a"? NASA CR-54352 mPWA-2530 TOPICAL REPORT ANALYTICAL STUDY OF LIQUID METAL CONDENSING INSIDE TUBES prepared for National Aeronautics and Space Administration January 1965 Contract NAS3 -2335 T echnical Management NASA Lewis Research Center Space Power System Division Nuclear Power Technology Branch Cleveland, Ohio Martin Gutstein Written b .R. Kunz, Project Engineer Approved S. Wyde, Program Manager .J. Lueckel, Chief, Space Power Systems E A S T HARTFORD 0 CONNECTICUT https://ntrs.nasa.gov/search.jsp?R=19650014412 2018-06-02T12:45:59+00:00Z
Transcript
a"? NASA CR-54352 mPWA-2530
TOPICAL REPORT
ANALYTICAL STUDY O F LIQUID METAL CONDENSING INSIDE TUBES
prepared for National Aeronautics and Space Administration
January 1965
Contract NAS3 -2335
T ec hnical Management NASA Lewis Research Center Space Power System Division
Nuclear Power Technology Branch Cleveland, Ohio Martin Gutstein
This report was prepared by the Prat t & Whitney Aircraft Division of United Aircraft Corporation, East Hartford, Connecticut, to describe the work conducted from May 15 to December 1, 1964 in fulfillment of Task IV of Contract NAS3 -2335, Experimental Investigation of Transients in Simulated Space Rankine Powerplants, Amendment 4. The report summarizes an analytical study of condensing flow inside tubes. Some of the material generated in this study is being used by the author as par t of a doctoral thesis at Rensselaer Polytechnic Institute in Troy, New York.
The author wishes to acknowledge assistance provided by S.S. Wyde, H. L. Hess, and H. L. Ornstein in computer programming and report editing.
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Foreword Table of Contents Lis t of Figures
~
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TABLE O F CONTENTS
I.
11.
111.
rv. V.
v I.
v 11.
Summary
Introduction
Previous Approaches
Theory
Comparisons with Data
Theoretical Predictions of Condensing Heat Transfer Coefficients for Potassium
Conclusions
Appendix A - Derivation of Shear Stress Distribution Equation
Appendix B - Derivation of P res su re Gradient Equation Appendix C - Derivation of Heat F l u x Distribution
Appendix D - Nomenclature Appendix E - References Appendix F - Figures
Equation
Page
i i i i i iv
1
2
3
6
18
2 3
26
28 34
44 46 49 54
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LIST OF FIGURES
Title - - Number Title Number
I
2
3
4
5
6
7
8
9
IO
I 1
12
13
14
Sketch o f Assumed Flow Model
ut vs y t for Fully-Developed Pipe Flow
Variation of TI r o and Developed Pipe Flow
Ratio of Eddy Diffusivities vs Reynolds Number a t y / r = 0.2
Variation o f Ratio of Eddy Diffusivities with y/ r and Rey- nolds Number froin Data of Beckwith and Fahien (Ref. 20)
Variation of Ratio of Eddy Diffusivities with y / r and Reynolds Number from Data of Sesonske, et a1 (Ref. 28)
Calculated Fi lm Thickness v s Measured Fi lm Thick- nes s for Charvonia 'sandChien 's Data. Experimental Data
Calculated F i lm Thickness v s Measured Film Thick- nes s fo r Charvonia 's and Chien's Data. d P / d Based on Dukler Correlat ion
Calculated Fi lm Thickness vs Measured Film Thick- nes s f o r Charvonia 's and Chien's Data. dP /dPBased on Lockhart-Martinell i Correlat ion
e H / u L with y l r f o r Fully-
dP /dPBased on
15 Calculated Heat T r a n s f e r Coefficient v s Experimental Heat T rans fe r Coefficient. dP /dPBased on Lockhart- Martinell i Correlation.
Calculated Heat T rans fe r Coefficient v s Experimental Heat T r a n s f e r Coefficient. Correlation. a Obtained from Equation 4b
Experimental and Calculated Heat T rans fe r Coeff ic ients v s Quality. NASA Data (Ref. 38)
Experimental and Calculated Heat T r a n s f e r Coefficients v s Quality. Carpenter Data (Ref. 37)
Calculated Heat T rans fe r Coefficient v s Experimental Heat T rans fe r Coefficient. dP/d&Based on Lockhart- Martinell i Correlat ion. a = 1. 0
Calculated Heat T r a n s f e r Coefficient v s Experimental Heat T rans fe r Coefficient. Correlat ion. a = 1.0
Variation of ut and t + with y t for a Low Quality Condensing Water Point
a Obtained from Equation 4b
16 dP/dO Based on Dukler
17
18
19
20 dP/dYBased on Dukler
21
22 Variat ionof r l ro, e, . , /uL, a, and e H / ( k L / P L C p L ) with y / r for a Low Quality Condensing Water Point
Variation of F i lm Thickness with Air and Water Flow Rates , Theoret ical Resul ts Based on Dukler 's P r e s s u r e Potassium vs Quality, Showing Ef fec t s of Total Flow Gradient Correlat ion Rate
Variat ion of ~ l l m Thickness with Air and Water Flow ~ ~ t ~ ~ . P r e s s u r e Gradient Interfacial Resis tance
23 Theoret ical Heat T r a n s f e r Coefficients for Condensing
24 Theoret ical Heat T r a n s f e r Coefficients for Condensing Potassium v s Quality, Showing Effect o f Liquid-Vapor Theoretical Resul ts Based on Charvonia 's Measured
2 5 Theoret ical Heat T rans fe r Coefficients f o r Condensing Potassium v s Quality, Showing Effect of Fr ic t ional P r e s s u r e Gradient
Calculated Fi lm Thickness vs Measured F i lm Thickness for Col l ier and Hewitt Data
Ef fec t o f Liquid Subcooling on P e r Cent Difference between W g / W t and Quality Based on Enthalpy
Variat ion of Heat T rans fe r Coefficient with Liquid Viscosi ty
26
27 Variation of r / r 0 , e M / Y,, , a, and e H / ( k , i P L C p , )
Variation of ut and t t with yt for Condensing Potassium
with y / r f o r Condensing Potassium
Condensing Heat T r a n s f e r P a r a m e t e r v s Fi lm Reynolds Number Showing Effect of Liquid-Vapor interfacial Res is tanc e
28
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P R A T T & WHITNEY AIRCRAfT PWA-2530
I. SUMMARY
An analysis is presented of annular, two-phase flow inside of c i r - cular tubes. thickness for one o r two-component flow and the heat t ransfer coefficient for condensing flow of a pure fluid. is given to condensing liquid metals.
This analysis enables an estimation of the liquid film
Special consideration
Results obtained using this analysis were found to be in good agree- ment with measured values of liquid film thickness in vertical upflow and downflow fo r air-water mixtures in annular adiabatic flow. Similarly, condensing heat transfer coefficients calculated for ver t ical downflow of s team were found to be in good agreement with measurements. Finally, results of sample calculations a r e presented for condensing potassium. These results indicate that i f no Liquid- vapor interfacial resistance to heat flow is considered, the local values of condensing coefficients for potassium in vertical downflow inside a tube a r e higher than those calculated using Nusselt 's theory fo r laminar condensing on a ver t ical surface with no liquid-vapor interfacial shear. result in significant reductions in heat t ransfer coefficients for con- densing potassium.
This interfacial resistance to heat flow could
In the analytical approach, a l l liquid is considered to be flowing in an annular film along the tube wall with no liquid entrainment in the vapor core. for the liquid film and a r e combined with empirical expressions for turbulent diffusion coefficients and w a l l shear s t r e s s to enable cal- culation of the liquid velocity and temperature profiles. sions for turbulent diffusion coefficients a r e obtained from data for fully-developed single-phase pipe flow. profiles enable the liquid film thickness and condensing heat t ransfer coefficients to be determined. No adjustment of empirical constants
between theory and two-phase data.
Shear s t r e s s and heat flux distributions a r e derived
The expres-
The velocity and temperature
obtained from single-phase data was necessary to obtain
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11. INTRODUCTION
F r o m July to December 1963, an analytical design study was conducted by Pratt & Whitney Aircraft on shell-and-tube condensers for use a s one of the compact condensers of a one megawatt (e lectr ic) nuclear Rankine cycle space powerplantl. c ient experimental and the0 r etical information w a s available concern - ing condensing heat t ransfer coefficients and liquid holdup in condenser tubes. This information may be required for condenser design and for determination of the variation of system fluid inventory with operating conditions for any given design. deficiency in theoretical information, an analysis of condensing flow inside of tubes was conducted. that study.
This study indicated that insuffi-
'
Because of the above-mentioned
This report presents the results of
The report begins with a brief review of the theoretical and experi- mental work conducted for the determination of the condensing heat t ransfer coefficients for flow inside of tubes where the condensate film is in turbulent flow. theoretical approaches a r e indicated and the derivation of the new method is presented. tions which a r e inadequate for condensing 1 iquid metals flowing inside of tubes. a digital computer:% in order to rapidly calculate liquid film thickness and condensing coefficients. made in this report with available data for conventional fluids. Finally, estimates a r e presented for the condensing heat t ransfer coefficients for potassium.
The primary assumptions of each of these
This new analysis removes most of the assump-
The final equations of this analysis were programmed on
Comparisons of theoretical values a r e
lNurnbered references a r e l isted in Appendix E.
*Copies of a manual describing this computer program, Report NASA,CR-54350, a r e on file a t the NASA office of Scientific and Technical Information, Washington 25, D. C.
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III. PREVIOUS APPROACHES
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A l a rge number of papers have been published treating the condensa- tion on vertical flat plates and inside ver t ical tubes where the liquid condensate film is in laminar flow. This l i terature is not reviewed in this report but is discussed in many heat t ransfer texts. sation with a turbulent film w a s f i r s t treated by Colburn2. a semi-theoretical relationship for predicting heat t ransfer coeffi- cients for turbulent condensate flow, considering that transition from laminar to turbulent flow occurs at a film Reynolds number of 2100 and liquid-vapor interfacial shear is absent. In a la te r study, Carpenter and Colburn3 included the effects of interfacial shear and momentum transport at the liquid-vapor interface due to the m a s s t ransfer , a s well as the effects of gravitational force and wall force. The shear s t r e s s at the liquid-vapor interface was obtained from a correlation derived by Bergelin, et a14. The assumption was made that the only resistance to heat flow is due to the viscous sublayer of the condensate flow and that this layer is of constant dimensionless thickness (yv’s/ v L at edge of viscous sublayer = constant). approach led to f a i r agreement between theory and data for a number of conventional fluids in turbulent condensing vertical down-flow inside a tube.
Conden- He derived
This
Seban5, Rohsenow, et a16, and Altman, et a17 included the effects of the resistance of the turbulent portion of the film in addition to the resistance of the viscous sublayer. A l l of these approaches used the universal velocity profile for turbulent flow to determine the turbu- lent eddy diffusion coefficients for momentum. made in these approaches that the turbulent eddy diffusion coefficients for heat equals that for momentum, thereby enabling the temperature profile across the liquid layer to be calculated. This approach w a s first used by Martinellis for fully-developed single-phase flow. In order for these approaches to apply to condensing inside a tube, all of the liquid must flow in an annulus along the inside of the tube wall . Seban assumed no interfacial shear. Rohsenow, et a1 considered that the interfacial shear can be calculated using the correlation of Bergelin, et al4. Altman, et a1 assumed that the w a l l shear s t r e s s could be calculated from the adiabatic pressure lo s s correlation of Martinelli and Nelson9. ment with appropriate data for conventional fluids.
The assumption was
A l l of these methods showed good agree-
PAGE NO. 3
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Dukler10 presented an analysis similar to that used by Deissler11 for fully-developed turbulent pipe flow. In this approach a shear stress distribution was calculated across the liquid film ignoring the effect of static pressure forces in the film. was calculated from an empirical correlation for adiabatic two-phase ver t ical downflow. The heat flux was assumed to be constant across the liquid film. expressions fo r eddy diffusion coefficients were used in each region. The ratio of eddy diffusion coefficient of heat to that of momentum was assumed to be equal to one and molecular conduction was neglect- ed in the region away from the wall. agreed very well with measured film thickness in adiabatic downflow and condensing coefficients for conventional fluids in vertical down- flow inside a tube.
W a l l shear s t r e s s
The film was divided into two regions and appropriate
The resul ts of this analysis
A l l of the above methods involve assumptions which make them in- adequate for the determination of the condensing coefficients for liquid metals. Fo r condensing liquid metals , the resistance of the entire condensate layer must be included because of the low Prandtl number of such fluids. is probably not equal to one, particularly in a thin condensate layer in close proximity to a wall. Also , frictional , gravitational and static pressure forces due to condensation should be included in the mo- mentum equation for a fluid element in the liquid layer .
In addition, the ratio of eddy diffusion coefficients
In addition to the resistance to heat flow caused by the liquid layer , interfacial thermal resistances between the liquid and the solid wall and between the vapor and the liquid may be present in liquid metals. Kirillov , et a l l2 found that wall-liquid interfacial resistances can be present in single-phase alkali metal systems but can be eliminated by adequate removal of oxygen, that significant vapor -liquid interfacial resistances can be present due to the high heat flux rates present in liquid metal condensation. They found this resistance present in mercury condensation.
Sukhatme and Rohsenow13 have indicated
Little data is available for liquid metal condensing. Bonilla obtained data for condensing mercury and sodium on a ver t i - ca l surface.
Misra and
This data indicated condensing heat t ransfer coefficients
P A Q E N O . 4
I A T T L WHITNCY AIRCRAFT PWA-2530
much lower than those predicted by Seban's theory. Lee16 suggested that this disagreement may be due to upward vapor shear on the liquid condensate layer. Rohsenow suggested that it is due to vapor-liquid interfacial resistance to heat flow. Condensing heat t ransfer coefficients for potassium flowing downward inside a vertical tube were obtained by Sawochkal'. This data also indicates measured values much lower than those predicted by Seban's theory.
Chenl5 and
Because of the disagreement between presently available theories and data, more experimental and theoretical work is required in the a r e a of liquid metal condensation. The purpose of this study is to present an analysis of the condensate layer resistance which includes additional factors not considered by previous investigations and which therefore would be more applicable for liquid metals. resistance can be used for design purposes. magnitude of additional resistances such as liquid-vapor interfacial resistance can be more accurately determined from experimental liquid metal data. Since the present approach also yields estimates of liquid film thickness, estimates of fluid inventory will also result.
The estimates of this Also, the presence and
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N. THEORY
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The physical problem to be analyzed is that of two-phase liquid-vapor flow inside a tube in which the liquid flows in an annular layer along the wall and the vapor flows in the core. erally present in condensing tubes in which the flow direction is vertically downward and is anticipated to be the flow regime present in condensers in zero gravity. In addition, this flow pattern i s common in two-phase two-component flow, including the case of 'vertical upflow. Therefore, the vapor in this analysis is considered to consist either of the same substance as the liquid o r a different substance to make the analysis more general. Fo r the former case, the condensing heat transfer coefficient and liquid layer thickness a t a local axial position a r e desired as a function of tube diameter, fluid, quality, p ressure , flow rate, heat removal ra te from the tube, gravity field, and tube orientation. In the la t ter case, the thickness of the liquid layer is desired as a function of the tube diameter, fluids, fluid flow ra tes , p ressure , temperature, gravity field, and tube orientation. The thickness of the liquid layer enables the amount of liquid present in the tube to be determined. these two-phase flow cases in which the core flow is considered to be zero.
This flow pattern is gen-
Single-phase flow i s a limiting case of
The basic approach to the problem i s to determine the thickness of the liquid film by determining the velocity distribution across this film, and then to determine the temperature distribution across the film to obtain the local heat t ransfer coefficient.
The major assumptions are:
The liquid film is annular and axisymmetric ( see Figure 1, Appendix F). Only gas o r vapor is present in the core. The flow is steady. Condensing mass t ransfer occurs at the liquid-vapor interface. Liquid properties a r e assumed constant ac ross the film. Sensible heat due tQ liquid subcooling is negligible. The eddy diffusion coefficients of momentum and heat a r e obtainable from empirical equations.
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8)
* 9 )
10) *ll)
12)
The wall shear s t r e s s is obtainable f rom correlations of two-component, adiabatic pressure loss data. Theacceleration t e rms can be neglected in the momentum equation of the liquid layer. The static pressure is uniform across the tube. Momentum fluxes can be evaluated using flow average velocitie s . The vapor i s a t saturation temperature.
The velocity and temperature profiles across the liquid film a r e obtained from the transport equations for turbulent flow which are:
These equations account for the transport for momentum and energy through the action of both molecular and turbulent transport mechanisms.
Dividing Equation (1) by the shear s t r e s s at the wall tion (2) by the heat flux at the wall qo the following equations a r e obtained
to, and Equa-
*Not applicable t o cases for adiabatic fully-developed flow
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Multiplying and dividing by PL pL' and rearranging terms
Noting that C P L P L = - PrL, p-- @L - - - "L.' d q =, v* and k L L
C H - I a the equations can be written as: M
Defining the dimensionless variables
dtt [c:o 'L ' L ) dt dut --- - Y L du and noting that 7 = dY dy and+ dY (V*O2 dy
1 CRATT .I WHITNLY AIRCRAFT
I
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the transport equations can be written in the following forms
An empirical expression of van Driest18 is used for the eddy diffu- sivity of momentum
This expression has been found to lead to good estimates of heat transfer coefficients in single-phase flow.
Van Driest 's expression for eddy diffusion coefficients was obtained by an extension of Prandtl 's mixing length approach to include viscous damping of turbulent eddies near a wall. in this expression were obtained from velocity profiles in fully- developed pipe flow in the region near the wall. found to equal 0.40 and A+ was found to equal approximately 26. Far from the wall, this expression provides values of diffusion co- efficient which a r e in e r ro r . However, since the diffusion coefficient is very high in that region, the velocity profile is affected only very slightly by such incorrect values of t M / Y For the case of a con-
the liquid-vapor interface, van Driest 's expression can be seen to 'permit zero values of eddy diffusion coefficient at the liquid-vapor interface. Since this occurrence i s considered unrealistic, an alter- nate option to using van Driest 's equation i s provided in the analysis. This option uses Van Driest 's expression out to the distance from the wall at which a maximum value of turbulent diffusion coefficient
Empirical constants
The constant K was
. densing film, *in which shear s t r e s s dis t r i i utions may go to zero at
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occurs. For greater distances f rom the wall, the option considers the diffusion coefficient to be constant and at its maximum value, This option was used for the calculations of all results presented in this report. is shown in Figure 2, where the velocity profile for single-phase flow calculated using this method and the present analysis a r e compared with the universal velocity profile which has been fitted to the data of Laufer l9. The calculated distributions of shear s t r e s s and turbulent diffusion coefficient of momentum are also shown in Figure 3. The Reynolds number used for these calculations corresponds to one of those of Beckwith and FahienZ0, who obtained data on the ratio of eddy diffusion coefficients of heat and momentum. The latter data will be discussed later in this report.
The validity of this assumption for single-phase flow
z, In the study of heat transfer to liquid metals in single-phase full developed turbulent flow inside tubes, a number of investigators 22, 23, 24s 25, 2 6 ~ 2 7 ~ 28 have derived expressions for a, the ratio of eddy diffusion coefficient of heat to that of momentum. of these studies considered that conduction of heat occurred between a turbulent eddy and i t s surroundings so that the effectiveness of the eddy to transport heat was reduced. values of a at low Prandtl numbers. average values of either of Reynolds number and Prandtl number, o r the maximum value of t M / Y L occurring in the pipe and Prandtl number. In the present study, an expression for the ratio of diffusion coefficients was desired that also included the variation with distance from the wall.
All
This effect resulted in lower In some of these approaches,
a across a pipe were calculated as a function
Subbotin, e t a129reported data obtained for mercury and sodium- potassium alloy. have a large effect on the ratio of turbulent diffusion coefficients for liquid metals. number is indicated at a value of y / r = 0.2. Figure 4. Sesonske, e t a128, Brown, e t a130, and Isakoff and Drew31 for mercury have been added to this figure. primarily a t liquid metal heat transfer, data obtained for air was not added to the figure.
This data indicates that Prandtl number may not
However, a trend of increasing a with Reynolds This data is shown in
The data of Beckwith and Fahien" for water, and of
Since this study is aimed
In general, air results in higher values of a.
Because of the absence of any apparent Prandtl number effect for liquid metals, an equation of the form
PRATT h WMITNRY A I R C R W T PWA-25 30
was considered. This expression results in values of a varying f rom ae ro for c M / u t equal to zero, to 1. o for cM. / u t equal to a very large value. of a near a pipe wall as either distance from the wall o r Reynolds number are varied independently. with c M / u L at constant Reynolds number, Figures 5 and 6 present the data of Beckwith and Fahien for w a t e r and of Sesonske, e t a1 for mercury. It can be seen that at a given Reynolds number, a tends toward zero as the wall is approached. also approaches zero. Therefore, the trend of 01 approaching zero as c M / U L approaches zero is indicated. The data presented in Figure'4 indicates that at a given Reynolds number increases. Since c M / ut at a given y / r increases as Reynolds number increases, the trend of a increasing toward one as
These trends a r e in agreement with the variation
Considering the variation of a
In this region, t Y / u L
y / r , a increases to near one as
cM / u t increases is demonstrated by this data.
The values of the constants a and n were determined using the data of both Sesonske, et a1 and Beckwith and Fahien for small values of y / r . The constants a and n were found to equal 2 . 0 and 0.5, respect- ively. i n Figures 5 and 6 based on measured values of seen to be in poor agreement with most data shown. values of a were also added to Figure 4. culated using Deissler's expression for eddy diffusivity variation22
Values of a using the resulting expression a r e shown plotted t M / v L and can be
Calculated These values were cal-
and the expression for friction factor
The agreement can be seen to be poor in this figure also.
Although the analytical expression for a does not agree with the data in general, inconsistencies a r e found in the data which appear to make the determination of an expression that will correlate all avail- able wdata very difficult. For example, the data of both Beckwith and
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Fahien, and Brown, e t a1 indicate an almost constant value of a for large y / r , but the data of Sesonske, e t al, and Isakoff and Drew in- dicate a decreasing a as y / r increases for values of y / r approaching 1. Also, the data of Brown, et a1 appears to be much lower than that of other investigators, as indicated in Figure 4.
Although the analytical expression does not result in good general agreement with the considered data, its general trends can be seen to be correct in the region near the wall. This region is most im- portant in its effect on heat transfer coefficients since most of the temperature gradient occurs in this region for conventional fluids. Fo r condensing flow of alkali metals the ratio of eddy diffusion co- efficients appears to be unimportant since most of the heat is calcu- lated to be transferred by molecular conduction in the condensate layer when the above expression for a is used. For these reasons the equation which is used in this study for determining a is
The variation of a with Reynolds number at y / r = 0.2 was calculated using the present method for the conditions of the data of Beckwith and Fahien. be in good agreement with the results obtained based on Equations (5) and (6).
These results a r e shown in Figure 4 and can be seen to
Equations ( Id) and (2d) can now be used to determine the velocity and temperature distributions in terms of the shear s t r e s s and heat flux distributions.
F r o m Equation ( Id)
Substituting Equation (7) into Equation (3)
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Transposing and rearranging t e rms
t -y+/A+ 2 K' y+' [I - e 3 = o
CM Solving for - vi
t M - " I . Since must be zero when y+ = 0, the cor rec t root is the
one with the plus sign.
Therefore
Substituting this equation into Equation (7)
Integrating from the wall to any radial distance y+ from the wall
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In a similar procedure the following equation for t t can be derived.
f- 90 - (13) I ’ a - 1 t d l t 4 2 KzysL [l-e -y+/A+
T O t- 1 -
o Fr, 2 L -
and - a r e derived in The following equations for - Appendices A and C using the Navier-Stokes and energy equations
T
70 90
The determination of the shear s t ress distribution from Equation (14) requires knowledge of the wall shear s t r e s s T~ and the pressure gradient dP/da. An expression for the pressure gradient is derived in Appendix B, a s follows
- - d P - dP -{!E-) dQ f+ic t ion - cosOL [ R L P , t ( l - R L ) P g ] t gC
where 2 7 0 s ( :; ) friction r 0 .
PAOCNO. 14
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The determination of the shear s t r e s s distribution therefore depends '
on known o r calculated values of local quantities except for d P and - dRL . (%)friction
dx
Three different assumptions were made concerning the frictional p re s su re gradient in order to determine calculated values of condens- ing coefficients and liquid film thicknesses. The first method of calculation used values of d P obtained experimentally for a given
data point. This enabled dP
mined from Equation (16a). derived by Lockhart and M a r t i x ~ e l l i ~ ~ for adiabatic two-component horizontal two-phase flow to determine directly. The
third method used the correlation derived by D ~ k l e r ~ ~ for adiabatic
to be calculatedand r0 deter- (F)
( 3 ) f r i c tion The second method used the correlation
two-component vertical downflow, to determine the sum of (LE)
(S) friction
dP friction and the gravitational pressure gradient which is the second t e r m in the right hand side of Equation (16).
The t e r m dRL
quality at the local point being analyzed. mination of this t e r m would require much more extensive computation, the liquid fraction correlation derived by Lockhart and Martinelli was used to determine the slope. The derivation of the resulting expressions used for this t e r m a r e presented in Appendix B.
is the slope of the variation of liquid fraction with - dx
Since the analytical deter-
32
When Equations (14), (15), and ( 16) a r e substituted into Equations (12) and (13), all t e rms under the integral signs become functions of y+ only, and the two equations can be solved independently.
Equation (12) can be solved to determine uf as a function of y+ by using a numerical integration method such as Simpson's rule o r the trapezoidal rule. the thickness of the liquid layer can. be determined using the known liquid flow ra te and the continuity equation derived below.
Once the velocity distribution has been determined,
PAQL NO. 15
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The differential flow rate through a differential a r e a of flow is
dWL = P L u d A (17)
A differential annular a r ea of flow in a tube is
dA= 2 7 ( ro -y ) dy
Thus
Integrating from the wall to the liquid-vapor interface located at a distance 6 f rom the wall gives the total liquid flow rate.
6
w,= 2 ?r P , 1 u ( r o - y ) dy (20)
Using the definitions of u t and y+ and defining Equation (20) can be rearranged to give
, r o t E ro v* VL
,
where &+= ** 'and i s the nondimensional film thickness. VL
From Equation ( 1 2 ) ut is obtained as a function of y+ and thus Equa- tion (21) can be integrated numerically. liquid film thickness then involves guessing a value of value i s guessed which gives the correct known liquid flow rate. Since the static pressure gradient, dP/dB in Equation (14) requires knowledge of film thickness, new values of this t e r m a r e calculated for each assumed value of of successive values of o rde r to a r r ive at a final value in a smal l number of tr ies.
The determination of the L+ until a
it used.in the calculations. The guessing 6+ can be handled by various methods in
PAQE NO. 16
PRATT L WMITNEY AIRCRAFT
Once the film thickness is known, Equation (13) can be solved to obtain the temperature difference from the wall to the edge of the liquid film (tv - to) and thus the heat transfer coefficient across the liquid film can be obtained through the following defining equation
This coefficient hfilm only involves the temperature difference due to the thermal resistance of the liquid film. If other thermal resistances a r e present a t the liquid-vapor interface o r a t the liquid- wall interface due to impurities, then an overall coefficient must be defined. In the case of the liquid-vapor interfacial resistance, an expression is presented in Reference 13 for the temperature drop a t the liquid-vapor interface in t e rms of the mass f l u x
In this equation u is the accommodation o r condensation coefficient which must be determined experimentally,
9 Since q t Am and hinterfacef
t,-ti
The overall heat transfer coefficient of the tube wall can then be found including both the liquid film and liquid-vapor interfacial resistance from
(25) 1
int e r face - = -
( r o - 6 1 rO
h t
1 1 h 'film
The t e r m ( '0 - lj r0
the wall to the liquid-vapor interface.
) accounts for the change in heat flux a rea from
PAQL NO. 17
CRATT WHlTNCY AIRCRAFT PWA-2530
V. COMPARISONS WITH DATA
The analytical method presented in the previous section was used to estimate liquid film thicknesses and condensing heat transfer co- efficients for a number of conditions for which data was available for comparison. The cases considered were the evaluation of water film thicknesses for air-water mixtures in vertical upflow and verti- cal downflow and the evaluation of the condensing coefficients for steam in vertical downflow. No horizontal flow cases were analyzed because of the possible large departure f rom the annular flow pattern in the experimental data. In each of the cases considered, the effect of different methods of predicting wall shear s t r e s s was investigated.
Liquid Film Thickness
Vertical Downflow
The analytical method was used for predicting liquid film thick- ness for comparison with the measured values of C h a r ~ o n i a ~ ~ and Chien35. ness of a water layer flowing along the tube wall when an air-water mixture w a s flowing in vertical downflow near ambient tempera- ture and pressure. The tube inside diameters were 0.208 f t for Charvonia and 0. 167 f t for Chien. Chien found that entrainment of liquid in the gas core occurred above certain values of liquid and gas Reynolds numbers. not consider entrainment, no data was used for comparison that indicated entrainment may be present, based upon Chien's cri terion for the inception of entrainment. Data points were selected to cover the range of the experimental variables for which no entrainment was expected to be present.
Both of these investigators measured the mean thick-
Since the present analysis does
Calculated values of film thickness using measured values of pressure drop a r e compared with measured values of film thick- ness obtained by Charvonia and Chien in Figure 7. ment can be seen to be good. Figure 8 shows the comparison between the values of film thickness calculated using Dukler's correlation for pressure drop in a vertical pipe and the measured values. Agreement can be seen to be good again. However, when values of film thickness were calculated using Lockhart and
The agree-
P R A T T L WHITNEY A I R C R A F T
Vertical Upflow
Collier and H e ~ i t t ~ ~ presented mean liquid film thickness data for the upward flow of an air-water mixture in a tube of 0. 104 f t inside diameter near ambient temperature and pressure. thicknesses were calculated f o r comparison with this data using
I
I Liquid film
I
I the method of Lockhart and Martinelli for estimating pressure I gradient. The comparieon between theory and values obtained I
f rom a mean line through the data is shown in Figure 12. Agree- ment can be seen to be good. to obtain pressure gradient because of its apparent inapplicability
Dukler's method was not used
I to vertical upflow. I I Condensing Heat Transfer Coefficients
Carpenter37 and Coodykoontaand Dorsch ,38 obtained measured values of local condensing heat transfer coefficients for vertical downflow
PWA-2530
Martinelli's correlation for pressure drop, agreement was found to be poor, a s shown in Figure 9. For some cases , this method of obtaining pressure drop resulted in unrealistic cal- culated velocity profiles in the liquid film so that no solution could be found.
The poor agreement between data and theory using Lockhart and Martinelli's method is probably due to the fact that their corre- lation is based upon pressure drops measured in horizontal tubes. Dukler's correlation which is based on data from vertical tubes with downflow resulted in good agreement. between data and theory using measured values of pressure drop indicates the validity of the analysis for predicting film thickness in the range of the data considered.
The agreement
In order to determine more clearly whether the trends indicated by the theory a r e correct , Charvonia's film thickness data a r e plotted in Figures 10 and 11 as a function of vapor and liquid flow rate. both measured pressure gradients and pressure gradients cal- culated using Dukler's method. The agreement can be seen to be good in both cases.
Theoretical values a r e presented which a r e based upon
PAQL NO. 19
P R A T T e WHITNCY A l R C R A f T PWA-25 30
inside of a tube. Carpenter used water, ethanol, methanol, toluene, and trichloroethylene as the working fluids in a tube of 0.459 inch inside diameter, and Goodykoontz'and Dorsch used water in a tube of 0.293 inch inside diameter.
In o rde r to determine the capabilities of the present analysis to pre- dict condensing heat t ransfer coefficients for flow inside a tube, cal- culations were made for comparison with data presented in both of the above references. the only working fluid considered from Carpenter 's data.
In order to limit the calculations, water was
Two adjustments were made to the analytical values to take into account basic effects not initially included. programmed on the computer includes the assumption that local qual- ity is equal to the ratio of vapor flow to total flow. effect of liquid subcooling in the condensate layer was ignored. Subsequent study of calculated results, indicated that at low qualities this assumption leads to significant e r r o r in the calculated values of vapor and liquid flow rates , based on a value of quality determined from a heat balance. Therefore, the temperature and velocity pro- fi les calculated by the analysis were used to determine more accurate values of vapor and liquid flow rates for each case using the equation
2 1 r & ~ ~ ~ ~ u , ~ i ~ r d r - WT iLgat - Wgigsat t 2r40-i PLuLiLrdr - WT iLlsat
The calculation procedure
Therefore, the
r0
X= - WT(igsat - i L s a t ' ) WT ( igslat - iLwt)
( 2 6 1 and iiLsat a r e the vapor and liquid where if3 sat
enthalpies, respectively, evaluated at saturated conditions. The computer program was then re run using the new flow rates. The importance of this correction for one particular case is indicated in Figure 13 where the percentage difference between Wg/WT and equilibrium quality based upon local enthalpy is shown as a function of quality based upon enthalpy. No adjustment was found necessary to the analytical values for the experimental conditions of Carpenter because values of flow rates including the effects of subcooling were presented in this reference and were used in the computer program initially,
P R A T T 8 WHITNLY A l R C R A f T PWA-2530
A second slight adjustment was made to take into account the varia- tion of liquid viscosity across the condensate layer. the initial analysis were used to determine the arithmetic mean film temperature based on wall and saturation temperatures. This mean film temperature was used to calculate a new liquid viscosity and the case rerun. condensing coefficient f rom the initial value. The importance of this correction is shown in Figure 14 where the calculated condensing coefficient for the conditions of one of Carpenter's data points is plotted a s a function of liquid viscosity and temperature a t which the viscosity is evaluated.
Results f rom
This correction resulted in only a small change in
Calculated values of heat transfer coefficients including these adjust- ments a r e compared with measured values in Figures 15 and 16. The calculated values presented in Figures 15 and 16 a r e based on Lockhart and Martinelli's and Dukler's methods for predicting pres- su re gradients, respectively. Agreement between data and theory is seen to be good in both cases.
In order to compare the trends of condensing heat transfer coefficient variation with quality as predicted by the theory and obtained experi- mentally by Coodykoontz and Dorsch and Carpenter, Figures 17 and 18 a r e shown. with the data over the entire range of experimental data.
The analysis can be seen to be in good agreement
The calculated values of condensing heat transfer coefficients for conventional fluids is very dependent on the ratio of eddy diffusion coefficients, is obtained indicates that the expression used for u which was obtained for fully-developed single-phase flow may be reasonably accurate in a condensing film.
a . The fact that agreement between data and theory
In order to show the importance of the assumption concerning a , calculations were performed assuming that a equals one. These resul ts a r e presented in Figures 19 and 20. The assumption of a equal'ling one can be seen to lead to values of condensing coefficient much higher than those rneagured experimentally. This result is surprising, considering the success found by previous investirators in correlating data using the assumption that a equals one. study of this resul t i s needed.
Further
PRATT & WHITNCY AIRCRAFT PWA-2530
Figure 21 shows calculated values of velocity and temperature pro- files across the condensate layer for the conditions of one of Carpenter 's data points,for which the quality was low. file is plotted for comparison. exists between the two velocity profiles. This result indicates that liquid film thicknesses and therefore condensing coefficients can be calculated using the universal velocity profile for this data. tion, much of the temperature drop takes place near the wal l ( y t < 10). to obtain correlations for conventional fluids using the assumptions that the velocity profile for this data is the universal profile and that most of the temperature drop occurs in the viscous sublayer. The distributions of shear s t ress , turbulent diffusion coefficients of momentum and heat, and ratio of turbulent diffusion coefficients a r e shown in Figure 22 for the same data point. The shear s t r e s s can be seen to decrease rapidly with distance from the wall. results in the velocity profile departing slightly f rom the universal profile.
The universal velocity pro- It is seen that only a small deviation
In addi-
These ias t k v o facts enabled previous investigators
This
The ratio of turbulent diffusion coefficients a can be seen to be much less than one for the entire film for this case. of the turbulent to molecular diffusion coefficients of heat is much grea te r than one for the entire turbulent region of the film.
Also, the ratio
Summary of Data Comparison
The present analysis results in good estimations of liquid film thickness for annular flow in a vertical tube where no entrainment is present. C h a r ~ o n i a ~ ~ and Chien35 and the upflow data of Collier and H e ~ i t t ~ ~ . The best correlation was obtained when the method of Dukler was used for predicting pressure drop in vertical downflow and the method of Lockhart and Martinelli f o r predicting pressure drop in vertical upflow. Also, the analysis provides calculated values of local condensing heat transfer coefficients for vertical downflow of s team which a r e in good agreement with the data of Carpenter37 and Goodykoontz and Dorsch38, when either Dukler's o r Lockhart and Martinelli's method a r e used for predicting pressure drop.
Agreement is good for both the downflow data of
PRATT L WHITNEY AIRCRAFT PWA-25 30 I
VI. THEORETICAL PREDICTION OF CONDENSING HEAT TRANSFER COEFFICIENTS FOR POTASSIUM
Because of the agreement between the present analysis and liquid film thick- ness and condensing coefficient data of conventional fluids, this analy- sis may enable the prediction of film thickness and condensing liquid film heat transfer coefficients for liquid metals. These two quantities a r e simply related for liquid metal condensation since the main mode of heat t ransfer is by molecular conduction.
In order to demonstrate the values of condensing heat transfer co- efficient that can be expected for potassium condensing in downflow inside a tube, calculations were made with quality, flow rate, and accommodation coefficient a s independent variables. These a r e plotted in Figures 23 and 24. ra te and of different assumptions concerning frictional pressure gradient were investigated.
In addition, the effects of heat flux
A basic condition was selected where
Wpotassium = 40 Ibs/hr Ppotassium = 6.6 psia tube inside diameter = 0. 625 in. Q = 60, 000 Btu/hr ft2 No liquid-vapor interfacial resistance Frictional pressure gradient obtained from Lockhart and Martinel li cor r e lation
For these basic conditions, the condensing coefficients were calcu- lated as a function of quality. The flow rate was increased to twice and reduced to one-half the standard flow rate with appropriate adjustments made to the frictional pressure gradient. calculations were repeated. a r e presented in Figure 23. can be seen to be very important, as was expected.
The effect of liquid-vapor interfacial resistance on the condensing coefficients at standard conditions is shown in Figure 24. Liquid- vapor interfacial resistances can be seen to be very important, e- qpecially a t high qualities where the condensing coefficients a r e high.
Then the The results for variation of flow rate The effects of both quality and flow rate
PAQL NO. 23
C R A W I WHITNCY AIRCRAFT PWA-2530
Similar calculations were made with the heat f l u x doubled and halved and everything else a t the standard conditions. cated that for the conditions analyzed, heat flux level has essentially no effect on the condensing coefficient.
These results indi-
The importance of the assumed values of frictional pressure gradient is demonstrated in Figure 25 for the standard conditions. Results a r e shown for values of frictional pressure gradient equal to that obtained from the Lockhart and Martinelli correlation and equal to one-half and twice this value. The effect of pressure gradient can be seen to be important. However, since the Lockhart and Martinelli method led to good agreement for condensing coefficients of conven- tional fluids, the e r r o r introduced by using this method for estimating frictional pressure gradient is not likely to be large. low quality may be somewhat in e r ro r since values calculated at a quality of 0. 1 using Lockhart and Martinelli's frictional pressure gradient were found to result in unrealistic velocity profiles. higher frictional pressure gradient would be required to give more realistic profiles.
The value at
A
Calculated velocity and temperature profiles for the case of standard conditions and 50 per cent quality a re shown in Figure 26. Again, the velocity profile is very close to the universal velocity profile. F rom the temperature profile, it can be seen that most of the tem- perature drop does not occur 'near the wall (y+ < 10) true fo r conventional fluids. is very nearly linear with distance f rom the wall. of shear s t r e s s , turbulent diffusion coefficients of momentum and heat, and the ratio of the turbulent diffusion coefficients of momentum and heat a r e plotted in Figure 27. The turbulent diffusion coefficients when plotted separately were put in dimensionless form by dividing each by its respective molecular diffusion coefficient. diffusion coefficient of heat is negligible when compared to the molecu- lar value because of the low Prandtl number for potassium. feature causes the temperature profile to be linear and is much differ- ent f rom results found with water, a s indicated in Figure 22.
a s is The variation of the temperature
The distributions
The turbulent
This
Since many investigators compare experimental values of condensing coefficient with values determined from Nusselt theory, the theoreti- cal values determined in this study for potassium a t the standard
P A ~ L NO. 24
1 CRATT L WHITNKY AIRCRAfT PWA-2530
conditions were plotted in terms of the dimensionless groupings used by Nus s elt with accommodation coefficient and quality as independent variables. These a r e presented in Figure 28. Seban's turbulent film results fo r a Prandtl number of 0.003 a r e also shown for com- parison. The results calculated in the present study can be seen to be much higher than those predicted by the theories of either Nusselt o r Seban when no liquid-vapor interfacial resistance is present, This result is due primarily to the presence of liquid-vapor inter- facial shear in the present analysis. shear s t r e s s was found by Rohsenow, e t a16.
A similar effect of interfacial
PRATT e WHITNCY AIRCRAFT PWA-2530
VU. CONCLUSIONS
The following conclusions can be drawn from the results obtained in this report:
A.
B.
C .
D.
E.
F.
The present analysis provides an accurate means of predicting liquid film thickness for annular two-component flow of conven- tional fluids inside tubes with vertical downflow orientation, if the pressure gradient i s known. This resul t indicates the basic validity of the analytical approach.
The analysis provides an accurate means of predicting liquid film thickness for annular two-component flow of conventional fluids inside tubes with both vertical upflow and downflow orien- tations, i f the pressure gradient is based on the method of Dukler for downflow and the method of L0ckhar.t and Martinelli for upflow. Therefore the present analysis can be used to pre- dict liquid film thicknesses even when no measured values of p re s su re gradient a r e available.
The analysis enables the accurate prediction of local condensing heat t ransfer coefficients for conventional fluids with ver t ical downflow inside tubes, i f the pressure gradient is based on either the method of Dukler o r that of Lockhart and Martinelli.
'
Because of the above agreement, the present analysis can be expected to accurately predict the thickness of condensate layers inside condensing tubes for vertical downflow orientation and under zero-gravity conditions.
F o r the cases analyzed, the local values of condensing coeffi- cient for potassium in vertical downflow inside a tube a r e higher than those calculated using Nusselt's theory for laminar con- densing on a vertical surface with no liquid-vapor interfacial shear , i f no vapor-liquid interfacial resistance to heat flow is considered.
Significant reductions in condensing coefficient for potassium can result f rom vapor-liquid interfacial resistance for accommo- dation coefficients as high as 0.9 o r greater .
PAQC NO. 2 6
CRATT L WHITNCV AIRCRAFT PWA-2530
G. The effect of liquid subcooling must be considered when calcu- lating condensing vapor and liquid flow ra tes for a flow of known enthalpy and total flow, if the quality of the fluid is low.
H. In many cases the thickness of liquid condensate layers and condensing heat t ransfer coefficients may be calculated using the universal velocity profile because of the small departure of the calculated velocity profiles from the universal velocity profile.
I. Condensing heat t ransfer coefficients calculated assuming that the ratio of turbulent diffusion coefficients of heat and momentum equals one m a y be in significant e r ro r .
CAOC NO. 27
C R A W L WHITNCY AIRCRAFT
~~
PWA-2530
APPENDIX A
Derivation of Shear Stress Distribution Equation
CAOL NO. 2.8
P R A l T L: WHITNCY AIRCRAFT
APPENDIX A
Derivation of Shear Stress Distribution Equation
The Navier-Stokes equation (See Reference 39) for laminar incom- pressible flow in the axial direction written in cylindrical coordinates is
Where:
v = velocity component in outward radial direction (r direction) u = velocity component in axial direction (z direction) w = velocity component in tangential direction ( 4 direction) F = the body force in z direction
ap a s - = static pressure gradient in z direction
Although this equation holds for laminar flow, it is assumed that turbulent flow can be treated using this equation when turbulent diffusion coefficients a r e added to the molecular terms.
a u a t Since the flow is steady - = 0 and since the flow is axisymmetric
- = O and a u ad
Thus the equation becomes \
CAaL NO- 29
P R A W 8 WHITNCY AIRCRAFT
a u , a n d - , the latter t e r m is - a r r a r a z U
alz Of the t e rms y , - very small compared to the f i rs t two t e rms since changes of the axial velocity in the axial direction occur at a much smaller rate than changes of the axial velocity component in the radial direction; for example, the ratio of e to e is in the order of the square
of the tube length to film thickness ratio which is a very large num- ber. Therefore, the e t e r m can be neglected, and
lar azz
a z Z
The shear s t r e s s component in the r plane and z direction is
a U
ar au << - 8 Since - az
and
Taking the partial derivative of Equation (A6) with respect to r
Dividing by r and rearranging
Substituting Equation (A8) into Equation (A3)
' (A9) a P 1 a ( r 7 ) = F - - -- a u t P,u - P,V - a r r a r a z a z
PAGE NO. 30
I PRATT L WHITNIY AIRCRAFT PWA-2530
The body force t e rm F consists of the gravitational force per unit volume on the liquid. Using the upward vertical direction as the
the gravitational force on the liquid is is the angle measured from the vertical upward direction.
-"L -$, gc C O S e , where 8
Thus
Substituting Equation (A 10) into Equation (A9)
The t e rms on the left hand side of Equation ( A l l ) a r e neglected in many boundary layer and condensing flow analyses 5 , 6, 7, l o , since these te rms greatly increase the complexity of any analytic solution and little e r r o r was found to be introduced. These te rms a r e both equal to zero for fully-developed flow since for that case v and
These terms may not be negligible for the case of condensing flow and must be evaluated from the results of the calculations to see if they can truly be neglected. Such a tes t is still not a positive proof of the e r ro r involved in neglecting these terms.
&u/az a r e both zero.
Neglecting these te rms , the Navier-Stokes equation becomes
o r
The term !.!? which could be measured with static taps) and thus will be written in the more familar fashion as dP /d%*
is the local static pressure gradient (i. e. the one a z
PAQC NO. 3 1
P R A T T L WHITNCY AIRCRAFT PWA-2530
Thus
Multiplying both sides by r ar
Integrating
At r=ro, T = to
Thus
dP - ro T o = (pL A- c o s e t - g C d l
Solving for C1
2
-52- t c , 2
Substituting Equation (A18) into Equation (A 16)
-1 Multiplying both sides by - 7 0
P A Q L NO. 3 2
PWA-2530 P R A ~ e WHITNLY AIRCRAFT
Noting that r =io -y radially inward
where y is the distance from the wall
or - - rO J (A221 - = - - 1 [ ( p L ~ cos e t 7
To ro-Y g C
Cancelling equal terms of opposite sign and factoring out ro from the right side gives
or
cos e t x ) ( 4 - *) r0 (A241 7 dJ! r0
Y r0
1 - -
Since = the final shear stress equation is 10
PAOL NO. 33
PRATT & WHITNLY A l R C R A f T PWA-2530
APPENDIX B
Derivation of Pressure Gradient Equation
PAQL NO. 3.4
P R A l T L WHITNCY AIRCRAFT
APPENDIX B
PWA-2530
Derivation of P r e s s u r e Gradient Equation
dP d T
i s obtainable f rom the momentum The static pressure gradient
equation written for the entire two-phase stream. su re is assumed to be uniform across the tube.
The static pres-
Consider the differential element along the length of the tube.
vertical
liquid
force The momentum equation is
forces = change in momentum flux
o r
wall force t pressure force t gravitational force = change in momentum flux
The wall force is - to ( Zrro 1, dll
Since for adiabatic flow without gravitational forces
the wall force is
PAQL NO. 35
PRATT e WHITNIY AIRCRAFT PWA-2530
d P is defined as a postive quantity. (,& friction Here
The pressure force t e rm is
(B2 1 d P - r r0 2 dP = - r r 0 ' -dR dP
The gravitational force t e rm is equal to the axial component of the weight of the fluid in the differential length; that is
where RL The change in momentum flux across the differential element is approximat e ly
is the fraction of the differential volume that is liquid.
a t Station Q t d i at Station P (B4)
where V, and Vg a r e flow average velocities defined by the continuity equations, W, = P L V L A R L and Wg = PgVgA( l - R L )
Therefore,
2 v, = ( w L )' PLARL
and
Substituting Equations (B5) and (B6) into (B4)
PAOL NO. 3,6
P R A l T & WHITNCY AIRCRAfT
-
PWA-2530
W t t W; PgA( l'RL lgc at Station P+d P PLARL gc
A\momentum flux =
which can be written in differential form as
The tube area A is constant with length and the density for condensing flow is considered independent of length. Thus
or
The differentials on the right side can be expanded to give
and
t dR, 2WgdWg '(&)= (1-RL) (1-RL)
PAQL NO. 37
PRATT & WHITNCY AIRCRAFT PWA-2530
Since Wg = X W T , then dWg = WT d x
and since W; = (1-x) WTr thendW, = -WTdx
Incorporating Equations (B13) and (B14) into Equations (B11) and (B12),
Incorporating Equations (B15) and (B16) into Equation (BlO),
Factoring out WT‘ and rearranging terms,
d(momentum flux) = - X
Pg( 1 -R,)
CAQC NO. 3 8
PRATT L WHITNIY AIRCRAtT PWA-2530
Since R, i s a function of WT,fluid properties, and x in the correlation of Lockhart and MartinelliS2., dRL=(%!d dx for this case.
dx
Therefore,
d (momentum
and
(B21) dx dA
and sensible heat has been neglected.
The local heat flux into the tube qo = WT where dA=2 I ro dP
X -
Thus
and
dx Substituting this expression for - into Equation (B20), it becomes de
X [ (= Pg(1-RL)
( m z l ( l - X ) * P g ( l - R , J 2 ) z X2 dRL ] dP (B24)
A l l t e rms in this equation can be evaluated from known local quantities
except for - dRL . In order to transform the -L dR
dx dx t e rm into
known local quantities requires use of an empirical method o r a more complex iterative procedure for solution. was used in this program. dR -, dx
An empirical expression The derivation of this expression for
in t e rms of known quantities'will be presented later. dRL ro, and WT.
dx
PWA-2530 P R A T T L WHITNLY AIRCRAFT
The t e rm RL is the fraction of the tube a rea a t E which contains liquid. through the equation
Thus, it is related to the thickness of the condensate film
or
Substituting the expressions from Equations ( B l ) , (B2), (B3), and (B24) into the momentum equation for the entire fluid stream,
- - l r 2 (-) d P d.Q 0 - l - 2 -dQ- dP -lr: cos 0 d& friction dll
dRL dQ ' ( P L R L ' ( l-x)z 0 P g ( l - R J ' ) K X2 ]
- - -($)(+) [ '(&- Pg(l-RL)
Dividing through by - r ri d.Q
d P t dP t COS B L ( R L P L + ( l - R L ) P g ) (,PI friction d Q
g C
2 w X
TgC
P g ( l - R L ) ' ) x X2 ~ R L ] Solving for - d P
dR - - dP - -(-) d P R L P L t (1-RL)p dQ dP friction
PAQL NO. 4 0
PRATT L WHITNLY AIRCRAFT PWA-25 30
This expression for dP/dP, together with empirical relationships for (dP/dP)friction and dR,,/dx can be substituted into Equation (A25) to give the shear s t r e s s distribution across the liquid film.
In order to determine the rate of change of liquid fraction RL with quality x using only local quantities at a given axial station in the case being analyzed, the empirical method of Lockhart and Martinelli 32 was used. X, which is the square root of the ratio of the vapor to liquid pressure drops i f these phases were flowing alone in the entire pipe. enables dRL /dx to be found since
They found that R L could be correlated as a function of
This
Lockhart and Martinelli's correlating line of R, versus X was fitted with the following equation
0.299 X O* 756
0.299 Xo* 756 t 1 R, =
Using this equation dRL /dX can be found by differentiation
0.226 - - - ~ R L dX x 0.244 ( 1. o t 0. 299X00 756)
The expressions for X to be used in this equation depend upon the flow regimes, laminar o r turbulent, in both the vapor and liquid st reams 32
laminar liquid- laminar vapor
("A) 0. 5 (e) 0. 5 xvv =
PL %
PAQL NO. 4 1
C R A W & WHITNIY AIRCRAFT
turbulent liquid- turbulent vapor
~-
PWA-2530
turbulent liquid- laminar vapor
laminar liquid- turbulent vapor
Transition is considered to occur when the Reynolds number of either fluid, calculated assuming it completely fills the pipe radius, is 1000. From these equations for X, the four expressions for dX - can be found by differentiation dx
0. 5
1-x -7 dXvv = - 0 . 5 dx
(F.37)
0 . 4 dXtv
0 . 9 - 0 . 4 ~ S(1-x) 0.5
dXvt - 18 .65 - - dx - Reg O - 4 bug
where the Reynolds numbers a r e the full bore Reynolds numbers mentioned above.
PRATT L WHITNLY AIRCRAFT PWA-2530
Combining Equations (B30) and (B32) with Equations (B37) through (B40) , the expressions for regimes. dx
dR, a re obtained for the four flow
0 . 5 0 . 5
where Xvv can be obtained from Equation (B33)
0. 5 0. 1 .O. 1
where Xtt can be obtained from Equation (B34)
0.5 0 .4x t 0. 5 0 . 4
-0 .0121 ReL %5 x0* 244( 1 . 0 t 0 .299 Xo* 756)2 ( 4, f i g ) ( 5( 1- 100 5
tv tv
where Xtv can be obtained from Equation (B35)
where Xvt can be obtained from Equation (B36) , through (B44) , together with Equations (B33) through (B35) , enable d R , / d x to be determined as a function of known local quantities.
Equations (B41)
PAOL NO. 43
CRATT & WHITNCY AIRCRAFT PWA-2530
APPENDIX C
Derivation of Heat Flux Distribution Equation
cAoc NO. 44
PRATT & WHlTNLY AIRCRAFT PWA-2530 ~
APPENDIX C
Derivation of Heat Flux Distribution Equation
The derivation of the heat flux distribution equation depends upon the assumption that the amount of heat convection in the liquid film and conduction in the axial direction a r e negligible compared to the total heat t ransferred in the radial direction ( see sketch below)
rvapor ,wal l
liquid J
The heat t ransferred at the wall and at any distance y from wall i n the liquid film a r e equal; thus Qo= Q Y
The a r e a a t the wall for a unit length is A,= 2 IT ro
The a r e a at the location y from the w a l l is .Ay= 2 r (ro-y)
Since qo = a and q - 9 - A0 AY
Thus
Since
PAQC NO. 45
PRATf & WHITNCY AIRCRAFT PW A- 25 3 0'
APPENDIX D
Nomenclature
P R A T T h WHITNCY AIRCRAFT PWA-2530
NOMENCLATURE
A+
g gc h hfilm hinter fac e
WT
Y+
dB
X
Y
d P
van Driest turbulent damping constant specific heat of vapor at constant pressure, Btu/lbm OR specific heat of liquid at constant pressure, Btu/lbm O R
acceleration of gravity, f t /hr2 gravitational constant, lbm ft/lbf hr2 local heat transfer coefficient, Btu/hr sq ft OR local heat transfer coefficient of liquid film,Btu/hr sq ft OR local heat transfer coefficient of interfacial resistance,
Btu/hr sq ft OR
enthalpy, B tu/lbm mechanical equivalent of heat, f t lbf/Btu Prandtl mixing length constant liquid thermal conductivity, Btu/ft h r OR axial length, ft
2 rate of condensation per unit a r ea lbm/hr f t molecular weight of vapor, lbm/lbm mole local static pressure, lbf/sq ft vapor pressure, lbf/sq f t Prandtl number of liquid heat flux a t any position y, Btu/hr sq ft heat flux at the wall, Btu/hr sq ft universal gas constant, ft lbf/lbm mole OR liquid fraction radial distance from tube axis, f t pipe radius, f t dimensionless pipe radius defined on page 16 local temperature a t position y, OR liquid temperature at liquid-vapor interface, OR temperature a t the wall, OR saturation temperature, OR dimensionless temperature defined on page 8 local velocity a t position y, f t /hr dimensionless velocity defined on page 8 average liquid velocity, f t /hr average vapor velocity, f t /hr - friction velocity =
gas flow rate, lbm/hr liquid flow rate, lbm/hr total flow rate, lbm/hr quality, Wg / WT distance from wall, f t dimensionless distance defined on page 8
static pressure gradient, lbf/cu ft
PAGE NO. 47
CRATT L WUITNLY AIRCRAFT
a eH
*M 6 a+ e
PR PL U 7
TO
ratio of eddy diffusivities eddy diffusivity for heat, sq f t /hr eddy diffusivity for momentum, sq f t /hr thickness of liquid film, ft dimensionless thickness of liquid film angle of pipe orientation measured from vertical
latent heat of vaporization, Btu/lbm vapor dynamic viscosity, lbm/ft h r liquid dynamic viscosity, lbm/ft h r liquid kinematic viscosity, sq f t /h r gas density, lb,/cu ft liquid density, lbm/cu ft accommodation coefficient shear s t ress , lbf/sq ft shear s t r e s s a t wall, lbf/sq ft
upward, degrees
CAOL NO. 48
~ P R A R L WWITNCY AIRCRAFT PWA-2530
APPENDIX E
References
CAQC NO. 4.9
P R A T T h WHITNLY A I R C R A F T PWA-2530
APPENDIX E
Ref e r enc e s
1. Hess, H. L., H. R. Kunz and S. S. Wyde, Analytical Study of Liquid Metal Condensers, Vol. 1, Design Study P W A Report 2320, NASA Report CR 54224, July 1964
2. Colburn, A . P . , Trans. AIChE 30, 187, 1933
3. Carpenter, E. F. , and A. P. Colburn, Proc. General Disc. Heat Transfer, Inst. of Mech. Engr. and ASME, pp. 20-26, July 1951
4. Bergelin, 0. P. , P. K. Kegel, F. G. Carpenter and Car l Gazley, J r . , Co-Current Gas-Liquid Flow 11. Flow in Vertical Tubes, Heat Transfer and Fluid Mechanics Institute, Published ASME, 1949
9. Martinelli, R. C., and D. D. Nelson, Prediction of P r e s s u r e Drop During Forced Circulation Boiling of Water, Trans. ASME, Vol. 70, 1948
loa. Dukler, A . E . , Chern. Eng. Prog. Symp. Ser ies Vol. 56, No. 30 pp. 1-10, 1960
lob. Dukler, A. E. , Appendix to 10a Document 6058, American Documentation Inst. , Photoduplication Service, Library of Congress, Washington, D. C.
PAQC NO. 50
P R A ~ e WHITNCY AIRCRAFT PWA-25 30
11.
12.
13.
14.
15.
16.
17.
18.
19.
. 20.
2 1.
Deissler, R.G., Heat Transfer and Fluid Friction for Fully- Developed Turbulent Flow of A i r and Supercritical W a t e r with Variable Fluid Properties, Trans ASME Vol. 76, No. 1, 1954, p. 73
Kirillov, P. L., V. I. Subbotin, M.Ya. Suvorov and M. F. Troyanov, Journal of Nuclear Energy, Part B: Reactor Technology, 1959, Vol. 1, pp. 123-129
Sukhatme, S. P., and W. M. Rohsenow, Heat Transfer During Film Condensation of a Liquid Metal Vapor, Third Annual High Temperature Liquid Metal Heat Transfer Meeting, Oak Ridge National Lab., Sept. 4-6, 1963, AEC Contract No. AT( 30- 1)-2995
Misra, B., and C.F. Bonilla, Chem. Eng. Progr . Symp. Vola 52, NO. 2, pp. 7-21, 1956
Chen, M. M., Trans. ASME, Journal of Heat Transfer 83, 48-60, 1961
Lee, J., Remarks on Liquid Metal Condensation, AIChE Paper 1, Seventh National Heat Transfer Conference, August 9-12, 1964
Alkali Metals Boiling and Condensing Investigations, Quarterly P rogres s Report 8, Edited by F. E. Tippets, NASA CR-54138, Oct. 20, 1964
van Driest, E. R., On Turbulent Flow N e a r a Wal l , Journal Aeronautical Science, November 1956, pp. 1007-101 1
Laufer, J., The Structure of Turbulence in Fully-Developed Pipe Flow, NACA Report 1174, 1954
Beckwith, W. F. , and R. Fahien, Determination of Turbulent Thermal Diffusivities for Flow of Liquids in Pipes, Ames Laboratory IS-734, November 1963
Jenkins, R., Heat Transfer and Fluid Mechanics Institute, p. 147, Stanford University P res s , 1951
PAQL NO. 51
C R A W WHITNCY AIRCRAFT PWA-2530
22.
2 3.
24.
25.
26.
27.
28.
29.
3 0.
31.
32.
33.
34.
Deiss ler , R.G., NACA Research Memo E52 F05, 1952
Lykoudis, P . A . , and T. S. Touloukian, Trans. ASME, Vol. 80, No. 3, 653, 1958
Rohsenow, W. M. and S. Cohen, The Influence of the Ratio of Eddy Diffusivities on Heat Transfer to Liquid Metals, MIT Heat Transfer Lab Report, June 1960
Mizushina, T., and T. Sasano, Paper No. 78 presented at 196 1 International Heat Transfer Conference, Boulder, Colorado
Azer, N. Z., and B. T. Chao, International Journal of Heat and Mass Transfer, Vol. 1, No. 1, 121, 1960
Dwyer, 0. E., AIChE Journal, Vol. 9, No. 2, p. 261, March 1963
Sesonske, A., S. L. Schrock and E. H. BUYCO, AIChe Preprint No. 25, Sixth National Heat Transfer Conference, Boston, Mass. , August 1963
Subbotin, V.I., M. Kh. Ibragimov, M.N. Ivanovskii, M. N. Arnol'dov, and E. V. Nomofilov, Turbulent Heat Transfer i n a Stream of Molten Metals, Soviet Journal of Atomic Energy, Vol. 10, No. 4, pp. 384-386, April 1961
Brown, H. E., B.H. Amstead and B. E. Short, Trans. Am. SOC. Mech. Engr. , Vol. 79, No. 2, p. 279, 1957
Isakoff, E. , and T. B. Drew, Heat and Momentum Transfer in Turbulent Flow of Mercury, General Discussion on Heat Transfer , Inst. Mech. Eng. and ASME, 1951, pp. 405-409
Lockhart, R.W., and R.C. Martinelli, Chem. Eng. Prog. Symp. Series, Vol. 45, No. 1, p. 39, 1949
Dukler, A. E . , PhD. Thesis, Univ. of Delaware, 1951
Charvonia, D. A. , A Study of the Mean Thickness of the Liquid Film and the Characteristics of the Interfacial Surface in Annular Two-Phase Flow in a Vertical Pipe, Jet Propulsion Center, Purdue University, Report Number 1-59- 1, May 1959
PAOC NO. 52
PRATT a WHITNLY AIRCRAFT
~ ~~
PWA- 25 30
35. Chien, S. F., An Experimental Investigation of the Liquid Fi lm Structure and P r e s s u r e Drop of Vertical , Downward Annular, Two-Phase Flow, PhD Thesis, University of Minnesota, April 1961
36. Collier, J. C., and G. F. Hewitt, Data on the Vertical Flow of Air-Water Mixtures in the Annular and Dispersed Flow Regions, Chemical Engineering Division, Atomic Energy Research Establishment, Harwell, Berkshire, England, Report AERE-R-3455, 1960
37. Carpenter, F., Heat Transfer and P r e s s u r e Drop for Condensing P u r e Vapors Inside Vertical Tubes at High Vapor Velocities, PhD Thesis, Univ. of Delaware, 1948
38. Goodykoonte, J., and R. G. Dorsch, Unpublished Prel iminary Data, NASA Lewis Research Center
39. Schlichting, H., Boundary Layer Theory, McGraw-Hill, New York, 1960
CRAW k WHITNCY AIRCRAfT
APPENDIX F
Figures
PAOL NO. 54
CRATT e WHITNLY AIRCRAFT
SKETCH OF ASSUMED FcloW MODEL
PWA-2530
LIQUID FILM MASS TRANSFER I- - %VAFOR CORE AT INTERFACE -
I DIRECTION OF FWW * I
\VAPOR LIQUID INTERFACE
Figure 1
PRATT e WHITNCY AIRCRAFT
0 (u
tu Q) k 1 M
G
I PRATT & WHITNCY AIRCRAFT
VARIATIW OF r/T',', AND €,/VL WITH y/f FOR FUUY-Dn/ELOP€D PIPE F W
1.0
0.9
0.8
0.7
0.6
f/ro 0.5
0.4
0.3
0.2
0. I
0 0 0.2 0.4 0.6 0.8 I .o
Y/r
PWA-2530
Figure 3
PRATT 8 WHlTNLY AIRCRAFT
CAOC NO.
P R A T T h WHITNLY AIRCRAICT PWA-2530
I .2
1.0
U I
v) W E 0.8 > J k &
8
E 0.6
W n
0 0.4 I- - a a
0.2
0 0 0.2 0.4 0.6 0.8
Y/r
Figure 5
CRAW L WHITNCY AIRCRAFT PWA-2530
VARIATION OF RATIO OF EDDY DIFFUSIVITIES . .
WITH y/r AND REYNOLDS NUMBER
FROM DATA OF SESONSKE, ET AL (REE28)
I .o
0
E Oo2
0 0 0.2 0.4 0.6 0.8 I .o
Y/r
Figure 6
CRATT & WHITNCY AIRCRAFT
CALCULATED FILM THICKNESS vs MEASURED FILM THICKNESS
FOR CHARVONIA'S AND CHIEN'S DATA - dP BASED ON EXPERIMENTAL DATA
d l CONDITIONS
1. AIR-WATER MIXTURES 2. VERTICAL DOWNFLOW 3. PRESSURE IATM 4. TEMPERATURE 55-85O F
B CHARVONIA (REF. 3 4 ) 0 CHlEN (REF. 35) TUBE l.D.=0.167 FT ,
TUBE l.D.=0.208 FT
.oo I
0 0 .OOl .002 .oo3 .004 .oos
FEET MEASURED'
Figure 7
CRAW L WCIITNCY AIRCRAFT
CALCULATED FILM THICKNESS vs MEASURED FILM THICKNESS
FOR CHAWONIA I S AND C w I s BASED ON DUKLER CORRELATION d l
CALCULATED FILM THICKNESS vs MEASURED FILM THICKNESS
FOR CHARVONIA'S AND CHIEN'S DATA BASED ON ~ K H A R T - M A R T I N E L L I COdRELATloN
d4 CONDITIONS
I . AIR-WATER MIXTURES 2. VERTICAL DOWNFWW 3. PRESSURE =: IATM 4. TEMPERATURE 55-8S°F
0 CHARVONIA (REF: 34 ) TUBE I.D.=0.208 FT
0 CHIEN(REF: 35) TUBE l .D~O.167 F T
,006
.OO 5
k
,004
& w 3 ,003
3 ,002
Qo
.001
0
PWA-2530
0 .0004 .OW8 ,0012 .006 .0020 0024
8 MEASURED FEET
Figure 9
I CRAW e WHITNCY AIRCRAPT PWA-2530
.00160
.00140
YARIATION OF FILM THICKNESS WITH AIR AND WATER FLOW RATES
CONDITIONS I. AIR-WATER MIXTURES 2.VERTlCAL WWNWARD FLOW 3. PRESSURE = I ATM 4. TEMPERATURE 55-800 F 5. TUBE I.D. * 0.208 FEET
.00120
t
I I I, I I I UAlER FLL)w RATE
I I I- NOTE: DATA OBTAINED BY CHLLRWNIA(ROF.
THEORETICAL RESULTS BASED ON DUKLER'S PRESSURE GRADIENT CORRELATloly
0 ' I I 1 I I 0 0. I 0.2 0.3 0.4 Q5 0.6
AIR FLOW RATE- LB/SEC
34;)
F i g u r e 10
C R A R WHlTNCY AIRCRA?T
VARIATION OF FILM THICK- YlTH AIR AND WATER FLOW =
CONDITIONS I. AIR-WATER MIXTURES 2.VERTlCAL DOWNWARD FWW 3. PRESSURE = I ATM 4. TEMPERATURE 8 5 5 - 8 0 O F 5. TUBE I. D. 8 0.208 FEET
I I I 1 WATER 1 THEORETICAL ,DATA,
-------- 0 Ww13LBBEC
DATA OBTAINED BY CtiARVONIA(REF.34)
6
AIR FLOW RATE-LWSEC
Figure 11
C R A W & WWITNSY AIRCRAFT
CAKULATED FILM THICKNESS VIS MEASURED F ILM THICKNESS FOR COLLIER AND HEWITT DATA
0 W
4 3 3 4
1 . 2. 3. 4. 5.
Inuv' -UR'TlU€UI W T K m
CONDITIONS AIR -WATER MIXTURE VERTICAL U P F U W PRESSURE = I A T M TEMPERATURE r 6S0 F TUB€ 1.0. .I04 FT
.oO02
0
a MEASURED * FEET
Figure 12
PWA-25 30 CRATT WHITNEY AIRCRAfT
ECT OF LlQU ID SUBCOOLING ON PER CENT MFFF,- Wa/WT AN0 QUqLlTY BASD ON ENTHqlW
CONDITIONS
I . FLUID-WATER 2. VERTICAL DOWNFLOW 3. PRESSURE L N E L s 25 PSI 4. SATURATIW TEMPERATURE z=24O0 F 5. TUBE 1.0. = 0.02445 F l 6. TOTAL FLOW RATE = 57.9 LWHR
+ NOTE: THESE CONDlTlONS ARE M E SAME A!5 THOSE IN FIG. I7
24
20
16
12
8
4
0 0.0 0.2 0.4 08 0.8 1 .o
X BASED ON ENTHALPY
Figure 13
PRATT WHITNSY AIRCRAFT
I20C
* t
N
a X \ 3
I I- z w
k 11oc
zi E
8 IOOC
8 Ir. v) 2
I- a a
s W r
9oa
VARIATION OF HEAT TRANSFER COEFFICIENT WITH LIQUID VISCOSITY
CON01 TI ON S 1. FLUID-WATER 2. VERTICAL DOWNFLOW 3. PRESSURE’ 16.8 PSlA 4. SATURATION TEMPERATURE= 219 OF 5. TUBE INSIDE DIAMETER:0.0382 FT 6. TOTAL FLOWS736 LB/HR 7. QUALITY =0.1276 8. HEAT FLUX DENSITY : 61,900 BTU/HR FT2 9. FRICTIONAL PRESSURE GRADIENT BASED ON
LOCK HART - MART I N ELL I CORRELATION
% L EVAWATED AT TSAT 1
665
pLEVALWTED AT TMFAN
PL EVALUATED AT
TWALL
0.95
Figure 14
PRATT & WHITNCY AIRCRAFT
~~
PWA-25 30
CALCULATE0 HEAT TRANSFER COEFFICIENT vs EXPERIMENTAL HEAT TRANSFER COEFFICIENT
@/dl BASED ON LOCKHART-MARTINELL1 -
CORRELATION. Q OBTAINED FROM EQ.4b CONDl T I ONS
1. FLUID-WATER 2 VERTICAL DOWNFLOW 3. PRESSURE RANGE 16.8- 25.0 PSlA 4. SATURATION TEMPERATURE RANGE 5. AVERAGE MASS VELOCITY RANGE
219- 240.F 9,~OO-IIO,OOO LWHR F t 2
2000c
a 12000
3 c I m
t
o.oc
1.D -0.02445 FT
TUBE I.D.-OD382 FT ,
1 I I I 4000 8000 12000 16000- 20000 1
Figure 15
P R A T T e WHITNLY AIRCRAFT PWA-2530
CALCULATED HEAT TRANSFER COEFFICIENT vs EXPERIMENTAL HEAT TRANSFER COEFFlCl ENT.
dp/dP BASED ON’ DUVLER CORRELATION OBTAINED FROM EQ.4.b
CONDITIONS
1. FLUID- WATER 2 VERTICAL DOWNFLOW 3. PRESSURE RANGE 16.8-25.0 PSlA 4. SATURATION TEMPERATURE RANGE 219- 240.F 5. AVERAGE MASS VELOCITY RANGE s~oo-~~o,ooo LWHR FT*
20000
IL 0
16000 I- LL
12000 3 I- a I
c- J
8000
a 3 0 4000 J
0 t a
0.0 0.0 4000 8000 12000 16000 20000
h EXPERIMENTAL -BTU/HR n2v
Figure 16
PRATT WHITNCY AIRCRAFT PWA-2530
EXQERIMENTAL AND CAFULA TED HEAT TRANSFER COEFFICIENTS v s QUALITY. 'NASA OATA (REF,3s),
_CO"S 1 . WID-WATER 2. VERTICAL DOWNFIJW 3. PRESSURE LWELz25 PSlA 4 SATURATION TEMPERATURE = 240.F 5. TUBE I.D. = 0.02445 FT 6 TOTAL FWW R A E = 57.9 LWHR 7. dWdJ BASE0 ON DUKLER CORRELATIW
QUALITY Figure 17
PRATT b WHtTNCY AIRCRAFT
P t a
w
I \
5
E 8 a b
t
I t z W 5
0
W
z E W 1:
COEFFICIENTS vs QUALITY CARPENTER :DATA (REF! 3271
CALCULATED HEAT TRANSFER COEFFICIENT vs EXPERIMENTAL HEAT TRANSFER COEFFICIENT.
dP/dlt BASED ON LOCKHART-MARTINELLI CORRELATION. (Y = 1.0
CONDITIONS
I. FLUID-WATER 2 VERTICAL DOWNFLOW 3. PRESSURE RANGE 16.8-25.0 PSlA 4. SATURATION TEMPERATURE RANGE 219-240.F 5. AVERAGE MASS VELOCITY RANGE 9,8OO-l10,000 LB/HR FT2
FT
Ox) 4000 8000 12000 16000 20000
h EXPERIMENTAL-BTU/HR F+F
Figure 1'9
PRATT e WHITNLY AIRCRAFT
CALCULATED HEAT TRANSFER COEFFICI ENT vs EXPERIMENTAL HEAT TRANSFER COEFFICIENT
dWd! BASED ON DUKLER CORRELATION
20000 0 8
N
16000
2000
8000 4 3
4 4000 Y c
0.0
a = 1.0 dONDlTlONS
1 FLUID- WATER 2. VERTICAL DOWNFLW 3. PRESSURE RANGE 16.8-25.0 PSlA 4. SATURATION TEMPERATURE RANGE 219-240.F s. AVERAGE MASS VELOCITY RANGE 9,eoo-iio,ooo LWHR F T ~
FT
0.0 4000 8000 12000 16000 2oooO
h EXPERIMENTAL - BTUIHR FT*OF
Figure 20
PRATT & WUITNLY AIRCRAFT
U+
QUAL ITY
18 CONDITIONS
I. VPrrlCAL 2 pREssuRE=16.8 PSlA 3 SATURATION fCM#RANRE=21S°F 4. TOTAL FLOW RATE = 73.6 LBMR
6 TUBE INSIDE MAMETOR = 0.0382 FT 7. HEAT FWX DENSITY 861,WO BlU/HR FT2 8. FRICTIONAL PRESSURE GRADIENT BASED ON
16 -5 QwuTY=0.128
LOCKHART-MARTINUI CORELATION
I - I I I I I I I I I
14 m T U R E PFKWllE m A I N FRoMOOMPVrOR-
UNNERSAL VELOCITY PFEOGllAM
W l T Y PROFILE OBTAINED FFIoM/ coMPuTERmoGRAM
12
56
32
20
24
t + 20
16
12
0
4
0 I 2 3 4 5 6 1 8 9 1 0
Y +
Figure 211
PRATT .I WHITNCY AIRCRAFT PWA-2530
WITH Y/r FOR A LOW QUALITY CONDENSING WA TER POINT CONblTlONS
I. VERTICAL DOWNFLOW 2 PRESSURE = 6.6 PSlA 3 SATURATION TEMPERATURE = I 2 W F 4. TOTAL FLOW RATE = 40 LB/HR
VARIATION OF U' AND t* WITH y* FOR CONDENSING POTASSIUM
U+
16
14
12
IO ~-
8 ,
6 ,
4
2
/
O i
\ TEMPERATURE PROFILE . 'OBTAINED FROM COMPUTER - PROGRAM
20 3 o I
0.16
0.14
0.12
0.10
t+
0.08
0.06
0.04
0.02
0 0
Y+
Figure 26
CRATT & WHITNLY AIRCRAFT PWA-2530
/b&J VARIATION OF TIrO C, /VL Qt AND e, -
WITH Y/r FOR CONDENSING POTASSIUM
I .o
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0. I
0
L SATURATION TEMPERATURE s1-F I, toTAL FLOW RATE 8 4 0 LB/HR L QUALITY=O.SO
0 .002 ,004 .006 .008 .OlO ,012
Y / f
10.0
9.0
8.0
7.0
6 .O
5.0
4 .O
3 .O
2 .o
I .o
0
Figure 27
P R A n L WHITNCY AIRCRAFT PWA-25 30
2 PRESSURE = 6.6 PSlA 3. SATURATION TEMPERATURE = 1250OF 4. TOTAL FLOW RATE 140 LBIHR 5. QUALITY = O.$ -.O*q 6. TUBE INSIDE DIAMETER =0.05208 FT 7. HEAT F W X DENSITY =60,000 BTU/HR FT2 8.FRlCTlONAL PRESSURE GRADIENT BASED ON
