and in each of these regions the value of effectiveness is given by

E = 1 in the regions i -+ 1 + 1 S - S i -+ 1 +- [; I hl [; I ; (137)

and finally, E decreases from unity to zero in the regions

i[(l/P) + 11 + (UP) 1/Ul (i + 1"1/P) + 11 (138)

(139)

with the value of the effectiveness given by

E = 1 - W/W - i[(l/P) + 11 - (l/P))

In Eqs. (134)-(139), i = 0 , 1 , 2 , 3 , 4 , . . ., m. Obviously the effectiveness of the longer generator (Al + m) is only a function of l /Ul and P, and Fig. 12

FIG. 12. Three-dimensional plot of the effectiveness of the long unidirectional regenerator against l/U, and l/p.


shows a three-dimensional plot of the effectiveness against 1/U, and 1/p. The oscillatory nature of the effectiveness and the plateaus of effectiveness values at unity for various unbalance factors are quite apparent.

D. COMPLETELY UNBALANCED REGENERATOR

The completely unbalanced regenerator ( p + 0) implies that the strong fluid period is of infinite duration or that the flowing heat capacity is of infinite size. For these situations, the temperatures of the fluid and the matrix remain constant at zero (i.e., the inlet temperature of the strong fluid) throughout the regenerator during the strong period:

e, = e,, = o (140)

E

FIG. 13. Effectiveness of the completely unbalanced unidirectional regenerator plotted against l /&.


The temperature of the fluid and the matrix during the weak period vary as

el(& rll) = Vl(AI5, UlA1771)

&1(5,7)1) = 1 - K(UlAlrl1, A151

E = E ( U ~ , A1 , 0, (+) = 1 - V,(A,, UiAi)/U,Al

(141)

(142)

(143)

The effectiveness of the unidirectional regenerator for this condition is

The temperature distributions [Eqs. (141) and (142)] and the effectiveness [Eq. (143)] are independent of the asymmetry factor (+, and only dependent on the dimensionless length A1 and utilization factor U, of the weaker period. The effectiveness is plotted against l /Ul for various values of A, in Fig. 13. The curves for all finite values of A, are to the right of the case of the long regenerator (A, + m) for p = 0.0 (i.e., the plot for p = 0.0 of Fig. 10).

The effectiveness for the range of l /Ul greater than 6 are shown in Fig. 14 where the abscissa is now Ul. For values of A, less than 5 the effectiveness appears not to be unity at Ul = 0. The values of the effectiveness at U, = 0 are given by Eq. (121), which is repeated here for completeness,

E = ~ ( 0 , A , , 0, (+) = 1 - exp(-A,) (144)

For small values of the dimensionless length, there is insufficient transfer of heat between the fluid and matrix during the passage of the initial plug to fluid through the regenerator to change the weak fluid temperature to that of the inlet temperature of the strong period. Hence, the fluid temperature at outlet of the regenerator for the beginning of the weak period is greater than zero. For larger values of Al the outlet temperature rapidly approaches zero as can be inferred from Eq. (144).

This special case is also equivalent to the situation encountered in the heating or cooling of a packed bed, whereby the packed bed is initially at a constant temperature throughout, and the fluid inlet temperature is changed instantaneously (i.e., a true step change). This problem has been the subject of many investigations in the literature starting with Anzelius [MI, Nusselt [l], and Schumann [19] between the years 1926 to 1929. The outlet response curves for a step change input are shown in Fig. 15 for a wide range of dimensionless bed-length values. It is quite apparent that for values of A1 less than 5, then the outlet response curves experience an instantaneous breakthrough at U = 0.0. These results substantiate the findings shown in the effectiveness chart in Fig. 14.

The effectiveness charts in Figs. 13 and 14 may be used to predict the amount of heat stored in the regenerator matrix during the weak period or

FIG. 14. Effectiveness of the completely unbalanced unidirectional regenerator plotted against Ul.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1 . 4 1 .6 1.8 2.0

DLMENSIONLES TIME RI)

FIG. 15. Outlet temperature response curves for a step change input into a packed bed.


for the packed-bed situation, the heat stored or removed from the solid packing up to the utilization factor value Ul :

Q = A A 1 - CP)P&~IE (145)

The effectiveness charts in Figs. 13 and 14 can also be used to predict thL effectiveness of cross-flow heat exchangers with both streams unmixed. The effectiveness formula [Eq. (143)] is identical to that for the cross-flow heat exchanger [20]. However, it is necessary to replace A, by Ntu and Ul by o.

E. BALANCED AND SYMMETRIC REGENERATOR

generators, the effectiveness is given by a reduced form of Eq. (97) as For the balanced ( p = 1) and symmetric (a= 1) unidirectional re-

The inversion of the Laplace transformation in Eq. (146) results in the use of the V2 function:

u, { rn=l (mulA1 A1

3c.

& = - 1 + 2 ( - l ) V ,

or by

cos((2n + 1).rA:U1/[(2n + 1)2n2 + A:U:]} (2n + 1)2

11 -(2n + 1)2~2A1 (2n + 1 ) 2 ~ 2 + A:U:

(147)

Both Eqs. (147) and (148) may be used to evaluate the effectiveness of the balanced and symmetric unidirectional regenerator. However, convergence is much quicker, if Eq. (147) is used for A1 < 5 and Eq. (148) is used for A1 3 5.

The oscillatory character of the effectiveness for fixed values of A, over a range of U1 values is very apparent in Figs. 16 and 17. As the value of A l is progressively increased, the effectiveness oscillation is bounded within the envelop formed for the long balanced regenerator (A1+%). The

174 BRANISLAV s. BACLIC AND PETER J. HEGGS

6 10 1 4

FIG. 16. Effectiveness of the balanced and symmetricc unidirectional regenerator.

oscillation is also around an effectiveness value of 0.5, which is the asymptotic value for Ul = 0 or l /Ul = 03.

F. UNBALANCED AND SYMMETRIC REGENERATOR

The effectiveness for the unbalanced (P # 1) and symmetric regenerator ( (T = 1) is given by

The effect of unbalance is to spread the oscillations observed in the effectiveness values for large values of A, and also to raise the asymptotic value of the effectiveness around which the oscillations occur. This asymptotic


FIG. 17. Three-dimensional plot of the effectiveness of the balanced and symmetric unidirectional regenerator.


1.0

E

i I 1

I j ~

I L 1 , 1 2 3 4 5

i/u, = A , i n ,

FIG. 18. Effectiveness of the unbalanced ( p = 0.4) and symmetric unidirectional regenerator.

value corresponds to 1/(1+ p). Figures 18 and 19 illustrate these characteristics for p = 0.4.

IV. Concluding Remarks

The most important conclusion that can be made from this discourse is that the oscillatory behavior of the unidirectional completely disappears by increasing the unbalance ( p + 0). Thus, the real unidirectional regenerator can operate over a wide range of l /Ul values at favorably high levels of the effectiveness, provided that a high unbalance (small p) is met at sufficiently high values of the dimensionless regenerator length Al .


b FIG. 19. Three-dimensional plot of the effectiveness of the (a) unbalanced ( p = 0.4) and

(b) symmetric unidirectional regenerators.

178

0

P Y &

77

e

A

A a Bi C

C*

CR

CR* cr

C

d KA

L M M m N

NTU,

act

circ f fc fh h in long

C

BRANISLAV S. BACLIC AND PETER J. HEGGS

NOMENCLATURE

area (m’) area per unit volume (m2/m3) Biot number fluid heat capacity rate, MC,,

capacity rate ratio, Cmin/Cmax,

ratio of minimum to maximum flowing heat capacities for recuperators

capacity ratio, Cr/Cmin, Eq. (8) rotor matrix heat capacity rate,

Wc,N, Eq. (11) (kJ/sec K) heat capacity (kJ/kg K) equivalent diameter (m) overall heat transfer coefficient

Eq. (10) (kJ/sec K)

Eq. (7)

area-product for regenerators (W/K)

length of regenerator (m) frequency factor [13] mass flow rate (kg/sec) mass velocity (kg/m2 sec) number of revolutions per

overall number of transfer units, second (l/sec)

Eq. (6)

NTU

Ntu

P

9r Q Q Re Ra St S

T 1 U

UA

V

W X

Greek Symbols

heat-transfer (W/m2 K) A

asymmetry factor, Eq. (56) 5 unbalance factor, Eq. (4) or (49) A*

thermal effectiveness, Eqs. (14) II

normalized time, Eqs. (39) and p and (15)

(40)

Eq. (41) normalized temperature,

dimensionless regenerator length, Eq. (1)

actual cold circumferental fluid frontal area for cold stream frontal area for hot stream hot inlet longitudinal

U

9

* Subscripts

m max min out Ph S

X 1 2

number of transfer units, aA/C,

number of transfer units for a

duration of a regenerator period

Prandtl number heat duty (W) heat stored, Eq. (145) (kJ) Reynolds number radius of matrix (m) Stanton number constant in the Laplace

temperature (K) time (sec) utilisation factor, Eqs. (3) and

recuperator overall heat transfer

Eq. (12)

recuperator UA/Cmi,,

transform

(48)

(W/K) coefficient-area product

interstitial fluid velocity (m/sec) mass of matrix (kg) distance of the entrance of the

matrix (m)

thermal conductivity (W/m K) equivalent conductivity (W/m K) normalized time, Eq. (38) dimensionless regenerator

density (kg/m3) asymmetry factor, Eq. (50) matrix voidage ratio of free

volume to total volume local effectiveness [13]

period, Eq. (2)

mean maximum minimum outlet phase solid cross sectional weak period strong period


ACKNOWLEDGMENTS

The bulk of this work was completed when both authors were in the Department of Chemical Engineering at the University of Leeds. The first author is indebted to the Science and Engineering Research Council, United Kingdom for providing a Senior Visiting Research Fellowship for one year. Finally, Jane Gibb deserves special praise for typing the manuscript.

REFERENCES

1. W. Nusselt, Z. VDZ3, 85 (1927). 2. H. Hausen, Z . Angew. Math. Mech. 9, 173 (1929). 3. J . E. Coppage and A. L. London, Trans. A S M E 15, 779 (1953). 4. J. E. Johnson, ARC Tech. Rep. No. 2360. HM Stationery Off. London, 1952. 5 . F. W. Schmidt and A. J. Willmott, “Thermal Energy Storage and Regeneration,”

p. 120. McGraw-Hill, New York, 1981. 6. R . K. Shah, in “Heat Exchangers-Thermo-Hydraulic Fundamentals and Design”

(S. Kakac, A. E. Bergles, and F. Mayingers, eds.), p. 495. Hemisphere, Washington, D. C., 1981.

7. P. J. Heggs, in “Low Reynolds Number Flow Heat Exchangers” (S. Kakac, R. K. Shah, and A. E. Bergles, eds.), p. 369. Hemisphere, New York, 1982.

8. S. J. Mitchell, Investigation of the factors affecting the operation of thermal regenerators. Ph.D. Thesis, Univ. of Leeds, 1982.

9. W. M. Kays and A. L. London, “Compact Heat Exchangers,” p. 31. McGraw-Hill, New York, 1984.

10. P. Razelos, in “Compact Heat Exchangers-History, Technical Advancement and Mechanical Design Problems” (R. K. Shah, C. F. McDonald, and C. P. Howard, eds.), HTD, Vol. 10, p. 91. ASME, New York, 1980.

11. H. Hausen, “Heat Transfer in Counterflow, Parallel Flow and Cross Flow,” p. 387. McGraw-Hill, New York, 1983.

12. A. Shack, “Industrial Heat Transfer,” p. 275. Wiley, New York, 1965. 13. A. Kardas, Int. J . Heat Mass Transfer 9, 567 (1966). 14. M. Kumar, Periodic response of a parallel flow, solid sensible heat thermal storage unit.

15. G. Theoclitus and T. L. Eckrich, Heat Transfer Conf. 3rd 1, 130. AIChE, New York,

16. E. P. Serov and B. P. Korol’kov, “Dinamika Parageneratorov.” Energiya, Moscow,

17. S . Parl, ZEEE Trans. Znf. Theory IT-26, 121 (1980). 18. A. Anzelius, Z . Angew. Math. Mech. 6 , 291 (1926). 19. T. E. W. Schumann, J . Franklin Znst. 208, 405 (1929). 20. B. S. Baclic and D. D. Gvozdenac, in “Regenerative and Recuperative Heat Exchang-

ers” (R. K. Shah and D. E. Metzger, eds.), HTD, Vol. 21, p. 27. ASME, New York, 1981.

M.S. Thesis, Pennsylvania State University, University Park, 1978.

1966.

1972.


Thermal Control of Electronic Equipment and Devices

G. P. PETERSON

Department of Mechanical Engineering, Texas A&M University, College Station, Texas 77843

ALFONSO ORTEGA

Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona 85721

I. Introduction

Increases in circuit integration along with efforts to reduce the operational execution time of individual components and overall systems have resulted in large reductions in the size of semiconductor chips. Although reductions in transistor switching energies have occurred, these reductions have not kept pace with the increases in gate density. Hence, the heat flux levels within these devices have increased by several orders of magnitude. With circuit densities continuing to increase at a rate of 30% per year, the problems associated with the thermal control of electronic devices will continue to multiply. Also contributing to these problems are increases in the power levels of other, larger devices. Diode ratings are now 2500 V and 1600 A, and thyristors with ratings in excess of 2500 V and 1000 A are being developed [l]. In these high-capacity power components, the heat flux or thermal dissipation required frequently reaches 50 W/cm2 and in some instances as high as 200 W/cm2 [2].

Thermal control of electronic components has one principal objective, to maintain relatively constant component temperature equal to or below the manufacturer’s maximum specified service temperature, typically between 85 and 100°C. Investigations have demonstrated that a single component operating 10°C beyond this temperature can reduce the reliability of some


182 G. P. PETERSON AND ALFONSO ORTEGA

systems by as much as 50% [3]. For this reason, new thermal control schemes must be capable of eliminating hot spots within the electronic devices, removing heat from these devices, and dissipating this heat to the surrounding environment. Figure 1 compares the thermal acceleration factor, defined as the ratio of the failure rate at temperature T and at 70°C, with the mean chip operating temperature. This figure illustrates the relationship between the reliability of bipolar digital components and their operating temperature and clearly demonstrates the need for a well- controlled thermal environment.

Thermal control schemes to remove the heat from individual devices and systems include the traditional means of free and forced gaseous and liquid convection as well as conduction and radiation or combinations thereof [4]. Several strategies have been developed within the past ten years for controlling and removing the heat generated in multichip modules. These include advanced air-cooling schemes, such as the one in the

TEMPERATURE (OC)

FIG. 1 . Effect of temperature on the failure rate of bipolar digital devices; P= failure rate at T failure rate at 70°C (Reprinted with permission of McGraw-Hill Publishing Company [1181.)

THERMAL CONTROL OF ELECTRONIC EQUIPMENT 183

IBM 4381 midrange processor [5], the Hitachi S i c RAM module [6 ] , and the Mitsubishi high-thermal-conduction module [7]; direct cooling, without phase change, such as that in the Cray-2 supercomputer [8] or with phase change such as the liquid-encapsulated module [9] where the electronic package is submerged in a liquid pool; and indirect cooling schemes, such as the IBM thermal-conduction module (TCM) [lo] in the IBM 3081/3090 processor or the liquid-cooled module (LCM) [ l l ] in the NEC SX. Other more advanced proposed techniques include miniature thermosyphons [ 121 or free-falling liquid films [13]. Incropera [14] has described these strategies and summarized the specific advantages of each type of application and the problems that must be addressed for each approach.

Although air cooling is the best understood, it is limited in the heat removal rate by the convection coefficient. Direct cooling methods are capable of attaining extremely high heat flux levels, but they present problems with contamination and are extremely expensive. Although an issue of considerable discussion, indirect cooling strategies are the best near- term solution for the thermal control of advanced computer architec- tures and as noted by Incropera [14], “The most fruitful approach to enhancing the performance of indirect cooling technologies is likely to be one which reduces the thermal path between the electronic packages and the (cooling fluid).”

Improvement of existing techniques for cooling individual devices or arrays similar to those in multichip modules, requires knowledge of the fundamental limitations and capabilities of each method. Although the need for additional development and research in the area of electronic thermal control has been highlighted in several articles [ 15-17], none has comprehensively reviewed the many ongoing investigations of the fundamental phenomena and new techniques under investigation. The following represents such a review, including a summary of the analytical, numerical, and experimental work as reported in the literature, in order to facilitate the improvement of existing schemes and provide a basis for the development of new ones. This review focuses on, but is not limited to, investigations performed over the last decade and includes information on the thermal control of semiconductor devices, modules, and total systems. The review is divided into sections that deal with fundamental research areas rather than specific applications.

11. Natural Convection

Natural convection has historically played an important role in the cooling of electronic equipment. Despite significantly lower surface heat- transfer coefficients, natural convection in air is preferred for low-end

184 G . P. PETERSON AND ALFONSO ORTEGA

applications, compared to forced convection or immersion boiling, because of its inherent reliability and simplicity. Communications switching devices, avionics packages, electronic test equipment, consumer electronics, and low-end computer packages are often cooled by natural convection in air. In air, device heat fluxes are limited to roughly 0.1 W/cm2 for allowable maximum temperatures of 100°C, as shown in Fig. 2, or on the order of 1 W maximum heat dissipation for a conventional cased chip module. Long-wavelength radiative heat transfer and substrate conduction may be as significant as natural convection in configurations with large temperature nonuniformity .

Use of natural convection in inert liquids by direct immersion has been investigated since the 1970s as one means of handling increased surface heat flux from small packages [14]. Heat fluxes on the order of 1.0 to 10 W/cm2 have been dissipated from small immersed heat sources [14,17], with an improvement of at least an order of magnitude over natural convection in air, and comparable to and in some cases superior to forced convection in air. Most of the work described in Section II,A, especially

FIG. 2. Temperature differences attainable as a function of the heat flux for various heat-transfer techniques. (Reprinted with permission of McGraw-Hill Publishing Company [1181.)


the experimental work, refers to natural convection in air. Section II,D specifically addresses natural convection in liquids.

A. VERTICAL PLATES AND CHANNELS

Large equipment frames or cabinets are typically cooled by the buoyancy-induced flow, or draft, that is created by the density imbalance between the inside of the cabinet and its environment. A typical configuration of in-line, rack-mounted, electronic circuit boards is shown in Fig. 3a. Each board typically has one or more electronic modules plugged into its surface.

Figure 3b shows a card-on-board configuration, where the cards carry power-dissipating chips mounted inside modules, Fig. 3c. Flow was induced into the enclosure and into the vertical channels formed by adjacent

a

C

b

\ BOTTOM SIDE

/ TOP SIDE CONNECTOR CONNECTOR

ALUMINIUM CAP

/(CHIP ,SOLDER PADS

SUBSTRATE

I I II - 1 I I rCARD

PIN

FIG. 3 . Typical natural convection cooled electronic packages. (a) Vertical cabinet with in-line array of circuit cards [23]. (b) Example of card with multiple electronic modules [108]. (c) Individual module showing paths for heat flow [108].


TABLE I

STUDIES OF BUOYANCY-INDUCED FLOW AND HEAT TRANSFER I N VERTICAL OPEN CHANNELS

Study Geometry Boundary conditions"

Aihara [18] Aung [20]

Aung et al. [22]

Bar-Cohen and Rohsenow

Bodoia and Osterle [24] Carpenter et al. [75]

Churchill [25]

Currie and Newman [26] Dalbert [28] Davis and Perona [29] Dyer [31] Dyer [30] Elenbaas [32] Engel [33, 341 Hetherington and Patten

Kettleborough [38] Lauber and Welch [39]

[761

[361

Meric [42] Miyatake and Fujii [45] Miyatake and Fujii [44] Miyatake et al. [46] Miyatake and Fujii [45] Nakamura er al. [48] Ofi and Hetherington [49] Ormiston et al. [50]

Ostrach [51] Sobel et al. [73]

Sparrow et al. [57] Sparrow and Prakash (711 Prakash and Sparrow [72]

Sparrow and Bahrami [56]

Sparrow et al. [74]

Quintiere and Mueller [53] Wirtz and Stutzman [59]

2-D channel 2-D channel

2-D channel

2-D channel


2-D and circular channel

2-D channel, open edges 2-D channel Circular channel Circular channel Circular channel 2-D channel, open edges 2-D channel Circular channel


Circular channel 2-D channel 2-D channel 2-D channel 2-D rhannel 2-D channel 2-D channel 2-D channel in large 2-D

2-D channel 2-D channels, in-line and

staggered 2-D channel 2-D channels, staggered 2-D channels, continuous and

2-D channel, open or closed

2-D channel

room

discrete (in-line & staggered)

edges


UWT symmetric UWT and UHF, symmetric and

asymmetric UWT and UHF, symmetric and

asymmetric UWT and UHF, symmetric and

asymmetric UWT symmetric UHF symmetric and asymmetric

with radiation UWT and UHF, symmetric and

asymmetric UWT, symmetric UHF, asymmetric UWT and UHF UWT, restricted entry UHF UWT, symmetric UWT UHF

UWT, symmetric, transient solution UHF, asymmetric, fully developed

flow UWT UHF, asymmetric UWT, asymmetric One wall UHF, other insulated One wall UWT, other insulated UWT, symmetric UWT UWT, asymmetric

UWT, symmetric and asymmetric UHF

One wall UWT, other insulated UWT UWT

UWT (mass-transfer analogy)

One wall UWT, other insulated, but

UWT, symmetric and asymmetric Uhf, symmetric

with radiation effects.

UHF,uniform heat flux and UWT, uniform wall temperature.


boards. The channels formed by adjacent boards had one relatively smooth wall and the other that was irregular, or hydrodynamically rough, because of the protruding modules and the spaces separating them. Heat was nonuniformly dissipated, partly from the electronic module surfaces and partly by conduction to the substrate board, then by convection through the exposed wall surfaces on both sides of the board.

Many basic studies have been performed on vertical round tubes, two- dimensional channels, and cabinets [ 18-77] to investigate effects such as variable properties [19], entrance conditions [18,38,48], asymmetric heating [20-23, 43-46], open or closed edges [56], flow restrictions [31, 61,621, radiation [74,75], and mixed convection [28,37,41,52]. Table I contains a partial list of contributions showing the geometry, boundary conditions, and other significant considerations. The earliest work on the two-dimensional channel is the experimental study of Elenbaas [32] on a channel formed by parallel plates at constant temperature. The developing flow problem was first studied analytically by Ostrach [51,52] and numerically by Bodoia and Osterle [24]. Significant early contributions were also made by Engel and Mueller [33,34]. The more recent studies of Aung [20-23,61,62], Miyatake [43-461, and Sparrow [55-57,70-72,741, and their co-workers are particularly extensive and relevant to the electronics cooling application.

The analysis of Aung et al. [22] illustrates the scaling employed for the general problem of symmetric or asymmetrically heated channels with constant temperature or constant heat flux at the channel walls. With the nomenclature shown in Fig. 4, the governing equations and boundary conditions can be written in the form:

bv X y,Y 6’

V = - , x=- LvGr’ v L G r ’

b2u u=-

and the dimensionless flow rate

(P - POP4 pL2 v2 Gr P =


To Po

FIG. 4. Nomenclature for vertical channel with symmetric or asymmetric heating [23].

For uniform heat flux (UHF), the boundary conditions are

U = M , V=O, 8 = 0 a t X = O , O<Y<1 (5) U = O , V=O, ae/aY=-rH at XrO, Y=O (6) u=o, v=o, ae/av= 1 at XrO, Y = l (7)

X = 1/Gr (8) P = o at X=O and

For uniform wall temperature (UWT), the boundary conditions are

U = M , v=o, f3=0 at X = O , O<Y<1 (9) U = O , V=O, 8 = r T at X r O , Y=O (10) U = O , V=O, 8=1 at XrO, Y = l (11)

(12) P = O at X=O and X = l / G r


In this formulation, the velocity was assumed to be uniform at the channel entrance, and the static pressure to be equal to the ambient pressure at x = 0. Aung noted that the uniform velocity assumption made this formulation inaccurate for large channel spacing, and Aihara [18] and others showed that the static pressure at the inlet had to be corrected for the pressure deficit incurred upon accelerating quiescent fluid to the channel entrance velocity. From this scaling, the relevant definitions of channel Rayleigh number are given by

Ra* = (g/3qb5/kv2L)Pr (13)

Ra = [g/3(Tw - To)b4/v2L]Pr (14)

for uniform heat flux and uniform temperature, respectively. Elenbaas [32] first demonstrated that a Rayleigh number for a channel modified in this way would allow the correlation of results over the entire range of wall-to- wall spacing, including the isolated plate. Figure 5 illustrates the excellent agreement between Elenbaas' measured plate-averaged Nusselt number and the more recent numerical computations of Aihara [MI, who used both uniform and parabolic velocity profiles at the entrance. Asymptotic behavior was achieved at small Rayleigh numbers, representing fully developed, buoyancy-induced flow in a channel with spacing much smaller

101 I ' ' " " " I ' ' """I ' I ' " " ' I ' ' ' " I

100

U, = PARABOLIC ----- I$

10-1

I

10-21 ' ' 1 1 1 1 1 1 1 ' ' 1 1 1 1 1 1 1 ' ' 1 1 1 1 1 1 ' ' ' ' 1 1 1 * 1 ' ' ' ' J I

10-1 100 101 102 103 104

Ra

FIG. 5. Average plate Nusselt number for isothermal plates computed by Aihara and compared to Elenbaas' data for air [HI.

1 .o

0.8

0.6

0.4

e - 4 n a x . 1

- - 099

U UO

0.2

0.0

1.5

1 0

0 5

0

0

0

0

0 0 0.5 1 .o

Y

- e ernax, 1

u UO

1 .o

0.8

0.6

0.4

0.2

0.0

2.0

I .5

1 .o

0.5

0

0

0

0

n 0 0 5 1 .o

Y

FIG. 6. Computed velocity and temperature fields for asymmetrically heated channel [22].

10-1 100 101 102 103 104 105

FIG. 7. Average plate Nusselt number for channel heated with uniform heat flux, differen! on each wall [22]; (-) numerical solution for all rH [22]; compared to the integral solution. rH = 1 (0) [33].


than its length and large Rayleigh numbers, where each channel wall approached the behavior of an isolated vertical plate. The interferograms of Wirtz and Stutzman [59] for uniform heat flux walls clearly illustrate the transition from fully developed channel flow to isolated plate behavior. Wirtz and Stutzman concluded that true isolated plate behavior is not achieved even at spacings that would be considered very large for electronic circuit board arrays. The computed local velocity and temperature profiles of Aung et al. [22] shown in Fig. 6 for one adiabatic and one UHF wall illustrate the fully developed behavior (small Ra) on the left, and the highly skewed, boundary-layer behavior on the right for large plate spacing. In the same study, it was demonstrated that results for asymmetrically heated channels could be correlated solely on the channel Rayleigh number when it was defined in terms of the average of the temperature (for UWT) or heat flux (for UHF) of each plate. Figure 7, for example, shows numerical calculations of the average Nusselt numbers for uniform heat flux as a function of the channel Rayleigh number. The modified Rayleigh number was defined in terms of the average of heat flux on each wall q and the Nusselt number in terms of ?j and the average of the midheight temperature on each plate, TLI2 - To. Similar results were found for the cases of isothermal walls. Results for the induced flow rate M pointed to a significant difference between the channel with isothermal walls (Fig. 8) and its counterpart with uniform flux (Fig. 9). When fully developed flow was achieved, the induced flow for the isothermal walls was not affected by further increases in channel length, depending solely on T, - To. This limit was achieved when the fluid became saturated, that is, when the fluid temperature approached the wall temperature, and the density ceased to change. For uniform heat dissipation on the walls, the fluid temperature increased continuously and the induced flow rate increased as (qL)”*, thus depending on overall heat dissipated.

Asymptotic limiting relations for Nusselt number and induced-flow rate, for the isolated plate and fully developed channel flow, are readily available and are summarized in Table 11. The fully developed results follow the approximate derivations of Miyatake et al. [43-461, which assume a parabolic velocity profile. Aung [20] developed exact results from direct analytic solution of Eqs. (1)-(12) that were identical to the approximate solutions except for the absence of the constants 17/70 and 13/35. These constants represent the inverse of the fully developed mixed mean Nusselt number and are identical to the values for a forced channel flow. The equations for the isolated plate generally follow the analysis of Sparrow and Gregg [78], although there is debate as to the appropriate values for the constants appearing in Eqs. (17), (23), and (24). Several investigators [25,55,59,76] have combined these limiting relations using the method of


FIG. 8. Dimensionless induced flow rate for isothermal plates [22]

100

10-3 1

I I I I

0.5

5 104 10-3 10-2 10-1 100

(-) FIG. 9. Dimensionless induced flow rate for uniform heat flux plates [22].

TABLE I1

ASYMPTOTIC LIMITING RELATIONS FOR LAMINAR BUOYANCY-INDUCED FLOW BETWEEN VERTICAL PARALLEL PLATES~

Fully developed channel Isolated place

M Nu,/z Nu, NUL/2 Nu,

UWT, 1/12 (15) Ra/24 (16)

UWT, 1/12 (18) Ra/12 (19) symmetric

- 0.519 Ra'/4 (17) -

- 0.519 Ra1'4 (17) - asymmetric

UHF, 4- (20) [(Jm) + (17/70)]-' (21) [(Jm) + (17/70)]-' (22) 0.596 Ra*'/' (23) 0.519 Ra*'l5 (24) symmetric

UHF, 4- (25) [(m) + (13/35)]-' (26) [(~'m) + (13/35)]-' (27) 0.596 Ra*'l5 (23) 0.519 Ra*'l5 (24) symmetric

~~

a Numbers in parentheses refer to appropriate equations in text.

TABLE 111 SUMMARY OF COMPOSITE RELATIONS AND OPTIMUM PLACE SPACING FOR VERTICAL PLATE ARRAYS".~

NU,/, Nu, Optimum spacing (bOpt)' ~ ~~

2.714P-' 25 (29) UWT, [ ( 576/Ra2) + (2. 873/Ra1/2)] '/' (28) -

U r n , [ ( 144/Ra2) + (2.873/Ra'/2)]-'/2 (30) 2.154P-' 25 (31) symmetric

asymmetric

symmetric

asymmetric

"From Bar-Cohen and Rohsenow [76].

-

U r n , [(12/Ra*) + (1.88/Ra*' 4)]-1/2 (32) [(48/Ra*) + (2.51/Ra*04)]-'/2 (33) 1.472R-'' (34)

[(6/Ra*) + (1.88/Ra*' 4)]-'/2 (35) [(24/Ra*) + (2.51/Ra*' 4)]-1/2 (36) 1.169R-" (37) UHF,

Numbers in parentheses refer to appropriate equations in text. For negligibly thick plates.

dBased on T, at L/2.


Churchill and Usagi [79] and have derived algebraic correlations for the Nusselt number that apply over the entire range of channel Rayleigh numbers. Bar-Cohen and Rohsenow [76] performed this procedure for the configurations given in Table 11. The results of this procedure are summarized in Table 111. These relations are useful in the design process and also for deriving quantities such as optimum spacing of PC boards for maximum heat dissipation.

1. Comparison with Actual Electronic Circuit Boards

Johnson [63] compared Eq. (33) in Table I11 and a similar result from Wirtz and Stutzman [59] to an ensemble of measured Nusselt numbers at a channel exit from a variety of experiments on actual printed wiring boards with and without discrete protruding electronic components. Figure 10 presents Johnson's data, reduced to conventional channel coordinates and compared to the limits for a uniform heat flux channel. Most of the data were for Ra* < 300, which represented applications with densely populated, narrow-channeled, and intensely powered assemblies. The data agreed well with the asymptotic equations, but not enough data were available in the transition regime to attempt any conclusions. Figure 11 compares the more extensive data set of Birnbreier [64], taken in vertical stacks of actual PC boards densely populated with heat-dissipating

lo2 7 I

FULLY DEVELOPED CHANNEL Nu = 0.144 Ra"35

ISOLATED PLATE Nu = 0.524 Ra.02

0 JOHNSON AND DUFFY Q SAXENA A SMITH 0 STARKEY

too 101 102 103 104 105

Ra'

FIG. 10. Experimentally measured Nusselt number at channel exit for actual printed circuit boards compared to asymptotic solutions for laminar flow in smooth channels [63].


FULLY DEVELOPED

10‘ 102 103 104 105 106

Ra’

FIG. 11. Experimentally measured Nusselt number at channel exit for a vertical array of printed circuit boards [64].

resistors. The data represent parameter ranges of 40 cm < L < 160 cm, 1.4 cm < b < 6.0 cm, and 0.5 W < Q 5.0 W, where Q is the total power dissipated from a board. Birnbreier’s data show significant departure from the fully developed channel result and a one-quarter power dependence on Ra*, compared to the one-fifth power expected from the isolated plate limit. It has been noted [80] that the Birnbreir data appear to lie somewhere between the expected results for smooth-walled channels and channels with very large protrusions. The effects of surface grooves and protruding components offer a possible explanation for the distinct behavior. Aung et al. [61,62] performed extensive measurements for in-line and staggered card arrays mounted in an electronic cabinet with relatively smooth wire-wrapped cards and cards with discrete, protruding components. This model is reproduced in Fig. 12, with representative data reproduced in Fig. 13, for an array of six vertically stacked cards in the interior of the cabinet. Reasonable agreement was apparent for the smooth channel results, despite the presence of flow obstruction from the card carriers and surface components. In particular, the maximum wall temperatures were well predicted for closely spaced boards. Coyne [65] performed an analysis that accounts for radiation and air infiltration from the channel “edges” for symmetric and asymmetric heat dissipation. This analysis compared favorably with the measured temperature rise in an actual PCB channel.


a 1'x l ' x 118'

CHANNEL Tantalum Reslstors

- COMPONENTS CARDS

1 2 3 4 5 6 ;

CARRIER

L

SPACING

4 1'

9 ' d 1

a b 1'x l ' x 118'

CHANNEL Tantalum -%

--..-.a I - m m -WIRE CARDS

- ALUMINUM PLATE

1-1116' r

f L

COMPONENT CARD \ Nichrome Wire Resistors

31 1

+ Thermocouple

7 Mylar Coated Epoxy Glass 6'xS'x 1116'

31 16'

WIRE CARD

FIG. 12. Experimental prototypes for natural convection studies of Aung, er al. [61,62]: (a) schematic of cabinet and (b) details of discrete component and wire-wrapped boards.

v a

u,

CI t-

Y

W

U W U 3 t- U U W n i5 t-

a

50 A TEST (with rear wall)

- 0 TEST (no rear wall)

4o b=7/16 in.

- = 5.7 win -

30 -

5

b

v 8oc

/THEORY [22]

0 TEST 1 A TEST 2

L=3.5 n

I I I I I I I I

0.25 0.50 0.75 1.00 1 5

DISTANCE FROM CHANNEL BOTTOM, f t CLEAR CHANNEL SPACING, in.

FIG. 13. Aung er al. [61] measured temperatures in an actual array of PCBs: (a) centerline temperature as a function of elevation and (b) maximum wall temperature dependence on clear channel spacing.

198

L

r v)

0.1

0.01.

G. P. PETERSON AND ALFONSO ORTEGA

BLOCKAGE ALONG I ONEL/ /. *-:g

/;-- BLOCKAGE ALONG -

BOTH LATERAL EDGES d'

I 1 1 1 1 I 1 1 1 1 1 1 1 1 ' I I

Air infiltration through the lateral (side) edges of a channel constructed from parallel planes may have a significant effect on the convective heat transfer and may explain some of the discrepancies between experiments on PCB channels and the two-dimensional analysis. Sparrow and Bahrami [56] performed an experimental study, using the naphthalene sublimation technique, of natural convection between two opposing square plates with all edges (top, bottom, sides) open and with one or both of the side edges closed. The plates were considered to be isothermal from a heat-transfer standpoint. The principal results, which are shown in Fig. 14, illustrate that the differences between blocked and unblocked lateral edges are important below Ra < 10 (i.e., for small channel spacing) with differences as high as 30% for Ra<3 . For large channel spacings (i.e., tending towards the isolated plate behavior) no effects of lateral blockage were detected.

2 . Staggered Vertical Plate or Card Arrays

Sobel et al. [73] first observed that in arrays of short vertical plates, the boundary layers from adjacent plates do not merge at the channel exit and therefore the fluid at the channel center remains relatively cool. By later- ally staggering the plates, as in Fig. 15, the cool fluid impinges on the leading edge of the downstream plate, begins a new boundary layer and produces an attendant increase in heat-transfer coefficient. They performed


T H

I 4

i L I

* * * I I

* * * I * * * I * * * I

* * *

I I I

I I I

I I I

I I

I Icy I 17

l t t FIG. 15. Arrangement of staggered vertical plates [71].

experiments with arrays of uniform heat flux strip heaters stacked three high. Each heater was 1.75 in. high. For Ra* > 200, where Ra* is evaluated using the channel half-width b/2, a 38% increase in NuLlZ was found. The staggered arrangement was found to enhance heat transfer for Ra* as low as 10. No advantage was gained by staggering for long channels, Ra* < 10. Sparrow and Prakash [71,72] undertook a numerical analysis for the case of staggered isothermal plates and found that the degree of enhancement depends on the plate spacing the total number of plates (N =H/L where H is the overall length and L the plate length). Figure 16 shows the ratio of overall heat dissipated in a staggered to in-line channel, &/&. . The ratio &/&. was equal to the ratio of the average channel Nusselt number or heat-transfer coefficient. For equal areas, temperature, and overall height, enhancement in heat transfer was found for values of the modified Rayleigh number, Ra > 2 X lo3, where the Rayleigh number was based on the hydraulic diameter, given by Dh = 4s. At small Ra, induced mass flow decreases producing a degradation in heat-transfer performance. As part of their experimental study, Aung et al. [61,62] also investigated the effects of staggering arrays of printed circuit cards. Their results, shown in Fig. 17, represent the percentage decrease in AT at midheight and at the exit due to card stagger compared to an in-line arrangement. They also found that staggering is most effective for large card spacing and small overall cabinet height with decreases as high as 30% obtained at the largest Ra*. Figure 17

I I I l l I 1 I l l 1 I I I 104 105 106

MODIFIED RAYLEIGH NUMBER, Ra (Bfi(Tw-To )Dt.p3 v2 H

40 c a z

4 ,,, v)

a

FIG. 16. Ratio of overall heat dissipation for staggered array of isothermal vertical plates to in-line array with some overall height, surface area, and temperature difference [71].

0

0

0

-

80--

ATMAX

MIDHEIQHT AT

) [731 ATMAX

MIDHEIQHT AT

MEAN LINE, PRESENT DATA

MIDHEIGHT AT -J

-101 ' ' , I I , # I 1 I , , I

10' 102 1 o3 1 o4 5

MODIFIED RAYLEIGH NUMBER, Ra*

FIG. 17. Reduction of temperature due to staggering an array of PCBs in a vertical cabinet

THERMAL CONTROL OF ELECTRONIC EQUIPMENT 20 1

shows that stagger may actually degrade heat transfer below a critically small Ra*.

3 . Interacting Convection and Radiation

The interaction of a developing laminar channel flow and radiation has been studied numerically by Carpenter et al. [75] for asymmetric isoflux channels and by Sparrow et al. [74] for a channel with one isothermal wall and one adiabatic wall. For symmetric uniform heating, radiation occurs at the inlet and exit and also from the hotter upper parts of one surface to the cooler entrance region of the opposing surface. For asymmetric heating, radiative transfer occurs across the channel between the two surfaces. Carpenter et al. [75] found that for E* < 2, the effects of radiation were negligible for both symmetric and asymmetric heating. As the channel aspect ratio L / b decreases (Ra* increases), end losses became important and produced enough cooling that the maximum temperature no longer occurred at the channel exit. Figure 18 shows the computed wall temperatures for asymmetric heating with no radiation ( E = 0) and =* = 5300 and for black surfaces ( E = 1). Referring to Fig. 7, it is apparent that =* of

1.125 I I I 1 I I 11 1 I El, €* = 0.0

1.000 ---- '1"2= 1.0

-

0.750 - - HEATED WALL (1) -

4--\

-

-

ADIABATIC WALL (2)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

AXIAL POSITION X/L

FIG. 18. Effect of radiation on wall temperatures for a channel with one adiabatic wall (wall 2) and one uniform heat flux wall (wall 1) [75].


5000 represents a large channel spacing, where isolated plate behavior is approached. The hot surface temperature decreased by as much as 40%, primarily because of transport to the opposing wall, and the temperature near the exit dropped because of radiative losses at the exit. These effects persisted for surface emmissivities as low as 0.3. Sparrow et al. [74] showed that heat-transfer enhancements in the range of 50 to 70% can be obtained because of radiation effects in a channel with one isothermal wall and one (globally) adiabatic wall. The effect of radiation was to raise the temperature of the adiabatic wall and thus transform it into a convectively active surface. The radiation-enhanced heat-transfer results fell between the case of pure convection with one isothermal wall and one adiabatic wall, and the case of two isothermal walls.

4. Optimum Spacing of Vertical Plate Arrays

When two-dimensional flow is assumed to prevail, the composite equations summarized in Table I11 can be used to derive results for the plate spacing that maximizes the heat dissipation from an array of plates. For an allowable maximum temperature rise, the heat dissipated from a channel decreases with decreasing spacing, but the total number of plates increases. Hence, there is an optimum number of plates per unit volume. Table I11 presents relations for optimum plate spacing developed by Bar-Cohen and Rohensow [76] from the composite relations presented in this same table, where P = Ra/b4 and R = Ra*/b5. Note that the parameters P and R do not depend on b. The optimum b for the isoflux cases are based on maximizing heat dissipation per unit average plate temperature elevation. An important result of this analysis is that optimized arrays of asymmetrically heated channels dissipated only 63% of the heat dissipated by the symmetric channel for isothermal surfaces [76] and 65% for isoflux surfaces as shown by Aung [21]. Levy [77] also found optimum spacing results for isothermal plates using a somewhat different approach. The analysis of vertical fin arrays closed on one edge followed closely that for the vertical parallel planes passage. Several studies, both numerical and experimental, have been performed [81-871. The reader is referred to the references cited, especially to the contributions of Aihara et al. [81-841, for an overview of this important area.

B. DISCRETE THERMAL SOURCES ON VERTICAL SURFACES

When discrete electronic components are deployed on a surface alone, or in a sparsely populated array, the thermal behavior is governed by the


convection generated by local buoyancy and by the convective transport due to the thermal plume or boundary-layer fluid impinging on the source. Both finite-size and line sources with and without a bounding surface have been studied [88-991. Interest in cooling discrete sources stems both from cooling conventional cased chip modules in air, as in Fig. 3, and more recently from immersion cooling of multichip modules where individual chips are more nearly flush heat sources on a substrate. Conjugate conduction effects into the substrate or board may play an important role in overall heat dissipation.

Carey and Mollendorf [88] measured the characteristics of single finite- size, flush-mounted thermal sources on an adiabatic vertical plate in water. A 0.47-cm-square source and a 1.3-cm-diameter round source were investigated. The spanwise growth of the plume was found to be weak, =x1I5, and the growth normal to the surface varied linearly with downstream position. The presence of the wall had a significant effect on the plume growth compared to unbounded point and line source plumes because the entrainment of cooling fluid was inhibited. The decay of the centerline temperature for these finite sources was found to lie between the decay for a wall-bounded point source and a wall-bounded line source. From a limited number of data points on the source, they determined that the source Nusselt number followed the familiar one-quarter power dependence on Grashof number for isolated bodies in natural convection.

Jaluria [89-91, 97, 981 has performed a number of studies, both numerical and experimental, to investigate the behavior of isolated and interacting horizontal line source and strip heaters mounted on adiabatic vertical surfaces. A typical result (Fig. 19), shows the computed dimensionless temperature 8 variation with dimensionless downstream distance for two heaters of length L spaced a distance D apart [91]. The heat flux was constant and the same on each heater. The degree of interaction was surprisingly weak, but degradation of heat transfer from the downwind heater occurred for small spacing, whereas heat transfer was slightly enhanced for large spacing, D / L = 8. Figure 19b shows the upper to lower average Nusselt number for various ratios of upper to lower heat flux. The result for equal heat fluxes shows that enhancement occurs for the upper heater for D / L greater than about three. It is significant that the results of Fig. 19 were computed using the boundary-layer form of the governing equations. In later studies [90] using the full governing equations, it was shown that nonboundary-layer effects were quite substantial at low values of the modified Grashof number Gr*, , where L is the height of the heated strip. In particular, the ratio of upper to lower Nusselt number was lower by as much as 16% for the full solution, compared to the boundary-layer solution.

204

15.0

10.0

_1 . 5.0

0.0


a b

0.0 1 .o 2.0

8 0

1.2

1 .o

-I 5 0.8 . 3 z

0.6

0.4

0.2

I I I

0.0 2.0 4.0 6.0 8.0

DIL

FIG. 19. (a) Surface temperature variation on two strip heaters on an adiabatic vertical surface: D is spacing between heaters and L is heater length in vertical direction. (b) Average Nu of upper to lower heater for various spacing [91].

Milanez and Bergles [92] performed experimental studies for a configuration similar to those of Jaluria, and also with two interacting cylinders placed on the vertical surface in both water and air. The results for the interacting cylinders are reproduced in Fig. 20. Focusing on the result for equal heating on both cylinders, degradation of heat transfer from the second cylinder was obtained for L (spacing) to D (diameter) less than 15, in both air and water, and above this spacing slight enhancement was achieved only for water. It was concluded that in an array of two heaters such as this, the heater dissipating more energy should be located at the top, and that by appropriate arrangement, the bottom component could in some cases enhance the heat transfer from the top element. In a related study, Park and Bergles [99] measured Nusselt numbers in R113 refrigerant for small (5 x 5 mm) flush-mounted and protruding heaters resembling uncased VLSI chips mounted on a vertical substrate. In contrast to previous findings for wide strip heaters [91], as in Fig. 19, it was found that for two flush-mounted heaters, the Nusselt number for the top heater was always less than that for the bottom heater, regardless of spacing, and the Nusselt number for the top heater increased with increasing spacing, up to a spacing equal to 3.5 times the heater height.


L I D

FIG. 20. Upper to lower Nu for two 2-mm-diameter cylinders spaced a distance L apart on a vertical adiabatic plate [92].

If discrete thermal sources are located on a plate that forms one wall of a vertical channel, the heat transfer is influenced by the strength of the forced convection flow that is induced into the channel by the chimney effect. This effect may not be sufficient to overcome the loss of uninhibited entrainment of cooling fluid. Ravine and Richards [93] experimentally investigated the heat transfer from a flush-mounted, horizontal strip source of heat on a vertical surface with and without opposing shrouded surfaces. It was found that for small channel spacing, the effect of the channel wall was to reduce the local heat transfer from the heater, as compared to a heater on an isolated plate, by as much as 30%. Discrete sources that protrude from the base surface, both two- and three-dimensional, have been studied almost exclusively by experiments [ 100-1121. Sparrow et al. [lo51 found significant enhancement of heat transfer from a short cylinder attached to a heated wall in the presence of a shrouding wall compared to the case of an unshrouded wall. Increases as high as 60% for conditions where the cylinder was located near the entrance of a duct with the narrowest spacing were reported. In a related experiment [103], it was shown that heat transfer from a long horizontal cylinder could be enhanced by vertical shrouding surfaces even when these surfaces were unheated.


\ ADIABATIC SHROUDING WALL

FIG. 21. Experimental apparatus of Ortega and Moffat [log]; an in-line array of cubical elements on an insulated channel wall.

14

P = 0.8 Wlelernenl

12

$.' 10

1 P i

8

6 2 4 6 8 10

ROW NUMBER

FIG. 22. Variation of heat-transfer coefficient with position on plate for an array of wall- mounted cubical elements with and without an opposing shrouding wall [log].


20 1 I I I

-

-

Row 6 b/B = 1.5 -

0 FORCEDFLOW - 0 NATURAL FLOW

I I 1 1 I ,

3 -

2

Ortega and Moffat [109-1121 performed a series of experiments on sparse heated arrays of cubical elements located on an insulated vertical plate, both with and without an opposing shrouding plate. The geometry used is shown in Fig. 21, where L = 35 cm and the lateral edges were sealed. Figure 22 shows the measured heat-transfei coefficient, based on element to ambient temperature difference, averaged across a horizontal row of elements, at each vertical position in the array. In the first six rows of elements, the heat transfer was enhanced by the induced forced flow, with the enhancement greatest for the narrowest channel and the row of elements at the lowest position, consistent with the findings of Sparrow et al. [102]. Heat transfer was degraded beyond row six because the beneficial effects of the chimney flow did not offset the increased temperature of the fluid. For large channel spacing, the heat transfer in the array was characterized by a complex plume-boundary-layer flow that resulted in a uniform temperature throughout the array. Although there was little doubt that the array temperature distribution depended on element spacing, this effect was not explored. For the narrowest channel spacing, it was found that fully developed conditions were achieved within a few rows and that the local heat transfer from an element was dictated by the buoyancy-induced channel forced convection. It was found [110,111] that the heat-transfer coefficient, when defined in terms of the equilibrium temperature achieved by the element when unheated, was the same for either buoyancy-induced forced convection or fan-induced forced convection. The data of Fig. 23


1.0

illustrate this comparison for an interior element at narrow plate spacing with corrections for variable property effects. It is noted that the flows shown in Fig. 23 represent mean velocities from 0.02 to 0.2 m/sec. The authors demonstrated that channel drag data, measured in forced flow over an unheated array, could be used to predict the channel flow rate in buoyancy-induced forced flow. The local heat-transfer coefficient and thermal wake dissipation function, both measured functions of the channel Reynolds number, could then be used to predict the temperature of an array element using linear superposition [110].

-

a 10 I

-

lo-’ ’ I I l , , , , l I I , ,

0

E

0

cu

2 r

0 J crn long

0.508

-11- 0.254 crn


b, cm

FIG. 24. (Continued)

C. IRREGULAR SURFACES

Laminar natural convection, by virtue of its viscous-dominated nature, tends not to be greatly affected by small irregularities in otherwise plane surfaces. Laminar boundary layers in particular are forgiving of small perturbations on the surface, and the only real effect of surface irregularities is to enhance the area available for heat transfer. These conclusions have been reinforced by the studies of Fujii et al. [113] on vertical cylinders with horizontal grooves and sparse, three-dimensional protrusions in oil and water. Relatively few studies have investigated surface roughness effects on natural convection [113-1171. Joffre and Barron [114] presented results that indicated very large enhancement of heat transfer due to surface roughness resembling repeating ribs, but there appeared to be


some ambiguity in their interpretation of the enhancement, which was due solely to increased area and that due to enhancement of the convective transport.

In cooling printed circuit-board-mounted modules, several regimes of what may be called surface irregularities, rather than surface roughness, have been identified. If electronic component arrays on cards or boards are densely populated, the space between adjacent components form grooves or channels roughly rectangular in cross section, usually both in the normal and streamwise directions. The dimensions of these grooves are normally small relative to the length of the exposed component surface, and they represent the characteristic surface irregularities. These grooves affect the heat transfer from the tops of the modules and increase the surface area available in the cavity surfaces. In contrast, if the component arrays are sparse, the exposed board area may be comparable to the component surface area for heat transfer, and the regime could change to that for a plane surface, the board, with two- or three-dimensional protrusions as its identifying characteristic. The protruding surfaces could affect the heat transfer on the protruding module itself [99] and on the surrounding substrate or board [loo]. For very large component spacing, the components may act as though isolated from each other. In summary, the effects of roughness may refer to heat transfer from the chip or module, from the substrate or board, or from the combined package.

Figure 24 shows heat-transfer coefficients as measured by Horton [lo71 on vertical isothermal plates with horizontal and vertical grooves cut into the surface. The resulting surface resembled a densely populated electronic circuit board. At large plate spacings, where boundary-layer behavior was obtained, the grooves had little or no effect on the average heat transfer from the plate, supporting the previous statements. At small plate spacings, where the heat transfer was dominated by buoyancy-induced forced convection, the surface grooves tended to enhance the heat transfer slightly. Although not discussed by the authors, it appears that in the forced convection regime, more flow is forced in the vertical channels, resulting in enhancement of local heat transfer from a component; the enhancement may be offset by increases in frictional drag and therefore decreases in flow rate. For a single plate spacing, the data of Fig. 24 did not necessarily compare the heat-transfer coefficients for smooth and grooved channels at the same flow rate. The enhancement of natural convection in vertical channels is a relatively new and unexplored area of research and contributions to it may have significant impact on electronics cooling.

Ortega and Moffat [lo91 compared the local heat transfer on a nearly isothermal array of cubical elements to the equivalent isothermal vertical


1 POWER, W/element

0 0 2 0 0.4

FIG. 25. Local Nusselt number for an array of cubical elements on an insulated vertical plate, compared to a smooth plate [109].

plate. As shown in Fig. 25, it was found that the average heat-transfer coefficients on the element surfaces were higher than those for the equivalent smooth plate at the same temperature. Since new natural convection boundary layers developed on each cube, and some degree of mixing of the boundary-layer fluid occurred between horizontal rows of elements, these results were not surprising; they again reinforced the conclusion that in natural convection enhancement, one of the key techniques is to interrupt boundary-layer growth [62,71]. Figure 26 reproduces the numerically derived results of Shakerin er al. [loo], which illustrate the effects of a large two-dimensional protrusion on one wall of a two-dimensional enclosure where the wall with the protrusion is at a temperature higher than that of the opposite wall. Nusselt number distributions are shown on the disturbed and undisturbed enclosure walls and the roughness element surfaces. The effects of the roughness elements on the plate heat transfer were localized to within two roughness heights before and after the element. The reduction of heat transfer on the plate was roughly balanced by the increased surface area of the roughness element. The heat transfer from the top and bottom lateral surfaces tended to be low because of recirculating flow in


Nu ---- -NU FOR WALL WITH

ROUGHNESS

70 0

NUL

FIG. 26. The effect of a two-dimensional protrusion on heat transfer from one wall of an enclosure [loo].

these regions. It was noted that the heat transfer from the lower surface would be significantly higher if a three-dimensional protrusion were used.

Park and Bergles [99] compared the heat transfer for flush-mounted and protruding heaters for both isolated heaters and interacting arrays. The chip models measured roughly 5 X 5 mm and protruded 1.1 mm from the substrate. For isolated heaters immersed in R113, protruding heaters were found to have heat-transfer coefficients approximately 14% higher than the equivalent flush-mounted heaters. This increase was attributed to the increased flow disturbance at the leading and trailing edges. In contrast to the flush-mounted heaters, the Nusselt number for the upper heater (in a vertical array of two heaters) was always greater than for the lower heater, and it was hypothesized that this was also due to the increased leading edge disturbance on the upper heater. Sparrow et al. [102,104-1061 investigated natural convection from single and multiple short horizontal cylinders mounted on a heated vertical plate and found that the degree of interaction depended on the Rayleigh number. Greater enhancement of heat transfer from downstream cylinders was found at larger intercylinder spacing,

213 THERMAL CONTROL OF ELECTRONIC EQUIPMENT

which is in general agreement with the aforementioned studies for flush sources.

D. LIQUID COOLING OF DISCRETE SOURCES

Direct immersion of electronic packages in dielectric liquid coolants has been utilized for some time for cooling high-powered components in specialized applications by natural convection and boiling [ 1181. The latter topic is reviewed in Section IV. Direct immersion of high-powered semiconductor devices, as on multichip processor modules in computers, is rare at present, but it is recognized that direct immersion may be required in the next decade and beyond to manage chip powers on the order of 50 W, with surface fluxes on the order of 50 to 200 W/cm2 [14]. Problems associated with boiling, discussed in Section IV motivated a recent renewed interest in

FIG. 27. Test assembly for experimental measurement of heat transfer from small flush heat sources simulating VLSI chips immersed in liquid (1191.


single-phase natural and forced convection from small heated surfaces in liquids.

The most significant studies on natural convection in liquids are those of Baker [119,120] on isolated flush-mounted heaters and Park and Bergles [99] on isolated and interacting arrays of flush-mounted and protruding heaters. Baker [119] performed a number of experiments on thin film tantalum nitride resistors on glass substrates. A reproduction of the test assembly is shown in Fig. 27. The resistors were rectangular with the vertical dimension, parallel to the flow, one-half as long as the spanwise dimension. Three heaters were used with areas of 2.0 cm2 (20 X 10 mm), 0.104 cm2 (4.6 X 2.3 mm), and 0.0106 cm2 (1.46 x 0.73 mm). Figures 28

FIG. 28. Heat transfer from a simulated chip, 20 mm wide by 10 mm high, in air (W) and Freon 113 (0) A = 2.0 cm2 [119].


200 -

100-

@4 5 0 - E 0 s x- 2 0 -

t- l o - a

3

u.

W I

5 -

2 -

1 2 5 10 20 50

FIG. 29. Heat transfer from a simulated chip 1.46 mm wide by 0.73 mm high, in air and Freon 113 [119].

and 29 reproduce the results for the largest and smallest heaters, respectively, in both air and Freon. Increases in heat-transfer coefficient on the order of five- to sevenfold were gained with the liquid compared to the air, even though the air data had not been corrected for substration conduction. At a nominal overheat of 20°C, the heat-transfer coefficient for the smallest heater was more than an order of magnitude greater than for the largest heater. Baker [119,120] apparently was the first to docu- ment this very large increase in heat transfer for small heat sources. He found that classical boundary-layer solutions for an isothermal suface predicted increases in the heat-transfer coefficient of this order for decreases in heater size in forced convection, but that the boundary-layer solutions underpredicted the heat transfer. It was concluded that the leading edge effects, the three-dimensional convective effects at the lateral edges, and conduction all contributed to this observed underprediction,


I I 1 I I I I 1 1 1 1 I 1 I 1 1 1 1 1 I I I I IIIII I 1 1 1 1 1 1

1 o5 lo6 1 o7 108 log

FIG. 30. Natural convection data for simulated chips, 5 and 10mm high, and various widths, in refrigerant R113, T b ~ 2 7 ° C and Pr = 6.9-8.3 [99].

1 OOL I

HEIGHT

2.5 rnm

11 I I I I I I I I I I I , 1 1 , , 1 I I 1 1 1 1 1 1 I I I I , I ,

1 o3 l o4 1 o5 lo6 10

FIG. 31. Natural convection data for simulated chips, 5 and 10 mm high, and various widths, in water T b = 27°C and Pr = 2.7-5.6 [99].


and that these effects became more significant with diminishing source size. Similar conclusions were made regarding natural convection.

More recently, the experimental study of Park and Bergles [99] has shed light on the observed departure of natural convection data for small flush heat sources from results for two-dimensional plates. The main variable in these experiments was the width of the heated surface. Two different heights were used, 5 and 10 mm, with widths varying from 2 to 70 mm. Figure 30 shows the data for a single heater in a refrigerant (R113), and Fig. 31 shows the data in water. The base-line correlation for laminar natural convection from a vertical isoflux plate was that of Fujii and Fujii [121], given by

Here, x is the distance in the streamwise direction, taken at the source midheight where the temperature is measured. In both R113 and water, the Nusselt number, evaluated at the heater midpoint, increased as the heater width was reduced. Nusselt numbers for the narrowest heater were 80 to 100% higher than for the widest heater (70-mm wide), and the widest heater Nusselt numbers were roughly 20% higher than those predicted by Eq. (38). The width effect was more pronounced in the refrigerant than in water. For all heater widths, the authors observed an upstream velocity at the leading edge due to conduction to the substrate and fluid. For the narrow width heaters, an induced flow at the sides of the heaters was also observed. The authors concluded that there is ample documented evidence [ 1221, both experimental and analytical, to substantiate the observed increases for small heat sources.

E. DISCRETE SOURCES IN ENCLOSURES

Several studies have addressed the fundamental aspects of natural convection pertinent to cooling electronic components in sealed and vented enclosures [123-1331. The configuration that has attracted the most study is that of a horizontal heater mounted on one vertical wall of an enclosure [125,126,129,133]. Both flush-mounted [125,126,133] and protruding [ 1291 heaters have been investigated. Figure 32 illustrates the configuration of (a) Chu et al. [125] and (b) the experimental apparatus of Knock [129]. Turner and Flack [126,133] replicated the numerical configuration of Chu in an experimental apparatus, however they considered a cooled strip and a heated opposite wall.

Chu et al. [125] obtained numerical results for air, Pr=0.7 , using a conventional finite-difference formulation for laminar, two-dimensional


IT i : f

T C

W -1

a

Heat Exchanger

Standpipe Thermocouple an Heater Leads

I i

Heater Assembly b

FIG. 32. Configurations for discrete heaters located in sealed enclosures: (a) numerical study of Chu er al. [125] and (b) experimental apparatus of Knock (1291.

625 x lo4

T 2.5 x lo4

0.1 0.3 0.5 0.7 0.9 1

0.1 0.3 0.5 0.7 0.9

HEATER LOCATION (dh) HEATER LOCATION ( d h )

FIG. 33. Effect of position of heater on vertical wall on heater average Nusselt number: (a) square enclosure with cooled horizontal walls ( l /h = 0.2) and (b) square enclosure with adiabatic horizontal walls [125].


flows. Boundary conditions considered are shown in Fig. 32a. Results were obtained for various heater positions, heater size, and enclosure aspect ratios. As expected, the case of isothermal horizontal walls gave higher heat-transfer coefficients on both the heater and the vertical cooled wall than for adiabatic horizontal walls. Figure 33 shows the average Nusselt number on the heated strip for various heater positions. The Nusselt number was defined in terms of the heater height and (Th - T,) , and the Rayleigh number in terms of (Th - T') and the enclosure height h. It was shown that the maximum heat transfer was obtained for a heater located at the approximate center of the vertical wall ( s / h = 0.5) for both the isothermal and adiabatic cases. As to the effects of enclosure aspect ratio, it was observed that the single primary circulation region split into two cells for tall enclosures (h /w > 3.4) and that the secondary cell tended to degrade the heat transfer on the heated strip. This was verified by the Nusselt number obtained, which tended to be maximized for an aspect ratio of approximately 1.0. Knock [129] also observed a primary upper cell, driven by buoyancy, and a lower shear-driven cell in experiments with an aspect ratio ( h / w ) of four. In all cases, the average Nusselt number increased with increasing heater size ( l /h) .

Turner and Flack [133] performed an experimental study equivalent to the numerical study of Chu et al. [125] with air as the working fluid, but restricted the experiment to adiabatic upper and lower walls. No corrections for radiative heat transfer were presented. Here it was also found that heater positions near the middle of the vertical wall maximized the heat transfer from both the opposite cooled wall and the heater. For square enclosures ( h / w = 1.0) it was observed that the average Nusselt number on both the heater and the opposite wall increased monotomically for increasing heater size and that the rate of increase was small for l /h > 0.25. In a companion paper [ 1261 experimental correlations derived from the experimental results were used to predict the temperature of a heat- dissipating integrated circuit in an air-filled enclosure. Although Turner and Flack [133] assumed that the enclosure wall opposing the heater was at the temperature of the environment, in a real enclosure, the convective heat transfer from the interior of the enclosure is conducted through the enclosure wall and gives rise to a boundary layer on the external vertical surface. Sparrow and Prakash [131] considered this interaction of internal convection in an enclosure and an external natural convection boundary layer and found that the coupling of opposing flows across a conducting wall could have significant effects on the heat transfer.

Knock [129] used the thymol blue pH indicator technique for flow visualization in a water-filled enclosure with a fixed aspect ratio of four. A limited number of measurements of Nusselt numbers on the protruding


heater showed that the heat transfer decreased as the vertical position of the heater increased, however the heater Nusselt number was found to be greatest for the lowest position on the wall rather than near the center. In this experiment, the vertical walls were insulated, and the top and bottom walls were cooled.

The effects of multiple discrete flush-mounted heat sources on one wall of a tall vertical cavity have been investigated experimentally by Keyhani et al. [128]. Through flow visualization in ethelyene glycol, the recirculating cell structure of the flow was shown to exhibit primary, secondary, and tertiary flow zones, similar to those for differentially heated cavities. The flow was more vigorous, however, and resulted in a substantial increase in heat transfer. Increases in heat transfer from 67 to 90% above the case for both walls uniformly heated were reported, with the degree of enhancement depending on the heater location relative to the nearest recirculating cells.

Several investigators [ 124,127,130,1321 studied configurations resembling printed circuit boards oriented horizontally in an enclosure. Buller and Duclos [ 1241 considered both natural convection and radiation in an enclosure built of material comparable to that used in small systems such as typewriters, computer terminals, and copiers. Heated plates of various sizes, typically 16.7 x 10.5 cm, and different finishes representing surface emissivities from 0.15 to 0.9 were used. An empirical correlation was developed relating plate temperature to heat dissipation, enclosure volume, and enclosure bulk air temperature. It was found to agree reasonably well with data for an actual memory card module. Krane and Phillips [130] studied natural convection from a square, horizontally oriented board mounted in a shallow, square enclosure, a configuration motivated by passive cooling of desktop computers and their peripheral equipment. The effects of openings in the vertical side walls and in the top and bottom walls were investigated, and numerous recommendations were made based on an extensive series of experiments. Recently, Torok [132] and Johnson and Torok [ 1271 demonstrated the use of commercially available finite- element and finite-difference codes in the analysis of natural convection in electronic cooling applications. Qualitative results for a tilted printed circuit board in vented and unvented enclosures and for a vertical channel were presented.

111. Forced Convection

Only in this decade have fundamental issues in forced convection cooling of electronic devices and systems attracted the attention of heat-transfer researchers. Although, there is a considerable amount of proprietary data


b

a C d

interface

FIG. 34. Examples of chip-in-cavity electronic packages showing paths for conduction: (a) components of a typical package, chip facing up, attached to heat spreader, (b) chip facing down in cavity, (c) chip facing down and attached to metal stud base of heat sink, and (d) chip facing up in cavity with heat sink on case [135].

in industrial laboratories, one has merely to look in recently published books [118, 134-1371 on thermal control of electronic systems to note the paucity of fundamental data in the sections on convective heat transfer.

Recent research in forced convection has addressed almost exclusively the cooling of heat sources related to two packaging configurations: (1) chips packaged in rectangular modules that are mounted alone or in arrays to a printed wiring board (PWB) or card; and ( 2 ) direct liquid cooling of bare heat-dissipating chips mounted on a multilayered ceramic substrate as in multichip modules employed in high-speed computers. Nakayama [135] has presented a concise review of current technology, including a description of state-of-the-art packaging technology. Figure 34, from his paper, shows the primary components of a chip-in-cavity package with heat sink attached, typical of the first configuration. Such packages are commonly in use, for example, for high-speed memory chips, and are exclusively air cooled. Multichip modules, commonly in use for VLSI processors, are not typically cooled by direct immersion now, but direct liquid cooling is the area of greatest current interest in forced convection research because of its inherent advantages in handling high heat fluxes anticipated in the near future [14].

A. DISCRETE FLUSH HEAT SOURCES I N CHANNEL FLOW

Forced convection from flush-mounted rectangular sources was examined by Baker [119,120] in the early 1970s and more recently revisited


TABLE IV

EXPERIMENTAL STUDIES OF FORCED CONVENTION FROM FLUSH HEAT SOURCES IN

CHANNEL FLOW OF LIQUIDS

Investigator L(mm) S(mm) L’IL SIL S‘/S H H I L ReH

Baker [119] 10, 4.9, 2.3 - 2 - - - - - Incropera

Samant and et al. [140] 12.7 3.18 1 0.25 1 11.9 0.94 617-8.6X lo3

Simon [141] 0.25 - 8 - - 2.79 11.16 4 x lo3 - 5 x lo5

by Incropera and co-workers [138-1401 and by Samant and Simon [141]. Current interest is in direct liquid cooling of silicon chips, which are anticipated to dissipate from 20 to 50 W per chip, with corresponding fluxes from 50 to 200 W/cm2 anticipated for the coming decade [14]. A summary of the experimental work is given in Table IV. All of the studies employed water or a coolant such as R113, FC72, or FC77 as the working fluid.

Baker’s apparatus is shown in Fig. 27 in connection with his natural convection data. Figures 28 and 29 illustrate approximately an order of magnitude increase in the measured heat transfer by free convection in air to free convection in Freon, and another order of magnitude increase with forced convection in Freon. Just as significant is the 20-fold increase in achievable heat flux, or heat-transfer coefficient, in the smallest heater (Fig. 29) compared to the largest (Fig. 28). A maximum heat flux of about 100 W/cm2 was achieved at an overheat of 20°C for the smallest device, but neither the forced convection velocity nor the Reynolds number for these conditions was quoted by the author.

Ramadhyani, Moffatt , and Incropera have performed numerical studies of conjugate heat transfer from one or more two-dimensional flush strip heaters on the floor of a channel with fully developed laminar [139] or turbulent [138] flow. The results of these investigations were compared to data from single-square sources and an array of twelve such sources in water and FC77. The configuration and nomenclature used are shown in Fig. 35. Figure 36 shows the computed average Nusselt number (defined in terms of the heat flux from the heater only and the channel hydraulic diameter) over a single source in laminar flow, normalized by the value for an adiabatic substrate. Substrate conduction decreased the average Nusselt number over the source and was especially significant at low Peclet numbers. However, total heat dissipated by conduction into the substrate and


/

. . . . . . . . . . . . . ..> . . . . . . . l x . . " l . . ' . ' . . . . . . . . . . . tri-~,,-i-+ . . . . . . ...;Ti. . . . . . . . ' \ . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . , . '

FIG. 35. Schematic diagram for problem of conjugate heat transfer from small heat sources on a substrate [139].

0.5b -1 . ( I I I I I * I

10' 10' 10' 10'

Pe

FIG. 36. Computed average Nusselt number for single isothermal source, normalized on value for adiabatic substrate, showing effects of channel Peclet number and thermal conductivity ratio [139].

the subsequent convection from the substrate, increased for these conditions so that the net effect was beneficial. This is illustrated in Fig. 37a, which presents the local Nusselt number at the fluid-solid interface. Both upstream and downstream conduction were amplified for increasing k, .


a b

FIG. 37. Computed local Nusselt number on source, normalized on value for fully developed flow and heat transfer (Nu = 4.86) showing (a) effects of substrate conductivity for single heat source (Pe = lo4) and (b) effects of spacing between two heat sources (Pe = lo4, k,/k,= 10) [139].

The Nusselt number was greatly reduced at the leading edge of the heater, but was hardly affected at the trailing edge. Figure 37b shows the interaction of two heated strips at Pe = la4 and k s / k f = 1, for different spacing between heaters. Results for turbulent flow [138], using a standard eddy- diffusivity closure model, were consistent with those for laminar flow, but the magnitude of the Nusselt numbers was much greater and substrate conduction was reduced.

In a subsequent study, Incropera et al. [140] experimentally investigated the characteristics of 12.7-mm-square flush heaters, both isolated and in an array. Data for the isolated heater are shown in Fig. 38. Data in the range 5000 < ReD < 1400 were correlated by the equation:

- . NuL = 0.13 Re$64Pr0.38(po/ph)0.25 (39)

to within 6%, where L was the heater length and D the channel hydraulic diameter. The viscosities, po and p h , were evaluated at the free-stream and heater temperatures, respectively, to account for property variation effects. Excellent agreement was found with numerical calculations for two-dimensional heaters using the eddy-diffusivity model. In the laminar regime however, the data were 30% higher than the computations, and the authors suggested three-dimensional boundary-layer effects, and mixed convection as reasons for the departure. The authors found that the thermal conditions for their range of heater lengths and channel heights were more representative of external flow over a plate rather than channel flow,


100 0 0 -

7 0 -

80.

50 -

40.

$ /b 30-

A ?

2 0 -

00 J

~ ~ ~ 0 0 0 2000 ' 6000 ' ' ' 10,000 j ' 20.0

Experimental Results

0 FC-77 Correcled (PI-281. 221)

0 water (PI =8 3. 5 4)

lolooo 2000 ' ' ' ' ' a ' Re, Re L

FIG. 38. Measured (square source) and predicted (two-dimensional source) average Nus- selt number for single flush source in water and FC77, uncorrected and corrected for substrate conduction: (a) Reynolds number based on hydraulic diameter of channel and (b) Reynolds number based on length of source [140].

but the data were still significantly higher than predicted by common flat-plate correlations, as seen in Fig. 38b. The numerical model was used to extend the heat-transfer results for Lh/L, of 0.1,2, and 10, conductivity ratio of 0.5 and 50, Pr of 0.7, 7, and 25, and ReD up to lo6, resulting in a correlation given by

- NuL = 0.037 Re:'' Pr0~3s(Lh/L,)0~85(kf/ks)o~02

Results were also given for an array of 12 heaters. It was found that fully developed conditions were present by the third row of heaters, and excellent agreement was found with numerical predictions for two-dimensional strip heating using an eddy-diffusivity model. Andreopoulos [ 1421 performed extensive measurements of the response of a turbulent boundary layer in air to a double step change in wall heat flux, as, for example, in the two-dimensional strip-heating situation. In certain regions over the heated strip, the correlation, v ' T ' , had negative values, indicating that the eddy- diffusivity concept, which related the turbulent heat flux, - v ' T ' , to d T / J y could not be valid. No such data exist to indicate whether this situation is also found for higher Prandtl fluids similar to those used in the electronics cooling predictions.

Samant and Simon [141] conducted an experimental study on a very small, high heat flux, rectangular patch on the floor of a fully developed turbulent channel flow. A novel heater design was developed, utilizing thin-film technology to run both power leads and voltage taps to a thin


1000.

eoo

N 300- .- t - z

100-

eo

I 30

a 3.0 I I

Numerical

-

-

I ' ' " ' I

FIG. 39. Heat-transfer results for single high-heat-flux source in a turbulent channel flow: (a) typical heat fluxes achieved in R113 and (b) measured and predicted (two-dimensional) Nusselt number as a function of channel Reynolds number in R113 (A) and FC72 (0), [141].

Nichrome heater element, 0.25 mm long in the streamwise direction, deposited on a quartz substrate. The heater was used simultaneously as a resistance thermometer to measure average temperature. Figure 39a shows typical heat fluxes in R113 coolant where Tb is the undisturbed bulk fluid temperature. A maximum heat flux of 204 W/cm2 was achieved at a heater temperature of about 120°C and coolant temperature of 48.6"C. Nusselt numbers of both R113 and FC72, given in Fig. 39b, were represented well by the correlation

Nu = 0.47 (41) where both the Nusselt and Reynolds numbers were based on channel height H , and the properties were evaluated at the bulk temperature. The data were also in excellent agreement with the numerical predictions for a two-dimensional patch, also using an eddy-diffusivity closure model. Sub- strate conduction was handled in an approximate manner by adding a preheating length to the actual heater length to account for upstream conduction through the substrate.

It has been suggested [139] that local buoyancy effects may introduce secondary flows that augment heat transfer from a heated patch in a relatively weak laminar channel flow. Indeed, such effects were observed by Kennedy and Zebib [143] both experimentally and numerically, in a configuration similar to that of Fig. 35, and also using a single heater on the top surface and heaters on both top and bottom surfaces.


200

100- m

m lz - 3 Z 50

20

B. TWO-DIMENSIONAL PROTRUDING ELEMENTS IN CHANNEL FLOW

Arrays of three-dimensional electronic modules have frequently, and understandably, been treated as two-dimensional rectangular modules, that is, ribs on a channel wall. This is especially true of numerical studies related to these geometries [144-1461, which, coincidentally, are normally restricted to laminar flow conditions. Experimental work on ribbed channels with application to electronics cooling has been done by Lehmann and Wirtz [147,148] and Arvizu and Moffat [149].

Arvizu and Moffat [149] presented average heat-transfer coefficients from aluminum ribs measuring 1.27-cm high and 2.54 cm in the flow direction. Their nomenclature and typical data for fully developed flow are given in Fig. 40. For given channel spacing and channel Reynolds number, it was shown that the Nusselt number increased with increased spacing between ribs, as the flow between ribs changed from a driven-cavity-type situation, to a wake-interference regime, to an independent roughness regime. Lehmann and Wirtz [148] showed this transition by smoke wire flow visualization in similar ribbed channels (Fig. 41a). It was found that for close rib spacing [147], S / L = 0.25, the flow in the cavity did not have

-

-

-

10

SIB - H/B 2.5 3.0 4.0 5.0 8.0 100 - 2.0 0 0

7.0 A A + o 4.6 0 a

U-

Nu, = C Re: 75

1 I I I I

FIG. 40. Nusselt number for large two-dimensional ribs in a channel flow for various rib spacing and channel height [149].

228

140

130

120

110

100

90 T g 8 0 - 3 70

60

50

z

G . P. PETERSON AND ALFONSO ORTEGA

- - - - - -

-

-

-

a

150 I I I I I I I I ..J

blL bf0

0 0.500 2.0 A 0.625 2.5 0 0.750 3.0 - A 1.000 4.0 0 1.500 6.0

- - - -

-

-

-

-

-

0

0

0

S = 0 = 12.5 m m P A8

40

10

0

- - I I I I 1 I 1

0 4000 8000 12000 16000 20000 24000 28000

b

.06

.01 A 200

Graetz solution Eq. (42) Eq.(43) S/L =

400 600

1 .o

- 1 1000 2000 4000 5000

Re,

FIG. 41. Heat-transfer results for Lehmann and Wirtz study of two-dimensional ribs with dimensions similar to electronic packages: (a) Nusselt number dependence on component Reynolds number, for closely spaced ribs [147] and (b) average convective resistance showing dependence on rib spacing [148].


much influence on the heat transfer from the top surface of the downstream rib, but significant cavity-channel flow interaction occurred for the largest spacing, S / L = 1.0. It was also concluded from flow visualization that laminar flow was achieved for Reb < 1000, and transitional flow for Reb = 2000. Both groups of investigators found that fully developed behavior, where the adiabatic heat-transfer coefficient ceased to change in the flow direction, was achieved within three to five ribs, depending on conditions. The Lehmann and Wirtz [147] data for rectangular ribs, with dimensions of flat-pack electronic modules is shown in Fig. 41a. The top surface of a rib with ten upstream ribs was heated and the local heat- transfer coefficient was measured with a Mach-Zehnder interferometer. For small rib spacing the average Nusselt number on the top surface showed little dependence on channel height and was correlated well by a Reynolds number based on rib length in the flow direction, (Fig. 41a). In a subsequent study [148], Lehmann and Wirtz further investigated the effects of rib spacing and channel height and developed correlations for case-to- ambient convective resistance, R,, , which is the inverse of average Nus- selt number based on the rib length in the flow direction. Figure 41b illustrates the dependence of the thermal resistance on component Reynolds number for a fixed channel height and three rib spacings. The recommended correlations, including the effect of channel height, were given by

0 < S / L 5 0.25 b / L = 0.5 1 R,, = 5.5ReL0.65,

0.25 < S / L 5 1.0 0.25 5 b / L 5 0.75 (43) 1 R,, = 4.68(b/S)0.'5ReL0.65,

The data of Fig. 41b show that the heat transfer from the ribs with close spacing, S / L = 0.25, were virtually identical to that for the smooth wall case. Increases in rib spacing simply shifted the data away from the smooth wall behavior at larger values of component Reynolds numbers.

Numerical simulation of laminar flow and heat transfer in two- dimensional ribbed channels has been performed for fully developed conditions [145,146] and developing flow with conjugate effects [144]. The first two studies were conducted using the finite-difference methodology of Patankar [150,151]. Sparrow and Chukaev [146] performed extensive numerical experiments with the geometry shown in Fig. 42. They assumed that the heat dissipation was uniform per unit length in the flow direction (not necessarily uniform on all surfaces) and arbitrarily different on each wall. Furthermore, they constrained the boundary conditions to uniform


f / e / / / / / / A I I t I I I I

FIG. 42. Ribbed-duct geometry considered in numerical study of Sparrow and Chukaev [146].

temperature on all surfaces, including the plate onto which the ribs were attached, and also in the spanwise direction. Computations were performed for two subproblems, the first, with the top surface heated and the bottom one adiabatic, and the second, with the heating reversed. From these, any combination of top and bottom wall heating could be constructed using superposition. Numerous parameter variations were investigated, spanning the ranges HIP = 0.2, 1, 5 , W / P = 3, 4, 5, and h / H = 0.1 to 0.9.

Schmidt and Patankar [145] considered the same geometry as in Fig. 42, with essentially the same numerical method, but the upper and lower plates were assumed to be adiabatic. Figure 43a indicates the nomenclature, while Fig. 43b illustrates some interesting results regarding the effect of channel spacing. The following definitions were employed:

uH , Re=- hH Q A P ( H / L )

( 2 B + W ) ( T B - T o ) ’ f = +pu* V NU=-, h =

k

where u was the average velocity at x = 0 from y = 0 to H. For fixed rib length W and overall spacing L , it was observed that the Nusselt number Nu averaged over the rib reached a minimum as B / H increased from zero. When B / H increased from zero, there was a decrease in h because of the increased surface area of the sides that initially were not effective in transferring heat. At B / H of about 0.3, a minimum was reached beyond which Nu increased. For this fixed ReH, as H decreased, the flow over the top of the rib accelerated causing an increase in the heat transfer from the top surface, and increasing the strength of the recirculation region and the heat transfer from the vertical surfaces. The friction factor increased continuously as the channel became more constricted. The experimental results of Lehmann and Wirtz [147,148] did not show this minimum in Nu, since only the heat transfer from the top of the element was measured.


b

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

LlH = 1.5, WIH = 0.5

f

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

B/H

FIG. 43. Numerical results for a rib on an adiabatic plate: (a) geometry and nomenclature (dashed lines denote computational domain) and (b) Nusselt number (0) and friction factor (0) dependence on channel spacing for ReH = 100, Pr = 0.7 [145].

Davalath and Bayazitoglu [ 1441 considered the effects of conduction to the plate onto which the ribs are attached and computed the entrance region for this geometry. In addition, the heat transferred through the plate to an adjacent, identical channel was computed. Although each component had a fixed heat dissipation, the heat flux varied with position on each face of the rib. In increasing the plate to fluid thermal conductivity ratio, k , / k f , from 1 to 10, the Nusselt number over the rib was not affected


significantly, but the percentage of the total heat generated that was transferred through the bottom surface of the plate into the adjacent channel increased from 32 to 46%.

C. ARRAYS OF THREE-DIMENSIONAL PACKAGES

Several experimental studies of heat transfer from three-dimensional modules resembling electronic packages, alone or in arrays, have been reported 152-165. In addition, numerical simulation of a full three- dimensional module has been reported by Bullister et al. [166] and Asako and Faghri [154].

1. Regular Fully Populated Arrays

This section considers single elements and arrays withouf intentional perturbations, such as surface fences, missing modules, and elements of different heights. Studies of these effects are reviewed in the following section.

Forced convection data from single electronic packages mounted on a circuit board were presented by Buller and Kilburn [156], both with and without attached finned heat sinks. All of the data were correlated successfully to within 15% using the Colburn &factor and a length scale account- ing for three-dimensional effects. The definitions were given by

L, = [ (Af /Cf) (AT/L)] ' /* , 8 = ( h / p ~ v ) P ? / ~ , , Re = ULe/v

where A f and Cf are the frontal area and circumference, respectively, AT is the total wetted area, and L is the module length in the flow direction. Velocities considered were in the range from 0.5 to 4.0 m/sc. Chang et al. [157] also used this approach in correlating data for single modules and pairs of modules.

Experimental heat-transfer coefficients for arrays of modules have for the most part been measured with only one active element (heated, or sublimating in the case of mass-transfer studies) in an array of nonactive modules. Moffat and Ortega [80] found that the measured value was the adiabatic heat-transfer coefficient and its proper application in a fully heated array could be made by correlating the element temperature to the adiabatic temperature rather than to the local mixed mean temperature. These values were found to be quite different depending on the level of diffusion and turbulent mixing. Nakayama [135] defined a temperature scaling factor u relating the local adiabatic temperature rise to the mean temperature rise, and showed that it could be derived from the ratio of the heat-transfer coefficient from an element in a heated array, and one in an


unheated array. Alternatively, m could be determined by measuring the adiabatic temperature rise of unheated elements downwind of a heated element [153,155,159,165] giving the thermal wake function 0, and adding these contributions by linear superposition as proposed by Arvizu and Moffat [153]. It was thus possible to predict the temperature of an arbitrarily heated array [153,110].

Moffat et al. [153,159] investigated flow and heat transfer from arrays of 1.27-cm (*-in.) cubical elements mounted on adiabatic plates. Cubical elements have four-fifths of their total exposed surface area on the lateral surfaces, whereas electronic packages such as leadless chip carriers and flat packs have most of their exposed area on the top surface, hence, differences were expected in their respective behavior. In the relatively sparse array, Moffat et al. [159] found that heat transfer from an element was dependent on channel spacing and array spacing, and also on both the mean local velocity about an element, which was different than the mean channel velocity, and on the level of turbulence. Figure 44b shows the mean velocity profiles through the channel for an array with S/B = 2 and a large channel spacing, H / B = 4.62. Figure 44c shows the streamwise component of turbulence intensity normalized on local mean velocity. Note that the top of the array is at y / H = 0.22 in both of these figures. Because of the flow drag, there was an increasing mass decrement in the region below the tops of the elements, but the level of local turbulence increased. For H / B = 1, where no flow occurred over the tops of elements, the entrance region was characterized by an increase in heat transfer from the first row to the fully developed region some five rows downstream; for H / B > 1, as in Fig. 44, the heat-transfer coefficient was highest on the first row and decreased to the fully developed value. The authors [159] postulated that h responded only to increases in turbulence intensity when there was no flow bypass over the top of the array and that the mass decrement overwhelmed the increase in turbulent mixing for larger channel spacing accompanied by decreases in the heat-transfer coefficient. Figure 45 shows representative data for three channel spacings and two array spacings. Figure 45a shows both the Nusselt and Reynolds numbers based on channel height and illustrates a clear grouping of data based on channel height. Dependence on channel height was taken out to a large extent by redefining a Reynolds number based on an average array velocity, defined as

4 r r a y l U = (cp/cp*)1’2 (44) where C, was the local total to static pressure coefficient and Cp* was the coefficient for the case of H / B = 1. Figure 45b shows the representative data described in terms of element height and array velocity and indicates

a

b

0 SMOOTH DUCT U 7 1 m/S

0 1 ROW UPSTREAM U 5 9 mlS

I .

Ul U

C

0 SMOOTH DUCT U = 7 1 mls

0 1 ROW UPSTREAM U = 5 9 m/s

A 3 ROWS UPSTREAM U = 5 7 m/s I . h

FIG. 44. Hydrodynamic data for an array of cubical elements in a widely spaced channel, H / B = 4.62, S / B = 2.0: (a) geometry and nomenclature, (b) mean velocity profiles at four positions in the array, and (c) u component of turbulence intensity at four array positions r ? c m

.U .

100

- X

5 0 . 9 v

m z

20

2oo‘

b

’

-

-

- REPRESENTA T lVE OF ALL OTHER CASES

10

200 500 1000 2 0 0 0 5000 10000

ARRAY REYNOLDS NUMBER ( U O r r a y B I U )

FIG. 45. Nusselt number for an array of cubical elements for three channel spacings and two element spacings: (a) Nu and Re based on channel height, Re based on channel average velocity and (b) Nu and Re based on element height, Re based on array velocity [159].

235


200

1 0 0

10

a

U - 0 o o c 1

b I . I

1-110

I I I I

1-110 . 0 1.25 v 1.50

0 3.00 4.62

- 0 2 . 0 0

Sparrow et al.

103 2~ 103 5 x 103

ReL ( U L I v )

104 2~ 104

FIG. 46. Heat-transfer data for sparse array of flat-pack modules: (a) geometry and nomenclature and (b) Nu dependence on component Reynolds number [165].


that the data do not organize according to channel height but are more tightly correlated, with the exception of data for SIB = 2, H/B = 1. Piatt [ 1601 measured heat-transfer coefficients for arrays of short cylindrical elements on an adiabatic channel wall and found behavior similar to the Moffat data for cubes.

An indication of the difference in behavior between arrays of tall elements, such as cubes and cylinders, and short, flat elements, such as flat packs, was given by the data of Wirtz and Dykshoorn [165]. The heat- transfer coefficients on arrays of aluminum modules, resembling LSI packages, measuring 25.4 X 25.4 mm on the top surface, and 6.35 mm high (1 x 1 x +in), with equal spacing S to element length L were measured. The layout is shown in Fig. 46a. Although a strong dependence on channel flow rate was found, only a weak dependence on channel height was apparent. The heat-transfer coefficient h was found to vary only 10% for H/B varying from 1.25 to 4.6. Figure 46b illustrates the excellent collapse of all of the data when both the Nusselt and Reynolds numbers were based on element length L , and the Reynolds number was based on average plate-to-plate velocity U . This suggested that the thermal conditions resembled a developing external flow more than an internal one. Basing the Reynolds number on the array velocity did not improve the correlation of the data, and it was concluded that the model was not appropriate for heat transfer from arrays of flat packs where the lateral surface area was smaller than that for cubes. It was further concluded that the thermal wake dispersed much more rapidly for the array of cubical elements than for the array of flat packs because of the greater degree of flow disturbance.

Sparrow et al. [161] used the napthalene sublimation technique to measure the Sherwood number and infer a Nusselt number for modules similar to those of Wirtz and Dykshoorn [165]. The modules measured 2.67 x 2.67 x 1.0 cm were spaced 0.67 cm apart for S / L = 0.25, and had a fixed channel height of 2.67 cm. Figure 46b compares the Sparrow data with that for the sparser array of Wirtz and Dykshoorn, based on the scaling recommended by the latter. The Wirtz and Dykshoorn heat-transfer coefficients were greater in magnitude and more weakly dependent on component Reynolds numbers where the data overlapped; Wirtz and Dykshoorn credit increased interaction of the cavity fluid with the bypass flow as the source of the difference by analogy to their experiments with rectangular ribs [147,148]. The Wirtz and Dykshoorn data were correlated by the expression

NuL = 0.348 Re t6 (45) and the data of Sparrow et al. [161] by

NuL = 0.0935 Re:”


TABLE V

HEAT-TRANSFER COEFFICIENTS FOR SEVEN SITUATIONS OF TURBULENT FLOW WITH AND

WITHOUT MODULES BETWEEN PARALLEL PLANES [160]

Source Case Correlation equation h C, Row

Kays and Crawford [167] Smooth planes h = 0.023 14 -

Sparrow et al. [161] With barrier h = 0.112 Re".7" 43 - Wirtz and Dykshoorn [165] Sparse flat packs h = 0.324 48 - Moffat et al. [159] Sparse cubes (3/1) h = 0.600 Re".56 60 0.120

Sparrow et al. [161] Dense flat packs h = 0.078 Re".72 29 0.034

Moffat et al. [159] Dense cubes (2/1) h = 0.650 Re".56 65 0.160 Buller and Kilburn [156] Single flat packs h = 0.722 61 -

Moffat et al. [159] compared correlations like those of Eqs. (45) and (46) for seven situations of turbulent flow in channels with and without modules. The results of this comparison are repeated here in Table V. In this table, the Reynolds number was based on the plate-to-plate spacing and the mean velocity in this same space, and C, was pressure drop per row made dimensionless by velocity over the elements. The first entry is from an analytical expression given in Kays and Crawford [167]. The value of h was for Re = 3700; the authors note that h and C, increased together for increasingly intrusive protuberances and that the Reynolds number exponent decreased. Both tendencies were consistent with the increased levels of flow disturbance, or turbulent mixing, promoted by the protruding elements.

2 . Arrays with Missing Modules or Height Differences

A research group under the direction of Sparrow performed a succession of studies investigating various effects on heat transfer from an array of modules described in the previous sections. These included the effects of missing modules [161, 1631, the effects of fences or turbulators [161, 1631, the effects of height differences between modules [164], and the effects of an array with and without fences on the heat transfer from the opposite, smooth wall [162].

Figure 47 shows the effect of a single missing module in a channel of fixed height at three Reynolds numbers. The Reynolds numbers in this series of figures were based on the height above the elements and the average velocity in the space above the elements, assuming no flow below the top of the elements. In Fig. 47, the ratio of h with and without a missing module is shown for three values of Re. The shaded module was


FIG. 47. The effect of a missing module on heat-transfer coefficient for other modules in the array [161].

the monitored module, and the numbers shown on other modules correspond to the ratio of h on the shaded module, with a module missing in the location of the module showing the data. Because of the flow disturbance introduced by the missing module, there was enhancement of heat transfer on modules downstream, upstream, and even to the side. Enhancement was greatest on the module just downstream of the missing module, for the lowest Reynolds number.

Surface fences, sometimes referred to as turbulators in electronics cooling applications, have been used to enhance heat transfer by introducing large levels of flow disturbance and transverse velocity components to the channel flow. Figure 48 shows the ratio of heat-transfer coefficients with and without an implanted barrier, or surface fence, for the taller of two barriers investigated in Sparrow et af. [161]. Enhancement was greatest on the second row downstream of the barrier, presumably near the region of reattachment. A twofold increase in heat transfer was obtained at the lowest Reynolds number, and 30% increases were obtained for as many as five rows downstream. A modest reduction of heat transfer was found upstream of the fence. It was found that enhancement was greater, and less dependent on Reynolds number, for the taller of two barriers, further reinforcing the observation drawn from Table V that increased intensity of


n-1 0 89 2 0 0

0 pJ 167

0 0 o 0 0 FIG. 48. The effect of a surface fence on heat transfer from downwind modules for

h / H = 3, various Reynolds number [161].

turbulent diffusion could account for most of the observed enhancement of heat transfer. As expected, higher pressure drops were the penalty for increased heat transfer.

The effects of heat-transfer-enhancing fences, or barriers, were continued in Sparrow e? al. [163] where enhancement due to a pair of barriers separated by one or more modules were measured. Typical results are shown in Fig. 49, for a single barrier on the left, and for a pair of barriers with increasing number of separating modules increasing from left to right. Enhancement is readily apparent, and there was an obvious dependence of average increase in h for the interbarrier modules on barrier spacing. Interestingly, h for the elements between two barriers was very nearly the same as for the elements downstream of a single barrier, with the exception of the elements just upstream and downstream of the second barrier. Sparrow and Otis [162] measured mass-transfer coefficients on the smooth wall opposite the module wall and found a complex variation of Sherwood number, especially in the presence of an implanted fence, as illustrated in Fig. 50. The Sherwood number was observed to reach a minimum just beyond the location of the fence, followed by an increase to a maximum and a subsequent drop-off. The authors hypothesized that the separation and reattachment process on the lower wall downstream of the fence gave rise to a separation bubble on the opposite smooth wall. Such behavior has been documented in flows downstream of backward-facing steps. The maximum Sherwood number on the opposite wall of Fig. 50 apparently corresponded to the point of reattachment of the flow on this wall.

d- "

Y H I t W

10

.

FIG. 49. The effect of interbarrier spacing on Shenvood number, normalized on its value for an array with no harriers, for an array with two bariers, for h / H = 0.4 and Re = 2000 [163].

I - Re

v 1800

A 3200 -

- 0 5600

0 10000 -

V

A 0

-

- 0

1.0 L a

V

V

A A

0 0 0

0 " 8 A v 0

V

V

A

A

0 0 0

0

0 , 1 I I I I 0 8 10 12 14 1

X/P

3

FIG. 50. Shenvood number on a wall opposite a wall with modules and a single barrier for h / H = 0.183 and various Reynolds number [162].


3. Arrays on Actual Circuit Cards

Bibier and Sammakia [155] made extensive heat-transfer measurements in an array of 20 circuit cards, each populated with heat-dissipating modules that could be arbitrarily heated. Their channel is shown in Fig. 51a. Each circuit card channel was fairly short in length, comprising just four to five rows of modules since, in this investigation, the entrance and exit effects were considered to be important. Cards were populated with dense arrays of modules of uniform size, and three sizes were investigated, as shown in Fig. 51b. In order to predict temperatures, the superposition approach of Arvizu and Moffat [153] was used. Since substrate conduction effects were found to be important, the experimentally measured superposition kernel functions accounted for both the convective and conductive interaction between a heated element and its unheated neighbors. It was observed that in actual circuit cards, conduction effects were nearly always important because of the presence of copper power distribution planes within the board. Ashiwake et al. [152] measured external thermal resistance and pressure-drop characteristics for finned LSI modules on printed wiring boards and developed correlations appropriate for design. The effects of arranging modules in a staggered array were investigated, and large increases in heat-transfer coefficients and pressure drops were found.

COL

a b

I IT4

36 mm

1 looooool

1 0000000 2 0000000 3 0000000 Z 4 m m

4 0000000 5 El 0000000

FIG. 51. Experimental apparatus of Bibier and Sammakia 11551: (a) circuit card array in channel and (b) typical circuit cards showing relative size of modules.


Unlike the case of buoyancy-induced convection in vertical printed circuit board channels, very little data on actual printed circuit boards have been reported with which to corroborate experimental correlations and numerical results from simplified, but more tractable situations. The data reported are generally in the form of external thermal resistance versus other parameters such as channel velocity. Such a presentation does not allow the separation of convective and conductive heat-transfer effects and thus hinders progress toward developing generally applicable data bases and design principles.

D. SUMMARY

In closing, two important areas with growing levels of research activity should be mentioned briefly. Mixed forced and free convection in configurations important to cooling electronic packages have received attention. Both numerical [ 168-1701 and experimental studies [ 171-1731 have been performed. These represent a partial sampling of more recent work.

Work reported by the research group of Mikic and Patera [174-1771 may have an important impact on enhancement of heat transfer in single-phase forced convection cooling of electronic systems. This group demonstrated, by both numerical and experimental investigations, the concept of resonant heat-transfer enhancement, wherein shear-layer instabilities in separated flows may be externally excited at the system resonant frequency. The resulting instabilities produce large-scale motions and lead to dramatic heat-transfer enhancement. Since electronics cooling applications frequently involve complex, separated, internal flows, this type of enhancement may have useful application. Patera and Mikic [177] demonstrated the concept on internal flow in a grooved channel, similar to the type discussed previously, and found up to a threefold enhancement in heat transfer when the channel flow was modulated at its natural frequency.

IV. Boiling and Immersion Cooling

As illustrated in Fig. 2, applications involving high heat flux densities may require that thermal control be provided by direct contact with low boiling point, dielectric fluids. This type of thermal control can be divided into two major categories: total immersion, in which natural convection boiling takes place; or jet impingement cooling, in which a pressurized liquid jet propels a fluid into contact with the device to be cooled.

Immersion cooling applications in the electronics industry were introduced in the late 1940s [178], with early applications in thermal control of


klystron tubes and high-voltage power supplies [ 1791. Other industrial reports followed, describing ongoing investigations at IBM [ 180- 1821, Raytheon [183-1851, and Trident Laboratories “61. In 1968, Bergles et al. [187] published one of the first academic reports, which dealt with high-power-density computer components.

Early investigations indicated that fully developed nucleate boiling could accommodate very large fluxes and that once the onset of nucleate boiling is reached, surface temperatures remain relatively constant regardless of increases in the applied power [188]. Observation resulted in a rapid increase in the research activity, especially in the area of boiling and the associated phenomena in dielectric fluids.

Investigations into the fundamental phenomena in immersion cooling (i.e., boiling [ 1891 and condensation [ 1831) indicated significant differences in the pool boiling behavior of dielectric fluids. Although the surface tension associated with these fluids is typically lower, this could not account for the observed differences including the high gas solubility at moderate temperatures, the surface temperature excursion associated with delayed bubble nucleation [ 1901, the premature boiling at surface temperatures below saturation [191], or the rapid decline in the condensation heat-transfer coefficient [ 1921.

Throughout the late 1970s, industry continued to lead in the development of laboratory experiments with isolated thermally simulated components and in the design, test, and implementation of immersion cooling systems for prototype microelectronic equipment [190]. Many of the major computer corporations became active [ 193-1961 along with companies involved in the packaging of military hardware [197-1991. The early 1980s marked the active involvement of academia in applied research [200-2061 with a wealth of new data [190].

A. IMMERSION COOLING

Although liquid natural convection from arrays of immersed components (Section 11) offers the possibility of excellent thermal control at moderate heat fluxes, ebullient heat transfer provides a unique thermal transport capability and has attracted much of the immersion cooling interest.

Numerous investigations of simulated chips mounted on printed circuit boards have been conducted. In the earlier studies, midpoint heat-transfer coefficients were obtained for simulated chips flush mounted on a vertical substrate [119, 1201. The physical sizes of these devices, 4.6 X 2.3 mm, 9.8 X 4.9 mm, and 20 X 10 mm, were such that the aspect ratio (width to height) was two. Tests were conducted in both air and liquid Freon-113


1 o6

1 o6 cu

E 5 mO U

1 o3 1 10 100

Ts 8 K

FIG. 52. Baker's boiling curves for R113, Ts=47.6"C [119].

(R113). Figure 52 illustrates the results obtained for Freon-113. As shown, there is a departure from established boiling behavior with the variation increasing as a function of decreased chip size. Although it was hypothesized that these variations were due to leading edge and side flow effects, this phenomenon was not investigated until Park and Bergles [207] studied a similar configuration using foil heaters that protruded from the surface approximately 1 mm to simulate microelectronic chips. Two configurations were investigated, those flush mounted on the circuit board substrate and those protruding from the substrate about 1 mm. The heat-transfer characteristics were obtained with varying height (e.g., 5 to 80 mm) and width (e.g., 2.5 to 70mm) in refrigerant R113. For the arrays investigated, inception of boiling on the top heater took place at lower superheats than for bottom heaters. Observation of three types of temperature overshoot, showed that the inception of boiling depended strongly on the location of the boiling site on the heater and the location of the heater within the array


of heaters. For the arrays investigated, inception of boiling on the top heater took place at lower superheats than for bottom heaters. Once boiling was established, the size of the heater appeared to have no measur- able effect. The critical heat flux increased with decreasing heater and/or width. The double temperature overshoot illustrated in Fig. 53 was observed occasionally and was assumed to be due to the lack of active nucleation sites near the location of the thermocouples. In addition, a reversed overshoot was observed during periods of decreasing power.

Anderson and Mudawwar [208] conducted an experimental study of boiling from a simulated microelectronic component immersed in a stagnant pool of Fluorinert (FC72). Various enhancement surfaces were attached to electrically heated calorimeters in a vertical orientation. Sev- eral enhancement schemes including fins, studs, grooves, and reentrant cavities were evaluated in an effort to reduce temperature overshoot.

1 o0

1 o6

N E s mu U

1 0'

L

POWER

INCREASING DECREASING

FOIL HEATERS H = 5.6 rnm. W = 40 mm ti = 5.6 rnm, W = 60 mm

Laminar Natura Convection

1 10 100

ATs. K

FIG. 53. Comparison of boiling for different width heaters [207].


100

10

P

1 .o

0.1 1 .o 10 100

FIG. 54. Effect of surface roughness on the boiling curve for a flat, vertical surface [208]: mirror polish, TI = WC, T, < 1"C, CHF= 19.5 W/cm2 (A); sanded (600 grit), TI = 13.l0C, T, < I T , CHF = 20.5 W/cmZ (0); vapor blasted, TI = 9.2"C, T, = 1"C, CHF = 20.3 W/cmz (0).

Low-profile surface geometries significantly enhanced the boiling characteristics; drilled surfaces had only a small effect. In addition, as illustrated in Fig. 54, the critical heat flux (CHF) was not affected by surface finish or artificial cavities (Fig. 55) , but was dependent on the macrogeometry. Boiling heat transfer within a thin falling liquid film on a vertical surface was investigated by Mudawwar et al. [209]. With flow visualization techniques, it was apparent that nucleate boiling caused the film to thicken in the flow direction, resulting in a high void ratio near the downstream end of the heater. As a result, film breakdown and separation were more likely to occur in long heaters at relatively low heat fluxes. The CHF data were

248

t I I I 1 1 1 r u

I - I

I -

-

- CHF - - Incipience +

-

T, = 85 OC - I I I 1 1 1 1 I I I l ;


100

10

N- E 0

2 Y

U

1 .o

0.1 1 .o 10 100

FIG. 55. Effect of artificial cavities on the boiling curve for a flat vertical surface [208]: horizontal array, Tl = 12.3"C, T, = 1"C, CHF = 20.7 W/cm2 (A); vertical array, Tl = 18.7"C, T, = VC, CHF = 19.5 W/cm2 (0); inclined array, TI = 7.9"C, T, < 1"C, CHF = 25.5 W/cm2 (0).

nondimensionalized and correlated with a semiempirical model based on Helmholtz instability and microlayer dryout.

Yao and Chang [210] investigated boiling heat transfer in confined annular spaces. Although power limitations prevented reaching the CHF, other experimental results indicate that for the same heat flux, the wall superheat decreases as the annular gap increases. As a result, for narrow gaps, the boiling curve is almost vertical at low surface heat fluxes.

Mohenski et al. [211] summarized the results of several investigations and concluded that for this type of configuration the heat flux in the lower superheat region tended to increase as the gap size was decreased, but the


CHF tended to decrease. An explanation for these phenomena was presented and justified. In a separate investigation, a 3 X 3 array of heaters simulating microchips was mounted on a channel wall with the opposite wall subcooled. Experimental results indicated that heat flux levels of 2 X lo5 could be reached prior to dryout [211].

Several investigations involving porous coatings or amorphous surfaces have indicated a substantial increase in the total number of active nucleation sites per unit area relative to the bare surface and shifted the active nucleation site population toward larger radii by forming reentrant cavities [190]. Figure 56 illustrates the nucleate boiling characteristics for R113 with high heat flux coatings. These effects were accomplished, as shown, but only for decreasing thermal loads. As a result, it was concluded that neither amorphous nor structured-enhanced surfaces are presently capable

I I I I I I l l 1 I I I I I I I I I I I I I I

0, A Increasing 0 , A Decreasing - - - _ Data of Bergles - Incipient Boiling

Natural Convection Prediction

1 o2 0.1 1 .o 10 100

Tw- TsAT ("C)

FIG. 56. Comparison of high flux surface with data of Bergles and Chyu [204].


of eliminating the observed boiling incipience wall temperature overshoot. In many applications, however, the thermal excursion could be within acceptable limits, and its presence might only slightly diminish the overall advantages of immersion cooling. After review of the thermal mechanisms responsible for incipience superheat overshoot, a method was proposed to obtain an estimate of the magnitude of incipience superheat excursion for boiling in dielectric fluids [ 1901.

In addition to the problems associated with thermal excursion and temperature overshoot, the ability to predict the heat-transfer characteristics for both nucleate boiling and critical heat flux is important. Many studies that emphasize CHF in forced internal flow have been reviewed previously [212-2221; relatively few deal specifically with CHF in pool boiling [223,224]. Although enhanced chip surfaces improve control of the transition from natural convection to boiling, some investigators have had difficulty in preventing chip temperatures from exceeding the maximum chip operating temperatures [ 1941, Several experimental and analytical methods involving pool boiling and CHF have been reviewed [225, 2261. These fall into two categories: those dealing with hydrodynamic aspects [227-2301 and those dealing with nonhydrodynamic aspects [226].

Investigations to determine the CHF for vertical heaters with one side insulated have also been performed [231, 2321, and a semianalytical correlation for predicting the CHF with this geometry was presented [230]:

q$HF/q$HF, z = 1.4/(H')'l4 for H' < 6 (47)

q$HF/q&F , z = 0.9 for H'> 6 (48) whcie

q$HF,Z = 0.131 hfgPv[dPf- Pv)~/P:] ' /~ (50)

which is ihe CHF prediction of Zuber [228]. The data set of Adams [232] for widz heaters was used to justify the asymptotic value of the correlation for large H' [202].

Hwang and Moran I2331 described a cooling technique for cooling substrate-mounted silicon chips during electrical tests. Thermal tests were conducted in order to characterize the boiling process in an open bath of FC86 with the experimental apparatus illustrated in Fig. 57. Both chip powers and chip temperatures were measured throughout the nucleate boiling region and up to the CHF. The following equation,


MULTlCHlP SUBS

CONNECT

TRA

'OR

VAPOR LINE

OVERFLOW LINE

BELLOWS

OBE

CHIPS 8 SUBSTRA

__ COOLANT IN

TE

DRAIN LINE

FIG. 57. Open bath test chamber of Hwang and Moran [233].

was proposed to estimate the CHF for silicon chips in a subcooled liquid. In addition, several possible enhancement techniques were experimentally investigated, along with chip orientation and chip surface treatments.

B. JET IMPINGEMENT COOLING

Jet impingement cooling techniques are common in industrial cooling or drying and have been the subject of numerous investigations in thermal control of gas turbine blades, annealing of metals and nonmetals, temper- ing of glass, cooling of pistons in internal combustion engines, and drying of various materials. The technique employed involves the direction of one or more fluid jets at the surface to be cooled. The high-velocity fluid impinging on the surface results in a large heat-transfer coefficient in the vicinity of the stagnation point. The fluid jet may be gaseous or liquid and the impingement can occur in a gaseous environment, referred to as free jet impingement, or be contained in a bath of the same fluid as that of the jet, submerged jet impingement.

In free jet impingement, the effect of the distance between the jet orifice and the surface to be cooled is insignificant if the liquid layer flow on the solid surface is exposed to an ambient gas environment [234]. In submerged jet impingement, however, the enhancement effect is greatly affected by the distance between the jet orifice and the target surface because of the viscous hydrodynamic interaction.


In both submerged and free jet impingement, the magnitude of the heat-transfer coefficient depends strongly on location and decreases rapidly as the radial distance along the surface increases. Maximization of the heat-transfer coefficient, and hence the heat-removal rate, requires optimization of the jet configuration, velocity, location, and number [235]. Because of the small chip sizes and the desire for uniform surface temperatures, this optimization procedure is considerably different from that on larger, higher flow rate, industrial jets. These problems are compounded by the high heat fluxes, the limitations on the resulting force on the chip, and the need for electrically nonconducting liquids.

Submerged Jet Impingement

Submerged liquid jet impingement may include bubble nucleation and the associated boiling or it may occur without boiling. Despite the high values reported for submerged jet impingement cooling, > 100 W/cmL [201], jet impingement cooling, with or without nucleate boiling, has received only limited attention. Data for air jet impingement, however, is available and can be applied to submerged liquid jet impingement because of the similarities. For this review, those cases in which no bubble formation occurs are in the section on forced convection.

Early work on the hydrodynamic behavior of single circular liquid jets impinging on horizontal surfaces produced information about the effects of the thickness of the liquid layer in the wall region for a wide range of flow rates and fluid properties in the supercritical region [236]. Analytical solutions for the heat transfer from horizontal plates with free laminar liquid jets impinging on them have also been developed [237] using a previously developed hydrodynamic solution. In addition, with a similar geometry and a mass-transfer analogy, analytical solutions for the heat- transfer coefficient have been obtained for both laminar and turbulent flows [238].

This analytical work was followed by investigations of submerged jets for the cooling of finite heated surfaces [239]. A single water jet cooled a heated, horizontal, square surface. Similar investigations [240] followed with various jet sizes and configurations. An empirical equation was presented for jets comprised of water, R12, and R113 over a broad range of jet diameters, density ratios, and jet velocities of up to 26 m/sec [241]. A correlation for CHF was found for velocities in excess of 26 m/sec, at which the effects of increasing jet velocity are reduced significantly [241].

Ma and Bergles [201] constructed an apparatus to study the heat-transfer characteristics of chip-sized, electrically heated test sections with normally impinging circular submerged jets of saturated or subcooled R113. Fig-


1 o6

cu E

5 lo6 b P

1 o4

I f

/ lo

6 0

0

0

TEST SECTION No. 5

0 POOL BOILING h uo = 1.08 m/s 0 uo = 2.72 m/s

r/d = 0 z f d = 2

10' 1 o 2

AT = Tw- Ts OC

FIG. 58. Jet impingement boiling data compared with pool boiling [201].

ure 58 illustrates the boiling characteristics. At each velocity, the data exhibit a slight temperature overshoot before established boiling, and the data for both velocities are coincident in the region of established nucleate boiling. Burnout heat fluxes varied as the cube of the jet velocity, but were also weakly dependent on subcooling. It was demonstrated that chip

TABLE VI

STUDIES OF JET IMPINGEMENT BOILING [201] ~ ~

Investigator Fluid Jet Single Partial Developed

Test section phase boiling boiling Burnout

Katto and Kunihiro

Katto and Monde

Ruch and Holman

Monde and Katto

Katto and Ishii

Miyasaka and Inada

Miyasaka and Inada

Monde

Water saturated

Water saturated

R113 saturated

Water, R113 saturated and subcooled

Water, R113, triclorethane

Water subcooled

Water subcooled

Water saturated

d = 0.71, 1.165, 1.6 mm; z = 1-30 mm;

d = 2 mm; z =30 mm;

d = 0.21, 0.433 mm; z l d = 22.6;

d = 2, 2.5 mm; z = ?;

u,, = 1-3 m/sec; 0 = 90"

u,, = 5-69 m/sec; 0 = 90"

uo = 1.23-6.87m/sec;~B-45-90"

u,, = 3.7-12 m/sec; 0 = 90"

w = 0.56-0.77 z = ?; uo= 1.5-15 m/sec; 0 = 15", 60"

w = 10 mm; z = 15 mm; u,, = 1.1, 3.2, 15.3 m/sec; 0 = 90"

w = 10 mm; z = 15 mm; uo = 1.5-15 m/sec; 0 = 90"

d = 0.7-4.15 mm; z = ?; uo = 0.3-15 m/sec

D = 1 0 m m

8 x 8 m m X

D = 12.9 mm

D = 11, 16, 21 mm

15 x 10, 15 x 15, 15 x 20 mm

4 x 8 m m X

1.5-2 mm

D = 11.9-25.5 111111

X X

X X

X

X X

X X

X X X

X X X

X X


powers well in excess of 20 W per chip could be accommodated within the usual 86°C limit on junction temperature.

Martin [235] prepared a comprehensive review of single-phase impingement heat transfer, which is summarized in Table VI [201]. This table lists the investigators, the type of fluid, and information on the jet characteristics and dimensions, the test-section diameter, and the fundamental phenomena investigated. Most of the information from these investigations is for applications larger than those of interest to individuals involved in the thermal control of electronic devices. Lacking is information on impingement cooling in vertical orientations and information on the effect of operating vertically in a totally immersed environment.

Ruch and Holman [242] investigated a single submerged jet of liquid refrigerant (R113) impinging on a flat surface. A maximum heat flux of 73 W/cm2 was obtained with an excess temperature of 45°C. The results of this investigation were presented in the traditional Rohsenow form and also as correlation with an added velocity term. Ma and Bergles [201] systematically quantified the effects of velocity variations, subcooling, flow direction, and surface conditions for a single jet of liquid refrigerant (R113) impinging on a flat surface submerged in refrigerant. Maximum excess temperatures of 40°C for jet velocities of 10 m/sec were reported, but no values of the critical heat flux were reported.

Jiji and Dagan [234] investigated the characteristics of free liquid jet impingement on high-power heat sources under single-phase conditions. Three heat-source arrangements were tested: single, square sources, and 2 x 2 and 3 X 3 arrays. Jets of diameter 0.5 and 1 mm were utilized in configurations of one, four, and nine jets per source. The impingement was perpendicular to the heat sources, which were oriented vertically. With FC77 and water, a single correlation was obtained for all configurations. Jet to heat source spacing and cross-flow effects to have minor influence on the average surface temperature. The results of this investigation indicate that the thermal resistance in jet impingement cooling can be reduced significantly by increasing the number of jets and decreasing the jet diameter. Figures 59 and 60 present a comparison of one, four, and nine jets per source with d = 1 mm. Although a slight improvement in the temperature uniformity is apparent with increases in the number of jets, as illustrated in Fig. 60, at Q=O.25 gallons per minute (GPM), the average surface temperature is lower with four jets instead of nine.

In submerged jet impingement systems, the stagnation Nusselt number can be expressed in terms of the dynamic and thermal parameters as Re1/2Pr2/5, where Re is the jet Reynolds number and Pr is the Prandtl number [238, 2431. Determination of the average Nusselt number, although less well understood, is typically in the form Re" P P , where

256

0 0 155-160 1 12560 10 1.0 10.04 0.125

0 149-154 4 3140 10 1.0 2.51 0.125 I 1 I I l l l l l 1 I 1 1


N E 5 x' 3 J LL

k

W I

a

W 0

LL K 3 cr)

W (3 U K W >

a

a

1 o6

8 . 0 0

8 0 0 0

t DDD FC77 J d V (A)

T,, - Tmin Run NO. ( S i i ) RE Z/d (mm) (rn/s) source - - - - - _ _ -

1 oo 10'

(T, - Ti) AND (Tmex-Tmin), OC

FIG. 59. Comparison between one and four jets per source for source F in a 3 x 3 array [234].

0.5 < n < 0.75 and 0.33 < m < 0.42 [138, 164, 1651. Evidence indicates that this general description of single-jet impingement behavior cooling cannot be extended to an array of impinging free liquid jets, since the latter is influenced by the jet-array configuration and the hydrodynamic interaction of adjacent jets [234].

Thermal control of discrete heat sources with these arrays or multiple jets has been investigated, and several findings have been reported. In an investigation of FC77 without phase change, fluxes of 50 W/cm2 were removed from simulated chip sites with corresponding temperature differences of tens of degrees Celsius [234]. It was concluded that increasing the number of jets on a given site increases heat transfer In a separate investigation [244] of the effects of jet velocity, two significant conclusions were reported: (1) there is an optimum jet spacing for maximizing the heat-


5 x' 3 A LL

I- U

lo6

W 0

LL er 3 v)

W

a

Q a U W > n a a

\

15 lo4

r

- ~m O A

8 A O A - - - - - - m~ -

DO0 - 0

ODD - FC77 / d 'J - T,-T, T,, - T,,, Run No (S:%> Re Z/d (mm) (m/s) (source) - 0 143-148 4 6280 10 1.0 5.024 0.25 A A 137-142 9 2790 10 1.0 2.233 0.25

I I I I 1 1 1 1 l I I I 1

FIG. 60. Comparison between four and nine jets per source for source F in a 3 X 3 array [234].

transfer rate for a fixed mass flow rate; and (2) the heat-transfer coefficient depends on the square root of the jet velocity.

Goodling et al. [241] investigated a novel scheme for cooling silicon wafers in high-density integrated circuits [Fig. 611. A Freon-12 refrigeration system was modified to cool a 4 X 4 square array of simulated VLSI chips. Capillary tubes augmented by a manifold were directly below each chip, and compressed liquid coolant was pumped through the manifold ports, resulting in saturated liquid impinging on the underside of the simulated chips. The entire system was mounted inside a transparent package so it could be observed. In the final prototype, a maximum power of 430 was dissipated on the silicon wafer of 46 cm2 with chips having a total simulated surface area of 4 cm'. In this configuration, temperatures at the individual chip sites were less than 10°C above ambient.


JT

FIG. 61. Cooling package proposed by Goodling et al. [241].

V. Thermal Contact Resistance

Two smooth, nominally flat surfaces contact in only a relatively few discrete points as shown in Fig. 62, because of the individual surface roughnesses and microscopic asperities. Most practical surfaces also have large-scale errors of form such as waviness and flatness deviations.

Experimental investigations have demonstrated that the actual contact area for smooth nominally flat surfaces, is between 2 and 5% of the apparent contact area [245]. Because of this area reduction, there is a thermal resistance and, hence, a temperature drop at the interface. Heat can be transferred across the interface by conduction through the solid contacts, conduction through the substance in the gaps around the contacts, radiation across the gap, or a combination of the three. In most

FIG. 62. Real surfaces in contact.


electronic applications, conduction through the solid contacts is the dominant mode, but at low apparent interface pressures, gas conduction may play a significant role in the overall heat transfer. Because of the relatively low temperatures in electronic devices, radiation is usually considered negligible.

The thermal contact conductance is defined as

h = (Q/A.) /AT (52)

where Q / A , is the steady-state heat flux based on the apparent contact area. The thermal contact resistance, also frequently used, is defined as

R = AT/Q (53)

Since this thermal contact resistance results from most of the heat being constrained to flow through the actual contacts, the first logical step in determining the value is to estimate the resistance of a single contact spot. The constriction resistance of such a spot is a measure of the additional temperature drop due to the presence of the constriction. The specific shape of the asperities, which has been shown to be of significant importance in determining the contact resistance [246], and the resulting boundary conditions depend on physical parameters, such as the material, the surface roughness, the hardness, and the nature of the problem being considered.

In a wide variety of electronic packages the construction technique utilized involves a silicon wafer that has been bonded to a substrate or heat spreader with an organic glue, such as epoxy or polyimide loaded with metallic particles, a hard solder, such as eutectics or gold/tin alloys, or a soft solder composed of large percentages of lead, tin, or indium [247]. Once the chip is bonded to the substrate-spreader, the entire device is encased in a silica mold compound. The heat generated in the chip is transferred from the chip through the bonding material, to the substrate, and in turn to the mold compound [248], as shown in Fig. 63. These interfaces are examples of the importance of the resistances occurring at mechanical interfaces in determining the overall resistance of semiconductor packages.

Antonetti and Yovanovich [249], in a review of the thermal management in electronic packages, presented and discussed the many factors contributing to the high internal resistances in semiconductor packages. Yeman et af. [247] and Mahalingham et al. [250] experimentally determined the effect of voids or cracks on the thermal conductivity of the bonded interface. These two investigations indicate that small concentrations of random voids have little effect on the overall resistance, but large contiguous voids may result in significant increases in the chip-to-substrate resistance.


Mold Compound

Diebond Material Flyinq Leads

Substrate-Spreader Material \ Pin Connectors J

FIG. 63. Typical package construction.

An experimental investigation of the significance and magnitude of the thermal contact resistance at the bonded joints between the silicon chips and substrate materials in semiconductor devices was conducted by Peter- son and Fletcher [251]. Seven conductive epoxies with thermal conductivi- ties ranging from 0.27 W/m "C to 1.93 W/m "C in contact with ground Aluminum 6061-T6 surfaces were evaluated. The results indicated that the thermal contact resistances at the chip-bond and bond-aluminum interfaces can be a significant factor in the determination of the overall joint resistance. Although the contact resistances were found to be constant with respect to the mean joint temperature, the contribution of the contact resistance to the overall joint resistance increased by a factor of eight with respect to the thermal conductivity of the diebond materials. An empirical expression was developed for use in the prediction of the overall thermal contact resistance as a function of the thickness of the bonded joint, the thermal conductivity of the bonding material, and the void fraction present. Figure 64 illustrates the results of this investigation and the relationship between the measured overall thermal resistance, represented by the solid symbols, and the computed bulk resistance due to the thickness of the diebond material, represented by the open symbols. The measured thermal resistances are much greater than the thermal resistance associated with an equivalent thickness of the diebond materials, indicating the significance of the interface resistances. The results of this investigation provided insight into the thermal behavior of these interfaces and demonstrated that in instances where the thermal conductivity of the diebond material is high, the thermal contact resistances can comprise as much as 50% of the overall thermal joint resistance.


1 .o

0.8

0.6

0.4

0.2

I I 1 I I 1 I

- -

- -

- -

a - - - - a I - -

~~ ~~~

0.0 0.5 1 .o 1.5 2.0

Thermal Conductivity of Diebond Material (W/mo C)

FIG. 64. Comparison of equivalent thickness resistance (0) and measured (0) values of diebond epoxies [251].

Yovanovich et al. [252] developed an expression for predicting the theoretical contact conductance between similar metal-to-metal surfaces where the roughness of one surface was of the same order of magnitude or greater than the flatness deviation of the other. This expression is a function of the relative pressure, defined as P / H , where P is the apparent contact pressure and the Vickers microhardness H and can be written as

h,u/k,, = 1.25(P/H)0 .95 (54) where

(T = ,1(u*)2 + (a2)* ( 5 5 )

It has been shown to agree within 5 1.5% of the exact theoretical results for a range of Investigations by Yovanovich et al. [252] , Antonetti [253] , and Hegazy [254] verified the accuracy of this expression for similar metal-to-metal surfaces where one surface is nominally smooth and the other has been bead blasted to produce a normal

5 P / H 5


distribution of roughnesses of the same order of magnitude as the flatness deviation of the smooth surface. Eid and Antonetti [255], studied the thermal contact resistance at the interface of 2 X 2 mm aluminum test sections and similarly sized silicon specimens over a pressure range of 27 to 500 kPa. The results of this investigation indicated that the expression given inEq. (3) could be used to predict the thermal contact conductances for bare metal-to-silicon junctions of the size encountered in semiconductor devices. In addition, analysis of the gap resistance suggests that values of the accommodation coefficient agree reasonably well with the previously published values [256].

Peterson and Fletcher [257] experimentally determined the significance and magnitude of the thermal contact conductance at the interface of mold compounds and substrate-spreader materials used in the assembly of semiconductor devices. The interfaces of four mold compounds and three heat spreaders were evaluated over an interface temperature range of 20 to 70°C and an interface pressure range of 0.5 to 5.0 MPa.

The results shown in Fig. 65 indicate that the thermal contact conductance at the mold compound and substrate-spreader interface is relatively constant with respect to variations in the mean interface temperature, but changes significantly because of variations in the interface pressure. As illustrated in Fig. 65, data from the test program compared favorably with the values predicted by Eq. (3). The excellent correlation may be due in part to the method by which the samples were prepared, which resulted in two conforming flat surfaces.

From an analytical perspective, numerous standard thermal modeling techniques have been modified to accommodate electronic packages. In addition, a number of analytical models, such as those developed by Pogson and Franklin [258], Buchanan and Reeber [259], and Andrews et al. [248], were specifically designed to predict the overall thermal resistance of electronic packages. Most of these analytical techniques utilize either finite-difference or finite-element schemes to predict the temperature distribution in the package as a function of the boundary conditions and the physical properties of the package components. All of the existing analytical models neglect the effect of internal contact resistances, such as those at the mold compound and substrate-heat spreader interface, in the computation of the overall package thermal resistance [25 11.

A. ENHANCING THE THERMAL CONTACT CONDUCTANCE

In addition to the interfaces within semiconductor packages, the thermal contact conductance between semiconductor chips and the thermal piston on the IBM thermal control module 12601, between the joints of thermionic converters [261], and at interfaces of high-powered microwave components


I I I I I I I I I 1 I I I r r T

1 oo 1 oo

1 I I I I I I l l I I I I I l l

1 0' 1 o2 4

Relative Pressure (P/Hc) x 10

FIG. 65. Comparison for analytical and experimental results for mold compound to heat spreader materials [257]:SSKhl8N9T bare interfaces (A), SSKhl8N9T interfaces with copper foil (O), SSKhN789T bare interfaces (m), SSKhN789T interfaces with copper foil (e), molybdenum VM-1 bare interfaces (a), and molybdenum VM-1 interfaces with copper foils (+I.

[262] are examples of interfaces where a better understanding of thermal behavior could lead to enhanced thermal performance and improved efficiency and reliability.

Thermal contact conductance in these applications is commonly enhanced by increasing the apparent contact pressure, but this is not always possible because of design or load restrictions. When the applied load is limited, as in electronic applications, the thermal contact conductance can be enhanced with thermally conductive greases, thin metal foils, or a thin metal coating deposited on one or both of the surfaces. The interface material flows into the gaps between the two surfaces and increases the actual contact area, which in turn increases the thermal contact conductance.

Thermal greases are the most desirable of these three techniques for general applications, but they tend to migrate at high temperatures or


vaporize in low pressure or vacuum environments. Once vaporized, they may redeposit on adjoining surfaces or disappear [253]. Metal foils are theoretically attractive but must be very thin to be effective and are therefore difficult to handle. If improperly applied, these foils may decrease the thermal contact conductance because of wrinkles or folds.

Investigation of the phenomena that govern the behavior of interfaces with thin metallic foils has been limited. Koh and John [263] performed a systematic experimental investigation of the effect of thin metal foils of copper, aluminum, lead, and indium, sandwiched between a pair of mild steel specimens. Although copper and aluminum have high thermal con- ductivities, the insertion of these foils reduced the thermal contact conductance compared to the bare joint, as illustrated in Fig. 66, the lead and indium foils improved it. It was concluded that foil hardness was of greater significance in determining the conductance than the foil thermal conductivity. In a second set of experiments, Koh and John [263] investigated the effect of foil thickness and determined that an optimum thickness existed, and that this thickness provided the maximum enhancement of the interface thermal conductance. For surface roughnesses on the order of 4 x lop6 m rMS, the optimum thickness of the foil was found to be 25 X lov6 m. With foils of this thickness, the thermal contact conductance was three times that for the bare metallic joint; foils less than lop4 m thick did not measurably improve the contact conductance.

Cunnington [264] compared the contact conductance of both smooth and rough, bare aluminum-aluminum, and magnesium-magnesium junctions with junctions in which indium foils were present. The indium foil substantially increased the thermal contact conductance. With aluminum specimens and indium foil, the thermal contact conductance of the interface increased as the surface roughness of the metal surfaces increased, indicating that a near optimum thickness had been used.

Mal’kov and Dobashin [267] studied the thermal contact conductance of stainless-steel interfaces with thin copper foils between metallic surfaces made of two different stainless steel and one molybdenum alloy [Fig. 66, curves 1, 2 and 3, respectively). Insertion of the copper foil reduced the value of the contact resistance by a factor of 3-5 for contact pressures of 2.5 x 105-2.9 X lo7 N/m2. Variations in the mean interface temperature had little or no effect on the contact resistance, either with or without the foil.

Molgaard and Smeltzer [265] conducted a similar investigation with gold foils over a temperature range of 50-300°C and a pressure range of 2.5 x 10’-9 x lo7 N/m2. This work and that of Moore et al. [266], which investigated interfaces with indium foils, supports the previous conclusions of Mal’kov and Dobashin [267]. Yovanovich [268] studied the effect of


CONTACT PRESSURE (lb/in2)

a L

2 0 0

d

CONTACT PRESSURE (kN/m2)

FIG. 66. Effects of metal foils on the thermal contact conductance [257]: 1, bronze surface coated with tin-nickel alloy; 2, uncoated bronze alloy, 3, stainless-steel surface coated with silver; 4, uncoated stainless-steel surfaces; 5 , stainless-steel surface coated with aluminum alloy; 6, uncoated stainless-steel surface; 7, nickel surface coated with nickel alloy; and 8, uncoated nickel surfaces.

several soft foils and suggested that the enhancement performance of various foils could be ranked according to the ratio of the thermal conductivity to the hardness of the foil material. It was demonstrated empirically that the higher the value of this parameter, the greater the improvement in the contact conductance over a bare joint. In addition, it was hypothesized and verified experimentally that an optimum foil thickness exists.


An experimental investigation of the effect of surface roughness in the presence of thin metal foils has also been conducted [269] to determine the thermal contact conductance at the interface of a smooth Aluminum 6061-T6 surface and four surfaces of varying degrees of roughness. Both bare interfaces and interfaces in which one of four different thin metal foils was present were tested over a pressure range of 0.5 to 2.0 MPa at a mean interface temperature of 40 -t 5°C. The four foil materials, lead, tin, copper, and indium, were selected for a maximum range of hardness and thermal conductivity values. An optimum thickness for each foil was determined by a method discussed previously [268]. Three conclusions were made. (1) The enhancement of thermal contact conductance can be ranked accurately using the ratio of the thermal conductivity to the hardness. The higher this ratio the greater the enhancement. (2) With an optimum foil thickness, the thermal contact conductance increases if the foil is either softer or higher in thermal conductivity than the contacting surface, but probably not if it is both. (3) With one rough and one smooth surface in contact, there is an optimum surface roughness. Small increases in this roughness may decrease the thermal contact conductance, but continued increases eventually lead to an increase in the thermal contact conductance 12691.

It is logical to assume that the optimum roughness of the contacting surfaces is a function of the hardness and the thermal conductivity of the contacting surfaces. It is necessary to test additional materials to determine which of these two parameters is most significant and to establish the relationship between the material properties and the surface characteristics.

The results of these and other investigations [270, 2711 can be summarized as follows. (1) Foil hardness is of greater significance in determining the thermal contact conductance than foil thermal conductivity. (2) There is an optimum foil thickness for maximum enhancement of the interface thermal conductance. (3) With an optimum foil thickness, the thermal contact conductance of the interface increases as the surface roughness of the metal surfaces increases. (4) Small variations in the mean interface temperature, 20 to 30"C, have little or no effect on the thermal contact conductance. (5 ) The enhancement performance of various foils can be ranked according to the ratio of the thermal conductivity to the hardness of the foil material.

Although the physical phenomena involved in the use of thin metallic coatings to enhance the thermal contact conductance are not well understood, metallic coatings have several desirable characteristics. (1) They are relatively easy to handle once applied and do not wrinkle or fold. (2) Under normal operating conditions, they are stable in a vacuum environment. (3) The vapor deposition, sputtering, and/or electroplating processes for applying these coatings are well understood. Thin layers


of almost any metal or combination of metals can be deposited in the desired thicknesses.

Several experimental and analytical investigations have dealt with the enhancement of thermal contact conductance through the application of thin metallic coatings. Figure 67 presents the results of four separate experimental investigations and compares the results of the contact conductance as a function of apparent contact pressure between coated and uncoated joints. As illustrated, significant increases have been obtained for different combinations of materials.

CONTACT PRESSURE (lb/in2

n Y el'

E

W 0 Z a

n L 3

Z 0 0 I- 0

I- z 0 0

a

A

r4 a W I I-

1 o4

1 o3

W 0

i, lo3 2

n z 0 0 I- 0

I- Z 0

a

0 lo2

d 1 a W I I-

1 o2 1 o3 CONTACT PRESSURE (kN/rn2)

FIG. 67. Effects of metallic coatings on the thermal contact conductance (use key for Fig. 66) [253].


In Fig. 67, curves 1 and 2 compare the results of an experimental investigation [272] in which the thermal contact conductance of uncoated bronze surfaces was compared with that when one surface was coated with a thin layer of a tin-nickel alloy. Coating one surface increased the interface conductance by a factor of three. The magnitude increased slightly with respect to increasing interface pressure. Curves 3'and 4 compare the results of two uncoated stainless-steel surfaces with that when one of the surfaces was coated with silver [253]. These tests, which were conducted in a vacuum, again illustrate a threefold increase in the contact conductance, but the increase displayed a decreasing trend with respect to increases in the apparent interface pressure. Curves 5 and 6 compare the results of bare stainless-steel surfaces in a vacuum with those in which one of the stainless-steel contact surfaces was coated with a thin layer of vacuum- deposited aluminum [271]. In this investigation, the increase varied from a threefold increase at low interface pressures to almost an order of magnitude increase at higher pressures.

Antonetti [273] has conducted the most extensive investigation to date on coated metallic interfaces. The coated metallic interface was reduced to an equivalent bare metal interface by the concepts of effective hardness and effective thermal conductivity and the bare interface correlation developed previously. The predicted thermal contact conductance values were compared with the results of an experimental evaluation of one bare and one silver-plated nickel surface. Figure 68 illustrates the comparison of the predicted values with the experimental results. Again, the contact conductance increased but with a relatively constant increase, approximately twice as large as that in the bronze or stainless-steel surfaces.

As shown, the experimental values compared well and indicated that the contact conductance for this combination of materials can be predicted accurately. In addition, and perhaps more important, this work included a method for ranking layer substrate combinations. Although not verified experimentally [273], it was demonstrated that this parameter could be combined with a bare metal relationship developed earlier to create a ranking parameter defined as

k ' / ( (58)

where k' is the effective thermal conductivity and H' the effective microhardness of the softer material.

Al-Astrabadi et al. [274] developed an analytical model for predicting the contact conductance of tin-coated stainless-steel interfaces and compared the results with those of their own experimental investigation. As


1 o1

n = loo

3 2" 0 Y

W 0 z a I- 10-1 2

a v) W

1 o+

TIN LAYER THICKNESS (1 m)

FIG. 68. Comparison of predicted (-) and experimental (0) results for tin-coated stainless steel P = 4000 kN/m2 [275].

shown in Fig. 69, the comparison is poor. The methodology enabled prediction of the actual constriction resistance, the resistance resulting from the reduction in the actual contact area. In the experimental investigation, however, the values measured were the total interface resistance, that is, the summation of the constriction resistance and the resistance due to the metal coating. This could partially explain the disparity between the predicted and measured values.

Analytical investigations of the effect of thin metallic coatings on the thermal contact conductance of two metallic surfaces requires the simultaneous solution of three problems: (1) the thermal problem, questions of

270

2


5 ", m - L c

lo5 n Y

E (u

2 Y

W 0 z I- 0 3

z 0 l- 0

I- z 0 0

I U w I I-

a

n 10'

a

* 1 o3

CONTACT PRESSURE (lb/in2

1 o3 CONTACT PRESSURE (kN/rn2)

FIG. 69. Effect of coating thickness: 0.0 urn (O), 2.4 urn (A), 7.2 urn (O), 18 urn (m) [253].

constriction resistance and thermal behavior; (2) the mechanical analysis, the various physical properties such as surface hardness and elastic modules [275] and (3) the metrological problem, the effects of the surface characteristics such as roughness, flatness, and asperity slope. The thermal problem has been addressed by Kharitonov et al. [261] and Yovanovich [245]; the mechanical and metrological issues have been investigated by Hegazy [254] and Song et al. [276].

Most of this discussion has dealt with ideal interfaces. It is important to note that oxide formation may play a significant role in the magnitude of


any thermal contact resistances. Some similarities between metal foils and coatings and oxide films are apparent. A comprehensive review of the literature through the 1960s has been presented by Gale [277]. A generally accepted conclusion is that oxide films, unless sufficiently thick, do not appreciably increase the resistance, although they may have a major effect in electrical contact resistance. In addition, the experiments of Tsao and Heimburg [278] on Aluminum 7075-T5 surfaces in dry air showed expected trends, namely, the time of exposure increased the resistance and degas- sing of the surfaces decreased resistance. Opposite trends were noted for the specimens aged in laboratory (humid) air. This anomalous behavior was thought to be due to the decrease of fracture stress of the aluminum oxide films in the presence of absorbed gases, especially moisture.

In general, investigations into the contact conductance at coated metallic interfaces have been directed largely at reductions in the thermal contact conductance due to the formation of oxide films. Because of major differences in the behavior of oxide films, which tend to be brittle and uneven, this work cannot be extrapolated for use in predicting the behavior of thin metallic coatings.

B. SUMMARY There are several analytical and empirical correlations for bare inter-

faces, which, given certain constraints, are capable of accurately predicting the thermal contact conductance at bare metallic interfaces; considerable experimental data are available. There is no analytical solution, however, that can accurately predict the thermal contact conductance of metallic interfaces in the presence of thin metal foils, and only a limited amount of experimental data are available. There is only one analytical solution for predicting the thermal contact conductance of metallic interfaces in which one surface has been coated with a thin metallic layer. This expression has been compared with only a single metal-coating combination but compared favorably.

Experimental evidence suggests that optimum foil and coating thicknesses exist and that these optimum thicknesses are functions of the material hardness, conductivity, coefficient of thermal expansion, and interfacial pressure. The material hardness is of greater significance to enhancement of the thermal contact conductance than is the material thermal conductivity, and parameters for ranking candidate materials or combinations of materials for both metal foils and metallic coatings have been proposed. None of these parameters has been validated through extensive experimental investigations.


VI. Thermal Control Using Heat Pipes

In 1942, Gaugler first introduced the concept of a passive two-phase heat-transfer device for transporting heat [279]. Approximately 20 years later, Grover et al. [280] independently invented and patented a similar device called a heat pipe. Because of the capability of this device to transfer heat over moderate distances with very small temperature drops, heat pipes found widespread use in both spacecraft and land-based applications. Recently, a rapid increase in the application of heat pipes to electronic thermal control has occurred. These applications range from the cooling of individual semiconductor devices to the heat removal from entire systems.

A classic heat pipe, illustrated in Fig. 70, consists of a sealed container lined with a wicking structure. The container is evacuated and backfilled with just enough liquid to fully saturate the wick. Operating on a closed two-phase cycle with only pure liquid and vapor present, the working fluid remains at saturation conditions as long as the operating temperature is between the freezing point and the critical state.

As shown in Fig. 70, heat pipes consist of three distinct regions: the evaporator or heat-addition region, the condenser or heat-rejection region, and the adiabatic or isothermal region. Heat added to the evaporator region of the container causes the working fluid in the evaporator wicking structure to be vaporized. The high temperature and corresponding high pressure in this region result in flow of the vapor to the other, cooler end of the container where the vapor condenses, giving up its latent heat of vaporization. The capillary forces in the wicking structure then pump the

Wick Structure

Liquid Return by Heat Addition Capillary Forces Heat Rejection

FIG. 70. Heat-pipe operation.


liquid back to the evaporator. Similar devices, referred to as two-phase thermosyphons, have no wick but utilize gravitational forces for the liquid return [281].

As illustrated in Fig. 2, the capacities of the various methods of thermal control vary significantly as a function of the allowable temperature difference between the electronic components and the ambient temperature. Conventional modes of conduction and free convection are inadequate for large amounts of heat removal, and a forced single-phase liquid [282] or two-phase techniques are necessary to remain within the required temperature range. Heat pipes have been an important means for implementing two-phase mechanisms for heat removal and thermal control. This technique is common in avionics cooling and more recently in the thermal control of individual devices and entire circuit boards.

Heat pipes, because of their high thermal conductivity, provide an essentially isothermal environment with very small temperature gradients between the individual components. Hence, they are an acceptable alternative to the large, bulky aluminum or copper fin structures of complex geometries that are currently the industry standard. The high heat-transfer characteristics, the ability to maintain constant evaporator temperatures under different heat flux levels, and the diversity and variability of evaporator and condenser sizes make the heat pipe an effective device for the thermal control of electronic components. This review of applications includes recent advances and developments that affect the implementation of heat pipes in the thermal control of electronic devices. The applications fall into three categories: (1) indirect, where the heat pipe is placed in contact with the component or device and serves the same function as a heat sink; (2) direct, where the device is an integral part of the heat pipe and/or is in direct contact with the working fluid; and (3) system-level heat pipes, where a heat pipe is used to control the temperature in equipment cabinets or systems.

Ruttner [283] summarized the application of heat pipe thermal control systems in the cooling of electronic components developed before 1977, therefore, this work focuses on more recent applications. Marto and Peter- son [284] reviewed the operating principles, modeling, design, and testing of heat pipes as applied to electronic applications and provided a comprehensive review of specific applications. Several other reviews [285, 2861, some involving detailed design procedures, are also available; hence, this summary focuses on applications.

A. INDIRECT COOLING TECHNIQUES

Because of the high effective conductivity of heat pipes compared to that of conventional heat sinks, heat pipes have been proposed and selected for


I I

CONDE

? BELLOWS SECTION

EVAPORATOR

FIG. 71. Heat-pipe cooling of semiconductor chip [287].

thermal control of individual components, series of components, and entire printed wire boards. The simplest heat-pipe heat sinks are cylindrical with a copper or aluminum case and water or acetone as the working fluid. Using this configuration, heat can be removed from power transistors, thyristors, or individual chips. These components are often mounted on the evaporator portion of the pipe and attached mechanically. A series of fins attached to the condenser end of the heat pipe provides the mechanism for heat rejection to a coolant, either through free or forced convection to a gas or a liquid. An example of such a device was conceptualized by Eldridge and Peterson [287] but not constructed (Fig. 71). It was proposed that the back of the integrated circuit chip be bonded to the evaporator portion of a heat-pipe evaporator, which could be constructed from screen, sintered powder, or a series of axial grooves. A porous wicking material lining the inside circumference of the heat pipe would connect the evaporator and condenser.

In 1981, Wolf [288] built and tested a device for vibration sensitive electronics. This device, illustrated in Fig. 72, was capable of transporting up to 50 W at adverse inclinations of 2.0 cm and slightly over 100 W at

THERMAL CONTROL OF ELECTRONIC EQUIPMENT

Contact Piate, \ t Heat 'J /

Set - Point Adjust Nut

Bellows /

Capillary - Evaporator

Switch Mount

' Condenser

/::Ed Socket)

Heat Source \

275

Heat Pipe Switch

Heat In

FIG. 72. Heat-pipe thermal switch [288].

horizontal orientations. It had a source-sink separation distance of approximately 10 cm and a thermal footprint of 77 cm2. In 1984, Miyazaki and Sasaki [289] reported on the development of a heat-pipe thermal switch for spacecraft electronics that also used the bellows concept. This device, which consisted of a space radiator, baseplate, and a stainless-steel bellows vessel (Fig. 73), had a grooved wicking structure. The thermal footprint was approximately 61 cm2, and the transport capacity was 60 W in a horizontal orientation.

Recently, Peterson [290, 2911 reported on the analysis and testing of a similar device referred to as a bellows-type heat pipe for use with electronic components or multichip modules. This device, shown in Fig. 74, could be used to provide a thermal conduction path between the surface of a semiconductor chip and a liquid-cooled cold plate. Its advantages can be seen by comparing the overall thermal resistance paths between the bellows heat pipe and a solid metal rod. The bellows heat pipe would have an overall thermal resistance that is significantly less than that for a solid metal


BASE PLATE

f I B E L L M

RAD'AToR ' ON-OFF PLATE

FIG. 73. Thermal switch for space applications [289].

Cold Plate

FIG. 74. Bellows heat pipe [290].

rod. Increases in the operating pressure would result in increases in the internal pressure and hence, increases in the force applied to the contact, thereby reducing the contact resistance. In addition, the inherent flexibility of the bellows structure would compensate for misalignment, again reduc-


f 1600

Y

$ 1400 E

(?

a a a 1000

2 000 a a

3 f

w 1200 I-

w L

I- 600 I-

I 400

r 200 x

ing the contact resistance at the interface of the heat sink and the chip. The results of prototype testing on a similar device having a circumferential condenser have also been presented [292]. Several problem areas were identified, including noncondensible gas formation, blockage of the axial condenser, and boiling limitations resulting from the small evaporator surface area.

Murase et al. [293] and Yoshida et al. [294] reported on a more conventional design termed a heat kicker. This device consisted of a small (12.7 to 15.88 mm outside diameter) copper water heat pipe with aluminum fins attached to either one or both ends depending on the location of the evaporator. Figure 75 illustrates the maximum heat-transfer rate as a

- -

-

-

-

-

-

-

I I 1 I 1 I 1 I I I 1 0 10 20 30 40 50 60 70 80 90 100 110

Q

Heat Heat Heat out In out

HEAT PIPE VAPOR TEMPERATURE, T, (OC)

FIG. 75. Maximum heat-transfer characteristics (screen heat pipe) for a 15.88 x 300 mm copper container, with water as the working fluid and a spiral-grooved wick, evaporator zone 1, = 80 mm (center), and the condenser zone 1, = 40 mm (both sides) [291].

278

8001- X Q

700 0

600

a K

W L

z

2 300

5 0 0 -

400-

a


-

-

-

5 100 r

a

+ 11 ii Heat Heat

In out

(Bottom Heat Mode) 7 / (Horizontal)

(Top Heat Mode)

X

5 0 10 20 30 40 50 60 70 80 90 100 I I I I I I I I I I

HEAT PIPE VAPOR TEMPERATURE, Tv(O C)

FIG. 76. Maximum heat-transfer characteristics (grooved heat pipe) for a 15.88 x copper container, with water as the working fluid and a 150-mesh copper screen wick, 1, = 80 mm, and I , = 80 mm [291].

function of the vapor temperature where the heat source is at one end of the heat pipe. In Fig. 76 the heat source is in the center with heat-rejecting fins at either end. In the former, the wick was made of 150-mesh copper screen; the centrally heated version had a rectangular spiral-grooved wick. This concept has been expanded to a heat-pipe heat sink for high-capacity semiconductor devices (e.g., diodes, thyristors, GTO thyristors, or SI thyristors) as applied to power controllers for large-sized motors and railway substations. Other standardized thermal control devices are commercially available for thermal loads of 250 to 2500 W [295].

Research performed at the Institute for Mathematical Machines in Pra- gue, Czechoslovakia, demonstrated that flat heat pipes designed to rest underneath a standard dual-on-line package (DIP) are capable of removing and transporting significant amounts of heat from individual components [296]. A flat heat pipe with cross-sectional dimensions of 2.5 X

6.5 mm and 225 mm in length can dissipate up to 10 W while maintaining a temperature difference between the individual components of only 1°C. A more compact (1.2 x 6.5 mm) heat pipe with the same configuration transported 7 W of power while maintaining an isothermal bandwidth of

CIRCUIT BOARD HEAT PIPE

\n

Type 1

Type 2

Type 3

\ COOLED

1 Pipe Ai Ap semi product 1

mm

$4 x 0.5 5.0 4.0 3.6 2.3 1.3 0.9 2.7

$9.2 x 0.4 13.5 12.7 12.3 2.3 1.5 1.1 10.2

$15.8 x 0.4 23.4 22.6 22.2 2.3 1.5 1.5 21.0

LOW THERMAL RESISTANCE AT MECHANICAL JOINTS

SIDEWALL HEAT PIPES

FIG. 77. Heat-pipe-mounted dual-in-line packages (DIPS) [297].

4


5°C. Both of these were flat heat pipes with sintered-copper-powder wicks and water as the working fluid. The components were mounted on the heat pipe surface as illustrated in Fig. 77.

In a separate but related investigation [297], this work was expanded to include an experimental study on three different sizes of heat pipes. Figure 78 illustrates the general shape and dimensions of the test pipes. A trial printed wiring board was constructed using eight type 1 heat pipes, 230 mm long. Seven ceramic packages were mounted on each pipe and the board was assembled. Each package was supplied with power sufficient to generate a total of 8.7 W. A maximum temperature of 60°C was obtained at the interface of the heat pipe and the ceramic packages before the onset of dryout occurred.

Figure 79 illustrates a finned heat-pipe design for cooling of high-power semiconductor devices [298]. The heat pipes shown are constructed of

I I

L L

BASIC PARAMETERS:

Tube material

Type of wicks

Working Fluid

Tube Length L (m)

Tube Diameter +D (m)

Fin Dimensions A x B (m)

Performance Range Q (W)

copper

grooved, sintered

Water, Methanol

0.1

0.032 0.04 0.064

0.125 x 0.125 0.1 x 0.1 0.1 x 0.125 0.1 X 0.15

500 - 2000

FIG. 79. Power semiconductor coolers [298].


FIG. 80. Heat-pipe heat block [298].

copper with a sintered-powder wick on the end and axial grooves on the inner longitudinal surface. Two working fluids were evaluated, water and methanol. The pipe was 0.1 m in length, 0.064 m in diameter, and had a maximum performance level of nearly 2000 W when equipped with 0.1 x 0.15-m-rectangular fins. A similar device is shown in Fig. 80. In this arrangement, the heat is conveyed through a metal block perpendicular to the longitudinal axis of the pipe. Figures 81 and 82 illustrate the thermal resistance of the single- and double-pipe configurations, respectively.

Malik et al. investigated the cooling of a thyristor by mounting it on a commercially available heat pipe [299]. The thyristor had a rated value of 150 A and a maximum allowable junction temperature of 120°C. It was mounted on a copper-water heat pipe 300mm long and 25.4mm in diameter. The transient thermal response of the heat-pipe cooling system was far superior to the conventional methods of both free and forced convection. In addition, the thermal resistance, 0.20°C/W at an air flow

n

x

I- z

3 \

Y

w : 2 U

W 0 U L

3 (I)

W 0 z U I- u, (I) W a -I U I a W I I-

a

0.2 1

0.15

0.12

0.09

AIR FLOW (m3/s)

FIG. 81. Heat-pipe heat-block performance, single pipe [298].

n

\

Y Y

I- z

3

w m

f 2

2 w 0

a 3 (I)

W 0 z U I-

u) w 0:

J U

W I I-

2

5

0.15

0.10

4% I

0.05 0.00 0.04 0.08 0.12 0.16

AIR FLOW (m3/s)

FIG. 82. Heat-pipe heat-block performance, two pipes [298].


rate of 217 liter/sec, was greatly reduced under low-power conditions and tended to remain constant as the power increased.

Several other indirect heat-pipe devices have been proposed but not constructed. Among these are a self-regulating evaporative-conductive link, which would provide an improved thermal path from an electronic chip to the coolant [300], and a cooling system for VLSI circuit chips [301].

Although all of the previously discussed applications are relatively large compared to the size of most semiconductor devices, this need not be the case. In 1984, Cotter proposed the micro-heat-pipe concept (i.e., a wickless heat pipe) “SO small that the mean curvature of the vapor-liquid interface is necessarily comparable in magnitude to the reciprocal of the hydraulic radius of the flow channel” [302]. The proposed device had an equilateral triangular cross section with side dimensions of approximate 20 pm and a length of 10-20 mm. The theoretical transport capacity of a single micro heat pipe with these dimensions and an optimum amount of methanol as the working fluid was estimated to be approximately 0.03 W. Using this type of device in a solid array, approximately 10% by volume, it was demonstrated theoretically that up to a few tens of watts per cubic centimeter of cooling could be provided.

In an application of this concept, a combined analytical and experimental investigation was conducted on a micro heat pipe with a trapezoidal configuration [303]. Four trapezoidal test articles, manufactured by ITOH Research and Development Company (Fig. 83), with a length of 57 mm and a cross-sectional area of 1 mm’, were constructed from both copper and sterling silver with ultrapure water as the working fluid. Pre- liminary results indicate that the liquid-vapor interface changes contin- ually along the pipe; the heat pipe is very sensitive to the amount of charge, since proper wetted conditions must be maintained without flooding the microchannels. Figure 84 illustrates the principal limitations as predicted using conventional steady-state modeling techniques. A transient model along with a manufacturing method for constructing and charging 100 pm heat pipes in silicon has also been developed. Because of the extremely small cross-sectional area, these devices could play an important role in future semiconductor and microelectronic device technologies where thermal loads are high and space is at a premium.

Murase ef al. proposed the use of multiple, parallel heat pipes to cool a series of high-power thyristors [304]. In this investigation, several heat pipes were mounted in a common heater block. These heat pipes, mounted in parallel, conducted heat to a finned array where the heat was removed by natural convection. The heat pipes, constructed from copper with water as the working fluid and with an outside diameter of 15.9 mm and a length of 1200 mm, were capable of transporting significant quantities of power.


CASE MATERIAL: COPPER OR SILVER

WORKING FLUID: ULTRAPURE WATER

MICRO HEAT PIPE

f CASE

A - A 8-B - c - c

SECTIONS

A X I A L LlOUlD DISTRIBUTION

FIG. 83. Trapezoidal micro heat pipe [303].


+ W I a

OPERATING TEMPERATURE ("C)

FIG. 84. Micro-heat-pipe limitations: 1 , capillary limit; 2, sonic limit; 3, entrainment limit; 4, viscous limit [303].

In similar applications, Kolb demonstrated the cooling of a 500-W, gas- fired, thermoelectric generator with 13 gravity-assisted, copper-water heat pipes containing aluminum fins [305]. Aakalu and Carlen used cylindrical heat pipes to transfer heat from circuit modules to water-cooled heat exchanger units [306].

In addition to using heat pipes to cool individual components, several methods have been proposed for cooling multiple arrays of devices or entire printed wiring boards. Bonding flat heat pipes to the back of printed circuit boards is a common method of employing two-phase heat-transfer mechanisms. One particularly efficient utilization of this concept is a sandwich-type device bonded between copper or aluminum skins [307]. This flat structure provides a nearly isothermal condition that is over four times more efficient than a solid aluminurn plate of the same size. Heat from the individual components vaporizes the working fluid, which can condense either in the condenser portion or in other regions where the heat flux is not as high. This type of heat pipe is particularly useful in eliminating localized hot spots at individual components. Another recent development in this area is the bonding of integral heat pipes directly to large transcalent devices, which typically include 250-A rectifiers with blocking voltages of 1000 V, 100-A transistors, or 400-A thyristors [308].


3 WATT UNIFORM

HEAT T SINK

0 1 2 3 4 5 6 7

LOCATION (in.)

FIG. 85. Comparison of temperature rise of heat pipe to metal conductors [309].

Several applications in the thermal control of entire printed wiring boards have been reported in the literature, including the placement of the hot components directly un small, flat heat pipes, or entire wiring boards, which function as a heat pipe [309]. Figure 85 compares the thermal performance of the first of these two methods, a flat stainless-steel heat pipe 1.02 mm thick, 4.01 mm wide, and 178 mm long, with both copper and aluminum strips of similar dimensions. As illustrated, the heat pipe yields a maximum temperature rise of approximately 7°C; the copper and aluminum yield 42 and 97"C, respectively. Of greater importance, however, is the uniform temperature across the face of the heat pipe caused by minimizing temperature variations between the components.

Basiulis et al. proposed that the entire printed wiring board could function as a heat pipe as illustrated in Fig. 86 and reported on two separate schemes by which this could be accomplished [310]. The first of these was


PRINTED WIRING BOARD

FIG. 86. Edge-cooled heat-pipe printed wiring board [310]

to embed a series of flat heat pipes within the walls of the wiring board; the second utilized the entire wiring board as the vapor chamber. Separation of the wiring board provides the option of using several heat pipes with different working fluids to accommodate nonuniform power-dissipation requirements. This alternative resulted in a more uniform surface temperature and reduced the number and temperature of localized hot spots, because of the three-dimensional vapor flow occurring at the hot spots.

Adami and Yimer reported on a similar device (Fig. 87) constructed from 1-mm copper sheets, 100-mesh copper screen, and water as the working fluid [311]. Both steady-state and transient operation were evaluated, along with the effects of gravity, impurities, and amount of working fluid. The device operated with a high degree of stability between 30 and 95°C and there was good agreement between the analytical model and the experimental results. Ogushi et al. conducted an investigation on a similar device shown in Fig. 88 [312]. This device had a vapor space height of 2.5 mm with axial grooves as the wicking structure to distribute the working fluid, methanol or ammonia, throughout the evaporator, and a sharp- edged corner for communication between the top and bottom plates. Although some problems with slugging were encountered, the temperature

FIG. 87. Flat-plate heat pipe [311].

Holes for Attachment of Electronic Component

\ I

harp-Edged Corner

bV 'L Bottom Plate ,

FIG. 88. Flat-plate-type heat pipe with axial grooves and sharp-edged corner for enhanced evaporator and condenser communication [311].

THERMAL CONTROL OF ELECTRONIC EQUIPMENT

- WORKING FLUID METHANOL

- T, =50°C

TILT h, (mm)

FIG. 89. Maximum heat-transfer rate for flat-plate-type heat pipe with axial grooves and sharp-edged corner [312].

drop was half of that for an aluminum plate of the same dimensions when methanol was used, and one-fourth for ammonia. Figure 89 illustrates a comparison of the predicted and measured maximum heat-transfer rate for methanol. The liquid slugging resulted in substantial decreases in the maximum heat-transport capacity.

B. DIRECT COOLING TECHNIQUES

Where the electrical power is high and the heat rejection requirements large, it may be necessary to control the temperature by immersing the devices in a dielectric fluid. Fluid near the saturation temperature typically results in nucleate pool boiling and requires the use of a vapor space condenser. This two-phase loop (i.e., the boiling of the liquid, the condensation of the vapor, and the return of the condensate) is viewed as one form


of a two-phase, closed-loop thermosyphon [284]. Bergles et al. reviewed several aspects of direct liquid cooling but omitted applications dealing with heat pipes and the heat-pipe effect [313].

The generation of vapor bubbles imposes several problems on the thermal control of electronic devices [384]. First and most important among these is the critical heat flux, the maximum permissible level of the evaporator heat flux. Beyond this level, the vapor completely blankets the heat source and results in an increased temperature drop, leading to dryout and overheating. Second, the formation and collapse of vapor bubbles may generate dynamic forces on the chips and leads, creating high-frequency mechanical vibration and subsequent failure. Third, the presence of vapor bubbles may decrease the electric breakdown voltage of the dielectric fluid.

One disadvantage of nucleate pool boiling is the large temperature drop at the interface of the liquid and the surface to be cooled. Two techniques have been investigated to reduce this temperature drop. The first is to make the device an integral part of the wick structure to ensure that fresh liquid always remains in contact with the heat source. The second is the direct evaporation (with no bubble nucleation) of a very thin liquid film.

An example of the first of these two techniques was proposed originally by Dean [314]. Nelson et al. expanded this concept and investigated the

VAPOR FLOW CARRYING HEAT AWAY

TRANSISTOR CHIP POWDER WICK

WIRE BONDS MIC CASE

IMPEDANCE MATCHING BERYLLIA TRANSISTOR NETWORK SUBSTRATE

ALUMINA MIC SUBSTRATE SOLDERED TO MIC CASE

FIG. 90. Cross section of heat-pipe-cooled MIC rf transistor [315].


240

220

200

180

160

140

120

100

inclusion of electronic devices as an integral part of a heat-pipe wick, as illustrated in Fig. 90 [315,316]. This approach overcame the problems of material performance and electrical compatibility and was a means to substantially reduce the junction temperature. Several types of wicking structures, including glass fiber bundles (fiber glass strands 0.1 mm in diameter and spaced 0.5 mm apart to permit a high heat flux density and powder wicks (U.S. Patent No. 4,047,198) were investigated [317]. Figure 91 illustrates a comparison of the thermal performance of this type of device with conventional-cooled semiconductor devices. The thermal performance of this device is excellent with the junction to case resistance reduced by as much as 33% over a wide range of power levels. Figure 92 shows the reduction in junction temperature when a conventionally bonded transistor and heat sink are exposed to two-phase cooling with

-

-

-

-

-

-

-

-

I I I I 40

SO I 0 10 20 30

TRANSISTOR POWER LEVEL, W

FIG. 91. Cooling characteristics of heat-pipe-cooled chips and conventionally cooled chips [315].


70

60

50

40

30

20

10

0 - 1 0

TRANSISTOR POWER LEVEL (WATTS)

FIG. 92. Comparison of reduction in junction temperature achieved by fiber bundle and powder wick and nonwicked transistors immersed in heat-pipe fluid [315].

either a powder or fiber bundle wick. Clearly, the powder-wick heat pipe is far superior to either pool boiling or to the fiber-wick heat pipe, with the junction temperature reduced by 63°C at power levels of 40 W. Immersion cooling with pool boiling provided only an 11°C reduction, demonstrating the effectiveness of this cooling technique.

Kromann et al. proposed a device (Fig. 93) similar to others mentioned in that a wicking structure promoted the flow of liquid to the heat source [318]. Heat generated in the chip die was transferred to the liquid in the wick where it was evaporated. Parallel channels within the cover were a


HEAT OUTPUT

CONDENSER CAP

CONDENSER SURFACE

SUBSTRATE

WORKING FLUID

EVAPORATOR SECTION

FIG. 93. Integral heat pipe showing the fluid-vapor transport and the resulting heat transfer 13181.

path for the liquid to flow from the condenser surface to the evaporator wick by gravitational forces. The wick consisted of eight layers of polyester cloth and the working fluid was pentane. Figure 94 compares the internal thermal resistance of this device at various die heat fluxes. As illustrated, the total package resistance is a strong function of the percentage of working fluid present. The best performance occurred when 52% of the volume filled. At this level, the total package resistance was approximately 0.2 W over a heat flux range of 10 to 27 W/cm2.

Another method by which the temperature drop between the coolant and the heat source can be reduced is thin film evaporation. In 1980, Andros and Shay proposed a closed miniature thermosyphon that utilized this technique [327]. As illustrated in Fig. 95, heat added from a disk- shaped heat source vaporized liquid at the liquid-vapor interface. The vapor flowed across the bubble where it condensed, rejecting heat, and returning liquid to the evaporator by flow through the continuous liquid


TOTAL POWER (W)

50 100 150 I I I 1.20 P

- 6 WICKLESS 26% FILL

0.80 WICKLESS 46% FILL - 4

- 2 0.40 -

0.20 -

0.0 1 I I I I 0 5 10 15 20 25 30

DIE HEAT FLUX (W/cm2)

FIG. 94. Total package thermal resistance [318].

Condensate Cap Seal

Liquid . . . ' . . t t t t

- Q Out (Condenser)

,Cylindrical Hole

- Q In (Evaporator)

cm2 oc

W

FIG. 95. Thin-film evaporation module [301].

film. Wayner and Parks [319] and Kiewra and Wayner [320] have conducted fundamental investigations of this device and demonstrated that an evaporating thin circular liquid film in the shape of an extended meniscus combined with condensation within a small volume acts as an effective heat spreader. Experimental procedures to determine the optimum parameters were developed to clarify the basic mechanisms of heat, mass, and momen-


tum transfer where both body and interfacial forces were present. The effects of both noncondensible gases and fluid flow in ultrathin films in the presence of evaporation were investigated.

C. SYSTEM-LEVEL THERMAL CONTROL USING HEAT PIPES

The superior heat-transfer characteristics of heat pipes make them ideal for the cooling of electronics in applications where the circuitry must be

EVAPORATOR CONDENSER

COOL n COLD

I t - - - - I t

VAPOR HOT U WARM LIQUID

CONTAMINATED INLET (COOL AIR)

CONTAMINATED OUTLET CLEAN OUTLE (WARM AIR)

PARTITION

FIG. 96. Heat-pipe heat exchanger [284].


enclosed and isolated from dust, moisture, oil mist, or harsh environmental conditions. Since the point at which a heat pipe passes through an enclosure can be sealed easily, it is possible to obtain a closed, airtight system that prevents external contamination.

Johnson investigated heat pipes for the removal of heat from sealed enclosures [321]. Shah and Giovannelli detailed design information for a heat-pipe heat exchanger similar to the one illustrated in Fig. 96, including the important coupling of the heat pipe thermal performance to the heat exchange capability of the finned surfaces [322].

Figure 97a shows a design by Gerak et al. [323] consisting of a heat exchanger with eight rows of 17 finned copper heat pipes 16mm in diameter with R12 as the working fluid. The measured power level as a

/ Tube Plate = Upper Caelng

of the Box

Control - Electronics

D e v I c e 8

Heat Plpe Heat Exchanger

/

(a)

/Axla1 Fan

c- - 3.0 Q

- Blower Fan

2.0

kW

1 .o

0 I

r

I 0 10 K :

A t = tAl -tBl

(b)

FIG. 97. Heat-pipe heat exchanger to cool electronic equipment in a sealed cabinet [323].


function of the temperature drop is shown in Fig. 97b, demonstrating the dependence of the heat-transfer capability of the heat exchanger on the inlet air-temperature difference. Jakes et al. [324] described a heat exchanger equipped with a bank of finned thermosyphons for use in the cooling of a closed electrical cabinet. The heat-pipe system reduced the heat exchanger volume by 65% without changing the velocity of the air flow. Both the calculation procedure and the methodology used to op- timize the heat-pipe dimensions and exchanger configuration are presented.

It is sometimes necessary to add heat to a specific system or device to maintain the desired operating temperature. Larkin successfully controlled the electronic circuitry in harsh environments with simple heat pipes [325]. The objective was to add heat to the electronic cabinets at remote weather stations. Figure 98 is a schematic of the device tested. The electronic equipment was in a cabinet on the surface and connected to a 3.65 cm long, 7.9 cm diameter thermosyphon with one end buried in the ground. Power levels of approximately 30 W were maintained with 500 g of R22.

Insulation -

Electronic Equipment

Ground Surface

Heat Pipe 3.65 m long 7.9 cm 0.d.

Heat Flux Into

Heat Pipe

0.61 m

1.52 m

1.52 m

FIG. 98. Heat-pipe application in remote weather stations (3251.


D. SUMMARY

Increasing the density of electronic circuits requires a more effective means of thermal control. Heat pipes, because of the vaporization and condensation process, have extremely high (effective) thermal conductivi- ties and the capability of a nearly uniform temperature distribution in the evaporator. They are compact and lightweight and can eliminate localized hot spots at individual devices, printed wiring boards, or entire systems. Heat pipes enable higher density devices with higher reliability and lower cost and are an acceptable, proven alternative to their conventionally cooled counterparts.

NOMENCLATURE

a A D D h

d

G r t

H H ’

h

K

k ks

K’ L

m

Nrad

6 E

80

contact spot radius area channel diameter hydraulic channel diameter equivalent Vickers diagonal;

Grashof number of heated strip

microhardness Effective microhardness of soft

layer on harder substrate thermal contact conductance;

heat-transfer coefficient thermal conductivity ratio

(substrate to layer) thermal conductivity harmonic mean thermal

conductivity, 2k,kz/(k, + k2) effective thermal conductivity overall channel height; length of

electronic module in flow direction; strip length

combined average absolute asperity slope, = Jm: + m: (rad)

radiation number (Fig. 18), 4bT3 u / k

indentation depth

(Fig. 19), g&L4/kvZ

conductive layer thickness relative contact spot radius, ~2, or contact area,

Greek Symbols

U

7

JAJA, dimensionless temperature of a

heated strip (Fig. 19), (Tw - T,)/(qL/k)GrtO.’

pressure; apparent contact pressure

heat flow rate dimensionless heat dissipation rate

for discrete (staggered) plate channel, JGpuC,( T - To)dy/ k(Tw - To) Gr Pr

Q for a parallel-plate array heat flux average heat flux for

asymmetrically heated uniform heat flux channel, (4 , + qz)/2

resistance case-to-ambient thermal resistance

coefficient (Fig. 41b, k/Lh,-, temperature average absolute temperature

difference for channel with walls of different temperature (Fig. 181, {[TI + Tz)/21+ T01/2

average midpoint temperature difference (Fig. 8), [(TL/Z - To11 + (TL/Z - T0)21/2

combined rMS roughness, __ Boltzmann constant, & + uz

relative layer thickness, [ / a


a b

c-a f fg j

C

L

adiabatic bulk temperature contact or constriction case-to-ambient fluid latent joint layer

Subscripts

0 channel inlet S substrate, surface t total

W channel wall 1 2

V Vickers, constant volume, vapor

one side of contact or layer other side of contact or substrate

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Combined Heat and Mass Transfer by Natural Convection in a Porous Medium

OSVAIR V. TREVISAN

Department of Petroleum Engineering, University of Campinas, 13081 Campinas, Slio Paulo, Brazil

ADRIAN BEJAN

Department of Mechanical Engineering and Material Science, Duke University, Durham, North Carolina 27706

I. Introduction

The objective of this Chapter is to summarize an important subfield that has emerged in contemporary heat- and mass-transfer research. This subfield brings together the studies concerned with the combined heat- and mass-transfer (or double-diffusive) processes that are driven by buoyancy through porous media saturated with fluid. The density gradients that provide the driving buoyancy effect are induced by the combined effects of temperature and species concentration nonuniformities present in the porous medium. The topic of pure heat transfer by natural convection through fluid-saturated porous media, which has received so much attention during the past two decades, is only a special limit of the combined heat- and mass-transfer phenomena reviewed in this chapter.

The general subject of heat transfer through fluid-saturated porous media is an area of rapid growth in contemporary research. The fluid- mechanics component of this activity has been reviewed in monographs such as those by Muskat [l], Bear [2], Scheidegger [3], and Greenkorn [4]. The heat-transfer component has traditionally been reviewed as a support discipline for geothermal-reservoir engineering (Cheng [5 , 61 ; O'Sullivan [7]; McKibbin [8]) and thermal-insulation engineering (Bejan [9], Nield [lo]). More recently, the heat-transfer aspects emerged as a separate


A l l ,,<--"-,.A,.".;-- :.. --. I.--

316 OSVAIR V. TREVISAN AND ADRIAN BEJAN

chapter in two heat-transfer textbooks [ 11, 121 and one heat-transfer handbook [ 131.

The combined heat- and mass-transfer natural convection mechanisms described in this chapter have not been reviewed until now. This activity took place in the 1970s and 1980s, which makes it more recent than the pure heat-transfer work presented in the earlier reviews [5-131. The combined temperature and concentration buoyancy. effects that are discussed next are the porous-medium counterpart of the older chapter of double- diffusive convection known in the study of pure fluids [14].

11. Physical Model

As a brief introduction to the nomenclature of this chapter, we note that the phenomena of convection through fluid-saturated porous media are described usually in terms of volume-averaged quantities. Each volume- averaged quantity + is defined by the operation

where 12/ is the actual value of that quantity at a particular point inside the sample volume V. The velocity components (u , v , w ) , temperature T, and concentration C used in this chapter are defined in the same manner as I) in

There are four conservation principles to consider in the study of convection with more than one buoyancy effect, in order, the conservation of mass, energy, species, and momentum. The first of these is

Eq. (1).

(Dp/Dt) + p V-v = 0 (2) where v(u, v , w) is the volume-averaged velocity vector and p the density of the fluid mixture that saturates the porous matrix. In the literature, it is routinely being assumed that the flow field is to a sufficient degree constant density, such that in two dimensions Eq. (2) reduces to

(au/C?x) + (&/a) = 0

(p~p)l[[~(dT/dt) + I V * V T ] = ~ V ~ T + q ” ’ + ( ~ / K ) ( v ) ’

(3)

(4)

The corresponding energy conservation statement is (e.g., Bejan [ l l ] )

where k is the effective thermal conductivity of the porous medium containing the fluid, and cr is the heat-capacity ratio [5],

= (b + (1 - (b)[(PC)S/(PCP)fl (5 )

HEAT AND MASS TRANSFER BY NATURAL CONVECTION 317

The second and third terms on the right-hand side of Eq. (4) represent the volumetric heat-generation rate (e.g., electrical) and the viscous dissipation effect. These two effects are usually negligible, therefore, in two dimensions Eq. (4) reduces to

d T dT dT d 2 T d 2 T (+-+u-+v-=a d t dx d y ( d x 2 -+- d y 2 ) (6)

where a is the thermal diffusivity of the fluid-saturated porous medium, a = k/(pcp)f. The energy equations (4) and (6) are based on the assumptions that k is a constant and that in each volume sample V the fluid (f) and the solid (s) are locally in thermal equilibrium. The porous medium is said to be modeled as homogeneous and isotropic.

According to the same model, the equation for the conservation of a certain species (i) in the fluid mixture that saturates the porous structure [ll] is

(7) dC

v - V C = D V2C i- mi" +at+ where C is the concentration (kilograms of species i per unit volume of saturated porous medium) and 4 is the porosity. The coefficient D (assumed constant) is the mass diffusivity of species i through the porous medium containing the fluid mixture. In most applications, the number of kilograms of i produced (or consumed) per unit time and per unit volume by a chemical reaction (mi") is zero; therefore, the simplified two- dimensional version of Eq. (7) is

Similarly, the momentum equation on which most of the following work was based is much simpler than the most comprehensive model developed by Vafai and Tien [15]

(u/K)v + blvlv = - ( D v / D t ) - p-lVP + vv2v + g (9) in which the empirical constants K and b are the permeability (or Darcy's constant) and Forchheimer's constant, respectively. That simple model is the so-called Darcy flow in which the velocity (the first term on the left-hand side) is balanced by the pressure gradient (the second term on the right-hand side). For example, in a two-dimensional system ( x , y ) in which the gravitational acceleration g points in the negative y direction, Eq. (9) reduces to two equations.

( v / K ) u = - p - ' ( d P / d x ) (10)

( v / K ) v = -p?(dP/dy) - g (11)


Next, the elimination of P between the last two equations yields a single equation in terms of du/dy, &/ax, and dp/dx. For sufficiently small iso- baric changes in temperature and concentration, the mixture density depends linearly on both T and C,

p + (dp/dT)P,C(T- TO) + (dp/ac>T,P(c - CO) + * ’ ’ (12)

where the zero subscript indicates the properties of the mixture in a reference state. In the end, the elimination of P and dp/dx between Eqs. (10)-(12) yields

du dv gK dT dC --+-=- dy dx v ( P - + P - dx

d x )

where p is the volumetric thermal-expansion coefficient,

= -:(%)p.c

and pc is the volumetric concentration expansion coefficient [ll]

In this way, the momentum equation [Eq. (13)] becomes coupled with the energy equation [Eq. (6)], the species concentration equation [Eq. (S)], and the mixture mass conservation equation [Eq. (3)]. The expansivities p and 0, are being assumed constant. The temperature and concentration gradients on the right-hand side of Eq. (13) account for the two buoyancy effects that drive the flow and heat- and mass-transfer processes.

111. Heat and Mass Transfer in the Vertical Direction

A. ONSET OF CONVECTION

The first problem investigated in this new area was the onset of convection problem, that is, the porous-medium equivalent of the critical conditions for the occurence of Bknard-type convection under the influence of two buoyancy effects. This problem was solved by Nield [16]. Figure 1 shows the subject of Nield’s analysis, namely, an infinite horizontal porous layer of thickness H , which is saturated with fluid. This layer experiences net heat transfer and mass transfer in the vertical direction, as a result of the temperature and concentration differences that are maintained between the two horizontal boundaries.


H

TO C O

FIG. 1 . Horizontal saturated porous layer with linear distributions of temperature and concentration.

According to the classical method of linear stability analysis, it is assumed that initially the system is in a quiescent state where heat and mass are being transfered via pure diffusion. This initial, or base, solution (designated by subscript “b”) is represented by

The next step is the postulate that the base solution is disturbed by a sufficiently small perturbation (represented by the prime), in other words that

(u, v , w ) = (u ’ , v’, w’) (19)

T = Tb+ T’ (20)

c = c b + c’ (21)

Finally, substituting these expressions into the energy, constituent, and momentum conservation statements and dropping the higher order terms yields

( 2 2 )

0 2 v’ = - Kg ( p v:r + pc v:c’) V


where V: = (d2 /dx2) + ( d 2 / d y 2 ) . In these three equations the perturbation quantities are assumed to vanish at the two horizontal boundaries,

v r = O , T’=O, C ’ = O a t y = O , H (25)

meaning that the boundaries are being modeled as impermeable, isothermal, and isoconcentration.

The dimensionless counterparts of Eqs. (22) to (24) are (aT/a2) - D = v2T

(+/u)(d/& - D = Le-’ v2e (26)

(27)

(28)

(29)

V2D = Ra(V:f + NV:e)

D = v‘H/a

where the new variables are

(a , 9, 2 ) = ( x , y, z)/H, T = T’/(To - TI), e = C’/(Co - CJ, 2 = at/uH2 (30)

This formulation reveals also the three dimensionless groups that distinguish the phenomena assembled in this chapter from pure heat-transfer natural convection, namely, the Darcy-modified Rayleigh number,

Ra = KgPH( To - T,) /av (31)

the Lewis number,

Le = a / D

and the buoyancy ratio,

N = PdCO - Cd/P(Tcl - TI) (33) In order to verify the stability of the flow field with respect to all possible

infinitesimal disturbances, the perturbation variables are assumed to be of the form

(0, f, 6 = [ ~ ( y ) , q y ) , y(y)] exp(s2) sin(m2) sin(n2)

s e - v = ( d 2 - 2 ) e (+/u)sr - V = Le-’(d2 - a2)y

( d 2 - a2)V = -Ra a2(8+ Ny)

(34)

(35)

(36)

(37)

Substituting these expressions into Eqs. (26)-(28) yields

where d 2 = d2( )/dy2, and where a = (m2 + n2)1/2 is the horizontal wave number of the disturbance. In association with the boundary conditions V = 0 = y = 0 at 9 = 0,1, the system (35)-(37) constitutes a characteristic

HEAT AND MASS TRANSFER BY NATURAL CONVECTION 32 1

value problem. Assuming solutions of the form

(V , 8, 7) = ( A , B , C) sin(f7rj)

(-f2n2 - a2 - s ) B = - A

Le-’[-f2.rr2 - a2 - (+/u))s Le]C = - A

(38)

the system (35) to (37) becomes

(39)

(40)

(41) ( -127r2 - a2)A = -Ra a2(B + N C )

Without loss of generality, the coefficient A can be set equal to 1, and Eqs. (39)-(41) reduce to

(f2.rr2 + a2)(f2.rr2 + a2 + s)[f2,rr2 + a2 + (+/u)s ~ e ]

= Ra a2[f2.rr2 + a2 + (+/u)s Le] + Ra a2N Le(f2.rr2 + a2 + S) (42)

The marginal state of stability for stationary convection corresponds to s = 0, which means

Ra + Ra N Le = (f2.rr2 + a2)2/a2 (43)

The minimum value of Ra (1 + N Le) with respect to a2 occurs at a = IT, for which the smallest f value is f = 1. In conclusion, the “critical” Rayleigh number condition for the onset of convection reads [16]

Ra + Ra N L e = 4.n2

The same conclusion was reached by different methods of stability analysis, for example, the energy method [17]. Depending on the sign of the buoyancy ratio N , the onset of cellular convection can be enhanced or inhibited by the interaction between the two driving forces. The convection onset criterion [Eq. (44)] shows also the emergence of the group Ra N Le, which is the solutal counterpart of the Darcy-modified Rayleigh number (e.g., Bejan [ l l ] , p. 336),

(44)

RaD = Ra N Le = KgP,H(C, - C l ) / v D (45)

Relative to the case of pure heat transfer ( N = 0), the presence of a second buoyancy effect introduces a new feature, namely, the possibility of instability due to oscillatory motion. The domain of marginal instability via oscillatory convection is found by substituting s = it in Eq. (42), which requires then


or, after minimizing it with respect to u2 and I,

Both Eqs. (44) and (47) represent straight lines in the (Ra, Ra N Le) plane. These lines are parallel only in the special case Le = o/+. In general, their point of intersection' is Ra = 4 d [ 1 - (u/$Le)]-' and Ra N Le = 4.rr2[1 - (+Le/u)]-'. The domains defined by these lines are shown in Fig. 2 for the Lewis number range Le > o/+. In the shaded area, the combined effect of the temperature and concentration gradients is not sufficient for destabilizing the quiescent state. The upper right-hand domain of Fig. 2, that is, above the line corresponding to Eq. (44), is the domain of monotonic instability. The resulting flow is one of stationary (steady) cellular convection. On the other hand, in the range of Ra and Ra N Le values located in the wedge-shaped domain between Eqs. (44) and (47), the motion is periodic (oscillatory). The vicinity of the intersec-

t RaNLe

FIG. 2. Stability chart for thermohatine convection in a horizontal porous layer.


TABLE I THE EFFECT OF BOUNDARY CONDITIONS ON THE ONSET OF CONVECTION"

Hydrodynamic Thermal/solutal condition condition"

Upper Lower Upper Lower Critical bound a r y boundary boundary boundary value

Rigid Rigid Rigid Free Free Free Free Free

Rigid Rigid Rigid Rigid Rigid Rigid Rigid Free

Level Flux Flux Level Level Flux Flux Either

Level Level Flux Level Flux Level Flux Either

39.48 = 4 w 2 27.10 12 27.10 17.65 9.87 3 0

"From Ref. [16].

and uniform mass flux. Level, uniform temperature and uniform concentration; flux, uniform heat flux

tion of the two lines has attracted attention [18,19], because it indicates the presence of oscillations as the primary form of instability in a region of transition between stable and unstable fluid motions. Convection onset conclusions that are equivalent to Nield's [ 161 were obtained also by Taunton et al. [20].

Nield [16] repeated the stationary-mode part of the linear stability analysis for several other pairs of boundary conditions, and his results are summarized in Table I. The critical value listed in the right-hand column of the table replaces the 4,rr2 value that appears on the right-hand side of Eq. (44). The temperature and concentration boundary conditions on a given wall are assumed to be of the same kind in each case, for example, in the first row of the table, uniform temperature and uniform concentration on the upper boundary.

B. NONLINEAR INITIAL PROFILES

The stability results discussed until now refer to initial states with linear vertical gradients of temperature and concentration (Fig. 1). Considerable attention has been devoted to the convection onset problem in which the initial state is characterized by nonlinear distributions of temperature and concentration. Such distributions are found to persist for a long time in fluid-saturated porous layers, or in configurations in which a net horizontal flow is present.


Rubin [21] studied the stability effects of two types of nonlinear concentration distributions, namely, a periodic (cosine) profile and a step- function profile. The initial unperturbed state is described by Eqs. (16), (17), and

c b = CO + (cl - cO)(F/H) (48) in which the vertical concentration profile, F = F(y, t ) , satisfies the boundary conditions F(0, t ) = 0 and F ( H , t ) = H . Since the characteristic time of the onset of instability is shorter than the diffusion time of the chemical species, the initial concentration field may be regarded as quasi steady, in other words, F = F(y). Proceeding in the same way as in the case of linear profiles, it is found that Eqs. (35)-(37) are now replaced by

se - v = ( d 2 - a2)e

(qb/u)sy - vf’ = Le-l(d2 - u2)y

(d2 - u2)V = -Rau2(8 + N y )

(49)

(50)

(51)

where f ( j ) = F(y)/H and f’(9) = d f / d j . The associated eigenvalue problem can be solved based on the Galerkin method, and the marginal state of stability via stationary convection (s = 0) is obtained by minimizing the right-hand side of the expression,

Ra iSlm + Ra N Le 2 f ’ sin(l7rj) sin(rn7rjj) djj

= [T ( 7 2 1 2 + 2 2 ) + u2 8, 1 where 8, is the Kronecker delta. The marginal stability results depend on the nonlinear profile function f. In the case of a linear profile, f’ = 1 , Eq. (52) reproduces the result quoted earlier in Eq. (43).

In the case of a cosine-shaped profile, F(y) = (H/2)[1- cos(~y/H)], or f’ = (7r/2) sin(7rjj), the result of minimizing Eq. (52) is

R a + $ R a N L e = 4 n 2 (53)

Relative to Eq. (43), this result shows that the cosine-shaped concentration profile has a stronger stabilizing effect than the linear concentration profile [note the greater than 1 coefficient of RaNLe in Eq. (53)].

A similar conclusion is reached in the case of a step-function concentration profile ( F = 0 at 0 < y < H/2, and F = H at H / 2 < y < H ) , which can be expressed in terms of a Fourier series. The marginal stability criterion that emerges in place of Eq. (53) is then

Ra + 2RaNLe = 47r2 (54)


In summary, the step-shaped concentration profile has an even stronger stabilizing effect than the cosine-shaped profile, Eq. (53). It is worth noting that the derivation of both Eqs. (53) and (54) is based on the assumption that the absolute value of the concentration Rayleigh number (RaNLe) is small.

C. OTHER EFFECTS

If a net horizontal flow is present in the porous layer, it will influence not only the vertical solutal gradient but also the phenomenon of solute dispersion. The phenomenon of thermal dispersion can also be affected; however, in most applications LY is greater than D (i.e., Le > 1) and, as a consequence, the solutal dispersion effect is more sensitive to the presence of through flow. The ultimate effect of dispersion is that the concentration distribution becomes nonhomogeneous.

The stability implications of the anisotropic mass diffusion associated with an anisotropic dispersion tensor were examined by Rubin [22] and, later, by Rubin and Roth [23]. The dispersion anisotropy reduces the solutal stabilizing effect on the inception of stationary convection and, at the same time, enhances the stability of the flow field with respect to oscillatory disturbances (overstable motions).

The phenomenon of mechanical dispersion changes also the size and shape of the BCnard cells that form inside the saturated porous layer. This effect was commented on by Nield [24], and subjected to a three- dimensional study by Rubin [25]. The latter confirmed that the convection cells are roll shaped and that their axes are perpendicular to the horizontal direction of the through flow. In addition, overstable motions are triggered by the formation of oscillatory rolls oriented such that their axes are parallel to the direction of the unperturbed flow (i.e., horizontally). In the eyes of an inertial observer, the resulting flow is both oscillatory and helical.

There are examples of porous-medium systems in geothermal and petroleum reservoir engineering, where the solid structures contain pores and fissures of unusual sizes. In such cavernous media even very slow volume- averaged flows can deviate locally from the Darcy flow model. The size of larger pores brings about an intensification of the dispersion of solute and heat, and-because of the high pore Reynolds numbers (Re) that are involved-the effect of turbulence. Rubin [26] investigated the departure from the Darcy flow model and its effect on the onset of convection in a horizontal porous layer with longitudinal through flow. This study showed that in the case of laminar flow through the pores (Re l), the steady horizontal through flow destabilizes the flow field by enhancing the effect of solutal dispersion. A stabilizing effect is recorded in the intermediate


regime (Re = 1). In the inertial flow regime (Re >> l) , the stability characteristics become similar to those when the convection is due to a single diffusive component; then, the horizontal through flow exhibits a stabilizing effect.

The studies discussed until now refer to an isotropic porous medium, that is, to an idealized model that simplifies the analysis greatly. Natural porous media, however, are both anisotropic and nonhomogeneous. The onset of thermohaline convection in a porous layer with varying hydraulic resistivity ( r = p / K ) was investigated by Rubin [27]. Assuming that the dimensionless hydraulic resistivity 5 = r/ro varies only with the vertical position, and that these variations are relatively small, the linear stability analysis yields the marginal stability condition

Ra + Ra N Le = rr2(5ij2 + 5t/2)2 (55) in which 5 h and 5, are the horizontal- and vertical-mean resistivities, respectively,

Considering the proof that the vertical-mean resistivity is always greater than its horizontal counterpart [27], th 2 &,, Eq. (55) shows that its right- hand side can be larger or smaller than 47r2, depending on which of the resistivity means is regarded as characteristic. A similar conclusion is reached with respect to the criterion for the initiation of overstable motions: in this case, the right-hand side of Eq. (47) is replaced by ,rr2(&/2 + @)2.

The effect of pronounced property variations has been studied by using Galerkin expansions in order to account accurately for the heterogeneity of the medium. This method was used by Rubin [28] in an analysis of the effects of nonhomogeneous hydraulic resistivity and thermal diffusivity on stability. The effect of simultaneous vertical anisotropy in permeability (hydraulic resistivity), thermal diffusivity, and solutal diffusivity was investigated by Tyvand [29] and, in a subsequent paper, by Rubin [30].

Viscosity variations and their effect on the onset of convection were considered by Patil and Vaidyanathan [31], who performed a nonlinear stability analysis using a momentum equation containing both the Darcy and Brinkman terms [in Eq. (9), this means discarding only the second and third terms]. Assuming a vertical viscosity variation of the form

v = VO + (Av) COS( ~ j j ) (57)

where Av<< vo, these authors found that the variable viscosity has a destabilizing effect in both Darcy flow and Darcy-Brinkman flow.


The effect of porous-medium coarseness on the onset of convection was documented by Poulikakos [32]. For this, he considered Brinkman’s modification of the Darcy flow model, which amounts to discarding only the second and third terms from Eq. (9). Poulikakos showed that the stability problem depends on an additional dimensionless group, the Darcy number

Da = ( f i / P ) ( W H ’ ) (58) in which p and @ are the fluid viscosity and the effective viscosity, respectively, of the porous bed [32]. The critical Rayleigh number for the onset of the direct mode of instability is given by

(a: + n2)2 Ra + R a N L e = [(a: + n2) Da + 11

Q C (59)

The critical dimensionless horizontal wave number (a,) depends only on the Darcy number,

(n’ Da + 1)lI2(9n2 Da + 1)1’2 - n2 Da - 1 4 Da a: = (60)

and approaches the Darcy flow limit (a , = n) when Da << 1, and the classical (pure) fluid limit (ac = n/2) when Da zs- 1. The corresponding critical conditions for the transition to overstable motions are described fully in Poulikakos [32].

D . FINITE-AMPLITUDE CONVECTION

The applicability of the linear stability analysis reviewed so far is restricted to Rayleigh numbers that are sufficiently close to the critical values needed for the transition from pure diffusion to convection. Since the linear theory does not describe the amplitude of the resulting convection motion, it cannot be used in order to calculate the net heat- and mass- transfer rates between the horizontal boundaries (Fig. 1).

Rudraiah er al. [33] used nonlinear stability analysis in the study of finite-amplitude convection in a porous layer with rigid boundaries and constant temperature and concentration. Using only the first two terms of Fourier series to represent the temperature and concentration distributions, these authors reported analytical Nusselt and Sherwood numbers for Ra values up to 300, and Ra N Le values up to 70. Their study showed also that a finite-amplitude instability is possible at subcritical values of the Rayleigh number. The flow field may exhibit subcritical instabilities to either monotonic or periodic perturbations, which are not possible in a porous medium saturated by a single-component fluid (i.e., in a flow driven by a single buoyancy effect).


E. SORET DIFFUSION

In the case of extremely steep temperature gradients, the cross coupling between thermal diffusion and solutal (chemical species) diffusion may no longer be negligible. This coupling, or the tendency of a chemical species to diffuse under the influence of a temperature gradient, is better known as the Soret effect. Its influence on the stability of thermohaline convection in binary mixtures was investigated theoretically and experimentally by Law- son et al. [34] and Lawson and Yang [35]. In a study of thermosolutal convection in a porous layer bounded by two rigid walls, Patil and Rud- raiah [36] found two distinct Soret coefficient ranges, in which the flow is either more stable or more unstable relative to Nield's classical problem of thermosolutal convection. This conclusion is not surprising because, depending on the sign of the Soret coefficient, the chemical species can migrate toward either the hot boundary or the cold boundary.

Analytically, consideration of the Soret effect changes the outlook of the species conservation equation [Eq. (S)], which in two dimension becomes

ac ac ac at ax ay

4- + U-+- V - = D v2c + D' V ~ T

The Soret coefficient D' is treated as a constant. According to Brand and Steinberg [37], the marginally stable state occurs when

Ra = 4r2/[ 1 + S( 1 + Le)] (62)

S = D'PJDP (63)

(64)

in which the Soret parameter S is defined as

The marginal state of stability to oscillatory disturbances is

Ra = [4r2(1 + Le)]/[(l + S) Le]

These results have been obtained also in a subsequent study by Taslim and Narusawa [38].

The effect of Soret diffusion on the stability of a porous medium saturated with a binary mixture, in which one of the components undergoes a slow chemical reaction, was analyzed by Patil [39]. In this study, the flow through the porous layer is described by the extended Brinkman model (in which only the second term of Eq. (9) is discarded), and the effect of thermal diffusion is taken into account. A convective instability that is driven by a fast chemical reaction was the subject of a more recent study by Steinberg and Brand [40]. Neglecting the ordinary diffusion of chemical species as well as the cross-coupled diffusive effects, these authors showed


that similarities exist between reaction-driven instabilities and Soret-driven instabilities.

The onset of thermohaline convection has been analyzed in several additional configurations, namely, in the presence of a third diffusing component [41,42], and in the presence of rotation and suspended particles [43]. The effect of rotation alone had been investigated earlier by Cha- krabarti and Gupta [44] for both small- and finite-amplitude convection. The critical conditions for the onset of convection in a doubly diffusive porous layer with internal heat generation were documented by Selimos and Poulikakos [45].

F. HIGH RAYLEIGH NUMBER CONVECTION

The interaction between the heat-transfer and mass-transfer processes in the regime of strong convection was investigated on the basis of a two- dimensional model by Trevisan and Bejan [46]. Their study consisted of numerical experiments backed up by scale analysis. Figure 3 shows the main characteristics of the flow, temperature, and concentration fields in one of the two-dimensional rolls that form inside the horizontal porous layer. This particular flow is heat-transfer driven, in the sense that the dominant buoyancy effect is the one due to temperature gradients ( N = 0). The temperature field (Fig. 3b) shows the formation of thermal boundary layers in the top and bottom end-turn regions of the roll. The concentration field is illustrated in Figs. 3b-3d: the top and bottom concentration boundary layers become noticeably thinner as the Lewis number increases from 1 to 20.

b C

FIG. 3. Two-dimensional numerical solution for heat-transfer-driven ( N = 0) convection in a horizontal porous layer (Ra = 200, H / L = 1.89): (a) streamlines; (b) isotherms, also constant-concentration lines for Le = 1; (c) lines of constant concentration for Le = 4; (d) lines of constant concentration for Le = 20 [46].


Of interest in the domain of high Rayleigh number convection are the overall Nusselt number,

and the overall Sherwood number,

where q:vg and javg are the heat and mass fluxes averaged over one of the horizontal boundaries. In heat-transfer-driven convection, IN1 << 1, it is found that the Nusselt number scales as [46,47]

Nu = (Ra/4~’)’/~ (67) In the same regime, the Sherwood number scale depends on the order of magnitude of the Lewis number [46]:

Sh = Le’/2(Ra/4.rr2)7/8, if Le > ( R a / 4 ~ ~ ) ’ / ~ (68)

Sh = Le2(Ra/4.rr2)’/’, if (Ra/4~’)-’/~ < Le < (Ra /4~’ ) l /~ (69)

Sh = 1, if Le < ( R a / 4 ~ ’ ) - ~ / ~ (70)

The corresponding scales of mass-transfer-driven flows, IN1 s=- 1, can be deduced from the above by applying the transformation Ra- Ra N Le, Nu- Sh, Sh-, Nu, and Le-, Le-’,

Sh = (RaD/4.r’)’/’ (71)

Nu = Le-1/2(RaD/4~2)7/8, if Le < (RaD/4.rr2)-1/4 (72)

(73)

Nu= 1, if Le > ( ~ a ~ / 4 . r r ’ ) ’ / ~ (74)

Nu = Le-2(RaD/4.rr2)’/2, if ( R ~ D / ~ W ’ ) - ~ / ~ < Le < ( R ~ D / ~ T ~ ) ’ / ~

where RaD is shorthand for the product RaNLe, Eq. (45). Equations (68)-(74) agree well with the results of direct numerical simulations of the convective transport processes [46].

IV. Heat and Mass Transfer in the Horizontal Direction

A. ONSET OF CONVECTION

The study of combined heat and mass transfer in the horizontal direction has attracted considerably less attention than the vertical orientation reviewed in the Section 111. The first work was again devoted to the stability


,a 2

PORoUS I MED'UM

L

FIG. 4. Vertical porous layer with horizontal temperature gradient and vertical stratification of chemical species.

problem, that is, the necessary conditions for the onset of convective fluid motion. Gershuni et al. [48] and Khan and Zebib [49] considered the steady motion induced by the side heating of a saturated porous slab subjected to a linear concentration profile in the vertical direction (Fig. 4).

Assuming that the side walls are impermeable and isothermal, the boundary conditions in the horizontal direction are r(/ = 0, d C / d x = 0, and T = +AT12 at x = +L/2. The imposed temperature gradient induces a vertical counterflow, the effect of which is to distort the initial concentration profile. The unperturbed solution is described by

T b = x (75)

- ll

cosh(mx) cosh(m/2)

(crb = m-2

where m2 = n Ra Le, and where n is the dimensionless coefficient for chemical species stratification in the vertical direction. Equations (75)-(77) have been nondimensionalized by using L, L2/a Ra, a RaIL, AT, and PATIP,


as reference units of length, time, velocity, temperature, and concentration [note that L replaces H in the Ra definition, Eq. (31)]. In particular, the dimensionless concentration gradient has the same form as the buoyancy ratio employed in Section 111, namely, n = p, AC/p AT, therefore, the solutal Rayleigh number is m2.

The assumption of small perturbations of type exp(st + ipy) leads to the system

(78)

(79)

yr" - $q.I = 8' + y'

el' - p 2 e = se + ip Ra(q -

y " - p 2 y = (4/a)sy+ ip Ra Le(C6 q.I - &y) - m2$i (80)

in which q.I(x), @), and y(x) are the disturbance amplitudes for stream function, temperature, and concentration. The resulting eigenvalue problem is analytically solvable in the limit m + m, that is, in the case of strong solutal stratification. In this limit i j b vanishes in the central region of the layer, which means that the core is virtually motionless and linearly stratified. The gradient dCb/dx approaches -1 as m increases, except in the boundary layer regions near the walls [thickness O(m-')], where d c b / d X + 0 [see Eqs. (76) and (77)]. In other words, there is no density gradient in the horizontal direction in regions situated sufficiently far from the walls, as the temperature and concentration gradients negate one another. The undisturbed state is therefore in mechanical equilibrium. This limiting situation was analyzed by Gershuni et al. [50], who concluded that only monotonic instabilities are possible in liquid-saturated systems, where the Lewis numbers are usually larger than O(1). The minimum critical Rayleigh number is given by

Ra/(Ra n Le)3/4 = 277'/*/11- Let (81) The critical conditions for the onset of oscillatory motion-possible in

gaseous mixtures where the Lewis number is of order O(1)-are more difficult to establish, and require the use of approximate methods. Similar difficulties are encountered when dealing with finite values of the parameter m. The numerical solutions of Eqs. (78)-(80) showed that the simplifying assumption of strong solutal stratification ["m -+ m" under Eq. (SO)] is valid only if m2 is greater than approximately lo3 [49].

B. BOUNDARY-LAYER FLOWS

The simplest geometry for simultaneous heat and mass transfer from the side is the single vertical wall embedded in a saturated porous medium.


The uniform wall temperature (To) and concentration (C,) differ from the corresponding properties prevailing sufficiently far from the wall (T,, C,). The buoyancy driven flow that coats the wall can have one of the four two-layer structures shown in Fig. 5, depending on the order of magnitude of the buoyancy ratio and the Lewis number.

The boundary-layer heat- and mass-transfer problem was solved first via scale analysis (Bejan [ l l ] , pp. 335-338), and later based on the classical similarity formulation [51]. The conclusions reached at the end of the scale

(Heat transfer driven f l o w )

N >> I

f Mass transfer driven flow J

L e >> 1 L e <c 1

FIG. 5 . The four regimes of boundary-layer heat and mass transfer near a vertical surface ~ 1 .


TABLE I1

FLOW, HEAT, AND MASS TRANSFER SCALES NEAR A VERTICAL WALL WITH

COMBINED BUOYANCY EFFECTS~

Drivine mechanism V Nu Sh Le domain

Heat transfer

Mass transfer (IN1 < 1)

(IN1 1)

"From Refs. [11,51]

analysis are summarized in Table 11. Each row of results shown in the table corresponds to one of the quandrants of the (N,Le) domain delineated in Fig. 5 . The v scale represents the largest vertical velocity, which in Darcy flow occurs right at the wall. The overall Nusselt and Sherwood numbers are defined as

where qiVg and javg are the heat and mass fluxes averaged over the wall height H .

The similarity solution to the same problem [51] was obtained by selecting the dimensionless similarity profiles recommended by the scale analysis (Table 11)

v = - ( . /Y) Rayf 'h) (84)

u = ( a / 2 y ) Ra:l2(f-qf') (85)

(86)

(87)

e(q) = ( T - T,)/(T,, - T,)

c(q) = ( C - Cm)/(C, - Cm)

in which y is the distance from the tip of the boundary layer, measured along the wall, and Ray = Kgpy(To - T,) /av. The conservation statements for momentum, energy, and chemical species reduce to

f" = -8' - ~~1 (88)

# i = + f # (89) c" = 4 fc' Le (90)

f

- 4

- 2

0 0

a

------I

7 7

b i i i i i i 1

8. C

0.5

0 0

\n

7 7

FIG. 6. The effect of buoyancy ratio on the similarity boundary-layer profiles when Le = 1: (a) velocity profiles; (b) temperature and concentration profiles [51].


for which the boundary conditions are f = 0, 0 = 1, c = 1 at r) = 0; and f ' + 0, 0 + 0, c + 0 as r ) + m. This dimensionless formulation reinforces the conclusion of the scale analysis, that the phenomenon depends on two dimensionless parameters, N and Le.

Figure 6 shows a sample of vertical velocity and temperature (or concentration) profiles for the case Le = 1. The vertical velocity increases and the thermal boundary layer becomes thinner as IN1 increases. The same similarity solutions show that the concentration boundary layer in heat- transfer-driven flows ( N = 0) becomes thinner as Le increases, in good agreement with the trend anticipated by scale analysis.

The effect of wall inclination on the two-layer structure was described by Jang and Chang [52,53]. Their study is a generalization of the similarity solution approach employed in Ref. [51]. The heat- and mass-transfer scales that prevail in the extreme case when the embedded surface is horizontal are summarized in Table 111. A related study was reported by Jang and Ni [54], who considered the transient development of velocity, temperature, and concentration boundary layers near a vertical surface.

Raptis et al. [55] showed that an analytical solution is possible in the case of an infinite vertical wall with uniform suction at the wall-porous-medium interface. The resulting analytical solution describes flow, temperature, and concentration fields that are independent of altitude ( y ) . This approach was extended to the unsteady boundary-layer flow problem by Raptis and Tzivanidis [56],

C. ENCLOSED POROUS LAYERS

The basic configuration for the study of combined heat and mass transfer across an enclosure filled with saturated porous medium is shown in Fig. 7. The temperature and concentration are maintained at different levels along

TABLE I11 FLOW, HEAT, AND MASS TRANSFER SCALES NEAR A HORIZONTAL WALL WITH

COMBINED BUOYANCY EFFECTS"

Driving mechanism U Nu Sh Le domar

Heat transfer ( a / H ) Ra2l3 Ra1l3 Ral/3 Le1/2 L e B l (IN1 9 1 ) ( a / H ) Ra2I3 Ra1l3 Ra'l' Le L e a l

Mass transfer ( a / H ) Ra(RalN))2/3 Le-'13 (Ral~Vl)'/~ Le-'l6 (RalNI Le)L/3 Le -=c 1 (IN1 B1) ( a / H ) Ra(RalN1)2/3 Le-'I3 (RalNI)'13 Le-'I3 (RalNI Le)'l3 Le % 1

"From Ref. [53].


5 5

I H

1 adiabatic and impermeable

\\\\\\\\\\\\\\\\\\\'

p a r a u s

medium

FIG. 7. Enclosed two-dimensional porous layer subjected to heat and mass in the horizontal direction.

the two side walls, and the chief engineering question is the calculation of the overall heat- and mass-transfer rates represented by Nu and Sh [Eqs. (82) and (83)].

Relative to the single-wall problem (Fig. 5) the present phenomenon depends on the geometric aspect ratio L / H as an additional dimensionless group next to N and Le. These groups account for the many distinct heat- and mass-transfer regimes that can exist. Trevisan and Bejan [57] identified these regimes on the basis of scale analysis and numerical experiments. Figure 8 shows that in the case of heat-transfer-driven flows (IN1 << 1) there are five distinct regimes, which are labeled I-V. The proper Nu and Sh scales are listed directly on the Le - ( L / H ) 2 Ra subdomain occupied by each regime.

Five distinct regimes are also possible in the limit of mass-transfer-driven flows, IN1 >> 1. Figure 9 shows the corresponding Nusselt and Sherwood number scales, and the position of each regime in the plane Le - (L/H)* Ra INI. The Nu and Sh scales reported in Figs. 8 and 9 are correct within a numerical factor of order 1. Considerably more accurate results have been developed numerically and reported in Ref. [57].

--Id Ill \ N U ‘i: ( L / H ) R ~ ~

--I6

Shz I

S h Z I

FIG. 8. The distinct regimes that are possible when the buoyancy effect is due mainly to temperature gradients [IN/ 1).

I \ FIG. 9. The distinct regimes that are possible when the buoyancy effect is due mainly to

concentration gradients (IN1 > 1).


40

Sh

30

20

10

0 I N 4

FIG. 10. The effect of the buoyancy ratio on the overall mass transfer rate (Ra = 200, H / L = 1) [57].

The most striking effect of varying the buoyancy ratio N between the extremes represented by Figs. 8 and 9 is the supression of convection in the vicinity of N = -1. In this special limit, the temperature and buoyancy effects are comparable in size but have opposite signs. Indeed, the flow disappears completely if Le = 1 and N = - 1. This dramatic effect is illustrated in Fig. 10, which shows how the overall mass-transfer rate approaches the pure diffusion level (Sh = 1) as N passes through the value -1.

When the Lewis number is smaller or greater than 1, the passing of N through the value -1 is not accompanied by the total disappearance of the flow. This aspect is illustrated by the sequence of streamlines, isotherms, and concentration lines displayed in Fig. 11. The figure shows that when N is algebraically greater than approximately -0.85, the natural convection- pattern resembles the one that would be expected in a porous layer in which the opposing buoyancy effect is not the dominant driving force. The circulation is reversed at N values lower than approximately - 1.5. The flow reversal takes place rather abruptly around N = -0.9, as is shown in


C

FIG. 11. Patterns of streamlines, isotherms, and constant-concentration lines, showing the flow reversal that occurs near N = -1 (Ra = 200, Le = 10, H / L = 1) [57]: (a) N = -0.85; (b) N = -0.9; (c) N = -1.5.

Fig. l l b . The core, which exhibited temperature and concentration stratification at N values sufficiently above and below -0.9, is now dominated by nearly vertical constant - T and -C lines. This feature is consistent with the tendency of both Nu and Sh to approach their pure diffusion limits (e.g., Fig. 10).

A compact analytical solution that documents the effect of N on both Nu and Sh was developed in a subsequent paper by Trevisan and Bejan [%I. This solution is valid strictly for Le = 1 and is based on the constant-flux model according to which both sidewalls are covered with uniform distribu-


tions of heat flux and mass flux. The overall Nusselt number and Sherwood number expressions for the high Rayleigh number regime (distinct boundary layers) are

Nu = Sh = 3 ( H / 1 5 ) ~ / ~ R a q l + N ) 2 / 5 (91) where Ra. is the heat-flux Rayleigh number, Ra. = Kg/3H2q"/avk. These theoretical Nu and Sh results agree well with numerical simulations of the heat- and mass-transfer phenomenon [58].

Another theoretical result has been developed for the large Lewis numbers limit in heat-transfer-driven flows (IN1 << 1) [58]. In this limit the concentration boundary layer can be described by means of a similarity solution, leading to the following expression for the overall Sherwood number

Sh = 0.665(L/H)1/1" Le1I2 Ra:/lo (92) The mass flux j used in the Sh definition, Sh = j H / D AC, is constant, while AC is the resulting concentration-temperature difference between the two sidewalls. Equation (92) is also in good agreement with numerical experiments.

It has been shown that the constant-flux expressions [Eqs. (91) and (92)] can be recast in terms of dimensionless groups (Ra, Nu, Sh) that are based on temperature and concentration differences-this, in order to obtain approximate theoretical results for the configuration of Fig. 7, in which the sidewalls have constant temperature and concentrations [58]. Similarly, appropriately transformed versions of these expressions can be used to anticipate the Nu and Sh values in enclosures with mixed boundary conditions, that is, constant T and j , or constant q" and C on the same wall. Numerical simulations of the convective heat and mass transfer across enclosures with mixed boundary conditions are reported in Refs. [58, 591.

The studies reviewed in this subsection are based on the homogeneous and isotropic porous medium model described in Section 11. The effect of medium nonhomogeneity on the heat and mass transfer across an enclosure with constant-flux boundary conditions is documented by Nandaku- mar [60] and Mehta and Nandakumar [61]. They show numerically that the Nu and Sh values can differ from the values anticipated based on the homogeneous porous-medium model.

D. TRANSIENT APPROACH TO EQUILIBRIUM

Another basic configuration in which the net heat and mass transfer occurs in the horizontal direction is the time-dependent process that evolves from a state in which two (side-by-side) regions of a porous


a b C d

FIG. 12. The effect of Lewis number on the time-dependent concentration field of a heat-transfer-driven flow (N = 0, Ra = lo3, H / L = 1, 4/u = 1): (a) streamlines; (b) isotherms, or constant-concentration lines for Le = 1; (c) constant-concentration lines for Le = 0.1; (d) constant-concentration lines for Le = 0.01 [62].

medium have different temperatures and species concentrations. In time, the two regions share a counterflow that brings both regions to a state of thermal and chemical equilibrium. The key question is how parameters such as N , Le, and the height-length ratio of the two-region ensemble affect the time scale of the approach to equilibrium. These effects have been documented both numerically and on the basis of scale analysis in Ref. [62].

As an example of how two dissimilar adjacent regions come to equilibrium by convection, Fig. 12 shows the evolution of the flow, temperature, and concentration fields of a relatively high Rayleigh number flow driven by thermal buoyancy effects ( N = 0). As the time increases, the warm fluid (initially, on the left side) migrates into the upper half of the system. The thermal barrier between the two thermal regions is smoothed gradually by thermal diffusion. Figures 12c and 12d show that as the Lewis number decreases, the sharpness of the concentration dividing line disappears, as the phenomenon of mass diffusion becomes more pronounced.


In the case of heat-transfer-driven flows, the time scale associated with

i = ( $ J / u ) ~ ( L / H ) ~ Ra-', if Le Ra > ( + / u ) ( L / H ) ~ (93)

the end of convective mass transfer in the horizontal direction is

i = ( $ J / u ) ' ( L / H ) ~ Le, if Le Ra < ( $ J / u ) ( L / H ) ~ (94)

t^ = t( a/uH2) (95)

The dimensionless time 1 is defined as

Values of t^ are listed also on the side of each frame of Fig. 12. The time criteria [Eqs. (93) and (94)] have been tested numerically in Ref. [62] along with the corresponding time scales for approach to thermal equilibrium, in either heat-transfer-driven or mass-transfer-driven flows.

V. Concentrated Sources of Heat and Mass

A. POINT SOURCE

A third category of studies of combined buoyancy effects in porous media is concerned with the local fields around buried sources of heat and mass. The first study in this area was reported by Poulikakos [63], who considered both the transient and steady state of the flow near a point source, in the limit of small Rayleigh number based on the heat source strength q (in watts),

The interplay between the buoyancy effects due to temperature and concentration gradients is governed by the new source buoyancy ratio

Ra, = KgPq/ vak (96)

in which m (in kilograms per second) is the strength of the mass source. The single-buoyancy-effect limit of this problem (i.e., the point heat source, N , = 0) was described in Ref. [64].

Figure 13 shows the shape of the streamlines in the transient state. The curves correspond to constant values of the group &t.t;'/2 (1 - N s ) , in which

= $/aK112, t. = t(a/uK) (98) and where I) is the dimensional stream function (in units of cubic meters per second). The radial coordinate r] used in this figure is defined by

r] = r/2(t,K)'/* (99)

344 OSVAIR v. TREVISAN AND ADRIAN BEJAN

FIG. 13. The transient flow field around a point source of heat and mass ( A = 1) [63].

which means that, in time, the flow pattern expands at tl/’. Figure 13 corresponds to the special case A = 1; the A parameter being proportional to the square root of the Lewis number,

Poulikakos showed that the A parameter has a striking effect on the flow field in cases where the two buoyancy effects oppose one another (N, > 0 in the terminology of Ref. [63]). Figure 14 illustrates this effect for the case N = 0.5 and A = 0.1; when A is smaller than 1, the ring flow that surrounds the point source (seen also in Fig. 13) is engulfed by a far-field unidirectional flow. The lines drawn on Fig. 14 correspond to constant values of the group 2 T+L~/’.


FIG. 14. The effect of a small Lewis number on the transient flow pattern near a point source of heat and mass ( N = 0.5, A = 0.1) [63].

In the steady state and in the same small-Ra, limit, the flow, temperature, and concentration fields depend only on Ra, , N, , and Le. Figure 15 shows the migration of one streamline as the buoyancy ratio N, increases from -0.5 to 0.5, that is, as the buoyancy effects shift from a position of cooperation to one of competition. When the buoyancy effects oppose one another, N = 0.5, the vertical flow field is wider and slower. The curves


FIG. 15. The steady-state flow field near a point source of heat and mass (Ra = 5 , Le = l), and its response to increasing the source buoyancy ratio N, [63] .

drawn on Fig. 15 correspond to = Ra R . / 8 r 7 where R. = R/K'I2 and R is a reference radial distance. Asymptotic analytical solutions for the steady-state temperature and concentration fields are also reported in Ref. [63].

B. HORIZONTAL LINE SOURCE The corresponding heat-and mass-transfer processes in the vicinity of a

horizontal line source were analyzed by Larson and Poulikakos [65,66].


The source buoyancy ratio in this case is

where q' (in watts per meter) and m' (in kilograms per meter per second) are the heat- and mass-source strengths. All the features described in the preceding sections are also present in the low Rayleigh number regime of the line-source configuration. The line-source Rayleigh number is based on the heat-source strength q' (see Bejan [ll], p. 387),

K "l'gpq ' a uk

Ra,. =

The single-buoyancy-effect limit of the same problem was described earlier by Nield and White [67]. In addition to developing asymptotic solutions for the transient and steady states, Larson and Poulikakos [65,66] and Nield and White [67] illustrated the effect of a vertical insulated wall situated in the vicinity of the horizontal line source.

VI. Concluding Remarks

It is clear from this review that the field of natural convection with combined buoyancy effects in porous media has grown considerably, especially during the 1970s and 1980s. The earlier work was dominated by linear stability studies; however, despite the large volume of that research, several aspects remain to be clarified further. Among these is the phenomenon of fingering, which is best known in the realm of double-diffusive convection in fluids [14].

The more recent work focuses on the multilayer structure of flows of the boundary-layer or concentrated-source type. Both steady-state and transient flows were considered, usually by relying on simplifying assumptions such as the boundary-layer slenderness and the low Rayleigh number of the concentrated sources. The study of this newer class of problems was motivated by modern engineering concerns, most notably, by the disposal of thermal and chemical waste.

At this time, the field is characterized by a total absence of fundamental experimental work, even though laboratory experiments have been reported in the neighboring field of natural convection driven solely by


temperature gradients [6]. An even greater opportunity exists for numerical experiments (e.g. , finite-difference simulations based on the full governing equations), for which the work of Lai ef al. [68] and Reddy and Mulligan [69] represents a good beginning. The few numerical studies that have been reported (e.g. , Refs. [57,58] are limited to configurations and Rayleigh numbers that were selected for the purpose of testing isolated cases represented by approximate analytical solutions.

NOMENCLATURE

a A

A, B, C b C

CP

C D D' Da f> F

f'

g g

H

i k K L

Le m in' my

n N Ns

horizontal wave number Lewis number function,

coefficients, Eq. (38) Forchheimer constant, Eq. (9) similarity concentration profile.

specific heat at constant pressure

concentration mass diffusivity Soret coefficient Darcy number, Eq. (58) vertical concentration profile,

similarity velocity profile,

gravitational acceleration gravitational acceleration

vector height of vertical wall (length

of horizontal wall in Table

Eq. (1W

Eq. (87)

f = F / H

Eq. (84)

111) mass flux thermal conductivity permeability horizontal dimension of the

enclosure Lewis number, a / D strength of point mass source strength of line mass source volumetric mass production

rate of species i, Eq. (7) buoyancy ratio, p c A C / p AT buoyancy ratio, Eq (33) point source buoyancy ratio,

Eq. (97)

NI

Nu

P 4 4' 4" 4"'

r Ra

RaD

Ra,

Ra,.

Ray

Ra.

S Sh

t t

U

V

V

V

Y X

line source buoyancy ratio,

overall Nusselt number,

pressure strength of heat source strength of line source heat flux volumetric heat generation

rate, Eq. (4) radial position Darcy-modified Rayleigh

number based on H ,

mass diffusion Rayleigh

point heat source Rayleigh

line heat source Rayleigh

Rayleigh number based on

Eq. (101)

Eq. (65)

Eq. (31)

number, RaNLe, Eq. (45)

number, Eq. (96)

number, Eq. (102)

local altitude, Kgpy (To- T , ) / a v

heat flux Rayleigh number,

Soret parameter, Eq. (64) overall Shenvood number,

time dimensionless time scale,

horizontal velocity vertical velocity velocity vector velocity disturbance amplitude horizontal coordinate vertical coordinate

KgpH 2q"/a vk

Eq. (66)

Eq. (95)


P

P c

Y

7)

0

0

avg b f

Greek Symbols

thermal diffusivity dimensionless horizontal wave

thermal expansion coefficient,

concentration expansion coefficient, Eq. (15)

concentration disturbance amplitude

dimensionless radial coordinate, Eq. (99)

temperature disturbance amplitude

similarity temperature profile,

number

Eq. (14)

Eq. (86)

average base solution fluid

disturbance quantities dimensionless variables,

Eqs. (29)-(30)

Subscripts

S *

Superscripts

-

viscosity effective viscosity of the porous

bekl kinematic viscosity horizontal and vertical mean

density heat-capacity ratio porosity stream function stream function disturbance

amplitude

resistivities, Eq. (56)

solid dimensionless variables,

Eq. (98)

dimensionless variables, Eqs. (75)-(77)

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Applicability of Solutions for Convection in Potential Flow

STEPHEN R. GALANTE* AND STUART W. CHURCHILL

Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104

I. Introduction

Boussinesq (1903, 1905) derived perhaps the first theoretical solutions for external convection. These solutions were based on the postulate of potenrialflow, and also invoked certain other idealizations. His derivations and partial derivations included plates, circular and elliptical cylinders, spheres, ellipsoids, needles, and disks, and were for an arbitrary variation in surface temperature. In the intervening 85 years these solutions have been extended and refined somewhat. They have also been applied and misapplied for many situations including, in particular, moving bubbles and the flow of moderately low Prandtl number fluids (liquid metals) over solids (see, e.g., Grosh and Cess (1958) and Hsu (1964, 1965, 1967). This Chapter has the following objectives: (1) to review and correct, and then collate in a systematic form the solutions derived by Boussinesq and subsequent investigators for convection in potential flow; (2) to obtain solutions for convection in potential flow for additional important geometries and boundary conditions; (3) to identify the errors in these solutions resulting from various idealizations, and thereby determine their limits of applicability; (4) to analyze the applicability of solutions for convection in potential flow for moderately low Prandtl number fluids, such as liquid metals; ( 5 ) to analyze the applicability of solutions for convection in potential flow for rising bubbles; and (6) to investigate the utility of solutions for convection in potential flow in the development of correlating equations for all Prandtl numbers.

Pittsburgh, PA 15213. * Current address: Department of Chemical Engineering, Carnegie Mellon University,

353 _ . -n , . I n T

354 STEPHEN R. GALANTE AND STUART W. CHURCHILL

Solutions for the velocity field itself in potential flow in most of the geometries considered in this chapter are available in classical books on hydrodynamics, such as Milne-Thompson (1960), and hence are reproduced herein only when essential to a derivation. Likewise, the solutions for pure thermal conduction in rectangular coordinates, which are utilized in the method of Boussinesq for convection, are herein simply borrowed from standard sources such as Carslaw and Jaeger (1959). The derivations of both known and new solutions for convection in potential flow are given only in sufficient detail to illustrate the methodology and to identify the idealizations.

11. Derivations and Solutions for Convection in Potential Flow

Steady-state behavior (thereby excluding turbulence), irrotational motion, zero viscosity, constant thermal conductivity, constant heat capacity, constant density, and the absence of thermal radiation and of thermal sources and sinks within the fluid are postulated throughout.

A. GENERAL FORMULATION FOR CONVECTION IN PLANAR MOTION

Under the above constraints the equations for the conservation of mass and momentum for two-dimensional planar flow in Cartesian coordinates can be replaced by

(a’*/ax’) + ( d 2 * / a y 2 ) = 0

(a2+/ax2) + (a2+/ *2) = 0 (1)

(2)

where the stream and potential functions JI and 4, respectively, are defined by

and (3)

(4)

when up the unperturbed velocity in the x direction. The corresponding expression for the conservation of energy is

u,(dT/dx) + u,(dT/ay) = a[(a2T/aX*) + (a2T/ay2)]

U,(dT/d+) = a[ (J2T/d*2) + ( d 2 T / d Q ) ]

( 5 )

Replacing x and y in Eq. ( 5 ) with ${x, y} and +{x, y} gives, by a straightforward but somewhat detailed process,

(6) This is known as the Boussinesq transformation.

SOLUTIONS FOR CONVECTION IN POTENTIAL FLOW 355

B. GENERAL SOLUTIONS FOR CONVECTION IN PLANAR FLOW WITH NEGLIGIBLE CONDUCTION ALONG THE STREAMLINES

along the streamlines) is neglected, Eq. (6 ) reduces to If corresponding to conduction in the direction of flow (i.e.,

um(dT/d+) = a(d*T/d@) (7)

which can be recognized as analogous to the equation for transient conduction in the $ direction with time replaced by +/urn. The basic boundary and initial conditions are

T = T , for ++ -a (8)

T = T , for 4 - x (9)

(10)

(11)

(12)

T = Tw{+}

-k(dT/d$) = jJ+}

for $ = O , OS+I+, I or

d T / d $ = O for $ = O , + < O and + > + 1

Here, (+ = 0, $ = 0) corresponding to ( x = 0, y = 0), and (4 = +1 , 4 = 0), corresponding to (x = L , y = 0) represent, respectively, the forward and rear points of stagnation on the body.

The effect of conduction along the streamlines is obviously to reduce the heat-transfer coefficient. Increasing the velocity decreases the relative contribution of conduction along the streamlines. Hence, the solutions of Eq. (7), which neglect this mechanism of transport, can be considered to be upper bounding asymptotes for Pe + x.

1. Isothermal Surface

For a budy with an uniform surface temperature T,, the well-known solution of Eqs. (7)-(10) and (12) for the heat flux density to the fluid from the surface at $= 0 is (see, e.g., Carslaw and Jaeger, 1959, p. 61)

jw{+}=k(Tw- T=)(U=/T~+);%, 0 5 4 5 41 (13)

The relationship between the heat flux densities from the 4-4 and x-y surfaces is

jw{+} d+ = iWM ds

j,{x} = k(Tw - T,)[(u,/T~+)”~ d+/ds],=,

(14)

(15)

where s is the distance along the surface in the x-y plane. Hence,


Equation (15) represents the generalized solution developed by Bous- sinesq for forced convection in planar, potential flow over an isothermal surface with negligible conduction in the direction of flow (along the streamlines). As illustrated in the following, solutions for specific geometries can be obtained simply by substituting for 4 and d 4 / d s on the surface (J, = 0) from the solution for the potential function for that flow. Solutions for specific geometries may, of course, be derived directly from Eq. (3, using the velocity field for potential flow, and either considering or neglecting conduction parallel to the surface. Some of the solutions that will be given were actually attained by the latter procedure, because that process proved more straightforward or the results took a simpler form. The principal advantages of the Boussinesq method are its generality and simplicity.

2 . Uniformly Heated Surface

Equation (14) indicates that, in general, an uniform heat flux density along the surface (J, = 0) in the 4 coordinate will not correspond to an uniform heat flux density jwo along the surface in the x-y plane. Hence, to obtain the latter condition a variable heat flux density,

jW{& = jwo ds /d+ at 4 = 0, 0 I + I Cpl (16)

must be specified in the 4-4 coordinates. The solutions of Eqs. (7)-(9) and (11)-(12) for a specified variation of

the heat flux density can be expressed formally as (see, e.g., Carslaw and Jaeger, 1959, p. 76)

Equation (17) can readily be reexpressed in terms of the local Nusselt number. The corresponding solution for the mean Nusselt number depends on the arbitrary choice of a mean temperature difference. Herein the integrated mean value is implied.

3. Other Boundary Conditions

Analogous formulations could be developed for other boundary conditions, but herein, in the interests of simplicity and practicality only, the conditions of uniform surface temperature and uniform heating on the surface of the body in the original x-y coordinates are considered.


c. GENERAL SOLUTIONS FOR CONVECTION IN PLANAR FLOW TAKING INTO ACCOUNT CONDUCTION ALONG THE STREAMLINES

Taking into account conduction in the direction of flow as in Eq. (6) results in a much more complicated problem. Nevertheless, the following formal solutions were derived by Aichi (1920).

For an uniform temperature over the surface in the range 0 I 4 5 41, the solution is

(18) where Ko{l} = modified Bessel function of second kind and zero order (see, e.g., Abramowitz and Stegun, 1970, pp. 374-379). The heat flux density j w { 4 } can be obtained by differentiating Eq. (18) with respect to $and then setting +h to zero. The heat flux density jw{x} can in turn be determined from j w { 4 } through Eq. (14).

For uniform heating in the x-y coordinates over the range 0 I 4 I 4, , the solution is

where here again j w { 4 } is related to jwo through Eq. (16). Since in general, both the functions and integrals in Eqs. (18) and (19)

must be evaluated numerically, direct solution of either Eq. (5) or (6) by a finite difference method may be more attractive.

111. Convection in Potential Flow along a Flat Plate

Equation (5) is readily solved directly for this geometry. However, Eqs. (6) and (7) will instead be solved as a simple illustration of the generalized method of Boussinesq. For this flow, 4 = x .

A. ISOTHERMAL PLATE WITH NEGLIGIBLE CONDUCTION ALONG THE STREAMLINES

For 4 = x , Eq. (15) reduces to jw{x} = k ( ~ , - ~,) (u, / . rrcrx) ' /~

Nu, = ( X U , / T C Y ) ' / ~ = (Pex/r)1/2 Hence,


The mean Nusselt number based on the total heat flux over a plate of finite length 0 I x I L is then

- NuL = 2 ( P e J ~ ) l / ~ (22)

B. UNIFORMLY HEATED PLATE WITH NEGLIGIBLE CONDUCTION ALONG THE STREAMLINES

For 4 = x , Eq. (16) gives jw{4} =jwo, and Eq. (17) can readily be integrated to obtain

T,{x} - T, = (2j ,o/k)((~x/~~,)”~ (23)

NU, = 4 (rrPe,)1/2 (24)

NUL = 3 (rrPeL)l/* (25)

Hence,

where, as noted above, NUL is arbitrarily based on the integrated mean difference in temperature.

C. UNIFORMLY HEATED PLATE TAKING INTO ACCOUNT CONDUCTION ALONG THE STREAMLINES

Since 4 = x and jw{4} = j w o , Eq. (19) can be reduced and rewritten as

where

Hence,

King (1914) computed values of Nu, from Eq. (28) by numerical evaluation of the integral of Eq. (27). Jaeger (1943) computed and tabulated additional values of this integral. King also derived the following asymptotic expression for small Pe:


where

y = Euler's constant = 0.57722 . . . The numerical calculations of King converge to Eq. (24) for large Pe, and indicate that Eq. (29) provides a good representation (within 3%) for Pe, < 0.18. Grosh and Cess (1958) compared numerical solutions of Eq. (28) with Eq. (24) graphically for Pe, = 50 only and concluded that they differed significantly only for x + 0.

IV. Convection in Potential Flow across a Circular Cylinder

Convection from a circular cylinder provides an illustration of the application of the Boussinesq method to a slightly more complicated flow. The well-known solution for potential flow normal to the axis of a circular cylinder (see, e.g., Milne-Thompson, 1960, p. 233) gives for the surface of the cylinder

(b = D( 1 - cos{ 0}) (30) where here 0 is measured from the forward point of incidence (x = 0). Also, from purely geometrical considerations,

ds = (D/2) d0 (31) These two relationships permit the direct application of Eqs. (15)-(19).

A. ISOTHERMAL CYLINDER For an uniform temperature on the surface of the cylinder, Eq. (15),

with x replaced by s, (b substituted from Eq. (30), and 8 substituted from Eq. (31), gives

Hence, (1 +cos{~}) PeD 'I2 1 NuD = 2[

l r (33)

Here, as indicated, D is chosen as the characteristic length in Nu and Pe. Integration over the cylindrical surface gives

- NuD = (4 /r ) (2PeD/r) ' /* (34)

Equation (34) was first derived by Boussinesq.


Tomotika and Yosinobu (1957) derived a solution in series form, starting with Eq. (6), and presented a tabulation of values computed numerically from this series. They also derived the following asymptotic expression for small PeD:

- NuD = (2/S)[1+ (1 - 2S)(PeL/16)] (35)

where S = ln(8/PeD) - y

Their numerical computations indicate that Eq. (35) is a sufficient approximation for PeD<0.5. Curiously, the first term of Eq. (35) was also derived by Cole and Roshko (1954) for an uniform free-stream velocity. Also, Eq. (34) was found to provide a sufficient approximation (within 1.5%) for PeD > 8, which thereby is the criterion for negligible conduction along the streamlines.

B . UNIFORMLY HEATED CYLINDER King (1914) applied his solution for an uniformly heated plate

[Eq. (28)], which takes into account conduction in the direction of flow, directly for a cylinder, but misinterpreted this result as directly applicable for uniform heating in the 8 coordinate. The fortuitously close agreement of that solution with experimental data for forced convection from electrically heated wires to air in the boundary-layer regime has produced much subsequent confusion as to the applicability of solutions for potential flow.

The correct solution for an uniformly heated cylinder, but neglecting conduction in the direction of flow, as obtained by combining Eqs. (16), (17), (30), and (31), is

where K{(} is the complete elliptic integral of the first kind (see, e.g., Abramowitz and Stegun, 1970, pp. 589-592). Grosh and Cess (1958) erroneously specified the denominator of Eq. (37) as K{(8/2)2}. For the forward point of incidence at 8 = 0, Eq. (37) reduces to

NuD{O} = 2(2 PeD/?r)l/' (38)

which is identical to Eq. (33) for 8 = 0 . Hsu (1964) integrated the local temperature difference inherent in Eq. (37) numerically with respect to 8 from 0 to T to obtain the integrated mean temperature difference and


thereby

NuD = 1.145 Peg2 (39) The coefficient of Eq. (39) for an integrated mean heat-transfer coefficient was computed to be 1.2585 although both Grosh and Cess (1958) and Hsu (1965) reported a value of 1.340.

V. Convection in Potential Flow over Elliptical Cylinders

A. ISOTHERMAL SURFACE Boussinesq formulated a solution for convection from an isothermal

elliptical cylinder of arbitrary aspect ratio. A more detailed and complete derivation is given by Hsu (1965) who obtained the equivalent of

where a is the half axis in the direction of unperturbed flow (in meters); b, the half axis normal to the direction of unperturbed flow (in meters); z , the distance in direction of unperturbed flow, measured along axis from the forward point of incidence (in meters); and E{(}, the complete elliptic integral of the second kind (see, e.g., Abramowitz and Stegun, 1970 pp. 589-592). Equation (40) can alternatively be expressed in terms of x or y by noting that

z = a - x (42) and

(x/a)2 + (y/b)2 = 1 (43)

where x and y describe the surface of the ellipse with x measured from the b axis and y measured from the a axis.

Equation (40) is applicable for any value of b/a but Eq. (41) is limited to b/a 5 1. An analogous expression for NuD for a/b 5 1 can, however, readily be derived after rearrangement of Eq. (40).

For b = a , Eqs. (40) and (41) can be reduced to Eqs. (33) and (34), respectively, for a circular cylinder. By letting b + 0, they can be reduced to Eqs. (21) and (22), respectively, for flow along a plate. By letting a + 0,


Eq. (40), after being reexpressed in terms of y per Eqs. (42) and (43), can be reduced to the solution given in Eq. (47) for flow normal to an isothermal strip.

Boussinesq (1905) showed that the result expressed herein by Eq. (41) is independent of the orientation of the elliptical cylinder with respect to the direction of the unperturbed flow.

B. UNIFORMLY HEATED SURFACE

Cess (1956) derived a solution for convection from an elliptical cylinder of arbitrary aspect ratio with an uniform heat flux density on the surface. His solution can be written as

where a , b , and z are as defined previously and

For b = a , Eq. (44) can be reduced to Eq. (37), and for z = 0, as well, to Eq. (38). By letting b + 0, Eq. (44) can be reduced to Eq. (24) (with distance from the leading edge represented by z rather than x ) . By letting a + 0, Eq. (44), after being reexpressed in terms of y per Eqs. (42) and (43), can be reduced to the solution given in Eq. (49) for flow normal to a uniformly heated strip.

can be obtained for any particular value of b/a by numerical integration over the surface of the temperature difference implied by Eq. (44).

-

VI. Convection on the Front Face of a Strip of Finite Height

The potential function along the surface of a strip of half height b and infinite width on which an uniform potential flow is impinging is (see, e.g., Milne-Thompson, 1960, p. 167)

+= b{l - [l - ( ~ / b ) ~ ] ” ~ ) (46)

where y is the distance, as measured from the centerline, along the surface in the narrow direction (in meters).

A. ISOTHERMAL SURFACE

The solution for this thermal boundary condition can be derived directly from Eqs. (46) and (15), or can be adapted from Eq. (40) by letting a + 0.


The final result may be expressed as

Integration gives = 2(2 ~e,,/.rr)'/~

Reduction of Eq. (41) indicates that the overall mean Nusselt number for the front and back surfaces is decreased from that of Eq. (48) by a factor of 1/& It follows that this overall mean Nusselt number is independent of the orientation of the strip to the direction of the unperturbed flow, and necessarily coincides with Eq. (22).


The following solution can be derived for this condition by the method of Boussinesq, or can be adapted from Eqs. (44) and (45) by letting a + 0:

NU2b = ( r Pe2b)'/2/[2E{77) - K{7)}1 (49) Here

(50) 1 - [ l - (y/b)2]"2

2 7 =

and, as in Eqs. (37) and (49), K{(} and E{(} are the complete elliptic integrals of the first and second kinds, respectively. Numerical integration of the temperature difference implied by Eq. (49) over the surface gives

(51) - NU26 = 1.236 Pe:i2

VII. Convection in Potential Flow over a Wedge

The potential function for flow over a wedge can be expressed as

4 = CX"+' (52)

where c is an arbitrary constant (per meter); x the distance along wedge from the apex (in meters); n = 0/(.rr- 0); and 0 is the half angle of the wedge (in radians).


For this thermal boundary condition the solution can be expressed as

n + 1) Pe, 1/2 Nux=( ( 7T ) (53)


where here, for simplicity, Pe, is based on the local velocity at the surface:

UO, = (1 + n)cu,x" (54) The corresponding integrated mean value over 0 5 x 5 L is

where the velocity given by Eq. (54) for x = L is used as the characteristic value in Pe, . The equivalent of Eq. (53) was derived previously by Morgan et al. (1958).

For n = 0, Eq. (54) reduces to uox = cu,. Then, Eqs. (53) and (55) reduce to Eqs. (21) and (22) for flow along a flat plate if c is taken as unity.


rived: For this thermal boundary condition the following solution can be de-

where NUL is again based on the integrated mean temperature difference, and Pe, and Pe, are again based on the local velocity at the surface at x and L , respectively. Also p{& [} = r{(}-I'{[}/r{( + [} = incomplete p function and r{[} = complete I' function (see, e.g., Abramowitz and Stegun, 1970,

For n = 0, p{*, 1/(1+ n)} = 2, uox = cu,, and Eqs. (56) and (57) reduce to those for flow along an uniformly heated plate, Eqs. (24) and (25), if c is taken as unity.

pp. 255-260).

VIII. Convection in Planar Potential Flow Impinging Normally on a Plane

Letting n = 1, corresponding to 8 = n/2, in the solutions for a wedge gives solutions for planar, normal impingement of an uniform potential flow. For this limiting condition uox = 2cu,x and p{+, 1/(1+ n)} = n.

Equations (53) and (55) for an isothermal surface then reduce to

Nu, = (2 Pe,/7r)'/2 (58)


Since uOx is proportional to x and therefore Pe, to x2, the heat-transfer coefficient is constant (independent of x), thus explaining the numerical identity of Eqs. (58) and (59).

For n = 1, Eqs. (56) and (57) for an uniformly heated surface also reduce to Eqs. (58) and (59). The expressions for Nu, and NuL for an uniform heat flux density are identical to those for an isothermal surface since an isothermal surface produces an uniform heat flux density and conversely for this particular flow.

Equation (58) is equivalent to Eq. (47) at the midplane, if the different characteristic velocity and length are taken into account, but not elsewhere since potential flow over and around a strip of finite height differs fun- damentally from the ever-increasing velocity upward and downward along the surface of an infinite plate. The latter motion is purely hypothetical. Equation (58) with c = l / a is also equivalent to Eqs. (33) and (37) at the point of stagnation.

IX. General Formulation for Convection in Axisymmetric Potential Flow

The Stokes stream and potential functions, $ and 4 , for axisymmetric flow in polar coordinates, r and z, respectively are

u,/u, = - ( l / r ) ( d $ / d z ) = dr$/dr

UJU, = ( l / r ) ( d $ / d r ) = dc$/dz

(60)

(61) Application of the Boussinesq transformation in terms of these two functions gives for the energy balance

The third term on the right-hand side of Eq. (62) represents conduction in the direction of flow and the second term the effect of transverse curvature on radial conduction. For very large Pe, both of these terms can be expected to be negligible, reducing Eq. (62) to

U, = (dT/d&) = ar2(d2T/a$2) (63) The further transformation proposed by Boussinesq:

(64)


then reduces Eq. (63) to

Um(dT/df$’) = a(aZT/a$Z) (65)

which can again be recognized as equivalent to that for one-dimensional, planar, transient conduction.

Expressions for the conversion of the heat-transfer coefficient from 6’-$ coordinates to r-z coordinates were developed by Yuge (1956) and subsequently in the following vectorial form by Rigdon (1961):

jw{r, z}= -k(i,.VT)n=o (66)

jW{$‘, $} = -k(i4.VT)4=o (67)

iq-VT= (1/8$)(aT/d$) (68)

(69)

(70)

h{r, 2) = h{4’1/(84)4=0 (71)

From vector calculus

where 6s; is a scale factor, here represented by

1/64 = [(a$/az)’ + (a$/ar)2]1’2

jw{(r, z ) =jw{4’, $1 = -k[(l/64)(~T/a$)Is;=o

Then

The use of Eq. (63), and hence Eqs. (71) and (69), for axisymmetric potential flow implies the neglect of the effect of transverse curvature on the velocity field as well as on the temperature field, and also the neglect of conduction along the streamlines. As noted for planar flow, the effect of conduction along the streamlines is to decrease the heat-transfer coefficient. On the other hand, the effects of transverse curvature are to increase the tangential area for conduction and to decrease the tangential velocity. Increasing the area increases the heat-transfer coefficient, but decreasing the velocity has the opposite effect. These effects might therefore be expected to compensate somewhat, but it is not obvious a priori which will dominate, and hence whether the effect of neglecting transverse curvature will be additive or compensatory with respect to the effect of neglecting conduction in the direction of flow. Therefore, whether or not solutions of Eq. (65) provide an upper bound for convection in potential flow is not obvious. In any event, these solutions can be considered as asymptotes for Pe -+ m.

Solutions for axisymmetric potential flow over several shapes follow. All of these are based on Eq. (65), and hence on Eq. (63) rather than on Eq. (62), except as especially noted.


X. Convection in Potential Flow over a Cone

The potential function on the surface of a cone is

Pm+1{pI (72) 4 = - A x m + l

where here A is an arbitrary coefficient (per meter); p = -cos{8}; 8 is the semivertex angle of the cone, (in radians); x the distance along the cone, (in meters); Pm{ p} is the Legendre polynomial of order m. The relationship between m and p is given by

d (Pm+ I{ P } ) / ~ c L = 0 (73)

A tabulation of m(p) is provided by Hess and Faulkner (1965). Since noninteger values of rn are encountered, Pm+l{p} may be computed as a Gauss hypergeometric function: F{m + 2, -rn - 1,1, (1 + cos{8})/2} (see, e.g., Abramowitz and Stegun, 1970, pp. 556-565).


The Boussinesq method leads to solutions for convection from a cone in terms of the Gauss hypergeometric function. A solution in considerably simpler and more convenient form can be obtained directly from the equivalent of Eq. ( 5 ) with d 2 T / d x 2 neglected. With this latter method the components of the velocity along and normal to the surface of the cone are taken to be

u, = U l ( X / l ) ” (74)

where m is again fixed as a function of the angle of the cone by Eq. (73) and I is an arbitrary reference length. It follows from Eq. (74) that u1 is the value of u, where x = 1.

The solution, as obtained by first applying the Mangler transformation (see, e.g., Schlichting, 1960, p. 190), then deriving a similarity transformation by the method of Hellums and Churchill (1964), and finally solving the resulting ordinary differential equation is

Nu, = [(m + 3)/7r]1’2 Pe;” (76) where here, as in Eq. (53), Pe, implies that the local velocity u,, as given by Eq. (74), is used as the characteristic value in the interest of simplicity.


Integration of the heat flux density over the surface of the cone gives

where here PeL implies that the local velocity at x = L is used as the characteristic value.


Boussinesq gives For an uniform heat flux density on the surface of the cone the method of

{ [A(m + 3); + l)m? ][F: + (~)Zsin2{B}F:]Pe,} 1 /2

{l Z,m+3 2 l (78) Nu, =

where here Fl is the previously noted particular Gauss hypergeometric function, and

F2 = -F{m -t 3, -m, 2, (1 + cos{8})/2} Integration of the temperature difference over the surface gives

XI. Convection in Potential Flow along an Infinitely Long Pointed Needle

For 8 = 0, Eq. (73) gives m = 0. Then, Pm+l{p} = -1 and from physical considerations A = 1. Also, from Eq. (74), u, = u1 and, therefore, must equal u,. The error due to the neglect of transverse curvature would be expected to be significant even for fairly large Pe in this limiting case.

A. ISOTHERMAL SURFACE For these conditions, Eq. (76) reduces to

NU, = (3 ~ e , / r ) ’ / ~ (80) Equation (77) can be reduced or the heat flux density given by Eq. (80) can


be integrated over the surface to obtain - NuL = 4 ( P e J 3 ~ ) ' / ~


For these same conditions Eq. (78) reduces to

Nu, = 1.187 Pei/2 (82) Equation (79) can be reduced or the temperature difference implied by Eq. (82) can be integrated over the surface to produce

NUL = 1.484Pe;I2 (83)

XII. Convection in Potential Flow Impinging Axisymmetrically on a Plane

Here x has actually become equivalent to r , the radial distance (along the surface) from the axis of symmetry, 8 = r / 2 , and Eq. (73) gives m = 1. Also, following Homann (1936), if 1 is taken to be D, twice the outward radial distance from the axis of symmetry, u1 = 3u,, and Eq. (76) can be rewritten as

NuD = 2(3 PeD/r)ll2 (84)

Equation (84) indicates that the heat-transfer coefficient is independent of x , just as for planar impingement, again because of the proportionality of u, to x . It follows that

NuD = 2(3 PeD/r)'l2 (85)

The equality of the coefficients of Eqs. (84) and (85) implies that they are also applicable for an uniformly heated surface.

XIII. Convection in Potential Flow over a Sphere

A. ISOTHERMAL SPHERE

expressed as The solution of Eq. (65) for this thermal boundary condition can be

NuD = 3(1 + cos{8})[PeD/r(2 + C O S ( ~ } ) ] ~ / ~ (86) where 8 is the angle measured from the forward point of incidence. For


6 = 0, Eq. (86) reduces to Eq. (84). Integration over the surface gives - NuD = 2(PeD/..)'12 (87)

Equation (87) was first derived by Boussinesq. Modern detailed derivations of the equivalent of Eqs. (86) and (87) have been carried out by Yuge (1956) and Hsu (1967).

B. UNIFORMLY HEATED SPHERE The corresponding solution for this thermal boundary condition is

NuD = ( 2 T k ~ ) ' / ~ / G { e } (88)

where

[ F( (sin-'{a}), ;} - F{(sin-'{a}), A} (89) 1 G{ 6) = 1 /2 3'14( cos { - :})

= cos-l{[cos(e}(cos2{e} - 3)]/2}

a = (cos{(o/3) + (7r/6)}/cos {(0/3) - (77/6)})'12

and F { & 5) = elliptic integral of the first kind (see, e.g., Abramowitz and Stegun, 1970, pp. 589-590. Numerical integration of the temperature difference given by Eq. (88) over the surface of the sphere gives

N u D = 1.290 PeD'12 (93) A complete derivation of this solution is given by Hsu (1967).

XIV. Effect of Transverse Curvature on Convection in Potential Flow over an Isothermal Sphere

The effect of transverse curvature in axisymmetric potential flow over an isothermal sphere has been evaluated both analytically and numerically.

Hirose (1975b) derived the following perturbation solution for inviscid flow over a fluid sphere at very small PeD,

1 3P2 [: ( :) 224 N U ~ = N ~ ~ + -P I + - co~{e}---(i+3~0~{2e}) (i-p)-l12 (94)


(95) - ( P 13P2 P3 7P4 P') NuD= (1 - P)-1'2 2 160 20 200 40

where

P = PeD/( 1 + PeD) (96)

Watts (1972) derived a perturbation solution for the local value of the Nusselt number at the point of incidence and for the average value for inviscid flow over a solid sphere at large but finite PeD. The solution for the local value was corrected numerically by Hirose (1973) and then further by Watts (1973), to become

NuD{O} = 2(3 PeD/.lr)'l2 + (8/3.lr) + 9{Pe;'} (97)

and that for the mean value one further time by Hirose (1975a) to become

- NuD = 2(PeD/.lr)'/2 + 0.827 + D{PeE'} (98)

The constant terms in Eqs. (97) and (98) represent the first-order effect of transverse curvature. As PeD increases, Eq. (97) can be seen to approach Eq. (84) [i.e., the limiting behavior of Eq. (85) for 8 = 01 and Eq. (98) to approach Eq. (87).

These perturbation solutions are invaluable in that they provide a quan- titative theoretical indication of the relative magnitudes of the two compet- ing effects of transverse curvature. The constant 2.0 in Eq. (95) represents the effect of the increased area for conduction with radial distance from the surface. The constant 0.827 in Eq. (98) represents the net effect of the increased area for conduction over that due to the decrease in velocity. Apparently, in this geometry, the relative contribution of the increased area decreases with PeD but is dominant over the effect of the decreased velocity for all PeD. The effect of transverse curvature is so great for a sphere that it would be expected to overshadow the effect of conduction in the direction of flow, which is neglected in all of the previous solutions.

Rigdon (1961) investigated the effect of transverse curvature on the local Nusselt number using numerical integration. He concluded that Eq. (86) was not in error at PeD = 1000 and 100 by more than 1.4% and 4.3%, respectively, at any angle. These differences are less than those predicted by Eqs. (97) and (98). This discrepancy may be due to error in the numerical integration or in the failure of the latter expressions to extend to these values of PeD.


XV. Convection in Axially Symmetric Potential Flow over Isothermal Spheroids of Revolution

A. PROLATE (OVARY) SPHEROID

The solutions for axisymmetric potential flow over an isothermal spheroid formed by rotating an ellipse about its major axis can be expressed as

N~2a

- (2 - 2)(6(1- E2) P~,,/T)'/~

(1 - (1 - E2)(1 - 2)2)(3 - 2 )

(99) where 0 5 Z 5 2 , and

- 4(1 - E2)(6Pe2,/p)1/2 3[sin-'((l - E2)ll2} + E(l - E2)l l2]

NU^^ =

where a is the major half axis in direction of unperturbed flow (in meters); b, minor half axis normal to the direction of unperturbed flow (in meters); E = b/a , is the eccentricity ratio; z , distance in direction of unperturbed flow, measured along axis of symmetry from the point of incidence (in meters); Z = z /a ; and

A solution equivalent to Eq. (100) was formulated by Boussinesq and completed in somewhat different form by Lochiel and Calderbank (1964).

Equations (99) and (100) can be reduced to Eqs. (86) and (87), respectively, for a sphere by taking the limit as E + 1, and noting that 2 + 1 - cos{e).

B. BLUNT NEEDLE OF FINITE LENGTH

Letting E + 0 in Eqs. (99) and (100) gives the following expressions for convection in potential flow along a very thin isothermal ellipsoid of length 2a:

, o s z s 2 TZ(3 - 2 )

NU^^ =

8 2 Pe2, 1/2 ; (7)


These solutions differ significantly from Eqs. (80) and (81) owing primarily to the effect of the bluff front of the body on the streamlines, even in the limit of vanishing thickness.

C. OBLATE (PLANETARY) SPHEROID

formed by rotating an ellipse about its minor axis b can be expressed as The solution for axisymmetric potential flow over an isothermal spheroid

(2 - Z’)[6E(1 - E2) Pe2b/~]’/~

r2 ((1 - E2)(1 - Z’)’+ E2}(3 - Z’)

(104) NU2b = [( (1 - E2)l12

where 0 I 2’ I 2 and

(105) - 4(1 - E2)(6 Pe2a/p’)1/2

3[(1 - E2)1/2 + E2 tanh-’((1 - E2)1/2}] NU^ =

with

(106) p l = sin-1{(1 - ~2 112 1 } - W - E ) 2 1/2

where the same nomenclature is used but the unperturbed flow is in the direction of the minor axis, and hence now

Z’ = z/b

Again, a solution equivalent to Eq. (105) was formulated by Boussinesq and completed in somewhat different form by Lochiel and Calderbank.

Equations (104) and (105) reduce to Eqs. (86) and (87), respectively, for a sphere in the limit as E + 1, again noting that Z’ + 1 - cos{O}.

D. THIN DISK OF FINITE DIAMETER

Replacing z in Eq. (104) with r through the following expression for the surface of the ellipsoid:

[(b - ~)/b]’ + (r/a)’ = 1 (107) where r is the distance normal to axis of symmetry (in meters); changing from 26 to 2a = D as the characteristic length in Nu and Pe, and then letting E + 0 gives for the front surface an isothermal disk

] ‘ I2 (108) 2 3 PeD [ (1 - R2)[2 + (1 - R2)l/’] NuD =- [1+ (1 -R2)1/2] ?r


where R = r / u , is the fractional distance from axis of symmetry to the edge of the disk. Integration over the front half of the disk gives

NuD = (8/n)(2 PeD/3)’/’ (109) Reduction of Eq. (105) indicates that the combined mean Nusselt number for the front and back sides is reduced from that of Eq. (109) by a factor of ($)’/’, just as was Eq. (48) for a thin strip.

Equation (84) is equivalent to Eq. (108) (after the different characteristic dimensions are taken into account) at the point of incidence on the axis of symmetry but not elsewhere owing to the different streamlines for flow over and around a thin disk of finite diameter as compared to the ever- increasing outward radial flow along the surface of an infinite plane.

XVI. Convection in Potential Flow over an Isothermal Spherical Cap

Very large bubbles are known to assume the shape of a spherical cap, with the flat surface facing downward, as they rise through a liquid. The local rate of transfer on the curved surface is given by Eq. (86). Lochiel and Calderbank (1964) assumed that transfer through the flat bottom surface was negligible, and integrated the local rate over the curved surface to obtain the equivalent of

NuDc = [1.183(3 + 4E2)2/3E1’3 Pe2(,2]/(1 + 4E2)’l6 (110) where E = I/w, is the eccentricity ratio; 1, the height of the spherical cap (in meters); w, the width of the spherical cap (in meters); and D, , twice the radius of curvature of the spherical cap (in meters). For a typical total angle of lOOn/lSO rad subtended by the flat of the cap, E=0.2326, reducing Eq. (110) to

- NuDc = 1.346 Pegc’ (111)

XVII. Comparison of Solutions for Mean Nusselt Number in Potential Flow over Various Shapes with PeD + w

The solutions for Nu for large Pe and, therefore, for negligible conduction in the direction the streamlines and in axisymmetric flow for a negligible effect of transverse curvature as well, are summarized in Table I for discrete geometries and in Table I1 for objects of variable geometry (wedges, cones, and ellipsoids). In all of these situations Nu is proportional to Pel/’ and otherwise a function only of geometry and the thermal

SOLUTIONS FOR CONVECTION IN POTENTIAL FLOW 375 TABLE I

SUMMARY OF COEFFICIENTS FOR MEAN NUSSELT NUMBER OVER VARIOUS SHAPES IN

POTENTIAL FLOW WITH Pe -+

Boundary condition Characteristic

Shape length Uniform temperature Uniform heating*

Plate L 21 & 3 m Planar impingement' L Jz7;; Jz7;;

Circular cylinder D 4 6 1 ~ J;;

Rounded needle L 8 4 / x .,& -

Thin normal diskd D 8 h / v h -

Spherical cap' D f 1.346 -

Normal stripd L 2 s 1.226

Sharp needle L 24% 1.484

Axisymmetric impingement' L 2 m 2-

Sphere D 21 J;; 1.290

"In all situations 6i /Pe ' /2 .

'Based on local velocity at surface. dFor front surface only. 'For subtended half angle of 50x/180 rad. /Diameter of curvature.

Based on integrated mean temperature difference.

TABLE I1

SUMMARY OF COEFFICIENTS FOR MEAN NUSSELT NUMBER OVER WEDGES AND SPHEROIDS IN POTENTIAL FLOW WITH Pe+ = AND UNIFORM SURFACE TEMPERATURE

_____

Characteristic Shape length

Wedge

Elliptical cylinder

(0 I n = e / ( p - e) 5 1)

@/a > 0)

Prolate spheroid ( 0 s E = b/a 5 1)

Oblate spheroid (01 E = b/a< 1)

L 2 / [ n ( n + 1 ) y

26 2{[1 + (b/a)](b/~a)} ' /~ E{[1 - (b/a)2]'/2}

2E2 where p = 1 - tanh-'[ (

( 1 - E 1 l + E

4(1 - E * ( ~ / P ' ) ' / ~ 2a &a= 3[(1 - E2)lI2 + E2tanh-'{(l - E2)1/2} ]

where p' = sin-'{(l - E2) ' l2 } - E(l - E2)' l2


boundary condition. The proportionality of Nu/Re’/’ to Pr’/’ implies a negligible velocity gradient at the surface as compared to the proportionality of Nu/Re’/’ to Pr1I3 for Pr --* m, which implies a linear velocity gradient at the wall.

The numerical values of &/Pel/’ have a remarkably constrained range, thereby providing a basis for a first-order estimate for geometries for which solutions do not exist. The actual values of Nu/Pe’/’ are, of course, dependent on the arbitrary choice of a characteristic length, and in the case of wedges and cones on the choice of a characteristic velocity as well.

The local values of Nu/Pe’/’ for an uniformly heated surface are in all instances equal to or greater than those for an isothermal surface. The equality occurs only in impacting flows such that the velocity on the surface increases linearly with distance, thereby yielding an uniform temperature for uniform heating and vice versa. For an uniformly heated surface the definition of the mean heat-transfer coefficient is arbitrary, and some choices may lead to a value less than that for an isothermal surface. Thus, the mean values in Table I for uniform heating of a finite strip and a sharp needle, which are based on the integrated mean temperature difference, are less than those for an uniform surface temperature, which are based on the integrated mean heat flux density.

XVIII. Applicability of Solutions for Convection in Potential Flow

The solutions for convection in potential flow have been widely proposed and utilized as approximations for heat transfer to liquid metals and for mass transfer to bubbles. The validity of these applications is examined in this section.

A. CONVECTION FROM LIQUID METALS The application of the velocity field for potential flow for the prediction

of forced convection from immersed solid objects to liquid metals is based on three premises: (1) that the velocity field outside the momentum boundary layer is closely approximated by potential flow; (2) that the free-stream velocity is sufficiently large and the viscosity sufficiently low so that the momentum boundary layer is very thin, and hence does not displace the region of potential flow significantly from the surface; and (3) that the Prandtl number of liquid metals is sufficiently small, that is, the thermal conductivity is sufficiently greater than the viscosity times the specific heat capacity, so that the thermal boundary layer extends far beyond the


momentum boundary layer, and thereby results in almost complete development of the temperature field in the outer region of potential flow, rather than within the inner region of the momentum boundary layer.

1. Validity of Premise 1 .

The first premise fails totally in the region of separation that occurs on all immersed solids at the high rates of flow that are required to satisfy the second premise. Therefore, the applicability of potential flow for convection to solids is necessarily limited to that portion of the surface ahead of the point of separation.

Solutions for the mean rate of heat transfer are correspondingly limited to regions ahead of the point of separation. Thus, solutions for Nu for planes, wedges, and cones may have validity within that restriction; those for strips and disks are limited to the frontal surface; and those for cylinders and spheroids have no possible regime of applicability.

Furthermore, the wake displaces the region of potential flow outward and thereby changes the flow near the surface up to the forward point of incidence even for large free-stream velocities. The experimental data of Heimenz (1911) and Schmidt and Wenner (1941) for cylinders, and of Yuge (1956) for spheres, all at large ReD, indicate that the free-stream velocity at the point of incidence is decreased 9.2 and 6.8%, respectively, below the value for pure potential flow. Since the rate of heat transfer varies as ( U ~ { O } ) ” ~ the corresponding overpredictions of NuD{O} are 3.6 and 5.0%. From measurements of mass transfer, Sparrow and Geiger (1985) determined the equivalent overprediction of Nud{O} for a circular disk to be 6.6%.

2. Validity of Premise 2

The validity of the second premise is shown by the velocity field obtained by Brauer and Sucker (1976) from numerical integration of the general equations of motion for flow along a plate, to be limited to values of Re, considerably above 100. Experimental data for the drag coefficient indicate significant deviations from thin-boundary-layer theory even up to Re, = lo3, which again implies a significant thickness for the momentum boundary layer (see, e.g., Janour, 1947; Dennis and Dunwoody, 1966). Similar limitations are to be expected for other shapes. Such thickening of the momentum boundary layer decreases the rate of heat transfer, and thereby results in a further overprediction of Nu by the expressions for potential flow.


3. Validity of Premise 3.

The Prandtl number of liquid metals ranges from approximately 0.01 to 0.03 at ambient temperatures and may be slightly less at elevated temperatures. The Schmidt number for the transport of a gas of low molecular weight through another gas of high molecular weight may be as low as 0.04. The Prandtl number of ordinary gases is about 0.7. The error resulting from the use of solutions for Pr = 0 for such fluids is examined in the following.

Theoretically computed values of Nu for wedges, including the limiting cases of parallel flow over and impinging flow on a flat plate, and for the point of incidence on cylinders and spheres for selected values of Pr are compared with values for potential flow (Pr = 0) in Table 111. This comparison is expressed in terms of the percentage of overprediction of Nu for Pr = 0 with respect to that for several infinite values of Pr. These theoretical values for Nu for finite Pr are from thin-boundary-layer theory and thereby incorporate premises 1 and 2. Thus, the comparisons of Table I1 are for the effect of Pr only, that is, they are for asymptotically large Re.

For a representative Pr of 0.02 for a liquid metal, the overprediction is seen to range from 7% for an isothermal cylinder to 19% for flow along an uniformly heated plate. For Pr = 0.7, which is representative for air and similar gases, the corresponding range of overprediction is from 13 to 84%. Air may be concluded to be a fairly high Prandtl number fluid by this

TABLE 111

PERCENTAGE OVERPREDICTION OF Nu BY THEORETICAL SOLUTIONS FOR

POTENTIAL FLOW (Pr = 0)

0.01 0.02 0.03 0.70

Uniformly heated plate

Isothermal flat plate [Churchill and Ozoe (1973a)l 14.2 18.7 22.3 84.2

[Chen (1985)l 9.4 13.0 16.8 61.0 [Churchill and Ozoe (1973b)l 9.3 12.8 15.6 61.7

Isothermal wedge ( 6 = ~/2)

Planar impingement

Point of stagnation on cylinder

Point of stagnation on sphere

[Chen (1985)] 6.4 9.1 11.1 53.7

[Chen (1985)l 5.2 7.3 9.0 43.2

[Evans (1968)l 5.0 7.0 8.5 34.6

[Evans (1968)] 6.2 8.7 10.5 41.9


criterion, and liquid metals to be low but not asymptotically low Prandtl number fluids. These overpredictions of the Nusselt number for a finite Prandtl number are in addition to those for finite Re and for the displacement of the free-stream velocity by the wake.

The experimental data of Ishiguro et al. (1975) for the local heat-transfer coefficient for convection from a heated cylinder to liquid sodium (Pr = 0.0073) for 1530 I ReD I 10,000 fall approximately 50% below the theoretical values for an isothermal surface and P r = 0 . They conclude from these results that the use of the solution for Pr = 0 “for a liquid metal, cannot be justified.” The overall coefficients measured by Witte (1968) for convection from a sphere to liquid sodium for 35,600 I ReD 5 152,500 also fall approximately 50% below the theoretical expression for Pr = 0.

4. Assessment

Solutions for convection in potential flow are absolutely limited in applicability to local values in the region on a surface ahead of the point of separation, and, even in that region, overpredict Nu significantly for liquid metals and grossly for air. The relative overprediction increases as Re decreases owing to the increased thickness of the momentum boundary layer.

As indicated for spheres, these overpredictions are further increased when conduction in the direction of flow is taken into account, but are partially compensated for when the effect of transverse curvature on heat transfer is considered.

B. CONVECTION FOR RISING BUBBLES

The application of the solutions for convection in potential flow to bubbles rising in a liquid is based on three different premises: (1) that the bubble is completely mobile; (2) that the velocity field in an unbounded field of liquid of finite viscosity outside a completely mobile bubble is the same as that for potential flow; and (3) that the shape of the bubble is known and invariant with time.

1 . Validity of Premise 1

Complete mobility of a bubble is attained only for a very clean liquid since minute traces of a surface-active contaminant or solid impede the internal circulation. The mobility of small bubbles in water is usually so low due to contamination that their behavior approaches that of solid spheres. With very large bubbles the contaminants are swept to the rear and


discarded, increasing the mobility. Weiner and Churchill (1977) found that bubbles of COZ in water behaved as completely mobile only for diameters greater than about 2.5 mm, corresponding to ReD greater than 700. Some- what different critical diameters have been observed by other investigators with different gases in different liquids or water of different purity.


The velocity distribution derived by Moore (1963) for a completely mobile, spherical bubble rising in a viscous liquid has been utilized in a number of theoretical derivations for the thin-boundary-layer regime (Weber, 1975) to obtain the equivalent of

The term involving ReD represents the effect of the deviation of the motion from that of potential flow due to the finite viscosity of the liquid. This deviation occurs primarily at the rear of the sphere but influences the forward portion as well. Equation (112) does not take into account either transverse curvature, which effect is indicated (for potential flow) by Eq. (98), or conduction parallel to the surface.

For creeping flow (ReD << 1) over a completely mobile, spherical bubble, Hirose (1975a) derived

- NuD = 2(PeD/3.rr)'l2 + 1.654 (113)

Comparison of the first term of Eq. (113) with Eq. (87) indicates that the deviation from potential flow in this extreme results in a factor of 3-l/'. The constant term of Eq. (113) is a first-order correction for the net effect of transverse curvature.


For bubbles of air in water, Rosenberg (1950) observed a spherical shape and rectilinear motion for ReD < 400, oblate spheroids with a vertical axis rising rectilinearly for 400 < ReDv < 500 but helically for 500 < ReDv < 1100, irregular oblate spheroids rising almost rectilinearly for 1100 < ReD < 1600, and a transition for 1600 < ReDv < 5000 to irregular, horizontally oriented, mushroomlike shapes rising more or less rectilinearly and followed by a turbulent wake. Similar behavior but with slightly different points of transition have been observed by others for air and water as well as for other gases and liquids.

SOLUTIONS FOR CONVECTION IN POTENTIAL FLOW 38 1

Obviously, for ReDv > 400 the shape must be known if the solutions for convection for spheriods are to be applied. Moore (1965) derived a theoretical expression for the eccentricity ratio of rising bubbles as a function of ReDv and gp4/pu3, assuming they were completely mobile, oblate spheroids. Lochiel and Calderbank (1964) showed that when Eq. (104) is expressed in terms of the volume-equivalent diameter:

D, = 2a E 'I3 (114)

rather than 26, NuDv/Pegvz varies only slightly with E. However, Eq. (110) when similarly expressed in terms of D, rather than D, retains a strong dependence on E (owing to the postulate of no transfer from the flat surface).

From a practical point of view the expression for convection from mobile bubbles should be recast in the form of h d / 6 / ( k ~ ) 1 / 2 g ' / 3 as a functim of D g 1 / 3 / v 1 / 2 , with E and gp4/pu3 as possible parameters, in order to eliminate u, which is really a dependent variable. Weiner and Churchill (1977) plotted their widely scattered experimental data for mass transfer from bubbles of COz to water in the equivalent of this form, and found Eq. (87) in terms of D, to be applicable only for Dvg'/3/v2/3 > 60, which corresponds to ReDv > 700 or D, > 2.5 mm.

Calderbank and Lochiel (1964) plotted their own experimental data, as well as that of others, for larger bubbles of C 0 2 in water. Their plot of the equivalent of h versus D, indicates that Eq. (87) in terms of D, is a satisfactory lower bound for D, > 5 mm, and that Eq. (110) with E = 4 is generally an upper bound. Equation (105) with experimentally observed values of E was shown to provide intermediate values and perhaps the best overall representation for these widely scattered data. Transfer coefficients far above the predictions of Eq. (87) for 14 mm < D, < 18 mm were attributed to oscillations.

4. Assessment

The velocity field outside a bubble approaches that of potential flow only if the bubble is completely mobile and the Reynolds number is very large. Complete mobility requires a liquid free from even traces of surfactants or solids, and/or a higher Reynolds number. On the other hand as the Reynolds number increases bubbles are deformed, the degree of defor- mation depending on gp4/pu3 as well as on ReD,. The expressions for convection from oblate spheroids and spherical caps in potential flow may then be applicable if the shape is known. The eccentricity may be predicted


theoretically for small deformations, but otherwise must be determined experimentally. The velocity of freely rising bubbles is a dependent variable, and must either be determined from a second correlation or elimin- ated from the expressions for convection.

Overall, the application of solutions for convection in potential flow to bubbles is limited to large bubbles, for which the varying and irregular shape introduce considerable uncertainty.

XIX. Utility of Solutions for Convection in Potential Flow for the Development of Correlating Equations

As previously described in detail, the solutions for convection in potential flow are of almost no direct applicability for liquid metals or bubbles. Nevertheless, these solutions are invaluable for the development of correlating equations in two respects.

First, a solution for convection from an immersed solid can be interpreted as an asymptote for Pr -+ 0, and, as such, combined with an analogous asymptote for Pr -+ m, for any particular regime of flow and thermal boundary condition, to obtain a continuous correlating equation for all Pr with only one empirical constant. The general procedure for the development of such correlating equations was originally described by Churchill and Usagi (1972). Examples of this application for flow along isothermal and uniformly heated plates in the thin-laminar-boundary-layer regime are given by Churchill and Ozoe (1973a, b).

A new application is now proposed. The dependence on distance of the asymptotic solutions for the local rate of convection for Pr + 0 and Pr + 03 for flow along a plate in the thin-laminar-boundary-layer regime, and hence their combination in the previously mentioned correlating equations for all Pr, is observed to be identical, suggesting that this commonal- ity may exist for other geometries. This relationship can be expressed as

Nub, Pr) = fib) - f*{Pr) (1 15)

with fi{x) provided by the known relationship

Nub, 0) = fib1 * f*{OI It follows that


For example, for convection to an isothermal disk, Eq (109) indicates that

3 j1I2 (118) NuD{R, Pr} - [ l + (1 - R2)l12]

N U ~ { O , Pr) - 2 ((1 - R 2 ) [ 2 + (1 - R2)lI2]

Equation (118) agrees very closely with the angular variation of the experimental data of Sparrow and Geiger (1985) for ShD , which they recognized to be independent of ReD over the range of their experiments, namely, 5000 5 ReD I 42,000 and Sc = 2.55.

Similarly, for an isothermal sphere, Eq. (86) can be expressed as

NuD{ 8, Pr} (1 + cos { 8)) (119) (2 + cts {J”

- N U ~ { O , Pr) - 2

which differs only 1.6% for 8 < 7r/4, but significantly for 8 > 7r/3, from following distribution calculated by Yuge (1956) for the thin-laminar- boundary-layer regime at Pr = 0.733 using an empirical velocity distribution:

NuD{O}/NuD{O} = 1 - 0.1558* - 0.053d4 (120) Again, for an isothermal cylinder, the distribution given by Eq. (33)

differs less than 2.8% for 8< 7r/4 but significantly for 8> 7r/3 from the following expression obtained by Frossling (1938) for convection in the thin-laminar-boundary-layer regime on an isothermal cylinder at Pr = 0.7 using a procedure similar to that of Yuge:

NuD{8}/NuD{O} = 1 - 0.13490’ - 0.039413~ (121) These examples imply that the distribution of the relative local Nusselt

number for potential flow provides a good approximation for all Pr in the thin-laminar-boundary-layer regime on flat surfaces and on spheres and cylinders for angles up to 7r/4. This approximation is very useful since theoretical solutions for the spatial variation of the Nusselt number are relatively easy to obtain for potential flow whereas this functionality requires extensive analysis and/or computation for any finite value of Pr.

XX. Summary and Conclusions

The method of Boussinesq provides a simple, systematic procedure for the derivation of solutions for convection from bodies immersed in an unbounded potential flow. This method was utilized herein to compile a set of solutions for many common shapes.


These solutions neglect conduction along the streamlines and the effects of transverse curvature, and thereby constitute asymptotic solutions for Pe --* m. Solutions taking into account these effects for particular conditions were used to evaluate the error due to their neglect.

Solutions for convection in potential flow were shown to overpredict significantly the Nusselt number for liquid metals and to have only limited applicability for bubbles.

The solutions for convection in potential flow are, however, shown to be very useful as asymptotes for Pr-0 in the development of correlating equations for all Pr, and also for prediction of the spatial variation of the local relative heat-transfer coefficient for any Pr.

NOMENCLATURE

A

a

b

arbitrary coefficient in Eq. (72) (m-')

half axis of elliptical cylinder or ellipsoid in the direction of unperturbed flow (m)

cylinder or ellipsoid normal to the direction of unperturbed flow, or half width of strip (m)

arbitrary constant in Eq. (52) (m-')

diameter of cylinder or sphere (m)

diameter of curvature of a spherical cap (m)

volume-equivalent diameter, (6 V/T)'/' (m)

diffusivity (m2/sec) eccentricity ratio = b/a for

an elliptical cylinder or ellipsoid; = I/w for a spherical cap

the second kind

kind

function

half axis of elliptical

complete elliptic integral of

elliptic integral of the first

Gauss hypergeometric

-F(m + 2, -m - 1 , 1 , (1 + cos{6)/21

-F{m + 3, -m,2, (1 + cOs{e1)/21

function defined by Eq. (89) acceleration due to gravity

(m/sec2) heat-transfer coefficient

(W/m2 K) integral defined by Eq. (27) unit vector normal to

surface in 4-4' coordinates

unit vector normal to surface in x-y coordinates

heat flux density on wall

heat flux density on wall in x-y coordinates (W/m2)

thermal conductivity

mass-transfer coefficient in terms of concentration

complete elliptic integral of the first kind

Bessel function of second kind and zero order

total length along plate or wedge (m)

height of spherical cap (m) index for cone, defined by

index for wedge, B/(T - 0)

(W/m2)

(W/m K)

(m/sec)

Eq. (73)


normal distance from surface (m)

Nusselt number based on characteristic length s, hs/k

PeD/(1+ Pea) parameter defined by

parameter defined by

Legendre polynomial of

Pkclet number based on

Eq. (101)

Eq. (106)

order m

u, characteristic length s, s u d a

P6clet number based on characteristic length s and local velocity along surface at s, s us/.

Prandtl number, v /a fractional distance from axis

of symmetry to the edge of a disk, r/a

distance normal to axis of symmetry (m)

Reynolds number based on characteristic length s,

function defined by Eq. (36) distance along surface from

point of incidence on a cylinder, or characteristic length (m)

S U=/ Y

Schmidt number, v / S Sherwood number based on

characteristic length s k’s l9

T Ui

u,

uor

V W

X

Y

Z

Z

Z ’

Greek Symbols

a thermal conductivity 9

HI, 61 incomplete p function m i complete r function 9 Y Euler’s

4

(m2/sec)

constant = 0.57722. . .

Eq. (69) (m-’) scale factor defined by

I dummy variable .i dummy variable

A

temperature (K) component of velocity in i

unperturbed, free-stream

local free-stream velocity

direction, (m/sec)

velocity (m/sec)

along surface of wedge or on a plane normal to the unperturbed flow (m/sec)

volume of bubble (m3) maximum width of spherical

distance in direction of unperturbed flow along surface of a wedge or cone, or normal to the minor axis of an elliptical cylinder (m)

direction of the unperturbed flow, normal to the surface of a wedge or cone, along the surface of a strip, or normal to the major axis of an elliptical cylinder (m)

distance along axis of an ellipse in the direction of unperturbed flow, or along axis of symmetry of an ellipsoid (m)

fractional distance along axis of symmetry of a prolate ellipsoid, z / a

fractional distance along axis of symmetry of an oblate ellipsoid, z/b

cap ( 4

distance normal to the

dummy variable, 1 - [ l - (y/b)2]’/2/2 in Eq. (49)

angle measured from forward point of incidence on a cylinder or sphere, or half angle of a wedge or cone (rad)

function defined by Eq. (92) (rad)


i

1 2a 26 D DV

L

n r

dynamic viscosity, cos{O}

kinematic viscosity (m2/sec) specific density (kg/m3) interfacial tension (N/m) dummy variable potential function for planar

(Pa sec)

flow as defined by Eqs. (3) and (4) (m)

potential function for axisymmetric flow as defined by Eqs. (60) and (61) (m)

modified potential function for axisymmetric flow as defined by Eq. (64) (m3)

stream function for planar flow as defined by Eqs. (3) and (4) (m)

axisymmetric flow as defined by Eqs. (60) and

function defined by Eq. (91) function defined by Eq. (90)

4’

4

* stream function for

(60) (m2)

(rad)

n 0

Superscript

mean value based on integral of heat-transfer coefficient over surface for uniform

temperature and on integral of temperature difference over surface for

wall a uniform heat flux density

Subscripts

at rear point of stagnation s based on major axis based on minor axis W

based on diameter X

based on volume-equivalent

based on diameter of Y

based on total length of object 30

diameter

curvature of spherical cap z 0

normal to surface in r direction

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based on characteristic

on wall in x direction or based on x

length s

as a characteristic d i m e n s i o n

in y direction in z direction in &direction unperturbed, free-stream

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Subject Index

A

Acceleration-pressure drop, slug flow modeling, 114

Accurate film profiles, slug flow calculations, 99- 100

Adiabatic heat-transfer coefficient, forced convection, three-dimensional arrays, 232-238

Air cooling, thermal control of electronic

Aluminum-aluminum foils, thermal contact

Anisotropic mass diffusion, natural

Annular flow, steady-state slugs, 85-87 Aperiodic regenerator operation,

Artificial cavities, thermal cooling, boiling

Asymmetric-balanced regenerator, 137-l38 unidirectional operation, 157-158

Asymmetric-unbalanced regenerator,

unidirectional operation, 157-158 Auxiliary variables, slug flow, 95-97 Axisymmetric flow, potential flow

components, 182-183

conductance, 264

convection, 325-327

unidirectional regenerators, 154-158

curve, 247-248

137-138

convection, 365-366 isothermal spheroids of revolution,

planar flow, 369 372-374

B

Baker’s boiling curves, immersion cooling techniques, 245

Balanced-symmetric regenerator unidirectional operation, 173-174 three-dimensional effectiveness plot, 175

Balance equation, “boxed” regenerator,

Bellows heat pipe, indirect cooling with,

Benard-type convection, heat- and mass-

144-146

275-277

transfer, natural convection onset, 318-319

Bernoulli equation, Taylor bubble translational velocity, 101-102

Bessel functions, unidirectional regenerator operation, 152-153

Birnbrier data, vertical plates and channels, 196

Blasius correlation, hydrodynamics, slug flow, 89

Blowout process, severe slugging, 120-121 Blunt needle of finite length, potential flow

Boe’s criterion, severe slugging, 121-122 convection, 372-373

classical vs. quasi-equilibrium operations, 128

Boiling region diagram, narrow space

Boundary conditions nucleate boiling, 47-49

heat- and mass-transfer horizontal direction, 331-332 natural convection, 322-323

potential flow convection, planar motion, 356

Boundary-layer flows, heat- and mass- transfer, horizontal direction, 332-336

Boussinesq transformation, potential flow convection

389

390 SUBJECT INDEX

axisymmetric potential flow, 365-366 planar motion, 354

narrow space nucleate boiling, 36-40 Bubble behavior

coalesced bubble region, 36,38-39,

isolated bubble region, 36,40-41

agitation model, 3-4 liquid and vapor profiles, 33-34 liquid film thickness, 33-34 surface configuration, 23-27

42-49

nucleate boiling, 2

Bubble population density, nucleate boiling,

Bubble velocities 5-8,l3-15

dispersed in liquid slug, 107-108 surface tensions parameters, 104-105

forced convection, augmented heat

heat- and mass-transfer

Buoyancy effects

transfer, 226

boundary-layer profiles, 333-335 convection suppression, 339 enclosed porous layers, 337-339 horizontal direction, 333-334 horizontal line source, 346-347 multilayer structures, 347 vertical direction, 318-319 wall inclination, 336

Buoyancy force, nucleate boiling, 2-3 Buoyancy-induced flow and heat transfer

asymptotic limiting relations, 193, 195 vertical plates and channels, 185-187

Buoyancy ratio, heat- and mass-transfer, 320-321

C

Calculation procedures, slug flow, 97-100 Channel flow studies

forced convection flush heat sources, 221-226 two-dimensional protruding elements,

227-232 Channel spacing, forced convection,

Chebyshev integration, unidirectional

Chip-in-cavity electronic packages, forced

230-231

regenerator operation, 154

convection, 221

Churn turbulent region, bubble velocity, 108 Circuit card channels, forced convection,

Circular cylinder, potential flow convection, 242-243

359-360 isothermal cylinder, 359-360 uniformly heated cylinder, 360-361

Clearance, narrow space nucleate boiling, 37-39

coalesced bubble region, 41-49 isolated bubble region, 40-41 pressure and heat flux, 47-79

heat flux, emission frequency and, 49,52 low liquid level nucleate boiling, 56-57 narrow space nucleate boiling, 36,38-39,

Coalesced bubble region

42-49 emission frequency, 43-45

one-dimensional model, 49-54 unsteady heat conduction model, 49,53

Conductive greases, thermal contact

Cones, potential flow convection, 367-368 Conical cavity, augmented nucleate boiling,

Conjugate heat transfer, forced convection,

Constant-film-thickness, slug flow

Constant-flux expressions, heat- and mass-

Constants and exponents, nucleate boiling,

Convection. see Natural convection Copper foils, enhancement of thermal

contact conductance, 264 Copper surface

augmented nucleate boiling,

conductance, 263-264

64-67

222-223

calculations, 99

transfer, 341

9-11

70-72 wettability characteristics, 76-77

Correlating equations heat transfer generalization, 11-13 nucleate boiling formulas, 19-20 potential flow convection, 382-383

direct cooling with heat pipes, 290 thermal cooling of electronic components

heat transfer characteristics, 250-251 high flux surfaces, 249-250 macrogeometery, 247-248 narrow spaces, 248

Critical heat flux (CHF)

SUBJECT INDEX 39 1

Cubical element array, forced convection hydrodynamic data, 233-234 Nusselt number, 233,235,237

Cyclic equilibrium, regenerator theory,

Cyclic steady state (equilibrium), “boxed” 135-136

regenerator, 147-148

inclined case, 102- 103 surface tension parameter variation,

103-104 Driving force, nucleate boiling, 2-3 Dryout phenomena, narrow space nucleate

boiling, 39-40

E D

Darcy flow model, heat- and mass-transfer,

Darcy’s constant 325-327

heat- and mass-transfer natural convection, 317 vertical direction, 320-321

resistance, 260-261

pipes, 289-290

Diebond material thickness, thermal contact

Dielectric fluids, direct cooling with heat

Differential equations, hydrodynamics, slug

Dimensionless bubble propagation velocity,

Dimensionless parameters, “boxed” regenerator, 146

Direct cooling techniques heat pipes, 289-295 thermal control of electronic components,

flow, 91-92

103- 104

182- 183 Discrete thermal sources

forced convection, 221-226 liquid cooling, 213-217 natural convection

in enclosures, 217-220 multiple flush-mounted sources, 220 vertical surfaces, 202-209

cubical elements, 205-206 heat-transfer coefficient variations,

Nusselt numbers, 204-205,207-208 surface temperature variation,

205-206,208-209

203-204 Dispersed bubble velocity, slug flow, 95-97 Downward-facing surface

heat transfer models, 28-36 liquid and vapor periods, 29-31

dimensionless, 103-104 horizontal slug flow, 101-102

Drift velocity, Taylor bubbles, 100-101

Eddy-diffusivity model, forced convection,

Edge-cooled heat-pipe, 286-287 Effectiveness concept, “boxed” regenerator,

Ellipsoid, potential flow convection,

Elliptical cylinder, potential flow convection,

Elongated bubbles, see Taylor bubbles Emission frequency, narrow space nucleate

224

145-146

373-374

361-362

boiling clearance, 43-45 coalesced bubble region, 49,51 equivalent heat source, 54-55

Enclosed porous layers, heat- and mass-

Enclosures, natural convection, 217-220 Enthalpy changes, “boxed” regenerator,

transfer, 336-341

147- 148

F

Fiber wick heat pipe, vs. nonwicked

Film length, slug flow calculations, 98 Film zone, geometry of, 85-87 Finite-amplitude convection, heat- and mass-

Finite-difference methodology, 229-230 Finite height strips, potential flow

Finned heat pipes, indirect cooling

Fixed-bed regenerator

transistor, 291-293

transfer, 327

convection, 362-363

techniques, 280-281

“boxed” process, 139-149 dimensionless parameters, 133- 135 equivalence with rotary bed regenerator,

mathematical representation, 133 U6-137

392 SUBJECT INDEX

Flat heat pipes, indirect cooling techniques,

Flat-pack modules, forced convection,

Flat-plate heat pipe, 287-289 Foil thickness, thermal contact conductance,

Forced convection, 220-243

278-280

236-237

266

discrete flush heat sources - channel flow,

three-dimensional package arrays,

actual circuit cards, 242-243 fully populated arrays, 232-238 missing modules or height differences,

220-226

232-243

238-241 two-dimensional protruding elements,

Forchheimer’s constant, heat- and mass-

Fourier number

227-232

transfer, 317

coalesced bubble region, 54 narrow space nucleate boiling, 49

Fourier transform, heat- and mass-transfer,

Free convection 327

electronic packages, 243 nucleate boiling and, 2-4

single-phase conditions, 254-257 thermal cooling of electronic components,

Free jet impingement

251-252 Free-rise velocity, 107-108 Free-stream velocity, 376-377 Freons

nucleate boiling formulas, 18-19 values for, 22-23

operation, 150 Frequency factor, unidirectional regenerator

Friction factors, slug flow calculations, 98, 114-115

G

Galerkin method, heat- and mass-transfer natural convection, 324-325 property variations, 326

Gas-liquid interface, mass balance equation, 87

Gas-liquid slug flow, velocity profiles,

Global force balance, slug pressure drop,

Global momentum balance slug flow modeling, 114 slug pressure drop, 94-95

112-113

92-95

Gold foil, thermal contact conductance,

Grashof number 264-265

natural convection cooling, 203-204 nucleate boiling, 2-4

H

Hardness ratio, thermal contact

Harsh environments, heat pipe thermal

Heat- and-mass transfer natural convection

conductance, 266

control, 297

horizontal direction, 330-343 boundary-layer flows, 332-336 convection onset, 330-332 enclosed porous layers, 336-341 transient approach to equilibrium,

341-343 horizontal line source, 346-347 physical model, 316-318 point sources, 343-346 vertical direction, 318-330

finite-amplitude convection, 327 high Rayleigh number convection,

nonlinear initial profiles, 323-325 onset of convection, 318-323 soret diffusion, 328-329

329-330

Heat-exchanger system “boxed” regenerators

cyclic steady state, 147-148 Nusselt assumptions and governing

equations, 141-144 overall balance equation, 144-146 overall heat-transfer coefficient,

weak and strong periods, 140-141 148-149

Heat flux circuit integration, 181 coalesced bubble region, 49,52

SUBJECT INDEX 393

heat-transfer coefficient and, 77-78 low liquid level nucleate boiling, 56-57 narrow space nucleate boiling

clearance and pressure, 47-79 emission frequency, 43-45 isolated bubble region, 40-41

nucleate boiling, 5-8 Heat kicker, indirect cooling techniques,

Heat pipes 277-278

indirect cooling techniques bellows concept, 275-277 cylindrical structure, 274 finned heat pipes, 280-281 flat pipes, 278-280 flat-plate heat pipe, 287-289 heat kicker, 277-278 high-capacity semiconductors, 278 micro-heat-pipe concept, 283-285 multiple array cooling, 285-286 printed wiring boards, 286-287 self-regulating evaporative-conductive

single- and double-pipe configurations,

thermal switch, 274-275 trapezoidal configuration, 283-284

thermal control, 272-299 direct cooling, 289-295 indirect cooling, 273-289 operation schematic, 272 system-level control, 295-297

Link, 283

281-283

Heat transfer augmented nucleate boiling, surface

configuration, 70-77 generalized correlation, 11-13 natural convection cooling

simplified formula for, 22-23

low liquid level nucleate boiling, 55-58

vertical plates and channels, 185-187

slug flow, 84

in liquid film, 58-63 Heat-transfer coefficient

“boxed” regenerator design, 148- 149 coalesced bubble region, 49-50 forced convection

flat-pack modules, 236-237 high-heat-flux source, 225-226 missing module effect, 238-239 three-dimensional arrays, 232-238

turbulent flow models, 238 two-dimensional protruding elements,

227-228 heat pipe cooling techniques

grooved design, 278 vapor temperature, 277-278

jet impingement cooling, 251-252 inclination angles, results of analysis,

low liquid level nucleate boiling, 56-57

narrow space nucleate boiling

35-36

surface configurations, 61-62

coalesced bubble region, 43-49 isolated bubble region, 40-41 pressure and, 46-47

immersion cooling, 213-217 irregular surfaces, 208-210 two-dimensional protrusion, 211-212 vertical surfaces, 205-206,205-209

isothermal channel wall, 208-209

natural convection

nucleate boiling liquid levels, 7-8 pressure factor, 17-18 surface conditions, 2 temperature difference, 6-7 thermal boundary layer thickness, 9-10

pressure dependence, 20-21 surface inclination, 27-28 See also Adiabatic heat-transfer coefficient

Heat-transfer-enhancing fences, forced

Heat transfer models, downward-facing

Heaviside step function, 159-160 Height differences, forced convection,

High heat regions, surface inclination, 27-28 High-Rayleigh number convection, 329-330 Horizontal circuit boards, natural

convection, 220 Horizontal line source, heat- and mass-

transfer, 346-347 Hydrodynamic behavior, submerged jet

impingement cooling, 252-253 Hydrodynamic parameters

convection, 240-241

surface, 28-36

238-241

forced convection, cubical element array,

slug flow modeling, 114 liquid film, 88-92

233-234

394 SUBJECT INDEX

Hydrogen-bubble technique, velocity

Hydrostatic pressure, quasi-equilibrium

thin disk of finite diameter, 373-374 profiles, 112-113 transverse curvature, 370-371

slugging, 125 wedge, 363-364 spherical cap, 374

I

Immersion cooling natural convection, 213-217 thermal control of electronic components,

243-251 Indirect cooling techniques

heat pipes, 273-289 thermal control of electronic components,

182-183 Indium foils, enhancement of, thermal

Inert liquids, natural convection cooling,

Infinite long pointed needle, potential flow

Interfacial friction factor, hydrodynamics,

Interfacial instability, severe slugging, 116 Interfacial shear, hydrodynamics, slug flow,

90 Inviscid theory, Taylor bubble translational

velocity, 101-102 Irregular surfaces, natural convection

cooling, 209-213 Isolated bubble region narrow space

nucleate boiling, 36,40-41 Isothermal cylinder, potential flow

convection, 359-360 Isothermal surfaces

potential flow convection conic, 367-368 elliptical cylinders, 361-362 finite height strips, 362-363 flat plate streamline conduction,

infinitely long pointed needle, 368-369 planar motion, 355-356 spherical, 369-370

373

372

contact conductance, 264

184-185

convection, 368-369

Slug flow, 89-90

357-358

oblate axisymrnetric potential flow,

prolate axisymmetric potential flow,

J

Jet impingement boiling high-density integrated circuits, 257-258 vs. pool boiling, 252-253 study summaries, 254 thermal cooling of electronic components,

251-258 free jet impingement, 251-252 submerged jet impingement, 252-258

K

Kronecker delta, heat- and mass-transfer, 324-325

L

Laminar natural convection irregular surfaces, 209-213 Taylor bubble velocity, 105

balanced-symmetric regenerators, 173-175 unidirectional regenerator operation,

Laplace transform

151-152 Latent-heat transport, heat transfer models,

Lewis number 28,31-33

heat- and mass-transfer enclosed porous layer, 339-341 high Rayleigh number convection,

horizontal direction, 332 point source, 344-345 time-dependent concentration, 342-343 vertical direction, 320-322

transfer, 319

3 2 8 - 3 2 9

Linear stability analysis, heat- and mass-

Liquid cooling, discrete thermal sources, 213-217

SUBJECT INDEX 395

Liquid deficient region, narrow space

Liquid fallback, severe slugging, H8-119 Liquid film characteristics

nucleate boiling, 39

hydrodynamics, slug flow, 88-92 low liquid level nucleate boiling, 58-63 thickness, heat transfer models, 33-35

Liquid metal convection, 376-379 Liquid periods, heat transfer models, 29-31 Liquid slug

bubble velocity, 107-108 liquid holdup in, 109-110 length and slug flow, 95-97 zone

geometry, 85-86 slug pressure drop, 92-95

slugging, 125-126 Local liquid holdup, quasi-equilibrium

Long regenerator recuperative operation, 155, 157 unidirectional operation

temperature distribution - unbalanced

three-dimensional plot, 169-170 vanishing transfer potential, 158-170

Low heat regions, surface inclination, 27-28 Low liquid level nucleate boiling, 55-63

heat-transfer characteristics, 55-58 in liquid film, 58-63

regenerator, 161- 170

M

Magnesium-magnesium junctions, enhancement of, 264

Mass balances, slug flow, 86-87 Mechanical dispersion, heat- and mass-

Metallic coatings transfer, 325-327

thermal contact conductance enhancement, 263,266-270 oxide formation, 270-271

Metallic foils, thermal contact conductance,

Micro-heat-pipe concept, 283-285 trapezoidal configuration, 283-285

Missing modules, forced convection,

Mixed force convection, electronic packages,

263-266

238-241

243

Mixture velocity, Taylor bubbles, 104-105 Momentum balance

heat- and mass-transfer, natural convection, 317-318

severe slugging, 120-121 slug pressure drop, 94-95

N

Napthalene sublimation technique, 237-238 Narrow space nucleate boiling, 36-55

bubble behavior, 36-40 clearance factor, 37-39 coalesced bubble region, 42-49 heat transfer characteristics, 40-49

isolated bubble region, 40-41 thermal cooling of electronic components,

248 Natural convection

cooling of electronic components, 183-220 discrete sources - in enclosures, 217-220 discrete sources - liquid cooling, 213-217 discrete thermal sources - vertical

surfaces, 202-209 irregular surfaces, 209-213 vertical plates and channels, 185-202

vs. actual circuit boards, 195-198 convection-radiation interaction,

optimum spacing, 202 staggered arrays, 198-201

heat- and mass-transfer horizontal direction, 330-343

boundary layer flows, 332-336 convection onset, 330-332 enclosed porous layers, 336-341 line source, 346-347 transient approach to equilibrium,

201-202

341-343 physical model, 316-318 point source, 343-346 vertical direction, 318-330

convection onset, 318-323 finite-amplitude convection, 327 high Rayleigh number convection,

nonlinear initial profiles, 323-325 soret diffusion, 328-329

329-330

396 SUBJECT INDEX

Newton-Raphson method, slug flow

Nield’s analysis, heat- and mass-transfer,

Nondimensional correlating equations

evaluation of constants, 9-11

transfer, 323-325

augmentation, 63-77

calculations, 98

318-319

nucleate boiling, 2-18

Nonlinear initial profiles, heat- and mass-

Nucleate boiling

prepared surface heat characteristics,

vapor nucleus stability, 64-70 70-77

bubble behavior, 2 characteristics, 70-77 direct cooling with heat pipes, 290-291 heat transfer coefficient

organic fluids, 1-2 surface conditions, 1-2 thermal property similarities, 18-23

jet impingement vs. pool boiling, 252-253 low liquid levels, 55-63

heat transfer characteristics, 55-58 liquid film heat transfer characteristics,

58-63 narrow space, 36-55

bubble behavior, 36-40 heat transfer characteristics, 40-49 model characteristics, 49-55

nondimensional correlating equation, 2-18

constant evaluation, 9-11 elementary process formulation, 5-8 free convection, 2-4 generalized correlation of heat transfer,

nucleation factor, U-15 pressure factor, 15-18

surface configuration, 23-36 boiling curves and bubble behavior,

heat transfer model - downward-facing

latent-heat transport, 31-33 liquid and vapor periods, 33 liquid film thickness, 33-35 results analysis, 35-36 sensible heat transport, 29-31 surface inclination, 27-28

11-U

23-27

surface, 28-36

thermal control of electronic components,

heat transfer characteristics, 250-251 heater widths, 246 high flux spaces, 249-250 narrow space, 248 surface roughness, 246-247

243-258

Nucleation factor correlation equations, 13-15

estimated values, 12, 14 heat transfer formula, 22-23 surface factor, 15 values, 11-12

Nusselt number augmented nucleate boiling

surface configuration, 73-74 coalesced bubble region, 53-54 forced convection

cubical element array, 233,235,237 flat-pack modules, 236-237 flush heat sources, 222-223

computed local number, 223-224 heat-transfer results - high-heat-flux

measured (square source) and source, 225-226

predicted (two-dimensional) source, 224-225

napthalene sublimation technique,

small rib spacing, 228-229 two-dimensional protruding elements,

237-238

227-228 heat- and mass-transfer

enclosed porous layer, 340-341 finite-amplitude convection, 327 high Rayleigh number convection,

narrow space nucleate boiling, 49-50 natural convection

328-329

enclosure sources, 218-220 immersion cooling, 217 irregular surfaces, 210-211 vertical channels, 187-191,203-205,

asymptotic limiting relations, 191, 193,195

nucleate boiling, 4 potential flow convection

207-208

correlating equation development, 382-383

SUBJECT INDEX 397

isothermal sphere, 371 liquid metal convection, 378-379 various shapes, 374-376 wedges and spheroids, 375

regenerator theory, U8-139 governing equations, 141-144

submerged jet impingement systems,

unidirectional regenerator operation

vertical plates and channels

255-256

numerical solution, 150-151

vs. actual printed circuit boards, 195-196, 198

Nusselt’s IV model, regenerator design equivalence, U6-137

0

Oblate (planetary) spheroid, potential flow convection, 373

Open bath test chamber, thermal cooling of electronic components, 250-251

Optimum thickness, thermal contact conductance, 271

Oxide formation, thermal contact conductance enhancement, metallic coatings, 270-271

P

Parl’s calculation method, 152-153 Peclet number, forced convection,

Periodic (cosine) profile, heat- and mass-

Periodic flow theory, regenerator theory,

Perturbation solutions, potential flow

Planar impingement, potential flow

Planar motion, potential flow convection

222-223

transfer, 324-325

138-139

convection, 371

convection, 364-365

conduction along streamlines, 357 general formulation, 354 negligible conduction along streamlines,

355-356 Point source, heat- and mass-transfer,

343-346

Porous-medium systems heat- and mass-transfer

natural convection, 325-327 physical model, 316-318

Porous surfaces, augmented nucleate boiling, 72-77

Potential flow convection axisymmetric flow

general formulation, 365-367 planar impingement, 369

isothermal cylinder, 359-360 uniformly heated cylinder, 360-361

circular cylinder, 359-361

cone, 367-368 derivations and solutions, 354-357

planar motion, 354 planar motion - streamline conduction,

355-357 elliptical cylinder, 361-362

finite height strip - front face, 362-363 isothermal surface, 361-362 uniformly heated surface, 362

isothermal plate - streamline conduction, 357-358

uniformly heated plate - streamline conduction, 358-359

infinitely long pointed needle, 368-369 mean Nusselt number for various shapes,

flat plate, 357-359

374-376 summary of solutions, 374-376 wedges and spheroids, 375

planar flow, normal plane impingement,

research background, 353-354 solution applicability, 376-382

364-365

correlating equation development,

liquid metals convection, 376-379 rising bubble convection, 379-382

axially symmetric convection, 372-374 isothermal cap, 374 transverse curvature, 370-371

Power semiconductor coolers, 280-281 Power spectral density, slug velocity, 113 Prandtl number

382-383

spherical, 369-370

wedge, 363-364

nucleate boiling, 4 narrow space, 49-50

398 SUBJECT INDEX

potential flow convection, 353 correlating equation development,

liquid metal convection, 376,378-379 382-383

submerged jet impingement systems,

Preburnout region, narrow space nucleate

Pressure

255-256

boiling, 43-45

augmented nucleate boiling, 68-70 heat-transfer coefficient and, 77-78 narrow space nucleate boiling

clearance and heat flux, 47-79 heat-transfer coefficient, 46-47

nucleate boiling formulas, 15-18,20-21 Pressure drop

Slug flow, 92-95,114 Taylor bubbles, gas pocket propagation,

102-103 Printed wiring board (PWB)

forced convection, 221 heat pipe cooling techniques, 286-287

Probability density function, slug velocity,

Prolate (ovary) spheroid, potential flow

Protoslugs, defined, 85-86

113

convection, 372

Q

Quasi-equilibrium severe slugging, 124-128

R

Radiation, natural convection cooling, 201-202

Rayleigh number heat- and mass-transfer

finite-amplitude convection, 327 high convection model, 329-330 horizontal direction, 332 point source, 343-346 time-dependent concentration, 342-343 vertical direction, 320-321

enclosure sources, 219-220 two-dimensional protrusion, 211-212

natural convection

Recuperative operation, unidirectional

Reentrant cavity, augmented nucleate

Regenerator theory

regenerators, 154-158

boiling, 67-70

“boxed” process, 139-149 cyclic steady state (equilibrium),147-148

governing equations - Nusselt’s assumptions, 141-144

overall balance equation, 144-146 overall heat-transfer coefficient,

weak and strong periods, 140-141 classical model assumptions, l38-139 cyclic equilibrium and thermal

effectiveness, 135- 136 operation classification, 137-138 regenerator parameters, 133-135 rotary system-fixed bed equivalents,

unidirectional operation, 149-176

148-149

136-137

aperiodic or recuperative operation,

balanced-symmetric regenerator,

completely unbalanced regenerators,

general case solution, 151-154 long regenerator, 158-170 unbalanced-symmetric regenerator, 174,

Regression analysis, low liquid level nucleate

Reynolds number

154-158

173-175

170-173

176-177

boiling, 56-58

augmented nucleate boiling, surface

bubble population density, 7-8 forced convection

small rib spacing, 228-229 surface fences (turbulators), 239-240 three-dimensional arrays, 238

configuration, 74

slug flow calculations, 98 submerged jet impingement systems,

Ribbed-duct geometry, forced convection,

Rising bubble convection, 379-382

255-256

229-231

bubble mobility, 379-380 bubble shape, 379-381

SUBJECT INDEX 399

Rotary-matrix exchanger, mathematical

Rotar y-matrix regenerators, “boxed”

Rotary regenerator

representation, 133

process, 139-149

dimensionless parameters, 134-135 equivalence with fixed-bed regenerator,

136-137

S

Sandwich-type heat pipe, cooling

Semiconductor chip techniques, 285-286

heat pipe cooling, 272-274 vs. conventionally cooled chips,

290-293 Sensible-heat transport, 28-31 Severe slugging

Boe’s criterion, 121-122 cycle for, 116-121 quasi-equilibrium, 124-128 stability criterion, 122-124 transient phenomena, 115-116

Shear stress, slug flow, 89 Sherwood number

forced convection, interbarrier spacing modules, 240-241

heat- and mass-transfer enclosed porous layer, 340-341 finite-amplitude convection, 327 high Rayleigh number convection, 329

Single-phase impingement heat transfer,

Sintered surfaces 254-255

augmented nucleate boiling, 70-71 optimum thickness, 72-73

defined, 83-84 geometry, 85-86 hydrodynamics, 84 separator movement, 116-117 severe slugging, 115-129

Boe’s criterion, El-122 cycle dynamics, 116-121 quasi-eliquibrium, 124-128 stability criterion, 122-124 transient phenomena, 115-116

Slug flow

steady, 84 steady-state modeling, 85-115

auxiliary relations, 95-97 average void fraction, 88 calculation procedures, 97-100 dispersed bubble velocities, 107-108 liquid film hydrodynamics, 88-92 liquid holdup, 109-110 mass balances, 86-87 pressure drop, 92-95 slug length and frequency, 110-113 Taylor bubble translational velocities,

100-107 terrain-induced, 84 transient slugging, 84, 115-116 two-phase, 83-129

Slug frequency and slug length, 110-113 Solid-liquid interface, augmented nucleate

Solid-vapor interface, augmented nucleate

Solute dispersion, heat- and mass-transfer,

Soret diffusion, heat- and mass-transfer,

Sphere, potential flow convection, 369-370 Spherical cap, potential flow convection, 374 Spheroids, potential flow convection, 375 Stability criterion

boiling, 64-67

boiling, 64-67

325

328-329

severe slugging, 122-124 classical vs. quad-equilibrium

operations, 128 Staggered vertical platekard arrays, natural

convection cooling, 198-201 heat dissipation ratio, 199-200 temperature reduction, 199-200

Steady-state flow field, heat- and mass-

Steady-state slug flow modeling, 85-115 transfer, 345-346

auxiliary relations, 95-97 calculation procedures, 97-100 dispersed bubble velocities, 107-108 liquid film hydrodynamics, 88-92 liquid holdup, 109-110 mass balances, 86-87 pressure drop, 92-95 slug length and frequency, 110-113 Taylor bubble translational velocities,

void fractions, 88 100-107

400 SUBJECT INDEX

Step-function profile, heat- and mass-

Stirring force, nucleate boiling, 3 Stokes law, bubble velocity, 107-108 Stokes stream and potential functions,

Streamline conduction, potential Row

transfer, 324-325

365-366

convection Rat isothermal plate, 357-358 Bat uniformly heated plate, 358-359 planar motion, 355-357

Submerged jet impingement, thermal cooling of electronic components, 252-258

Surface configuration augmented nucleate boiling

experimental apparatus, 24 heat flux and boiling curve, 24 low liquid level nucleate boiling, 59-60 natural convection cooling, 209-213 nucleate boiling

heat-transfer characteristics, 70-77

boiling curves and bubble behavior,

downward-facing surface model, 28-36 heat transfer, 2 inclination angles, 28-29 liquid and vapor profiles, 33-35 liquid film thickness, 33-35 pressure dependence, 20-21 surface inclination, 27-28 thermal cooling of electronic

components, 246-247

23-27

thermal contact conductance, metal foil

Surface curves, nucleate boiling and, 23-36 Surface factor and nucleation factor, 15 Surface fences (turbulators), 239-240 Surface inclination

enhancement, 265-266

boiling heat transfer, 24-25 angle variations, 25-27

heat-transfer coefficient, 27-28

recuperative operation, 155-156 unidirectional operation, 150

Symmetric-unbalanced regenerators,

recuperative operation, 157-158 unidirectional operation, 150

Symmetric-balanced regenerators, U7-138

l37-l38

System-level thermal control, heat pipes, 295-297

harsh environments, 297 sealed cabinet electronic equipment,

296-297

T

Taylor bubbles defined, 83,85-86 severe slugging, blowout, 120-121 translational velocities, 100-107

as function of heat flux, 184-185 gradients, natural convection, 347-348 low liquid level nucleate boiling, 59-60 nucleate boiling formulas, 18-23 thermal control of electronic components,

unidirectional operation, long regenerators, 160- 161

Temperature

182-183

Temperature overshoot, immersion cooling

Terminal equilibrium level, slug Row, 97-98 Terrain-induced slugging. see Severe

slugging Thermal boundary layer thickness, nucleate boiling, 9-10

techniques, 245-246

Thermal contact conductance defined, 259-260 enhancement of, 262-271

conductive greases, 263-264 metallic foils, 263-266

interface, 262-263 mold compound and substrate-spreader

Thermal contact resistance chip-bond and bond-aluminum interfaces,

chip design, 259-261 diebond material thickness, 260-261 electronic component cooling, 258-271

Thermal control - electronic components boiling, 243-244 forced convection, 220-243

260-261

discrete Rush heat sources - channel

three-dimensional package arrays, flow, 220-226

232-243 actual circuit card arrays, 242-243 fully populated arrays, 232-238 missing modules or height

differences, 238-241

SUBJECT INDEX 40 1

two-dimensional protuding elements - channel flow, 227-232

heat pipes, 272-299 direct cooling, 289-295 indirect cooling, 273-289 system-level thermal control,

295-297 immersion cooling, 244-251 jet impingement cooling, 251-258 natural convection, 183-220

discrete sources - enclosures, 217-220 discrete sources - liquid cooling,

discrete thermal sources - vertical surfaces, 202-209

irregular surfaces, 209-213 vertical plates and channels, 185-202

vs. actual electronic circuit boards,

interacting convection and radiation,

optimum spacing, 202 staggered plate or card arrays,

213-217

195-198

201-202

198-201 theoretical background, 181-183 thermal contact resistance, 258-271

conductance enhancement, 262-271 Thermal dispersion, natural convection, 325 Thermal effectiveness, regenerator theory,

Thermal energy conservation, augmented nucleate boiling and, 78-79

Thermal properties, thermodynamic similarity, 18-23

Thermoexcel, augmented nucleate boiling, 74-77

Thermohaline convection, heat- and mass- transfer

135-136

porous model, 326 Soret diffusion, 328-329 stability chart, 322

slug flow calculations, 98 slug pressure drop, 95

Thickness of film

Thin disk of finite diameter, axisymmetric

Thin film potential flow, 373-374

evaporation, direct cooling with heat

narrow space nucleate boiling, bubble pipes, 293-294

behavior, 37-38

Three-dimensional package arrays, forced convection, 232-243

actual circuit cards, 242-243 fully populated arrays, 232-238 missing modules or height differences,

238-241 Thyristor

heat-pipe cooling of, 281,283 thermal control, 181

Tin-coated stainless-steel, thermal contact conductance enhancement, 268-270

Tin-nickel alloy, thermal contact conductance enhancement, 268

Total package thermal resistance, direct cooling with heat pipes, 293-294

Transient equilibrium approach, heat- and mass-transfer, 341-343

Transient flow field, heat- and mass-transfer,

Translational velocity 344-346

slip parameters, 105-106

Taylor bubbles, 100-107 slug flow, 95-97

liquid slug dispersion, 107-108

convection, 370-371

operation, 154

Transverse curvature, potential flow

Trapezoidal rule, unidirectional regenerator

Trapezoid heat pipe configuration,

Turbulators, forced convection, 238-241 Turbulent flow, Taylor bubble velocity, 105 'ho-dimensional model, heat- and mass-

'ho-phase slug flow characteristics, 83-84

283-284

transfer, 329-330

U

UC High Flux, augmented nucleate boiling,

Unbalanced regenerators, temperature

Unbalanced-symmetric regenerator

effectiveness plot, 176-177

75-77

distribution, 161- 170

unidirectional operation, 174, 176-177

Unbalanced unidirectional regenerator, 170-173

dimensionless parameters, 171 effectiveness charts, 170-172

402 SUBJECT INDEX

Unidirectional regenerator operation,

aperiodic or recuperative operation,

balanced-symmetric regenerator,

completely unbalanced regenerator,

long regenerator-vanishing transfer

unbalanced-symmetric regenerator, 174,

149-176

154-158

173-175

170- 173

potential, 158-170

176-177 UNIDRE double precision function, 154 Uniform heated surfaces, potential flow

convection, 358-359 conic, 368 cylinder, 360-361 elliptical cylinders, 362 finite height strips, 363 infinitely long pointed needle, 369 planar motion, 356 spherical, 370 wedge, 364

Utilization factor, unidirectional operation, 161-164

V

Vanishing transfer potential, unidirectional

Vapor dome, low liquid level nucleate

Vapor-liquid interface, augmented nucleate

Vapor nucleus, augmented nucleate boiling

operation, 158-170

boiling, 61-63

boiling, 64-67

conical cavity, 64-67 pressure and stability, 68-70 reentrant cavity, 67-70 stability, 64-70 variation in vapor pressure, 65-67

Vapor periods, heat transfer models, 29-31 Vapor space condenser, direct cooling with

Velocity field heat pipes, 289-290

natural convection, 316-317 potential flow convection

liquid metal convection, 376-377 rising bubble convection, 379-382

Velocity profiles, liquid slugs, 111-113 Vertical plates and channels, natural

convection cooling, 185-202 buoyancy-induced flow and heat transfer,

card-on-board configuration, 185, 187 comparison with electronic circuit boards,

composite relations, 194-195 convection-radiation interaction,

experimental prototypes, 196-197 isothermal wall flow rates, 191-192 Nusselt number, 189-191 optimum spacing, 194-195,202 Rayleigh number, 189-191 staggered arrays, 198-201 symmetric-asymmetric nomenclature,

uniform flux flow rates, 191-192 uniform heat flux (UHF) boundary

conditions, 188, 191 uniform wall temperature (UWT)

boundary conditions, 188, 191 velocity and temperature fields, 189-190

Viscosity variations, heat- and mass-transfer,

Void fraction

185-187

195- 198

201-202

187-188

326-327

liquid slug zone, 109-110 slug units, 88 surface inclination, heat transport models,

32-33 Volumetric concentration expansion

Volumetric thermal-expansion coefficient, coefficient, 318

318

W

Wall inclination, heat- and mass-transfer, 336 Wallis correlation, hydrodynamics, slug flow,

Wedge, potential flow convection, 363-364

Wick design

90

Nusselt number, 375

direct cooling with heat pipes, 290-293 fluid-vapor transport and heat transfer,

292-293

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