Steady-State Applied Design by Wayne C. Edmister and Joseph M. Marchello 4-1. Genwal. The basic equation for the steady le eonduction of beat is obtained by taking Eq. 3-2, separating the variables, substituting for the th&mal conductivity, which vari% with tempera- ture, the functional relationship k = f(t), and integrating qs;” = -l;:(t)dt : Eq. 4-1 Multiplying and dividing the right si,de by (tr - t,) given s t2 - f iti dt s x2 dx t1 -~ = 9 A te - t1 ct2 - w x1 Noticing that the tenn[{z f(t) dt] /(tz - 111’ is the mean value of k over the temperature range tl to tJ and designating it by k, give for the rate *of he:t flow q = k,,, (tl - tz) / J x2 xI dx,‘A Eq. 4-2 It should be noted that the use of k,, does not introduce any error regardless of the relation be- tween 5 and A. 4-Z. Homogeneous piane z@Z. For a piane mall, SS shown in Fig. 3-1, with psrallel’surfaces z ft apart, the area A is constant and independent of the diatance 5. With a steady state, the surface temperature tl and tz on the warmer and colder sides are also constant. Because of the homogeneity of the wall, the temperature gradient dt/dx is eonstsnt throughout its thickness if it is assumed that nny changc of k with temperature may be neglected. In Eq. 4-2, x1 = 0 and xp = x, Fig. 3-1, and q=Ak,,,(tl-tz)i’x .Eq.4-3 Petro/Chem Engineer March 1965 43 where: A is the area, sq ft, tiormal to the direction of heat Aow IC,, is the mean thermal conductivity, Btu/ hr ft OF, for the temperature range tz.to tl tl - t% is the temperature differente, “F Eq. 4-3 can be presented in another form, namely. q=AC(tl-tal =A(t,-tz!/R Eq.4-4 in whieh C = k,,;‘x Eq. 4-5 is the thmmal cmrd~uctmce of the wall, or the ability of the wall of thickness a to conduct heat and is expressed in Btu/sq ft “F hr. The reciproca1 of C is designated by R and is the themd n%~&wxe which the wall offers to the ,flow of beat, R = x/k, Eq. 4-6 Eq. 4-3 applies to a homogeneous body with a constant cross section in which heat flows only in one direction, perpendicular to the surfaces, and is referred to as u,nidirectional flow. Therefore, Eq. 4-3 may be used to c~alculate the heat conduetion through a piane wall only when there is no heat flom through the edges or if the beat-transfer area is so large compared with the perimeter that neglecting the heat conduetion through the edges will not result in a large error. Eq. 4-3 may be used also to calculate the axial heat conduction through a prismatic or cylindrical bar with parallel ends having temperatures tl and ti if the bar is insulated SOthat beat does not flow through the sides. As mentioned in Section 3-l and shown in Fig. Z-1, when calculating the beat slow from a hotter fluid to a colder one, separated by a wall, the temperatures to be used in Eq. 4-3 are those of the surfaces of the wall, and not of the fluids. EXAMPLE 4-l. A brick building wall 12$$ in. thick has an inside surface temperature of 53 F and an Outside surface temperature of 15 F; the wall is 9 ft high and 32 ft long. Find: (a) the beat trans- fer by conduetion through the wall per hour, (b)
Transcript
Steady-State
Applied Design by Wayne C. Edmister and Joseph M. Marchello
4-1. Genwal. The basic equation for the steady
le eonduction of beat is obtained by taking Eq. 3-2, separating the variables, substituting for the th&mal conductivity, which vari% with tempera- ture, the functional relationship k = f(t), and integrating
qs;” = -l;:(t)dt : Eq. 4-1
Multiplying and dividing the right si,de by (tr - t,) given
s
t2 - f iti dt
s
x2 dx t1 -~ = 9 A te - t1
ct2 - w
x1
Noticing that the tenn[{z f(t) dt] /(tz - 111’
is the mean value of k over the temperature range tl to tJ and designating it by k, give for the rate
*of he:t flow
q = k,,, (tl - tz) / J x2 xI dx,‘A Eq. 4-2
It should be noted that the use of k,, does not introduce any error regardless of the relation be- tween 5 and A.
4-Z. Homogeneous piane z@Z. For a piane mall, SS shown in Fig. 3-1, with psrallel’surfaces z ft apart, the area A is constant and independent of the diatance 5. With a steady state, the surface temperature tl and tz on the warmer and colder sides are also constant. Because of the homogeneity of the wall, the temperature gradient dt/dx is eonstsnt throughout its thickness if it is assumed that nny changc of k with temperature may be neglected.
In Eq. 4-2, x1 = 0 and xp = x, Fig. 3-1, and
q=Ak,,,(tl-tz)i’x .Eq.4-3
Petro/Chem Engineer March 1965 43
where: A is the area, sq ft, tiormal to the direction of heat Aow IC,, is the mean thermal conductivity, Btu/ hr ft OF, for the temperature range tz.to tl tl - t% is the temperature differente, “F
Eq. 4-3 can be presented in another form, namely.
q=AC(tl-tal =A(t,-tz!/R Eq.4-4
in whieh
C = k,,;‘x Eq. 4-5
is the thmmal cmrd~uctmce of the wall, or the ability of the wall of thickness a to conduct heat and is expressed in Btu/sq ft “F hr. The reciproca1 of C is designated by R and is the themd n%~&wxe which the wall offers to the ,flow of beat,
R = x/k, Eq. 4-6
Eq. 4-3 applies to a homogeneous body with a constant cross section in which heat flows only in one direction, perpendicular to the surfaces, and is referred to as u,nidirectional flow. Therefore, Eq. 4-3 may be used to c~alculate the heat conduetion through a piane wall only when there is no heat flom through the edges or if the beat-transfer area is so large compared with the perimeter that neglecting the heat conduetion through the edges will not result in a large error. Eq. 4-3 may be used also to calculate the axial heat conduction through a prismatic or cylindrical bar with parallel ends having temperatures tl and ti if the bar is insulated SO that beat does not flow through the sides.
As mentioned in Section 3-l and shown in Fig. Z-1, when calculating the beat slow from a hotter fluid to a colder one, separated by a wall, the temperatures to be used in Eq. 4-3 are those of the surfaces of the wall, and not of the fluids.
EXAMPLE 4-l. A brick building wall 12$$ in. thick has an inside surface temperature of 53 F and an Outside surface temperature of 15 F; the wall is 9 ft high and 32 ft long. Find: (a) the beat trans- fer by conduetion through the wall per hour, (b)
the conductance of the wall, (c) the thermal re- sistance of the nall.
SOLUTION: From Table 3-5, Eor brick with l?& moisture, k = 0.40; the area A = 9 x 32 = 288 aq ft; thickness z = 1X5/12 = 1.04 ft.
(a) by Eq. 3-3, Q = 288 x 0.4(53 - 15)/1,.04 = 4200 Btuihr
(b) By Eq. 3-5, C = 0.40/‘1.04 = 0.385 Btu/sq ft hr “F Btu,!sq ft hr ‘F
(cj By Eq,. 3-G, R = 1iC = 1.04/0.40 = 2.6 sq ft hr “F/Btu
4-3. Composite @me ~wall. A wall made up of three layers iu illustrated in Fig. 4-l. Each layer consists of a homogeneous plane wall. Therefore, by Eq. 4-6 the thermal resistances of the Isyers are
R, = xl/k,; Ri = xJk2; RB = xg’k, Evidently, the resistance R of the composite wall ia equa1 to the sum of the separate resistances
R = RI + R, + RS = x,/k, + xz/kr + xajk:; Eq. 4-7
and the thermal conductance of the wall
C = l/R = l/(x,/k, + x/kz -lm xa’kJ) Eq. 4-8
Witl! C or R thus detevmined, the rate of beat flow is found by Eq. 4-4 using for the temperature differenee the extreme surface temperatura of the composite va11 (tl - t4).
Figure 4-l. Heat transfer by conduction through a composite wall.
The interior surface temperatures may be cal- culated, notieing that with steady-state conditions the heat flow Q through each layer is the same and is equa1 to the tota1 beat flow,
If tl and tl are knonn, ts and t3 may be calculated Erom Eq. 4-9.
Sometimes, in engineering constructions, com- posite walls are employed in which the surfaces of the layers are not planar oi the layers are not homogeneous, BS concrete blocks with holes. For standard types of such nalls, tests are made of
the wall as a whole, and its conductance C is given in tablca. The value of the reciproca1 of C from such data, in a,ccordance with Eq. 4-5 and 4-6, mut be used for the corresponding ierm in finding the over-z.11 conduetance by Eq. 4-8.
EXAMPLE 4-Z. If the wall in Example 4-l is 12$5: jn. thick and built of 8 in. concrete blocks made of sand and grave1 aggregate with three ovrl COT~S with :$-in. stucco finish in the outside and :(Y-in. plsster on the inside, find: c.aj the conduct~ance of the wa~ll a,nd (h) the tntnl heat Mow, assuming the same surface temperntures.
SOLUTION: From Table 3-5, assume for stucco li1 = 0.40 because the outside wa,ll may have ZL high, moisture content, for inside plsster & = 0.25 consideri,ng dry plaster; for the ccmcrete blocks C = 0.781.
Ca) By Eq. 3-9
c = 1/(0.375/12 x 0.40 + liO. + 0.375/ 12 x 0.25) = 0.675 Btu/sq ft hP “F
(b;) By Eq. 3-4,
q = 288 x 0.675(53 - 15) = 7400 Btu/hr ai
With the 6urne temperature drop, the heat 10% is considerably higher because of a higher conduct- ance of this composite wall. Actually, due to a higher heat Row, the inside surface temperature will be slightly lower tha,n 53 F and the outside surfaee temperature will be slightly higher than 15 F; thus the temperature drop will be less than 38 F and therefore the heat loss will be somewhat~ smaller than 7400 Btu/hr.
The following example will bring out the in- fluente of ‘a small air space between two meta1 surfaces which cannot be avoided even with the most careful machining.
EXAMPLE 4-3. Determine the conductance at room temperature of: (aj a l-in. Monel-clad niild-steel plnte having a Monel layer of 0.10 in. and (b) the same plate if the two lapers are not welded to- gether but separated by an air space 0.0005 in. thick.
SOLUTION: From Tsble 3-8, the eonductivity of 0
Monel at 70 F is 14.4 Bt.u/hr ft “F; the conductivity of mild steel at 70 F is 25.4 Btu/hr ft ‘F.
isj By Eq. 4-6 and J-7, the tota1 resistance of the meta1 piate is
0.10 R = i2 x 14.4
~- -t$x+m = 0.00058 + 0.00212 -
= 0.00270 sq ft hr~“FjBtu
and by Eq. 3-5
(bi The additional resistance of the air space with k = 0.0151, Table 4-1, is by Eq. 4-6
R, = L!!!!L~ 12 x 0.0151
0.00276 sq ft hr “F/Etu
and the conducta~nce of the two layers of meta1 becomes
March 1965 Petro/Chem Engineer 44
1 c = o.00270 + 0.00276
= 183 Btu/hr sq ft “F
or less than one-half of that at condition (a). 4.4, Thick eylindrical wdl. Fig, 4-2 shows a
cylinder with a thick, homogeneous wall with heat flowing radially from the inside outward. In the cross sectional plane, the heat flow is in al1 direc- tions and is referred to as a tuo-dimensionai flow. If a cylinder with the length L is eonsidered at a radius r, its area is 27vL and the general Eq. 3-2 becomes
q = k,,, (tl - tz, s
5, r; dr/2rrL
Integrating, pive.7
q = Z,rLk,, (tl - ti) ,‘ln(r&~)
Using the more convenient diameters instead of radii giva
q = 2~Lk,,(tl-t~)/In(DI/D,) Eq. 4-10
Multiplying both the~numerator and denominator
a ‘- of Eq 4 10 by & and noticing that rDzL represents the outside area A-, Eq. 4-10 may be presented SS
Figure 4-2. Homogeneous cylindrical wall.
Ah,, (ti - td q= 0.5D,ln(De/D1)
Eq. 4-11
0 A comparison of Eq. 4-11 with the general Eq. 4-3 shows that the term 0.5DMD~/DJ is the equivalent thic!mess of the cylinder wall, z,, when the heat flow is computed referred to the outside cylinder surface.
Similarly, using the inner surface, the heat flov may be comppted from the expression
AI km (LI - 12) ’ = o;5D, In(D,/D,)
Eq. 4-12
In eonneetion with Eq. 4.11 and 4-12 it should be mentioned that while the tota1 heat flow through every concentric surface is constant in a thick- walled cylinder, the heat Aow per square foot de- creases from the inner surface toward the outside.
Instead of using Eq. 4-10 or its modifications, Eq. 4-11 or 4.12, one may use the basic Eq. 4-3 with a mean area A,,, for A. Equating the right sides of Eq. 4-3 and 4-10, substituting for the wall thickness z = CD2 - DJ/2 and far the ratio of the diameters, the ratio of the inner and outer cylindrical surfaces
Petro/Chem Engineer March 1965
and eancelling out k (tl - t2) giva 2A,j(D, - D,! = 2nLlln (Az/Al)
Eq. 4-13 is the logwithmic-mean wea designated by AI
A, = (A, - Ali/ln(An,/A~)
As may be easily shown, AL is always smaller than the simple arilJ~maf.ic-vkcan area
A,= (Ap+A1j/2 .Eq.4-14
However, if D2/D, = A2/A1 5 2, the error of using A, instead of Al is less than 4%; this securacy is sufficient far most problems in heat conduction.
4-5. Composite cylind~rical .wa~ll. Fig. 4-3 shows a composite cylindrieal wall sueh &s a meta1 pipe with a heat conductivity kl covered with a heavy layer of heat insulating materia1 having a con- ductivity ki, Under steady-state eonditions. assum- ing that a hot fluid is flowing in the pipe, the heat flow rate through the inner pipe and the insulating Shell will be the same, or by Eq. 4-10
4= ZrLk,(tl- tz) = 2?rLkz(tt - trj
In (DA’I) ln(Dw’Di)
from which
t _ t- = q ln(Dw’D~) q In Oh’Dz) 1 1 2nLkr
andtz-t? =p 2?rLkz . Eq. 4.15
Therefore, the over-al1 temperature differenee
tl - t3 = (tl- t*) + ctz - t31 =
&ln(D2/D’),/k, + ln(Ds;Dz)/kz]
Solving for q giva q = 2rrL(tl-t3j/ [ln(D2/DlVk, +
In(D/Dx)/kzl Eq. 4-16
, It-L- Figure 4-3. Composite cylindrical wall.
Generalizing this relation for u case of n layers giva
q = Z?rL(t, - t,+l)/~lln(Di+~/Di)/k,l . Eq. 4-17
Please turn to l>age 46
45
EXAMPLE 4-4. A 5-in. standard steel steam pipe is insulated with a 3.in. layer of 8571, magnesia. The surface temperature on the inside of the pipe is 400 F; on the outside of the insulation it is 100 F. Calculate: (a) the rate of beat flow pei hour per foot of pipe length; (b) the temperature on the outside of t,he steel pipe; (cl the conductance C of the pipe and insulation, based on the inside surfaee area.
SOLUTION: (aj For a 5-in. standard steel pipe, from Table 4-2, D1 = 5.047 in. and D, = 5.563 in. In Eq. 4-16, Da = 5.563 + 3 x 2 = 11.563 in.; the heat conductivity of steel (from Table 3-S) extra- polating for 400 F is
(b) By Eq. 4-15, tl - t2 = 108.2 In (5.563/5.0472/
Zn 25.0 = 0.056 F
and therefore,
ta = 400 - 0.06 = 399.94 F (cj From Eq. 4-4, with A = m x 5.047 x 1112 = 1.32 sq ft C = 108/1.32 (400 ~ IOOj =
0.27 Btu,/hr sq ft “F
4-6. Tlwee-dinwnsinnd beat flozo. When heat flows from a space in al1 directions, such as in the case of a furnaee or a refrigerator box, it is called a three-dim~ens%onaZ heat flow, but Eg. 4-3 is still applicable.
Hollo~o sphere. If heat is flowing radially in al1 direetions at a uniform steady rate through a hollow sphere with an inside radius ~1 and outside radius Q, Eq. 4-2 may be rewritten as
s ; dr/A = k,,,(tl - ti)/q Eq. 4-18
Substituting for A the expression far the surface at any radius 4vr’ and integiating between the limits ri and Q gives
Wr - l/rz) /47r = k,n(tl - tz)/‘q
from which, eliminating the fractions l/rl and 1/ i2, gives
4 = 4?rk,,, (t> - ta) il rJ(rz - il) Eq. 4.19
Noticing that the yeonzetrica.hean area is
A, = \/A, As = 4m, rz Eq. 4-20
and that vz - rl = z, the thickness of the wall of the sphere, substituting these values gives an ex- pression in the form of the basic Eq, 4-3:
46
q = A,k,,, (tl - tz) /x Eq. 4-21
Various practieal shapes. Eq. 4-21 in connection with Eq. 4-20 may be applied with sufficient accuiacy also to other shapes if they more or lesa approach a sphere, as the beehioe shape of com- mercial coke ovens.
4-7. Hollow reetangul~ar ywallelepifleds. In the case of hollow cubes or generally of rectangular paiallelepipeds with thick walls, BS encountered in furnaces, only parts of the surfaces receive the heat flow at right angles; and Eq. 4-21, in connec- tion with Eq. 4-20, wou,ld give an exaggerated oslue for the heat flow Q.
Unfortunately, the use of the Eq. 4-2 is also impract~ical because of the complexity of integra- tion of the term dz/A. In many cases however, the calculations may be made by Eq. 4-21, using instead of A,, the mean area A, computed from various empirica1 equations.’ The empirica.1 equa- tions giva below agree closely with equations whieh Langmuir obtained from theoretieal analysis and which are also confirmed by lata tests.
The first four equations apply to hollow ree- tangular parallelepipeds with al1 walls having the l same uniform thickness and with al1 faces inter- secting at right angles. Five cases mi11 be dis- cussed, differing in respect to their relative inside dimensions L,, Lz, and Ls.
Case 1. Al1 three dimenuions L 2 215:
A,, = Al + 2.16 x (L, -I- L, -l- La) + 1.2X2 Eq. 4-22
where A is the inside surface of the parallelepiped. The second term in the right side of Eq. 4-22 is a correction for the edges and the third term - for the eight corners.
Case 2. Two dimensions LI 2 z/5, La 2 x/5, and the dimension L,, < x/5: The corrections for the mal1 side can be neglected and Eq. 4-22 becomes
A,,, = A, + 1.86 x (L, + L2) -+ 0.35x’ . Eq. 4-23,
Case 3. One drnensmn L,,,,, 2 ~15, the other two LS < x/5 and Li < x/5: l
A, = 6.4 L ,n_ x/ln (A,/Al) Eq. 4-24 where Al is the inside and Az is the outside surfaee of the parallelepiped.
Case 4. Al1 three dimensions L < x/5:
A, = 0.79vA1Ai Eq. 4-25
EXAMPLE 4-5. A corkboard box has inside dimen- sions of 12 x 15 x 18 in. and walls 6 in. thick. The temperature of the inside surfaee is -26 F and of the outside surface is 50 F. Find: (a) the rate at which heat must be removed from the inside of the box in order to maint,ain these temperatures using the proper empirica1 equation; and (b) com- pare the results with those obtained by using mean surfaces.
SOLUTION: Iivnin = 12 in.; z = 6 in.; L,,, = 22, therefore Eq. 4-22 should be used: LI = 12/12 =
March 1965 Petro/Chem Engineer
1 ft; Lz = 1.25 ft; L, = 1.5 ft, 5 = 0.5 ft.
Al = 1 x 1.25 x 2 + (1 x 2 + 1.25 x Z! 1.5 = 9.75 sq ft
(a) By Eq. 4-22.
A,,, = 9.75 + 2.16 x 0.5(1 + 1.25 + 1.51 + 1.2 x (0.512 = 14.1 sq ft
From Table 3.4 for corkboard, k = 0.025 Btu/hr ft “F and by Eq. 4-21
4 = 14.1 x 0.025(50 + 25),‘0.5 = 52.3 Btu/br
(b) As = 2 x 2.25 x 2 + (2 x 2 + 2.25 x 21 2.5 = 30.25 sq ft
By Eq. 4-14, the aritbmetic-mean area is
A, = (9.75 + 30.25)/2 = 20.0 sq ft
which would~ pive an error of (20 - 14.1)/14.1 = 0.418 or t- 41.8%.
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Petro/Chem Engineer March 1965 FO, F”,,h ,n,o,m.,io” on Advertimd P,od”r,r sae Faode, servire hd 47
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This eorresponds to an error of (17.2 - 14.1) /14.1 = 0.22 or 22.0%.
By Eq. 4.13, the logarithmic-mean area is
A, = (30.25 - 9.75),/ln(30.2a/9.75) = 18.1 sq ft
This c”rresp”nds t” an error of (18.1 - 14.111 14.1 = 0.284 or 28.476. With a greater relative thiekness the errors would be still greater.
Case 5. Different wall thieknesses. If “ne or weral of the six walls have a different thickness the tota1 beat flow should be calculated by adding together the heat flows through each separate wall; A, should be calculated for each wall and the” substituted separately in Eq. 4-21 ~instead of A,,. The edge, corrections thus will be figured twice and the corner corrections three time% therefore, to eliminate this duplication Eq. 4.22 should read.
A, = Al + l.O&(L, t L, + La) +~ 0.4~~ ..? Eq. 4-26
and Eq. 4-23 should read
A,, = A + 0.93x(L1 + L,) + 0.12~~. Eq. 4-27
EXAMPLE 4.6. Compute the amou,nt of heat that must bc furnished in Exnmple 4-5 if the thickness of the two largest sides is increased to 9 in.
The inside area of the four thinner walls is
Ai = 1 x 1.25 x 2 + 1 x 1.5 x 2 = 5.5 sq ft The corresponding mea” area, by Eq. 4-26, is
Am = 5.5 + 1.08 x 0.5,x 3.75 t 0.4 x 0.52 = 7.62 sq ft
and the corresponding heat flow is, by Eq. 4-21,
Q’ = 7.62 x 0.025 x 7510.5 = 28.6 Btu/hr For the two walls with a thickness of 9 in. =
0.75 ft, the corresponding figura are
Al = 1.5 x 1.25 x 2 = 3.25 sq ft
Al= 3.75 t 1.08 x 0.75 x 3.75 + 0.4 x 0.75% = 7.02 sq ft
Q” = 7.02 x 0.025 x 75,‘0.75 = 17.6 Btu/hr
and the tota1 heat flow is
4 = 28.6 + 17.6 = 46.2 Btu/hr
a decrease of (52.9 -46.2)/52.9 = 0.127 or 12.7%. Using the principles laid down in this section,
it is possible to handle almo& any problem of threedimensional beat flow encountered in practice.
References
‘Heatin.g, Ventilating, Air Conditioning Gwide, Vol. 29. New York: American Society of Heating and Ventilsting Engineers, (1951) p. 178.
NEXT MONTH - Applied Heat Transfer for
&l
48
Wax solidifies and shatters as it travels. Dottor biade scrapes it into granulator at discharge end.
Water-Cooled Steel-Belt l Conveyor Solidifies
Wax Polymers . A “Water-Bed” steel-belt conveyor cools and so- lidifies wax polymers for flaking and packaging at the At,lantic Refining Co. refinery, Philadelphia.
The relatively new wax product could not be flaked on drum fakers used for ordinary pet,roleum wax, because the drum flaker would not cool the high-viscosity product rapidly enough for flaking during the short period it would be in contact with the drum.
Wax polymer blends, used primarily for coating food packa,ging papers, are piped to the conveyor at temperatures of from 150 to 250 F in well- insulated steam-traced lines ,and flow onto the steel belt at “ne end.
They solidify and partially shatter in transit,~ are scraped off by a doetor biade at the discharge é
end, fa11 into a granulator, and are elevated to hoppers which discharge into automatic weighing and paekaging equipment.
The load-carrying strand of the stainless steel belt floats on water eontained in four shallow pans. Chilled water at 45.50 F or water at atmospheric temperature may be circula,ted in any of the four pans, depending on the polymer blend being pro- cessed.
Films of wax range in thickness from paper thin to l/lO in. thick, the thicker films beina more com- mori since they result, in greater produetion. Up to 3000 Ib/hr can be proeessed.
The conveyor wa~ supplied by Sandvik Steel, Inc., Fair Lawn, N. J.! and meaaures 58 ft between ter- minal centers. The stainlesn steel belt is 48 in. wide by ,040 in. thick. Automatic tracking contro18 pre- vent the belt fr”m tracking off center beqond pre- set limits.
March 1965 Petro/Chem Engineer
1
2
F Figure 8. Nomograph Used to Find Liquid Surge Height.
I
t
m,, H,=------ .ICq, G
where :
Shell length required far surge capacity, ft. 0 H”I t ~ LIquid residence time, min.
Qi, = Rate at which liq,uid is heing withdramn from the tower, hot gpm.
The quiekway sto wzt this dimension is by the use of Fig. 8. The stepu involved are:
Step 1. Enter the nomograph at the left with the tower diameter.
Step 2. Draw a line from the diameter through the liauid Row rate. Extend it ti11 it cute the IGdex Line.
Steu 3. Enterine with the design residence time, draw a-line from it through the Inde; Line. Extend it to the surge height scale. Read the required surge height.
Example 5: (a) One-fourth of the baffle liquid (300 gpm) is withdrawn from the tower in the examples. (The remainder is recirculated.! What shell length is required for 4 minute residence time?
(b:) The tawer has a 4.in. liyuid inlet, an B-in. manhole, and a 16.in. vapor inlet. What is the tangent lenpth?
Petro/Chem Engineer Apri1 1965
Solution: (a) See Fig. 8 for the calculation. A 4.ft, 6.in. length is required.
Tangent Length =18 ft, 3-in. This phase of tower sizing is now complete -
diameter and tangent length are known. As sized, this tower is not necessarily the best oi most economie, design. Alternate designs would be looked at (different hole sizes, seal dam heights, number of liquid flowpaths, etc.) before the size is con- sidered final.
References 1. Davia, J. A. and Gordon, K. F., PETRO/CHEM
ENGINEER 33, 54 (Octobei- 19Slj 2. Scheiman, A. D., Hydrocarbm Proceasiny and, Potml.eum Refiwr, 42, 165 (October 19631.
Erratum Equntion 3 in the first part of EIow to Size
Shower Deck Raffled Towers Quicker which read: 0.29 Q
n := Tfig should have read n = -~,-? I 15 h,
79
big Heat Conduction in the Unsteady State - Part 5
Applied Heat Transfer Design for the klPl
by Wayne C. Edmister and Joseph M. Marche110
5-l. Introduction. When the temperature dis- tribution within a body changes with time the beat conduction is said to be in the unsteady state. Unsteady-state conduetion takes piace in annealing, quenching or otherwise beat treating meta1 parts, vulcanizing rubber, thermosetting plastics, burning bricks, steaming lumber and many other industrial PrOCWSeS.
As an illustration of this transient phenomena, Fig. 5-1 shows diagrammatically the temperature ehanges taking piace in a homogeneous solid body 12 in. in dism with a temperature tl : 60 F exposed to surroundings at a uniforni higher temperature tz = 260 F. As heating takes piace, the temperature at a certain point asymptotically approaches the temperature t, of the surroundings. Points near the surfaee quickly approach t2, whereas points further away show a time Jag depending upon their distante from the surface. Fig. 5-lb shows a replot of curves of Fig. 5.121 indicating how the temperature of the body inereased’ from the uni- form leve1 t1 - tl to the temperature distribotion tz - a, after 2 hours, tz - b, after 4 hours, and SO on unti1 after 36 hours when it has effectively reached a uniform level, tp - tz. In this case heat fhws ih the body from the circumference to the eenter and is stored in the body, with a subsequent increase in temperature. As the temperature at any point in the body increases, the temperature differenee and henee the rate of beat flow decrease unti1 a nem steady state is reached.
In the general case of heat transfer into a solid, liquid or gas, the accumulation of heat in the sub- stante may be represent,ed by the volume integra1
I”
0 4 8 12 16 20 24 j 28 32 36 Time, hO”II
Figure 5-18. Temperature4ime relation for heating round timbers in a steam bath.
The net amount of heat carried across the boundary of this volume nmy be represented by the area integra1
Distante from surface, incha
Figure 5.lb. Replot of Fig. 5.la showing how the temperature of the body increased from the uniform level doring several time spans.
A heat balance over the volume st,stes that the in-
{
put minus the output is equa1 to the accumulation of beat, that is i
A
where ris t,he flux of heat e xms the boundary. Since
-+ the area veetor, A, points outward norma1 to the sur- face, heat flowing into the volume will have a negative value and heating flowing out of the volume will have a positive value. Then the ’ area integra1 of the’ flux across the boundary repre- sents the net output minus the net input of heat for the volume.
n(pC,tl ~-. dJ7 = - aT ?.;ia Eq. 5.1.
v A l Applying t,he divergente theorem frequently termed Green’s or Gauss’ theorem,, Eton ~the area integra1 Eq. 5-l may be written as
1 dV = 0. Eq. 5.2
Since Eq. 5-2 mnst, hold for any volume, it will applg far a small volume 6V
spc,t, 8.7
+ V. ; 6V - 0 Eq. 5-3 1
The volume under consideration cannot be zero, (sv#o!.so:
d $m) T+yy.j+=o .Eq.5-4
The heat flux is made up of severa1 terms repre- senting the various mechaniums of beat transfer.
Apri1 1965 Petro/Chem Engineer 80
-‘1_ For comection and eonduetion
P J=pc,tZ-kvt Eq. 5-5
where the convection term pC,t v represents heat transfer due to macroscopie or bulk motion and the conduction term -k Vt iu Fourier’s, law. Com- bining Eq. 5-4 and 5-5,
J !pC,t j -+
a7 + ‘J. (pC,,t O) = V. (kDt) Eq. 5-6
Using the wntiw&, eyuation discussed in Part 6 :
~-l-v.(pa)=o . . . . Eq. 5-7
and Eq. 5-6 together with the assumption of con- st,ancy of heat capacity yields the Fourie?=Poisson, equation
at -+ xT + 1). vt =
1 pc [V . (kvtj] Eq. 5-Sa
P
In terms of the Stokes derivative
[V . (kVt/l Eq. 5-8b
This is the general equation for combined con- veetion and conduction for which the velocity components must be obtained from the equatioms of fiuàd tiotion.
If therè is heat genemtion or wleaue within the system as there would be if a chemical or nuclear reaction tiere occurring, it is neeessary to introduce a beat souree or sink term. This term would appear in the volume integra1
il (pC,t)
M a7 -+v
v
where Q iS the rate of beat production per unit time per unit volume. The final expression similar to Eq 5-3 containing this source term would be
$++=‘[V.kVt+Q] .Eq.5-9 PCI,
Solutions to this equation for a number of eases are available in the 1iterature.‘,2
When beat is transferred by conduetion only the velocity term in Eq. 5-8 is negligible. If the sub- stame under consideration is isotropie the con- ductivity is independent of direction and
1 at k V’ t 37 PC,
Eq. 5-10
l Conduction. The temperature distribution in a body for the unsteady-state condition is giva by Fowier’s general Inzu of th,ree-dimensiona1 heat
1 conduction, Eq. 5-10. This law presented in the form of a partial differential equation is3
Petro/Chem Engineer Apri1 1965
where: t is temperature at any poifit giva by the coordinates, z, y, z
,r is time
k is the coefficient of thermal conductivit,y
c is the specific heat of unit mass
p is the mass per unit volume.
The term k/pc is involved in al1 problems of unsteady heat conduction and ,Eq. 5-1 indica& the manner in which the rate of temperature change in time is dependent upon this term. Actually, it simply means that t,he rate of temperature change is more rapid in a body that has both a high thermal eonduetivity and a low specific heat per unit volume. Thus B body with a high value of k/pc heat,s and cools off faster than a body with a low k/pc value. This term is giva the nane thermal d~ifxsivity and is designated by the symbol a and has the~ dimen- sions LVT usually expressed as square feet per hour.
The solutions of Fourier’s Eq. 5-11 by advanced caleulus are quite involved and even after the solutions have been found, extreme care must be exercised in selecting the proper solution that will fit the conditions of t,he prob1em.l Example 5-l is an illustration uf this type of analysis using Laplace transforms t,o obtain the solution.
Example 5-1. Obt,ain the temperature profile and heat transfer rate for a solid whjch occupies the upper half of the y-z plane. Assume Fourier’s second law is obeyed in the form
Eq. 5-12
Boundary conditions are t (0, y) = t,, t (7, 0) = ti and t (7,“) = t, where t, and ti are consta&.
Solution: Before applying the Laplace transfor- mation to Fomier’s second law, it is convenient to make t,he translation
2’ = t-t,> Ey. 5.13
Then, Fourier’s seeond law becomes
Eq. 5-14
and boundary conditions are now
v(0.y) = 0;t!(7,0) = tj - t,
and
From tables of Laplace transforms, one obtains the expressions
= Su- u (0,~) Eq. 5.15
Eq.. 5-16
Applyjng the Laplace t,ransformation to Eq. 5-14,
81
using Eq. 5.15, 5-IG and the initial condition Vjo,~) = 0 giva
d2; s- ~--
dY2 ci ?’ = O Eq. 5.17
Remaining boundary conditions must alsa be trann- formq. Using the fact that the transform of o, constant is simplg the constant divided by S, we have
V(S.0) = +tj - t,/ and ;(,s,3cI) = 0
Eq. s-18
In making the transformation, we have gene from the 7-u domain to the S-v domain and reduced the partial differential equation to an ordinary differen- tial equation. Eq. 5.17 is a second-order linear differential equation with constant eoeffieients. The solution is
s
J- -. y
Tl=& a 2
-i-c+ - Ly J-- . y Eq. 5-19
Applying the transformed boundary conditions, we obtain c1 = 0 and
ti - t, c2 = 77 Eq. 5-20
Combining these equations, the transformed solu- tion is
;=(~)+ -45. Eq. 5-21
New, the solutipn must be inverted or transformed back into Ehe r-y domain from the9y dbmain. Fr~m tables of transforms or by using the inversion theorem, one obt,ains the expression
In OUP case, k -d= SO the inverse Laplace \h
transformation of Eq. 5-21 is,
(&+) = L~/c~$~ , Eq. S-23
This is the solution to our problem in terms of the eomplementary 6rror function, erfe.
The error function and the complementary error function are related by the equation
m
Due to the complexity of engineering problema, it is frequently useful to employ an analog or mode1 ’ to pwdict the behavior of the aetual system. The thermal-eleetrical nnalogy is based on the similarity of the equations describing beat and electrical con- ductioms Far beat cenduction a comparison is mi betwtien the partial differential equation for the temperature distributien and the partial differentiaJ equation for the distributimn of eleetric potentml.~ In the analogy electric current cnrresponds to 4 heat flow, eledric resistance to thermal resistance, electric vwltage differences to temperature differ- enees and electric capacity corresponds to thermal l capacity.
Since it is not usually possible to have distributed electrical resistanee and
wfc(n) = 1 - e,j(n,) 2
=- e-““dn \!?7 i
Eq. 5.24
n
Thus, using Eq. b-24, it is possible to put the snlution into the form
(G) = 1 ’ erf ($z ( Eq. 5-2.3
Series expansion of the error functian is
t, ”
capacitante, these items are
-El
replaced by “lumped” re- sistances and capacitances. 1
i,
This restriction corre- c
: sponds to resolving the 1
e
4
differential equation into
PG~~~ an equation of finite differ- 1
e r, enFit; scale factors are ~
Dirtmre I - only one kind of restriction under which an analoane
Figure 5.2. Tempera- tura in a thick wall.
must operate. The actual analogue is made up of physical eomponents, and these must be capa!& of closely approximating the idea1 charactsristics of the
82 Apri1 1965 Petro/Chem Engineer
r +.... Eq. 5-26 .,_
Using tables of error function or the serie8 expan- sion, we now can determine the temperature at any ~~ -4 point in the slab and any time from Eq. 5-25.
It remains to determine the rate of heat transfer into the face of the slab. This is dnns hy sub- stituting the temperature pradient at the sui-face obtoined from Eq. 5-X into Fonriei’s first, Iaw.
The derivative of temperature with respect to position at the wall is most easilu obtained by expanding aceording to the serie8 forrn and differentiating
1
Substituting yields the rate of heat transfer ;rt th~e wall.
16 q wall = --{Ti 7 T,,) Eq. 5-29
j/Tr CZ0
k where a> = -
PC, rate of heat transfar is inverselv related to the square root of time.
5-2 EleCtric andog method. Two general com- puting devices are often used in the solution of engineering problems, the digita1 and the analog computer. The digita1 computer deals only in num- bers while the analog computer operates with con- tinuous physical variables.
I
P elemento of the lumped thermul circuit. Eleetrical eapacitors do not leak, and this factor often influences analogue design and accuracy. Changing and nonuniform temperature can produce undesirable extraneous effe& if the component resistances and capacitanees are temperature dependent. The aeeuracy of the analogue results depends upon how well the initial and boundary conditions are applied. Following these conditions closely may be dificult. Elaborate instrumentation is required in the electrical analogue beeause of the high speed of the process and because of the small currents and high resistanees used.
5-3 Dim.&ional. ilnal~~sis. A number of practical problems can be worked rather simply by applying dimensiona1 analysis. 4s in al1 solutions by dimensiona1 analysis the numerica1 values of the dimensionless functions must be established either bv experiment or by some other t,ype of analysis. In the following problems the values of the funetions used are either given analytically or graphically as obtained from the solutions of Fourier’s equation.
the number of dimensionless combinations designated by n must
rnbY3= 2, or two ~‘8, ‘in> and ,rz
4 These two n’s may be composed of any two sets of four quantities chosen from the five variables. Therefore one can write
971 = (t, - t)” (b, - t,,)” zc Ct’J
as
as
[ml = [tl” [LID [Tl’ [PT-‘]”
5. The equations for finding the exponents are’ for m:fort <,___ a+b=Ofor~~: fort a=O
L c + 2d = 0 L b -I- 2d, = 0 T.....-d=O T.....c-cl=0
6. In the left column, assigning to a the value a = 1 will simplify the expre$sion for V, and gives b = - 1, c = 0, and d = 0. In the same manner in the right colnmn, assigning the value b = 1, gives d = - 1/2 and c = - M.
5-4 Thick wa~ll, szcdden ehunge of the swfacs temperature. Fig. 5-2 shows a portion of an infinitely thick wall which initially was at a uniform temperature, t,, as shown by the line, t, - t, The temperature of the surface is suddenly ehanged to t,, and at some time 7, the temperature at any piane parallel to the surfaee and a distance z away from it is t. The temperature distribution is represented by the curve t, - t.
Temperatthre inside the urall. The analysis will be conducted following the procedure given in Section 2-2.
1. The temperature differente t, - t depends upon change in surface temperature t, - t,, distante z, time’ 7, and thermal diffusivity a.
2. The variables involved and their dimensions are given in Table 5-1.
,,,,. ---- -- TABLE 5.1. Variables Involved In the Temperature Change, Thick Wall
Quantity Symbol Dimensions
Temperature differente between the sudace and plane n-n at time t,-t t
Change in rurface temperature t, -‘o t Distante bom surface to plane n-n x L Time elapsed from surface-temperature change 7 T Thermal diffusivity of waff materiaf a LT- 1
3. The number of quantities involved is n = 5 and the number of dimensions by wbich they are expressed is T = 3. Therefwe,
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‘7. This eives
According to $heorem D, of Part 2
t,= +L) t, - t,
The funetion f (x/‘\/or~) can be replaced by f,, (z/ZdG) from which it differs only by a constant factor 2. The advantage of x/27/= known as Gauss’ en-m integral, is that values for 2/2\/~‘7 from 0 to 3 have been accurately computed.7 From this modified expression ~the following equation far the temperature t at any piane n-n and at any time 7 is obtained:
t = t, - (t. - tm) f ($=). Eq. 5-30
This discossion of the use of dimensiona1 analysis has been extended to & variety of physical situa- tions. Typical among these 8~: thin walls with sudden change of surfaee temperature; thin walls exposed to a fluid at a constant temperature; and similar prohlems in cylindrical and spherical geometry. In the thick wall case treated bere Eq. 5.30 becomes Eq. 5-25 of Example 5-I. In similar fashion one obtains results that dea1 with Fournier serie6 Bessel functions and other mathematical expressions just as we used the error function.
5.5 Gmphical and numerica1 methods in tmstead-u- state beat eonduction problems. Consider a wall of uniform thickness L, with thermal properties in- dependent of temper+ture, the sui-face temperature on one side is suddenly increased from t, to t, while that on the other side is held at t,. The problem is to determine the temperature distribution at any
Figure% S-3. Graphical deter- mination.
Assume that the wall is divided into i equa1 thicknesses AL = L/i. If the surface resistance is negligible, l/h = 0, after u Finite time ar the tem- peratures at the planes 0, 1, and 2 will have the values of t,, t1 and tg, re- spectively, shown by the points a, b and c in Fig. 5-11. Imagine the first two layers each bi- sected by planes D-D and E-E. The a.veK+ge tempera-
ture gradient at D-D is equa1 to (t, - tl)/nL nnd at E-E it is equa1 to (tl - t&/AL. At this instant, the rate of beat eonduction into a unit area of D-D exceeds the beat conduction out of a unit area of E-E by the amount k(t, - tl/@L - k(tl - tY) /AL. During a time increment AT, this exeess beat will ra,ise the average temperature of the wall between the planes D-D and E-E from t, to tl’. The corresponding beat balance may be written as
di._ _ ,..
k(t, - tl) _ k(h - tr) _ ALpCiti - tl) AL AL A7
Eq. 5-31
Rearranging Eq. 5-31 and noticing that k/pc is the diffusivity, ,o; gives
t, + t2 -t = AL’ (tl’ - t,) 2 i 2aA7 ’
Eq. 5-32
If the ratio aL2/(2aA) is made aqual t,o unity by selecting suitable values far the time interval AT and distane AL, Eq. 5-32 gives
t, + t2 tl’= ~
2 Eq. 5-33
Eq. 5-34
the new temperature at piane 1 is equa1 to the arithmetic mean of the temperatures that &isted before~ at planes 0 and 2. The game rule can be applied to any three planes AL apart. This may be expressed by the general eqnation
t’ = t,, -1 + t,,, + 1
n 2 Eq. 5-35
Equation 5-35, in connection with Eq. 5-31 forms (I
the basis of a graphieal or numerica1 solution of unsteady-state conduetion problems.
Graphical, nwthod. Figure 5-3 illustrates the graphical procedure: temperatures t, and t, are given; tl at the end of the first time interval must be either given or determined analytically; the temperatures tz, tg, and t4 at the end of the Ii& seeond and third time intervnl, respectively, are equa1 to t,. A line connecting points 2 and a determines the temperature Cl’, and a line conneeting points 3 and b gives point c and determines tz’, both temperature at the end of the second time interval. A line conneeting e and a determines tl!f, a line connecting 3 and d gives t/, and a line connecting 4 and e gives t,“, al1 three occurring at the end of the t,hird time interval.
References 1. H. S. Carslaw and J. C. Jaeger. Conduetion of
West in Solids. New York: Oxford University Press (1959).
2. 111. Jakob. Heat Tmnsfw, Vol. 2 & b. New 0 York: John Wiley & Sons, Inc. (19571.
3. M. Jakob, op. cit. pp. 9 and 252. Vo1 1. 4. H. S. Carslaw and J. C. Jaeger. op. cit. 5. M. Jakob and G. A. Hawkins. Elements of
Heat Transfer. New York: John Wiley & Sons, Inc. (1957) p. 80.
6. H. Schenck Jr. He& Twansfer Engineering. New York: Prentice Hall, Ine. (1959) p, 160.
7. M. Jacob. He& Transjer. p, 254. 8. G. A. Hawkins and J. T. Agnew. The Sohtion
of Transient Heat Cmdmtion Problems by Finite Differences. Purdue University Engineering Bulle- tiri Research Series, Wo. 98 (1947). n
NEXT MONTH - Applied Heat Transfer for HPI, Part 6.
Heat Convection & Fluid Motion
84 Apri1 1965 Petro/Chem Engineer
Heat Conuection and Fluid Motion - Part 6
Applied Heat Transfer Design for the’ HPI
by Wayne C. Edmister and Joseph M. Marche110
6.1. Gene&. Heat convection is the transfer of heat by fluids in motion. It is often visualized BS an exchange of heat between moving particles of the fluid itself and between these particles and surfaces which hnve a diffcrcnt temperature.
When the motion of the fluid particles is caused by a differente in density resulting from tempera-
* ture ehanges, convection is termed free or na~tural. When the motion is produeed by some mechanical means, such as a, pump or a blower, convection is called forced.
Convection is present in a large proportion of problems of heat transfer and is used in various devices both far heating and cooling gases, liquids and solids. The present tendency toward smaller and lighter equipment has increased the use of forced convection and its importance in the process of heat transfer. Since convection is based on motion of fluids, a, brief review of the laws govern- ing the ilow of fluids will be useful far a better understanding of the prineiples of conv&ion.
6-2. FZow of f&ds. When a fluid, liquid or gas is fowing in a pipe or duct, the tgpe of flow depends upon severa1 faetors of which the most important is the velocity. At low velocities and at constant temperature the fluid partic,les flow in paths parallel to the axis of the pipe or duct and the flow is
0 termed variously as streamlined, niscous or lan~inar. Iri laminar flow, the velocity of the fluid particles changes from a maximum in the center to zero at the walls of the pipe or duct. For isothermal flow in a pipe, the velocity distribution is parabolic, Fig. G-l, and the average velocity ‘va” is equa1 to one-half of ~n,rx at the center, D;>” = 0.5 e,,,.,.
The general equations which govern the Aow of fluids are derived from Newton+ laws and are known as the equations of motion. Tbep may be expressed veet,oriallyl by the relation
pmDt =-VP- V.7. ipg Eq. 6-l [ 1 3
where ; is the stress tensor. The lefthand term in Eq. G-l represent,s the mass per unit volume time the acceleration. Similarly the terms on the right represent: the pressure force on an element per unit volume; the viscous force on an element
Petro/Chem Engineer May 1965
per unit volume: and, the gravitational force on an element per unit volume.
The equation which expresses the conservation of mass for a fluid element is known a8 the equation of continuity
Equations 6-l and 6-2 serve as the starting point far the description of fluid slow problems. When the fluid exhibits Newt.onian flow properties and the viseosity and density are eonstant we obtain the Navier-Stokes version of Eq. 6-l
where ,u is the viscosit,y. As an example of the application of Eq. 6-3 eon-
sider the c&w of steady state lamina’ fow in a horizontal pipe. Equation 6-3 becomes for the axial z dirMion:
With the boundary conditions u = 0 at T = R and &./d~ = 0 at T = 0, the solution is
Eq. 6-5
where
aP AP - ~~_ i?x =r
When u liquid is being heated, the viscosity at the wall is less than in the center of the pipe and the velocity at the center tends to increase; this results in a radia1 flow from the center of the stream toward the wa,ll, and pure laminar flow no longa exists. Heating of the liquid in a pipe tends to flatten the puabolic form of the velocity traverse, Fig. G-1, while cooling has the opposite effect of stretching the parabola. For gases, the temperature effect on the viscosity is in the opposite direction and therefore t,he ehange of the velocity traverse is reversed.
If the velocity is increased, radia1 components
73
of velocity appear and eddies, or vortexes, are formed near the center line. With a further in- crease of the velocity above a critica1 value, the eddies spread over the whole cross section of the pipe. This type of tlow is ealled turbulent. The average velocity over the whole cross section be- cornea approximately 0.8 to 0.9 of the maximum velocity. In general, the ratio increases with the equivalent diameter of the conduit and smoothness of iti walls.
The existence of a laminar or turb111~nt flow depends not only upon the velocity but on other characteristics of the system. The flow condition of the wstem may be expressed in terms of the Reynalds number Nxe = Dup:~. The value of Nnr below which streamline Aow exists ovcì the entire cross section is called the critica1 value, and the corresponding velocity is called the criCcaI velocity.
Based on experiments with pipes conducted under different eonditions, variotu authors give far critica.1 Reynolds number, N~eo values ranging from 2100 to 2300. Far certain conditions laminar Bow can
Figure 6-l. Effe& of heat transfer on velocity flow traverse in laminar flow.
exist at a much higher NRe. On the other band, turbulent flow may appear to be less turbulent in a smooth pipe for N,. < 4500. Flow between Nn, = 2100 and Na. = 4500 is often considered to be in 8 transition stage.
The absence or presenee of eddies, or turbulenees, affeets the beat transfer by convection and also the friction resistance to the Bow of the fluid. Fig. 6-2 shows cuives plotted through average values of the coefficirnt of frietion, f, found from a larga number of experiments with different fluids and pipe siza2 The left curve, a-b, presents data for laminar flow; under certain conditions the curve can be extended to point c with a gradua1 increase of the flow veloeity and a corresponding increase in Nn.. The path c-d is the most likelg curve for an increase of i in the transition stage with Nn. between 2100 and 4500. For a further increase of the Row velocity, the eoeficient f gradually decreases curve d-f. Since the main purpose of Fig. 6-2 is to show the presente of two different types of flow and the transition from one to another, only lower values of Mnc are shown. Far experiments conducted by decreasing velocity u and Nne, turbulent flov is observed along the path d-e with a transition path e-b. The forma- tion of eddies in the fluid during the transition stage is rather irregular depending upon many factors, such BS roughness of the pipe walls, vibra- tian of the pipe, etc.
6-5. tleynokzs mmmer. ‘The origina1 form of the dimensionless ratio, termed the Reynolds nu%bsr in bonor of Osborne Reynolds who first established and used it, is
NR~ = vx/v Eq: 6-6 where: ‘u is the velocity of flow of the Auid
z is a characteristie distante v is the kinematic viscosi& of the fluid.
In this arti& Reynolds number is used primarily in conneetion with Aow of fluids in tube and other 4 closed conduits. Also the dynamic visto&’ F, rather than the kinematic viscosity Y, is used extensively. For such cases, the general expression for the Reynolds numher may be written as
N ,Q=~~~vP/,u .Eq.6-7 where ih is the hydraulic radius defined as the ratio of the cross section ares S of a conduit over its wetted perimeter P
t‘,, = SjP . Eq. 6-8 For a round cross section
r,, = D/4 Eq. 6-9 Since Revnolds number is used far the most nart in con&tion with round pipes, it is conv&ient to write mi
Nnc = Dvp/p , Eq. F-10 For a non-eircular cross seetion the value 4~ in Eq. 5-2 is called the squivalent diameter and desig- nated by L>,. The value of D, is determined from Eq. 6-8 as
D, = 4S/P . Eq. 6-11 For an annular conduit with D1 and D-, the small and larger diameters, respactively, Eq. 6-11 gives
D, = D% - Dl Eq. 6-12 In many ealculations it is convenient to substitute
the value G for the product vp G = vp = m/S Eq. 6-13
where G is the wznss velocity and is equa1 to the rate of flow m, Ib/hr divided by the cross area S, sq ft, of the stream. With this change far a. round duct, the Reynolds number may be written
Nsc = DG,‘fi Eq. 6-14 and far a duct of any other cross section
Nm = D,G/p Eq. 6-15 where the equivalent diameter n, is obtained from Eq. 6-11.
0
The use of G instesd of up not only simplifies caleulations but hss the additional advantage in that G is independent of changes in temperature and pressure.
6-h. PresSure drop in svmotk pipes. In lamina fio".', the ~rmmre drop may be calculated from the Poiseuille formula
p =~32&LVlg,D2 Eq. 6-16 where: ‘p iu the pressure drop, psf,
p is the dynamic viseosity, Ih/ft-sec,” L is the length of the conduit, ft, ‘~2 is the average velocity, fps,” g, is the dimensiona1 constant, 32.17 Ibm-ft/
Ibf-sec,” and D is the conduit diameter, ft. *Any other unit of time may be used, pro- vided the same unit is applied to al1 three quantities ,u, g, and se
May 1965 Petro/Chem Engineer
B - CommercIoI Steel. Cast-imn. etc.. Pipa C - SmoothTubes. Glass. Cococr. Drown TubinQ
Reynold’s Numbar, NR.
c,
AP=(fl/D)(pVz/2g,h Af,= ( fl/D) where:
D = inslde pipe diometer. (11) AP,,= pressure drop, (number of pipe f = Dorcy frlction facto,. dimenkvtless velocity heods 1
& = pravltational constont. v m average fluld velocity in pipe 32.17 (Ib mass/lbforce)(ft/?ie~) (ft/aec)
L m length, ot straight pipe. (ft) p = fluid density. (Ib mass/ft3)
AP m pressure drop, (Ib forcc/ft’) JJ - fluld uiscosliy (Ibmoss/firecl.(ce~~~)
Figure 6-2. Darcy friction fa&? for isothermal flow.
The pressure drop both in laminar and turbulent flow may be calculated by the Darcy formula
ap = fLpV”z,‘8rhg, Eq. 6-17 where: &J is expressed in psf,
f is the friction coefficient. Far flow in a round pipe, Eq. 6-17 may be presented more conveniently as
np = fLpVz/ZDg, Eq. 6-18 or, expressing the pressure drop as loss of head H in feet of the fluid,
H = fLVz/ZDg, Eq. 6-19 For laminar flow, f may be computed with
sufficient accuracy from the theoretical value A f =,64/Nne Eq. 6-20
For turbulent flow in smooth pipa, there exists a number of empirica1 expressions which all ‘give the value of f in terms of the Reynolds number and al1 are in comparatively close agreement. One of the beat known is the Nikuradse formula”
f = 0.0032 + 0.221/(NRr)0.237 Eq. 6.21 A simpler expression, obtained from Koa’s
formula,4 is f = 0.184,/(N~e)~.~ Eq. 6-22
In the range from NRe = 2100 to 2,000,OOO. it giva results which are only 2 to 4% higher than those obtained from Eq. 6.22.
Another formula for computing pressure drop is the Fanning equation; the form most convenient for flow in pipes is5
me-fourth of that used in the Darcy formula. This important differente should be remembered when finding f for use in Eq. 6-18.
EXAMPLE 6-l. Water at 120 F flaws in a Z-in. clean steel pipe, 75 ft long, with a velocity of 100 fpr+ Determine the pressure drop.
SOLUTION: First N,, must be foond, D = 2.067 in. = 0.172 ft, p = 61.73 lb/cu ft, and p = 1.36 lb/ft-hr. Therefore by Eq. 6.16
Nne = 0.172 x (100 x 60) x 61.730.36 = 46,900 The flow is turbulent and either Eq. 6-21 or Eq. 6-22 must be used to find f. By Eq. 6-21
f = 0.032 $ 0.221,/46,900°~23’ = 0.0205 By Eq. 6.22
f = 0.184/46,900°~’ = 0.0214 The differente in f by Eq. 6-17 is only 4% and is on the safe side.
AP = 4fL,, V2j2geD Eq. 6-23 Th% formula is basically identica1 with the Darcy formula, Eq. 6-17 but refers only to a circular sectian and therefore, uses Di4 for t-h. The Fanning equation uses a friction factor f whose value is
When using the Fmning formula, Eq. 6-23, the friction eoefficient found by Eq. 6-21 or 6-22 must be divided by 4, or simply use in this case 4f = 0.0214, and with gC = 32.17 Ibm ft/lbf-se? the velocity must be expressed in fps. Thus,
ap = 0.0214 x 75 x 61.73 x (100/60)~,‘(2 x 32.17 x 0.172) = 24.9 psf, or 0.17 psi.
6-5. Resistances to fimo in ?ipss - Roughness of walls. The condition of the wall surface does not influente the resistance to flow of a fluid in pipes if the flow is laminar, but influences it considerably when the flow is tnrbulent. Eq. 6-21 and 6-22 pive values for f in smooth pipa; in the presente of roughness, f will be greater. However, it is veiy
diffieult to express the increase of f numeridly, especially sinee in praetice the degree of roughness varies and is rather indefinite. Some basi8 for
Petro/Chem Engineer May 1965 75
I
energy: potential, kinetic and floio-work energy. Potential energy is due to the elevation of a mass m at the iiuid z ft above an arbitrarily selected referente piane; this energy, expressed in beat units, is equa1 to mz/J. Kinetic energy is due t” the velocity v, feet.per unit time, usually per second, and is expressed as mu?/2m’gJ: where d aceounts for the velocity profile, 01’ is $4 far laminar flow and approximately unity for turbulent flow.7 The slow-work energy is necessary to main- tain the flow and is obtained as the product of the force, ps, which is pressure p times the cross sectional area S, and of the distante, mv/S, which is the tota1 volume, ww, where 5 is the specific volume divided by t,he cross seetional area S. Then flow work, expressed in beat units, is (PS) (rnv/S) / J = mpv/J.
In addition, the fluid has interna1 energy E, Btu. The heat added between the sections 1 and 2 is
Y
/ Laminor I - Buffer
f I
L-----X Tmnsition Paint
Figure 6-5. Boundary layer flow over a flat piate.
Ir& and the energy IWz, ft-lb, added by other means, such as a blower or pump, is IWz/J.
For a time interval during which the mass m. of the Auid passes from section 1 to section 2, the energy balance may be written as
Noticing that the sum of interna1 and flow-work energies c”nstit,utes enthalpy i, Eq. 5-29 can be rewritten as
The values of the enthalpies il and i, must be taken from tables available for most ‘liquids, vapors and gasa. However, it should be remembered that enthalpy has a relative value that depends on the temperature taken as basis and different tables use different temperature as basis. For perfect gasa, with PV = NRT.. the change of enthalpy may be take” as
ip - il = c,(tr - tJ Eq. 6-36
where cp is the man value of specific beats between the temperatures tl and tz.
6-8. Film thsory. When a fluid is flowing in a
pipe or duet, regardless whether the velocity is low or high, whether the flow is lnminar or tur- bulent, there is always a film or layer of the fluid that adheres to the solid surface of the conduit. Experiments have ehown that the film is not en- tirely stagnant; only particles touching the wall are completely wit,hout motion while the next laser of particles is moving alowlq under the influente of shear with the adjacent particles flowing parallel to the wall.
The existence of an effeetive film is very im- portant in the consideration of beat transfer from a solid wall to a fluid, whether liquid or gas.
6-9. Boundary Layem. If a fluid flows with an initial uniform oelocity “wv a, Rat, plata, the velacity decreases unti1 it is zero at the surface of the piate. This’ decrease in veloci@ takes plaee in a small region near the wall called the “boundary lager.”
At the leading edge the boundary layer is of zero thickness and becomes thicker as the flow progresses along the piate.” Flow in the layer is laminar at the leading edge of the piate and may become turbulent us the flow progresses along the piate. (Fig. 6-5.) The transition between laminar and turbulent flow “ccurs at a. Reynolds number, Vxlv, of 500,000.
The thickness of the laminw bound~ary layer, 8, which eorresponds to a point (,r/ = 81 where approximat~ely 99% of the velocity is attained is
8 = 4.96 (&) Eq. 6-37
where: ,v is the kinematic viseosity of the fluid x is the distante from the leading edge V is the velocity of the main fluid relative
to the piate
N vx
Re = - v
The value of the Reynolds number formed with the distante from the leading edge may be used to determine whether the boundary layer is laminar or turbulent. Far Reynolds numbers in exce% of 500,000 the boundary layer is turbulent, Fig. 6-5. The thickness of the turbulmt boundav$l laer corresponding to a point where approximatelq 99% of the full velocity is attained is
8 = 0.13 (=&)
When fluid flows past u two-dimensiona1 body, such as u cylinder, the fluid is accelerated as’ it passes over the forward portion of the body and is the” decelerated after it passes the thickest part of the body. This effe& cuuses separation of the boundary layer from the curved surface. Actual separation of ,fluid as it flows around a cylinder has been observed experimentally by stu,dy- ing flow patterns.‘2
The second section of Part 6 will appear in the June issue.
May 1965 Petro/Chem Engineer
Heat Convection and Fluid Motion,:, Part 6 (continued)
Applied Heat ~Transfer Design for the HPI
by Wayne C. Edmister and Joseph M. Marche110
6-10. Tnrimlent fluw ~CILP‘ Q solid wall. The mechanism of beat transfer bet,ween u solid wall and a turbulent fluid has been the source of a considerable amount of investigation. The problem is, for the most, part, one of fluid mechanics since it is neees~~ry to describe the fluid behavioz in order to evaluate the heat transfer characteristics.
The main differente between viseous and turbulent
b!e ow fields is in the method by which shear is
‘.~ t,ransmltted. In viseous flow, shear is transmitted uniformly, while iti turbulent How it is transmitted irregularly. These irregular deviations of the loca1 flow from the average flow pattern haw. a random- ness whieh becomes apparent over long time in- tervals or distances. The limiting diatance over which this randomness is noticeable depends upon the scale of ~the turbulenee. Alth,ough the scale of turbulence might be considered to be the size of the prevalent eddies, it has bearne convenient to define the scale of turbulence by use of the correla- tion coefficient.
It is a characteristic of turbulent flow that it must be described by statistica1 quantities. The amplitude$ have a random distribution and there is no characteristie frequency or wave length of the fluctuations. The early work of Prandtl, Taylor’” von K&“rn&~ and Howarth’4 laid the basi8 for the statistica1 theory of turbulenee.‘5
mie. The turbulence prese& in the bulk of the fluid
1s suppressed in the vicinity of a solid surface.‘6 Earlier theories assumed that a laminar sub-layer existed at the surface. A number of recent investi- gatiix# have been made which present the velocity and temperature patterns very close to the solid surface. Deissler’8 and vari Driest’!’ have presented analyses which give continuous velocity and shear distributions for t,urbulent, flow near solid walls.
The eddy eoefficients for momentum transfer, E”, and heat transfer, E=, may be defined directly by the flux equations:
dU 7gc = p(Ev + VI-- dy “.” Eq. 6-39
dT Q/.4=&p(wta)G Eq. 6-40
Velocity and temperature profiles and momentum and hent transfer :rates may be determined by
Petro/Chem Engineer June 1965
int,egrSng these flux equations. In~, order to carry out the integration it is necessary to know the dependeney of the loca1 flox and eddy eoefficient upon position.
6-11. Viscositl/. As brought out in’:this chapter, viscosity is one of the most impeitant properties of a fluid. Viseositv enters in.~most calculations of heat transfer and floid ‘motion either directly oi througb the Reynolds and Prandtl numbers.
Figure 6.7. Viscosities of liqoids at 1 atmosphere pressare. See Table 6-3 far coordinates X and Y.
Viscosità oi liqxids. In spite of the fact that there exists a tremendous amount of experimental data on the viscosity of liquids, the theory is not sufficiently developed and ealculations involv- ing viscosity of liquids have t,o use empirica1 information.
In the practieal range of beat-transfer, the in- fluente of pressnre is rather small.
The. viscosities of some commonly owzd liquids and of a few mixtures and solutions are presented in Fig. 6-6. Viscosities of othei Iiquids ma- be determined using Table 6-3 and chart, Fig. 6-7.
In industry the specific viscosity 6, is expressed in seconds and is measured by a Sa,ybolt Universa1 Viscometer at two temperatures, usually at 100 F and 210 F. These two specifie viscosities are plotted on a special semi-log type ASTM chart such as Fig. 6-8 and the two points connected by a straight line. The specific viseosity t, at any temperature for this oil can be read at the intersection of this inclined line with the corresponding vertical tem- perature line. The inclined line can be extended to higher and lower temperatures within the liquid state of the oil.
The specific viscosity t, is converted to kinematic viscosity Zn in centistokes by means of the equation
z, = 0.22 t 5 - E!! ts
Eq. 6-41
The specifie gravity yt of the oil at the same tem- perature t can he calculated from the eqtiation
yt = ~60 - 0.00035(t - 60) Eq. 6-42
where yRO is the specific gravity of the oil at 60 F.
44
The absolute or dynamic viscosi,ty Zt, in centipoises, at the temperature t is ohtained from the velation
Z = Z1, yt Eq. 6-43
The viscosity may also he converted to Ib-per-ft- per-hr anits hy multiplying Z by 2.42
/x = 2.42 Z Eq. 6-44
Table 6-4 gives the specific grwities of some lubricating oils and in conjunction with Fig. 6-8 may be used to determine their dynamic viscosities.
Table 6-5 givcs conversion faetors for the
3000
2000 l \ l / I I Y\ l
400, 200 210
Tempemture, “F I
Figure 6-8. Viscosity-temperature chart far xnne oils. See Table 6-4 for oil characteristics.
~0.044t~
“0036 8 .$
c 0.026 3 2 z 0.020
-40 0 40 60, 120 160 200 0.012 l-Lu-Lu
Temperature OF
Figure 6-9. Viscosities of some common gases.
~June 1965 Petro/Chem Engineer
I
Table 6-3. Viscosities of Liquids” Coardinates far use aith Fig. 6.7.
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June 1965 Petro/Chem Engineer
Forced Convection Inside Tubes-Part 7
Applied Heat Transfer Design for the HPI
by Wayne C. Edmister and Joseph M. Marche110
Y-1. Swfacc cnnd~ucta~n~ce. The exchange of beat between a fluid and the surface of a solid is known
Quantitatively the beat exchange depends upon the characteristics of the fluid, shape of the solid and the relative motion of the fluid. The numerica1 vslue of surface conduc- tante, expressed ,in Rtu per hour per sq ft of surface for n differente of 1 F between the temperatures of the fluid and surface, is called the swface coeficient. In Fig. 7-1, the distanee AL represents the thickness of the effettive fluid film through whieh heat travels by conduction, Seetion (i-4. Applying Eq. 3-2 to this case, with dt = & - t, and dz y AL, giva
q = (k/nL) A(tl - to) Eq. 7-I
where: A is the su-face in contact with the fluid. The heat flow Q can be ealculated from Eq. 7-1, however, engineering praclice has been Lo group the terms k/nL together rather than use the effettive film thiekness, AL, alone. The value k/&L is the film hm&transfw cocj$Gent designated
w by h. Thus, Eq. 7-l becomcs”,,.~ ;~.,.
i)
Figure 7.1. Heat flow from a fluid to a solid wall.
PETROKHEM ENGINEER July 1965
q = hA (t, - t2j Eq. 7-2
This equation was first proposed by Newton. Its form is 80 convenient that it is used also for hest transfer by radiation with the corresponding sub- script h.,. In radiation there is no film, therefore the beat transfer coefficient is more appropria% called the ~&mx~, cpq@i&. This~ nane will be applied to both radiation and convection coefficients, designating them as h, and h,, respectively.
The fluid’temperature profile does not change as abruptly as is shown by the solid lines in Fig. 7-l but is more gradua1 as indicated bg the dotted line. The temperature, tl, is the i@ or c~p-mizin%~ tem~perature of the fluid ~~,~~~~.~~~ .~..~~.~ .,--. -,--”
where :
For a pipe: ii = VR” and da e %vàr ,~Li:: The bulk temperature is the fina1 iiuid temperature i, ,: t,hat one would measure if~ the tluid were allowed ,-t to flow without heat loss into a container and mixed. Formally, as expressed by Eq. 7-3, the bulk temperature is the velocity weighted area average temperature of the fluid.
Equation ‘7-2 may be spplied to find ,the beat Rux if the snrfaee coeflieient h, and the temperatures tl, of the fluid bulk and ta of the wall are known. It may also be usad to find the temperature tx of ihewaUf,J.he-te 0 @t f&w. q~~+&h.e tempera- ture tl of the fluid are known. In either case it is neces.qary to evalvate the surface coefficient 11,.
7-2. ,%-face, coeficient. Foreed conveetion or- dinarily takes piace in turbulent flow. In doveloping an expression far the surface coefficient of beat
101
must be found experimentally. This equation is known as Nuss~lt’s expression and the thrcc dimen-: sionless groups are giva the names, as indicated in Table 2-1, of Kusselt, Reynolds, and Prandtl numbers or moduli. Thus Eq. 7-5 may be written also as
From Eq. 7-5. the general expression for the surface coefficient inside of pipa may be written as
In order to obtain couect numeri&1 values for these and al1 other dimensionless groups, it is extremely important to use consistent units in al1 terms. The danger of making an error is con- siderably lessened if al1 length dimensions are expressed in feet. avoiding pressures per sq in. or conductivity values per inch thickness. Al1 time measurements must be in t,he same units, preferably referred to Ihe hour. Paitieular care must Ire erercised when dealing with viseosities because in technieal literature viscosity is expressed in a number of units based on different basic units of mass or force, time und distante. In ,the English system, the units of viscosity which are most con- venient far calculations in beat-transfer problems are the pound mass per ft-hr oi the Ib-force-hr per 8q ft. Table 6-5 may serve for conversion of diff&ent unita to the desired system.
Twbulent Flow. For turbulent flow of liquids such as water, acetone, alcohol, kerosene and light oils, when N,, $ 2100, the recommended surface coefficients are, in uccordanee with Eq. 7.6. for heating :’
and far cooling:’
A more conservative expression, recommended for heating liquids which have viseosities lover than those of water and gases’ for slow in horizontal pipes, is:
Far liquids of high viscosity and Reynolds num- bers exceeding 10,000 the use of Colburn’s equation is suggested.z This equation differs from Eq. 7.9 only in the exponent of the Prandtl number cpL/k. Its fa-m for both heating and cooling of liyuids is
104
0.02 -7 / IO.2
c o-0 c 0 200 400 600
Temperature, deg. F ,
Figure 7.2. Properties of saturated steam
<i ““I G3.0 I I I 53
, , 100 200 300 400 500
temperature, OC.
Figure 7-3. Specific beat of superheated statii.
convection and radiat,ion. Therefore, it is desirable to have an expression for LVp, which avoids the use of k. By substitoting for the thermal eonduc- tivity k, Eq. 3-5 and eliminating the eonstant al by means of Eq. 3-6, the following equation is obtained.
Specifie beat. An important factor, in Nusselt’s equation is the specific beat of the fluids. Except far vapora near the state of saturation, it is influenced by the temperature of the fluid but not by its pressure.
Gmes. Speeific heats of some commonly used gases and vapors may be calculated by means of
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0.070 0.044 0.033 0.024 0.017
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3,o 2.5 2.1 1.7 1.4
0.10 - / 1 Mercury
~OO&pyJ 0 I 1 +:t&erotSufe,
120 160 I 200 "F.
Figure 7-4. Specific beats of some liquids
equations given in Table 7-4. In addition, specific heat values for air may be taken from Table 7-3, for saturated steam from Fig. 7-2 and far super- heated steam from Fig. 7-3.
Liquids. Speeific beats of some commonly used liquids may be taken fiom Fig. 7-4 and for water from Table 7-2.
Specific beats of petroleum produets, including lubricating oilS, may be computed with an aceuracy wit,hin 5% from the equation:h
c = (0.388 + 0.00040 t)/r~,, Eq. 7-12
where t is the temperature of the liquid and ysO is its specilic gravity at 60 F referred to water also at 60 F.
7-5. Fhid tem,pwa.tures. As pointed out in Section 7-4, the beat-tra,nsfnr coeficient h, depends on the fluid eharacterist,ics c,k, p and p. These quantities. in turn, are functions of the fluid temperature. When a fluid flowing in a tube is heated or caoled, its temperstnre ehanges both in the direction of t,he flow and radially frnm the surf;~ce of the tube toward its center. 4 knowledge of the radia1 tem- perature distribution is required in determining the bulk temperat,ure of the fluid and the values of fluid properties. Fig. 7-5 shows an example of velocity and temperature distributions aeros a vertical brass pipe, 1.92.in. inside diam, in which air is being heat& by forced convection. The velocity and temperature at the center of the pipe were v< = X7.4 fps and t, = 75.9 F and the wall temperature was t, = 109.4 F. From Fig. 7-5, taking into aceount the differences in air flow, CU ft/sq ft- hr, at different distances from the center line, the mean or bulk air temperature hy Eq. 7-3 i8
t = 86.5 F. The flow was turbulent, NE, = 75,800; there is a large temperature drop in the outer layel of the air stream; and a slow temperature increase from the center toward the outside of the tube.
Film Temperature. The numerica1 values of C, a, and b in the basic Eq. 7-6 which appear in Eq. 7-7 to 7-10 were determined with the fluid properties c, k, fi and p evaluated at the average temperature of the main fluid body of the stream and not at, the mean temperature of the stagnation film. However, in these tests the differente between the measured temperature of the fluid stream and the estimated man temperature of the film was small. When the differenee between these two tempera- ture is greater than 10 F far a liquid or greatei than 100 F for a gas, the fluid properties should be evaluated at the temperature of the film. Un- fortunately, this temperature cannot be meaaured and must be estimated. In the case of turbulent flow, the film temperature t, may be taken as
TABLE 7-4. Specific Heat Equations for Gasa and Vapors
9
Acetylene 0.288 + 2.38T x IO-’ 1,100a Ammonia
g* 3 0 102 + 0.277 x 10-d 1,lOOb
Benzene CA 0.079 + 5.747 x 1ov 1,100’ Butane CAo 0~079 + 5.407 x 10~4 1,100~
0.368 148J-’ - Carbon dioxide C@, + 3.2 x 104 x T2 6,900=
Carbon monoxide CO 0:338 - 117 x T-1 + 3.82 x 10’ x J-2 9,OOOd
Chlarine C’I 0.106 + 0.023 x T-1 x 10’ 85Ob Ethane CA 0.133 + 0.545 x J x 10” 1,100~ Cthylene CA 0.151 + 0.42 x T x 10-1
CCI,F 1,100=
Freon-l1 0.079 + 0.11 x7x 10-l 76Ob Freon-21 CHCI,F 0.090 + 0.11 x Tx 10-a 76Ob rreon-22 CHCIF, 0 073 + 0.17 x T x IO-1 760b Freon-113 CCI,F-CCIF, 0.107 + 0.10 XTX lo-’ 760b Gasoline (octane) CA, 0.126 + 4.65 x T x 104 1,100~ Heptane VI, 0.061 + 5.29 x J x 104 1,100~
Hydrogen “2~ 2.86 + 2.86T x 10-d + 9.92 x T~O.5
Hydroxide CH 0.435 - 0.437 x lo-’
Kerosene (dodecane) ;;H,, + 2.2E.T’ x 10-n
1” @!4
IlMlane Nitric oxide
Nitrogen
Oxygen
Propane Steam (water)
Sulphur dioxide
0.051 + 5.21 x J x 10-a 1,100~ 1.8004
NO4 0.263 + 5.127 x T x 104 0.218 + 0.0295 KTX 10-1 1,620d
N? 0.337 - 0.124 x T-’ ,- 4 13 x T-2 x 10’
5,000~
02 0.360 + 5.37 x T-o.6 + 47.8 x J-1
1,100a
22 0.093 + 6.45 x T x 104 1,100’ L 1.1-33.lxJ-0.5 5.4005
+415xT-’ 9 0.176 + 0~57~ JxlOA ‘1,210b
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t, = t -l- 0.5(t;--ti = 0.5C.C, i-ti Eq. 7-13
where t iu the mean bulk temperature of the fluirl and t, is the mean surface tempeiat,we of t,he conduit:
Irregular temperat~m%s. In a long pipe when the temperature varies considerably and irregularly nlong the length of the pipe, the average tempsra- ture should be obtained by integration.
L 1
t,,, = -i- s
tdl Eq. 7-14
D
Radio1 distane from center line
Figure 7.5. Temperature and velocity ~distribution across a tube with turbulent air flow.
A,simple graphical integration is usually adequate. Plot the measured or estimated liquid temperatures as ordinates against pipe lengths as abscissae; connect the points by a sinooth curve; measure the area under the curve and divide it by the length of the diagram to obtain the average temperature along the pipe length.
References 1. F. W. Dittus and L.M.K. Boelter, “Heat
Transfer in Automobile Radiators of the Tubular Type,” %iv. of Calif. Publications in Engineering 2, 443-61 (1930).
2. MeAdams, Beat T?ansmission. op. cit. p. 219.
3. A. P. Colburn, “4 Method of Correlating Forced Convection Heat Transfer Data and Com- parison with Fluid Friction,” Trans.Am.Inst.of Chemical Engineers 29, 174.210 (1933.34)
4. R. T. Xent, Machnnic~l h%gi?f.err’s Hmdbooli, 12th ed. II, 12-65.
5. J. R. Pannell, Experiments on Heat Trans- misiion in the Case if Air Flowing Through a Heated Pipe,” Twlinira~l Repwt of the AdrinwZ, Co»i?nittee lor .&w~~wwtics, London. Reprint Memo No. 243; Vol. 1 (1916-17) p. 22: also M. Jakob and G. A. Hawkins. Elments of Heut Tmnsfer and I%sulat~io?t, 2nd ed. Xew York: John Wiley and sons, Inc. (1950) p, 213.
PROGRESS IN HPl..TECHNOLOGY
Petrochemicals ~@ E Furt,her d&ails have beeome available on Dow’s Toluene-to:Phenol process which was commercialized 2-3 years ago.
As is already Bnown, the method involves oxi- dation to benzoic acid which is then air oxidized to phenol. The first step, carried out by liquid phase air oxidation in the presente of a soluble cob& salt, allows formation of benzoic acid in approximately 90% yield. The air oxidation of benzoic acid is carried out, also in liqaid phase, in the presence of copper and magnesium salts, with simultaneous introduction of water vapor. The reaction mechanism involves intermediate formation of benzoyl salicylic acid which is con- verted to benzoic and salieylic acids. The latter is decarboxylated to phenol. A yield as high as 95% is reported when the reaction is carried out at 240 c. *
q A list, of synthetic polymers authorized far use in contact with food has been published by the U S. Depar:nxnt of Ilealth,, Education and Welfare’s E. B. Detwiler (SPE J. Jan. 1960, 61-63).
Severi major use classes are involved: 1) Syn- thetic polymers, generally recognized as safe far the intended ,purpose; 2) Synthetic~ poly- mers regulated as direct food additives, 3) Syn- thetic polymers for which prior sanctions have been granted. 4) Synthet,ic polymers far which prior sanctions haue been granted for use as films. 5) Synthetic polymers regulated as indi- rect food addit,ives; 6) Materials containing synthetic polymers regulated as indirect food additives; and 7) Adjuvants far synthetic poly- mers regulated as indire& food additives.
PETROKHEM ENGINEER July 1965
Coking onit at the Baroni Refinery in Bihar, India
participation, tn separate the aromatics which cause kerosine to burri with a smoky flame. The extraetion of aromatics is effected from the feed- stock tiith liquid sulfur dioxide by the Edeleanu solvent extraction process.
The extract and raffinate solutions thus obtained are proeessed through separate sulphur dioxide recovery systems where the solvent is boiled off leaving pure extract and raffinate. The solvent vapors are compressed, condensed and recirculated through the unit. The pure extract if further fractionated to yield a light extract which is blended with gasolirie and heavy extraet disposed of in furnace oil and cutbsck bitumen.
The pure raffinate which is superior kerosine is pumped directly to storage. Fractions for aviation turbine fuel and from coking unit are similarly treated in this unit to obtain quality produets wit,h improved propert,ies.
Coking Unit
In the coking process the feedstock, which is a mixture of short residue, wux vacuum distillates and the overheads from the bituma unit, is heated to a high temperature where the destructive de- composition process transforms part of the feed- stock into hard petroleum coke. The vapora leaving the top of these chambers contain the light cracked
Petro/Chem Engineer August 1965
products gas, LPG, gasoline, kerosine, light diesel oil and furnacc oil fractions which are aeparated in a fractionating column and pumped to the respective storage tanks. The coke in the coke chambers is removed by a hydraulic decoking proeess where water at very high pressure cuts out the hard coke whicb is then sent to the storage yard.
The lube oil fractions from the atmospheric vacuum distillation unit contain such undesirable components as aromatics, wax, asphaltenes, and sulfur compounds. These are eliminated in three different units, in succession, i.e., phenol extrnction, dewaxing and clay contact units.
In the phenol extraction unit the lubrieating oil cut iS subjected to a solve& extraetion proeess using phenol as the solvent to remove aromatics, and the separation and recovery of phenol from the extraction column is effected by distillation. The pure extract is used for bitumen and the pure raffinate further refined in the dewaxing unit.
In the dewaxing process both wax and oil are separately passed through a solvent reeovery system where the solvent is removed and recireulated in the dewaxing unit. The wax is fed to the coking unit. The lube oil fraetion afta dewaxing is then passed on to the clay contact unit to improve ita color by a process of ph>-sica1 adsorption using activated elay. The elay is filtered away and treated lubricating oil fractions sent to storage tanks for disposal as base oils.
Bitumen llnit
The short residue from the atmospheric vaeuum unit is not suitable for use as road asphalt and has to be further processed. TO achieve this the mixture of short residue and phenol extract is transferred into vessels called oxidizers where it is blown with air. The overhead vapors are condensed and pumped into cokitig unit feed tanks and the blown bitumen filled into drums. Part of this is also thinned down to produce cutback. bituma.
BY processing Z,OOO,OOO metrie t,ons of crude oii the following product,s in metrie tons will be produced by the Baruni refinery.
According to the revised project it has been decided to expand the refinery capacity from 2,000,OOO to 3,000,OOO metric tons. When the expan- sion scheme is completed the product pattern in metric tons would be as follows:
Forced Conuection Inside Tubes----- Part 7b
Applied Heat Transfer Design for the HPI
by Wayne C. Edmister and Joseph M. Marche110
7-G. Surface coe.ficient caleulations. Turbulent flozo. Using Eq. 7-3, the formulas for surface coeffieient may be rewritten in a more convenient form for praetical application: for heating fluids, Nne > 2100, from Eq. 7-7
h, = O.O243(k/D! (DG~~)00-8(c~,‘k)0~1 Eq. 7-15 for cooling fluids, NR, > 2100, from Eq. 7-8
h., = 0.0265 (k/D) (DG/,~)“.B(~~/k)0.3 Eq. 7-16 far heating of gases and low viscasity fluids, NI(= > 2100, from Eq. 7-9
h, = O.O23(k/D) (DG//A)~~~(c@)~.~ Eq. 7-17 for heating or cooling of high viscosity fluids, NRe > 10,000, from Eq. 7.10
where ti is estimated from Eq. 7-13. Oil in tuba. For the heating of oils the following
empirica.1 formula may be used :g h, = 0.034~/@~~ Eq. 7-22
where u is the veloeity in ft/hr and p is viscosity in lb/ft-hr at the average temperature of the oil. If in a long pipe the temperature varies irregularly, the oil viscosity must be referred t,o the average temperature obtained by integration, as explained in Section 7-5.
If the oil is being eooled, the surface coeffieient found by Eq. 7-22 should be reduced by about 25%.
EXAMPLE 7-l. Determine the average surface coefficient of beat transfer at the water side of a single-pass stream condenser. The tubes have a
50
va-in. OD and 0.049.in. (1RBWG) wall thickness. 4 the average water velocity is 7.5 ft/sec; the water
enters at GOF and leaves at 68F. SOLUTION: Eq. 7-Y 1s most suitable if IV,, > 2100. The average water temperature t = (GO + 68)/2
= 64F. By interpolation from Table 7-4, ‘c = 0.999; p = 62.31; p = 2.57; and k = 0.346. The inside diameter D = (0.875 - 0.049 x 2)/12 = 0.0648 ft ; IJ = 7.5 x 3600 = 27,000 ft/hr.
The Reynolds number is found as: NR., = 0.0648 x 27,000 x 62.31/2.57 = 43,400
and therefore, the use of Eq. 7-9 is justified. The Prandtl number is compnted and is:
Np7 = 0.999 x 2.57/0.346 = 7.42 and, by Eq. 7-9,
h, = 0.023(0.34G/0.0G48j43,400°~* x 7.42°.4 = 13Y5 Htuihr-sq ft OF
r-7. Special cases of fluids ix tubes. Vertimi tubes. The Nusselt equation for heat transfer and other equations of the same type are based on tests with flow in horizontal pipa, nevertheless they may be applied satisfactorily to flow in inclined and *
aertical pipes, with the results being on the con- servative side.‘” However, in the case of a liquid mar the boiling point, where the position of the pipe may influente the liberation of vapor bubbles from the surfaee, other equations should be used.
Curved tubes. If straight tubes are conneeted by semi-circular bends or if tuba are bent to form helical coils, turbulence of the liquid will be in- ereased; which will result in an increase of the surface coeffieient~. Tests by Richter” with a helically coiled double-tube water-to-water beat exchanger~ showed that with water velocities of 6 to 8 ft/sec the surface beat transfer coefficient was about 20% higher than in an exchanger with straight tubes. Far gasa and low viscosity liquids sueh as water, a fair approxmation may be ob- tained by multiplying h, determined for straight flow by a factor’2
August 1965 Petro/Chem Engineer
~b = 1 -!- 3.5 DJD, . . Eq. 7-23
where D,i is the inside diameter of the tube and D, is the mean diameter of the coi1 or curvature.
Amulw spacea. In double-pipe beat exchsngersl one of the fluids is flowing in an annular SPK formed by two concentric tubes. In determining the hydraulic radius T,~ or the equivalent diameter D, that enters in the expressions for Nne and h,, there will be a differenee depending on whether the outside of the smaller tube, or the inside of the lager tube, or both are participating,in the hcat transfer to or from the fluid in the annular space.
If heat flows through both surfaces, the value of the equivalent diameter is given by Eq. 6-12 BS D, = De - Bl. If heat flows only through one of the two wetted ~urfaces, the hydraulic radius is based on the perimeter of this surface. Thus, if heat flows through the inner surface, by Eq: 6-8 and 6-11,
D’, = 477, = (D2s ~ DV,‘D : Eq. 7-24
If beat flows only through the outer surface:
e = (PZ - D”i);Dv Eq. l-25
@~ D1. _ Eq ‘7 24 and 7-25 give lager values far the equivalent diameter than Eq. 6.12 and, therefore, give more conservative values for h,. They were used by Nusselt in correlating experimental data. However, even if the outside surface is beat in- sulated, it will participate in the heat exchange to a certain extent, if not by convection then by radiation. On the other band, the inner surface is seldom ever insulated and in equations for IL, the diameter enters only in the 0.2 power and Dy2 found by Eq. 7-22 or 7-23 numerically does not
F ‘etro/Chem Engineer August 1965
A QUANTITY SURVEY . . . We’ve received numerous letters in- dicating that it will be advisable to reprint the complete APPLIED HEAT TRANSFER FOR THE HPI series as soon as it is concluded.
It would help us greatly if we had an indication as to whether you might purchase a copy of the entire 19 in- stallments, in handbook form, at a nomina1 cast.
TO indicate your interest, simply circle item #60 on the postpaid Reader Service Center reply card bound in this issue.
You will not obligate yourself in any way and it will insure your re- ceiving full details on how to obtain
:P%%t Hc% TRANSFER FORI of the complete
THE HPI the moment it is available.
Heat Exchangers designed and built by THERMAL TRANSFER CORP. mean new materials, new engineer- ing, concepts and exact fulfillment of customer requirements and specifi- cations.
Heat Exchangers designed and built bv THERMAL TRANSFER CORP. piovide increased eficiency, greater durability and produce guaranteed results.
Why not give your next heat ex- changing problem to the THERMA~L TRANSFER CORP’s. design engin- eering staff far an easy solution.
Monroeville, Pennsylvania 15146
51
differ much from U;~* by Eq. 6.12. Therefore, the simpler Eq. 6-1213 is recommended.
7-8. Heat transfer and fluid friction. Both theory and experiments show that there is a dire& relation- ship between the film conductance and pressure loss for flow in tubes and pipe% Experiments show that beat and momentum in a fluid are transferred in the same manner. A simple equation expressing this result was developed by Reynolds:
h, = fvc p/8 <~ Eq. 7-26
where f is the friction coefficient discussed in Sections 6-3 and G-4.
Eq. 7-26 was derived by Reynolds assuming that the Prandtl number -Vp, = cp,& = 1, therefore this relation gives accurate results only for gases having four or more atoms per molecule (Np, -‘ 1). Far other gases that have a smaller number of atoms, N,, < 1, and for liquids the inaecuraey of Eq. 7-26
Or I 1 1 I 0 0.25 0.50 0.75 1.0
Rodia1 disfance, y/R, from the woll of the tube
Figure 7-6. Temperature variation with distante fmm the wall.
is very large. A more general equation may be presented in the form:
h., = fvc p$/S .~ Eq. 7-27
where the function + takes into account the in- fluente of the Prandtl number. Unfortunately, the various expressions that have bea developed for $I are rather eomplicated and are often based, on questionable assumptions.
A simple expression relating h, with the friction coefficient f, whieh applies for turbulent flow of various fluids in long tube3 or ducts, may be derived from Eq. 7-9.
(1.9 “.S h, = O.O23vcp/N~, Np, Eq. 7.28
noticing that for smooth pipa Eq. 6.22, 0.023/ U.?
IVR,~ = f/8, gives:
h, = fvep/%A$y Eq. 7-29
A comparison of Eq. ‘7-2’7 with Eq. 7-2Y shows,
for flciw in smooth pipa, $ = l/Ni,t 7-9. Ana.logy between beat and momentum trans.
52
fer. Sine Reynolds first suggested that beat and momentum are transferred by identica1 mechanisms, a number of investigators have proposed analogies between heat and momentum transfer in conduits. The temperature profile and transfer rate may be determined by integrating the Au equatians, Eq. G-39 and 6-40. In order to earry out the integration it is necessary to know the dependency of the loca1 flux and eddy coefficient upon position.
In their early analogy calculations, Prandtl and von Karman point.ed out t,he importance of physieal properties and the need for accurate velocity data. Mart,inelli’4 employed the generalized velocity profile to develop an analogy solution for the temperature profile and Nusselt number. His analogy is based open the assumptions that the eddy transport coeffieients for heat and momentum are equa1 and that the flux and shear stress have the same de- pendency upon radia1 position. Martinelli’s solution fits the heat transfer data fairly ~11 for small values of the Prandtl nomber (Iiquid m&ls).
The recent extensions of Martinelli’s work have eliminated the laminar region by developing ex- pressions for the eddy eoefficients which vanish at the ~11. The analogies of Deisslerlj and 0 ReichardV” are typical of this extension.
Somewhat more sophisticated analogia have ben developed by Hunziker” and Sleicher and Tribus.18 Their methods remove the assumptions that the eddy coefficients are equa1 and that the heat flux and shear stress variations are similar. However, neither of these c%lcul&ons were extended for very large values of the Prandtl number.
Results of the more recent analogy ealculations’g agree well with Eq. 7-Y and 7-28. These calculations also predi& temperature profiles that are in good agreement vith Fig. 7.6.
A number of investigators have measured tem- perature profiles for mercury, air, and water being heated in turbulent flow. There is good agreement. among the various investigators, and profiles for al1 three systems have been measured at Reynolds numbers of approximately 40,000. Fig. 7-6 shows the variation of temperature wit,h position for these three eommon fluids. As Prandtl number in- m ereasea the fluid temperature rises steeply and tends to approach the centerline value more rapidly.
7-10. Swfaee eonditions. Surface deposits. Al1 expressions far the surface coefficient h, giva in Sect~ion ‘i-6 apply to turhulent flow of fluids in cbm- mercial steel pipa and meta1 tube in clan condi- tion. Any deposit on the inside surfaee of a pipe or tube of oxides, scale, grease, or gas or steam bubbles will appreeiably reduce the film eonductanee. For gas flowing through relatively short tubes, a light carbon film from smoke has been found to reduce beat transfer by lo%?” Also for gas flowing in brass tubes, the differente between h, far 8 clean polished tube and LC for u tube with an oxide film, normall~ found in brass tuba. is sbout 16%,.
Far water oi other liquids flowing in a tube, the effect of & film of foreign matter on the tube surfaee is ewn greater because the resistance of
August, 1965 Petro/Chem Engineer
a film of liquid is relatively mueh xnaller than the resistance of a gas film.
Actually, deposits of foreign matter on a tube rv+ll~ do not at&xt the convection of beat but the beat conduction through the wall. The deposit acts as an insulator because the beat eonductivity of oxides, scale, and the like, is considerably lower than of the meta1 of the tube. This prohlem will be discussed in more detail later ixi discussion of the overall beat transfer coefficient.
Surfaee roughness. Scratches and roughness of a tubesurfaceincreaseturbulenee, decreasethethick- ness of the stagnant fluid film and inerease the heat t,ransfer coefficient. This is particularly apparent for flow in small tuba in which the roughness, expressed SS the average depth of roughness divided by the tube diameter, is relatively larga than in larger tubes. Increasing the roughness of tuba intentionally is not practical, because the increase of the hea,t transfer coefficient due to increased roughness is smaller than the increase in frietional resistanee to flow and resulting inerease in power requirements.
l 7-11. Velocity eh.ange in short pipes. The abrupt
change in the velocity of & fluid stream as it enters a tube produca turbulence near the tube enti-ance due to a contraction in the cross seetion of the stieam caused by boundary-l&yer separation. This contraction increases the fluid velocity. Both effects extend some distante along the pipe and serve to increase the beat transfer coefficient. According t,o A. D. Mo0re,~1 far short t,ubes, L/D < 40, the sur- face coefficient h, determined by the usual procedure for long tubes should be multiplied by 8 factor C. The factor C may he determined from the following empirica1 expression:
C = 2.07,‘(L/D)“.2 . . Eq. 7-30
Latzkoz2 has analyzed boundary layer build up in the inlet region for bellmouthed entry. His analysis
predicts that the mean heat transfer eoefficient from the point of entry to the point L is given by:
B,* b’, -=l++D .Eq. h cc
731
where the factor C’ is related to hydrodynamics by the equation
C’ = O.l44N,,% Eq. 7-32
Data indicate that the factors C or C’ should be applied whether the tube inlet is sharpedged or rounded. Although the rounded inlet does not cause a eontraction of the stream, it increases the fluid velocity near the tube wall to almost the average veloeity of the stream.
7-12. Swface cot?fJicients with l,arnina7 j%w. In most heat exehangers with forced positive flow of the fluid, the velocity is above the critica1 value and the flow is turbulent. However, for viscous liquids, the fluids often move at velocities below critica1 values. In this case the flow will be laminar, or streamline, if no heat exchange takes piace. When beat is being added to or removed from the
Petro/Chem Engineer August 1965
liquid, convect,ion current,s are set up and the flow ceases to be completely laminar. However, if the velocity is below the critica1 value, it is usually still referred to as laminar.
In view of the limited use of streamline flow in heat exchangers, there are eonsiderably less ex- perimental data on which to base a surface coefficient equation. From tests with heating water, glycerine and lighter oils in lamina flow, the following formula has been developed23
h, = 2.53(kjDj (mc/kL)s Eq. 7-33 where: m is the rate of floid flow, lb/hr
D is the actual or equivalent diameter of the conduit, ft L is the length of the beat exchange sur- face. ft
The dimensionless group ,me/kL is the Graetz number and is designated by Nel. Eq. 7-33 may be applied for NGz ranging from 8 to 3500.
Eq. 7-33 may also be used for cooling low viscosity liquida However, for cooling very viscous oils, it is recommended that k, be reduced by ahout 30%. This gives
lì, = 1.78(k/D) (.m.c/kL)” Eq. 7-34 The reduction of h, iti caused by an appreciable inwease in viscosity, which decreases the convection flow when the oil is cooled.
Film temperotwe. If the differente between the mean pipe-surfaee temperature t,’ and mean liquid temperature t is over lOF, then the viscosity f-~ and the densi& p must be determined at a weighted film temperature t’,>z( where
t’, = t + %(t, - t) Eq. 7-35
EXAMPLE 7-2. An economizer in a boiler plant is built of seamless steel pipe Z-in. OD, 0.120.in. wall thickness and lo-ft long. The average tempera- ture of the water is 140F and the inside surface of the pipa is at 273F and the water veloeity is 3 ftisec. Determine the probable surface coefficient of heat transfer.
SOLUTION: The inside tube diameter D = (2-0.120 x 2) 12 = 0.1467 ft. The weighted temperature of the water is, by Eq. 7-35, t, = 140 + (273 - 140)/4 = 173.3F. At 173.3F, from Table 7.4: p = Gq.72 lbi’eu ft, c = 1.001 Btu/lb-deg F, ,/J = 0.88 Ib/ft-hr and k = 0.387 Btujhr-ft-deg F. The water velocity is v = 3 x 60 = 180 ftihr.
The Reynolds number is: Nne = Dv p,‘p = 0.1467 x 180 x 60.72/0.88 = 1824
This shows that the flow is laminar. The rate of flow is found SS:
,m. = 0.7854 x 0.14672 x 180 x 60.72 = 184.8 lb/hr and the Graetz numher is:
‘Nca = mc/kL = 184.8 x 1.001/0.387 x 10 = 47.8 This shows that Eq. 7-33 can be used and therefore:
where : fi is viscosity of the liquid at its bulk temperature t and ps is its viscosity at the temperature t, of the inside surface of the tube. For preliminary caleulations the empirica1 simple Eq. 7-27 may also be used for laminar flow of oil.
r-13. Non-Newtonian Raids. Newtonian fluids exhibit a proportionality hetween shear stress and shear rate in lamina flow. This is usually stated as
‘=TcdT. Eq. 7-37
where: p is the viscosity of~ the fluid. Non-Newtonian fluids aré materials which do not ohey this relationship.
All gasa and liquids of low ~molecular weight exhibit, Newtonian behavior. High molecular weight polymeric materials and suspensions of solida in liquids comprise the major portion of the materials t,hat do not obey Eq. 7-37.‘6
At the present time the flow conditions have not been clearly defined for the various types of non- Newtonian fluids and it is difficult to make accurate predictions of the heat transfer problem. It is recommended that the equations discussed in Section 7-13 be used t,ogether with average values of the effettive viseosity.
7-14. Liquid meta1 h.eat transfer. Perhaps the most important heat transfer application of liquid metals is SS, coolants in nuclear reactors. The basic reasons for using liquid metals as coohmts are :
(1) The reactor can be operated at high tem- perature and low pressure. This is due to the low vapor pressures of liquid metals.
(2) The reactor coolant must have a high heat transfer eapability. This property of liquid met,al& is indicated by the very low values of the Prandtl number which they exhibit.
A principal phase of any nuelear heal transfer design is prediction of the loca1 and average values of the heat transfer eoeffieient.27 A number of analytieal and numerica1 solutions have heen ob- tained,28,2Q some taking into account the variation of temperature and physical properties. However, the bulk of the usefu,l information consists of generalized dimensionless correlations of test values of h.
Lubarsky and Kaufma# have summarized the work of over 40 investigators and recommehd the equation
U.4 N NEL = 0.625 NP@ Eq. 7-38
where: Nr. is the product of the Reynolds and Prandtl numbers and is called the Peclet number.
54
References: “An Investigation on Aircraft Heaters, National Advisory Commi&?e for Aeronautics, ARR5A06 (1945).
7McAdams, Heat Trammission, op. cit. p. 226. 8A. P. Colburn, “Mea” Temperature Differente
and Heat Transfer Coefficient in Heat Exchanger,” Imi Eng. Chem. 25, 874 (1933).
9F. H. Morris and W. G. Whitman, “He& Trans- fer for Oils and Water in Pipes,” Jour. Ind. Eng. Chem. 20, 234-240 (1928).
IoMcAdams, op. cit., p. 228. 1’G. A. Richter, “Douhle-Pipe Heat Interchang-
em,” Trans. A.Z.Ch.E. 12, Part 11, 147-185 (1919). l2D. Jesehke, “Die W&neuhertragung und Druck- verlust in Rohrschlangen,” Zeitschrift Vereines deutschw Ingenieure, 69, 1526 (1925). ‘3McAdams, Op. cit., p. 241. laR. C. Martinelli, Il’yans. ASME, 69, Y47 (1947). ‘jR. G. Deissler, Nat. Adv. Comm. Aero. (now National Aero. Space Adm.) T.R. 1210 (1955). IaH. Reichardt, Nat. Adv. Comm. Aero (now Nat. Aero Space Adm.) Report, N-411947 ,‘(1956). IrP. R. Hunziker, Journal of the Franklin Institute 265, 205 (1958). IsC. A. Sleicher Jr. and M. Tribus, Trsns. A. S. M. E. 79, 789 (1957) ; “Heat Transfer and Fluid l Mechanies Institute, Preprints, Stanford Univer- sity (1956). 19J. M. Marche110 and H. L. Toor, Ind. Eng. Chem. Fundamentals, 1, 97, (1963) % R. Parsons and D. R. Harper, “Radiators for Ai,rcraft Engines,” National Bureau of Standards, Tech. Paper 211, 326 (1922). “A. D. Moore, “He& Transfer Notes fo,r Eleetrical Engineers,” Ind. Eng. Chem. 28, 704-708 (1936). =A. Latzko, Nat. Adv. Comm. Aero. (now Nat. Aero. Space Adm.;l T.M. 1068 (1944). i3W. J. King, “The Basic Laws of Heat Trans- mission,” Mech. Engr. 54 412 (1932) “McAdams, op. cit., p. 149. z5E. N. Sieder and G. E. Tate, “Heat Transfuand Pressure Drop of Liquids in Tube&” Ind. Eng. Chum. 28, 3,429-36 (1936). z6A. B. Metzner, “Non-Newtonian TeehnOlogy: Fluid Mechanies, Mixing and Heat Transfer,” Advances in. Chemical Engineering, Vol. I, edited by T. B. Drew and J. W. Hodges, Aeademic Press Inc. New York (1956). *% F. Bonilla. Nuclear Enginewing. New York: McGraw-Hill Book Co., Inc. (1957). =R. G. Dressler, Nat. Adv. Comm. Aero., (now Nat. Aero. Space Adm.) RM-E52F05 (1952). zsR. W. Lyon, “Forced Convection Heat Transfer Theory and Experiment with Liquid Metals,” ORN L-361, Tech. Div. Eng. Res. Sec. Oak Ridge National Laboratory (1949). aoB. Lubarsky and S. J. Kaufman, Nat. Adv. Comm. Aero. (now Nat. Aero Space Adm.) T. N. 3336 (1955).
Part 8, Heat Transfer Obtside of Tuba.
1
WIII appear m the September ~ssue.
August 1965 Petro/Chem Engineer
Heat Transfer Outside of Tubes-Part 8
Applied Heat Transfer Design for the HPI
by Wayne C. Edmister and Joseph M. Marchello
S-l. Fwced eo?mection. WWT external swfaees. Experimental data for beat transfer by forccd convection outside of tuba may be correlated by using a dimensionless equation of the same Nusselt type as was obtained and used for fluida flowing inside of pipa:
h,D& = C(D,~p~/~~cii~(cç~f/k~)~ _‘Eq. 8-l where: D, designa& the diameter of the outside
norma1 to a tube oi rod wit,h NI<, = Dapf/p~~ from 0.1 to 50, the recommended numerica1 values’in Eq. 8-l are C = 0.9, a = 0.39 and b = 0.31.’ The resulting expression far the surface coefficient is:
h, = 0.9(kriD,) !D,w~~//L~I”.~~ (c+,)“-~~ Eq. 3-z For fluids flowing with Nne from 50 to 10,000,
the recommended faetor and exponents we: C = 0.6. a = 0.5 and b = 0.31. Thus:
l .surface, jLC = 0.6 (kt/DJ (D&pf//Lf) O.5 icp&) o.31
21 is the free-stream velocity, and Eq. 8-3 f indicata that k, p, and p must be
evaluated at the film temperature 1,. However,~ the numerica1 values of the constant C and exponent,s 0: and h are different from those in Part 7.
For fluids flowing with Nn. from 1,000 to 50,000, the recommended factor and exponents2 are: C = 0.38, a = 0.56 and b = 0.3. Thus:
Flozu along pipes. Far fore& convection and turbulent flow along pipa, us in a shell with a bundle of pipe or tuba in it, the sane equations as in Part 7 may be used. However, the term D must be replaeed by an equivalent diameter D, as determined by Eq. 6-11.
EXAMPLE 8-l. Water moves axially in connterflow inside an Il-in. Shell that contains twenty-two l!$ in. OD tube; the water veloeity 1s 50 ft/min, its average temperature is 75 F and the average temperature of the tuba is 85 F. Determine the su-face coefficient far these eonditions.
SOLUTION: The flow is turbulent and from Part 7, the film temperature is estimated to be tf = (75
) ai- 85),‘2 = 80 F. The properties of the water, at t = 80 F, are: c = 0.998 Btu,/lb-“F; p = 62.17 lb/cu ft; ,L = 2.03 lb/ft-hr; k = 0.355 Btu/hr-“F; and N,,,, = 5.84.
By definition G = 150 x 60 x 62.17 = 559,500 lb/hr-sq ft,. D = 1.5/1!2 = 0.125 ft. The Shell diameter is 11/12 = 0.917 ft and the flow area. is:
s = 0.7854 x 0.9172 - 0.7854 x 0.1252 x 22 = 0.315 sq ft and, the equivalent diameter:
D, = 4 x 0.315 ~-~ = 0.1095 ft
11.51
Checking far the Reynolds number, NRE = 0.1095 x 599,500/2.08 = 29,400.
From Part 7:
h,. = 0.0243 (g$)’ x 29,400”.8 H 5.x40.”
= 662 Etul;hr.-sq-ft-‘F
Petro/Chem Engineer September 1965
Eq. 8-4 Flow ILCTOSS a tube bank. Tests indicate that beat
transfer from fluids to banks of tuba with tur- bulent flow is considerably higher than to single tuba. For flow of gases and liquids normal to banks of staggered tubes, not baffled, and Reynolds numbers ranging from 2100 to 32,000 the recom- mended values in Eq. 8-l are? C, = 0.33, a = 0.6 and b = ‘:b. This results in the following expression for the mean beat-transfer coeffieient:
Far banks of tuba in Z&e, not, staggered, the use of Eq. 8-5 is recommended with the reduction of the factor C from 0.33 to 0.26.a
Mechanism. of h.eat trans.fei. Experimental in- vestigations of t,he nature of fluid flow across single tuba and tube banks have shown that the fluid film does not have a uniform thickness. It is thinnest at, the point directly facing the stream and grows in thickness on the sides and at the downstream section where eddies are formed Fig. 8-l. For this reason the coefficient of heat transfer between the nuid and pipe is not uniform around the perimete?. Fig. 8-2 shows loca1 rates of beat
57
58
Figure S-2. Pattern of heat flow ar0una a
pipe with air flow normd to one tube.
flow plot,ted radially st various points around n steam heated pipe in an air stream.
For design purposes, it is easier to dea1 with average, cr mean heat transfer coeffidents, desig- nated h,,,, rather thsn with loca1 coeficients. Sine equations of the Nusselt type csnnot represent average coefficients with a sufficient accuracy, in engineering practice i,t is oft,en necessary to apply empirica1 formulne bnsed on tests with speeific corresponding fuids as given in the following.
8-S. Fiozu of pxes a.cioss tpdws. Single tube. An empirica1 expression reeommended~ for flow of air at right angles to a tube or rod, for IN.+ from 0.1. to 1000 is:
h,,, = (/c;/D,) 10.32 + 0,43(D,G//.~,)~-“~] Eq. S-6 and for .NRe from 1000 to 50,000 the expression is:’
h,,, = 0.24(7~,/D,) (D&//L,)‘~~ Eq. 8-7 Finally, for NR, from 40.000 to 400,000 the formula is:”
h ,,,, = 0.0239(kf/D,j (D,G/p,)“.*“S Eq. 8-8 ,Flw acrosg~ a Tebe ba,z7~ Far turbulent, flow of
air at right, angles to a,~ tube bank, foor or more rows deep, with .stnggered tube spacing, the film c&fficient may be comput.ed from the expression :O
h,, = 0.0032ct + 460)“.VY’/U,” Eq. 8-9 where G is the mass velocity, lb/hr-sq ft, through the ,+ninirn.um free area S between the tubes; t iu the average air temperature, deg F; D, is the outside diameter of the tuba, feet. For banks of t,ubes in. Iine, the factor 0.0032 in Eq. 8-9 must be reduced to 0.00’25. Eq. 8-6 to 8-9 mayy be npplied to any gas nhose 7~ and ,L do not differ greatly from those of air.
EXAMPLE 8-2. Atmoapheric air with an avernge temperature t = 120 F is ,Oowing at 25 fthec across Z-in. OD tubrs whose surface is kept at a temperature of 212 F. Determine the surface coefficient for these conditions.
SOLUTION: In order to select the proper’ ex- pression for l~.~,,,,. first one must determine the Reynolds number nr,,. The film temperature is:
t, = 0.51120 i- 212‘1 = 166 F Interpolating from Table 4-l (see March 1965, p, 43) giva /c, = 0.0173 Btuihr-ft-“F, pLi = 0.050 lb/ft-hr, p = 0.0636 Ih/eu ft, N,>r = 0.70, v = 25 x 3600 = 90,000 ft/hr and G = 2’ = 90,000 x 0.0636 7 5724 lb/hr-sq ft. D, = 2/12 = 0.167 ft and IV,, = 0.167 x 5724/0.050 = 19,100. Thus, Eq. 8-7 can be used and:
8-3. Flow of li&k aeross tubes. Single tuba. ~For liquids flowing with Nne from 0.1 to~200, an empirica1 equation similar to Eq. 8-6 may be used:7
Ii ,,/ = (k,lD,l (cp#~,) OJ
IO.35 + 0.56(D,G/~j)“.5”] Eq. 8.10 For flow wilh NRe over 50 and up tu 10,000 the
use of Eq. 8-3 is recommended. Flow aeross a tube bmlc. Heat exchangers with
liquids bot,h inside and outside the tubes are of’ a great variety of design. Many have baffle plates to increase the heat transfer by increasing the flow velocity. In many designs the liquid is moving partly acro~s and partly along the tuba. It is impossible to give any special formulae for the different cases. Usually the mean beat-transfer coefficient lt.,,, is eomputed using Eq. 8-5 and, if baffles are used, h,,,, is multiplied by 0.6 to allow for leakage, or short-circuits in t,he flow.
Far water, for Nnc > 2000, the reeommended simplified eqoation far staggewxl tnbas is:R
Ih,,, = (1 -;- 0.0067 td iv,,,,,)0.F/(D”)0.4 Eq. 8-11 where v,,,,~,, and D, are in ft,-hr units, as in al1 other expressions. For tubes in Cine, there is a reduction of 20% in the value of h,,,. In baffled exchangers, there is a reduct,ion of 40% in IC,,, due to leakage.
EXAMPLE 8-3. Find the surface coeffìcient using the same data 8s giva in Example S-1, assuming that the water Bow across the tubes is guided by snitable baffle plates. l
SOLUTION: The Reynolds numbei in this case is Nrt, = 0.125 s 559,500/2.08 = 33,600 and by Eq. 8-5:
880 Btu/hr-sq ft-“F
The surfnce coefieient is considerably higher tha,n with flow along the tube because of a betta turbulence. Even if a correction eoefficient of 0.8 is used to allow for buffle leakage, h, = 880 x 0.8 = 704 Btuihr-sq ft-“F. This is stili higher than found in Example S-1.
Checking by the simplified Eq. 8-11 gives:
= 830 Btu/hr-sq ft-“F:
Thiu is a satisfactory agreement with the result~ obtained hy the more elaborate Eq. R-5.
8-4. Flozo /dong plo»e s~wfa~~;zs. From the theory of analogy between heat transfer and frietion faetor, with the constant exponent found from air data, the follorving equation has been established:
NN,, = 0.055Nng0.= Eq. 8.12 Substituting for .NNrL and Ni,, their oalues gives for the surface coeffieient expression:!’
h, = 0.055(k/Lj (Lv&LLJ”.~~ Eq. 8-13 If the length L of the surfnce along which the
air flow~ exceeds 2 ft, t,he turhulence will become fairly constant and the use of L = 2 ft is rec- ommended for longer surfaces. Eq. 8-13 applies to smooth surfeees. With rongh surfaces Il., will be highei; however, at the present time there are not enough data available to erpress this influente analyt~ically.
A satisfact.ory correlation of lata tests,‘0 also with air, wau obtained in the form of the equation:
September 1,965 Pet’ro/Chem Engineer
The Barton Flotrac Meter was designed with high pressure water metering applications in mind, but offers excep-
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Flotrac accuracy is 20.5% of reading, with an operating pressure range up
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(Request Bulletins 306-l and 309-l
from BXiZi4 Instrument Corporation, a subsidiary of International Telephone and Telegraph Corporation.)
N ,.qu = O.O49ZN$f Eq. 8-14 where Np. is the Peclet number. From Eq. S-14:
h, = 0.0492(k/L)“~2 (v~c)~.~ Eq. S-15 However, h, by Eq. 8-15 is 15% to 25% higher
than by Eq. 8-13, probably because of certain eon- ditions in the tests.”
An empirica1 expression, convenient for pre- liminary caculations for standard air of 70 F and atmospheric pressure, has the general form:”
h, = a ,t bv” where far air velocities below 16 ft/sec, a = 0.9, b = l/l?OOO, n = 1.0; for uii velocities fiom 16 to LO0 ft/sec, n = 0, h = 1/1180 and n = 0.78. However, for more dependable and conservative calculations, especially if the air temperature differs appreciably from 70 F or the pressure from at- mospheric, Eq. 8-13 is recommended.
Eq. 8-11 can he used also for fluids other than water if the Peclet number is introduced as in Eq. S-14. YofldT -
Figure 8-3. Thermal and hydionamic boundaiy layers on a flat pl&.
S-5. Thermal bow~dary layers. As fluid flows past a solid body which is at 8. different temperature, a thermal boundary layer BS well as a hydrodynamic boundary layer, forma. In the thermal boundary layer the fluid temperature at the solid surface equals that of the surface. The thickness of the thermal boundary layer may be different fiom that of the hydrodynamic boundary layer, and it starts at the point where the beat transfer begins, which may oi mag not be at the leading edge.
The thermal and hydrodynamic boundary layers on a flat plate are shown schematieally in Fig. 8-3. ‘Far a Prandtl number of unity, both boundary layer~‘~ grow at the same rate. For Prandtl numbers less than one, the thermal layer grows more rapidly than the velocity layer. On the other band, for Prandtl numbers greater than unit-, the thermal layer grows slower than the velocity lwer.
The differential equation far the temperature distribution is :
where <y = k/C,p, the thermal diffusivity. The velocities, u and v in Eq. S-16 must bc
obtained from the equation of motion and the continuity equation, and will be fwxctions of the independent variables z and’ g.”
The major portion of the early work with thermal boundary layers w&s done by PohlhausaGS who solved Eq. X-16 to obtain the temperature profile and beat transfer coefficient for laminar flovj parallel to flat plates. A considerable amount of work1”J7 has been done recently on the problem of heat
60
transfer taking piace during flow past two- dimensiona1 bodies auch as cylinders CIP airfoils. For the most part, the results are in agreement with the empirica1 correlations of Section 8-l and 8-4, when the hydraulic +adius copcept is employed.
X-6. High velocity f!mo. The design of rock&, nirplanes and gas turbina requires a knowledge of beat transfer at high fluid velocities.‘s At high velocities frctional heating beeomes appreciable and density ehanges in the gasa become significant.‘g A classica1 problem involving friction heating in compressible Aov is that of re-entry of racket nose cones into the earth’s atmosphere.
Ka fluid streaxn of velocity V and temperature t is brought to rest without heat loss (adiabatically), the rise in temperature due to’ the conversion of flow energy to ~thermal energy will be:
P t, -- t = -- ZSJG
Eq. 8-17
In the boundary layer surrounding a high velocity vehicle, the air is brought to rest hy viscous forces. The air temperature near the wall is inereased by both stagnation and by transfer of momentum to the wall. If no ,heat is transferred through the wall l l of the vehicle the system will be adiabatic. The temperature rise in the boundary layer causes beat conduction back into the surrounding air. The adinbatic waT1 of the vehicle assumes a temperature, t,,,, somewhere between the stagnation temperature and that of the surroundings. The degree to which stagnation ia achieved is expressed in terms of a tempemture mcove~~ factor, N,t:
.h;, = “TL+? Eq. 8-18
The ratio of the actual velocity V to the acoustic velocity V, is the Mach Number, N,,z,. For an idea1 gas the Coustic velocità is piven by:
v,, = Eq. 8-19
where R is the gas constant, M is the molecular weiglit and T is the absolute temperature of the gas.
By combining ‘Eq. S-17, X-18 and S-19, the re- covery factor may be expressed a~:~‘l ea
Johnson and Rubesi+ have reviewed the theary and data on recovery factors. They predict that for the range of Mach numbers from 0 to 10 in the laminar boun.darzJ lager :
N,! = NZ Eq. 8-21~
and foi a ~turbulent boundaiy layer :
N,,= .$ Eq. 8-21b
Eq. 8-20 and 8-21 may then be used ta caleulate the adiabatic wall temperature.
In high velocity flow the effect of stagnation is to make the temperature differente between the wall and the fluid finite, even though heat may not be transferred to the wall (adiabatic). This diffi-
September 1965 PetrojChem Engineer
culty is resolved by t,he use of an effettive tempera- ture differente. t,, - t,,. If the wall temperature is different than the adiabatic wall temperatuw, beat transfer will “ccur :
where h., is the effect,ive beat transfer coefficient. In general, the data far high velocity flow agree
well with the eorrelations presented earlier in this ehapter when the conventional beat transfer coeffi- cient, h, is replaced by the effretive coefficient. h,. For example, beat transfer data” far Mach numbers in the vicinity of uniti have been cos- relat,ed bu :
h,D -~ = 0.478 N;; -i”R EQ. 8-23 k
which is of the form of Eq. 8-1. There are two important faetors to be eonsidered
in eompressible flow calculat.ions. First, average values~ over the flow length must be used whcn ehanges in pressure and density are importa& as is often the case in high velocity flow. Second,
0 velocity limitations “ccur in duets and throats when the exit velocity is sonic. An example of thi,s is the sonic throat “i & roeket erigine. The com- bustion gasa flow through the throat at Mach “ne and accelerate in the expansion section unti1 they pass out int” space: however, ang small change in the snrrounding pressure (space) does not ap- preeiably change the flow rate through the sonic throat. 8-7. Selection of insuiation. Selecting the insulating materia18 most suitable for a eertain application is an important problem whieh involves faetors sueh as: (a) effectiveness, (b] initial cast, (c) ease of application, (d) maintenance and Ce) life.
Effectiueness. The effectiveness of a heat-insulat- ing materia1 is in inverse proportion to its thermal conductivity. Values of conductivity for some com- monly used insulating materials may be fou,nd in Tables 3-4 and 3-6 (see Feb. 1965, pp. 36-37). Some materials have ai tendency to absorb moisture; “thers are verq porous and permit atmospheric
Da mcnsture to penetrate inside the insulating layer. In porous materials, when used at ordinary but fluctuating temperature, such as for buildings, or ;lt low temperatures for refrigeration and air conditioning equipment, atmospheric moisture in- side the insulating layer may condense and inerease the eonductivity of the material. The condensation lowers the materials’ effectiveness, sometimes to the point of almost destroying t,he, insulating effe& Measures can be taken to prevent the penetration of moisture but, they will increase the cast, of the insulation accordingly.
Initi& eost. The cast, of insulating materia18 varie cansiderably from materia1 to materia1 and with t,he improvement in their manufacture. The only safe way to determine insulation costs is bg obtaining qnotations from manufacturers or the? representatives.
Petro,)Chem Engineer Sept,ember 1965
sheets or bats and is simply la,id on or attacbed t” t,he surface that must be protected. Other insula- tions come in powder form and must be made int” paste nnd applied with, a trowel. ,After the paste dyies, the insulation is usually wrapped uith cloth, and the cowr pairited with an oil paint. The method of applying an insulation affects the initial cast considerably and must be taken into consideration.
Mainteimnce. Keeping the insulation in good working condition also may affect the cast. Some jnsulation, such as cork slabs, requires very little maintenance work; other types of insulation, such as blankets made of rock-wool, must be covered with a proteetive cover that requires periodie in- spection and possible repair.
Life. The life of insulating materials depends “11, the ch,emic,al ntabilitg nf the raw materia18 and the combination of these materials in the finished state. If a materia1 consists of a fiber with a binder the latter should be nonsoluble in water and the materia1 should not have u tendency to deteriorate when expoed to humid or dry, hot or cold ntmospheres. Moisture is an enemy~ of most insulating materials. Ordinarily, materia18 with low absorption and capillwy coefficients are preferred. However, highly absorbent materia18 may be satis- factory in applications whcre they will rcmain dry, sueh as those built into a household or commercia1 refrigerator a,nd properly sealed from inside and outside air.
5-s. Heat 1.0s~ tiwortgh insulatioa. Fiat swfece. Heat 108s through an insulated wall can be eom- puted by the general Eq. 8-24 in which bl designates the temperature of the surface~eovered by insula- Con and tx designate8 the temperature of the sur- rounding air.
The over-al1 coefficient U may be presented as l/(l/h! + z jki, where li’ = h, + h, and h, is the surfaee caefficient for convection and h, is the eoefficient of radiation, 5 the thickness of insula- tion, and k its beat conductivity. The corresponding expression far th,e beat flow has t,he form:
Eq. 8-24
The combined coefficient h’ and conduetivity k depends upon the temperature t,Y of the outside surface of the insulation which usually is not known and must be estimated. However, the in- fluente of an error ‘in estimating 5, is not very great. If u more accurate result is desired, the exact surface temperature ma,y be found by a trial and error method as shown in the following CGUllpl”.
Por comparison, the beat 108s without insulation may be fotmd from Eq. 5-24 with z = 0 and h’ = li,: + ir,, a,s :
/ i ’ Ii
Offices Thraughout The Southwest
l : 1
1 1
, 1 1
1
Pipe inszdatiovz. The expression far t.he beat loss through the insula- tion on a pipe or other cylindrical object may be derived similarly to Eq. S-24, a,nd is:
Z?rLk (tl - tz:l h
Q = &(D,/D,~) + 2k/h:~~) 0
Eq. 8-26 For comparison, the heat 106s
from a pipe without insulation may be found by Eq. 8-26 with Do/Di ’
,hiCk”WS
= 1, as: Figure 8-4. Economica1
R = dL(h, + h,j (tl - tsi thiekness of insulation.
Eq. 8-27
Example 8-4. A 6-in. IPS pipe earries super-heated steam of 800 F and must be covered with 85% Magnesia insulstion. Determine the beat 108s per 100 ft of pipe length to still air st 70 h’ using the~reeommended thiekness of insulation. h, + h,, = 1.65.
Solution: Neglecting the temperat,ure drop through t,he pipe wall, assume tl = 600 F. From Tahle Z-1, use 3-in. insulation. The OD of the bare pipe, from Table ,1-2 (see Mach 1965, p. 471, D,i = 6.625 in. or 0.552 ft; the value of D, in Eq. 8-27 is 6.625 t 3 x 2 = 12.625 in., or 1.052 ft. For such a heary insulation, a.wume the outside temperature t, = 100 F. The average temperature of the insulation is then 350 F and from Table 3-6 by interpolation, 7~ = 0.039 Btu/hr-ft-“F. Eq. S-26: 0
&=- 2Tr x 100 x 0.039 (600 - 70)
ln(1.052i0.552) i- (3 x 0.039/1.65 x LOS] = 18,230 Btulhr
Cheeking the assumed value of t, by an equation similar to Eq. 8-27 gives : 18,230 = * x 1.052 x 100 x 1.65 (t, - 70)
From ~this equation t, = 103.5 F. This is SO close to the value assumed before that a recalculation would chnnge the fmal figura less than the expected accuracy of the involved data.
8-9. Tlkiclcness of inmlation. Hot s2wfaces. In considering insulation of surfaces losing beat, it, is customary to compare the cast of beat loss per year from the bare surface wit,h that from the insulated one. The differente will give the basis for calculating the amount, of money that can be spent economically far the insulation. The ieduced hent loss with the surface inoulated mny he computcd using the thickness recommended hy the manu- facturer of the selected insulation in his catalogue. A typical sample of these recommendations is giva in Table 8-l. The loss of beat, far such an in- sulation is relatiwly small and amore accurate nnalysis is seldom justified.
Cold swfaces. In the refrigerating industry and in other applications of ,e4
low temperatures, the beat infiltrating from the surroundings is the main xat load and a more detailed analysis is justified, especially in a large in- stallation. The eonditions can be illustrated hy Fig. 8-4. As the thickness of the insulation is increased, t,he smount of beat tha,t must be ahsorbed hy the refrigerator per year decreases a+d with it the yearly eost decreases as’ shovn by curve >M. At the same time the eost of the insulation itself in- xeases, as shown by the curve N. The most economieal, or optimum thick- ness z,, is given hy the point 0 where the sum of the ordina& M and V has a minimum value.
Analytically the value of z,, can be determined as follows.‘” Designate ihe yearly c&t of refrigeration, or beat that must be taken away, in Vsq ft, by &’ and let:
a zj’k + R Eq. 8.28
where: a = T AtC,, T is hours per year of operation of the installation t is the temperature differente, cold medium to surroundings
62 September 1965 Petro/Chem Engineer
Solving Eq. 8-31 for 2 gives:
Eq. 8-32
C, is the cast of heat absorbed, $/Btu :z is thickness of insulation, in feet 7; is conductivity of the insulation, Btujhr-
ft-“F R is the sum of al1 thermal resistances
aside from the insulation, hr-sq-ft- “F/Bto.
Designate the cast of insulation per sq ft per year by N and:
N = hz + C Eq. 8-29 where: b is cast per year of insulation in $/ft-sq ft,
C is the yearly eost of maintenance of the insulation. $/sq ft.
The sum of M and N is the tota1 cast per square foot per year and can be written, after rea~rranging Eq. 8-28, 88:
M + N = g& + bz + C Eq. S-30
Differentiating Eq. S-30 and equating the derivative to zero results in:
dII _ - a,lc dx (z + Rh)”
+ h = 0 Eq. 8-31
Eq. S-32 shows elearly that al1 other things being equal, the economica1 thickness is directly proportional to the square root of the number of hours of operation and inversely proportional to the square root of the first cast of insulation:
8-10. Pipe md vessel insuhtion. The recom- mended, most economical, insulation thicknesses are given in the eatalogues of insulation manufacturers in form of tables. Some manufacturers give the recommended thicknesses simply as u function of temperature; others, in their tables, take into aecount both temperatures and pipe sizes.2a These latter tables are more accurate as they consider both eonvection and radiation.
Useless insulntion. If insulation with a relatively high conductivity is applied to a small-size pipe, the effeet, of the lower outside surface temperature
Ba.’ may be offset by the increase of the surface that dmslpates beat. Also wrapping thin asbestos paper around a galvanized pipe or tank may result in an increase of beat loss due to radiation sinee the emissivity of galvanized steel is approximately 0.28, whereas asbestos paper has an emissivity of about 0.94.
S-11. Imula.tion of buildiwgs. Recently air con- ditioning insulation of the walls and eeilings of residences, offices, and stores has become a large business. However, sinee t,he temperature differenee between the air inside of buildings and on the outside is not very great, the problems of in- sulation are relatively simple. The cast approach is simplified by the fact that under average climatic conditions, the beat loss through the ceiling to the attic space is about twice BS great as through the walls. This is a vel’y important consideration in the insulation of single-story buildings. Insulation should be installed SO that it will not be subject
Petro/Chem Engineer September 1965
to condensation of atmospherie humidity and it should last for the life of t,he building without nny maintenance expenses. The cast of insulating n building for human comfort usually is belo,w 174 of the tota1 construction cast. Therefore, the thicknesses recommended by the manufactwers of insulation for the part:icular locality can be used and no elaborate calculations of an optimum thick- ness are justified.
References ‘W. H. McAdams, Heat Transmiusion, op. cit.,
p. 267. ?W. J. King and L. C. Kraus. Refrigerating Da,ta
Book. 5th Ed. New York: American Society of Refrigerating Engineers, ~(1942) p. 144.
sA. P. Colburn, “A Method of Correlating Forced Convection Heat Transfer Data and u Comparison with Fluid Friction,” Trans. A.I.C,h.E. 29, 174-210 (1933) ; also, McAdams, op.cit. P. 272.
4McAdams, op. cit., p. 260. “E. R. G. Eckert, Intmd~uction to Transfe~ of
Heat and Mass, op. cit. p. 142. “W. J. King and L. C,. Kraus, Refr%gerating Data
Book, op. cit. p. 145. rMeAdams, op. cit., p. 268 RMcAdams, op. cit., p. 276. “W. J. King, 1. c., p. 413. IaL. Slegel and G. A. Hawkins, Heat Transfer
from a Vertical Piate to an Air Stream, Engineer- ing Bulletin, Purdue University, Research Seria, No. 97 !Ma~- 1946) p. 4.
,“M. Jakob. Hmt Tmnsfer. 01). cit,.. D. 558 =w. Jii,g&, “Der W&rme~berg& an einer
ebener Wand,” Gesundheits-Ingenieur, Beiheft 19, Reihe 1 (November 1924) p. 42.
laM. Jacob, Hent Tra~nsfw, Vol. 11. New York: John Wiley & Sons, Inc. (1957j.
l,IJ. G. Knudsen and D. L. Katz, Flhid D~~mwnics and Hmt Transfer. New York: McGraw-Hill Book Co., Inc. (1958).
‘“E. Pohlhausen, 2. Angeir. Math,. U. Mech., 1, 115 (1921).
‘“D. R. Chapman, and M. W. Rubesin, J. Aeronaut. Sci., lG, 547 (19491,
liS. Levy and R. A. Sebsn, J. Appl. Mechanics, 20, 415 !1953).
lhJ. Kaye, “Survey of frietion eoefficients, re- covery factors, and beat-transfer coefficients for suuersonic flow.” T. R. 6418-5. Office Nava1 Re- seàrch (1953).
InW. H. McAdams. Heat T~mnsmission. 3rd ed New York: McGraw-Hill Book Co., Inc. 1954)
rnM. W. Rubesin and H. A. Johnson. Tram. A.S.M.E. 71, 383 (1949).
=M. 0. Scadron and 1. Warshawsky,~ Nat. Adv., Comm. Aero. (now Nat. Aero. Space Admj T. N. 2599 (19523.
“P..Nichols, “Eeonomic Thickness of Insulation in the Refrigerating Field,” R&igera,ting Eng%- neering 9, 152 (Nov 1922); 1. B. McMillan, Heat Transfer Through Insulation in the Moderate- and High-Temperature Fields, Tra%s, A.S.M.E. 48, 1300 (1925).
=W. H. McAdams. Heat Tmnsmissior~. New York: McGraw-Hill Book Co., Ine. (19543 p. 282.
Parf 9, Free, Or Natural Convection, will appear in the October issue.
PR(lGRESS IN HPI--TECHNOLOGY
Petrochemicals
w Polyether-based urethanes are about 10 times more resistant to hydrolytic degradation th,an polyester-based types, according to a re- port by Du Pont.
In polyesters, water attacks the ester group itself; in polyethers, water attacks the urea and urethane groups, which are mare high& resistant to degradation. Other factors regard- ing urethane degradation: Curing syystems used to produce polyether-based urethanes have a significant effe& on the rate of degradation (polyol cure resists degradation more t,han diamine cure by a factor of at least 2); rate of loss of physical properties of the urethanes is often accelerated when the samples are placed under a slight stress; polyester-based urethanes are more resistant to water than to dilute acids; polyester-based urethanes are much less resistant to wet oil than are polyether-based materials.
q Glass fiber cords that can be used to reinforce severa1 types of elastomers for ap- plication as belting, hose and tires have been jointly developed by PPG’s Fiber Glass Div. and Esso’s Chemicals Research Div.
The fibrous glass cords are manufactured with an elastomer coating composition applied around the individua1 glass filamenta. The coat- ing becomes chemically bonded to the glass during vulcanization and protects the glass filaments against interfilament chafing fail- me; the coating also acts as the adhesive to bond the c,ord into the rubber product. These
64
rubber-coated glass cords are already being produced by PPG for use in nitrile rubber, SBR or neoprene. Cords for other rubber types are expected to be available shortly.
m Petroleum sulfonates of high molecular weight provide outstanding characteristics as lube oil additives, according to Phillips Petrol- eum Co.
The new sulfonates, which cover R range from C.10 to app Cl00 involve a complex of paraffinic, aromatic and naphthenic structures. Reportedly, the additive combines low-temper- sture sludge dispersancy with viscosity index improvement and reduction of octane re- quirement increase. Phillips indicates good performance both, in passenger automobiles and in diesel-and heavy-duty gasolme erigine oper- ations. The materia1 is produced by the sulfona- tion of a solve&extracted, high viseosity-index oil with an average molecular weight of about 800, and is used in the form of its calcium salt. The sulfonation step is carried out by reacting the feedstock with 10 wtycx SO3 in SOg. Afte@ the sulfur dioxide has been expelled from the reactor effluent, the liquid batch is neutralized by aqueous lime slurry. This is followed by evaporative dehydration, filtration of the naphtha-diluted sulfonate oil and finally by recovery of the solvent naphtha.
m Optimum conditions for sulfating a series of alcohols and ethoxylates with sulfur trioxide have been studied by investigators at Con- tinental Oil Co.
Working with a small laboratory reactor they concluded that mixtures of alcohols from Cs to 60 gave best results Ata temperatures around 5 F above the freezing point of the reaction mixture; SOJair wt% 1.5-15; SOs/alcohol mol ratio, 1 to 1; mass velocity of SOdair mixture above 120 lb/sq ft-sec and reaction time 0.5 to L5 hr. Best conditions for sulfating ethylene oxide adducts of these alcohols were, SO-90 F; Od
SOs/air wt% below 10 ; SOJadduct mal ratio 0.99/1.0; gas mass velocity greater than 120 lb/sq ft-sec and reaction time 0.5 to 1.0 hr. They also desc~ribe apparatus and conditions for neutralizing lhe sulfated mixture, emphasizing need for prompt processing to minimize hydrolgsis.
m Chlorostyrene as il diluent for polyester resins leads to faster molding cycles, lowered cast and superior products than can be achieved with styrene, according t,o Dow Chemical Co.
Parts were molded at cycles as low as lo-15 sec. Flexural strenpth, flexural modulus, impact strength and beat distortion temperatures were al1 significantly improved over aalues shown
Plfmse twn to paye 66
September 1965 Petro/Chem Engincer
Free, or Natura1 Convection - Part 9
Applied ‘Heat Transfer Design for the HPI
by Wayne C. Edmister and Joseph M. Marche110
9-I. Introduetion. When a fluid cornea in contact with a hotter surfaee, it begins to expand. The redueed density of the heated particles of fluid causes them to rise, and new particles come in eontact with the hot surface (Fig. 9-l). If the surface is colder than the fluid, fiuid particles will begin t,o fa11 due to the inereased, density. The heat exehange between a surfaee and a fluid set in motion only by temperature differences in called free or natura~l eon~vectiow.
Convect,ion currents usually have low velocities, the eorresponding Reynolds numbers are very low and the flow is laminar. Tests with a vertical heated
Ob -‘pIate and air have shown that the highest upward velocity exists at a distante of about 3/32 to li8 in. from the surface of the plate. The velocity in- creases along the height of the piate up to about 2 ft afte which no marked increase in veloeity is found.
At
Petro/Chem Engineer October 1965
The temperature of the air increases alnng the height of the plate also only to about 2 ft. The temperature is highest close to the surface and decreases away from it along the linea shows in Fig. 9-l. The t,hiekness z of the, layer of air par- ticipating in free eonvection generally a0f3 not exceed $5 in.
The temperature differente hetween the hot piate and the air moving upward gradually decreases along the height of the plate and there is a corresponding decrease in the surface conductance. However. from a height of 1 to 1% ft the increase in oelocity offsets the deerease in temperature differente and the surface conduetance begins to increase slightly up to a height of about 2 ft at which point, the temperature and velocity of the air and the surface conductance reaeh essentially constant values.
9.2. Application of dimensiona1 analysis. A gen- eral expression far the free-conveetion surface eoefficient of a fluid in contact with a hot vertical surface may be obtained from dim?mional analysis. The analysis follows the method of Part 2.
1. Corre& selection of the variahles is difficult for an involved prohlem of thin type. From the analysis of Pwt 7 it may be expected that, with certain modifieations, the same variables will be involved as were for forced convection. For instanee, the height L of the surface should enter instead of the diam D. In the piace of the fluid velocity V, the factors that produce the eonvection currents: temperature differente, At, between t,he main body
59
pressure p, psia, a must be multiplied by (p/14.‘7)a. Ali fluid characteristics, including a, should be referred to the film temperature tP
The esponent, rl in Eq. 9-9 and 9-10 has been determined from the. slope uf the curve obtained bg pIoMing on logarithmic paper values of NY,, = Ir.,L/k BS ordinate against N,, x NP, = L” &jgAte/@ as abscissae. In the range of Nc, x NP< from 10:’ to lo”, the curve is prsctically a strnight line with a slope d = 1/4. At values of NGr x Np, > 10”. which are mainly encountered in free convection of liquids, the slope is steeper and d = 113 may be used with sufficient accuracy. For values of NU, z Npv < 1000, whieh will occur only for very fine wires and small At, d gradually de- creases with a decrease of Nc, x Np7 and approaches zero far Ncy x NP, = 10 +.
Constmt C. The analysis of a number of experi- menta1 results show that the constant C depends on the product Nm x Ne. For the range of NG, z Np? between lOa and lo”, C has a general value of 0.55 for a vertical piate and Eq. 9-10 becomes :
eonstant C are giva in Table 9-2. The accuracy of these values is within * 20%.
Sizes. When the charaeteristic dimension L, either vertical or horizontal, in the Grashof nnmber is greater than 2 ft, its effeet upon the film con- ductance is negligible; the effect of increasing the diameter of u horizontal eylinder over 8 in. is also small. These conclusi,ons were reached from datu for free convection of gas.% but seem to be also generally correct far liquida. For Nc, x NP, < 10” this is also confirmed by the fact that if L > 0.7 ft the value of (L”)*/L in Eq. 9-9 or 9-10 becomes elose to 1. According to Eq. 9-10, for Nu, x Np7 > 10R the film eonductance is independent of the di- mension L. These faets lead to the conclusion that the value of L to be used in t,he expressions for the film coefficient of free convection should be limited to 2 ft.
h, = 0.55!7c/L) (aL”At)Qi Eq. 9-11
For Ncr x Np7 > 10” which will oceur for gases if the height L of the surface is 2 ft or greater, and in a liquid, if the height L of the surface is more than 0.15 ft, the use of the general value C = 0.13 with the exponent il = 1/3 is recommended for a vertical plate’. Eq. 9-10 becomes:
h, = o.l?jk(aAt)” Eq. 9-12
While the correlation of the results for different shapes and positions of surfaees is not very dose,
nevertheless the, differences are sufficiently pro- nounced to permit the recommendntion of different, values for the constant. Various values of the
In the case of short vertieal platea, small and short cylinders, either horizontal or vertical, and spheres, free convection is influenced both by the horizontal and vertical dimensions’Lh, and L,,. The effettive value of L can be determined from the expression :
l/L = l/Lp,, + l/L, Eq. g-13&.
For a sphere, Eq. 9-13 becomes:
1jL = l/D + l/D = Z/D == l/R Eq. 9-14
The values of the physical properties of the fluid entering in the equations for film coefficients must be determined at a temperature that is the average between the mean temperature of the surface and the temperature of the main body of the fluid.
The coefficient of thermal expansion p far gases may be taken as the reciproca1 of the absolute temperature of the gas, “R. For 1-a~ow and liquids p must be determined from its definition:
p = (v2 - u,) lvl (tS - t,j Eq. 9-15
where the specific volumes v1 and &. CU ft/lb, at the temperat~ures t1 and tr, respectively, must be obtained from tabulated data. Table 9-3 gives p for a few representative liquids.
Consistent Units. All terms must be in consistent,. units and with pound, foat and haur as basie units. uc
Throughout this book the valve of g is 4.17 x 10R ft/%+.
EXAMPLE 9.1. A horizontal, bare, S-in. standard steel pipe carries saturated steam at 10 psig and is exposed to still air at 68 F. Determine the coeffieient of free eonveetion to the air.
SOLUTION: From steam tables the temperature of saturated steam at 10 + 14.7 = 24.7 psia is 239.5 F. The temperature drop through the wall thiekness is very small, about 0.1 F and may be neglected. The film temperature t, is 68 + (239.5 - 68!/4 = 110.9 F. From Table 4-1, for 110 F:
30 50 70 90 110 130 150 170 190 temperature, “F
Ligure 9.2. Modurus a and hsat conductivity k for water. =thyl ahhol and transformer oil at various temperatures.
a, = 1.15 x 10°/cu ft-deg F L = D = 5.563/12 = 0.464 ft At = 239.5 - 68 = 171.5 F
Please turn to page 67
62 October 1965 Petro/Chem Engineer
Hence : aL3At = 1,150,OOO x 0.464” x 171.5 = 1.97 x ,107 < 10”
Therefore, Eq. 9-11 must be applied. From Table g-2, C = 0.35; from Table 4-l far air at 110.9 F, k = 0.0160 Btu/ft-hr-“F and with these values, Eq. 9-11 giva:
12, = 0.45 (0.016/0.464) (19,700,OOOj ‘A = 1.04 Btu/‘sq ft-hr-“F.
EXAMPLE 9-2. A heating coi1 made of 2% in. ID coppa tubing carrying saturated steam of atmos- pherie pressure is immersed in water kept at 138 F. Determine the ontside surface coefficient of free convection.
SOLUTION: The temperature of saturated steam at 14.7 psia is 212 F. Negleeting the temperature drop through the tube wall, which is less than 0.1 F, the mean film temperature at the tube surface will be:
tt = 138 + (212 - 138)/4 = 156.5 F
From Table 3-9 for t = 156.5 F, by interpolation
,@a = 145 x lOa l/cu ft “F the OD of a 2% in. copper tube IS 2.625 in112 = 0.219 ft; t = 212 - 138 = 74 F and:
a,YAt = 14.5 x lOe x 0.21g3 x 74 = 1.13 x 10” therefore, Eq. 9-12 must be used. From Table 9-2, Cl = 0.11 and from Table 3-9, for t = 156.5 F, k = 0.383 Btu/ft-hr-“F:
9-4. Boundary lager. In free convection, the velocity of the fluid rising due to thermal buoyancy is rather small, and as a result, the thickness 6 of the boundary layer is much greater than with forced conveetmn.
Vertical piane swface. Sinee the derivation of t,he equation for detwmining 6 is rather involved2 only its fina1 form will be giva bere:
b + s = 3.93(~)-y952 +T)U
(y) * x ‘” Eq. 9.16
where: At is the temperature differente between the surfaee and fluid z is the distane from the lower edge of the plate.
Since the Prandtl and Grashof numbers are used repeatedly in dealing with natura1 convection, it is convenient to present Eq. 9-16 in the form of:
6 = 3.93N,-” (0.952 + NrdY’Nc,.z-“, Eq. 9.17
where the Grashof number Ncr,z is built up with
I the distanee, z,NGbz = gpAtSpZ,‘$. As mentioned in Section 9-3, if z > 2 ft, the
limit value of 2 ft should be uaed. Film coe.@zient. From the definition of the film
Petro/Chem Engineer October 1965
TABLE 9.2. Constants for Equations 9-9 to 9-12 ~~~ x 14~~. OI aL3At
-703 ta 109 Ove, 10Q Shape and PO”i,iO” Of c Heat~Exchanger s”rfacBs 5
Vertical plate 0.55 0.13 - Horizontal plate, face up 0.70 0.17 Hotizontal plate. face down 0.35 0.08 Horizontal cylinders, pipes, wires 0.45 0.11 Long vertical cylinders 0.50 0.12 Sphere, L taken as its radius 0.63 0.15
TABLE 9.3. ,Cwfficients of Thermal Expansion of Some Liquids’
heat transfer coefficient in free convection based on heat conduetivitg and the approximation of the temperature profile in the boundary layer by a parabola” it follows that:
,?, = E 6
. . Eq. 9-18
This can be presented in the dimensionless form:
&=N 2x k N”, =s Eq. 9-19
BY introducing the boundary-layer Equation (3.17) :
The thickness of the boundary layer gradually increases from the lower edge of the plate. This is also show” by an examination of Eq. 9-17. There- fore, according to Eq. 9-18, the loca1 film coefficient h decreases with an inerease of the distante CZ. BY integration over the distante 5, the average heat- transfer coeffieient h, is found as:
h, = 413 h _, Eq. 9-21
If the t~emperature differences are small, p = l/T,, where T, is t,he absolute temperature, “R, of the gas outside the boundary layer. For atmos- pherie air Np7 = 0.71, and Eq. 9-20 becomes:
NN*, ì = 0.378 Nc,, ,‘h Eq. 3-22
‘EXAMPLE 9-3. A verti& heating surface is main- tained at a temperature of 212 F and is exposed to still a,ir at 72 F. Determine the thickness of the
67
boundarylayer at a distante of 3 ft from the lower ,edge of the piate.
SOLUTION: The mean film temperature of the air is 72 + (212 - 721114 = 10’7 F. The properties of air at this temperature are, from Table 4-l: e = 0.240 Btu/lb-“F, ,L = 0.046 lb/ft-hr, k = 0.0159 Btu/hr-ft-“F, and p = O.O?‘O%lb/cu ft. The coeffi- cient of expansion p = lj(105 + 460) = 0.0017 R-‘; At = 212 - 72 = 140 F. Using these values with the limit value of .i: = 2 ft, as men- tioned above, giva:
Np, = 0.240 x 0.046 = o,694 0.0159
Nc,= 4.17 x 10s x 0.0017 x 140 x 2” x 0.0703’ 0.046”
= 1.86 x 108
Now, by Eq. (9.17) :
S = 3.93 x 0.6946”(0.952 + 0.694)” (1.86 x lO#) -% x 2
= 0.052 ft = 0.625 in.
TABLE 9-4. Explanatory Data for Figure 9.3
Horizontal cylinder. From a theoretieal analysis” which agrees well with experimental results” Her- mann has shown that for boundary-layer thicknesses that are small compared with the tube diameter, the average film beat-transfer coefficient has the sane value &s the average film beat transfer coeffi- cient on a vertical ~~11 when, the height is 2.5 time the diameter. The Nusselt number, NnuiB. can be calculated from Eq. 9-20 using z = 2.5 D in the Grashof number. In’this manner the point value of h may be obtained from Eq. 9-19 and the average value of the coefficient, h,, may be obtained from Eq. 9-21. Finally, the boundary layer thicknesses may he found by Eq. 9-18 using h for h, and for &.
EXAMPLE 9-4. From the boundary-layer approaeh, determine (a:) the film coefficient and (b) the boundary-lager thickness of free convection using the data of Example 9-l.
SOLUTION: (a) The Nusselt number, NN,“>,J> must be found by Eq. X-20 using z = 2.5D = 1.16 ft. The Prandtl numher frum Table 4-l is lvp, = 0.70. The Grashof number with y = 0.46/0.703 = 0.654, is computed as:
Narld = 1.163 x 4.17 x 108 x 0.0017 x 171.5 x 0.654-” = 4.43 x 1OR and, by Eq. 9.20:
- 0.508 x 0.70”(0.952 + 0.70)” <4f4?;i;+ = 54.2
68
with k = 0.0159 Btuj’hr-ft-“F, from Eq. 9-20:
h= 54.2 x 0.0159
~~ = 0.742 Btui’hr-sq ft-“F 1.16
and tiith Eq. 9-21:
IL, A 4 ,x 0.742 = 0.990 Btu/hr-sq ft-“F
This is in satisfactory agreement with h,, = l.OG Btu/hr-sq ft-“F obtained from Eq. 9-11 in Example 9-l.
(b) The thickness of the boundary layer is found, from Eq. 9-18 as:
S= 2 x 0.0159
0.742 = 0.0429 ft. or 0.514 in.
The average boundary-layer thickness is, by Eq. 9-18,
S= 2 x 0.0150
0.990 = 0.0321 ft, or 0.385 in.
D-5. @nplified ‘3qil~diOn8 far air. Most authors agree on the general type of the equations but differ on the numerical constants. The followin simplified expressions for the su-face coefficien @ are recommended.“J
Vertical. &ne surfaces, up to 2’ft high, H < 2, airatOto250F:
h, = 0.28 (Ab/H)” . Eq. 9-23
Vertical surfaces more than 2 ft high:
h, = 0.23 AP . Eq. 9-24
Horizontel swfaces, heated with face up or cooled with face down, an approximate value may be obtained as:
IL., = 0.30 Atu , Eq. 9.25
and for horizontal surfaees, heated with face down or cooled with face up:
h, = 0.18 At’A .~ Eq. 9-26
Vertied cylindur, more than 1 ft high, for a mean temperature hetween the air and surface t,,, = 120 F: I,d
h, = 0.27 (At/D,)‘h . Eq. 9-27
for t, = O”F, the numerica1 coefficient will be about 0.28 and t’or t, = 250 F, it will decrease to 0.25.
Horizontd cylindem. The recommended expres- sion for long cylinders is:
h, = C(At,UIo,)%G Eq. 9-28
Far a man temperature t, = O”F, C= 0.25; for tm = 120 F, C = 0.24; and for t,, = 250 F, c = 0.22.8
Eq. 9-28 may also be used for short eylinders, both horizontal and vertical; the influente of the ends may be taken into nccount by using Eq. 9.13 to fìnd the effettive dimension D,.
Pressare Influmce. Al1 above equations are in- tended for air at atmospheric pressure. For any
October 1965 Petro/Chem Engineer
Y
L
r
BETZ PETROMEEN’ CONTROLS CORROSION IN GAS CONCENTRATION UNIT
The Thermal Cracking Unit in a Midwest refnery suffered costly operational difficulties due to failure of the Admiralty tube bundles serving the Debutanizer and Depropanizer. A study of the problem
0 by Betz Fngineers pinpointed the dificulty as duincification of the Admrralty metal initiated by nitrogenous gasa hberated in the crack- ing process and aggravated by the presente of organic sulfides. The
~8~ history of condenser tube replacements indicated that corrosion was locahred m the upper tube bundle of the condensa banks, in the area of the water dew point and initial point of hydrocarbon condensation.
A specially formulated BETZ PETROMEEN, designed to function in high pH environments and in the presente of light hydrocarbons, was chosen for application. During’ start-up, a pretreatment dosage of PETROMEEN was fed to insure thorough flm formation, followed by a continuous treatment leve-l of 10 ppm.
Prior to the use of PETROMEEN, the Debutanirer and Depropanizer bundle life had been restricted to 7 months. With the application of BETZ PETROMEEN. this unit completed 22 months of operation with- out downtime due to tube failures. The refnery Inspection Department reported no corrosive attack in those areas treated with PETKOMEEN.
The efktiveness of BETZ PE.I.ROMEENS as corrosion inhibitors and anti-foulants augmented by a thorough approach of monitoring per- formance was responsible for reducing costs involved in equipment replacement and loss of production. A Betz Program tailored to your plant’s corrosion problems is available by contacting the Beta Engilieer in your area or writing us directly.
BETZ PETROMEENS
A complete line of corrosion inhibitors and anti-foulants, specially formulaed to handle the problems of corrosion and fouling encountered in petroleum rehning process equipment is ot7ered by BETZ. Known as I’ETROMBENS. these products havu been thoroughly tield- tested and i>r comprririuc sfrrdies bave proved m~sf cffcctivc and economical,
BETZ PETROMEENS ore available in a wide range of formulations, depend- ing on the particular process problem involved. PETROMEEN corrosion inhibi- tors have accomplished ~rcellent resu//s in horb bydrocarbon and MEA .~yrmm. These inhibitors are equally applicable in process streams containing either light or heavy hydrocarbon fractions. PETROMEENS are miscible in hoth OI’D- ,naric and aliphatic pelroleron s,reo»rs.
The inhibitors are designed to pro- vide an impervious film, which adheres to the meta1 surfaces. The molecular film, which serva BS a barrier between the corrosive media and the meta1 is characterized by high ~bennal sfnbility o>,d is eflclive over a wide ra,w o/pH. Of equal significante, is the absence of adverse effects on proccss catnlysts and product purity. Nor is foaming or poly- merization encountered wilh the use of these inhibitors.
PETROMEENS exhibit excellent de- tergency properties enabling processes to operate with improved hent transfer and designed slow rates. The anti-foulant formulations are fortilied with special dispersants and solvents to assure maxi- 1mum throughput of process equipment.
A complete corrosion contro1 pro- gram is availablc with the use of BETZ PETROMEEN, which includa such moni- toring fxilities as retractable corrosion test coupon probes, corrosion meters, water and deposit nnalyses, etc. Com- plete details on BETZ PETIKMEEN corro- sion inhihitors UIK/ service are nvailablc.
other pressare i>, psia, the surfaee coeffieient cal- culated by one of the above equations must be multiplied by the ratio i71/14.l!“~j.
R-F. Inflzwncti of ot/zw surfaces. Parallel surfaces kept at the sane temperature begin to affect the surface coefficient of free conveetion only wben the separating distante is decreased to :!/4 in. If this distanee is reduced further, the convection rapidly deereases, due to friction of the air and reduction in convection eurrents, unt,il at a distanee of !:S in. the sui-face coefficient becomes almo& zero.
Parallel surfaces at different, temperatures with air between them will transfer heat across the air space from the hottei to the colder surface. The beat transfer will be a combination of con- duction, eonvection and radiation. With a decrease of the distante 6 between the surfaces, the heat tiansfer by conduction increases faster than the beat transfer by convection decreases and 8s a result, the combined beat transfer increases. For an air space thickness s from 0.02 to 0.25 ft, the beat transfer coefficient by eonduction and convec- tion, leaving out radiation, may be approximated by the expression:”
U, = 0:009/S + 0.09 (t,, -i- 90j”.“. Eq. 9-29 where t, is t.he average between the temperatures tu and ti of the~ two surfaces. Eq. 9-29 may be used far t, from 20 F to 150 F. The heat transfer per sq ft of the surface area is obtained by multiplying the value of U, by the differente tt ~ tg.
9-7. Free conwction in Ziqtiida. For surfaces immersed in liquids, the coefficients for free con- vection are’considerably higher than in the case of gases. There are comparatively few experimental data and no empirica1 equations except for bodies of average size immersed in stili water above 40 F Tor which h, may be calculated by the equation:‘O
h, = 0.165 (tw i- 30)&“,S Eq. 9-30 where t,, is the temperature of the main body of water, deg F.
For oils the coefficient may be estimated by the equation:
h., = 24 At0.=,/#4 . Eq. 9-31
where pf is the dynamic viscosity of the oil al the mean or film temperature.
Fig. 9-3 giva values of the eoefficient as found by different experimenters for a few liquida as indieated in Tabe 9-4, and may be used as a basis for estimating l&,. These and other data indicate that the coefficient of free convection for fine wires is considerably higher than for other bodies.
Table 9-5 contains the viscosities of the light oils corresponding to, the data giva in Fig. 9-3. The oil M had viseosities simila to oil 21-SW.
TABLE 9-5. Dynamic Viscosities of Oils in Figure 9-3, Ib/ft.hr
Figure 9-3. Free convection data for various liqoids
Estimates of the surface eoefficient for other liquida have to be made by using either Eq. 9-11 or Eq. 9-12. First the physical properties of the liquid must be obtained. Then, depending upon the value of NGr x I\‘?~, the suitable equation is selected and used in connection with Table 9-Z.
Since the numerica1 value of the modulus a is about one thousand time larger for liquids than for air, the coefficient is not appreeiably affected by the significant dimenaion L when L > 0.15 ft.
Aero. Space Adm.! T. N. 2635 (1952). 8Brown and Marco, Introduction to Heat Tram-
fw, op. cit. pp. 138-39. “Derived from data giva by F. B. Rowley and
A. B. Algrem, “Thermal Resistance of Air Space,” Trans. A.S.M.E., Vol. 95, 165 (1929).
‘OW. J. King~, op. cit,. p, 353.
l’art 10, “Radiation Principles,” will appear in the November issue.
October 1965 Petro/Chem Engineer
Radiation Principles - Part 10
Applied Heat Transfer Design for the HPI
by Wayne C. Edmister and Joseph M. ~Marchello
IO-I. Beuic ooncepts md dejinitions. Radiation may be defined aa loss of heat energy contained in a body to the surroundings by means of eleetro- magnetic waves. These waves transmit heat from one body to another without ehanging the tem- perature of the fluid separating the two bodies. The phenomenon of heat transfer by radiation consists of three different components: first, ther-
e mal energy of the hot source is converted into an electromagnetic wave motion; swond, the WBVPS trave1 through the intervening space’ even if it is a vacumn; and third, the wave motion is reconverted into thermal energy by absorption at the colder body.
The radiant-heat waves have the same charac- teristics as other rays such as x-rays and light waves and differ from other rays only in length, their lengths varying from 0.3~ to 10~, and more. Thus the transfer of radiant heat energy is governed by the same laws as light: it travels in straight lines, obeyn the laws of reflection, suffers refraction, may be polariaed, and is weakened with t,he inverse square of the radia1 distante from the source of radiation. The velocity of radiant-energy maves is the same as that of light, about 186,000 miler, per seeond.
If radiation impinges upon a surface, part of the heat energy penetrates into the interior of the body and the rest is reflected from the surface.
* In regu1,a.r reflection the angle between the reflected ray and the norma1 to a smooth surface equals, the angle between the impinging ray and the normal. If the surfaee is rough, the incident ruy, after refleetion, is dispersed in al1 directions. Reflection from rough surfaces is termed diffuse reflection. Of the radiant energy that penet,rates into the body, part is absorbed by the body and the rest leaves the body through another surfaee.
The ratio of the reflected radiation Q~ to the t,otal incident radiation q is called wfkctivity, 1’ = q>/4; the ratio of the absorbed,radiation qn to the inei- dent radiation is called a.bsorptivitv, a = W’u; and the ratio of the transmitted radiation qr to the incident radiation is called the transnLissi%‘it!/, t = q,Jq. From the Law of Conservation of Energy:
qr + qo + qr = q . Eq. 10-I
Dividing both sides of Eq. (10-l) by 4 gives:
Petro/Chem Engineer November 1965
T + a 4. t = 1 Eq. 10.2
Solid and liquid bodies absorb practically the entire heat radiation that penetrates through the surfaee in a very thin outer layer. In electric conductors the thickness of the layer absorbing radiation is the order of lp, whereas in non- conductors, with very few exceptions the layer absorbing the entire radiation is approximately 0.05 in. thick. If the thickness of an opaque materia1 is greater than this, t = 0 and:
T i- (I = 1 Eq. 10-3
10-Z. Bhck body. A body whose surface does not reflect any radiation, 1’ = 0 and in accordante with Eq. 10-3 ha.8 a = 1, is called a black body. The basis for this nane lies in the fact that a body which absorbs al1 light rays appears black to the eye. However, a surface can absorb praetically al1 thermal radiation withoot absorbing light rays and therefore not appear blaek to the eye. A typical example is a white-washed wall. No totally blaek body exists in nature, but the concept of a black body is useful because it helps to establish laws of radiation from actual surfaces and bodies.
A practically perfect black body for experiments in radiation can be made by taking u hollow body b provided with a small hole in its side, Fig. 10-l. The walls of the body must be kept at a uniform temperature. Any ray cl of radiant energy which enters the hollow space will be repeatedly reflected. If the inner surface of the body has a great absorptivity, then at every reflection a great part of the radiation is absorbed, so that after severa1 reflections almost al1 radiation is absorbed. Thus the opening of the hollow space acts like a perfeet black body.
Theoretically complete absorption will oecur only when the area of the hole is infinitely small compared with the inter- nal area of the hollow space. Actually for ex- perimental work a suffi-
Figure 10.1. Hollow body ciently accurate approxi- for measuring radiation. mation is obtained with
43
a Z-in. cylindrical tube, S-in. long with a %-in. hole in one of the closed ends. When such a hollow cylinder is heated., the inside walls will radiate energy and some of this radiation will pass out through
Figure 10-2. Determination of the opening. He& radiation in different directions. energy flowing
from this opening i,s ealled black-body mdiation..’
Intensitu of rarliabion. The beat radiated from & unit surface in a certain direction is called the intensity of mdiation, i. The value of i can be obtained from the beat flow dq radiated from a unit area of the radiating body and passing through an element ds of the surface of a hemi- sphere of radius i = 1 sround the radiating sui- face element, Fig. 10-Z. The area ds of such a ring-shaped element numerically is equa1 to the solid angle do and therefore:
i ~= dq/do Eq. 10-4
For a blaek body, the radiation intensity .iD in any direetion at an angle p from the norma1 is equa1 to the norma1 intensity .&., SO that the rate of beat emission is proportional t,o the projected area, where B is the angle between the directions of ih and iu,,. This relationship is known BS Lav- bfd’s cosine lnw:
dq = ih,, cos /3 do Eq. 10.5
Emissive power. A black body radiata beat energy in al1 directions away from the body. The radiant heat flow Erom a unit surface ara per unit time is called emissive power and designated by E.
The tota1 beat radiated from a unit area of a black body, its emissive poter Eo, may be found by integrating the elementary beat dq, Eq. (lo-5), and notieing that the solid angle do is equa1 to 2~ sin pdp, Fig. 10.2. This giues:
7r/2 Eb =
1~ 297 i, ,,,, (cosp j (sinp) dp = 7r ima,,,
0 Eq. 10-G
Thus the tota1 beat iadiated from a unit surface area of a black body, its emissive power El,, is T time.! its norma1 radiation intensity.
Monoehrowmtic radiation. The radiant beat emitted by a body always comprises rnys of different WBY” length. The amount of beat, dq, radiated within an infinitesima1 wave-length range dh is als” nn infinitesima1 quanti,ty. However, t,he rati”:
ih = dq/dx Eq. 10.7
Tota,1 emissiuity, e, of a body is the rat,io of its
L t I , I !
I \ 2700’ R
0 2 4 6 8 IO
Wave lengfhh, micmns, p Figure 10-3. Mooochromatic intensity of radiation black body at various temperatures.
tota1 emissive power E to the emissive power E,, of a blaek body at the same temperatur’e:
e = EIE, Eq. 10-Y
Monochromatic emissi~uitg, ek, is the ratio of the monochromatic emissive power of a body to that of a black body at the same temperature and for the same wave length.
0.a = E,JEl>,, Eq. 10-9
10-S. Fundamen.tal hm. Planck’s law far the monochromatic norma1 thermal radiati”” intensity of a black body is:
iAi, = Cl/[A~(&LkT~) 1 .’ . Eq. lo-10 e
where: C, = 1.17 x 10R~4 --- Btu/hr-sq ft, CZ = 25,740 p - “R, X = wave length, microns, and T = absolute temperature, “El.
Intensities, ih,,,, eomputed by Eq. 10.10 for severa1 different temperatures and for~wave lengths from 1~ to 10,~ are plotted against X in Fig. 10-3 which shows that st any temperature beat is rsdiated in a wide range of wave lengths. The area under an intensity curve represents the normal emissive power at this temperature or tota1 heat radiated normally by u unit area, of the black body far the corresponding wave-length range.
Wien’s law. Fig. 10-3 indicate that nt any tem- perature the radiation first increases with an in- crease of the wave length, reaching a maximum at
November 1965 Petro/Chem Engineer 44
a certain wave length, and then begins to decrease assymptotically approaching zero for vwy large wave lengths. As the temperature decreases, the maximum value of ~the radiation moves toward larga wave lengths. The wave length of hm,, c”- responding to the maximum intensity may be found from Wien’s displacement law :
x m”Z = 5,19O/T Eq. lo-11
Thus at the temperature of the sun, about 10,700 ‘R, hm, = 0.485~~ whieh is the wave length of blue light. At T = 2,700 “R, A,*,, = 1.92p and only a small part Of the energy is visible since the human eye range is from about 0.4 to 0.7~.
Kirchh,ofs law, based on theoretical and expei% menta1 information states that for radiation at the same temperature and the same >vave Irngth, the ratio of the emissive power E to the absorptivity a is the same for al1 bodies including the blaek body. This principle applies to both the tota1 and monochromatic radiation. Since the absorptivity of a black body is unity :
E/a = Exj’ah = Eb Eq. lo-12
Eq. lo-13
This is the mathematical expression of KirchhofYs law.
In connection with Eq. 10-S:
E = eEa . Eq. lo-14
From Eq. Lot-13 and 10-14:
a = e . Eq. lo-15
whieh States that emissivities and absorptivities of a body at the same temperature are equal. As shown later, for many materials this relation is only approximate, Buti often very useful. Eq. lo-14 indicate that at the same temperature and wave lengths a good absorber is also a good radiator. However, a giva surface may have a low absorp- tivity at high temperatures, and for solar radiation. but a high absorptivity and emissivity at low
I~mperatures. Gray body. If a body emits radiant energy over
a continuous band of wave lengths with a distrihu- tion curve similar to those of Fig. 10-3 and its monochromatic emissivity at any wave length is a eonstant fraction of the corresponding blaek-body value, it is called a gray body. This is approximately true for most non-metallic surfaces in the range of ordinary temperature.% The tota1 emissivity oi absorptivity of a gray body is equa1 to its mono- chromatic emissivity at the same wave length. Good approximations of a gray surface are siate, tarboard, and dark linoleum. In the range from 0.5~ to 9p. These materials absorb from 85 to 92% of the incident radiation.
Stejan-Boltzmmm law stata that the tota1 radia- tion from a blaek body is proportional to the fourth power of the absolute temperature:
Eb = UT* Eq. lo-16
Petro/Chem Engineer November 1965
Temperature, OR Key to Figure 10-4
1 Slate composition roofing 9 Cork 2 Linoleum, red-brown 10 White dutch title 3 Asbestos slate 11 White chamotte 4 Soft rubber, gray 12 MgO, evaporated 5 Concrete 13 Anodized ~luminum 6 Porcelain 14 Aluminum paint 7 Vitreous enamel, white 15 Polished aluminum 8 Red brick 16 Graphite
The arem bounded by 17, 18 and 19 represent limits of data on gray paving brick, abestos paper, wood, various cloths, plaster of Paris, lithopone and papa. Figure 10-4. Variation of absorptivity and reflectivity of various materials with temperature.
where <r = 0.1713 x 1O-8 Btu/sq ft-hr’R aecording to the latest determination’ based on very accurate experimental data. From Eq. 10-6 the value of Eb is represented by the area under the intensity curve, Fig. 10.3, owr the interval X = 0 to h = m. Combining Eq. lo-14 with Eq. lo-16 giva:
E = e r T4 Eq. lo-17
This ‘expression may be used to caleula$e the tota1 emissive poter of a body if its temperature 7’ and emissivity e at this temperature are known. For actual application a more convenient form of Eq. lo-17 is:
E = e(0.1713) (T/100)4 Eq. lo-18
10-h. Radiation properties of materials. Al1 solid and liquid surfaces reflect a certain part of the incident radiation energy and therefore radiate less beat than a blaek body. As mentioned in Sec- tion 10-1, there is a pronounced differente in the behavior of electric conductors and noneonductors. Conductors reflect a major part of the incident radiation and radiate relatively little beat. Non-
51
conductors absorb a major part of the incident radiation and henee radiate a relatively large amount of beat. For both groups the absorpt,ivit,y and emissivity change with the temperature, as shown in Fig. 10-4, but again there is a marked differente between electric eonductors and non- conductors.
For conductors, as represented by aluminum and carbon, the absorptivity increases with temperature, for noneonduetors it decreases. Industrial radiators hsve temperatures between 520 and 2500OR. For these t&nperatures the absorptivity of noncon- ductors is much higher than that of metals.
For surfaees which have a monochromatic ab- sorptivity ah, independent of the surface tempera- ture, the integrated tota1 absorptivity a will be exaetly equa.1 to the tota1 emissivity e of the surface when it is heated to t,he temperature of the black
0.3 0 0.2 0.4 0.6 0.8
Norma1 emissivity, en
Figure 10.5, Ratio of average to normal emissivity far different zurfaces.
radiator. However, far. most surfaces the absorp- tivity depends upon its temperature. Eq. lo-15 does not apply and for these surfaces the values of the tota1 absorptivitl a as shown in Fig. 10-4, are only a first approximation of the value of the emissivity e. Far nonconductor surfaces the in- fluente of temperature usually is small and can be neglected if the temperature is not very high. The absorptivity of a meta1 surface at temperature TI for blaek-body radiation at a temperat,ure T3 is equal to the emissivity of the same surface at the temperature Ta = \/TlTz.
Industrial materials do not follow Lambert’s cosine law exactly, Eq. 10-5. As a result, the average or hemispherical emissivity e for radiation in al1 direetions differs somewhat from the emis.Cvity c?,~ in the direction norma1 to the surface. Fig. 10-5 shows theoretical values2 which agree very elosely with observed values.
Tables 10-l and 10-Z give values of norma1 emissivities at various temperatures of surfaces made of metals and other materials, encountered
TABLE 10-l. Norma1 Emissivities of Metal Surfa&
Temperature t Emissivity e” Metal and Condition of Surface of radiation, “F at t
Aluminum, highly polished plate polished plate rough plate oxidized at 1110 F
Bismuth, bright Brass, highly polished
polished plate rolled plate, natural du,, plate oxidized at 111OF
oxidized at 750 F 750 0.110 galvanized sheet iron,
bright sane, gray oxidized
82 0.230 75 0.280
November 1965 Petro/Chem Engineer 52
p
Figure 10.6. Relative positions of radiant surfaces.
most often in industrial radiation problems. For metals, the emissivities increase with an increase of temperature, but for nonconductors, including meta1 oxides, this rule is not always true.
The average emissivity e may be found by multiplying the values of e, giva in Tables 10-l and 10-Z either by the ordina& of Fig. 10-5 oi, with sufficient aceurucy, far m&als, by 1.2. For other substances and oxidized meta1 surfaces, the values in Tables 10-l and 10-Z must be multiplied by 0.95, if the surface is smooth, or by 0.98 if the surface is rough.3
10-S. Radiation between swfaces. The French scientist Prevost proposed the theory of exchanges
which stata that there is a continuous interchange of heat energy among bodies by the processa of
a ” radmtlon and absorption. Al1 bodies at any tem- perature above absolute zero are radiating beat to their surroundings and at the same time are absorbing heat from than. If two bodies are in an enclosure, isolated from the rest, the hotter body, will emit by radiation more energy to the
TABLE 10.2. Noimal Emissivities of Various Surfaces
Temperature t Materia and Condition of Surface Of Radiation, oF Emizfty e
colder body than it will receive and absorb from the lattei-, its temperature will decrease and that of the coldei body will increase. However, even afta’ their tempwatures become equalized, the process of interchange continues. Only each body radiata as much energy as it receives.
Surfaces sepwated by ~mnabsorbing media. Sup- pose that the areas Al and Az of two surfaces are small compared to the distane T between their centers, Fig. 10-6. If n1 and 12% are the normals to the surfaces, then the solid angle subtended at A1 by AS is equa1 A2 cos Oa/+, and applying Lam- bert% cosine law, Eq. 10-5, the heat flow for the black-body radiation from AI to Az may be writ- ten as:
ql2 = ial Al(cos &j&(eos &).r” Eq. lo-19
According to the theory of exchanges t,he radiation from A, to A, will be
qzl = ibs A,(cos &)Al(cos 01)/r2 Eq. lo-20
Ratio, side or diameter Dirtance between plqnes, D/r
Figure 10.7. Values of Factor F, for parallel pfanes directly opposed.
Substituting for ibl and ib2 the value from Eq. 10-6 and lo-16 giva for the net exchange of radiation between the two surfaces:
4 = 412 - 421 = dr; - 2,;’
(cosf?~) (cos&)A~A~/ar~ Eq. 10.21
If the term CO& c~sb’~A~/rn~ is designated by F:
F = (CO&,) (cos&jA2/d Eq. lo-22
the expression for the net beat exchange becomes:
q = u(!P - T’iFA, Eq. lo-23 i ?
or, similarly to Eq. 10-18:
g = 0.174 [(Z’llOO)i - (To,‘lOO)‘]FA1 Eq. IO-24
Variable swjace temperutures. 1-E the tempera- ture of a surface changes from one end tu the other, then the mean temperature must be used in Eq. lo-24 for the corresponding surface.
10.6. Geometricnl factw. The term F, Eq. 10.22 ,is alled the geometrieal factor because it depends upon the geometry of the two surfa&s. their shapes, orientation, and relative sizes. F is always small& than 1 and the symbol FI2 is used to designate the
Radiation between adjacent rectongles in perpendiculor planes.
I r=o.i
/-
l 1 I I , I Il
0.2 0.4 0.8 2 4 6810 Dimension ratio Z =I/X
Figure 10-g. Values of Factor F. for retangles norma1 to each other. Radiation between adiacent rectangles in perpendicular planes.
fraction of the radiation leaving the surface AI in al1 directions which is intereepted by the sui-face Az. Evaluation of the value of F is necessary to use Eq. 10-24. Values of F have heen calculated foì different surface arrangements with the simplify- ing assumption that emisswity e does not depend upon the angle 8, Fig. 10-7. These results are exact for black surfaces and very nearly SO for most nonmetallic and oxidized, tarnished, or rough meta1 surfaees.
Legend 1 - Direct radiation between planes. 2 - Squares. 3 - Disks. 4 - Radiation between planes eompleted by
noneonducting but reradiating wells. ,%mwrieal’ values of F. For two infinite parallel
planes 01‘ infinitely long cylinders one inside of the other, F = 1. This is also true if one body is com- pletely surrounded by another body, regardless of their shapes and relative sizes. The values of F for opposed parallel planes may be found from Fig. 10.7, curve 1; for squares from curve 2; for discs from curve 3. The faetor F for a rectangle with sides a x b may be found from:
F = \/F&'a Eq. 10-29
where F, is found from curve 2 for a square with sides a and Fb, for a square with sides b.
For two adjacent rectangles in pe*pendicular planes, the factor F may be found from Fig. 10-S. For B. piane parallel to u syst,em of iows of parallel tuba, F may be found from an”es 1 and 3 of
54
0
Figure 10.9. Radiation from a plane to one or two’row of tubes parallel to the plane. Referenze: Hottel, Tram. ASME 53, 265 (1031).
Fig. 10-9. Values of F for other case? may be found in the, 1iterature.l
Additimal relations. An important equation that is of use in evaluating t,he factor F is :
FEzA = FzIA~. Eq. lo-26
If Eq. lo-26 did not apply there would be a net heat flow between Al and As when they are both at the same temperature. If Al and Ar are at different temperatures T,l and TS, combining Eq. lo-28 and lo-26 giva:
v(T'~ T4)F12A1 = r(T* ~ T")FuA2 1 2 1 2
Eq. lo-27
This relationship indicate that interchange of radiation may be determined by evaluating the one * way radiation from either surface to the other, whichever is more convenient.
Another useful equation is:
FI1 + Fp2 + F,a = 1 Eq. lo-28
and if AI is flat or convex and therefore cannot “sei? any part of itself :
FI1 = 0 Eq. lo-29
EXAMPLE 10-l. Determine the net radiant heat interchange between two black parallel piane sur- faces 4.ft wide, 1%ft long, 4-ft apart, kept at temperatures of 1000F and 400F, respectively.
SOLUTION: Eq. lo-24 must be applied: TI = 1000 + 460 = 1460 R; T = 400 + 460 = 860R; Al = 4 x 12 = 48 sq ft. The factor F is found from Fig. 10-7, curve 2: for a square 4 x 4 ft, the ratio D/r = 4/4 = 1, and F, = 0.20; for a square
November 1965 Petro/Chem Engineer
PREHEAT EXCHANGERS SET ONSTREAM RECORD . . . with FI FTO L Anti-Foulant
Since 1958, fouling problems in a midwestern refinery had plagued its hottest crude unit preheat exchangers. A desalta xas delivering uude with a salt content of 1 to 3 pounds per thousand bar&. Residue in thesc exchangers was 60% organic and 40% inorganic. Severa1 anti-foulants were tested, but on- stream time was not improved.
Then .~FWL* Anti-F‘oulant was injected at the rate of 12 ppm into a clean system. In comparison with the besl previous run, the heut transfer elficiency conrinued at a considerably higher leve1 while treating costs were much loaer, BS shown in the graph. Savings ira EXC~SS fud ps COSLT ulone nmomted to $5/M bbls.-or three times the COSI of the AFTOL! CUntinued use of A~TOL maintained this efficiency, and surpassrd al1 rxisting refinery records far throughput and onstream time.
Results like these are possible in your refmery, tao with AFTOL. TO prove it, calt the Man in the Red Car, or write .
Petro/Chem Engineer November 1965 io, F”,cw ,nformotion 0” Adre‘,irsd Prad”rlr set Raoder lervi‘s Cd 55
12 x 12 ft, Dir = 1214 = 3, and Fb = 0.54. There- fore, by Eq. 10-25: ~.
F = VO.20 x 0.54 = 0.33 With these data, by Eq. 10-24:
Il = 0.1713(14.V ~ 8.64) 0.33 x 48 = 108,000 Btu/hr
is represented by the combined area AR at u uniform temperature TR: the” the factor F,, may be ex- pressed in terms of direct geometrica1 factors F as:
F,, = FI2 : FIKFRI/(l -F,& Eq. lo-31
Influente of refractory rwfrfaces. If the enclosure consists in part of black sorfaces t,hat radiate or absorb beat and in part of refractory surfaces, from whieh there is no net new flow but only re-radiation without ar.y absorption, Eq. lo-23 and lo-25 can be rewritten in the general form:
If there are ~two radiating surfaees AI and Az, then by applying the principles expressed in Eq. lo-26 and 10.28, Eq.~ lo-31 reduces t,o the more convenient form:
iq. IO-30
where s is the geometrica1 factor which takes into aceount the eombined mechanisms of direct radia- tion and re-radiation from the refraetory surfaces,
Eq. 10.32
If there are only two radiating surfaees, Al and AB, and neither of them has a negative curvature, then using Eq. 10-26, lo-28 and 10-29, Eq. lo-32 is reduced to:
?
EXAMPLE 10.2. Determine the heat,exehange for data of Example 10-1, assuming that the peripheries of the surfaces are connected by refractory walls.
SOLUTION: In this case Eq. lo-30 must be applied. It differs from Eq. 10.23 and 10.24 only by the factor 3, whieh is found from Fig. 10-7, using the same procedure us in Example 10-l :
F = v’o.53 x 0.75 = 0.63 rl = 108,000 x 0X3/0.33 = 206,000 Btu/hr
Eq. 10.33
This case covws the majority of problems of radiant-heat interchange in an encloaure. It must be noted that Eq. lo-33 assumes a uniform refractory temperature whieh for most casa corresponds snfficiently to the actunl condit,ions. l
The equilibrium temperature of the refractory surface, for the c,ondit,ions assumed in Eq. 10-33, may be computed from the expression:
The re-radiating refractory walls inerease the 4, CA, - A#‘,dT4 + (A, - A&)Th heat interchange eonsiderably. This influente in- TR = J 1
cieases with u deerease~ of the ratio D/T. (-4, - AIFIZI + (As - AIFIZ)
EXAMPLE 10-3. A vertical boiler has a grate 8.ft wide and 7-ft deep; the bundle of tuba connecting the~loru-er cylindrical drum with the upper one has an over-al1 diameter of 8 ft and the opening through which the tubes are exposed to radiation is 4-ft 9-i”. high and 8.ft wide. The temperature of the coal-bed surface is about 2400 F; the temperature of the tube surfaee is 390 F; the three vertical sides and the roof of the duteh-oven furnace are of refractory brick and may ~be assumed to, be non- conducting and re-radiating. Determine: (a) the theoreticai amount of radiant heat transferred from the fuel bed to the tuba assuming that both su- face8 are black and (b) the approximate tempera- ture of the refractory surfaces.
. Eq. 10.34
TO use Fig. 10.8 in this case, y = 7 ft and v = 7/8 = 0.875; z = 4.75 ft and Z = 4.7518 = 0.595; with these data FI2 = 0.175.
Ne& AI = 8 x 7 = 56 sq ft; Az = 8 x 4.75 = 38 sq ft and this giva by Eq. 10-33: -
F,s = (38 - 56 x 0.175”)/‘(56 + 38 - 2 x 56 x 0.175) = 0.488 The temperature 7’1 = 2400 + 460 = 2860 R and TS = 390 + 460 = 850 R, and by Eq. lo-30 modified similarly to Eq. 10.24:
q = 0.488 x 56 x 0.174(28.6& - 8.5d1 7 3,160,OOO Btu/hr.
The temperature of the refractory surfaees is, *~
by Ea 10-34.
SOLUTION: (a) The amount of heat exehange may be found using Eq. 10-30. The geometrica1 factor ,F,, may be eomputed by Eq. lo-33 in which the factor F1, may be found from Fig. 10-8 assuming that the projected area of the tube bundle can be considered BS a surface perpendicular to the grate surface and having a eommon side z = 8 ft.
,,,~~ 175) 2860” -(38-56x0.175) 8504
56 x 0.175) i- (38 - 56 x 0.1751 = 2540 R, w 2080 F
References
‘W. H. McAdams, Heat Transmission, OP. cit., p. 59.
The faetor 3 has been determined far a few geometrieally simple cases presented by curves 3 and 4, Fig. 10-7. and cuives 6 and 6, Fig. 10.9. For reetangles it can be determined osing Fig. ~10-7 and Eq. 10-25. For other cases it must be approximated. Thns if an enclosure consists of severa1 black surfaces AI, Az ., and the rest of the enclosure, re-radiating refractory surfaces,
ZM. Jacob, Heat Transfe~. op. cit., p, 52. 3Eckert, Introduction tu the Transfer of Heat. and
Mass, op. cit., p. 277. ‘W. J. King, Mechanical Engineering, Vol. iiL,
loc. cit., (19321 pp. 699-704.
I Part 11, Radiation Applications, will appear in the Llecember issue. I
56 November 1965 Petro/Chem Engineer
Radiation Applications- Part 11
Applied Heat Transfer Design for the HPI
hy Wayne C. Edmister and Joseph M. Marche110
Ro@, -. 11 1 Nonblack swjaces. Radiating characteris- tlcs of surfaces dealt with in engineering problems differ from those of a blsck-body surfacc. In most cases it is permissible to consider al1 surfaces as way which simplifies considerably the treatment.,
If~ a gray-surface body has an emissivity el and an absorptivity al at a temperature T,, then it will both emit and absorb at this temperature proportionally less energy than & black body. The net heat exchange between the gray body and blaek surroundings of temperature Tz may be written, in accordante with Eq. 10.24, as:
With sufficient aecuracy the emissivities mag be considered equa1 to the absorptivities for the same temperatures.
If two gray surfaces Al and As have emissivities of el and e2, resp&ively, the” ei and ez must be ‘ntroduced into Eq. lo-23 or 10-24. The simplest
m vuy of doina this is by introdueina a factor F, which is a finction of é, and e2 &d also of the
enclosed body whose surface AI is not
For a completely enclosed body whose surfnee Al is small eompared to Ac ‘F, = el . . .b Eq. 11-5
For many other case when the area AI of two parallel surfaces is small compared to the distance
Eq. 11-6 c A
Eq. 11.6 also applies to parallel surfaces con- nected by non-conducting reradiating walls. It also applies to two rectangles with comnion sides in perpendicular planes, Fig. 10-8, and for radiation from a piane to t,ube banks, Fig. 10-S.
In intermediate cases it may be advisable to use for F, a value interpoluted between values found by Eq. 11-4 and 11-6.
A useful equation, based on the same reasoning as Eq. 10.26 and lo-30 is:
FemAr = Fez,Az. . Eq. 11-7
11-2. Refmctory sr~~faces. For the case of two radiating surfaees Al and A2, the combined effect geometrica1 relation, between the area AI and Ap.
Thus Eq. 10.24 may be written as: q = O.l713A,[ (T1/100)~ - (TJ100)4]FF,
A b& of the geometrica1 and emissivity factors may he r&. ,~,~, ~, -prese”tyd =s :
Eq. 11-Z gu F’, - l/[l/Flz f (l/el - 1) +
EmissivitU factor F, has been determincd for (A1/AZ) ‘(l/er - 1):I Eq. 11-8
several, more eommon eases. For the general case where F,, is the geometrica1 factor which tskes of a complete radiant convex surface AI surrounded into account the influente of refractory surfaces by a concave radiant surface Az: and for most problems may be calculated from
F, = l/[l/el + (All’A,) (l/ez - l)] Eq. 10.33. . . . . . . . Eq. 11-3 Since the factor F’, by Eq. 11-8 already contains
For large parallel surfaces with AI = AP, Eq. 11-S the goemetrical relations between the surfaees in
becomes : the faetor FI%, Eq. 11-2 far, this case becomes:
Eq. 11.4 applies also to two equa1 parallel su- EXAMPLE 11.1. A furnaee with a chamber of a face if the distance T between them is small eom- rectangular parallelapiped shape is heated by the pared with the area~. AI = Az. It also applies to a combustion of gas inside of vertical radi& tubes
Petro/Chem Engineer December 1965 43
placed alongside the two longa walls. The tubes are 5-in. OD on lo-in. centers: and their mean surfaee temperature is 1540F; the floor. sidewalls and roof are of refraetory brick with dimensiom shovm in Fig. 11-l. The furnace is used to anneal steel plates at a temperature of 1100F: The plstes come in sizes up to GO-in. wide and 144.in. long and the depth of the furnace is G in. greater. Assume the emissivity of the radiant tubes 0.75 and of the scale-covered plates 0.92. Find the net heat exchange between the furnace tube und steel p1ates.
Fu = Fa + l/Il/Fz~ -1 A,i(AaF,,, ] Eq. 11.10
i ,_li” 10 ff
& . -.- :_ Figure 11-l. Furnace chamber. , n <
SOLUTION: First the emissivity of an imagina:- piane through t,he centei li,nes of the tubes and having an area iil = 6 x (144 + 6)/12 = 75 sq ft must be found by Eq. 11-7. This plane must be equivalent in radiation to the radiation area Al of the tuba. The number of tuba on each side will be in = 12.0/(10/12:~ = 15 and:
AI = 71(5/12)G x 15 = 118 sq ft F,, is found from curve 5, Fig. 10-9, with 7 = 1015 = 2 as 0.88. By analogy with Eq. 11-8, assuming the emissivity of the refraetory ez = 0.97, Table 10-2 :
This means that the area Al including the action of the refractory behind it can be replaced by u plain wall Az with an emissivity el = 0.724 and the same temperature as the tuba
The new simplified furnace chamber is enclosed by two G x 12.5 ft radiating side walls, 2Az, a horizontal piane A8 = 5 x 12 ft receiving radiation, and refractory surfaces AE. The heat exchange may be calculated by Eq. 11-2 introducing F instead of F because of the refractory surfaces. The factor Fz3 is equa1 to F1 from Az to (A, + str1
^.. of n,j mmus fil from AB to the strips. Both F’s
: ,may be evaluated from Fig. 10-8, for Y1 with 2: = 12 ft, Y = 6112 = 0.5, Z = 6.5/12 = 0.54 and
_ this giva F, = 0.25; for FS. Y = 6/12 = 0.55.
In this case A, = F x 12.5 x 2 = 150 sq ft and it “se& AR, Aa, and pari! of itself - the opposite 6 x 12.5 ft piane - and t,herefore F,, = 1 - FZ2 -& is the direct geometrica1 factorbet&en the two parallel rectangles 6 x 12.5 ft separated by 8 ft. Using Fig. 10-7, curve 2, and Eq. lo-25 giva for the 6 x 6.ft sq FI = 0.13 and for the 12.5 x 12.5-
ft sq Fz = 0.30 and ~FL~ = ,/.13~-!0197;1~ This gives for Fzn = 1 ~ 0.197 - 0.15 = 0.653.
The other factor, FSR = 1 - Fa2 = 1 - FzaAzJA8 = 1 - 0.15 x 150/60 = 0.625; and with these data:
Fm = 0.15 + 1/[1/0.653 + 150/(60 x 0.625)] = 0.331 and by Eq. 11-8, in accordante with the changed designations :
11-3. Simplified radiation calcula.tions. In many engineering problems, especially if the heat trans- fer by radiation is of the sane order or smaller than by eonvect,ion, it is desirable tu determine the radia&beat, exchange by an expression similar to equations used in conv&ion problems. Sueh an equation may be written SS:
~7 = hvA,(t, - tz,) Eq. 11-11 where IL, is called the coe.ficient of radiant hat transfel’.
44
~,-i Z, = 1.5/12 = 0.125 and this &es F2 = 0.10. Thus "1
Fm = 0.25 -~ OJO = 0.15. Now Fz3 may be cal- culsted by Eq. lo-32 changing the subscripts in aecordanee with the designation applied to the new furnace which becomes: A eompsrisim of the general Eq. 11-2 with Eq. 11-11 gives:
Ir, = 0.174 FF,C(T,,‘100)4 - (Tx/100)4],‘t, - tz)
. Eq. ,11-C? For blaek surfaces F, = 1 and if one su-face com- pletely encloses the other surfaee. F = 1. The values of h, with these simplifying assumptions are plotted in Fig. 11-2 which covws the most frequently encountered ranges of temperatures. The deviation of the surfaee properties from those offe
ea
Figure 11.2. Coefficient of heat transfer by radiation.
December 1965 Petro/Chem Engineer
Many complete refineries around the world are included ently meet the highest performance standards: on speciii- in the more than 500 Procon projects now operating. In cation, on budget, on time. Regardless of where your next Africa, far example, Procon has completed three grass petroleum, chemical or petrochemical engineering and roots refineries, in Southern Rhodesia, Kenya and Mozam- construction assignment is located-in Europe, Africa, bique; and another is well tinderway in Nigefia. There Asia, Australia or the Americas-it will pay you to talk are many other examples of Procon projects which consist- to Procon and to utilize its experience in each locality.
a blaek-body can be taken int” aecount by eom- puting the emissivity factor F, by “ne of the Eq. 11.3, 11.6 “r 11.8, as the case may require.
11.4. Nodunrinmm gnscs. Gases do not emit radiati”” at al1 wave lengths, as do sulids, but they emit and absorb radiation in certain bands of wave lengths. Of particular interest are hydro- gen and carbon eombust,ion produets. These games may be divided into two groups - those with symmetrica~l mole&des. such as “xygen, nitrogen, hydrogen, and those with unsymmetrical molecules of whicb~ the more important are carbon dioxide, water vapor, and hydrocarbons.
Gsses of the first group do not absorb radiant energy at temperatures met in industrial prxtice and therefore do not emit it. On the other hand, when blaek-body rndiation passes through a gas ma.ss containing carbon dioxide “1‘ water vapor, absorption of heat energy takes plsce in certain regions of the infrarea spectrum. Cunversely, if such a gas mass is heated, it wiil radiate energy in the same wave-length regions.
The radiant-beat intercbange between a gas, such as carbon dioxide, at a temperature TQ, and a partial pressure p, and an element of a blsek surface at a temperature T, is determined in the following manner. Suppose that the gas mass has the shape of a hemisphere of radius L and the blaek-body.element is located at the center of the base of the hemisphere. TO the surface A, the gas radiata the amount of heat uT”,e,A, where the gas emissivity e, is the gas-to-surface radiati”” divided by the black-body radiation at the same tempeia- ture. The gas ala” absorbs an amount of beat d’,“a,A radiated by the surface, where ay is the absorptivity of the gas for the black-body radiation of the surface. The net radia&heat interchange is then:
Figure 11-3. Emissivity of carbon dioxide.
l 2.0
1.5
ci- $- II 4”
0.8 1.0
- I” em +0.6 - 0.5
0.4
0.3 0.05 008 01 0,2 0.3 0.5 0.8 1.0 2.0 3.0 5.0
Figure 11-4. Correction for effect of tota1 pressare on carbon dioride radiation.
,J = (rT,“e, - <rT,“a,)A Eq. 11-13
Cwbon Diosido. For carbon dioxide the emissi- vity designated by e,, depends on the gas tempera- ture, T,, the total pressure P, and t,he produet p,L. The product p,L is a measure of opacity of the gas - the smaller &L, the more transparent the ga,s and the lower its emissivity e,. Values of er far different T, and p,L may be read from Fig. 11-3 for a pressure P = 1 atm. Values of absorpti- vity a, may be obtained from the same Fig. 11-3 BS follows: If T, > T,, t,hen a, may be considered to be equa1 e, at the same value of p,L but at the temperature T,; if T, < T,, then a, is obtained by rcnding e, at T, but at pcL (T,/T,I and multiply- ing the result by (T,/T,)“,65. If the tota1 gas pres- sure p differs from 1 atm, both emissivi& and absorptivity values det,ermined using Fig. 11-3 must be multiplied by a factor C, read from Fig. 11-4; C, - ecl,/ec, where cc is the emissivitp a,t 1 atm, as found from Fig. 11-4 and ecP is the emissivity at any “ther pressure P.
Watw ?fa~“r. Far water vapor the emissivity depends on T <,, P,,>L and tota1 gas pressure y, as far COy, and in addition on the partial pressure p,,; of the water vspor. Emissivities of water vapor as a function of T, and p,,L for a tota1 ga,s pressme of Figure 11-S. Emissivity of water vapor.
46 December 1965 Petro/Chem Engineer
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PetrojChem Engineer December 1965
1 atm and the imaginary case of p, = 0 are pre- sented in Fig. 11-5. The correction for the actual conditions of p, atm, and p,,,, atm, is made by multiplying the value e,, from Fig. 11-5 by a factor C,, read from Fig. 11-6 as a function of (p,, + p)/? and p,,L. The procedure of finding the ab- sorptivity G,~ from Fig. 11-5 iS similar to that for COz. Approximately a,, = e,,,: read from Fig. 11-5 at &,L and T,; more accurately a,,, = e, read at T, and puL (T,,/T,j and multiplied by C,(T,/T,)O,4j.
Carbon Diozide and Water Vapor. The tota1 radiation due t,o carbon dioxide and water when they are both present is slightly smaller than the sum of their effects ealculated separately, be- cause each gas is somewhat opaque to radiation from the other. The correct,ion far opaeity ae is resd from Fig. 11-7 and deducted from the sum of e, + e,, each evaluated independently from Figs. 11-3 and 11-5, with the necessary corrections. The emissivity eg of a gas containing both COS and Hz0 is evaluated as:
eg = e, + e,, - A, . Eq. 11.14
The same kind of correction aa, read from Fig. 11-8, must be applied to the absorptivity giving:
a, = a, + a,, - na Eq. 11.15
Combustion products of most hydrocarbons used as fuel, both liquid and gaseous, have the ratio of partial pressures p,J(p, + p,) very close to 0.5 and for such a gas it is more convenient to read
the corrections ne and Aa from Fig. 11-8 which is a crossplot of Fig. 11-7.
Nonblack surfaces. If the surface bounding a radiating gas is not black, multiplying the right side of Eq. 11-13 by its emissivity e, would take into account the Iosa of energy in the primary beams from gas to surface and surface to gas,
500 750 1000 1250 ,500 1700 Temperotures, ‘F
a Figure 11-8. Correction for opacity for P,,./(Pw f Po) = 0.5.
1.8
1.6
( 1.4
h 1.2
-2 1.0 4, II “0 *-oa % e v 0.6
0.4 AND PARTIAL PRESSURE
0.2
0
respectively. However, due to opacity of the gas, part of the gas radiation initially reflected from the gray surface will be refleeted by the gas and fìnally absorbed by the surfaee. As a result, the factor by which the right side of Eq. 11-13 must be multiplied is greater than the surfaee emissivity e, but smaller than unity. The more transparent the gas is the smaller its pL. For most industrial surfaces e, = 0.7 for which the effettive emissivity eJ niay be determined with a sufficient accuracy from :
e’, = (e. + 1)/2 Eq. 11-16 The fina1 form of Eq. 11-13 for radiant heat ex-
ehange between a gas containing CO2 and Hz0 and its bounding surfaee becomes:
Q = 0.1713e’,[e,(T~/100)4 - a,(Tz/100)4]‘4
Figure 11-6. Correction far effect of tota, and partial pressure.
Figure 11-7. Cowction to gas emissivity due to special overfap of water and carbon dioxide.
11-5. Beam length L. Eq. 11-13 and 11.17 were established for radiant-heat interchange between a gas hemisphere and a small area on its base. In this case the length of the path L of the radiant beam is the same in al1 directions. For gas-filled spaccs of industrial importancc it has bccn found that any space may be approximately represented by an equivalent hemisphere of a proper diameter, that is, far any space there is a mean beam length L which can be used in evaluating emissivities and absorptivities from Figs. 11.13 and 11-15 As pL approaches aero, the mean beam length approaches as a limit, the value 4V/S, where V is the gas volume and S is the bounding surface. For values of pL encountered in praetice, L is always less than the limiting value, and generally a satisfactory approximation is :
L = 3.4V/S Eq. 11.18
48 December 1965 Petro/Chem Engineer
Table 11-1 gives some frequently encountered shapes and values of the numerica1 factor by which the governing dimension of the space must be multiplied to obtain the beam length L.
11-6. Variable tempemtures. If the temperature of the gas and of the surface change continuously from one to the other, the temperature differences governing the heat exchange may be determined by integration over the length. A simpler, usually satisfactory procedure, is to use an arithmetic mean of the surface temperatures for the surface tem- perature:
t, = (t.1 -l- t,r)/Z Eq. 11-19
and the average surface temperature, t,, plus the logarithmic man of the temperature differences between the gas and the surfaee at the two ends, for the average gas temperature:
EXAMPLE 11-5. Flue gas containing 9.2% CO? and 10.0% H20, wet basis, flows through an oil still eonsisting of bands of 3-in. tubes on S-in.
9 equilatera1 triangular centers. The gas enters at 1750F and leaves at 1100F. The oil flows counter- current and the tube temperature rises from ‘750 to 950F. Calculate the average beat-input rate per square foot of tube surface due to radiation only. The tube-surface emissivity is 0.82.
-A. 1. Haslam and H. C. Hottel. Tranractionn ASME, FSP. 50, 9 (1928).
Figure 11-9. Chart for determining absorption strength of luminoos flames.
t,, = Red brightness Temp., ‘F (X = 0.6651~) t, = Green brightness Temp., OF (X = 0.5553~) A = t, - t, tr = True Temp., “F
k,Lr = Absorption strength
The effettive emissivity of the tube surface, by Eq. 11-16 is:
e: = (0.82 + 1)/2 ‘= 0.91 The temperature terms: T,/lOO = (1394 + 460)
/lOO = 18.54; TJlOO = (850 + 460)/100 = 13.1
and by Eq. 11.17: q = 0.1713 x 0.91 (0.191 x 18.54”) -
0.222 x 13.14) = 2490 Btu/sq ft-hr. 11-7. Oaer-ali emissivity. Sometimes it is helpful
to know the emissivity of an equivalent gray body which can be used instead of both the emissivity of the gas and absorptivity of the surfaee. This emiasivity also takes into account the shape of the surface and therefore is called ovw-al1 missi- vity and designated by er. In this manner, Eq. 11.13 becomes:
q = <~ep(T,4 - 7’84) Eq. 11-21
Equating, the right sides of, Eq. 11-21 and Il-13 and solving for eF gives:
eF = e,(T,/Ts)4 ( T,/‘T,/) 4 - 1
Eq. 11.22
11-a. Radiation from, flames. In general flames may be divided into two different classe& namely, nonluminous flames and luminous flames. Non- luminous flames are either invisible or appear as transparent bluish flames and their radiation may be determined by the same formulas as giva in Section 11-4 for gases.
Luminous flames are ‘made visible by incandescent solid particles of soot or carbon Iiberated due to dissociation of hydrocarbons at high temperatures or of powdered coal particles suspended in the com- bustion gasa. Accurate eomputations with luminous flames are difficult because their emissivity and absorptivity decrease with an increase in wave length and the tota1 emissivity is 1~s~ than the emissivity in the visible spectrum.’ For this reason dire& visual determination of luminous-flame emissivity is often misleading. However, by using an optical pyrometer with red and green color
Petro/Chem Engineer December 1965 49
screens it is possible to determine the true time temperature tr, “F, from the observed red-bright- ness temperat~ure t, and green-brightness tempera- ture t,. The flame temperature is obtained from Fig. 11-9: the point of intersection of a vertical line through the observed t,. and an inbxpolated curve corresponding to At = t, - t,., referred to the two nearest curves of~ t,;, giva true flame temperature f,r. A horizontsl line drawn through the point of intersection to t,he left-hand scale gives the absorption strength klLr. In the product klLB. the fìrst term is a function of the soot concentration in the flame; and LI,. is the thickness of the flame through which the pyrometer is sighted. This absorption strengt,h must be multi- olied by a factor from Table 11-1, depending upon
Figure Il-IO. Emissivity of Iumiflous flames
the shape of the Rame. Using this corrected absorption strength kL, the total emissivity is found from Fi,g. 11-10 the ordinate corresponding to the point of interseetion of a vertical line through kL and the corresponding temperature curve.
The rate of beat transfer by radiation from a luminous flame wit,h a temperature ta to a surface at a temperature t, enelosing the flame is giva bv a formula similar t,o Ea. 11-2:
where: A = the area of the flame ermelope, sq ft. Tr == the absolute true flame temperature,
t, + 460, “R, T, = the absolute surface temperature,
t, + 460, “R, t?p = the flame emissivity found bg Figs.
11.9 and 11.10, e’, = the effettive emissivity of the surface.
Approzimate cahlations. The burning of liquid hydrocarbons generally is accompanied by some smoke formation, at least in the furnace near the burner. If an optical pyrometer with red and green screens is not available or cannot be used, the radiant heat released may be estimated by Eq. 11-23. Using the gas temperature t, for tF and the gas emissivity en for er, from Eq. 11.14 and multiplied by a factor Cr. to take into account
50
luminosity of the carbon particles, results in: e,>. = Cu?, Eq. 11-24
The main difficulties are in estimating the value of Cc which depends upon the amount of excess air present, temperature of combustion, and fine- ness of atomization of the liquid fuel. From avail- uble dat,a it is snggested that for good atomization and avel’age combustion temperatures, n-hm there is no excess air present, CL = 4, with 100% excess air CL = 3, and vith a, oery large excess of air, about 9 time the theoreticul amount, CL = 2.
Il-8. Steam boiler jirrnaces. A determination of the radisnt heat absorbed directly from borning fuel by the heating surfaees of a boiler is neeessary in order to find the hest given up by the hot gases by convection.
Theoretical computations are rather involved and it is convenient to use an empirica1 formula. One of the best knawn and simplest is, for slag-free heating surfaces ?
m = l’(1 -l- O.O37r,, \/Wi.A:) Eq. 11-25 wh&e: ‘m = the fraetion of the tota1 heat released which is absorbed by the boiler surfaces by radintion, l
I’,~ = the air-fuel ratio, lb air per lb fuel, 1s’ = the weight of the solid fuel actually burned,
expressed as equivalent weight of bitumin- ous coal, lb/hr,
A = t,he boiler surface exposed to radiat,ion, sq ft. Far boiler surfaces, rvhich may become coated
with Ag, a modifìed formula is reeommended:3
0.572 (0.5 -l- 1.1s) (1 - 0.8s) m = -=- Eq. 11.26
1 + O.O37v&W,‘A
where s is the proport,ion of the absorbing surfaee covered with slag and ash, and ,ma,y vary from
~0 to 0.4. The value of m increases wit,h increase of s, because the amount ot heat absorbed by con- vection deereases.
Far oil and gas-fired furnaces the following equationd may be used for a box-type fornace with a single row of absorbing tube:
m = 1111 + (r,r,/Q/A)/3200] Eq. ll-27-~ ei
where: & ,= the tota1 net heat released by the fuel, Btuihr.
References ‘II. C. Hottel and F. P. Broughton, “Determina-
tion of True Temperature and Tota1 Radiation from Luminous Gas Flames (Use of a Special Two-color ryrometer) “, Imi. Eng. Chem.., and. ed., Vol. 4, 699-704 (1932).
‘Orrock-Hudson formula diseussed in Trans. ASME. Vol. 47 (1925’1 n. 1148.55: also .74arks’ Handb~ok, op. cit.; p. 362.’
%I. 0. Croft, Tlierwmd~nnmics, Flzrid Flhu and Heat Tmnsmissim, op. cit. p. 195.
“D. W. Wilson. W. E. Lobo and H. C. Hottel. “Heat Transmission in Radiant Seetions of Tube SMls,” Znd. Eng. Chem. 24,486 (1932).
Part 12, “Conduction, Convection and Radia- tion,“~ will appear in the January 1966 issue.
December 1965 PetroKhem Engineer
Technology in 1966
data in the low temperature region, and the experi- ente gained in gas liquefaction and other cryogou- ic processes will provide valu,able data for “8e in new gas processing plant design. Adsorption pro- cesses will continue to find limited application on small volume streams.
Increased emphnsis will be placed on power and wnst,e heat recovery to attain better fuel economy. Both gas expander and fluid drive” equipment Will be “sed more ext,ensively. Continued improvemeut in both equipment and technolog?for power and
g’ ~’ wastc beat, recovery can be anticipated. The increasing movement of mixed product
streamu by pipeline affords new opportunities for economie processing of streams which eannot sup- port a complete plant with fractionation, storage and loading facilit,ies. Even in larger plants there in a trend away from on-site fractionation.
As greater emphasis is placed on minimizing la- bor costs and improving efficiency, process contro1 becomes an increasingly important faetor in de- sign and operation. Improvemerrt is evident in the continuing development of instruments for moni- toring and/or contro1 of the composition of critica1 plant streams, and in the wide range of contro1 hardware available to the industry. TO date, prog- ress in mntching instrument application and pro- cess design for maximum benefit has bee” limited. This area, however, will eommand increasing at- t.ent,ion and offws promise of affording substan- tial brnefits.
Plant layout and construction concepts are also
undergoing change. The keyword in layout is eom- pactness, with its inherent benefit,s of minimum piping and insulation req,uirements. The trends to- ward reduced valving and minimum provision of sland-by equipment also contributes to more com- pact 1ayout.
Plant constructors are seeking new ways of ra- duciug plant costs. Along with compact plant lay- out t,here will likely be continued and improved “se of prefabrication techniques, and further de- velopment of unitized equipment syst,ems.
Modernization and Consolidation Processors are giving increased attention to ex-
isting plants in seeking’ opportunities for improv- ing product rec~ovrrirs xnd/or reducing operatmg costs through modernization of individua1 plants or consolidation of separate plants into larger,
Petro/Chem Engineer Januarg 1966
more profitable operations. This aetivity is stimu- lated ‘by n combinntion of factors,~ including: 1) increased demand far propane and ethane, prod- uds which m;my older plants were dcnigned t,o re- cover either in nomina1 quantity, or not at al1 ; 2) opportuuities for sale of mixed produet st,reamn; 3) increasing labor and maintenanc,e costs; 4) de- cline in gas volumes, or, availability of new sources of gas for processing, and 5) increased fuel and “tility costs. Modernization and consoli- dation opportunities offer a rea1 challenge to the industrg, for such projects require careful evalua- tion and planning. Processors, however, will eon- tinue to find this challenge a renarding one.
Plant Safety Severa1 of the present trends in the industry
have implications directly related to the matter of plant safety. Far example, reduction in operating and maintenance personnel, more compact plant layout, and redueed provision of stand-by equip- ment al1 impose factors which can adversely affect, safety. The gas processing industry prides itself in its safety conxiousness, and will undoubtedly meet the challenge of applying its changing tech- nology and methods in a manner whieh will not sacrifice, but rather maintain and strengthen its safety standards. n
Correction:
“Economie Alternatives in Petro- chemical Feedstocks,” Peter W. Sher- wood, PKE, Nov. 1965, page 26.
During typesetting, severa1 lines were left out at the bottom of the first column on page 26. The corre& version is:
The bulk of capacity utilizes Linde’s “IsoSiv” and Universa1 Oil Products’ “Molex” processes. (In addition, Esso, Texaco, and British Petroleum have developed different commercia1 versions of molecular sieving) Of the two main commercia1 processes, the former is operated entirely in, the vapor phase; desorption is carried out at reduced pressure. By contra&, the “Molex” process calls for liquid-phase operation
in both the absorption and desorption parts of the cycle; desorption is, effected by use of low-boiling paraffinic solvents.
Conduction, Convection and Radiation- Part 12
Applied Heat Transfer Design for ,the HPI
by Wayne C. Edmister and Joseph M. Marchello
n 1.2-I. Introduction: The basic principles of heat transfer by conduction, eonveetion and radiation were discussed in the preccding chapters separately. Usually heat transfer from one substance to another takes piace not by one of these mannew alone but simultaneously by two or al1 three manners. When the beat which has entered a body is given up by convection,~ either the full amount or a considerable part must be conducted through the body. Thus, the combined action of conveetion and conduction must be considered. In many problems, in addition to heat given up by eonvection, it is also given up by radiation.
12-2. Rod heated at me end. Consider beat, flow through a rod connected at one end to a heated wall and cooled on its entire surface by a fluid. The constant cross-seetional area of the rod is A, its circumference is C, and its length is 1. The temperature of the wall is tl and that of the fluid is t,. Assume that the surface coeffieient h has a constant value over the whole surface. Consider a different,ial element of the rod with the length dz at a distante z from the wall,~ Fig. 12-I. Under steady state conditions, the amount g, of heat energy coming byeeonduction from the wall must, be equa1 to the amount ql> of energy passing through the element to the free end and heat ,IC lost by conveetion to the swrounding fluid:
q<I = q~ + qc Eq. 12-1
The heat flow ria by conduetion is:
q CL = -,&dt dx
Eq. 12-2
where t is the temperature of the rod at the distance z. The heat flow qb is smaller than qz,, by
the amount 2 dz or, using for & Eq. 12-2:
4a=-kA$+kA$dz. Eq. 12-3
and the heat flow g< is:
rlc = hC dz (t - tf) Eq. ,X-4
Substituting qn, Q and qc from Eq. 12-Z to 12.4 in Eq. 12-1 results in:
kA g = bC (t - tt) Eq. 12-5
50
If the symbol 7 = t - t, is used for the tempera- ture differente, t( = const, d’tjd9 = d2r/dx2, Eq. 12-5 beeomes:
Introducing the symbol m, where:
Eq. 12-6 can be simplified and written as:
D = m% Eq. 12-8
The general solotiun of this differential equa- tion is:
T = CI~“‘~ + Cpe~‘“~ Eq. 12-9
where Cr and CZ are two constants of integration which can be found from the boundary eonditions. The first boundary condition is taken at z = 0, when the rod has the temperature t,. This gives:
The second condition, is taken at z = I, when the free end transmits heat to the surrounding fluid. This process is presented b+ the equation:
If the cross-seetion area A, is mm11 compared
Figure 12-l. Steady heat flow in a rod.
January 1966 Petro/Chem Engineer
Many complete refineries around the world are included ently meet the highest performance standards: on specifi- in the more than 500 Procon projects now operating. In cation, on budget, on time. Regardless of where your rat Africa, for example, Procon has completed three grass petroleum, chemical or petrochemical engineering and roots refmeries, in Southern Rhodesia, Kenya and Mozam- construction assignment is located-in Europe, Africa, bique; and another is well underway in Nigeria. There Asia, Australia or the Americas-it will pay you to talk are many other examples of Procon projects which consist- to Procon and to utilize its experience in each locality.
with the length 1, the heat flOw at the end can be disregarded. This results in the equation:
Soloing the simultaneous Eq. 12-10 and G-15 for C, and Cp giva:
6, = TI- 1 + #” / and Cr z -~~.,.L 1 + e-a”’ Eq. 12.16
Substituting these values in Eq. 12-9, simplifying it by using u common denominator and the symbol At for temperature differences, results in:
cosl~ m. ii -1; + e-“” l-
A t, -cos,Lml .,.. Eq. 12-17
The temperature differenee of the free end, with 2 = 1, is:
* t2 = Ah1 coslr ““”
Eq. 12-1X
The heat flow through the base of the rod’ with Eq. 12-13 is:
:L kAm(CI- CI) Eq. 12-19
Subatituting the values for Cl and Ca from Eq. 12-16 and simplifying the equation by using a common denominator results i,n:
(1, = kA‘m t, e”l” ~ e-“l”
3-+ 1 = kAm.at, tmh n1.1
Eq. 12-20
Eq. 12.20 is derived assuming that the heat fow from the free end is negligible, as giva by Eq. 12.12. The derivation of an equation without this simplifying ansumption given:
Figure 12-Z. Heating surface with rectangolar fins.
Comparing Eq. 12-20 with Eq. 12-21 shows that the incurred error becomes zero when the dimen- sionless number (lk/hj is a very small fraction, or h is small compared with kC/A, and at the sane time ml is not too small.
12.3. Finned awfaces. The beat-transfer eoeffi- cient between a gas and meta1 wall usually is rather low and the tota1 amount, of heat transferred from a giva area is correspondingly low. The amount of heat flow can be consi.derably increased by increasing the heat transfer surface area. Finned surfaces are used extensively in many apparatus sueh as automotive radiators, cylinders of air-cooled combustion engines, economizers in steam-power plants, eleetric transformers, con- denser eoils of household refrigerators, etc.
Straight +Y. For straight~~ fins of constant thickness IJ, and height 1, Fig. 12-2 and the equa- tions derived in Section 12-2 apply. With the deaignations of Fig. 12-2, the cross sectional area A of the fin is bL and the perimeter C is 21, sinee b is small eompared with IL. Introducing thene values in Eq. 12-7 givas:
Eq. 12-22
The hest loss of the fin is gioen by Eq. 12-20. Under what conditions can the heat flow from
a wall be increased by the use of fins? Evidently, fins are useful if the heat flow t,hrough a fin-~ increases with an increase of the fin height. If the beat flow dccreascs with an increase of the fin height, the fm is retarding beat flow and should probably not be used. The limit of this condition is found by setting the first derivative of Eq. 12.21 equal to zero. This condition States that there i,s n,n rhange in ,heat flow rate mith change in fin length. The resulting equation may be sihnplified to the expression:
li’ w12 = T<Y. Eq. 12-23
Substitut,ing for m from Eq. 12-22 and rearranging terms gives : -
1 b -== Eq. 12.24 ti
‘The left side of Eq. 12.24 is the surface heat-
January 1966 Pet,ro/Chem Engineer
transfer resistance; the right side is the heat- conduetion resistanee of a piane wall with a thiekness equal to b/‘Z. If both resistanies are equal, the fins are useless. In the preceding csl- eulation, it was assumed that the temperature varies only in the direction of 1: and is constant on normals to 1. Actually the temperatures decrease from the center of u fin toward its faees. This influences the numerical relations in Eq. 12-24. ,In general, it is advantageous to use fin8 if :
$> 5 Eq. 12-25
EXAMPLE 12-l: Determine whether cast iron fins l/s in. thick are useful on a heating surface when heat is transferred: (a,) to air and (b) to water. Use representative h values sueh as those employed in previous examples.
SOLUTION: The thermal conduct,ivity of cast iron may be taken as k = 28 Btu/hr-ft-“F.
(a) For air in motion, h = 9.25 Btu’hr-sq ft-“F and :
-m g = gyy=; = 581
Sinee 2Wh.b 1 5, fms will be useful in this case. For stili air, h = 1.04 Btu/hr-sq ft-“F.
Evidently fins will be even more useful. Ib) For laminar flow of water inside pipa,
h = 24.2 Rtu,‘hr-sq ft-“F and:
2k xX=
12 x 28) (12:) = 222 ~~ 24.2 x 0.125
4s 2k/hb is greater than 5, the advantsge of ,fins will be significant.
For turbulent flow of water outside pipes, h = 600 Btu;hr-sq ft-“F and:
2k -= hb
(2 x W,,pJ = 8,12 600 x 0.125
Therefore, fins are useful. For non-agitated water, h = 200 Btu/hr-sq ft-“F
-and:
2k 11 b
(2 x 281 (12, = 26,2 200 x 0.125
Again fins would be useful. These examples show that, in general, fina are
useful for air and other gasa where h is low, and are of less value wit,h water and other liquida, when h, is high.
Spacing of fins. In order to take proper advantage of fins, they should be placed as close to each other as possihle. However; the fins must be SO spaeed that the boundary laqers which are built up on their surfaces do not influente each other. There- fore, the diutance between two fina must not be appreclably smaller than twice the boundary-layer thickness. This distance may he computed hy equations developed earlier.
i2-4. Advmtages g&ed b,v fins. Afta it has
Petro/Chem Engineer January 1966
been determined that fins will increaue the beat flow in accordante nith inequality, Eq. 12-25, the fina1 ‘selection depends on which of the possible savings, in cast, wright or space, is the deciding factor.
The savings in eost and space can be determined best by comparing two alternate designs, one with fins and the other without fins.
Swing in weight. The problem of exchanging the ~greatest amount of heat wi~th the least amount of weight of the beat, exchanger can be solved analytically by the following procedure. The weight W of each fin, with the designations of Fig. 12-5, is:
W = bZLw = AILw Eq. 12-26
where w is the specific weight of the fin materia1 and AI is the cross-sectional area norma1 to L. The length L is given, and the fin dimensions b and 1 must be selected to give a, maximum beat Aow q1 for a giva area Ar = bl. By introducing m. =
\,‘2h./kb, and I = AJb in Eq. 12-20:
This hecomes a maximum for dq/db = 0. Intro- ducing the designation:
Differentiating Eq. 12.27, setting the different,inl quotient equa1 to zero, results min:
Solving this equation by trial and error giva for the dimensionless number u, a value of 1.419. Thns, the maximum heat flow through a fin of a giva
weight is obtained if 1x/-L> = 1.419. This re- lationship may be presented in the form:
21 - = 1.419 ,/ll Eq. 12-30
b
The ratio of height to half the thickness is a function of the same characteristic ratio (2k/hbj that enters into inequality Eq. 12-25 determining the usefulness of fina.
The temperature differente b%tween the tip of the fin and the surrounding gas is obtained from Eq. 12.18:
For an exchanger in operation, measuring the temperatures Atl and Atl will show whether the fins have the optimum height.
Fin eflectiveness. Far a heating surface with fins, the heat flow from the fins is, by Eq. 12.27 and Eq. 12.28:
q1 = L,/2h,kb Atl taniru Eq. 12-32
If the surface has no fins, the heat Aow from the
53
Figure 12-3, Effectiveness of straight fins.
area equal to the fin base is q’ = hbL ntl. The ratio of the. beat flow qi to q’ shows the effective- ness of the fins:
Eq. 12.33 determines far giva h, k, and b, the maximum possible beat-flow inerease by the addi- tion of fins to the area coverai by them.
1.2-5. Eficiency of fin surface. In discussing the performance of fins it is useful to introduce the concept of effectiveness of the fin surfaee.’ This term is defined 88 the ratio of the amount of heat actually dissipated with and without fins when the smne film coefficient h, is used and the fin surfaces are assumed to be at the temperature of the fin base. Using Eq. 12-32 for the heat actually dis- sipated, the effectiveness 7, after reducing the terms and using Eq. 12-28, heeomes:
Ld2hkb Atl t.anh u fAm7l~ 21 ?1= 2Lth At,
- ~ Eq. 12-34 ‘L‘
This equation is represented by the lower curve vV in Fig. 12-3.
Required area. The optimum cross-sectional fin area +t,, required for a giva beat flow ql, is bob- tained from Eq. 12-28 and Eq. 12-32 by mtroducing the numerica1 values tanh ?c = 0.889 and u = 1.419 and eliminating the thickness b. Under these conditions :
AI = g($$. Eq. 12-35
Consideration of Eq. 12-35 indica& that it ie advisable to make fins as urna11 as possible and increase their number. Far example, in order to doublethe~heat flow vl, it is better to use two fins of the origina1 size and area, A,, than t,o use one fm with an eight time larger section and weight.
Eq. 12-35 also afforda a means of comparing different materials far use in fins. It shows that the cross-s&ional xea Al is inversely propoitional to the heat eonductivity k. Hera, the ,fin weight is proportional to w/k. Table 12-1 giva average values of heat conductivity st 212 F and of weights of various metals and alloys and permits a direct comparison on the basis of w/Jc ratios. The right column gives a eomparison with the weight of aluminum fina taken as 1. It shows that by using aluminum instead of copper, it is possihle to save 78% in weight in spite of the higher heat con- ductivity of copper. Of the ferrous materials, cast iron is better than carbon steel, and stainless steel on the weight basis; should never be used for fins.
12-6. Va~iotcs fin shapes. The fin with a rec- l tangular section is the simplest shape available and is ued quite extensively. However, due to u gradua1 heat dissipation by the sides of a fin and due to R decrease of the fin temperature, the heat flow toward the fin end decreases. Under such eonditions a betta utilization of the fin materia1 will be ohtained, if the thickness of the fin decreases toward the end. The calculation for other fin sections is eonsiderably more involved and only the result,s will be giva here.2
Triangulw fG1z.s. For an optimum triangular fin, Fig. 12-4, the ratio of the height, Z, tp the half- thickness b/2 at the base is giva by:
21 b
= 1.309 4% Eq. 12-36
which is smaller by 22.576 than the same ratio for a rectangular fin, as giva by Eq. 12-30. The temperature differente on the fin tip is:
Ata = 0.277 At, Eq. 12-37
which is smaller by 40.6% than the temperature differente for a rectangular fin, by Eq. 12.31. The
TABLE 12-1. Comparison of Materials Far Use in Fins
ratio of the thickness bt of the triangular fin at its~base to the thickness b, of the rectangular fin with~ equa1 beat, flow is 1.31. However, ,the:ratio of the cross sectional areas is 1/1.44, therefore the triangular fin requires 44% less material.
The efficiency vt of a triangular fin is also slightly higher than that of a rectangular fin and is show by the eorresponding curve in Fig. 12-3.
Fin of sm.aZZest weight. The determination of a fin having the smallest weight for ,a given heat flow is of considerable practical interest. In sueh a fin, every part should be utilized to the same extent, which means that the specifie rate of beat flow must be constant throughout the fin height.3 The beat-flow lines, therefore, must be equally spaced straight linea, parallel to a’ plane passing through the center line of the fin, as shown in Fig. 12-5. For a constant specific beat-flow rate the temperature decreases linearly along any flow line from tl at the base of the fin. The limit of utilization of the fin will be reached if the tip temperature equa18 the temperature t, of the sui= rounding gas, as shown in the upper part of Fig. 12-5. The differente between the fin temperature
% expressed 8s: t at a distance z and the gas temperature t, may
t, ~ t, = (tl - tn) $. Eq. 12-38
ConsideT a surface element of the fin at the distance % which forma an angle OL with the center piane. The specific beat-flow rate through this element is q siri 01, where Q is the specific beat-flow rate in the direction of the fin height. This heat flow must be equa1 to the heat 1~ (t. - ta) trans- ferred to the gas, or:
q siri 01 = h (t, - t,) Eq. 12-39
Introducing Eq. lo-38 for the temperature differ- ente, results in:
Sind _ h(t1-G :z Eq. 12-40
) :@yonstallt .alu,c The fraction on the right side of Eq. 12-40 has a
Therefore, the contour of the mvestlgated fin is a circle, because a circle is presented by the equation sina = x/?’ which is identica1 with Eq. 12-40 if T = qdh (tl- tu). HOW- ever, it is not necessary to make the fin with an infinitely thin edge shown in Fig. 12-5. Any pakt of the cirele can be used to form the contour far a fin with a constnnt beat-flow rate, us shown by dashed lines.
The ~difference in weight between a fin with a circular contour and a triangular contour is small. The triangular contour is easier to make, therefore, practically it may be considered as the best shape.
12-r. Cirmmferential fina. Circumferential fins attached to a cylindrieal surface cannot utilize the fin materia1 as effieiently 8s fins norma1 t0 a fiat surface.4 The efficiency Q of a eireumferential fin decreases with an incresse of the relative height of the fin as expressed by the ratio Y = UZ, + Z)/R,, where RI, is the radia1 distante of the fin base from
Petro/Chem Engineer January 1966
Figure 12.5. Fin with smallest weight
the axis of the cylinder and 1 is the height of the fin. Numerically the effeetiveness r), can be ex- pressed by the eficiency 7v of a reetangular straight fin as:
q = T,~ -- AT, . Eq. 12-41
where the value vr is giva by Eq. 12-34 or Fig. 12-3, and the correction APJ may be taken from Table 12-Z.
TABLE 12-2. Correction Values Aq For Circumferential Fins
Special. equation There is available a more dire& procedure for eomputing beat 108s from a medium- diameter cylindrical surface with fins to an air stream.” This procedure eomists in finding a special overall heat transfer coefficient U’ referred to the outside surface of the cylinder A and com- puting the heat flow from the equation:
Q = A U’(t,, - t,i Eq. 12-42
where t, is the temperature of the cylinder wall, t, is the temperat,ure of thn conling air and U’ is found from the following equation expressed in units and symbols used herein:
Figure 12-6. Film coefficient for various fin spaces and ai,-stream "docities.
where: 11, = a film coeflieient to be taken from Fig. 12-6
,T = the space between t,wo fins, oi s + b is the pitch of the fins
»i = determined by Eq. 12.22 I = the height of the fins
D, = the diameter of the cylindricnl surface
The flow of air between the fins is retarded by their frictional resistance; therefore, the film coefficients giva by the cuives of Fig. 12-6 are considerably lower, than formed far smooth cylin- ders without fins.”
,f%S. Combimd, convection and .radia~t%on. In some practical problems, where heat is lost simultane- ously by convection to the air and by radiation to the enclosing surroundings, it is convenient to take advantage of the fact that sir does not absorb
radiant energy at the temperature met in in- dustrial practice. As a result, the effe& of con- vection and radiation are simply superimposed and the combined beat, losr. q mny be presented as:
q = A(h., -II IL,) (tl ~ tl Eq. 12.44
where t, is the temperature of the hot surface, t2 is the temperature of the air and of the sw- roundings.
For a body completely enclosed by the surround- ing surfaee, F = 1. If the surroundings are as- sumed to haue the properties of a black body, F, = c, where e is the emissivity of the body. In this case it is giva by:
h,. = 0.00696e (&)" '. Eq. 12-45
where T,, is the average of the temperature TI of the surface and of the temperature TP of the en- closing walls, T,,, = (Tl + T%j/z. Eq. 12-45 giva a reduced value forh, but if T1 5 1.5 Ti, the error is less than 4%.
Sinzplified. procedwe. For preliminary calcula- tions when the beat-loss conditions, particularly th
9 temperature differente, are not known with any accuracy the combined coeffieient (h, + h.,,j for still air may be taken directly from Table 12.3. Values in Table 12-3 are determined foti bare steel pipes and flat surfaces. Sinee values~ of F for insulating materials do not differ much from e ,from oxidiaed steel, they may be used also for insulated pipa.
For small temperature differences, BS exist in buildings, the experiment,al values of the combined coefficient (!l., -1 h,) for still air in cantact with various materials are :7 cement plaster finish, 0.9; concrete or corkboard, 1.3; brickwork, sheet as- bestos, or finished wood surface, 1.4; magnesia board or window glass, 1.5.
For air in motion, the above combined coeffi- eients should be multiplied bg a ‘factor taken from Table Y-4.%
TABLE 12-3. Combined Coefficient (h, + h,) in Stili Air At Roo,m Temperature, Btulhr-sq ft-“F
Loss of beat from the sorfaee of a rotating cylinder to air, BS in flywheels or armatures, can be calculated from values giva in Teble 12.5.”
EXAMPLE 12-2. Calculate the heat loss per linea foot from a 5.in., horizont.al bare standard steel pipe whieh carries saturated steam at 10 psig and is exposed to stili air at 68F.
The temperature differente, t = ) .,,:.:‘u”“:= 171F The outside diameter, D =
5.56 in. Interpolatinp data from Table 12-3 gives:
= 2.4G Btu/hr-sq ft-“F as compared’ with Ir,, = 1.04 Btu’hr-sq ft-“F found in Example 9-l for conoection alone. The outside area of 1 ft, of the pipe A = v x 5.56/12 L 1.455 sq ft. The beat, 10s is:
g = 1.455 x 2.4F x 171.5 = 614 Btu/hr
12.9. Combined conduetion, convection, and mdia- tion. When a surface is insulated, the beat mast pass through the layer of insulation by candurtion before being dissipated by convection and radia- tion.
Consider the important case of an insulated pipe shows in Fig. 12-7. Let t, br the temperature of the Dioe surfaee. %. the temnernture of the outside
temperature of the air. The a,mount of heat con- ducted radially from the surface 1 to surfacr 2 and is equa.1 to the amount dissipated through con- vection and radiation and giva by Eq. 12.44:
Often t,he temperature t2 of the ootside surface is not known. The simpiest way of finding the heat losi by Eq. 12-46 is to use a trial-and-error pro- cedure for finding the temperature tr and then to use Table 12.3., For this procedure it is convenieut to present Eq. 12-46 in the form:
EXAMPLE 12-3. Find the heat loss far the, pips of Example 12-2 if the pipe is covered with V!&in. insulation that has a thermnl conductivity Ii = 0.035 Btu/hr-ft-“F.
SOLUTION: The inside d&meter, D1 = 5.56 in., and the outside diameter. & = 8.56 in., or 0.713 ft. Substituting al1 known quantities in Eq. 12-47 results in:
Assume first that tz = 90F, then Eq. rl giva it,,: + h., = 1.54. Extrapolating ill Table 12-3, the Lione far the nearest diameter Dl = 8.63 to t = 22F gives h.,: + h,, = 1.66. This shows t,hat ta must, be slightly lover. Assume next, tY = 89F. Then Eq. a giva h, + h,- = 1.63 and from Table 12-3, Ir, + il,, = 1.65. A further decrease of t% will make the differente in the opposite direction and lager. Thuti t2 = 89F and from Eq. 12-4G:
y’ = 1.G3 x 7i x 0.713 (89.68’1 = 76.ti Btu/hr
References
ID. R. Harper and W. B. Brown, “Mathematical Equations for Heat Conduction in t,he Fins of Air-Cooled Engines,” Nat. Adv. Committee for Aeronauties, Technical Report No. 158 (1923) pp. l-30, 2.
3E. Schmidt, loc. cit., p. 948. ‘Hnrper and Brown, loc. cit., p. 2. “A. E. Biebermnnn and B. Pinkel, “Heat Transfer
from Finned Meta1 Cylin,der in an Air Stream,” IVACA Repwt No. 488 (1939i pp. 15 and 11.
“Ibid, p. 11. rA. C. Willard and 1,. C. Liehty, “A Studi of the
Heat Trnnsmission of Building Materials,” Univ. of Illinois Bulletin, No. 12 (Nov. 19, 1317).
BA. C. Willard and L. C. Lichty, loc. cit. “Marks’ Handbook, op. cit., p, 378.
I Part 13, “Overall Heat Transfer,” will appear in the February issue. l
Petro/Chem Engineer January 1966
PROGRESS IN HPITECHNOLOGY
Petrochemicals n A potential answer to the detergent problem in water treatment may be provided by foam fractionation techniques.
Operating on principles similar to those ap- plied in ore flotation, feed water is passed con- tinuously through a unit where it is caused to foam by air introduced through spargers. Dis- solved detergents, along with other contami- nants, both solid particles and dissolved impuri- ties, are concentrated at the foam interface,. The foam is spilled over a baffle or through a pipe, the foam broken and the sludge disposed of. The improved water is drawn off continu- ously ;at the bottoni. 70% removal of ABS is reported. It is suggested that the method may be useful in purification even when only “soft” detergents are present. For a 100 MM gal/day installation, capita1 eost has recently been esti- mated at 1.26 gal/day and operating cast at 1.4$/1000 gai.
w Polymerization of methyl methacrylate in situ by irradiation improves and stabilizes the properties of wood.
Investigators at the University of West Vir- ginia bave processed wood by first saturating with nonomeric methyl methacrylate, styrene or vinyl acetate (other monomers may be used) and polymerizing within the strutture of the wood by exposure to gamma radiation. The wood is vacuum treated to withdraw air and impregnation is carried out under reduced pres- sure. Irradiation is done in an atmosphere of nitrogen, because the effe& of irradition take piace more rapidly in the absence of oxy- gen. Work has been done principally with white
8 In-line chromatographs continuously monitor ultrapure hydrogen at National Cylinder Gas Div. of Chemetron Corp. in Dallas Tex.
With tight limits of 2 ppm Nz, 2 ppm HzO, 0 ppm hydrocarbons and O,, the unit samples from 2400 psi gas which has been purified by ,Palladium diffusion. One analysis is made automatically every 12 min and automatically checked against a standard every 2 hr. The earrier gas is also hydrogen purified by a small Palladium diffuser. This avoids time consuming purges of equipment between analyses. A 5 cc sample is used. The equipment has performed satisfactorily sinee 1962. A similar unit is used ,for monitoring Ns. On the
ed basis of performance another hydrogen unit is going into NCG plant in Barherton, Ohio.
n Modification in method of determining Reid vapor pressure for gasoline can improve aceuracy.
An investigation,at Phillips Petroleum Co. com- pared a number of vapor pressure measurement methods as to correlation with dire& vapor to liquid ratio measurements and automobile ac- celeration performance. A modification of the Reid method increasing air t,o liquid ratio from 4:l to 25:l and test temperature from 100 Fé to 130 F gare the best correlation with direct V/L measurements at 120 F, 140 F and 160 F. It was also found to be suitahle to characterize a fuel with regard to acceleration of automobiles in hot weather. Values determined, were satisfactory for adequately precise blending of fu&, equiva- lent to the standard Reid values.
pine? sugar maple and birch, but redwood, beech, yellow poplar, yellow pine, red, oak, white oak, ponderosa pine and spruce have also been treated.
Hydrocarbon Processing
M Production of synthesis gas from cellulosics is under study as a means to supply synthesis gas in countries which do not hsve ready access to fossi1 fuels but are rich in agricultuml wastes such as rice hulls, bagasse, sawdust, etc.
Recent studies by M. W. Kellogg Co. have demonstrated the feasibility of such gasification in a reactor in wh,ich feed mamrial ~was dropped continuously on a packed bed maintained at 1000 C. In the gasification of wood, for example, the gas formed on steel wool (at heating con- tract surface) contained approximately 50 mol % each of carbon monoxide and hydrogen, with only minor formation of carbon dioxide and methane. In the case of sugar gasificatio
a on the other band, formation of the latter two products was more substantial. Mention might be made bere that one version of a wood gasification’ process has been commercially operated in Frante during the early 1950’s.
January 1966 Petro/Chem Engineer
formation on fluids and gasa iu listed in alphabetical order for eauy referente. Jordax Valve Div., Rich- nrds Industries, Inc.
25 EQUIPMENT for tempere ture and pressure correction in- strumentation is show” in an Spa@ brochure. Detailed are Temcorector types 1, T and W, which are adaptable t,o large ca- pacity positive displacement gas meters. Applieations, specifications, temperature and pressure ranges are listed. Roekwell Mmufactwriny CO.
W6 GENERATOR SYSTEMS- A brochure deseribing the nev 750 kw “Saturn” indust,rial gas,turbine eontinuous duty generator set, in- cluding a discussion of system eco- nomics using exhaust heat recover, is offered by this firm. Dimension- al drawings, curves and other teeh- nieal data are included. Solar Div., International Harvester Co.
27, METALLURGICAL AND Meehanical Prnpertirs of Tit,anium Alloy Ti-679 is the title of a 20. page manual with 19 graphs, 15 microstructure~ photographs and 8 tables. The manual notes that the metal has eharacteristics desirable in applications requiring prolonged exposure to high stress at tempera- tura in the 650-900 F range. Titaniuna Metals Cm-p. of America.
l -..“+& ~~~~~~~.
28 SHOT CLEANING SYS- tems-This 12.page brochure de- seribes shot cleaning as one of the best methods of cleaning tubular air heaters and other boiler sec- tions which operate below the ash fusion temperature. It gives details Y” how the system operate5 and shows what the system has accom- plished in various plants. Diamond Power speciazty cwp.
Ani~rher Unique Verti-Line Pump Advantage . . .
Although there are other verti4 turbine pumps using stainless steel bearing sleeves between the line shafts and their Ruid-lubricated bearings, only orve-“Verti-Line”-has developed stai&% steel sleeves that can be fìrmly locked nnywhere on the shaft without threads, wirhout welding or solder- ing, and without undercutting CIT in any other way weakeninr! the line shaft!
What This Verti-Line Adwncement Means to You. . . ~:‘~::,
OI othenuise interfere with quid<, easy release when replace- ment or re-locadon are desired!
Troublefree bearing sleeve lockr are iust one more of the many unique advancements built into Verti-Line Pumpr -advancemen+s pioneered and develaped ovei more than a holfxentury of leadership in the verti& turbine pump field. May we send you more de+& explaining why Verti- line Pumpr are the atondard of the industry?
13-I. Introtluction. As brought out in the previous part, heat transfer from one medium to another usually takes piace simultaneously by al1 three mechanisms - convection, conduetion and radiation.
4 large number of the problems met in engineer- ing work deal with steady-state heat transfer from one fluid to another fluid separated from .the first one by a solid wall. Generally spesking, the heat from the hotter fluid is conveyed tn the solid wall by convection and radiation, travels through the wall by eonduction, and is transferred to the eolder fuid by radiation and conveetion. Even at low temperatures when radiation can be neglected, there stili remain two different manners of heat transfer to be considered - convection and conduetion.
Far ?hese problems the logica1 procedure would be to ealculate the temperatures on both sides of the wall and, knowing the fluid temperatures, cal- culate the corresponding heat transfer rate for eonvection radiation and conduction. However, for practic+ purposes it is often desirahle to simplify the calculatiions and calculate the rate of heat flow directly from the equation:
q ~ AU(tl - tz) Eq. 13-1
where tu and tl are the temperatura of the two fluida and U is the so-called over-ali beat tmnsfer coefficient, or simply the over-al1 coefficient.
For some practieal conditions the over-al1 eoeffi- cient has been determined experimentally. However, in general it is determined by taking into aecount the separate faetors affeeting the beat flow. These factors are physieal and thermal properties of the fluids, slow conditions of the fluids and thermal and geometrica1 eharacteristics of the wall.
13.2. ‘I’kenr~al Resistance. The application of the ‘over-al1 eoefficient to heat transfer problems is made easier by the introduction of the concept of resistance t,o beat transfer :
R, = 1 7r
Eq. 13-Z
where R, is the resistanee to heat flow by convec- tion. Similarly, R,, the resistanee to heat flow by radiation is :
1 Rv = x Eq. 13-3
56
Over-Ai1 Heat Transfer- Part 13
Applied Heat Transfer Design’ for t,he HPI
by Wayne C. Edmister and Joseph M. Marche110
Since thermal conductance is defined aa the re- ciproeal of thermal resistance, the tot,al resistance to beat flow may be obtained from Eq. 13-1 as:
Eq. 13-4
On the other hand, the tota1 resistance R evi- dently is equa1 to the sum of al1 separate resistances in seria : *~~~
R-RykRgf.<.R,,. .Eq.13-5 ~~~
From Eq. 13-4 and 13-5:
LI= 1
RI -I- Ro -II R,, Eq. 13-6
In design prohlems it is important not only to determine the over-al1 coeffieient U for given con- ~~~ ditions but also to know how to increase the coefficient U for beat exchzmgers and decrease U for parts thst must be protected from heat 108s or heat gai”. Depending upon the seleet,ed condi- ~~ tions, the values of the different resistanees RI, Rs can be vsried by design modifications. According :~ to Eq. 13-6, in order to change the value of U :~ : appreciably, it is necessary to change those resi& _~ anees t,hat have the higher values. Thus, in design- ~~ ing a heat exchanger between a liquid and a gas, ~~~ the resistance to eonductivity ususlly is small com- pared to the resistances to convettive beat AO a from and to the two fluida. Therefore, changing the meta1 wall from steel to brass which has a much higher conductivity, or decreasing the thick-
Figure 13.1. Heat transfer between two fluids.
February 1966 Petro/Chem Engineer
ness of the wall, will not change U appreciably. Also, since the film coefficient of a liquid is usuaUy much higher than that of a gas, the over-al1 coe- fficient can be increased primarily by deereasing the thermal resist,ance between the gas and wall, i.e., by inereasing the gas-film coefficient.
13-3. Over-dl eoeffrcient. An expression for the over-al1 coefficient U can be derived for heat trans- fer from a fluid with a temperature, t, to u second fluid having a temperature td. The physical situation is shown in Fig. 13-1, where tz and ta are the surface temperature6 of the wall and z is its thickness.
The flow of heat from the hotter fluid to a unit area of the hotter wall surface is :
* = hi(tl - b2) Eq. 13.7a
ahere hl is the film coeffieient on the hot side. At steady-state conditions the same amount of heat will flow by conduction through the same area to the other side of the wall:
k q = Z(ts ~ t3) Eq. 13.7b
where k is the thermal conductivity of the wall
8 material. The flow of heat through the same area to the other fluid is:
q = h:s(ta ~ tr) Eq. 13.7~
where ha is the fdm coeffieient on the corresponding wall surface. On the other hand, the same heat flow may be expressed by Eq. 13-1 as:
q = U(t, - ta) Eq. 13-8
where U is the over-al1 coefficient of heat transfer. Substituting for Q in Eq. 13-S the three temperature differences from Eq. 13-7~1 to 13.7~ may be written ZJ:
tl - ta = u (tl - ti) Eq. 13-9n 11%
tz-tj=+t-tnl Eq. 13.9b
t:, - t4 = $(tl _ td Eq. 13-9~
l . 3 Addmg these three Eq. 13.9a to 13.9c gives:
Eq. 13-10
from which:
LI= 1
&+G+&
Eq. 13-11
For a case where there are more than three resistances, Eq. 13-6 may be applied dire&. In the case shown in Fig. 13-2 when the beat-exchanger wall is covered by scale, U may be expressed as:
13.4. Air spcms. When ai wall has air spaces. its
PetrojChem Engineer February 1966
thermal resistance is increased by a film coefficient at each su-face facing the inner air space. For a narrow air space, beat is earried aeross the space by convection currents and by radiation. The re- sistanee of such an air space may be expressed approximately as the sum of the two surface re- sistanees, or as Uh,. Fig. 13-3 shows such a wall for whieh the over-all coetlicient is:
Eq. 13.13
For relatively large vertical air spaces with the distane between the walls s > 0.125 ft, h.r may be determined from earlier discussion and depends on the height of the air space.
Enclosed Air Spaces. For narrow enelosed air spaces, as encountered in hollow walls and double windows, Eq. 13-13 may be modified by introdueing the resistance of the air space l/C, where C is the conductance:
u = ~~i---~.~~~~~,~, 1 1
or 1 ‘. Eq. 13.14
-+x+c+c+1L, h.2
While convection~ decreases with a decrease of the air space thickness, from 11/ in. down, the total beat transfer coefficient inereases due to an in- crease of heat transfer by conduction through a thinner layer of relatively stagnant air.
The effwtivanasti oi &n air space used for heat insulation can be increased by cutting the space up both vertically and horieontally. Sectioning the vertical height reduces the eonvection currents by limiting the air buoyancy differente. Subdividing the space horizontally into severa1 narrower spaces by layers of aluminum foil or thin sheets of other
Figure 13.2. Heat transfer through a composite wall.
Figure 13.3. Heat transfer throogh wall with air spa&.
61
bright material, parallel to the main vertieal sui- faces, decreases both convection and radiation.
EXAMPLE 13-l. Determine the over-al1 heat trans- fer coefficient of a brick wall 8 in. thick. The inside surface is beat insulated by celotex sheetg 7/16-in. thick fastened to the briek wall with u ?-in. air space and covered with u ::/b-in. layer of plaster. The room temperature is 70 F, the outside air temperature is 10 F. Assume a nind, velocity of 15 mph.
SOLUTION: In order to facilitate the working of the problem, the main data are shown in Fig. 13-4. Designate the outside surface coefficient by FI,%, the inside surface coefficient by ho, and the con- ductanee of the air space by C. The thermal con- ductivity of briek rnay be taken as k ‘= 0.40, assuming 1% moisture. The thermal conductivity of eelotex is: k = 0.028. The thermal conduetivity of the plaster may be taken as k = 0.40. to be on the safe side. With these data, the equation for computing the over-al1 coefficient is similar to Eq. 13-14, only for composite walls on both sides of the nir space:
u= 1 0.0304
+0.028+
Eq.~a
The value of hl may be determined from the observation that at an air velocity of v = 15 x 5280 = 79,200 fph, the flow is turbulent. k = 0.0135 Btu/hr-ft;“F, p = 0.0845 lb/cu ft, and ,L = 0.040 lb/ft-hr, and taking for L the limit value L = 2.0 ft, ,gives:
0.m 0.0135 (2) (79,200:)
hz = 0.055 ~ [
i:o.o84oj
2.0 0.040 1 = 5.15 Btu/hr-sq ft-“F
The value of the conductance C may be found by Eq. 8-29 (see Sept. 1965, p. 63). As a first approximation t, may be taken as an average between the outside and inside temperatures. How- ever, Fig. 13-4 shows clenrlg t,hat the ?esistance to heat flow from inside is much greater than to the outside; therefore, it is advisable to loner the average temperature, sxy by ZOy&, or:
t, = (70'.l- IOj (0.5) (0.8) = 32 F
and :
C = 0.009/0.052 -I- 0.09 \/32 -II 90 = 1.17 Btu/hr-sq ft-“F Eq. b
If lata it is found that t, differs appreciably from the assumed value of 32 F, the value of C should be recaleulated.
The value of 11.6 may be calculated from eorrela- tions. Unfortunately, there is no sound basis for estimating the temperature At, in this case (70 - te). The simplest way is to entimate this tempera- ture, say as At = 20 F compute hs, find the value of C and the corresponding heat flow 9 by Eq. 13-l ;
62
Figure 13.4. Data for wall of Example 13-l
the temperature differente At is then found from v and from it the approximate value of hr,; t,he value of hg is then recalculated with the value of At found as a first approximation and a new, cor- l rected value of U is cnlculatcd. With At = 20 F:
hs = (0.23) 20% = 0.625 Btu/hr-sq ft-“F Substituting the calculated values of h2, C, and ?L, in Eq. (ai gives:
u=-- 1 0.20 + 1.67 + 0.85 + 1.30 + 0.08 + 1.60 1
=5.70 = 0.175 Btu/hr-sq ft-“F
By Eq. 13-l:
Q = 0.175 (70 ~ 10) = 10.50 Btu/hr-sq ft
This hent flows through every separating surface and the temperature differente, with the approx- imate value of hG, is:
70 - ti; = 10.50,‘0.623 = 16.8 F Eq. c
The corrected value of the over-al1 coefficient is now changed to:
1
1 5.79
- 0.173 Btu/hr-sq ft-“F Eq. d
and the new value of Q is:
rl = 0.173 (70 ~ 101 = 10.38 Btuj’hr-sq ft
This giva a new temperature differente:
70 - ta = 10.38iO.591 = 17.3 F
This is suffieiently close to the previously assumed value of 17.0 F and no further adjustment of the value of li.6 is r~ecessary. However, before consider- ing the value of L’ = 0.173 found by the Eq. (di) us final, the value of C should be checked by finding the temperatures ts and tp. At the same time it is advisable to check the correctness of U by finding tl. The proee&~re is based on the eonstsnt flow of
February 1966 Petro/Chem Engineer
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beat, as used in Eq. (c). With t6 = 70 - 17.3 = 52.7 F:
tj = (52.7 - 10.38) 0.08 = 51.9 F t4 = (51.9 - 10.38) 1.30 = 38.4 F tp, = (38.4 ~ 10.38) 0.85 = 29.6 F t2 = (29.6 - 10.38) 1.67 = 12.3 F tl = (12.3 - 10.38) 0.20 = 10.2 F
The temperature t1 = 10.2 F is sufficiently close to 10.0 F m that this check may be considered as satisfaetory. Now the conductance C, Eq. (b), may be checked. The man temperature t, = (t4 f td/2 = (38.4 +, 29.6)/2 = 34.0 F, whereas it was as- sumed as 32 F. The differente is SO mm11 that it will not affect the value of conductance C. Thus the value of U = 0.173 Btu/hr-sq ft-“F may be taken as the fina1 answer.
Accwacy. It should be remembered that the aecuracy of the over-all coeficient eannot be very high because SO many factors enter in, the caleula- tions and eaeh factor usually earries a probable erro* of i-5 to 10%.
13-5. lJnequ.aZ surfaces. The foregoing discussions dealt with piane walls with equa1 surface areas on both sides. When the outer and inner surfaces are not equal, BS in a box with relatively thiek walls or in a pipe with thick walls, the value of U~must be applied to a square foot of either the outer or inner or mean surface. If the ,subseript o is used for the outside terms and the subscript i for the inside terms, A, is the area of the outside surface, Ai is that of the inner surface, and A, is the area of the mean surface. Then the value of U, referred to the outer surface, may be presented by modifying Eq. 13-11 as:
u”=AD. ;, 1. Eq. 13-15 0
A& +Yi;j;+x
In Eq. 13.15 the thermal resistance of the inner surface, l/hi, is multiplied by A,/A< to take into account the greater resistame to heat flow of the smaller inner surface. Likewise the term x/k, thermal resistance to eonductance through the materia1 of the wall, is multiplied by the ratio A,/A, to take into account the greater resistance ,of the smaller mean area as compared with the outside area. For & thick-walled cylinder or pipe, the ratio A,/Ai = D,/Dl. Substituting far the mean area results in :
u, = 1
2% + g In L!!? + .,+ Eq. 13.16
D,$Li 0
However, if A$A1 < 2, the arithmetic mean area A, = (A, + Ai)/2 may be used with suffi- ciant aeeuracy and Eq. 13-16 then beeomes:
u, = 1 Eq. 13.17
DO D;hi
+ +(D, - DJ ~+ ,,JJ (Do + Di) k lf.,,
If it is desired to refer U to the inner surface, U, from Eq. 13-17 must be multiplied by D,/Di, or Ui may be obtained direetly from the equation:
64
EXAMPLE 13-2. Superheated steam at 450 psia and 800 F flows at a velocity of 12,000 fpm through a 6-in. horieontal extra-strong non-insulated steel pipe suspended in a large tunnel with air at 100 F. Calculate the over-al1 coefficient U, referred to the outside surfaee of the pipe.
SOLUTION: The outside~diameter of the pipe is D, = 6.625 in., inside diameter Di = 6.761 in. The ratio D,/D; = 1.15 and therefore the simpler Eq. 13-17 may be used. The data are:’ k = 0.0443 Btu/hr-ft-“F; Di = 5.761112 = 0.48 ft; the specific volume of steam v = 1.6074 CU ft/lb and therefore:
G = 12,000 x 60 1.6074
.-- = 404,800 lb/hr-sq ft
Viscosity p = 7.09 x lo-’ Ih sec/sq ft or, fi = 7.09 x 1O-i (115,813) = 0.082 lb/ft-hr; specific heat c = 0.54 Btu,ilb-“F. From these dat,a and Eq. 6-14 (see May 1965, p. 74) :
= 307 Btu/hr-sq ft-‘F.
The temperature of the pipe can be assumed to be equa1 to the temperature of the steam. The thermal conductivity of mild steel at 800 F is k = 23.6 Btu/hr-ft-“F.
The outside beat transfer coefficient k, is the. result of the simultaneous action of free convection and radiation, or h, = h, + h,. In order to obtain the modulus a the temperature of the air film may be calculated using for a first approximation the steam temperature as the pipe mrface tempera- ture, or:
t <” = 100 + (800 - 100) x 0.25 = 275 F
and hence a = (0.365) 10”; L = D, = 6X25/12 = 0.552 ft. The approximate temperature difference~# is 800 - 100 = 700 F. Thus the term:
No, x Nrr = aL3 At = (0.365) 10” ,(0.552”j (700) = (4.3) 10’
Therefore C = 0.45. For t = 275 F, the conduc- tivity of air is K - 0.0198:.
h, = 0.45 gg (4.3) (107) “.X
= 1.31 Btu/hr-sq ft-,“F.
For smooth, strongly oxidized steel, using t = 800 F, by interpolation, e = 0.84. Since the tunnel eneloses the pipe completely, F = 1. For a first approximation T1 = 800 + 460 = 1260 R and T, = 100 + 460 = 560 R. Thus:
The hbat flow per lineal foot of the pipe is, by Eq. 13-I:
q = ,r (0.552) (6.08) (800 - 100) = 7,380
Btu/hr Using this value the temperatura of the inside and outside pipe surfaees may be determined using the pertinent areas :
800 - t,j = 7,380/(r) (0.48) (307) = 15.9 F.
In determining the temperature drop through the pipe wall thiekness the avenge area must be used:
A = ?i (0.552 - 0.48j x 0.5 = 1.62 sq ft
and with the wall thickness z = (0.552 - 0.48)/2 l = 0.036 ft,
t, _ t = 7,380 x 0.036 ” ~~,,.- = 7.0 F
1.62 x 23.6
With these data the average temperature of the pipe wall becomes 800 - 15.9 - 7.0/2 = 780.6 F and this giva /c = 23.9 and:
t; ~ t,> = 7,380 x 0.036 = 6,g F 1.62x 23.9
Thus the outside pipe temperature becomes:
t, = 800 - (15.9 + 6.9) = 777.2 F
The corrected film temperature is:
t,, = 100 + (777 - 100) x 0.25 = 269 F
By interpolation the correeted modulus is found as a = 0.38 x 100 and aL’ At = (0.38) 10’ (0552” (777 100) = (4.33) 107 By interpolation, the corrected conductivity of the
l ’ au at 269 F is k = 0.0197 Btu/hr-sq ft-“F and:
Ir.,: = 0.45 gg; L(4.33) (lo’)] “45
= 1.30 Btu,/hr-sq ft-“F
The corrected radiation coefficient, with T, = 777 i- ~460 = 1237 R:
,~~, = +713 x 0.84 ([12.37]4 [5.6]“) 777 - 100
= 4.78 Btu/hr-sq ft-“F
and :
11, = 1.30 + 4.78 = 6.08 Btu,‘hr-sq ft-“F
With these corrected values Eq. 13.17 giva:
LT0 1 = -.. 1.15 0.552 x 0.0’72 1 307 + 1.03 x 23.9 + 6.08
= 5.89 ,RlhL;~h,-sq ls”F
Thus the seeond approximation of the surface tem- perature resulted in a lowering of U, hy less than 4%. A third approximation would give a very slight increase of U,. However, this increase can be only a small fraction of the per cent found above and therefore, much below the probable accuracy of the final result. Therefore additional calculations are not justified.
Composite umll. Consider a wall composed of severa1 layers of different materials of appreciable thickness, as in the case of a pipe covered with beat insulation. The middle term in the denominator of Eq. 13.15 whieh expresses the resistance to con- duetion, must be repeated for each layer using corresponding values of k and of the inner and outer diameters D,i and Da.
13-L;. 3’oding. In operation of heat exchangers, dirt, scale, oil or other foreign materials are de- posited on the tube surfaees, and products of COT- rosion also may be formed. Such deposits are re- ferred to as fotdhg. They ,increase the thermal resistanee of the tube nalls and thus lower the over-all heat transfer eoefficient.
The fouling resistance is directly proportional to the thickness of the deposit and inversely propor- tional to its thermal eonductivity. Fouling is pro- duced by most fluids, liquide., vapors, and gases. Figure 13-5 may be used to evaluate the fouling resistance R, of some eommonly~ encountered de- posits.
The influente of fouling upon the over-all coeffi- cient’ U may be taken into account by eombining Eq. 13-6 with Eq. 13-4, RI = l/U. Designating the resistance of the fouling film by R, the new over-al1 eoefficient~ U’ may be found from the expression :
U’ = 1 1,X’ -l- R,
Eq. 13-19
As may be seen from Eq 13-6 and 13-19, the influenee of an additional resistance in the denomi- nator depends upon the sum of the other resistances. Far a small combined resistance and hence a high
- -~~~. TABLE 13-l. Thermal Resistances of Scale Deposits
From Water R. Hr-Sq Ft-” FIBtu
source and cordition 0, heatad watsr
Distilled water 0.0005 0.0005 0.0005 0.0005 5ea water 0.0005
over-all coefficient U, the addition of a eertain fouling resistame will affect the coefficient U much more than the same fouling resistance will affect a~smaller U which has a larger combined resistance.
EXAMPLE 13-3. Compare the relative influente of a fouling resistance R, = 0.005 hr-sq ft-“F/Btu upon two clean tubes which have (a) U = ZOO, and (b) U = 50 Btu/hr-sq ft-“F.
SOLUTION: (a) l/U = 1/200 = 0.005 and by ~Eq. 13-19,
1 u’ = 0.005 i- 0.005
=lOO Btu/hr-sq ft-“F
(b) l/U = 1/50 = 0.02 and by Eq. 13-19,
, u
1 ~ o.oi + 0.005
- = 40 Btu/hr-sq ft-“F
The same fouling resistance R, = 0.005 reduce6 the clean-tube, over-al1 coefficient U by 50% in case (a) and by only 20% in case (b).
TABLE 13-2. Thenal Resistances of Deposi& -- From Some Industrial Fluids
bottoms, over ZOoAPI Refrigerant wpors. dusty air Gas-oil feed over 500F. vegetable oils %3 Naphtha feed over 5OOF. quenching oil 0.004 Fuel oil, residua1 bottoms, below 2O”API 0.005 Cokeaver and other manufactured gas, diesel exhaust 0.01
66
TABLE 13-3. Ovwall Coefficients for Building Constructions
Type Of construction u. [ ,,;.:;fi.oF~
Brick vaII, 8 in. thick. plain 0.50 Brick wall. 8 in. thick. % in. plaster inside 0.46 Brick wall, 8 in. thick. furred, plastered inside 0.30 Brick w.4, 12 in. thick, plain 0.36 Concrete wall, 6 in. thick. solid. plain 0.79 Concrete wall. 6 in. thick. furred and plastered 0.39~ Frame wall. plaster inside typsum sheathing. wood siding 0.31 Frame waII, plaster inside, wood sheathing, stucco outside 0.30 Hollowtile wall. 8 in. thick, furred. plastered. stucca outside Ceiling. plaster on wood lath, no floor above
;:X?;
Ming, ,plarter on wood lath. 1 in. pine floor above 0.28 Ceiling, plastered 3% in. rock wood fili between joists 0.08 Roof, wood shingles on wood strips, no Ceiling 0.46 Window, ordinary Window, double or daub+glared i:E ~0
In service the thickness of the fouling deposit on beat-exchanger surfaces increases gradually up to a certain limit which is still permissible and is called nownal. fouling. The amount of norma1 foul- ing depends upon the fluid and varie8 from Rf = 0.0005 for distilled water or for sea water below 125 F, and up to Rf = 0.01 for dirty river or pond water at low velocities and higher temperatures. For various industrial oils Rf varies from 0.001 to 0.005; for lubricating oils it is much higher, as shown in Fig. 13-5.a For manufactured gases and far diesel-erigine exhaust R, is about 0.01; for organic vapors and clean steam, R, = 0.0005, and for refrigeratory vapors and air Rf = 0.002. More data are given in Tabler; 13-1 and 13-Z.
19-r. Values of ouer-dl coefGwLt. Values of ll for various components of building constructions
February 1966 Petro/Chem Engineer
made up of different materials and thicknesses have been calculated and are available in the tech- nical literature~in the form of tablmJ These tables pive rather ‘conservative values far heat losses to air but have been ehecked experimentally and may
be safely used in designing buildings. A few repre- sentative values for an outdoor wind velocity of 15 Ales per hour are reproduced in Table 13-3. Far other combinations of materials the values of U may be determined by the methods illustrated in Examples 13-Z and 13-3.
Over-a.11 cuellicienls of beat trsiufer for meta1
-.
TABLE 13-4. Over-AH Coefficients For Various Heat Exchangers
Type of Heat Exchanger
Air Preheaters 1-4 Ammonia condensers. atmoapheric. drop type 125.200 Ammonia condenaerr, double pipe 150-250 Ammonia condensers, sbell and tube 150-300 Brine coolers, double pipe 150-300 Brine coolers, Shell and tube 90-100 Brine to freezing water in canE Caoling ccils, brine ta rtill air Cooling CO+ boiling refrjgerant to aid
l ~;;;%%?:5 ta “r
2.5-5 2-2.3 4-8 5-9 z-10
ZOO-1500 Oil coolers. water circulation. transverse flow 50-150 Steam boiler tubes 3-13 superheaters, c,ose to turnace 8-15 Surface steam condensers 200-1000 Water coolers. Shell and coi1 15-25
Petro/Chem Engineer February 1966
tubes and surfaees as encountered in seine power- plant and refCgeration equipment are given in Table 13.4. However, the values for i7 in Table 13-4 must be considered only BS limit values for the different conditions. The coefficient U depends upon many variable faetors, such as velocities and temperatures of the fluida, cleanliness of the mi- face& and shape and arrangement of the elements that influente the degree of turbulence of the flow of fluida. Therefore, the data of Table 13-4 may be used only for rough estimates. If the factors affecting heat transfer are known or specified in a new design, the values of U should be calculated using the methods out,lined in the preceding sections.
References
‘J. H. Keenan and F. G. Keyes. TI~ermodymmie Properties of Stemn. New York: John Wiley and Sons, Inc. 1936.
‘Standards of Tubular Exchanger Manufacturers Association (abbreviated TEMA) 3rd ed., ,New York (1952) p. 83.
3Heating, Ventilation and Air Conditiming Guide. Published each year by the ASHVE, New York. Giva a very complete tabulation of values of U as encoutitered in praetice computed on the assump- tion of dry materials and an outdoor wind velocity of 15 mph.
Part 14, “Over-All Heat Transfer Applica- tions,” will appear in the March isbe.
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Natura1 Gas Odorizing’s New Plant Uses
Direct Synthesis for Mercaptans
Drum-filling equipment with scale provides accurate weighing of each drum of prodoct shipped.
n Who would think it “ecessary or eeonomic to make ethyl mercaptan from hydrogen sulfide and ethylene? Natura1 Gas Odorizing didn’t 20 years ago when they first separated mixtures of mer- captans and hydrocarbons obtained from oil refinery chemieal treating. But with growth of hydrogen treating in refineries, mercaptan mixtures have bee” in short supply and the price has increased. SO they developed their ow” methods of direct synt,hesis for ethyl mercaptan and heavier mercaptans that con- stitute their full line of odorants.
The first problem met by Natura1 Gas Odorieing was location. They were too far from sources for the two main raw mnterials, hydrogen sulfide and ethylene, to receive them economically by pipeline. Nor was receipt in t,he liquid form economie, and more plant capacity and greater flexibility was “eaded for their relatively seasonal products. SO “Mohammed had to go to the mou”. tain,” in this case, Humble Oil & Refining CO.‘s
68
refinery at Baytown, Tex., 20 mila cast of Natura1 Gas Odorizing’s old plant in Houston. The “ew loeation is adjaeent, t,o the refinery from whieh high-purity hydrogen sulfide and ethylene are delivcred by pipeline. Propylene and isobutylene are received by pressure tank ears.
Natura1 Gas Odorizing spent almost $2 million for “ew equipment and plant site. They reloiated a subst,antial amount of equipment and al1 heavy pressure st,orage tanks 20 mila, had delays due~ to weather and equipment delivery in the 1965 0 boom-year of construction, and stili were in full operation far the winter senso”. It took a lot of doing, but this is the history and reaso” for suc- cess of Natura1 Gas Odorizing, division of Hel- merich and Payne, Inc. They are and always have bee” the world’s largest producer and seller of odorants and other warning agents to the natura1 gas industry. Over 7 million lb of odornnts are sold annually under the trade “ame of CAPTAN to more than 1300 eustomers throughout the U.S.A. and abroad. They are aold in quantities ranging from 1 lb containers to 65,000 lb tank car loads.
The research facilities of Natura1 Gas Odor- izing, plus their modem plant, puts them in a position to expand their odorant line and to mann- fatture other spticialty chemicals for themselres and others. They started their first contract to custom manufacture specialitv mercanta” sulfur compounds as soo” as- they &oved in their ne-7 p1ant 1ast fall.
New Plant Details The “ew plant can manufacture up to 13
million lb annually of merca,ptan sulfur com- pounds. Hydrogen sulfide and the appropriate olefin (ethylene, propylene, ete.) are charged con- tinuously at high temperature and pressure through fixed-bed catalytic reactors. The resultant product is purified in 60.ft tal1 by 20-i”. diam st,ainless steel fraetionation towers using sieve-
-type trays. Fabrication of “ew preusure vessels and trays was by ABCO Fabricators of Hous- ton, Ta.
The latest in pneumat,ic and electronic controls are used. Data is transmitted continuously to recorders and remote eontrollers in the centra1 contro1 building. A modem laboratory with a chromatographie analyzer provides around-the- clock quality control.
Over 250,000 gai of storage in a dozen pressure
February 1966 Petro/Chem Engineer
7
/
x / / e
Figure 1. The system is comprised of an insertion type gas burner which can be moved from boiler to boiler and the flame detector equipment.
Figure 2. The flame relay is assembled in a rectangular casing which is mounted io a mobile panel. Auxiliary components far the flame detector are stacked on a tray at the botto,,, of the pane,.
Petro/Chem Engineer March 1966
by E. Villanueva and E. J. Bardouille La,go Oil & Transport Co., Ltd.
Aruba., Nethehmds Antilles
Portable Flame Dectector IS
Safeguard for Boiler Startups n A 7%hoor warm up on gas reqoired by pitch fired boilers presented the Lago Oil refinery -4th severa1 re.4 and costly problems. Safety consider- ationn in preventing flame-outs dictated con- stant operator vigil during the long startup. This limited the operator’s nctivity around the power- house.
Recently, lue11 Flamr
Lago devised a means to put a Honey- Detector on wheels. New. one uortable .
unit safeeuards al1 boiler startups and the opera- t.or’s costly “starting time” has been eliminated.
On ‘the process side, the system comprises an inscrtion t.ype gas burner of our own design (see Fig. 1). This burner has a 3,i,-in. flexible hose extension terminating in a %-in. Asco solenoid ralve. The valve is of the normally elosed con- struction type and is explosion proof. The sole- n,oid end of the hose is tied into the main gas sup- ply header with a regular plug-in connection. The eombined burner. hose and valve can be dis- eonn,ected and moved easily from one boiler to another as required.
The flame detector uses a C7012A Honeywell Flame Radiation (ultraviolet) Detector as t,he pri- mary element. The detector activates an R7023A Honeywell Flame Relay. This sends a signal t,o the solenoid valve in the gas supply line. The Radiation Detector has been modified. It has been provided with a special adapter flange to suit the peephole over the burner. The adapter, which has a single bolt connection, enables quick and easy mounting of the detector.
The flane relay is assembled in a Foxboro rec- t,angular casinp. The casing is mounted in a mo- bile pane1 as shown in Fig. 2. The Foxboro easing also incorporates two indicator lights, alarm beli, a transformer, a wiring terminal and the main switch for the electric power supply. The use of flexihle conduit for the eomponents’ Signa1 and power supply lines permits storage of all compo- nents external t,o the reetangular casing.~ The components are stacked on the tray at t,he bottom of the mobile panel.
In the presente of a flame and with switch S,
41
closed, a rectified current is permitted to flow through the electronic network of Fig. 3, energizm ing the flame relag. The relay closes conta& be- tween terminals 6 and 8 and bet,ween terminals 3 and 5. The green light is on as current flows through its circuit, indicating that the flame is on. The closing of conta& 3 and 5 allows current flow through the solenoid, which, in its energized condition, maintains the valve in open position.
When a flame-out occuis the relay breaks contacts 6 and 8 and contacts 3 and 5. while con tacts G and 7 and 3 and 4 are made. The red light come on with the closing of G and 7, setting off the alarm bel1 which is wired in parallel. Switch Sa is used to acknowledge and silence the bell. At the same time, the breaking of conta& 3 and 5
Figure 3. Diagram showing the electronnz network of the flame detector system.
cause8 the solenoid to be de-energized, thus clos- ing the valve in the supply line to the burner.
TO supply the initial gas for ignition, a problem is presented. In the absence of a flame, the valve would be in closed position (de-energized.) Some means of energizing the solenoid valve had to be devised.
“1 thought we ought to have an interchange be- tween engineers, teehnical people, research and op- erating management. That.‘s what led to the Oper- ating Practices Committee, which was designed to be an open forum, an int,erchange between the var- a ious oil companies of case history atudies of acci- dents. It’s pretty hard to get up an,d te11 about your mistakes, but this is the only way that we can ae- eomplish ou1’ goal-m exchange of ideas.”
The solution was found in spring-returned, push button switch SB, wired in parallel with ter- minals 3 and 5. Depressing SS energizes the sole- noid and opens the gas supply to the burner. The push button is held unti1 the green light cornea on, indicating the flame is on and the relay has closed the eont,scts between terminals 3 and 5.
Jesse Ducommun’s three-decade drive to im- prore process safety in oil refineries reaehed a high-point last year when American Oil’s 12 refin- eries reeeived the Award of Honor, the highest award recognition of the Sational Safety Council. Ducommun, receiving the whitr peunnnt of snfety on behalf of American’s 7’700 refinery cmployees, summed up his approach to process safety this way :
In operation, the det,ector system has performed “Winning the award is impressive, but not as
well. When flameouts occurred, the solenoid valve important as the safe-working practices that shut off the gas supply within two to four earned it. That’s th,e rea1 purpose of our safety seeonds. n program.”
42
PROFItE/continued
was drained, the water in the line to the sparc pump V7as not.”
“TO make the drain system fool-proof, we cut off the second nozzle and carne down with one line from the bottom head of the vessel SO that it would be connected to, the same drain line. This helps the operators by halving the number of points they have to check for collection of water.”
Jesse Ducommun’s safety manual series has re- eeived wide acclaim throughout the industry for pointing out hazards in refinery and related oper- ations and suggesting ways of correcting or elimi- nating them. It is a singular tribute to their practi- tal usefulness and accuiacy that most eompeting, oil companies have included them in their own em- ployee education programs.
As Ducommun continued to work his way up through his company’s corporate management strueture, he became more attive in promoting process safety on an industry-wide basis through the American Petroleum Institute’s Division of Re- fining. In Janoary 19G0, just, before Dueommun bc- carne a director and vice presidmt of manfactur- ing of American Oil, a group dedicated to the p& vention of refincry processing accidents carne into a
being in API’8 Division of Refining. Called the Op- eiating Practices Conmittee, its creation’ mpped u two-year drive by Ducommun to set up a forum in which refining executives and operatmg supqvi- sois could interchange process-safety ~knowledge and experiencou for common good.
“1 saw the need for the Operating Practices Committee,” Dueommun says, “because theie was a segment of the industry not attive in API-theop- erating supervisors. If you were a vice president. you were attive in API. If you were a staff engi- neer or a research man, you were attive in API committee work. But the operating management of ihe refineries mainly attended socia1 functions ut night.
March 1966 Petro/Chem Engineer
Over-AU Heat Transfer Appl&atìons - Part 14 “I
1 ,,,,,
~~’ Applied Heat Ttyansfer Design for the H PI
I)y Wayne C. Edmister and Joseph M. Marche110
ie, 14-1. Mean temperatum differente. Whenever
beat exchange takes piace between two fluids, . their temperatures change; the temperature of
the colder fluid goes up, and that of the hotter goes down, Fig. 14-1~~ and lb. However, in cases where condensation of a saturated vapor or evapora-
4 tion against a constant pressure take place, the temperature of the eorresponding fluid remains
2 constant, SS shown in Fig. 14-1~ and Id. The lines I % ^
representing the ehanges of temperature are b actually cuives because the temperature gradient Jong the path of the fluide changes. Thus the
.- beat transfer rate is a fur$ion of variable tem- p&ure differences (ti ~Y t2) and Eq. 13-l may be changed to:
where At is the mean temperature differente. The two com~only used expressions for determi&g the mean temperature differente are:
(a) the arithtiktic-mean temperature differente, At, and
tb) the loga.rithmio-man temperature differente, At,.
The arithmetic, or average, temperature differ- ente is obtained finn the expression:
At, = (Ati - Atyj’z ‘< Eq. 14.2
where At, = t’l - t”l, the differente between the temperatures of the hot and cold medium8 re- spectively at the beginning of heat exchange and ~b2t’~ - t”?. at the end. Since the temaeratures of
q=AUAt. _/. .Eq.14-1 the Auids change along curved lines, Eq. 14-2 gives
l surfoce A: Heat exchangeparallel flow.
Surface C: Condensation.
Evaporafing Liquid
I L Surfoce Surfoce
8: Heat exchanger-counter flow. D: Evaporation
Figure 14-l. Temperature differences in beat exchange.
Petro/Chem Engineer March 1966 43
NEW PENEX @~ produces balanced blend gasoline
Here is a commercially-proven process which enables the refiner to economize and upgrade pentane and henne fractions from naphthas and natura1 gasolines. It provides a high octane product with improved performance qualities, good sensitivity and warm-up, clean burning, betta lead susceptibility and improved ignition properties.
The Penex’a process is specifically designed far the continuous catalytic isomerization of norma1 pentane, norma1 hexane and mixtures of both. Reaction conditions are mild because of the nature of the reaction and the catalyst employed. Conditions selected are such that regeneration is not required, thus reducing capita1 investment costs, simplifying operation and minimizing maintenance. The non-corrosive nature of both catalyst and reaction provides additional econorny in installation.
Yields are exceptional, with a high rate of conversion. Operating manpower reguirements are minimum due to simplicity of process
0
flow and optimization of design features. There are no by-product, sludge, catalyst or chemical handling’problems as may be encountered in other processes, and which can reduce yields and pose disposal problems.
TO more fully apgreciate the profits which may be realized with a Penex unit designed to your specific needs, cali or write taday far further information or a preliminary analysis-at no obligation. Contact your nearest UOP area representative located in New York, Houston, Los Angeles or Des Plaines.
-
an exaggerated vahe for the temperature differ- ente At. It is necessary to use the logarithmic-mean temperat,ure differente:
If the temperature differenee nt, at one end of a beat exchanger does not differ from Ati at the other end hy mo’re thsn 50%, the arithmetic average temperature differente nt, from Eq. 14-2 will be higher than the true mean temperature differente by less than 47%.
14-Z Pamllel .fiow and cozmterflow. The direc- tions in which the fluids flow in a beat exchange? materially influente its performance. Counterflow whic,h is obtained when the hotter and colder fluida move in opposite direetions has two important ad- vantnges : (aj for counterflow the end temperature of the colder fluid can be raised nearer to the initinl temperature of the botte? Auid than~ with paralle flow; (b) for exchange of a giva amount of beat, al1 other conditions being equal, a smaller surfaee area is required with count,erflow than with parallel AOW.
The first advantage may tie easily seen from Fig. 14-la and 14-lb. Regardless of how far the area is inereased in the case of parallel Aow, Fig. 14.la, the fina1 temperature of the hotter fluid can never become equa1 to the fina1 temperature of the colder fluid because ii Atv would become zero, no beat exehange could take place near the outlet end of the heat exchanger. On the other hand, for counterfìow, the fina1 temperature of the hot liquid tends to approach not the fi?.al temperature of the colder fluid bu,t its initial temperature and the tìnal temperature of the colder fluid may not only equal the fina1 temperature of the h?tter fluid but even exceed it.
The expression for the mean temperature differ- ente may be written, as:
The second advantage of counterfow, smaller sorface area, is explained hy the fact that between the same temperature limita the mean temperature differente far counteriiow is greater than far parallel flow. The separate resistances and hence the over-al1 coefficient do not depend upon the direetion of the beat flow. If the temperatures of both fluids vary, for the same beat flow and the same initial and fina1 temperatures and hence, the same value of U, the surfaee A in Eq. 14-l must be smaller for counterflow, Fig. 14.lb than for paralle flow, Fig. 14-1~1.
Double-pipe beat ezehmyeis. An example of a counterflow exchanger is a small pipe positioned cancentrically inside another, larger pipe. One fluid flows through the inner pipe, the other through the annular space, Fig. 14-Z. Usually it is desired to have approximately equa1 pressure drops in both circuits and the lager mass flow is put through the larger cross-sectional area. In order to obtain
46
Figure 14-Z. Doublepipe exchanger.
+d
Figure 14.3. Diagram of a I-2 exchanger.
an adequate beat-exchange surface, pipe sections 8s shown in Fig. 14-2, usually are eonnected in series. For large mass flows, two or more ex- changers made up from unita connected in series art! connected ta headers and th,us work in psrallel.
14-3. Mdtipass and CTOBS Am. In a large numher of heat exchangers one fluid passes through tuba that are located inside a shell through whieh the ~@ second fltiid passes. Such equipment, called shell- and-tube exchangers may have the flow paths arranged SO that either or both Buids redense directions of flow une or more time8 and trave1 back and forth through the exchanger. As a result a combination of parsllel slow and counterflow, called rwsrsed jZvu>, iu obtained. Reversed flow is osed to obtain velocities that give higher beat transfer coefficients. Exchangers with reversed flow are also ealled multipass exchangers. The simplest arrnngement of this kind is shown schematically in Fig. 14-3. It is called a i-2 uzchanyer because it has one shell pass and two tube passes. Usually the hotter fluid trave18 inside the tubes and the eooling fluid passes through the Shell outside the t,ubes. In the first,, upper tubes the hot fluid travels in parallel with the colder fluid while in the lover tuba it travels in counterflow. ‘The expression for the mean temperature differente Atm, even for this comparatively simple arrangcment, beeomes very cumbersome. It is customary to use the Iogarithmic 0 mean temperature differente atz calculated for counterflow multiplied by a. correction factor F = At,,Jat, to take into account the lesa efficient parallel flow. For Fig. 14-3 the logaiithmic differ- ente At, calculated by Eq. 14-4 and the effettive or actual mean temperature differente is:
t, = FAt, Eq,. 14-5 The eorrection factors F al‘e found from diagram plotted in terms of two dimensionless parameters, X and Y:
y = Pi '- t", -2 Eq. 14-7
t'a -- t’l
The psrameter .X repiesents the ratio of the temperature change of the cold fluid to the differ-
ence between the initial temperatures of the two fluids and therefore may be called tenzpemture l TyLcY.~
Fig 14 4 shows a few typical srrangements used in multipass heat exchangers. Fig. 14.5 gives dia- gratis for determining F BS a funetion of X and Y far the three basie schemes of Fix. 14-4a. 14.4b. _ and 14-4~. Some exchangers have more than two
passes for each Shell pass, as shown in Fig. 14.4d. The grcater numbcr of tube passos simply increnses the heat transfer area of the exehanger but does not affect the parameters X and Y and the correction factor F: In order to obtain better utilization of the fluid passing through the Shell by mixing it more thoroughly, baRle plates may be inserted as shown in Fig. 14-4~ Such an arrangement affects the temperature differences fd - t’:, and Y2 - VI but does not affect the parameters X and Y and the factor F.
In some heat exchangers, called wowjiow ex- changers, the directions of the fluid ase at right angles as shows in Fig. 14-6. In these exchangers the mixing of the Shell fluid lowers the value of the correction factor F.2 Thus, for & one-tube pass nxchxnger, the factor F may be obtained from Fig. 14.7~3~ if both fluids are unmixed. If the Shell fluid is mixed, more’ accurate, results may be obtained from Fig. 14-5s. With two-tube passes, Fig. 14-7b may be used. For the Shell fluid mixed, the diagram of Fig. 14.7b mag be used. If in Fig. 14.6b, the direction of flow of one of the fluid is reversed, SO thst the Shell fluid first flows over the seeond tube psss and then over the first tube pass, t,he factor F may be obtained with fair aceuracy from Fig. 14-5b. Far a drip-type beat exchan&r, Fig. 14.6~ as is often used in cooling towers, the factor F may be obtained from Fig. 14-k. Diagram for finding F for other exchanger types may be found in the literature.3
A: 1 Shell PSSS and 2, 4 o, any even nomber of tube passe%
B: 2 Shell passes and 4 or any multiple of 4 tube passes.
Figure 14-5. Correction factors far mean ,temperature differences in
EXAMPLE 14-l. Determine the mean temperature differenee for a. Shell-and-tube exchanger with two tube passes, Fig. l-1.4a, operating with the hot water flowing through the tubes. Water at the rate of 6000 lb/hr must be cooled from 160F to 1lOF and cooling water at the rate of 10,000 lb/hr will be used. The cooling mater enters at 55F and leaves at XEF.
SOLUTION: In Eq. 14-5, F must be taken from Fig. 14-5a and with the designations of Fig. 14-4~1, by Eq. 14-6:
and by Eq. 14-l: ~ = X5-55 --. = 0.67
160-115
With these parameters, Fig. 14.5a gives F = 0.93. The value of At, wa,s found to be 65F. Thus:
At,, = 0.93 x 65 = 60.5F
This is praetically the average between parallel flnw and counterflow, as cuuld be expected.
I.@. Vnriable ovewdl~ coe.fieient. In the above discussions and formulas it was assumed that the over-s11 coefficient U is constant. If U varies only slightly with temperature, an average value of U may he used. If U varies considerably but the variation is linear in temperature, and hence ii nt, or U = U1(l i- b nt), the recommended average expressio+ is derioed by a similar proce- dure BS was used for the derivation of At,,. This enuation is:
4 = A u1 aa,- uy @tl
G.(U, At2/U2 at, 1 Eq. 14-8
where each product conta& U at one end of the exchanger and At at t,he other. Eq. 14-8 applies both to parallel and eonnterflow and also to cases l where the temperat,ure of one of the fluids is constant, BS when a liquid is heated by eondensing steam.
48 March 1966 Petro/Chem Engineer
If the over-al1 coeffieient~ U varia considerably throughout the exehanger but not linearly, the use of Eq. 14.8 is possible. However, more accurate results w-ill be obtained by considering the ex- ehanger divided into several exchangers in series, in eaeh of whieh U varies more or less linearly in temperature, and calculating the beat transfer fol each exchanger separately from Eq. 14-17.
EXAMPLE 14.2. A double-pipe beat exehanger made of a Z-in. standard pipe jacketed by a 3.in. pipe must be used to cool 20,000 Ib/hr of aniline from, 250F down to 120F; the average specific beat of aniline is 0.53 Btu/lb-“F; water enters at G8F and flows, in counterflow through the jack& at a high veloeity of 300 fpm; the values of U, referred to the outside of the 2-in. pipe are U1 = 540 Btu/hr-sq ft-“F at the hot end and Ux = 320 Btuihr-sq ft-“F at the cold end. Determine the necessary length of the pipa.
SOLUTION: The amount of beat to be taken from the aniline is 4 = 20,000 (260 ~ 120) x 0.53 = 1,378,OOO Btu/hr. The amount, of water is found, assuming an average water temperature of lOOF, p = 61.99 Ib/cu ft; the cross area of the annular space is S = 0 7854(3.068’ -~ 2.3752:)/144 = 0.206 l sq ft ; and t,he ‘flow per hour :
TV = 0.026 x 300 x 60 x 61.99 = 23,000 Ib/hr.
The fina1 water temperature will be:
t2 = 68 + 1,378,OOO
23,000 x 0.997 = 128F
The mean logarithmic value of Unt is: 320(250 - 128j - 540(120 - 68)
= 33,100 Btu:hr-sq ft
and the nece~sa~v area, br Eu. 14-8: A _-1,378,~0~ ^
33,100 = 41.6 sq ft
The outside surface of 1 ft of a 2-in. pipe is A’ = ‘ii x 2.375112 = 0.62 sq ft and the required length will be 41.G;O.W = 67 ft.
EXAMPLE 14-3. Dry ssturated steam at 2.5 psig pressure must be used to beat 10,000 lb/hr of crude petroleum from 80 to 210F t,hat flows inside straight tubes. The speeific beat of the oil may be assumed as 0.49 Btu/lb-“F; the over-al1 coefficient U from steam to oil varies with the bulk temperature of petroleum as shown below. Find the required area of the inside beat transfer surface.
Temperature t, “F Coefficient U,
Btu/hr-sq ft-“F
80 120 160 210
29 51 70 89
SOLUTION: The exchanger is divided into three parts inside of which U is assumed to vary linearly with the temperature. The~ beats absorbed by petroleum in each part are found as
A 1966SHOWCASE OF LATEST INNOVATIONS. IMPROVED TECHNIQUES AND ECONOMIC BREAKTHROUGHS
Petro/Chem Engineer March 1966 Far hr,har Informolion 0” Adwrlirad Pmdur,r les RWdW IPWkP hd 49
From stesm tables the temperature of the steam is found ai t’ = 220F.
Auvlying Eq. 14.8 and solving it for A gives for ._. - the area of the first part:
A I
= 106,000 h j29(220 - 120)/51(220 - 8011
29(220 - 120) - 51(220 - 80) = 41.6 sq ft
fm the second part:
As = ~96,000z~[5i~220~160j~70(220-120)1
51(220 - 160) - 70(~220 - 120) = 41.2 sq ft
for the third part : A
3 = 245,000 Zn [70 (220 - 210) 189 (220 - 160) 1
70(220 - 210) ~ S9(220 - 160) = 107.5 sq ft
Thu,s, the tota1 required surface is 190.3 sq ft. Alternate proeedwe. Sometimes, even if U varia
considerably, it may be preferred to use the log- arithmic temperature differente At, and to intro- duce a specially eorrected value of the over-al1 coefficient which may be designated by L’,. The basic Eq. 13-1 then assumes the form:
q = AU, nt, Eq. 14-9
The value of the over-al1 coeffieient is found from the expression:
u, = u,, + (U, - UJF Eq. 14-10 where the factor F’ is found from Fig. 14-8 as indicated on it. This method may be used only if the change of the over-al1 eoefficient from U1 to llr can be considered to be linea with temperature.
14-5. He& transfer in cliemicd reactors. In- variably ehemical reactions are aecompanied by thermsl effe& whieh aie ari important feature of practieal reactor design. Since most commercially
important reactions must be carried out at tempera- tura other t,han ambient, the problem is often one of heat transfer as well BS kinetics.
The heat evolved or absorbed by a chemical reaction may be obtained from thermodynamic considerationas Since enthalpy is a state function, the heat of reaction may be evsluated by adding the beat of reaction values for individua1 reaetions which make up the over-all reaction.O The heat of reaction depends upon the reaction temperature because the enthalpies of the products and reactants depend upon this temperature. I’ressure also must be specified to eompletely defme the States of the products and reactant,s from a thermodynamie oiewpoint.’
The deyree of completion of a. chemieal reaction is dependent upon u number of factors. The maxi- mum conversion that can be obtained for a chemical reaction is t,he equilibrium value. The rate of con- version is obtained from chemical kineticss and is used to determine the operating conditions neces- sary to produce a desired amount of product.
EXAMPLE 14-4. A reaction is conducted iso- thermslly at 180F in a jacketed, batch reaetor.g The rate eauation is:
7 = - $ = 40 C’ lb molesihr-gal 0
Temperature is controlled by regulating the rate of water supply which enters +t 90F. Other data are :
AH, = - 5000 Btu/lb mole at 180 F v = 100 gal li = 80 Btu/hr-ft’-dee F A = 125 ft’ C, = 0.2 lb mole/gal
Please turn to page 57
Figure 14-8. Moditied Colburn Chart tor evaluation of over-dl coetticients.
50 March 1966 Petro/Chem Engineer
HEAT TRANSFER TOPICS/ISSUE NO. 8
BECHTEL/ FLORIDA POWER AND LIGHT SELECT FINTUBES
Far the demanding reliability essential far the four new. centra1 station power plant,s being erected far Florida Power and Light, Brown Fintube tank suction heaters, pipe line heaters, and fintube hairpin section burner fuel oil heaters were combined to efficiently heat Bunker “C” oil. The BFT heating system reduces pumping costs and achieves the necessary XO’F temperature without coking, far boiler burner requirements at al1 firing rates. Relying on Brown Fintube heaters to handle fuel oils, asphalt, and viscous chernicals, far every job from refinery to end use, results in these bene&:
LESS FOULING-Fintube heaters foul less as fins keep metal temperatures of heaters lover. This virtually eliminates coking and residue build-up. Fintube heaters commonly operate far years without cleaning. Far 15 year case his- tory, write far data sheet 6634.
ONE-STOP SOURCE-Fintube tank heaters, suction heaters, line heaters, high presare heaters, fired fintube line heaters, fìred fintube immersion heaters-whatever the need, BF2 has the xight equipment to me& your tota1 heating needs.
LONGER LIFE-BFT’s oxclusive method of welding fins to tubes creata a full weld between ,the entire base of the fin and ,tubing. No amuunt of honting/coaling cycles rvill wark the fin loosa. There are no gaps Tor sludge to collect and start corrosion. Heauy 10 gage tubes add corrosion allowance.
VERSATILITY -If you beat viscous chemicals, heavy lubricants, resins, paint products, heavy cru,des, asphalt, Bunker “C” “il, molpsses, etc., ouì enginrrrs iwkonLe LiK! 0pportunrL.y of helping you sdoe youi problems. Ca11 us far process and high pressure exchangers, too.
USER ORIENTED DESIGN-RFT heaters have the design features users need far year-after-year profitable operation, including heavy tube walls, grooved tubesheets to eliminate leaks, studded tubesheet far removable tubeside bonnet, full penetration welds, and ASME Code construction.
NO SLUDGE DEAD SPOTS-Fintube constructiun enables straight through flow with no,dead areas far sludge buildup behind cross baffia. Also tube bundle removal is easier.
BROWN FINTUBE COMPANY ELYRIA, OHIO 44035
BROWN FINTUBE IS WORLDWIDE
Petro/Chem Engineer March 1966 Far Furlhar Infomlolion on IdWli$Pd Produrlr SP. Reader lorvica Cd 51
We have sa,id YE!$ to the oil Industry
on more than s2ti billion in loans
rive us an opportunity to say YES! to YO” /
First National Bank in Dallas MEMBER FEDERAL DEPOSIT INSURANCE CORPORATION
IN DALLAS
52 March 1966 Petro/Chem Engineer
It is sufficiently accurate to take the logarithmie mean-temperature differenee. Establish the exit- water temperatare and flow ra,te as functions of the concentm.ti«n amI the tirmi.
SOLUTION: The wte of he&t evolution by reaction equa16 the rate of beat pickup by the water and also the rate of beat transfer:
Heat transfer in batc’h reactors hss heen cor- related for hoth the walls and tube coils.“’ The results may he represented by the following dimen- sionless equations.
Far jack& heat transfer:
For coi1 heat transfer:
where: d = diameter of the vessel
hk = kettle side coefficient at inside surface in Eq. 11:30 and at outside coi1 SUI’- face for Eq. 11.31
pS = viscosity at surface temperature p = viscosity at bulk temperature P = length of paddle agitator N = agitator shaft speed, revolutions:unit
time p = density of mixture at bulk temperature C = speeifie heat at bulk temperature k = thermal conductivity at ‘bulk tempera-
ture.
The correlations presented earlier Cane be used to estimate the heat exchange with the surroundings for tubular j?ow reactors. The heat transfer rate at the wal! of the reactor is giva by the dimen-
Petro/Chem Engineer March 1966
sionless correlations for the heat transfer coefti- cient, using ai temperature differente uased on the bulk mean temperature of the reaction mixture.
The presente of solid particlesl such ai catalyat inereases the heat transfer coefficient in packed and fluidized beds to severa1 times that in an empty tube at the same fluid flow rate. Leva 11,L2 and others have correlated fissd-bed heat transfw coefjT- cients for air over a wide iange of variables. The c!mpirical expressions for air data are:
rvhere: h, = the beat transfer eoefficient in packed tubes
(1, == diameter of the tube d, = diameter of the packing particles,
Dow and Jacob13 proposed the following correla- tion far fiuidized-bed beat transfer coeficients.
Eq. 14-15
wbere: ki = heat transfer, coefficient at the inside tube wall
L = length of tube kg = conductivity of gas pg = density of gas pS = density of solid particles C, = specifie heat of gas C, = specific heat of solid partieles.
References:
‘R. A. Bowman. A. C. Mueller, and W. M. Nagle, “Mean Temperature Differente in Design,” Trans. ASME, Vol. 68, 283-94 (1940).
=K. A. Gardner, “Mean Temperature Differerxe in Multipass Exehan&vrs,” Ind. Eng. Chem. 33, 1495-1500 (1941).
31bid, pp. 1215-1223. lA. P. Colburn, op. cit., Ind. Eng. Ch,em. 25, 876
(1933). “J. M. Smith. Ch.emical Engin.eer&g Kinet.ics.
New York: MeGraw-Hill Book Co., Ine. (1956) p, 8. “American Petroleum Institute, Research Project
44, U.S. Bureau of Standards, Washington, D.C. (1946).
iO. A. Hougen and K. M. Watson, Chemical Process Principles. New York: John Wiley and sons, Inc. (1943).
sJ. M. Smith, loc. cit. “S. M. Wales, Reaetim Kinetics for Chemical
Cngineers. New York: McGraw-Hill Book Co., Inc. (1969).
“‘T H. Chilt,on, T. 8. Drew, and R. H. Jebens, Ind. .&.g. Chom $6, 510 (1944).
“M. Leva, Ind. Eng. Chem. 39, 85’7 (1947). lzM. Leva. M. Weintraub. M. Grummer. and E. L.
Clark, ha. Eng. Ckem. 40. 747 (1948). laW. M. Dow and M. Jakob, Ch,em. Eng. Preg. 47,
637 (1951).
Part 15, “Evaporation,” will appear in the Apri1 issue. 1
WHAT’S GOING ON/cootinued -,
Air Praducts Ends Merger Talks
Mergw negotiations betwee” ,Air Products & Chemicals and Souther” Nitrogen have been ter- minated. No raso” was giva far the de&io”.
Grate Sells PE Units to Allied
w. R. Grate & Co. has sold its high-density polyethylene plant at Baton Rouge, La., and its research laboratory at Clifton, N.J., to Al- lied Chemical Corp. The Baton Rouge plant has a capacity of 125 million Ib/yr. A reason ciled far the move was a lack of basic ethyl- ene position by Grate, while Allied is building a “ew et,hylene plant at Geismar, La., in cooperatio” with Wyandotte Chemical. Bllied also makes polyethylene at Orange, Tex., a”,d Tonawanda, N. Y.
Phillips to Make Nylon in Puerto Rito
Phillips Petroleum a”d Rhone- Poule”c SA of France will form a jointly owned company to make nylon ti6 fiber in Puerto Rito. As yet unnamed, the “ew fir”i will build a 40 million Ib/yr fiber plant, in the Guayama area of Puerto Rica An eventual capacity of 120 million lb/yr is planned. CompIe- tion is due late 1967. Rhone-Pou- lene is said to be the largest pri- vately owned industrial eorpora- tio” in Frante, with a current pro- duction of 450 million lb/yr of non-cellulosic fibers in Europe.
CSC.Reichhofd Merger Called Off
By mutua1 agreement, Commer- cial Solvents Corp. and Reichhold Chemicals have terminated “ego- tiations for a proposed merger of the two firms. Au it was discussed, the merger would have set up H “ew firm-CSC-Reichhold, 1”~.
NPRA Meeting Schedule for 1966
The meeting schedule for the remainder of 1966 ha been an- nounced by the National Petro- leum Refiners Association: March 28-30 - NPRA Annua2
Meeting, The St. Anthony Hotel, San Antonio, Tex.
June 7-9 -. Mi&conti~w~t Regional Meeting, Lassen Motor Hotel, Wiehita, Ka,“s.
Oet. 11.13 - Q&A Session. on Re- fining TBcII.?AoZO~~J, Marriott Motor Hotel, Dallas, Tex.
Dec. 6-8 - Conqmtw Cmfemnce, Jack Tar Motor Hotel, Sa” Francisco, Calif.,
Poerto Rican Firm Formed by Phillips
Phillips Pet~roleum~has filed pa- pers in Delaware to incorporate a “ew firm t,o handle the company’s petrochemical activities in Puerto Rito-Phillips Puerto Rica CORE, 1”~. The firm will be 75% owned by Phillips and 25% by the Puerto Rito Induatrial Development Co. (PRIDCO).
Dow Chemical Sets Up Energy Unit
Dow Chemieal is setting up a “ew Hydrocarbons and Energy Dept,. to “formulate company stra- tegy for gaining and maintaining supel’iorit,g in the acquisition, of hydrocarbons, fuels and energy s&ces.~~
Macaulev Whitinp. nresident of Dow Che&cal I”t,&l”atio”al, has bee” give” the responsibility for establishing the “ew department.
WHAT’S GOING UP/continoed
Front& Chemical, diviaion of Vulcan Materials, will incresse methylene chloride capacity at its
‘Newark, N. J:, plant by 30%. Com- pletion is due the third quarter of this year.
General Tire & Rubber Co.% Chemical Div. is expanding facili- ties at Odessa, Ta., and Moga- dore, Ohio. The present 170 mil- lion Ib/yr lata and rubber output at Odessa will be increased by ZOy&. Output at the Ohio plant will be increased by 25%. Both expan- sions are due far completion late t.his year.
‘BT F. Goodrich Chemical Co. plans to expand polyvinyl chloride resi” and compounding eapacity at its Avori Lake, Ohio, plant. Com- pletion, ia due the first quart,er of 1967.
Herculeu Puwer Cu. bau col”- pleted a 40% increase in facilities for “Parlo”” chlorinated rubber at, its Parlin, N. J., plant.
Sequoia Refining Corp. has re- ceived a license to build a Uni- cracking-JHC plant at Hercules, Calif., The unit will have a capaci-
Cyanamid Expands Catalyst Units
American Cyanamid Co. w-ill ex- pand facilities for production of petroleum refining and chemical processing catalysts by more than 10 million lb/yr.
Work at the Michigan City, Ind., and Woodbridge, N. J., plants is essentially compl&d. Construc- tion has started to expand hydro- ge” processin,g catalyst production at Fort Worth, Tex., with comple- t,io” due mid-qear. During the last half of 1966, expansion of facili- ties at Willow Island, W. Va., will be completed.
Cities Service Expands NGL Stake
Cities Service Oil has formed two “ew divisions in a move to ex- ~” pand its natura1 gas liquids busi- ness. A “ew LP-Gas Marketing Div. will be responsible for all LP- ~.~ i gas sales. 4 “ew Project Develop- m ment Div. will be resuonsible for increased NGL supplyethrough ac- quisition, construction and expan- sion. Fred H. Ramseur Jr. has u
bee” named, manager of the Natu- ral Gas Liquids Dept. A. R. Olsen will manage the “ew Marketing Div. and W. G. Osborne will head the Project Development Div.
ty of 2580 b/sd and will use the single-stage version of the Uni- cracking-JHC process. Feedstock t,o the plant w-ill he a blend of heavy atmospheric ,and vacuum gas oils which will be co”verted entirely to gasoline.
Staffa Chemical Co. has ca. tracted for Wellman-Lord t,o de- *Y.~
sign and construct a 2000.ton/day sulfuric acid plant at Houston, Tex. The proposed plant, capacity is said to be more than 25th larga than a”v existine “lant. Comole- tion is die late 1366:
Air Product,s & Chemicals, Inc., is “ow operating its “ew 600 ton/day anhgdrous ammonia plant at New Orleans, La. The plant was engineered and construeted by M. w. Kellogg.
Union Carbide Corp. has eom- pleted a major expansion of aectic acid facilities at Brownsville, Tex. The “ew facilit,ies add more than 200 million Ib/yr to Carbide’s capacity for aeetic acid equiv- alent, as well as additional capaci- ty for bi-products, including me- thyl ethyl ketone.
March 1966 Petro/Chem Engineer
Evaporation- Part 15
Applied Heat Transfer Design for the HPI
by Wayne C. Edmister and Joseph M. Marche110
15.1. Clmnye of State. When a liquid is evapo- ratea, a ehange of state to vapor or gas occurs. This is connected with a change of its physieal properties such as density, viscosity, specific beat and thermal conductivity. During the change of state, a quantity of thermal energy called latent heat of mporization is absorbed. All these effects must be taken into account in problems of beat transfer during the process of evaporation.
Boiling. Evaporation of a liquid from a free surface int,o a space filled with vapor of the same
Q. bquld by the addition of heat from an outside source and with the formation of bubbles is called hoiling, or ebullition.
Boil,in~g point. An, important factor in analyzing problems concerned with boiling is the boiling point temperature. The boiling point of a liquid is defined as the temperature at which its vapor pressure is equa1 to the total pressure on an, en- closed vessel containing the liquid and its vapor. There is no definite relation between the boiling point and the temperature of the liquid subjected to boiling, except that the latter is always higher than the boiling point. Ususlly the differente is small. However, under certain conditions the tem- perature of the liquid may be consid&ably higher than the hoiling point,before ebullition starts. Even when a liquid is boiling vigorously, its tempera- ture may remain higher than its boiling point.
Superheating. An inerease of ‘the temperature of the liquid above its boiling point is called
@ : s~perheatz?zg of the liquid. In an industrial process liquid superheating is undesirable because iti in- terferes with oraerly evaporat,ion ana means adai- tional beat losses. The influente of various factors that proauce superheating cannot~ yet be expressed numerically. However, it has ben established that the tendency of a liquid to beeome superheated is reducea by: (ai the presene 0f ais50lvea gases or small particles that form nuclei for the forma- tion of bubbles, (b/ bubbling vapor through the liquid, (e) localized hot spots, such as presented hy electric heaters, and (dl raising the pressure.
15-e. ?&echanism of boiling. Vapor rising from a free surface is invisible when the beat input is small. but becomes visible due to entrained drop- lets as the heat input is inereased. When the heat input is increased above a certain amount, vapor bubbles begin to form somewhere below the surfaee.
The bubbles increase in siae due lo evaporation of the enclosing liquid rise to the surface, break
Petro/Chem Engineer Apri1 1966
through it, overcoming the surfaee tension of the liquid, and join the vapor above its sui-face. This kind of boiling takes piace due to the, presente in the liquid of some solid particles, impurities, dissolved gases or other nuclei that set as ebul.latoru on which hubbles are formed. Hence such boiling sometimes is ealled nuclear, or nucleate, boiling. However, it is often referred to simply as ordinary boiling.
Bubbles are also formed on the metal surface of the vessel where the surface temperature is higher than the boiling point of the liquid. The formation of sueh bubbles is also enhanced by the presente of nuclei, SO t,hat this phenomenon does not differ appreciahly from ordinary boiling. The bubbles that are formed first cling to the meta1 surface. Vaporization of liquid into, the hollow space of a bubble inereases its size and buoyaney unti1 the latter force breaks it from the meta1 sui- face and forces it to rise to the free surface and burst. In this case the rate of formation of the bubbles, t,heir shape, and their eonnection with the metal surface depends upon the surface tension at the interface between the liquid and the heating surface, as illustratea in Fig. 12-l.’
Fig. 15-la shows u bubble on an unwetted sur- face, such BS would occur when the liquid is water and the meta1 sui-face is covered with a thin layer of oil. The free edge of the bubble is drawn as a fine ~wedge between the heating surface and the liquid and thus is held fast against the action of buoyaney. As a result the steam hubhles get com- paratively large, up to 5’16 in., before they become detaehed and are at once replaeed by new ones. The ‘bubbles spread themselves out densely on the whole unwetted sui-face. The evaporation of pure mercury in a mercury boiler probably follows a similar pattcrn.
Fig. 15.lb shows evaporation from a smooth half-wetted surface. The bubbles originate at only a few spots of the surface and are distrihuted over it irregularly. The buhbles rise from these points
0 b c
Figure 15-l. Typical shapes of steam bubbles on heat- ing surfaces.
77
in the form of columns thut become larga ns they rise. These columns eonsist of a great number of small bubbles rising uninterruptedly. The number of colunms increases with an increase of heat input.
Fig. 15.1~ shows a bobble that is formed when the heating surface is completely wetted. The bubble rests only at one point; the Huid tends to shear it off and to puh it away. The bubble has a globulau or oval shape and leaves the heating surface while still very small. The whole wetted surface is cooered with these small bubbles.
The existence of types of boiling WBB first clearly presented by Nukiyama,2 although earlier investi- gators had been avare of the existence of maximum and minimum rates of evaporation. The various regimes of boiling are sbown in Fig. 15.2 for the atmospheric boiling of water.
In Fig. 15-2, the region 4-B corresponds to the low rate of evaporation which occurs when there is no bubble formation in the liquid. Heat is trans- ferred from the heated surface by natura1 convee- tion and evaporation occurs at the free surface of the water. Nncleate boiling t,akes pluce in the region B-C. The outstanding feature of this type of boil- ing is that bu,bbles form at speeific preferred points on the hot surfuce.” In other words, the vapor phsse is nucleated at attive site on the solid suiface.
A maximum or peak heat flux occurs at the c~it%ca.l-te?n,psrat,Ll~~e diffewnse designated by the symbol C in Fig. 15-2. Beyond this point the heat flux from the heated surface to the boiling liquid decreases as the temperature differente is increased.~ In the tiansition wgion at least part of the surface is insulated by a vapor film which ser’ves to retard the rate of beat, trsnsfer.
At the point D in Fig. 15-2, a minimum occurs in t,he heat flux.“~~ Further increase in the tempera- ture differente between the heated surfacr a,nd the liquid increases the rate of hest transfer. In the region D-E and beyond. jX?n boili?ty occuis. Heat is transferred through a vapor film at the solid surface hg conduction and radiation.
15-S. Facto+s %n~henci?zy boil%ng hcat trmzsfev. Quantitatively, heat conductance to a boiling liquid is influenced by the following factors, which are stated in the approximate order of their importance : (aj intensity of beat flow, (bj pressure and/or
temperature, (c) nature, of the heating surface, (d! additional agents in the liquid and (e) agitation.
Heat flw. The intensity of the specific hert flow q, Btu per hr per sq, ft, ha8 a great influente upon the coefficietit of heat transfer h and on the tem- perature differente At = t, ~ ti, where t, is the temperature of the heating surface and ti is the temperature of the boiling liquid as measured by a thermometer. As the heat flow 4 is increased, al1 other conditions being the sane, both the coefficient 71. and the temperature differente Lt increase up to a certain point, called the cviticai tcmpwatwe diffey- w,ce. When the critica1 point is reached, a futher in- crease of the specific heat flow will continue to increase At, but t,he coefficient h will begin to decline as may be seen from Fig. 15-3b.$ For other liquids and conditions, the peak may be less pro-
78
nounced, Fig. E-4.” Obviously, an evaporator should not be operated st greater than t,he critica1 value. ha Fig. 15.3b shows that for water the critica1 value of At was about 35F. Fig. 15-4 shows that for vsrying conditions, different materials and surface conditions of the heating plate, the value of the critica1 Ot may change considerably. Thus, for ethyl alcohol the critica1 At may change from 45F to about, GOF, and even higher.
Tempwa.t~we and ~ressure. For a pure substance or a mixture of a volatile and a non-volatile sub- stante, the boiling point is a function of the vapor pres8ure, and temperature is regulated by contro1 of the pressare. Fig. 15.3a shows data obtained by severa1 experimenters for liquids boiling at atmospheric and reduced pressures. For a given temperature diffeience Ot, the coefficient 71. de- creases very fast with a decrease of piwsore and temperature of the liquid. In other words, in order to maintain a certain coefficient 11, with a decrease of the pressure and temperature, At must be in-
Apri1 1966 PetrojChem Engineer
NEXT TURNAROUND-1970!
Crude Preheat Exchangers On Way
TO Record FourYear Runs With New, Low-Cost Nalco Antifoulants
Nalco’s Petroleum Research Center conducts a cantinuous search far salutions 10 the industry’s processing problems. Here, Nalco Researchers developed antifouling addilives to control deposits formed by organic binders, salts and other inarganic matler that foul crude preheat exchangers, bath before and after desalters.
New you em break through the production bottlenecks caused weiy year or two by fouled crude preheat exehangers. Naleo has developed new antifouling additiva that work better to extend time between turnarounds. In some cases they have already doubled norma1 exchanger rum and the units are still going strong with wery indication of reaehing four years on-stream.
And, Nalco’s new antifoulants provide these results at a eost that can be easily justified in your operating budget. especially when you add up the dividends reeeived from increased throughput, improved heat transfer efficieney and important fuel savingi. In addition, the additiva will help minimize maintenance labor and parts replacement.
Your Naleo Representative has the complete details and can assist you in setting up a program to meet your speeific needs. Contact him today. Or, if you prefer, mite Nalco direct.
NALCO CHEMICAL COMPANY
,...Serving the Petroleom lndustry fhrough Practical Applied Science
ereased. The required increase may be quite appreciable.
Data showing the effect of reducing the pressure and boiling temperature upon the over-al1 coefficient U, obtsined from an experimental evaporator con- densing steam inside a copper coil, are given in Table 15-Lr
TABLE 15-l. OvwAll Coefficient U From Steam TO Water
Boilinipoint Temperature differente At’F
20 30 40 50 60 70 80 90
212 390 490 500 187 520 600 158 510 600 660 720
Nat,we of heating mrfaco. A rough meta1 mrface has a tendency to absorb air when exposed to the atmosphere. The absorbed air assista formation of bubbles in the range of moderate heat input and results in a. high heat conduetance and evaporation. However, the air is soon boiled out and the con- ductance drops and approaches the conductance for a smooth surface.
The met,al of the heating surface bus a gr& influenee.8 Fig, 15-4 shows surface caeffieients found for boiling ethyl alcohol, CnHoO, at atmospheric pressure on a horizontal piate provided with severa1 different surfaces.Y With a temperature differente At = 40”F, il is 3400 Btu/hr-sq ft-“F for a freshly polished copper plate, Ir is about 2000 for a fresh gold plate, 1500 for a fresh chromium plate, and 500 for an aged chromium plate. Fig. 15-5 shows over-al1 coefficients from steam to. water for a horizontal submerged-tube evaporat,or when eva- porating water boiling at 167F by condensing steam at different pressures.‘” The use of a coppa tube more than doubles the coefficient obtained with an i,ion tube.
Fieure 15.3b. Effect of temperature on pressure on h:-singk? horizontal tube%
Ethanol boiling at 1 atm: a - Freshly polished copper plate. b-Fresh gold plate. c- Fresh chromium piate. d -Aged chromiom plate.
Scale deposits. In evaporators for boiling aqueous solutions heated by condensing-steam, the heat- transfer coefficients on both the steam and liquid sides are large, if the surfaces are clan. Therefore, even a thin layer of scale or rust mi11 affeet the over-al1 caeffieient materinlly, ns shomn in Fig. 15.5. The data from Fig. 15.5 may be used to determine the influente of the scale in these tests.” Table 15.2 may be used far estimat,ing the thermal resistance of scale from water, and Table 4-2 (see p. 4’7, P,/CE, Mareh 1965) for’resistance of scale from some industrial fluids.
SJmpe and~size of mrfa.ce. The shape of a heating surface seemingly does not affect the eoefficient h, provided the vapor bubbles can detach themselves and rise to the free surfaee. Vertical and horizontal platea show approximately the same values of 11.~~” In the case of tuba submerged in boiling liquids, neither the number nor the diameter of the tubes seem to influente the coefficient h.
IO 20 30 40 50 60 80 100 Temperature difierence n t, ‘F
Figure 15-4. Effe& of meta, and’condition of surface,
surface tension by 47% and increased h hy 23%.11 In another experiment the addition of Triton W-30 reduced the surface tension by 277% and inereased h by 2076.‘~
Agitation. Experiments indicate that in the range of moderate specific heat flow, agitation increases the beat-transfer eoefìicient h becsuse of a more vigorous formation of bubbles.‘” However, during agitation, the coefficient h does not increase for art increase in the heat flux and a resulting in- crease of AP
Wetting agents. Substanees that decrease the Nature of liquid. The beat-transfer coefficient 1~ ,surface tension of the liquid, or wetting agents, is a, function of the thermal characteristics of the increase the film coefficient 11.. For water boiling at liquid; therefore, h. varies for different liquids as atmospheric pressure, the addition of less than 1% may be seen from Figs. 15-3. Table 15-3 shows the of, an organic solution, Nekal BX, redueed the influente of dissolved solida in water. The solutions
KW;;
6000
4000
2000
‘% 600
400
200
Apri1 1966 Petro/Chem Engineer 80
were boiling at atmospheric pressure in the same evaporation with which data of Table 15-l were obtained.17 Salt slightly increases the overall coefficient whereas organic malasses decreases V very appreciably.
15.4. Eqmtions far fi1.m condmtanee. Attempts to develop a general equation by means of dimen- sional analysis have not been successful because of the complexity of the phenomenom and lack of dependable, eonsistent experimental data. Jakob and Linke have developed an empirica1 dimension- less equation based on theoretical c,onsiderations, whieh giva satisfactory values of the heat transfer coefficient h for specifie beat flows q, ranging from 9000 to 50,000 Btu,‘hr-sq ft.
This equation is?
where: (r is the surfaee tension of the Iiquid, Ib per ft,
p, is the density of the liquid, Ib per CU ft, pn fs the density of the vapor, lb per CU ft,
l q 1s the speeific heat flow, Btu per hr
per sq ft, h is the latent heat of vaporization, Btu
per Ib, ub is the average veloeity of bubbles, 9i8
ft per hour. The left side of Eq. 15-l represents an effettive
Nllsselt number NN,. Eq. 15-l gives :
There exist a number of other empirica1 correla- tions of the heat transfer coefficient for nucleate boiling.‘” These expressions are more involved and merely serve to emphasize the difficulty of selecting a satisfactory empirica1 correlation.
The theoretical methods of Rohseno@ and of Forster and Zuberz’ are improvements on the emPirica approach. These investigators were al80 forced in the end t,o the use of dimensionless groups with empirica1 coefficients and empirica1 exponenttj. The principle difficulty is t,hat at least two processes
•~ . ae nwolved during nucleate boiling: The forma- tion of bubbles and the subsequent growth of these bubbles. Difficulty arises because the impor-
IO 20 30 40 50 60 70 Over-oll lemperature ditference, Af, OF
Figure 15.5. Effect of surface condition on beat transfer in horizontaf.tube evaporaor.
PetroKhem Engineer Apri1 1966
TABLE 15-2. Representative Values Of Heat Transfer Coefficients
Material and k or h Made of Transfer Conditionr Btu/hraq n-v
Condurtion Aerogel, silica, opacified at 24BF 0.013 Concrete, dry at 70F 0.5 Steel. mild. 70F to 4OOF Copper, pure. 70F to 300F 2:;
Radiation
3.7
11.3
NStURI Convection Plane surfaee in air
plane surface in water 2.in. pipe in water
Forced Canwtion, Tsrbulent Ffov Air moving in a duct 4
8te.m flowing in pipe 160 Alcohol and other light organic liquids ,;;; Water in pipe
Boiling Liquids Ammonia. evaporatine. 200 Alcohol on copper PIate.
maximum flow 1200 Water on steel plate, maximum flow 2700 Water on chromium plate,
maxinwrn flow 4200
Benzene. film condensation 300 Ammania. film condensation 1600 Steam, film condensation 2000 Steam. droewise condensation 15000
TABLE 15-3. Ovet=All Coefficient u From Steam TO Solutions
tant variables for these processes are not necessarily the same and either one or both of the processes may be rate eontrolling.
Surfaee tension. The surface tension of a fluid is the tangential force per unit length appearing in the surface layer of the fluid when it is in contact with another fluid. This tangential force at the interface of the two fluids is caused by the interna1 molecular forees in t,he fluids. Surface tension is measured in pounds per foot and, like viscosity, decreases with an increase of the tem perature of ,the fluid. Values of surface tension for a few liquids in contact with air or with their own vapors are giva in Table 15-4 and Fig. 15-7.
Influente of ~ressure. Eq. 15-l was developed for standard atmospheric pressure p,. psia. The actual value of the eoefficient h, eorrected for the influente of the pressure p may be found approximately by using the expression:22
“’ = ’ (5.) “‘: Eq. 16-3
where h is the value found by Eq. 15-2.
81
Simplified~ eqr~otions. The following formulas are recommended for water boiling at standard at- mospheric pressure p, under free-convection eonditions:“”
Ca.1 Far’ horizontal heating surfaces in a ivide vessel with a mal1 specific beat slow, up to Q = 5000 Etujhr-sq ft:
Ir = 4Sf',Ys Eq. 15-4 and for a heat florv from 5,000 to 75,000 Btu/hr- sq ft:
11 = O.txqO.‘j ‘; Eq. 15.5 (bi For vertical heating surfaces in a wide
vessel with a very small heat flow, up to 4 = 1,000 Btu/hr-sq ft :
il = 50q”-125 Eq. 15-G and for a heat flow from 1,000 to 20,000 Btuihr- sq ft,
h = O.?‘IJ~.~” Eq. 15-7 For the inside of vertical and slightly inclined
tuba, it, is recommended that values of 71 found by Eq. 15-4 and 15-5 be increased by 25%.
The~influenee of pressure is taken into account ‘by using Eq. 15.3.
As iti the natura1 convection correlations where At was expressed in (degrees Fahrenheit] /(l de- gree Fahrenheit,), q in Eq. 15-4 through 15-7 is expressed as iBtu,/‘hr-ft”)/il~ Btu/hr-ft?). The con- stants in these equations ae not dimensionless and have, been selected to give IL in terms of (Btu/hr- ft’-“F).
15-5. Vaporizin.g equipnzent. If it, heat exchanger is used to evaporate water or aqueous solutions, it is conventionally termed an evapomtw When used to supply the heat requirements at the bottom of a distillation column, whether the vapor is steam or some other substame, it, is a mfioiler. Used in connection with a power-aenerating system it is a ~mue~-plant evaporator and when used to con- centrate a chemical solution by evaporation of solvent, it is a clr~m~~ical em~oioratw.
Most vaporizing equipment is designed on the bais of over-al1 heat transfer data obtained from experimental studies on proto-type or pilot models. This is due to the high transfer coefficients with which evaporators operate and the diffieulty of i,dentifying and correlating each of the individua1 resistances whieh make up the over-al1 resistance.
MzdtipTe-e.fcct evapo,.&m In the chemical in- dustry the manufacture of heavy chemicals such as caustic soda,, table salt and sugar often starts with dilute solutions from which large quantities of solvent must be removed before the crystalliza- tion process can be carried out. ,In poner-plant evaporators the unevaporated portion of the feed water is residue, blowndown, Fig. 15.Za. In the chemic~l evaporator the unevaporated feed is prod- uct, Fig. 15..%b.
In many evaporation unitu the vapor formed is useful steam. If a pound of steam is supplied to a process as shown in Fig. 15-Sa and the vapor produced in the first evaporator was then used as a heat source in a second evaporator operating at a lower pressure than the first, a greater utilization eould be made of most, of the beat.= When the vapor
82
TABLE 15-4. Surf& Tension Of Liquids In Contact With Vapor
formed in the first effe& is reused as the henting medium in a second effe&, the combined unit is called a double-effect evaporator. In order to main- tain temperature differences for heat transfer be- tneen the vapor from one effect and the ~boiling liquid of the next effe& the pressure on each succeeding evaporator must be lower than its predecessor.
Boili?~g Point R%sr (BPR,J. The boiling point of t,he liquid in an evaporator is effected by both composition and hydr&atic head.
A concent~rated aqueous solution undergoes a hoiling point rise above the saturation temperature
Apri1 1966 Petro/Chem Engineer
I ! 1 1 I I I l 1 I
Procon completes third Cosden installation Cosden Oil and Chemical Company recently asked Procon to design and build toluene hydrodealkylation and cyclohexane facilities at its Big Spring, Texas refinery. The project was completed and producing specifi- cation products eleven months from the contract date. Design capacity of the HydrarB unit is 510 BSD: the Hydeal@ unit, 2220 BSD. When Procon does the job, it’s done right: on time, on budget. on specification.
Feed Steam
Condenser
Produci Condensate
Figure 15-8~3. Doubfe effect evaporator with paraffel feed and horizontal tobes.
zwall vopor vapor
Feed
Figure 15.8b. Triple effect chemical evaporator with farward feed and vertical tube
corresponding to pure water at the same pressure. S,uppose steam enters the ehest of a chemical evaporator at 45 psia and is to evaporate water from a salt solution. The steam temperature is 274F. If pure water is evaporated at 18 in. Hg, the temperature of the vapar farmed would be 169F. However, the dissolved salt mises the boiling point to 206F at 18 in. Hg instead of 169F. The tempera- ture differenee across the heat transfer surfaee is only 68F and the BPR of 37F represents lost potential which eannot be attained owing to the dissolved salt. The BPR can be calculated only for dilute so1utions2j and for most rea1 casa must be evaluated experimentally.
The effeetive boiling point is also retarded by the depth of liquid above heat transfer surface. The effect of hydrostatic heat may be estimated from:
T Ri At, = 0.03 A- AP Eq. 15-8
5
where: At, = elevation of the boiling point due to hydrostatie head, “F
TR = boiling point of the solution, “R 2) = specific volume of vapor at Tnz ft3,0b.
As = latent beat of vaporization, Btu/lb. Ap = hydrostatic head, it.
In most casa, L)p is taken aa one-half the indicated liquid level. The influente of liquid head will be greakr as the condensa pressure is lowered, since v varies considerably with pressure while hs is only slightly premure dependent.
EXAMPLE 15-1.26 It is desired to concentrate 50,000~ lb/hr of a chemical solution at 100F and 10.0% solids to a product which contains 50% solids. Steam is avsilable at 12 psig, and the last effect of a triple-effect evaporator with equa1 heat transfer surfaces in each effect will be as- sumed to operate at a vacuum of 26.0 in. Hg referred to a 30-i% barometer. Water is available at 85F for use in a barometrie condenser.
Assume a negligible BPR, an average specific heat of 1.0 in al1 effe&, the condensate from each effect leaves at its saturation temperature and that
there are negligible radiation losses. Calculate: (aj steam consumption, Ws, (b) heating surface re- quired for each body, Al, Ai. Aa, (cl cOnden8er water requirement. The aceepted over-al1 eoeffi- cients of heat transfer for the different effe& will be Ui = 600, Uo = 250, and Uz = 125 Btu/hr- f+“F.
Material: wl, + w* i- LC~ = IL’,.;] t, = 100F TS at 12 psig = 244F Ts at 26 in. Hg oacuum (1.95 psiaj = 125F Tota1 temperature differente = 119F
When a forward-feed multiple-effect evaporator l employs equal surfaees in each effect experience indicates that the differenees in the pressures be- tween effe& will be nearly equal. This will rarely be entirely true, but it forms an exeellent starting point for the caleulation of the pressures in the effects. Any discrepancies can be adjusted later.
26.70 ~ 1.95 Average premure differente := -m3~,~-
= 8.25 psi per effect
Breakup of The Tot.4 Pressure Differente
Prersure Ap .steam or Psia psi Vapor, ‘F h, Btu/lb
(Use 1600 ft*/effect) Heat to condensa- = iuaha = 14300 x 1022
= 14,610,OOO Btu/hr
Apri1 1966 Petro/Chem Engineer 84
Three Armco Steels ,Team Up to
Save Money in Refrigerated Ammonia Barges
Refrigerated ammonia barSe is launched at Port Houston Shipyards. Critica1 tank StrrJCt”re* are Of ?.h&Lo-Temp stee,.
Al1 major elements of this new refriger- ated ammonia barge are of a modem economica1 Armco steel. For the critica1
storage tanks, Shef-Lo-Temp plates we~e used. This normalized carbon steel has excellent impact prop- erties down to -75 F well below the storage tem- perature for liquid ammonia.
Armco ABS hull quality plates were used far the barge strutture. Both Shef-La-Temp and ABS plates are members of a full rage of carbon and allay plate products available from the Armco plant in Houston. The strategie location of this mi11 saves shipping costs for fabricators throughout the South-
west and West. In addition, Armco ZINCGRIP@-EI hot-dipped,
zinc-caated steel-provides a durable outer skin protection far the insulation materia1 surrounding the tanks. Fabricator of this barge-and many more like it-was the Port Houston Shipyards, Houston, TCCFiS.
Helpful Data Available
Armco can provide convenient, useful data to help your design and fabricating work. Write us, outlin- ing your needs. Armco Steel Corporation, Dept. W-1046, P.O. Box 1367, Houston, Texas 77001.
During operati”n the equal pressure-drop dis- tribution may not maintain itself. This uill occul’ if there is undue scaling in “ne of the effe&, if a body is gas bound, or if liquor levels are not properly maint,ained. Another factor may be the wi~thdrawal of a large quantity of steam from “ne of the effe& as a source of low-pressure heating steam. Any deviation from an equa1 pressure-drop distribution does not mean that the entire multiple- effect assembly will fai1 to operate but instead that the unit will a-sume a new pressure distribution and operate with a xduced capaeity and steam economy.
xhe?-~,,~oeo~r~pr.euuion. In an evaporator the steam genera& oapors which have nearly as much heat content BS was originally present in the steam. If the vapors from the evaporator are compressed to the saturation prrssure of the steam, the tem-
Liquid LeveI
Bottam Product
perature of the vapors would be raised to that of the origina1 steam. The compressed vapors can then be mi,xed with the st,eam to improve the evaporator economy. During compression a small portion of the vapor will be eondensed and enter the stesm chest BS liquid.
The operation of recompressing a vapor t,o per- mit its reuse is called thermocompression. The compression may be accomplished with a centri- fuga1 eompressor, or, where steam is available at a higher pressare than that needed in the erapora- tor, the recompression can be carried out in a steam jet booster.
Relmilols. The heat exchanger used t,” supply heat at the bottom of a distillation column is called a reboiler, Fig. 15.9. If a liquid is il mixture of compounds it does not ordinarily boil isothermally. Instead it has an initial boiling temperature (bubble pointi and a fina1 boiling temperature (dew point:) This means that throughout the reboiler there is a temperature range over which boiling oeeurs.
The beat, requirements for a reboiler can be determined from heat balances over the distillation col~rnn.?~ In most columns a portion of ,the con- densed product from the t,op of the column is returned as reflux. Assuming that enthalpp data are a,vailable, the heat duty of the reboiler can be determined if the distillate, feed and bottom quan- tities and temperatures are known and if the reflux ratio and heat losses are aleo given.
Reboilers are usually operated with either forced or natura1 circ.nlations.z8 Tn general forced circula- tion is used only on small installations and in casa in which the liquid is very viseous FO that natura1 circulation is impeded.
References
‘Max Jacob, “Heut Transfer in Evaporation and C,ondensati”n,” fifech. Enginewing SS, 643-660, 720.736. Reprint No. 10, Univ. of Ill. Eng. Exp. Station i:193G) 24.
‘Nukiyama, J. Soc. .ifech. Engm. (Japen) 37, 367, S53 (1934).
“J. W. Westwater, “Boiling of Liquids,” Advwxes in Clmnical En.gi~~xwing, Volume 1, Academic Press Inc., New York (195Gj.
“This is the Leidenfrost point representing the “ecurrence of a “repulsion” between u liquid and a very hot solid. (see Westwater, op. cit.‘)
W. A. McAdams, Heat Tvansnksion, op. cit. p. 380.
“Ibid., p, 375. .?V. A. McAdams, Heot T.yaarm,issio??., op. cit.
p. 382. RH. M. Xurihara and J. E. Mgers, A. 1. CkE.
Jmmml 6, 83 (1960~). “C. F. Bonilla and C. N. Perry, “He& Trans- l
mission t” Boiline Binarv Liauid Mixtures.” Tmnu. A. I. Ch. E., 37,685.705’ (1941).
‘OW. L. Badeer and G. E. Seavov. He& Tmnifer and Cv@dL&tiox, Harvey, &“is : Svenion Evaporation Company (1945) p. 33.
“L. A. Pridgeon and W. L. Badger, “Study in Evauorator Desien.” Ind. Eno. Clmn. 10. 473.78 ci9i4j.
I
lrM. Jakob and W. Linke. Fwrlr. Gefiieted In- geniew, 4, 75 (1933).
lsM. Jakob and W. Linke. “Der Warmeuberzanz v”n eine urangerechten Plat,ten siedender Wasser,’ Forsh..a..d. Gebieted Ingmiewr Wesm 4, 75.81 !1933).
“T. H. Insinger and H. Bliss, “Transmission of Heat to Boiling Liquids,” T,rans. A.1.Ch.E. 56‘ 401-516 (194Oj.
‘%D. B. Robinson and D. L. Kata, Ctmm. Eng. Preg. 47, 317 (1951/.
,lGWestwater, Op. cit. 17MeAdams. 00. cit. D. 382. lsM. Jscob,‘H& T~&s.fer, Op. cit. p. 642. ‘“Westwater. On. cit. 0
?OW. M. Rohs&“w, “He& Transfer and Fluid Mechanics Institute.” D. 123. Stanford Univ. (1953).
=H. K. Forster;&d N: Zuber, J. Appl.‘ P&. 25. 474 (1954’).
=M, Jacob, ‘Heat T.rnnufw, op. cit. p. 698. ‘“Jakob and Hawkins. Heat Tmnsfer and Imula-
tim, op. cit. p, 164-65. =D. Q. Kern. Process Heat Twwfe?. New York:
McGraw-Hill Book Co., (1950). =D. F. Othmer, Ind. Eng. Clwn. 32, 841 (1940). ?D. Q. Kern, op. cit. %?C. S. Robinson and E. R. Gilliland. El~emcnts
o.f Fra.ctional Dist%lla.t%m. New York: McGraw- Hill Book Co., Inc. (1950).
l*J. W. Pala and W. M. Small, “Hydrocarbon Processing and Petroleum Refiner”, 43, 199 (19641.
Part 16, “Condensation,” will appear in the June issue.
Apri1 1966 Petro/Chem Engineer
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Condensation- Part 16
Applied Heat Transfer Design for the H PI
by Wayne C. Edmister and Joseph M. Marche110
61-l. Introduction. Heat transfer during con- densatian of steam 01 other vapors differs from heat transfer during ordinary cooling of a fluid in two rape&. First, the cooling fluid has to absorb and carry away the latent heat liberated by the condensing vapor, and second, the beat transfer has to overeome an additional resistanee in the form of a liquid film which covers the cooling surface either completely or at least psrtially. The thickness of this film depends on its viscosity and on the condition and position of the cooling 8urface. The thinner t,he film, the smaller is its heat re- sistance. Roughneus of the surface increases the film thiekness. Drainagc is more rapid from a vertical than from a horizontal or even inclined surface, and the film is accordingly thinner. On a vert,ical surfaee, the film thiekness will gradually beeome greater toward the lover end, due to addi- tionsl condensation of the vapor in contact with the surface. Hence, the effectiveness of a vertical or inelined eooling surfaee decreases toward the lower end. surface they will be most effieient if they are smooth. in horizontal nosition with the condensation taking piace on the *outside of the tube SO that the condensate can drip off readily. In this respect,
In film condensation, the vapor is in contact with a fluid of nearly saturation temperature. The temperature differente and hence the condensing effeets are relatively smoll. In dropwiee condensa- tion, the vapor is in direct contact with t,he cooling surface. The effettive temperature differente is much higher and the formation of droplets takes piace in a much more intensive way than in the case of film condensation. In addition, the coetli- cient of heat transfer to droplets with u small radius of curvature is much higher than to a plane surface. As a result the surface coefficient for dropwise condensation is severa1 times jfor steam 15 to 19 times) larga thnn for film condcnsotion.’
Therefore, when tubes are used as condensing for severa1 horizontal rows of tuba the arrange- ment of Ginobat, shown in Fig. 16.lb, w-ill pive thinner films on a considernble portion of the sur- face and will be more effettive than the arrange- ment of Fig. 16-1~
The rate of condensation may also be increased by increasing the velocity of the vapor over the surface covered with a film of condensate.
Mechanism of condensation. Two types of con- densation are known: one called film condensation, and the other called dro~uise condensation. A combination of the two and may be called mized~ condensation.
However, it mnst be remembered tha,t the pres- ente of film-t,ype or dropwise condensation affects only one of the coeffieients entering in the make-up of the over-all heat t,ransfer coefficient V. Aceord- * ing to Eq.~ 13-11 the coefficieht U is controlled by the smallest individua1 eo- efficient, which case iti on the side of the cooling Rquid. Thus, the influente of dropwise con- densation upon U is mach smaller than the influente of fouling or scale on the liquid side of the cooling surface.
In film condensation, the saturated vapor coming in conta& with a, cooler meta1 surface eovers the cooling surface with a thin layer of condensate. The condensate flows down due to gravity and is
16%. Fih condenmtion. According to Nusselt’s liquid-film theory,’ sat- urated vapor forms a con- tinuous liquid film on the wall, and the beat flow is
eontinuously renewed by the condensing vapor, so determined by the thermal 0 b
that the cooling surface is never free of liquid. &resistance of this film. The Figure 16-l. Film condensation on *uiEs, one above anottler. I
Film condensat,ion takes piace on a suiface that is metted by the liquid.
On u nonwettable sui-face, that is, on a surface’ in which there are large interfacial forca, the vapor is condensed in drop form. From the prin- ciples of capillarity, when~ the droplets originate they are very small. As the droplet,s grow by further condensation, t,hey run down, under the a
aetion of gravity and vapor friction, and unite with other droplets on the surface.
June 1966 Petro/Chem Engineer
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Figure 16.2. Condensate - film thickness. velocity and temperature profile far film condensation on a vertical Wall.
film t,hickness is zero at the upper edge of the wall, increases donnward as indicated in Fig. 16-2, and reaches thickness 6 at the bottom. The temperature of the wall t, is assumed constant and the temperature, of the film increases lineally frqp t, to t,, the temperature of saturation of the vapor on the outside film surface. The velocity ‘v of the film is zero at the wall and its inerease within the film is assumed to follow a parabola. Under these conditions the mean velocity- v,, at the lower end of the wall, where the film thickness is 6, is found as:
Eq. 16-l
where: p is’ weight density, lb per CU ft, g is acceleration of gravity, 4.17 x lOa ft per hr sq, fl is dynamic viseosity of the condensate, lb per ft per hr.
The values of p and p must be taken zit the mean temperature of the film, (t, + t,)/2.
The mass flow rat,e at 2 is obtained from the eontinuity equation. For unit width the area at 5 is (8) (1) and the mass flow G per unit width beeomes :
G = p’s%? ~ Eq. 16-2 3P
In a small distante dx, the mass-flow rate will have been inereased by dG, which from Eq. 16-2 is pWg/p. This additional mass added to the layer is the, result of condensation. If the latent heat of condensation is h. then:
dG = F (t, - t,,)dz Eq. 16-3
The left-hand side of this expression is the tota1 amount of beat that must Aow for the, mass dG to be condensed. Substituting in the expression for dw pive8
s 6 0
s”fi = -$ (t, - &J /- ds Eq. 16-4 ”
where the integration is to be peiformed from the top of the plate t,o some z value.
If the temperature differente (t, - tw) is desig- nated by Ab, this analysis, due to Nusselt gives foi the film thickness the expression:
50
where: k is the beat conductivity of the condensate, z is the distante from the upper edge of
the wall, and h is the Iatent heat of vaporization
In this case the film coefficient h is the same BS beat conductance C, therefore, it can be pre- sented as:
In dimensionless form, Eq. 16-6 may be written:
It should be remembered that Eq. 16-7 is derived far laminar flow; therefore, before applying it, this condition must be ehecked by comparing the corresoondinx Revnolds number with its critica1 value. For sheet flow the film thickness must, be used for the hydraulic radius :i
The critica1 value of NRe usually is giva as 2000 to 2300.”
~16-3. Hm-tran?fer coeficients. Eq. 16-6 giva the loca1 heat transfer coefficient at a distante z from the upper edge of a vert,ical surface. The mean value alang the distante L, designated by /&,,, is found by integration, namely :
h .,,, = ; h Eq. 16-9
and, by Eq. 16-3, for s = L, the theoretical equa- tion derived by Nusselt results in:
h .,,,, = 0.943 !%f!! ‘f Eq. 16.10 (/AL)
Published test datai in general confirm the coi- rect,ness of the theoretical Eq. 16-10 but pive, for m
Figure 16.3. Influente of steam velocity on h,; con- densing *team flowing downward inside a vertical pipe.
June 1966 Petro/Chem Engineer
the constant, values that are in many cases 34 to 53%, highcr than 0.943. There cxist. two explano~ t,ions far this diserepancy. The first is the occu- rence of turbulence in the film condensate. The second is the occurrence of partial dropwise condensation.
Ammnt of condensa.te. The weight G of liquid flowing through a imit width of the surface is found, in lbs per hr per ft’ as:
G = v,,Sp = $! Eq. 16.2
By using the rate flow G, the temperature differente At can be eliminated from Eq. 16-10. The fact that the tota1 heat liberated by condensation, GA, passes through the sheet of condensate to the sur- face of unit width and lengt,h L, can be presented as a heat balance:
GA = h,,,LAt Eq. 16-11
Substituting At from Eq. 16-11 in Eq. 16.10 and reducing the results gives:
Eq. 16.10 and 16-12 are the basic equations for surface coeffieients with film condensation on various condensing surfaces.
Reynolds nwmber. The influente of the Reynolds number Nne can be brought out in the following manner. A substitution of G for v,,& from Eq. 16-8 giveu :
4G NE, = T Eq. 16-13
By rearranging Eq. 16.12 and substituting NJ4 for G/p from Eq. 16-13 in the right side of the resulting equation, the following dimensionless equation is obtained:
%($)” = 1.4S(&)s Eq. 16-14
0
where 1.47 is a pure number. Theoretical dat,a plotted” as :
against log (NRrj give a straight line AB shown in Fig. 16-4 for laminar flow, N,<, from 0 to about 2100. Experimental data correlated by the same procedure pive a straight line A’B’ with values of h,,, about 20% higher than obtained theoretically.
Vertimi su~fa~es. Far a vertical reetangular sui- face of height L, the film coefficient may be com- puted directly by Eq. 16-10, if the temperature t,,, of the condensing surface is known or can be estimated. k,,,, can also be caleulated from Eq. 16-12, if the amount of condensate G is known or can be estimated. In either case the value at h, should be multiplied by 1.2. For practical use it is convenient to substitute 4.17 x 10R for g and move it out of the parentheses. Then from Eq. 16-10 the design coefficient becomes :
Petro/Chem Engineer June 1966
Eq. 16-15
Another form of the design coefficient is. from Eq. 16.12:
h, = 829,( $)” Eq. 16-16
Incli~md surfaces. For an inclined surface OF tube. the term inside the parentheses in Eq. 16-15 must be multiplied by sinq, where 1Ir is the angle be- tween the surface and horizontal. Thus:
h, = 161,8 @p’Xsinv % ( p AtL
) Eq. 16-17
Vertimi tubes. For vertica,l tuba with condensa- tion on the outside surface, approximate calculations may be made using the same Eq. 16-15 and 16-16 as for vertical surfaces.
Hwrizontd tubes. For condensation outside a single horizontal tube the value of h, may be found by substituting 2.81 D, where D is the outside tube diameter, for L in Eq. 16-10. Thus, the cor- responding equation becomes :
h,, = ,03.7( pG)’ Eq. 16.18
Eq. 16-18 may be applied with sufficient wcuif~cy to a vertical bank of horiaontal tuben, as shown in Fig. 16-la, by using Di instead of D, where i is the number of tube in a vertical bank. The increase of the film thickness on the consecutive tuba, due to the accumulation of condensate from the upper tuba, is to a gr& extent offset by the agitation produced by the condensate when it drips from one tube to the next tube below it.
Vapor ueloeity. When moleeules of vapor come in contact with the film of condensate flowing on the outside of a tube, they are absorbed by the con- densate and a vacuum is created into which vapor from the main .body rushes in. The velocity hesd of vapor that fills the vacuurn is very high eom- pared with the veloeity head of t,he vapor in a condenser.” Therefore, the influente of vapor velocity on the beat-transfer coefficient of con- densation usually is negligible.
For vapor flowing down the inner wall of a vertical tube, Nusselt developed a dimensionless correction factor to be applied to the constant in Eq. 16-10 or Eq. 16-12. Tests eonfirmed this theory
51
if a small fractionl up to one-tenth,‘of the steam is con,densed.7 Comparison of data from a” extended theory with experimental dat,a are shoan in Fig. 16.3.” The tests confirm the general trend and also show’ that the influente of the temperature differ- ente (t.?,> ~- t,j is much smaller than eould be expected.
16.4. Twb~den.t film flnw. Oceasionally, for film condensation of vapor on u tal1 vertical pipe, the film of flowing condensate mai become rather thick, giving a high condensation rate and n Reynolds number NRe grenter tha”, its critica1 value. These conditions give rise to turbulent, flow.
In turbulent flow the beat t,ransfer is influenced by the Prandtl number Np4, = cp/k. A “umber of test da&” correlated in accordanee with the general Eq. 16.14 for N1+ = 1 and Np, = 5, gave the cur+es BG’ and BD, respectively, in Fig. 16.4. However, test data obtained with condensation, of diphenyl oxide and Dorotherm A, on vertical tubes’O are represented by line EC. Line EC corresponds to the equation :
0.8 / I ‘1
0.7 I I 0 ,004 .008 ,012 ,016 .02 ,024 026 ,032~ 036 .04
C, oir concentrotion by weight
$( sg)‘: = 0.0077 (NR<,)‘Vi Eq,. 16.19
iyhich is recommended for the determination of the mea” heat transfer coefficient il.,,, for turbulent flow in film condensation.
16-~i. Su~awlreated vapws. Equations established for condensation of saturated aapor may be used for soperheated vapor by substituting (i - i,) for X, where i denotes the enthalpy. or heat content, of the superheated vapor and i,: denotes the en- thalpy of the condensate. Since (i ~ i,) > h and the temperature differente at in the film remai”3 unchanged, the heal-transfer coefficient h,,,,. in aceord once with the general Eq. 16-6, will be slightly increased. Physically the process consists of two parts: first. the vapor loses the heat of superheat by convection, aeting as a dry gas with a relatively low beat-transfer coelìicient unti1 its temperature drops t,o the saturatio” temperature t,. Th,e” the process of condensation begins with its characteristically high beat-transfer coefficient. Rased on the over-al1 temperatme differente, super- heated steam does not present any appreciable advant,ages~ for process heating.ll
16-6. Dropwise condensation. As stated in Section 16-1, dropwise condensation is obtained only if the surface is not wetted by the condensate. Drop- wise condensation of steam occws on meta1 surfaces cont,ami”ated by organic substances which prevent the condensate from w&ing the surface and thus act as drop yro*mobers.” Many substances make the surface nonwettable t,emporarily; however, only those that are adsorbed or otherwise firmly held by the surface have practical significante BS pro- moters. Some promoters are specific for certai” surfaces, such as mercaptans on coppa And its alloys; others can be used with a number of sui- faees, such as oleic acid on copper, brass, chromium and nickel. Usually a very thi” film of the promoter gives the best results. A heavy film of contaminant
Figure 16-5. Constant to be used in Eq. 16-18 and Eq. 16-19.
I
substance may result in the formation of a con- tinuous condensate film. Some promoters seem to be influeneed by the presente of noncondensable gas. Drop condensatio” is more easily maintained a o” smooth surfaces than on commercially rough surfsces.
Carefully conducted tests with steam hnve give” I average surface conductance oalues for dropwise condensatio” of 3,4,000 Gtu,/hr-sq-ft”F BS wainst about 2000 with film condensation.‘” In general, test data show that drop condensation cu” pive
I
8 to 15 times, and up to 19 times, higher surfac~e coeffieients t,han film condensation. However. since dropwise condensation can be obtained only under
I
firmly controlled conditions, condensing equipment ordinarily is designed assuming film condensati”” I
16.7. Ejffrxt of Iznn.eo?~densa.ble gases. When a vapor is mixed with a noncondensable gas, the mecha,nism of heat transfer to a surface by con- densati”” is complicated by the phenomenon of diffusion. The condcnsation of vapor on the cold surface removes it from the vapor phase im- rnediately in contact with the surface and results in a gas layer between the condensat,e film and l main body of the vapor-gas mixture.
In the gas layer the partial pressure of t,he vapor is low and the partial pressue of the ga8 is high. The vapor in the main body of the mixture moves toward the cooling surface due to the vacwm formed by condensation and the differente of par- tial press”rrs. It. must. pass through the gas layer and overcome its resistance. At the same time, the gas from~ the layer starts to diffuse into the main body of the misture where its partial pressure is lower than in the laser in contact with the condensate. The additional resistance of the gas film lovers ~the heat transfer coefficient of con- densation considerably. 4 theoretical treatment of this complieated phenomenonli is not suitable for use in design problems. A method for the numerica1 computation of the beat-transfer coefficient for the condensation of vapors in the presente of “on- condensable gases has been giva by~ Colbur” and
June 1966 Petro/Chem Engineer
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Petro/Chem Engineer June 1966 Fo, hrhar ,nrmlolion on *d.er,iied Prod”rlrlae Reah IM”h bd 53
Hougen.‘” However, this method requires a long and tedious series of gradua1 approximations.
Smmll air contmt. For vertieal tubes with con- densation on the outside surface, there exists a simpler method based on Nusselt’s theoretical Eq. 16-10 which is accurate within + 1O%.10
In this method the empirical value of 0.670; iCiO-” is substituted for the constant in Eq. 16-10 where C is the part, by weight, of air in the steam- air mixture. This results in the equation:
It, should be noted that the constant 0.670 already contains an inerease of 53$X as found from experi- ments, over the theoretical value. The range of use of Eq. 16-20 is C - 0.001 to 0.04, or 0.1 to 4%.
For condensation outside a horizontal tube with an outside diameter D, Eq. 16.20 must be ehanged by substituting 2.810 for L. Thus :
h,, = 0.515($3°~25 (+)“” .’ Eq. 16-21
For a vertical bank of horizont,al tuba, i tubes high, the value of D in Eq. 16-21 must be multi- plied ,by i.
Instead of computing the value of the constant 0.670;‘(C)0~11 or 0.F15:‘C0.“, respectively, the con- ‘stant may be taken direetly from Fig. 16.5.1r
Larger aiv content. For an air content of 0.2 to 20% of the steam weight the eoeffie&mt ir,’ in condensation can be obtained with sufficient ac- curacy by using Fig. 16-6,1R which giva h, as a
0 20 40 60 80 Temperature differente /s-fw, deqF
Figure 16-6. Influente of air content on the beat transfer coefficient of steam contained air.
function of the air content C, and of the tempera- turc diffcrcnce At = t, - t,,. First il,, is calculated for the giva conditions when there is no air present, C1 = 0. This value is multiplied by the ratio h.,/ho taken from Fig, 13-6, where h, is the heat- transfer coefficient, corresponding to the air/steam ratio C,, and ho is taken from the curve C, = 0. Both values are taken at the existing temperature differente At.
16-8. Surface emdensem. Surface condensers used in steam power plants to obtain low back pressure in steam engines and turbina eonsist of a bank of horizontal water tuba enelosed by a steel shell containing the steam. Water admitted to the tubes flows through them in a single or double pass arrangement. Steam is admitted at the top side of the shell and flows over and between the tubes. The condensate flows by gravity to the hot well from whieh it iS pumped out to the outside atmosphere. The cooling-water outlet is always mar the top in order to keep al1 the tubes filled with water.
The single-pass arrangement is used where the water supply is large and t,he power to deliver it, is small. The double-pass eondenser permits a bet- 0~~ ter utilization of the water and is preferred where the water supply is limited.
Fa.ctors aff‘ef2in.g eondenser perform.ance. The main factors affecting the performance of a, con- ~, denser are: differente in temperat,ures of the two fluids, water velocity, cleanliness of tubes, steam distribution, and air removal.
Temperatwe diflevence. According to the rules set by the condensa section of the Heat Exchange Institute, no guarantees should be made for (a) operation below 0.7 in. Hg absolute premure, which carresponds to a steam temperature of, 68.4F and (bi a differenee of less than 5F in the temperatures of steam and the outgoing water.
Water velocity. In single-psss eondensers with good water, a velocity from 420 to 450 fpm is good practice; however, higher velocities can be used. With corrosive or salt water, it is recommended not to exceed 400 fpm. In two-pass condensers common practice is to use 360 to 420 fpm. Lover wat,er velocities decrease the beat transfer eoeffi- 0,
cient too mueh and water velocities below 180 fpm may not give a uniform distribution of water to al1 the tuba.
Cleanl%ness of tuba. Even water free from foreign matter gradually forms a deposit on the inner surfaee of the tuba. This deposit increases the thermal resistance of the inner surface which is the eontrolling faetor in the over-al1 heat- transfer coefficient. In operation the tubes are cleaned at certain intervals. TO insure proper condenser operation with the desired yacuum it is advisable, in designing a cond&ser or finding its rating, to reduce the correlation constant by about 20%. This results in the equation:
)O.“(-hTp,+ Eq. 16-22
Lubricating oil contained in the exhaust steam of I
54 June 1966 Petro/Chem Engineer
Petro/Chem Engineer June 1966
reciprocating engines gradually forms a film on the outside surface of the tuba that further rc- duca the condensa capacity. However, numerica1 data are too nieager to apply a correction faetor to the outside surface coeiiicient. Therefore, it is recommended that the capaeity rating of the con- denser as used mith steam turbina be simply re- duced 35% mhen it is used with reeiprocating engines.
Vapor distdmtion. Because of the large specific volume of km-pressure vapor, its entering velocity is always very high. This is useful’ in obtaining a high rate of heat transfer but prevents the use of baffles far betta distribution because they would increase the pressure 10~s. Vapor distribu- tion is obtained by arranging the water tuba SO that lanes or passages are formed between bundles of tuba as illust,rated by Fig. 16.7. Fig. 16.73 shows an ariangement with veitical and hoiizontal lz&s for steam distribution; Fig. 16.7b shows radia1 and eylindricsl lanes and Fig. 16.7~ shows tubcs arrangcd in converging rows. The longi- tudinal distribution of steam is obtained by ex-
I - firrf poss ll- SKO”d pss
IO) ib) cc 1 Figure 16-7. Tube arrangements in steam condensers.
tending the ~steam admission opening over a large part of the condensa length and also by providing a large steam dome.
Air remwoaC. Effeetive removal of air entrained with the steam is important since air reduce the heat transfer rate, as brought out in Section 16-7. In order to reduce the work of the vacuum pump, the air must be cooled as much as possible. This is done by bringing it in contact with cold water tuba in the first pass, as shown in Fig. 16.7a and 16-7b or providing separate tube for air cooline, Fig. 16-7~.
Strmtwal factoss. Admiralty meta1 tubes are commonly used for fresh water and aluminum brass tuba for salt and brakish water. Standard tube sizes are %, %, Ti8 and l-in. OD; the “sua1 wall thickness is 0.049 in. for (18.WBG), although ?/a and %-in. tuba are sometimes used with 20 BWG ~valls, and 17 and 16 BWG are used witb high-corrosive water. Tube lengths vary depending on the condensa size, as shown in Table 16.1.
Oakside-swface cor.fi.eient. In a commercial con- denser, the outside coeffieient h,o varia considerably. It is highest in the upper rows, where the steam
velocity is high and eficiently removes the film of condensate from the tuba, and gradually de- creases toward the bottom where it is also affected by the inereasing air concentration.
No satisfactory data for 71” under different op- erating eonditions are available. On the other band. it is not difficult to obtain values for the over-al1 coefficient UU by testing a condensa of similar design.
Fog fommtion in cwr&n.sws. When t,here are large temperature gradients bet,ween the surface and the condensing vapor iog may be iormed.lu It is difficult to collect and interferes with the norma1 operation of the condenser.
In order for fog to form t,he gas mixture must be cooled below its dew point. Fog formation in corldenseru occuru in the gas film near t,he cold wrface.
26.9. Condenser design. The safest design of a new surfsce condenser would be one based on the value of ho determined from experimental values of Un for tests of a similar condenser. Expressing the value of UO by Eq. 13-17, substituting the values of
Un, DO, D, ho IC, and solving the resulting equation for hi will give a more dependable numerica.1 valne 0 for hi than a value obtained from a theoretieal formula with rather indef?nite correction coeiiicients.
However, in the absence of experimental values of UO. the design should be ~based on Eq. 16.18. Since the inside thermal resistance, Ri = l/h.i, usually is higher than the outside resistance, Ro = l/ho, a moderate uncertainty in the value of ho twill not affect the design tao much.
Design equations. If the beat eontent of steam entering the eondenser is 4, in Btu per hour, then the beat balance far the outside tube surface A, is:
q = A,lJqnt, Eq. 16.23
Substituting for Ao its va,lue expressed by the outside diameter of the tubes Do, their length L, and their total nlunber i an,d solving far i gives:
4 vD,LlJ&t,
Eq. 16.24
The beat carried awvay by the water may be pre- sented as:
q = W,,e (tr. ~ t,i Eq. 16.25
The weight of water W, can be expressed in terms of its velocity ‘v, density p, and inner cross ,area of the tubes, S = (rD1’/4jzi,, where i, is the num- her of tuba in a pass; for a single-pass condensa i, = % and for double-pass condensa i, = i/Z. The specitic heat c of t,he eooling water can be taken as 1.0. Substituting these values in Eq. 16-25 and solving it for tr - tl, results in:
tz-t,=-4-_ /XTD&V~
Eq. 16.26
Design procedwe. The following steps give the quickest solution : 1. Select the tube size, Do and Di, and its length
L, based on Table 16-1.
56 June 1966 Petro/Chem Engineer
2. Assume the water outlet temperature tz and compute the temperature differenee Ah.
3. Find the number of tubes i by Eq. 16-24. 4. Find the temperature rise tz ~ ti by Eq. 16~26. 5. Compare this value of tz with the assumption
made in step 2. Assume a new value for ts about halfway be-
tween these two ralues and repeat the procedure unti1 At, and tz -- ti agree. If necessary, change the length L.
EXAMPLE 16-1. A condenser is to be designed for a turbogenerator with a maximum exhaust of 42,000 Ib/hr. The desired condensa pressure is 1.5.in. Hg. The available cooling water is untreated river water and its maximum temperature is 78F. Find the required: ia) size and number of water tubes, (b:) cooling surface and (e) amount of cooling water.
The design must be similar to a double-pass condensa with a surface of 3719 sq ft. Operating data are: steam to condenser, 28,000 lblhr; con- denser pressure, 1.25.in. Hg; corresponding steam temperature, 86F; ,tubes %-in. OD with 18 BWG walls (clan surface) ; eooling water, 4100 gpm; velocity, 5.5 fps; and temperature, 67.OF in and 80.OF out,.
SOLUTION: First U0 of the tested condenser must be determined. Heat removed by the water: 4100 x (231/1728) x 60 = 32,900 cu ft/hr; at 67F, p = 62.29 Ib/cu ft and:
q = 32,900 x 62.29 x 0.999 (80 - 67) = 26,600,OOO Btu/hr
A check from the amount of latent beat in t,he uteam, assuming 950 Btujlb, giva:
q = 28,000 x 950 = 26,600,OOO Bt,u/hr The mean temperature diffcrence is:
This giva :
l U” = 26,600,OOO 3719 x 11.28
= 634.0 Btu,‘hr-sq ft-“F
Next the inside coefficient Il1 must be determined. The tube diameters are: Do = 0.875/12 = 0.0729 ft; Di = (0.875 - 0.049 x 2) = 0.0647 ft. For turbulent flow, the film temperature is equa1 tti the mean between the wall temperature and bulk temperature of the fluid. The bulk fluid tempera: ture is (67 -1,~ 80) /2 = 76.5F. The wall temperature is approximately (86 + 73.5112 = 79.8F and the film temperature will be approximately (73.5 + 79.81/2 = 77.7F. At this temperature, from Table 3.9 (P/CE, Feb., 1965, p. 39), c = 0.998 Btu/lb-“F; p = 62.19 lb/eu ft; p = 2.15 Ib/ft-hr; k = 0.354 Btu/hr-ft-“F; and ?J1+ = 6.06. Now, wit,h v = 0.5 fps:
0.0647 x 5.6 x 3600 x iz.2 = 0.0243 (;;)(- ‘x1
2.15
x 6.06°.4 = 4233 Btu/hr-sq ft-“F
Petro/Chem Engineer June 1966
Substituting the above values in Eq. 13-17 giva:
634 = 1
0.0729 0.0729(0.0729 - 0.06472 1 ~.-- 0.0647 x 1233 + (0.0729 t 0.0647) 65-+11,
(a) The tube siae must be selected in order to find Un for the specified operating conditions of the new coudenser. Tht: capacity of lhe new condenser is only 50% lwger than thst of the tested con- denser, therefore the same size, %-in. can be used. The water velocity may’ be taken us the average reeommended in Section 16-8, for two-pass con- densers, or 390 fpm = 23,400 fphr. From the steam tables, t,he steam temperature is 91.71F. A pre- liminary assumption of the water temperat,ure rise, t. - tl, must be made and later, when the required quantity of water is determined, the assumption must be checked. Since tO should not be less than 5F below the steam temperature, assume t2 = 86F. This gives tr - tl = 8F. Thus the average bulk temperature is t = 78 i- 8/2 = 82F and the ap- proximate film temperature tr = (82 + 91.71)/2 = 86.86F.
With theue data and those from Table 3.9, the Reynolds number of the water is found as:
The tota1 heat removed is 42,000 x 950 = 39,900,OOO Btu,/hr. In accordante with Table 16-1, selecting the number of tubes is found, from Eq. 16-24 as:
39,900,000 ’
= 1714 tubes. m x 0.0729 x 18 x 620 x 9.1
Eq. a
For a double-pass condensa i,, = 171412 = 857, and by Eq. 16.26,
t. - t, = 4 x 39.900,000 -.,-~., ~.~-,.~
62.13 x T x 0.0647” x 857 x 23,400
= 9.73F. .Eq. b
Before assuming a, new value for ta and re-
calculating At, in order not tu make tz too close to t,, it is advisable to slightly reduce the length of the tubes from 18 to 1G ft. This ,will give, Eq. (a), i = 1714 x 18/16 = 1928 and Eq. (b), tV. - tl = 9.73 x 857/964 = S.64F.
Now assuming tl ~ tl = 8,40F, or tY = 86.4, gives :
with this At, Eq. (a) gives i = 1928 x 9.10/8.86 = 1980 tuba and Eq. (b) gives tl - tl = 8.64 x 964/990 = 8.4ZF. This is close enough to the as- sumed 8.40F.
Thus the condenser must have 19X0 tubes, 7/s-in. OD, and 16 ft long.
(b) The required cooling surface is:
A
0 = 39,~00,000
620 x 8.42 = 7650 sq ft.
(c) The amount of water required is found by Eq. 16-25, with c = 0.997 Btu/lb-“F, as:
wu> = 39,900,000
8.42 x 0.997 = 4,750,OOO lb/hr
TABLE 16-1. Maximum Recommended Tube Lengths, Feet
500 1,500 10 1,500 2,400 10 14 i-4
2,400 3,800 3.800 6,000 :4 :i
:i :8 :B
6.000.10.000 - :i 20 over 10,000 -~ 20 22
TABLE 16-2. Representative Values Of Film Conductance’
or, with the aeight of ~1 gallon st 78F equa1 to 62.19 x 231/1728 = 8.31 Ib/gal, tbe required water uupply is:
w w
= 4,750,000 8.31 x GO
= 9520 gpm
16-10. A~prozkxte vnlues of conductmce. In practice, the operating conditions of condensing equipment often are not known with sufficient accuracy to emnpute the surface coefficient. Rather than using an average over-a11 coefficient V given in technical references, it is betta to calculate LT from approximate, oi representztive, values of t,he film conductance h,,. Representative values of film conductance IL, may be taken from Table 16-2.
Other representative values of heat transfer coefficients for various situations are giken in Table 15-2 (P/CE, Apr. 196G, p.).
Transfer unit mnethod. The~temperature differente varies from point to point in the condenser as the coolant is heated. B heat balance ‘en the coolant over on increment of heat exchatige area is:
dq = WC,dt, Eq. 16-27 0
In f:erms of an overwll hent trunsfer caeffieient:
dq = U(t,,-t<)dA. Eq. 16.28
Combining Eq. 16-27 and Eq. 16-28 and integrating:
For a condensation of a pure substance, the condensation temperature, th, does not change throughout the exehanger and Eq. 16-29 becomes:
The dimensionless group UA/WC, is called the number of transfer units, NTU. Eq. 16-30 may be rearranned to the form:
t,, out - t,, im. = 1 - e-NTU . Eq. 16-31
m th - t,, in.
where: NTU = UA/WC,.
The left-hand side of Eq. ,lG-31 is the temperature efectiveness,“” E, since ‘it represents the ratio of the actual temperature ehange in the coolant to the maximum possiblc ehange. The value of NTU is a measure of the dimensionless size of the heat exchanger and sei-“BS as a measure of the esse oi difficulty with which a unit may be fitted to speeifications.
References
‘M. Jakob, “He& Transfer in Evaporation and Condensation,” ~echam~cal Engineering 68, 643-60, 729-36 (19363. Reprint No. 10, Univ. of Illinois Engineerina Experiment Station (1937) p. 24.
21bid, pp. 50-54 and E. R. G. Eckert, Introduction tu Heat and Mass Tmnsfor, op. cit., pp. 172-75.
June 1966 Petro/Chem Engineer
I whichvG* features are important t6 you?
PetrojChem Engineer June 1966
“R. C. L. Bosworth, Heat Tmnsfe? Ph~enomena. New York: John Wiley & Sons, Inc. (1952) p. 147.
lBosworth, op. cit., p. 147; and MeAdams, op. cit., p. 260. Eckert, op. cit., p. 175, giva for NRe, computed on the basi6 of Eq. (13.5), 1,200 to 1,600; Jakob, Heat Transfer in Evaporation and Con.densa- tim. op. cit,, p. 71, gives for NI<,,,, from 1,500 to 2,300; U. Grigull, “Warmenbergang bei Kondensa- tion mit turbulenter Wasserhsut,” Forsch.ung a..d. Gebiete d. Inge?~.ieurzuesen,” B. vol. 13 /1942), p. 53, gives a tabulation of NRe,c determined by various authors with values ranging from 2,100 down to 1,000 and States that the discrepancies are due to experimental diffieulties and indicate the necessity of additional research work.
X Jakob, S. Erk, and H. Eck, “Der Warmeuber- gang beim Kondensieren Stromenden Dampfes in einem Verticalen Rohr,” Forschung, ad. Gebiete In~e~v~ewwesen 3 (1932) pp. 161-170; M. Jakob, “Heat Transfer in Evaporation and Condensation,” loc. cit. pp. 61-66.
GTinker, loe. cit. p. 5. rJakob, Erk, and Eck, loe. cit. RM. Jakob, S. Erk, and H. Eck, “Verbesserte
Messungen und Bereehnungen des Warmeubergan- ges beim Kondensieren Stromenden Dampfes in einen vertikalen Rohr,” P1l~zikalisch.e Zeitschrift 36, 73-84 (1935).
“A. P. Colburn, “Relation between Mass Transfer (Absorption) and Fluid Frietion,” Ind. Eng. Chem. 22, 967-70 (19301 ; and A. P. Colburn, “Calculation of Condensation with a Portion of Condensate Layer in Turbulent Motion,” Ind. Eny. Chem. 26, 432.34 (1934j.
l”C. G. Kirkhride, “Heat Transfer by Condensing Vapor on Vertical Tuba,” Trans. A.I.Ch.E. 30, 170-186 (1933.34) ; Ibid <condensed), see Ind. Eng. Chem.. 26, 425-28 (1934) ; W. L. Badger, “He& Transfer Coefficients for Condensing Dorotherm Fibra,” Tram. A.I.Ch,.E. 33, 441.46 (19381, Znd. En,g. Ch.em.. 29, 910-12 (1937).
“0. Lyle, The E,.(i?eient Use of Steam. London : H. M. Stationery Office (194’7).
lzT. B. Drew, W. M. Nagle, and W. Q. Smit,h, “The Conditions for Dropwise ,Condensation of Steam.” Tram A.1.Ch.E. 81, 605-21 (1935).
laW. M. Nagle, et al, op. cit. pp. 593-604. I’“P. Grassmann, “Gegenstrom Koudensation bei
Gleichzeitigem Stoffaustausch bei Zweistdffgemis- cha,” A>an.alen der Physik 7, 54.62 (1950).
15A. P. Colburn and 0. A. Hougen, “Design of Cooler Condensers for Xixtures of Vaporu with Noncondensing Ga,ses,” Ind. Eng. Ch.em. 26, 117% 82 (1934).
‘%. J. Meisenburg. R. M. Boarts and W. L. Badger, “The Influente of Small Concentrntions of Air in Steam on the Film Coefficient of Heat Trans- fer,” Tmns. A.1.Ch.E. 31, 622 (19343.
lrTEllfA Standwds, op. cit. p, 85. lRM. Jakob. Heat Transfer. OD. cit. u. 692. ‘“T. K. Sh&nood and R. i. Pigford Abswptiow
and Estioction. New York: McGraw-H’ill Book Co., Inc. (19523 p. 111.
%OH. Sehenek, Jr. Heat Transfe? Engineering. New York: Prentice Hall, Inc., 211 (1959).
l Part 17. “Design of Heat Exchangers,”
will appear in the July issue. l
PROGRESS IN HPI TECHNOLOGY
Petrochemicals l
q DangerS of dependence on tests of finished plastic parts are pointed out, by appliance engineers at G. E. Major Appliance Laboratories in Louis- ville, Ky.
Such tests, it is emphasized, must be accelerated and are therefore hazardous unless considered along with other tests. Predictions based on specific time and temperature related parameters, as well as effects of geometry, mechanical and chemical en- vironment and processing history must also be eonsidered. As an example, creep measurements of a high impact polystyrene used for a fan biade predicted no ereep at 0 F. Allowing a maximum dimensiona1 change of iO%, a life of well over 20 yr is predicted at 73 F. Field t,ests SO far have confirmed this. Testing procedure8 are directed, therefore, to a tempering of aecelerated, finished parts test results by consideration of directly evaluated monerties of the materia1 itself.
n 18.21% yields of 1,4-dichloro-2.butene are achieved bv va~oi Dhase ehlorination of 2-but,ene in work ai M<dwe& Resenrch ,Institute, using a single pass system.
Conditions were st,udied to determine the effect on addition and substitution rea~ctions. It was con- cluded that the only significant variable studied was the ratio of reactants and that between 2.3 and 2.0 ratio of chlorine to butene the maximum wus realized. Yields of neither product nor by- products were affected by wxiations of preheat temperature or space velocity. Greater yields of HCI, 2-chlorobutane, 3-chloro-1.butene’ i,3-dichloro- 2.butenes and dl-2,3-dichlorobutane were measured at higher temperatures but lower yields of meso- ~Z&dichlorobutane. Oxygen inereased substitution reactions only between 260 C and the optimuml 375 C. Attempts to chlorinate 1-chloro-2.butene gave ‘very low yields of 1,4-dichloro-2-butenel in- dieating that the 1-chloro compound is not an in- termediate step.
June 1966 Petro/Chem Engineer
oi exeeeds production capacity at which time high prices are established. Any misjudgment of the time necessary to develop markets and any overop- timistic estimata of volumes and prices can have a major effect on the profit return patta-n. In the example, a depressed short term benzene price, a reduced initial volume and higher delivery costs al1 would lower the early profit dollars and create a need for additional profit make-up in later years.
As noted, the benzene project shows a reason- ably attrattive return which obviously must be related to some corporate tax strutture and cast of capital. But, a margin of error in early profits could offset the expected justification and perhaps break even with a company’s cast of capita].
Any of the pitfalls discussed above, could elimi- nate or reduce profits during the early years of op- erations. If the first year’s profits are eliminated, one-fourth of the rate of return would be lost and ouì project would show a 19% Discounted Cash Flow before taxes, at best, a borderline venture. The payout increases to 41/4 years. Figure 3 illus- tra& the cash flow which cumulata only $4,140,. 000 after ten years. If the fina1 year of the earning
Y: Ife showed no profit, in other words, a nine-year
hfe project, the return would decrease only one percent to 24%. This illustrata the time value of money coneept and the need to concentrate on early profits.
There is always the possibility that the full product volume cannot be sold immediately as ex- peeted and some market development required. If the full product volume were not sold unti1 the third year with one-third of the full volume the first year and two-thirds the second, the reduetion in the profitability of the project would be equiv- alent to that of the first year at no profit (19% rate of return).
Aecovering Lost Profits There are several possibilities to ,be considered
for recovering a 105s in early profits. Let’s exam- ine a few and note their effectiveness.
Earning life of the venture might possibly be ex-
*. ‘. tended If we double the life of our hydrodealkyla- tlon umt to 20 years, the 19% return would only increase to 21%. We have recovered only one-third of ouì 108s by doubling the life. Tripling the life to 30 years would only add 5% to the 21%. The eon- clusion is that increased project life is not a prac- tical approach to recovering lost profits.
Are reduced operating eosts a reasonable profit recovery meehanism? In oui example, no. A lg/gal produet cast reduction out of l.Z$/gal di- rect operating eosts would be required to recover the 108s of early earning, as discussed above. Ob- viously, this is an impraetical requirement. As a general rule, produetion costa are usually accurate and well defined from the available process de- sign, leaving little room for further reduction. There are, however, many instances where the op- erating costs are a major percentage of the tota1 product cast, and an area of cast reduction is prac- tical.
PETROKHEM ENGINEER July 1966
About the Author - Louis R. Bechtel is wakation xupewisos in th.e Atlmtie Diri. of Atlmtic R%cl&ld Co. His darties include preparing capital. jlLstifi- cations, conducting profìt- ability studies and per- formance analyses. Bechtel received bis bachelor’s d.e- gres in chemical engineer- “g from case institute of Teclmolog~. He is chairm Comty Ama of D&mure
cm, AICHE, Dela:mwe Valley section.
Imorovements in moduct vields or efforts di- rected toward by-product upgrading are potentials for profit improvement. These should be reason- ably well-known for the authorieation evaluation, but in instances of integrated and eomplex pro- cessing some potential in by-product upgrading can be parleyed into profits. Our toluene to ben- zene example is extremely limited in this regard sine an increase in yield t,o stoichiometric propor- tions could only reeover about one-fourth of the early earning life profits which were lost as de- scribed. In addition, ow process offeru 110 signifi- cant volume of by-product to upgrade.
Additional product revenue from either more volume or higher sales realizations could provide the needed additional income for the remaining years to recover lost expected earnings. For oui profect an additional 250 b/d of benzene, a 25% increase in plant eapacity, must be sold at incre- menta1 profit to bring back ow depressed 19% rate-of-return venture to the planned-for 25%. Today’s technology in proceus drsign might not allow for this required inerement,al capacity with- out more capita1 cast.
Supply and demand will establish the pl-oduct value which can be commanded in the marketplace and, therefore, any ability to enhance the later years’ profits above predictions is external to an individua1 company project. Highly speeialized chemicals moving into limited markets may bave some freedom in price strutture to offset unfore- seen process or planning difficulties; but commod- ity values, such as that of benzene, are established by major suppliers.
Summary In our study, we have ~eeu that one-lourtb of
the predicted rate of return of our capital projeet can be lost if either the first year’s profits are lost or some unexpected market growth pattern is needed. Assuming that the foundation work for the initial evaluation was adequately done, the practical means available for recovering lost prof- ita are not necessarily effettive. TO avoid calling on contingent mechanisms for profit recovery, take time to define more precisely the early in- come pattern and thoue pitiallu wbich might delay initial earnings. n
95
H. Verkoren” and Dr. G. H. Reman Koninklijki Shelr-Laboratorium,, Amsterdam, Holland
Gas/ Air Turbulence Vaporizer Giues . . .
Quiet, Ground Flare far m Shell h3s developed a quiet ground flare for re- finery waste gasa, based on the Bunsen principle. ‘rhe essential feature is that t,he gas/& mixture passcs through a turbulence vaporizer and is ignit- ed immediately behind this.
When the gas to be burned is under pressure, the Bunsen prineiple can be used. With the aid of a jet, the pwssure is eotiverted into velocity. Where the mixing chamber is suitably designed, pressure differences will result from this velocity and will make it possible to draw in the combus- tion air and mix it with the gas (Fig. 1) Though the flames of Bunsen flares have low luminosity and produce little soot, they can be quite noisy.
F’or the Shell flare, the turbulenee vaporizer consists of a grid with circular or square ducts, the dimensions of which must satisfy eertain re- quirements. A ratio of 0.25 length to diameter is feasible, but best results ape obtained with an L/D nbove 3.0.
The influente of the turbulence vaporizer on the
*Rased on paper presented by H. Verknren at the IRtl, Annua1 Conference of the Deutsche Gesellschaft fur Mineralolwissenschaft und Kohlechemie, Cologne, West Germany.
noise produeed is shown in Fig. 2. There is consid- erable improvement in the frequency ranges with the greatest disturbances.
A multijet is used to keep the mixing chamber BS short ,as possible. Severa1 burners of this tgpe are fitted into’a combined combustion chamber. The wall of this chamber is lined with beat- resistant. materia1 SO that the flame temperature remains high and th6 outside wall is protected. The inlet of secondary air is possible. When in operation, only a short blue flame emerges from the combustion 0 chamber. There is no vertical beam of light. The wall temperature does not exceed 1OOOC (1832F).
The contro1 range of a Bunsen burner is limited, and flame flahback can oecur below 25% of the norma.1 load. However, since for t,he most part, large contro1 ranges must be dealt with, the neces- sary adjustment can be effected by switching the burners on and off. An automatic system can be used, with gas pressure controlling the necessary number of burners.
Gas stream turbulence can be increased by in- troducing an auxiliary gas under pressure into the waste gas. This gives a more rapid blending of the gas with the ambient air. The auxiliary gas can be
Refinery Gases either compressor air o~steam. An advantage of steam is the water gas reaction (carbon plus water produca carbon monoxide plus hydrogen) which suppresses the formation of soot during combus- tion.
It is important, of course, to keep consumption low. Improved air intake with higher gas speeds for greater turbulence will help reduce steam con- sumption. Too high a gas speed, however, will pro-
In a scale mode1 flare in which 200 kg propane were burned off, ,several types of throttle appara- tus were tested. The best results were obtainrd with the equipment show” in Fig. 3. Afta three- phase expansion, the st,eam is led sideways into t,he gas stream. The resulting noise cari be ignored. Even in the case of a distane twice that of the flare diameter between the throttle and the flare tip, a good degree of mixing was achieved.
Soot-free combustion of 200 kg propane/hr in the mode1 required 1 kg steam/k,g. This had to be in- creased to 1.5 kgjkg to reduce the luminosity. In a Ixter full-scale flare, the quantity of steam had to be adjusted aceording to the composition of the gases to be burned. 0
Figure 3. Madel of flare tip with steam throttle.
STEAM
HPI TECHNOLOGY/PETROCHEMS continued from Page 22
I Polyolefin products are rapidly supplanting.jute and sisal in Britain and Germany.
Fibrillated tape of polypropylene. made by IC1 Fibres, is finding use in baler twine. fish nets, packaging twine, hose and electrical usa. 25% cheaper than filament yarn and 21/2 time8 stKmger than sisal, it is soft and has high resistance to torque and stress, giving it excellent knot stability. Because of resistance to stretch and kinking, it is finding preference in slings, mooring lines and net mount,ings. A 3-i”. rupe of fibrillated yarn, 4 g/denier, is ~said to be stronger than one of 8 g/denier multifilament. Farbwerke Hoechst in Germany produces a yarn made by twisting strips of stret,ch oriented, 0.04mm high density polyethy- lene. The stretched strip is about 0.013 mm thick and 3.10 mm wide, twice ns strong as jute and with an elongation of 20.25% eompared to 3% for jute. A sack of this yarn weighs one-fifth as mueh as one of jute and withstands a drop test from twice as high.
w New liquid hydroxyl terminated resins offer promise for urethane and ureaurethane products.
Developed by Sinelair Petrochemicals, Inc., two all.butadiene polymers, Poly B-D Rl5M and Poly B-D R45M. a butadiene-styrene, Poly B-D CS15 and a butadiene-acrylonitrile, Poly B-D CN15 are said to react with diisocyanates to produce elasto- mers, foams,, etc. or first with amines and then with diisocyanates to produce urea-urethane poly- mera. Products are said to show high hydrolytic stability. Retention of 75% of physical properties afta 10 days in boiling water is claimed. The poly- mers are extendihle with hydrocarbon and synthetic oils, and compatible with many general purpose elastomers. Pot live are short, but the polymers go first into’a pasty state which can be co-milled with other elastomers such as polyisoprene, GR-N rubber, GR-S rubber, etc. and even injection molded. Cured produets resist swelling in lubricating oils and such solvents as benzene, acetone, and dimethyl formamide.
w Further information has become available on the electrolytic dimerization of acrylonitrile to adiponitrile.
Work at Badische Anilin and Soda-Fabrik has resulted in better understanding of the process. Yields as high as 82.5% are reported. Both cur- rent and materia1 yields are said to be independent of current density hetween 2 and 30 amps/dm2. Optimum value of pH was found to be about 9, with higher values producing more ether strutture and lover leading to propionitrile. Cathode poten- tial was found to be independent of pH above 2.5. At concentration of, acrylonitrile lower than 10% (2.0 molar) yields fa11 off. Solution aids such as acetonitrile or dioxaue are used. One electron and one hydrogen ion is required for each moleeule of ucrylonitrlle aceording to the overall reaction:
2CHs = CH-CN + 2Ht + 2e = NC (CH%) &N
PETROKHEM ENGINEER July 1966 w
Design of Heat Exchangers- Part 17
Applied Heat Transfer Design for the H PI
by Wayne C. Edmister and Joseph M. Marchello
17-l. Geneml. In the design of industrial heat transfer equipment many factors are encountered which are difficult to an,$.lyze accurately. The more important design faetors are: degree of roughness and.cleanliness of the surfaces; thiekness of possible scale deposita and tendency of solid-liquid surfaces to, beeome wetted. The influente of these factors must be taken into account by means of assumptions, and a thorough understanding of the underlying principles will help in their evaluation. The equip- ment designer must frequently rely on test data, which~ when properly obtained and interpreted, will show mhere inaeeuracies lie, will indicate how to improve the equipment, and will serve as a basi8 for future more accurat,e designs.
Intelligent and appropriate assumptions for first approximation are possible only when based on an understanding of the principles of heat transfer. They will often Salve considerable time and cast in the successful development of new equipment.
Another factor that will affect the design of in- dustrial equipment is the use of standard, or stock, materia~ls: standard tube diameters and wall thick- nesues, standard tube lengths, and standard ~thick- nesses of pkates.
The important economie balance between effi- eieney, first cast and operating expenses will be discussed in a separate chapter.
In most industrial heat exchange equipment, the main part of the heat involved is transferred throu,gh the walls of tubes. Accordingly, this chap- ter will deal vvith the design calculations of various tubular heat-transfer, apparatus-heaters, coolers, heat exchangers, craporators and condensers.
Electronic computers are beginning to be used in design of heat exchangers at present and it is expected that their use will be ‘tionsiderably wider in the future. Both digit& and analog2 have been employed in the design of cracking furnaces, con- densa and multipass exchangers.
17-2. Quantitative relations. In al1 of the above mentioned apparatus, beat is transferred from one fluid to u meta1 wall and to the second fluid by the combined action of convection and conduction. The effect of radiation is usually negligible. When fluids which move inside or outside tube8 without change of phase are cooled or heated, the coeflicient of heat transfer increases with an inerease in velocity. However, at the same time, the pressure drop increases and the designer must find the most
90
economica1 velocity. Sometimes a fixed pressure drop is prescribed and the equipment must be de- signed to meet this condition. In other cases the tube diameter and length may be fixed and velocity and pressure drop must be determined accordingly. Relations between these quantities are giva in the preceding chapters but will be summarized here for conveniente.
Heat bahnee. The heat q transferred through the wall of a tube from a hot fluid to a cold one is expended primarily in inereasing the enthalpy of the colder fluid. However, part of the heat energp may l also be used in increasing the kinetic energy of the colder fluid and in doing work against gravity. Some of the heat may be lost to the surroundings. Disregarding the secondary beat expenditures the main heat flow p from the hot fluid is:
q = m'c'(t~'- t,') = SG(tl- t2) Eq. 17-1
where S is the cross section norma1 to the slow. For flow inside tuba, Si = ND,“/4, where N is the number of tuba with parallel flow. For flow- outside tubes, So = nyoL, where n is the number of rows of tuba, L their length, and 7~~ the cl~arance be- tween two tuba.
If~n&her fluid changes in phase, the heat bal- ance is:
q = m'c'(tl' - tr') = m”c”(t,” - tl") Eq. 17-Z
where m’, c’, tl', and tz' are the weight-flow rate, specific beat, and temperature of the hot fluid and m”, c”, tl", and tz", the respective data for the cold fluid. The heat balance is very helpful and fixes the 0’~
relation between the temperature changes and weight flow rates. As indicated in Section 14-3, the ratio Y = (ti" - tl")/ftl' - t2') = m’c’/n$‘c” has an import,ant influenee, on the mean over-al1 tem- perature differente At,, in multipass and cross- AOW beat exchangers. Because of this relation for multipass exchangers it is not posuible to select Value of the four temperatures which merely satisfy the he+t bala,nce of Eq. lw and t2" - t%'.
A selection of the four temperatures must be cbecked with the value of F to determine whether the proposed heat exchange can occur in an appara- tus with the selected type of fluid flows.
-Heat-flow Me. The hourly heat flow from a bundle of i tubes can be expressed as:
q = UA At, = Ui(rDiL)F Atl Eq. 17-3
where Di is the inside tube diameter, L is the tube
PElRO/CHEM ENGINEER July 1966
: temperature
1
ed from Eq. nt for At,. As
;pass and eross- un one of the
<r.+s with tempera-
1 1 D,(Do-Di) D”
Eq. 17-5
1,+ (Do+D<jk+m
and referred to the inside surface of a tube, it is:
l ui = l 1 Di(Do-Di) Di ; Eq’ 17-6 ,? (Da + DJ k + D&,
Therm.al resistanca The last two equations may also be presented in the form of resistance equa- tions as:
and
& ZR+; : .Eq. 17-7
1 - = ZR +; <, Eq. 17-8 ut
where ZR represents the sum of al1 resistances except the resistanee l& or l& of the outside or inside film, respectively, under consideration. Obviously, all resistances must be based on the same surface area.
Fouling. As discussed in Section 13-6, the beat transfer performance of exchangers in operation is often lowered by fouling. In order to provide
0 a surface sufficiently large enpugh to maintain the requlred performance with a reasonable servite time between cleaning, TEMA standards for the design of heat exchangers recommend that an allowance should be made for norma1 fouling. The amount of resistance R, to, be added to the re- sistances in ealculating the over-al1 coeficient by Eq. 13-7 or Eq. 13-8, us the case may be, can be taken from d&iled TEMA tables.” A few represen- tative values from these tables are reprodueed in Tables 13-l and 13-2 (P/CE, Feb. 1966).
Another procedure, would be to first calculate a theoretical over-al1 coefficient assuming clean tube surfaces and then to find the reduced design value Ud by means of Eq. 13-12, using an appropriate fouling resistanee RI.
Surface eoeJj%ients. Equations for computing the surface co&eients h,i and h,, to be used in determin- ing the value of the over-al1 coefficient U, were giva earlier. A simple procedure for taking
PETROKHEM ENGIN,EER July 1966
radiation intu account is given in Section 11-3. For turbulent flow inside tubes without phase
change, the values of hi and ho may be conveniently calculated from the dimensionless equations:
2/8
(&)(F) +(&)I:h ,,Eq. 17-9
($&~= ~(f$-:. Eq. 17.10 0 0
where, for ordinary purposes, the constants are ai = 0.023 and ad = 0.33, the exponents are 1zi = 0.8 and n,, = 0.6, and F,,is u safety factor. The necessary value of F, depends “pori the problem; however, in general it may be taken as 1.25 for gases and water, and 1.5 for viscous liquids. The safety faetor is introduced t,o take care of uncer- tainties in the determination of Ir. Thus, 1.25 is used if the possible error is within i 20%, and 1.5, if it is within t 3370.
Eq. 17.9 can be rearranged a,nd presented a,s:
where: h.i = a,jG;n Eq. 17-11
In a similar manner, from Eq. 17.10:
ho = anG,,“‘~,q Eq. 17-13
Bafles. Eq. 17-10 applies to flow norma1 to tuben that are not baffled. When baffles are used, the fluid will leak from one compartment to the next so that the average velocity and hence ho will be redueed. Furthermore, in baffled exchangers, the fluid flow is not entirely norma1 to the tubes, which reduees ho additionally. Finally, some parts of t,he tuba are not swept by the Aow, which reduees the effectiveness of the corresponding suì- faces. It is difficult to state by how much the value of ho found from Eq.~ 17-10 should be reduced. Sometimes the correetion coefficient is taken as 0.6. However, it is better to estimat,e it from considera- tion of the eonstruction of the exchangers. Spacing baffles closer together has a tendeney to increase both the velocity and ho in spite of leakage.
T,ube tw~peiabzcre. The properties of the effec- tive films of the fluida pa,ssing through heat ex- changers depend to a great extent “pon the tempera- tura of the meta1 tobes that separate them. The temperatures tY and tr, Fig. 17-1, of the inside and outside surfaces of a tube can be calculated fram beat balance equations if the heat surface CO&- cients hi and ho, thermal conductivity IC of the tube metal, its thickness z, the corresponding diameters, and mean fluid temperatures tl and ta, are known. Thus, far steady Row and a unit length of tube:
q = ?rDihi (t, - tzl = s-Dm (3
(t2 - 83) =
,rD,&(ta -- ta) . . , Eq. 17-15
99
In most cases, for liquida and relatively thin walls, the temperature ~difference, t2 - tB, is negli- gible, tr = fa. Ina this case, Eq. 17.15 yields the relation :
t 2
= Dihitl ” D&& D& + Doho Eq. 17.16
Eq. 17-16 is very convenient both for estimating the surface temperature and for subsequent check- ing of these assumptions.
Pressure drop inside tubes. For beat-exchanger calculations, it is usually convenient to modify the premure-drop equations by expressing the velocity v in terms of the mass velocity G from the relation G = pv. For flow inside straight tuba:
AP,’ = fiB,LG?
2cq~iDi Eq. 17-17
The influente of the additional friction losses due to contractions is taken into aeeount in Eq. 17.17 by using an equivalent length L, far the length L. This equivalent length L, is obtained by multiplying the actual lensth L by a factor B,, where:
B. = API + API + AP, + AP5 AP,
Eq. 17-18
The value of the basic friction loss, AP,, is obtained from Eq. 6-7; the values of AP,, AP,, and AP, are computed hy Eq. 6-25, 6-27, and 6.28, respectively (P/CE, May 1965)
The general form of the expression for eal- culating the coefficient of frietion fi in Eq. 17-17 follows from Eq. 6-22 with the substitution of DG:p for NRe:
fi = ai (Gr. Eq. 17-19
where the exponent mi is 1.0~ for streamline flow and 0.2 for turbulent flow. For DG/p in the range of 2100 to 200,000, the dimensionlcss constant ai is equal to 0.184 for smooth tuba, and .22$ for steel tuba. The value of 9 is 1.0 for isothermal flow; for nonisothermal turbulent flow, J, is equal to (plp.>OJ4, where JL is the viseosity of the fluid at the mean bulk temperature t and /Li is its viscosity at the surface temperature.
Flow aw”s8 tlbes. The pressure drop AP, out- side tubi% may be calculated from Eq. 6-24 which may also be presented as:
The mass velocity Ga must ,be referred to the minimum free area. The friction coefficient f. is computed from Eq. 6-30 or Eq. 6-31, whiehever fits the case. Generally, fa is lager than fi and depends on the Reynolds nurnber DoGo/fio and on the arrangement and spacing of the tuba. The effect of leakage of fluid through the clearances between the tubes and holes in the haffles is un- certain. Unless exact dimensions are available, it is advisable to neglect the effect of leakage on
Wall thiekness. The thiekness of tu’be walls is select,ed to withstand working presaures and tem- peratura. An allowance~ must also be made for corrosion. Tuba are specified by the outside diameter and nomina1 thickness of the wall, from 1.8 to 10 B.W. gages. For average-wall tuba, the variation in wall thickness from the nomina1 size
may be i 5%; for thin-wall tubes the variation usually is positive. These variations affect the in- side diameter and must be kept in mind, especially far smalldiameter tuba and turbulent flow since pressure drop varia inv&ely as the diameter raised to the 5.2 power, in accordante wit,h Eq. 17-17 and Eq. 17.19.
Tube Zengths. Standard over-al1 tube lengths are 8, 12, and 16 ft.” The cast of the expensive head
100 PETRO/CHEM ENGINEER July 1966
and Shell walls atid of from considerations o allowable values are glven m
Tube diameter. The tuba used in exchan 1 specified by their outside diameters. The com used diameters of carbon-steel, admiralty-metal, and coppa tuba are s, %, 1 and 1Q in: and , , l?& in. far steel tubes. The sizes reconnnended by TEMA are shown in Table 17-l. The lager di- ameters are used far viscous fluids that have a tendency to foul the tubexrapidly. In some casa, even for viscous fluida, it is possible to use tubes of modeiate diameter by increasing considerably the flow velocity and thus reducing the tendency e to foul. In a plant where a number of exchangers are installed for different duities, it is desirable to use one or as few as possible tube diameters in order to reduce .serv~e and replacement costs appreciably.
Using a smaller tube diameter usually increases the heat transfer coefficient and the cast per square foot of the cooling area. However, as may, he seen from Eq. 6-18 or Eq. 17-17, for a smaller diameter t,he pressure drop increases and with it the power cast. As a result, for giva heat transfer opera- tion, there is an optimum diameter at which the sum of’ fixed charges and power eost reaehes a minimum. Also, the difficulty of eleaning the in- side of the tuhes increases with a deerease in diameter.
Tubes of admiralty meta1 and copper are made in smaller siaes of 1% 96 and ;&in. diameters as well as the above listed sizes. Pipa of from $& to l%-in. in diameter, made of manv snecial ferrous l ,_ and nonferrous alloys such as stainless steel and monel metal, are also available.
cast,ings and of machining the tube sheets, ex- panding the tube ends in the tube sheets, and assembling the whole exchanger is practieally the same for both short and,~long exchangers. An in- crease of the heet exchanger surface and capacitp can be obtained by using longer tube and Shell. Therefore, it, is advisable to use longer tubes, whenever possible. Thus, the use of 12-ft tuba instead of 8.ft tuba lowers the cast of a beat exchanger for a ,given capacity by about 16% and the use of 16.ft tuba lowers the cast by 24cc.s
If necessary, tube lengths from 4 to 22 ft can readily be obtained. The shorter lengths are used: (a) when the exchanger is loeated high above the floor, to reduce the cast of supporting platforms and hoisting xquipment for removing the tube bundles for cleaning; (bj where the length is limited by available space; or (c) where very large shell diameters must be used. Location on or near the groun’d should be, preferred, especially when exehangers require frequent servicing.
Tube arrangemmt. The recommended pattern is the square pitch, as show” in Fig. 6.4a and 6.4b;
a
and tube lanes should form continuous lines throughout the bundle. (P/CE, May 1965.)
Tuba on a triangular pitch, Fig. 6.4~ and d, may be used when the Shell-side fluid is non-fnuling.
Tube pitch, or center to center distante, usually should not be less than 1.25 D,. When tubes are on
Fi.wre 17.2. Types of tranwerse baffles.
17-5. Fixed tem~peratures. If, in addition to the main data, the terminal temperatures are slso de- termined by proeess requirements, the beat balance and hence the values of the beat flow rate Q and Atl are fixed and can easily be computed. The mass- velocities G, and G,, can be determined at first from the pi-ocess requirements and later corrected when the flow areas are established. The outside tube diameter Do and the wall thicknesn that determines the inside diameter Di are selected in accord once with data given in Section 17.3. For given mass flows, m’ and nl”, and giva or selected mass velocity G,i and diameler Di, the tota.1 number N of the tuba is found. Sinee the temperatures are
PETROKHEM ENGINEER July 1966 101
bz~Y$J6 c3 a square piteh, a minimum eleaning lane of ?/G in. should be provided.
I Ba.fies. Baffles are used 0000
1
to obtain better mixing on 0000 the Shell side either by
changing the direction of Figure 17-3. Single pass the flow or by loca1 in. tube arrangement.
creases of the velocity of the fluid. The three most commonly used types of transverse baffles are show” in Fig. 17-Z. The segmentai baffles, Fig. 17.2a, serve mainly to pro- vide good mixing by a complete reversal of the flow. The disk-and dmghnut ba.ffle combination, Fig. 17.2b, also provides reversal of the flow but does not affeet the centra1 part of the shell space as well. The wifice baffles, Fig. 17-2c, cause mixing through an increase of velocity in the ring-shaped open- ings, or orifices, where t,he tuba pass through the baffles. Orifice baffles give the most effettive, uniform mixing, however, at the expense of a greater pressure drop than the other baffle types.
The holes in t,he segmenta1 and disk-and-dough- nut baMes are drilled %Y in. larga than the out- side diameter of the tuba. TO avoid vibration of the baffles and scoring of the tuba, the holes must be slightly countersunk from both sides to remove burs. The baffle piate thickness should be at least $6 in. for a Shell diameter not over 13 in., hut in any case not less than twiee the tube-Wall thickness, if t,he support plate spacing is not over 24 in. For an increase of the Shell diameter and support spacing the plate thickness should be increased to 5/~ in. More explicit details are ,given in TEMA statiwas.
When longitudinal baffles are used, they may be welded to the shell or may be of the removable type, with provisions to prevent excessive leakage.
17-4. Fixed desQn data. In addition tu the main data, the amount of heat to be transferred for gioen initial temperatures of the two fluids, the beat exchanger design may have to meet eertain other requirements, such BS terminal, or initial and fmal temperature? of the fluids, mass-flow rates, velocities, pressure drop or t,ube length dictated by space limitations.
Naturally, the same equations apply in al1 cases, but the approach in each case may be slightly different and the equations can be rearranged to fit the speeific conditions.
specified, the physical properties of the fluids are fixed and can be caleulated. After this, the de- signer can calculate the coefficients, ih; and ho, and find the over-al1 eoefficient U. With these data the tota1 required length L of tubes is found from Eq. 17.3, and dividing L by the number of tubes N, the tube length is found. This will determine the number of passes and the tube, arrangement, and in case of cross-flow on the Shell side, the number R of tuba in a horizontal row and number i of tuba in the vertical banks. With these data, the inside diameter of the Shell is found. Finally the friction coefficients fi and fu and t,he pressure drops L$< and Avo are ealculated.
17-6. Fixed piessure drop inside the tuba The pressure drop inside tuba usually is higher thsn that outside tuba and eontrols the design. Com- bining Eq. 17-1, 17-4, 17.9, 17-15 and 17-17 and eliminating L giva :
where the symbol ei stands for :
2,s
+,i = (t’,tit’j” (T) Eq. 17-22
and Abi is the temperature drop acrosa the inside fluid film, or the mean temperature differente be- tween the fluid and the inside tube surface. The dimensionless factor Bi is giva by Eq. 17-18 but more eonveniently may be eomputed from the equation:
Lli=l+* Eq. 17-23 / 5
where ntn is t,he number of tube passes and the numerica1 factor k, is 0.55 for single-pass and about 1.0 for multipass fl,ow if the flow in the tube is turbulent and its average velocity is about twice as high as in the headers, which usually is the case.
Thus far a giva rrtD, Bi can be computed by means of Eq. 17-22 and 17-23 without determining the heated tube length L,,. The procedure of finding Lh depends upon the inside thermal resistame.
Zmide resistance eontds. If the beat resistance on the inside is the controlling factor, as in an oil cooler or a steam-heated gas ha&&, the pro- cedure is as follows:
1. The faetor 4~~ is calculated from Eq. 17-22 and Bi is found from Eq. 17-23.
2. Gi is computed from Eq. 17-21 as for a tur- bulent flow.
3. Di is selected and the Reynolds number DiGi/pi is found; when pi is giva, DiGJpci usually exceeds 2100 and Eq. 17-27 applies.
4. hi/cGi is eomputed from Eq. 17-9 and the heated length Lh of tuba in one pass is found by combining Eq. 17-1 with Eq. 17-3 and noticing that S = r Di2/4 and L = Lhn,. This gives:
Lh, = D;G$i(tg,- t,‘)
41~s At,n, Eq. 17~24
102
5. The number of tubes in a bundle is found from the flow rate m.< as:
7% ’ = 0,7854D&
Eq. 17-25
6. The length of each tube L,,’ is found by divid- ing Lh by i from Eq. 17-25. If Lh is not tao long, a single pass is used, n, = 1 in Eq. 17-24, other- wise a multipass arrangement must he used.
An alternate procedure for the 4t,h step is to find fi from Eq. 17-19 and then to find Lh from Eq. 15-17 as:
2gc b&‘i LI, = fiBiG;2nl, Eq. 17-26
EXAMPLE 17-1. Determine the dimensions and the number of steel tuba for an air preheater to heat 80,000 lb/hr of air at 125 psig from 65 to 195 F by condensing steam at 2 psig outside the tubes. The permissible pressure drop in the tubes is 5 psi. Readily availahle tuba have 1 in. O.D. with 12 BWG walls; however, both smaller and larger tubes may be obtained, if necessary.
SOLUTION: The steam temperature, from st,eam tables, is 218.5F. The tube surfaee temperature will not differ appreciably from the steam tempera- @ ture and the mean temperature differente is found by Eq. 14-4:
nt 1
= (218.5-65) - (218.5-i95i= 69,2F
In 218.5 - 65
( 1 218.5 - 195
This gives for the mean air temperature t = 218.5 - 69.2 = 149.3F and the air film temperature, with~ turbulent flow will be tf = (218.5 + 149.3)/2 = 183.9F. At this temperature and atmospheric pres- sure the air properties are, from Table 4-l (P/CE, Mar. 1965, p. 47), e = 0.241 Btu/lb-“F; p = 0.0617 lb’cu ft; JL = 0.051 Ib/ft-hr; k = 0.0178 Btu/hr- ft-“F. The higher premure will affect only the density p, increasing it to p = 0.0617(125 + 14.7) /14.7 = 0.587 lb/cu ft.
The mass velocity G, may be found from Eq. 17-21, in which a3 = 0.023, as stated in conneetion with Eq. 17-10. In connection with Eq. 17-19, ai is equal to 0.22 $ and $ = QLL/~~)~.~~ where 0 p must be taken at the bulk temperature of the air, t = (65 + 195j,/2 = 130F, or ,p = 0.048 lb/ft-hr, and ps at the surface temperature, t, = 218.5F, or ps = 0.053. Thus:
0.14
a/, = (E) = 0.9854
and hence ai = 0.22 x 0.9854 = 0.217. For air a safety factor F, = 1.25 is used. The factor & is found from Eq. 17-22 as:
ei = ‘“;,“” (0.69)W = 1.467
and Bi must be found by Eq. 17-23, where in this case, 1~~ = 1 and ki = 0.55. Thus:
Bi = 1 + 0.55 x 0.023
1.467 x 1.26 x 0.21’7 = 1.032
PETROJCHEM ENGINEER July 1966
with these values and Api = 5 x 144 = 720 psf, by Eq. 17-21:
Gi = 2 x 0.023 x 4.17 x lo8 x 0.587 x 720
0.217 x 1.25 x 1.467 x 1.032 = 140, 300 lh/hr-sq ft
If l-in. tuhes are used with 12 BWG walls, from Table 17-1, D; = 0.0652 ft and the Reynolds nuinber is found in order to check whether Eq. 17-21 applies:
NR, = 0.0652 x 140,300
0.051 = 179,500
New (tL/cG)i is computed from Eq. 17-9:
h
( )
0.023 -= CG i 1.26 x 179,5000.2x 0.69 218
0.023 = 1.25 x 11.29 x 0.781
= 0.002096
By Eq. 17-24, noticing that in this case At, = At,, the length of each tube is, assuming ntP = 1,
L h
= 0.0652 x (195 - 65) = 14,6 ft 4 x 0.00209 x 69.2
In order to check this value by Eq. 1’7-26, the a ” fnetlon coefficient fi is found from Eq. 17-19:
fi = 0.217
179.5000-2 = 0.0193
and from Eq. 11-26:
L, =720 x 2 x 4.17 x 108 x 0.587 x 0.0652 2 h 0.0193 x 1.032 x 140, 300a
14.6 ft
Thus, 14.6-ft long tube8 are sufficient, howeoer, prohably 16.ft standard lengths will be used.
The number of tuba is fo’und from Eq. 17-25:
80,000 ’ = 0.7854 x 0.06522 x 140,300
= 171 tubes
The tubes will be spaced with a 1.25-L pitch, so that each tube will require 1.25% = 1.6625 sq in. and 171 tube requiie about 267 sq in. The inside shell diameter must be about D = d267/0.7854 = 18.5 in. or slightly under 20 in.
Inside resistance does not control. If the inside
0 heat resistance is appreciable hut is not controlling, the following procedure may be used:
1. A preliminary value of lJ/h< is aasumed and At, is evaluated as At, = At,,,,U/h,.
-- TABLE 17-l. TEMA Standard Dimensions
‘Far Heat Exchanger Tube9
PETROKHEM ENGINEER July 1966
2. The value of Gi is eomputed from Eq. 17-21 as in the first case.
3. Di is selected and h; is found from Eq. 17-9 and U, is found from Eq. 17-8.
4. The ohtained ratio U/hi is compared with the assumption in Step 1. If the differente is con- siderable, the procedure is repeated, using the obtained ratio lJ/h,,, unti1 satisfaetory results are obtained.
5. The length Lh is computed from Eq. 17-24. 17.7. Fiaed ;oressure drop outside of th,e tubes.
Taking Eq. 17-l in which SO = ‘vo%, where y0 is the clearance between the tuba in a bundle that corresponda to the minimum free area, and com- bining it with Eq. 17-4, 17-10, and 17-20 and eliminating the number of rows n, the result is:
where the dimensionless factor +0 stands for:
Eq. 17-28
Again two eases must be considered. Shell-side vxistance eontrols. If the heat re-
sistance on t,he Shell side is eontrolling, U0 = ho and A t. = At,,. The factor & is computed from Eq. 17-28, G. is found from Eq. 17-27, ho is found from Eq. 17.10, and SO and ~0 are found from, Eq. 17-l. Naturally, n must be a whole number;
Shell-side resistame does not control. If the beat reaistance on the outside of the tubes is appieciable but not controlling, G. may be found for & fixed Ap, from Eq. 15-27 using u trial and error proee- dure similar to that outlined for the fixed pressure drop inside the tuba.
17-8. Fised tube Zength. If the tota1 length L = nsLh is fixed, and the desired fina1 temperatures are specified, the corresponding value of Gi may be found by combining Eq. 17-1, 17-4 and 17-9. This giva the weight rate of flow:
Gi= ($($$)’ Eq. 17-29
If the inside beat resistana is eantrolling, di is found directly from Eq. 17-22 and Gi is found from Eq. 17.29. Howevei, the term with the exponent 5 must be known precisely. Thus, ehanging the safety factor F, from 1..25 to 1.35 lowers Gi by 32%.
If the inside heat resistanee is considerable but not controlling, Eq. 17-29 may be used by a trial and en-or procedure but & must be approximated very closely.
If the outside beat resistance controls, G. may he found by combining Eq. 17-1, 17.4, and 17-10. The result is:
Eq. 17-30
where y, is the clearance in feet between the outer surfaces of tube8 in a bundle that corresponds to the minimum free area, and:
103
It should be remembered that Eq. 17-29 and 17-30 apply only to turbulent flow.
EXAMPLE 17-2. A vertical closed-type feed-water heater must be designed to heat 50,000 lb/hr of water from 56 to 18OF, using saturated steam having a temperature of 228F. Maximum length of tuba cannot exceed 10 ft.
SOLUTION: With sufficient accuracy, the tem- perature of the inside tube surface can be taken BS ~228F and At, = 228- (56 + HO)/2 = 1lOF. The prop.erties of water at the average temperature of 118F are c = 0.997 Btu/lb-“F, /L = 1.39 Ib/ft-hr, k = 0.371 Btu/hr-ft-“F, and Np? = 3.73.
By Eq. 17.22:
180 - 56 ?+Jr = 110
x 3.7321” = 2.71
Select Da = V-in., admiralty metal, 16 BWG tuba, or from Table 17-1, Di = 0.0517 ft, and BS stated in connection with Eq. 17-9, for water F, = 1.25.
Therefore, by Eq. 17-29:
(& = l.xJ +
4 x 0.023 x 10 0.0617 0.0517 x 2.71 x 1.25 i
= 90,500 Ib/hr-sq ft
Before going any further, it is necessary to check whether the Aow is turbulent. By Eq. 6-10:
Nnr = 80,500 x 0.0517 = 2ggo
1.39
The neeessary tota1 flow area S = 50,000/80,500 = 0.621 sq ft and the number of tuba:
0.621 i = 0.7854 x (0.0516)2
= 297 tube
with a square pitch p = 1.00 in., the cross seetion of the, Shell must be:
s = 297 x 1.0” = 297 sq in.
and the, inside diameter of the Shell is:
D= = 19.45 in.
A shell diameter of 20 in. can be used. 17-9. Fized velocities. The tube dimensions Di
and Do must be selected and the corresponding values of 9, II.<, h,O, and Ll calculated. Then the product FL is found from Eq. 17-3. If L is such that~ a single pass can be used, F is 1; otherwise the type of tube arrangement and fluid flow out- side of tuba must be selected and F evaluated from Fig. 14-5 or 14-7, as the case may be. The corresponding pressure drops ase found from Eq. 17-17 with the corresponding auxiliary equations and from Eq. 17-20 or Eq. G-29 with Eq. 6.30 or Eq. G-31 (P/CE, May 1965).
If the velocities are not fixed definitely and the designer has a. choice, he may first select velocities
104
far the inside and outside flows and go through al1 eomputations required to operate it us outlined above, then find the size and estimate the cast of the exchanger and of the power. After this is done, another velocity is chosen and the whole procedure is repeated, resulting in a new size and cast of the exehanger and cast of the power to operate it. The second solution will indicate the trend and facilitate the selection of optimum velocities.
In general, higher velocities are desirable in order to imure turbulent flow and higher surface coefficients. The limits for velocity are set by pressure drop and power consumption.
For air and other gases optimum linear veloci- ties are in the region of 20 to 100 fps. For liquids reasonable velocities are 3 to 6 fps.
Ir-10. DetermGuztion~ of outlet tempemtures. Far fixed inlet temperatures tf and tl”, weight-flow rates m’ and m”,, and giva physical properties of the fluids, the procedure for finding the outlet tem- peratures ts” and tz”, henee the heat capaeity q which would be obtained in a given heat exchanger is as follows: The over-al1 coefficient must be evaluated in the usual way and the temperature ty’ found from an equation obtained by combining l Eq. 17-2, 17-3 and 14.4. The result is:
tu = tl’(.l - Y) + (ai - 1) b” aj - Y
Eq. 17-32
where :
a5 = e “AP Cl-Y,,ln’C” Eq. 17-33
The other terminal temperature tz” is found from the beat balance, Eq. 17-2.
If the value of Y given by Eq. 14-7 is unity, At is constant throughout the counterflow ex- changer and the over-al1 terminal temperature differences are equal.
In this case:
tZr = tl’ + a&”
l+as Eq. 17-34
UAF G=- - m’e’
EXAMPLE 17-3. A horieontal Shell-and-tube ex- ehanger in an oil refinery has the following dimen- sions: 320 3/4-in. 14 BWG steel tube 20.ft long are arranged on a triangular pitch of 15116 in.; the Shell has an I.D. of 22 in. and 22 baffles in the shape of disks mith alternate 5.in. segments cut off from bottom and top. The petroleum fraction that must be cooled flows in t.he Shell side, in a single pass, at 91,000 Ib/hr and has an average specific gravity y = 0.74 and a viscosity of 0.30 cp, and enters with a temperature of 285F; the colder fraction flows in a quadruple pass at 140,000 Ib/hr, enters the tubes at 190F, and has an average viscosity of 0.38 cp and & gravity y = 0.69. The thermal cpnductivities of both hydroearbons are 0.0765 Btu/hr-ft-“F, at 86F, and 0.0745 Btu/hr- sq ft-“F, at 212F. Determine the termina1 tem-
PETRO/CHEM ENGINEER July 1966
WASTE HEAT BOILERS, incorporating the combined skills of oor boiler and heat exchanger divisions, offer a wide range of designs for the recovery of heat from gases. 1 They give efficient, low maintenance service in ammonia plants. methanol plants, petroleum refineries, petrochemi.
L HIGH PAESSURE EXCHANGERS. Vogt heat exchanger desiens. far sawicas which involve- orassores oo @O&~til& &lf:sealing gaskets such as double’cone and delta types as well as the more conventional types.
peratures of both fractions and the heat capaeity of the exchanger at the giva eonditions.
SOLUTION: First, the physieal properties must be established. The hotter fraction has p = 0.30 x 2.42 = 0.726 Ib/ft-hr. Its conductivity k is, found by extrapolation, assuming an average bulk tempera- ture of 260F, as k = 0.0737 Btu/hr-ft-OF. Its specific heat of 260F is found by Eq. 7-12, estimat- ing yeo = 0.808, as c = 0.625 Btu/lb-OF. The colder fraction has p = 0.32 x 2.42 = 0.775 lb/ft-hr; its conductivity, assuming an average temperature of 200F, is found by interpolation as k = 0.0747 Btu/hr-ft-OF and the speeific beat, by Eq. 7-12, with yso = 0.737, as c = 0.648 Btu/lb-OF. The tube diameters are: D, = 0.75/12 = 0.0625 ft and Di ,= (0.75 - 0.083 x 2)/12 = 0.0486 ft, and the shell diameter is D, = 22112 - 1.833 ft.
The cross area of one tube is 0.7854 x 0.0486z = 0.001855 sq ft and the flow area through the tuba, with four passa IS found as S = 0.001855 x 320/4 = 0.1484 sq ft. Thus the weight velocity Gi = 140,000/0.1484 = 944,000 lb/h+?sq ft,.
The Reynolds number, in each tube. with Gi = 944,000/80 = 11,800 lb/hr-sq ft, is by Eq. 6-14:
N Re = 0.0486 x 11,800 = 740
0.775
The flow is laminar and Eq. 7-33 must be applied. It giva:
= 34.75 Btu/hr-sq ft-“F
For the Shell side the smallest cross area will be the cut;off seynent in the baffle plates, for which k/D - 5122 = 0.227; S/D’ = 0.134; and S = 0.134 x 1.8332 = 0.45 sq ft. Therefore in the cut-off sement, there are 320 x 0.45/2.64 = 54 tubes and the clear area, is:
S = 0.45 - 0.7854 x 0.0~?25~ x 54 = 0.285 sq ft This gives GO = 91,000/0.285 = 319,000 Ib/hr-sq ft.
The equivalent diameter must be computed by Eq. 6-11. The length of the art is7 0.9693 x 22112 = 1.775 ft; the length of the chord is 0.8377 x 22/12 = 1.535 ft and:
D, 4 x 0.285
= = 1.775 + 1.535 + T x 0.625
0.079 ft x 54
and, by Eq. 6-14:
NE, = 0.079 x 319,000
0.726 = 34,700
The flow is turbulent and Eq. 6.17 must be applied.
It giva:
ho = 0.0625 (E) 34,700’,“( o~5g;,;~) ‘-’
= 179.0 Btu/hr-sq ft-“F
The conductivity of steel for these tubes may be taken from Table 3-8 (P/CE, Feb. 1966, p. 38) as 25.9 Btu/hr-ft-“F.
New, the over-all beat transfer coefficient is found by Eq. 13-18:
1 0.0288 + 0.0002 + 0.0043
= 30.0 Btulhr-sq ft-“F
By Eq. 14-3, remembering that m’ and c’ refer to the fluid in the tube:
y = 140,000 x 0.606 91,000 x 0.592
= 1.575
The inside area Ai = r x 0.0486 x 20 x 320 = 977 sq ft. The faetor F should, be taken from Fig. 14-5a. However, since the temperature efficiency X is not known, as a first approximation F may be taken a8 equa1 to 1.0; later F may be determined and the results corrected. With these data, the exponent in Eq. 17-33 beeomes :
From the ,middle and right parts of Eq. 17-2: ‘, tz” = 1.575(190-213) + 285 = 248.8F
New, by the right side of Eq. 17-2, the beat ca- pacity is found a8:
q = 91,000 x 0.592 (286-248.8) = 1,950,OOO Btu/hr
TO check the assumption of F = 1, X is determined by Eq. 14-6:
x = 213-190 285-190 = o’242
From Fig. 14-5a, with Y = 1.575, F is read as 0.98. The differente is negligible. In fsct, in- troducing this value in the expression for the ex- ponent gives 1~ = -0.19, as ,= 0.822. and finally, ti = 212.4F instead of 213.OF.
References
‘D. L. Katz and D. E. Briggs, “Chemical Engi- neering Progress”, 61, 91 (1965).
3Standards of TEMA, op. cit., pp. 85-87. ‘Standards of TEMA, op. cit., pp. 7-77. %‘Wmiar& of TEMA, op. cit. pp. 8, 25, 49. BE. N. Sieder, “Cost Relations of Heat Ex-
chingers,” Chemical and Metallurgica1 Engineering, 46, 322-25 (1939).
?Marks’ Handbook, loe. cit.
I Part 18. the concludine oart. I
; Combined percentage* of “Should Have” and “COUld “Ee” rerponses
1 5gure 2. Job-oriented subjects that at least 50% of the tngineers in the Pennsylvania study thought they “Should
i I iave” or “Coold “se.”
Petro/Chem Engineer August, 1966
E. Management
Understanding Individua1 & Group Behavior in Wark Situati0
needs rather than on just achieving a degree. How did this group keep up-to-date on new tech-
nological advances? Fig. 1 represents graphically the use that was made of each medium. It also illustrates quite pointedly the number of engineers wh” responded that they do not use any other soui-ces. Note the large area of “Do Not Use,” which indicata the number of engineers who do not utilize ~“uices of information that could be used far t,heir professional updat,ing.
Training Needs - Each engineer indicated his degree of need for further training and education in 1’72 job-oriented subject arem by marking “Should Have,” “Could IJse,” “ì “Don’t Rally Need” for each. Fig. 2 tabulata t,he professional training needs of the engineers that rated 5O:a or better. Not too surprisingly, the highest per- centages were chalked up by subjeets in the com- munication skills category, such as rapid reading, public speaking and technical report writing.
Perin Sta,te’s Dubin and Marlow offer a number of recommendat.ions for the company, professional society and the engineer to improve this area of contiming professional study in the refining industry:
For the Company
l Management, to be creative, should initiate and foster the updating of its employees. This process requires a three-pronged attack (a) personal moti- vation of t,he individua1 employee, (b) the strong, support and encouragement of supervisory per- some1 and (e) company policy that insists on up- dating personnel as a part of the daily work. l More released time should be granted to engi- neers to take advaneed degree and noncredit work. A sabbatica1 leave should be provided BS o regular practice.
l Long-range educational programs should be initiated and financed to parallel long-range com- pany programs and polieies.
l Top management should review periodically the updating needs of its practicing engineers and initiate correttive action where necessary.
For the Professional Society
l In conjunction with universities, societies should seek t” motivate engineers t” pursue eon- tinuing professional education.
l Engineers should be encouraged to beeome attive in the professional societies in order to help keep abreast of current technieal developments.
l Professional societies should expand their offerings of short c”urses, utilizing feasible varied media of instruction to fulfill the needs of practicing engineers.
For the Engineer’.
l The engineer has a responsibility to himself and to his company to undertake contiming pro- fessional education to keep abreast of current de- velopments in his field. n
35
Economie Factors in Heat Transfer ~~ Part 18 (Conclusion)
Applied, Heat Transfer Design for the H~PI
by Wayne C. Edmister and Joseph M. Marche110
18-i. Intnxhetion. The general definition Of economy in any industrial installation is: g$tins the most for the money spent “ì paying the least to obtain eertain results. In order to achieve the best economy in heat transfer, it is necessary to analyze the factors that enter into the fina1 cast of any commodity eonnected with heat transfer.
The costs ~of owning and operating heat trans- fer eqnipment mny be grouped under two main hendings: Ca) fixed eharges and ib! operating charges.
Fized cl~wges. Fixed eharges are the eosts re- sulting from ownership of equipment and include: il) interest on the investment, (2) amortization, (3) insuranee and (4) taxes. If the equipment is “nly rented, rental charges take the piace of in- terest and amortization.
From t,he above considerations, it is evident that if equipment can he rented at a cast nmaller t,han the interest on the capita1 required for its purchase plus the c~ost of its amortization in a reasonable time commensurate with its wear and expected life, it will be a mistake to purchase the equipment.
Operati~ng dmges. These charges include cast of: (lj power, (2:) fuel, (3j cooling water, (4) labor to opera,te the equipment, (5) maintenanee cast, which in turn includa such items it8 lubricating oil, replacement of worn and broken parts, and the lahor required far making repairs, cleaning, paint- ing, etc.
Final cast. The final cast of operating industrial equipment is arrived st 8s the sum of fixed and operating charges.
Fixed charges can he lowered by purchasing a simpler, less expensive equipment. H6wever, such equipment usually is less efficient and its operating charges may overbalance its smaller fixed eharges. Thus, before deciding on the purchase or rental of equipment to obtain cert,ain cooling or heating re- sults, it is best t” compute the final cast for severa1 types of equipment: elaborate and efieient, but expensive; and, less efficient but also less expensive. The questipn of amortization is also very important. If an efficient but relatively expensive piece of equipment can be expected to be useful over a long period of years, its amortization may be lower than that of a piece which is less expensive. Being built
36
of eheaper material, the less expensive equipment may deteriorate or become obsolete considerably s”“ner and therefore must be amortized in a fewer number of years, resulting in a higher amortiza- tion cast.
The main faetors that affect heat transfa effi- ciency are: (a), type of heat exchangers, Cb) beat exchanger area, (cj fluid velocities, (d) tempera- ture differente and (e) kind and thiekness of the l heat insulation.
18-Z. Types and, sizes of ezchangws. Al1 beat ex- “bange equipment may be classified into three main groups: (5) tube exchangers, (b) double pipe ex- “hangers, and (cj shell-and-tube exehangers.
Examples of tube exehangers aie water co”li,ng coils in cooling towers and al1 tubular steam gene- rators. Fintube automobile radiators and air-cooled condensers in househpld refrigerators also belong tu this class. The use of such heat exchangers is ususlly predetermined by the service wjuirements.
Double pipe exchsngers are used BS ammonis condensers, brine eoolers, air dehydrators, lubricat- ing oil eoolers and in various industrial processes. Double pipe exchangers are quite efficient and ere used successfully far smaller installations. For large amounts of beat transfer, their cast usually be- come prohibitive.
Shell-and-tube exchangers eomprise the largest group or type of exchanger in industrial usage (I, today. They are used for heat exchange between two liquids; between a liquid (usually water) and a gas; between steam and water in surface steam condensers; and, between two gases, as in air preheaters. The type of shell-and-tube apparatus used for beat exchange between two liquids may influente not only the first cast, but also the mainte- narice cast as well. One of the most important factors is fnuling of the tube surfaces. Since it is easier to clan t,he inside of,tubes, that liquid whieh gives more depdsit, should be put, through the inside of the tuba. The arrangement of the flow, counter flow, parallel tlow or mixed flow, and the number of tube and Shell passes infuenees the fixed charges and the maintenance charges in opposite direetions: the simpler the arrangement, the lower the first cast, but the higher the operating cast. The optimum condition can be determined only by making severa1
Please turn to page 38
August 1966 Petro/Chem Engineer
Petro/Chem Engineer August 1966 FN i”,liw Inhwtion on bdW,id hdur,r sas bodsr h-4<r h,s, 37
-
different selections and ealcolating the combined or fina1 cast per year far each selection.
18-5. Heat erehmnge arm. A greater heat transfer area per unit of heat load in general gives a higher efficiency. However, it also inereases the size of the exchanger and hence the fixed charges. The tota1 cast, C, in dollars, of .a beat exchanger depends: (a) on its size or heat exchange area; (b) on the basie cast, Gli, dollars per sq ft of area, which in turn depends upon the type, construction and metals used for the tuba, tube sheets and Shell; (c) on the working pressure as expressed by a factor kp; and (d) on the size, or area factor, IL. The tota1 cast can be presented as a product of these values:
C = AC, . k, * K.4 .~ Eq. 18-1
With the present rising costs of materials and labor, cach manuf&urer must determine the value of CII in accord with his operating conditions and the prevailing market prices. The value of the factor kg may be taken from the curva of Fig. 18-1, depending upon which of the pressures, in the tuba OI’ in the Shell, is higher. The size factor Ka may be taken from ,Fig. 18-Z.
Figure 18-Z. fnfluence of tota, area on coWof shell-and- tube exchangers.
18-h. Fluid velocities. Determination of the opti- mum flow velocities in exchangers for the general case, in which the surfaee coefficients /z; and hu on each side of the tube vary with ,velocity, is rather involved because within certain limits the veloci- ties and eorresponding power~costs can be varied independently. Sine the derivation of these equa- tions does not contribute to a clearer understanding of the heat transfer problems, only the fina1 results will be giva bere.
The two symmetrical equations for the optimum i.elocities inside of the tubes and on the shell side are:’
c,i (Ce;, RI) wls G; = Eq. 18-Z
2.5 -t 2.76 At,/At, ,I cai/(c,” Ir”)
1
Il.:lil Eq. 18-3 3.76 + 3.39 Atg’At,
In these equations C, is the hourly fixed charges on the exchanger, in dollarsjsq ft-hr, based on Ai. Tf C is the tota1 cast of the exchanger, then:
Cd = C(1 + mEi
&nT Eq. 18-4
where E is the yearly amount of fixed charges exelusive of the first cast, meaning interest, in- summe, and taxes, n is the number of years of useful life of the exchanger, and T is the numbw of hours per year of exchangei’ operation.
Ce is the eost of mechanical energy supplied to the fluid, dollars per foot-pound, Cei, for fluid inside tube, Ce, for fluid outside the tuba
Ki is a dimension’term defined by the equation:
K, = Bi% (/dDi) “i 2g&
Eq. 18-5
where Bi is a dimensionless correction factor given by Eq. 17-18 and Eq. 17-23 and the values of ai and m are the same SS in Eq. 17-19; pi is the mean density of the fluid inside the tuba.
Similarly :
K,, = 2Boao (po/Du)“u(~o/DJ
mp% Eq. 18-6
where B,, is a correction factor discussed in connec- tion with Eq. 17-20 u0 is a dimensionless factor which must be caliulated from Eq. 6-30 oì Eq. G-31, depending on the tube arrangement; % is the minimum elearance between the tubes ; the exponent ntO = 0.15; and pLo is the~mean density of the fluid outside the tuba.
The terms Gti and Ath in Eq. 18-2 and 18-3 are temperature differences: At, - between the mean temperature of the tube surface and mean fluid temperature inside the tubes; and At,, - between the mean temperatura of the tube surface and fluid outside the tuba.
Eq. 18-2 applies when the Reynolds number (DG/‘+L)~ exceeds,lO,OOO and Eq. 18-3 when @G;‘@o exceeds 1000, which is almost always the case.
18-5. Practical application. The procedure far finding the optimum mass velocities Gi and G. is as follows. The unit eosts Ca,, Cei, and Ceo must be eomputed from available cast, ,data. The terms Ki and K,, are evaluated by Eq. 18-5 and 18-6 from the physical properties of the fluids and diameters and spacing of the tuba. The values At, and Ata must be measured or estimated. Trial values of G; and G. are calculated from Eq. 18-2 and 18.3. The eorresponding values of h; and II.” are eal- culated from Eq. 17-9 and 17.10 respectively. The surfaee temperature is calculated from Eq. 17.16 and the corrected values of At, and At, are found. With these new values revised values of Gi and GO are found~from Eq. 18-2 and 18-3. Since con-
38 August 1966 Petro/Chem Engineer
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Petro/Chem Engineer August 1966 Tor Furlher Info,“Olim on Idvar,ired Pmd”Cll lae Radar scni<a cantsr
siderable variations in the values of the terms in parentheses in the denominators of Eq. 18-2 and 18-3 bave only a small effect on the values of Gi and Go, the equations will converge rapidly.
EXAMPLE 18.1. Design an air preheater for the conditions given in Example 17-1, however, using flue gasa to be eooled from 750 to 390F for the heating medium outside the tuba and taking into account the following cast faetors: Annua1 fixed charges are estimated to be $1.05 per sq ft of ,inside surface and the equipment will be in opeiw tion al1 the time, allowing four days per year for cleaning and servicing; cast of power delivered to the air and flue gasa is $0.006 per kw-hr. Assume that the gas properties are the same as those of air.
SOLUTION: Det,ermine the properties of the air and flue gasa and the rate of flow of t,he flue gases. The mean temperature differente is found by Eq. 1,4-3 :
t, = ~(700 - 195) ~ (390 ~ 65) _ .QJOF 7Fdl - 1%
In --- 390 ~ 65
The air temperature ehanges less, therefore its mean temperature may be assumed as t, = (195 + F5i/2 = 130F and the mean flue gas temperature is then t, = 130 + 430 = 560F. This t, differs appreciably from the average temperature t = (760 + 3901’2 = 57OF. Thus, from Table 4-l (PICE, Mar. 1966, p. 471, the properties of air will he: c = 0.240 Btu/lb-“F; p = 0.048 lb/ft-hr; k = 0.0165 Btu/hr-ft-“F; and e,& = 0.70. ,The densit,y is found from the characteristic equation, since p = 16 Ib/cu ft, for p = (125 + 14.7) = 139.7 psia and the gas constant R = 53.34 ft/“F:
139.7 x 144 x 1 P= - 58.34 (130 + 460)
= 0.G40 lb/cu ft.
Far the fue gases the corresponding values are: c = 0.249 Btuj’lb-“F; ,, = 0.0390 lb/cu ft; /L = 0.070 Ib/ft-hr; k = 0.0262 Btu,/hr-ft-“F; and cpL/k = 0.66.
The rate of flow ?n of the flw gases is found from the heat balance Eq. 17-2:
The various terms to he nsed in Eq. 18-2 and 18-3 aie found as follows. The numher of hours of operation ia 7 = (365 -- 4) x 24 = 8664 hr/yr. Therefore C, = 1.05,‘8664 = 1.212 x lo4 $/sq ft-hr; C,!, = C,” 0.006 x 0.746/33,000 x FO = 2.25 x 1O-9 $jft-lb. The value of K,i must be computed by Eq. 18-5. The faetor Bi must be found by Eq. 17-23. As stated in eonnection with Eq. 17-23, if nt, = 1, 8s in this problem, K; = 0.55, also, us stated in con- nection with Eq. 17-9, a3 = 0.023, and in connection with Eq. 17-19, ai = 0.22 $ and $ = (CL/&) OS where p is taken at the bulk temperat~ure of the air, t, = 13OF, and /.L# at the surface temperature t, which at first must be estimated. The estimate can
40
be made using Eq. 17.16 and assuming that sinee the rate of flow in the tuba is severa1 times greater than outside, that 11,~ = 211,0. Then:
t, 0.0652 x 2h,, x 130 + 0.0833 x _ Ir” x 560
0.0652 x 2h,, + 0.0833 x h,"
= 298F
With this t,emperature, fiS = 0.057 lb/ft-hr and:
This gives ai = 0.22 x 0.9739 = 0.214. New by Eq. 1’7-22:
+== (1195 - 65) x 16.70) K =. o,61
(248 - IdO)
By Eq. 17-23:
Bi i 1 x 0.55 x 0.023 = 1 = O.Gl x 1.25 x 0.214
1,078
and, by Eq. 18-5, with pI = 0.640 Ibleu ft:
K,, = 1.0’78 x 0.214 (0.048/‘0.065Z)a~2 _ 6.34 x lO.‘o 2 x 4.17 x 10” x 0.640’
At, = 298- 130 = 168F; At,, = 560 - 298 = 262F. l
Now, when al1 the values entering in Eq. 18-2 are found, the latter giva:
Ci =
[
1.212 x 101
1
0.357
2.2G x lo-!’ x 6.34 x lO~‘“(2.j tu 2.76 x 2621168)
= 47,500 lb’hr-sq ft.
In order to find Go by Eq. 183, K,, must be com- puted by Eq. 18.6. As stated in eonnection with Eq. 17-20, if baffles are used, as is necessary~ in this problem, the value of Bu should be known from prac- tical experience. An average value Bo = 1.25 may be assumed. The value of ao depends upon the tube arrangement. If a squaw pitch is assumed, a0 is giva by Eq. 6-30, and the pitch p, = p, = II0 + vo = 11,. in.; z,, = 1.25 and zt = 1.25; m = 0.43 i (1.13/1.25) = 1.334:
0.08 x 1.25 0 ao = 0.044 + (1,25 _ li 1,9R4 = 0.68
The clearance was already assumed as the minimtim recommended, vo 7: 0.25 in. = 0.0208 ft, and TQ = 0.15. Therefore, by Eq. 18-G:
K” =
2 x 125 x 0.68(0.068;0.0833j”~‘~(0.0208/0.0652) 77 x 4.17 x 108 x 0.0394’
= 2.65 x IO-’
New, by Eq. 18-3:
GO =
1.212 x 10-1 li.%il .,~
2.26 x 10~” x 2.62 x 10-r(~3.75 A 3.39 x 168/262 1 = 4910 lb/hr-sq ft.
August 1966 Petro/Chem Engineer
The mass velocity Gn is rather low aria a check of A’R~,o should be made:
N RB.0 = 0.0833 x 4,910 = 5840
0.070
Since NP,,, > 1000, Eq. 18-3 applies. For flow in- side the tuba:
NR%i = 0.0652 x 47,500 =
0.048 G4 5oo
2’
Now the surface coefficients can be found. From Eq. 17.9:
!l.i = 0.023 x 0.240 x 47,500 1.25 x 64,500”‘x 0.70”
= 29.05 Btu/hr-sq ft-“F
Also, from Eq. 17-10:
h,, = 0.33 x 0.244 x 4,910
1.25 x 5,8400-“x 0.66”
= 13.18 Btulhr-sq ft-“F
The eorreeted mean surface temperature b, can be found by Eq. l’i-16 and al1 other values that, depend upon it should ‘be recalculated. By Eq. 17-16:
t, =
0.0652 x 29.05 x 190 + 0.0833 x 13.18 x 560 0.0652 x 29.05 + 0.0833 x 13.18
= 288F
This giva ps = 0.057 lb/ft-hr and:
0.14
0.048 $ = 0.057
~ = 0.0762
Then ai = 0.22 x 0.9762 = 0.215 and, by Eq. 17-22:
& (195 - 65) x 0.70” ~_ o.65 288 - 130
By Eq. 17-23:
Bi = 1 + 1 x 0.55 x 0.023
0.65 x 1.25 x 0.215 = 1.073
l By Eq. 18-5:
k~ _ ,1.073 x 0.215 (0.048i0.0652j”~” .,~:, 2 x 4.17 x lOa x 0.640’
= 6.35 x lo-lo
The corr&ted temperature differences are now At, = 288 -- 130 = 158F and At,, = 560 - 288 = 272F. With the new values, Eq. 18-2 gives:
Gi =
[
1.212 x 10-d U.857
2.26 x IO-” x 6.3 x lo-‘” (2.5 + 2.76 x 272/158! 1
= 45,500 lb/hr-sq ft
The numerical value of Ro is not changed and, by Eq. 18-3:
G,, =
[
1.212 x 10-4 1 0.351 2.26 ~~10.~ x 2.65 x 1O-i (3.75 + 3.39 x 1581272)
= 4050 lb/hr-sq ft.
Petro/Chem Engineer August 1966
The new Reynolds numbers are X,,r,; = 61.800 and Nmo = 6010. The corrected surface coefficient,s become :
hL = 0.023 x 0.24 x 45,500
1.25 x 61,8OO”.?x 0.70”s
= 28.05 Btu’hr-sq, ft-“F
li ”
= 0.33 x 0.249 x 5,050 1.25 x 6,010”.” x 0.66’h
= 13.33 Btu;hr-sq ft-“F
and the nem mean surface t,emperature becomes:
t, = O.OG52 x 28.05 x 130 i- 0.0833 x 13.33 x 560 0.0652 x 2X.05 t 0.0833 x 13.33
L 292.2F.
A third repetition of these calculations would give a value for t, between 288.0 and 292.2F. In viem of the converging tendency, the valu$ will be approximately 291.OF.
The next step is to check t,he assumption made in the derivation of Eq. 17-16, nemely that the differenee in temperatura ht on the inside and outside tube surfaees is small. The heat conduetivity of steel at 291F, k = 24.80 Btu/hr-ft-“F ; thickness of the tube wall, L = (0.0833 - 0.0652)/2 = 0.00905 ft; and taking into account that the heat conductivity in the tube is referred to the mean tube diameter, in this case to (OF0832 + 0.0652) /Z = 0.0742 ft, the beat balance equation for the heat flow through a squaw foot of the inner surfaee may be written as:
(291 - 130) 28.05 = At x 24.80 x 0.0742
o,oogo5 x C0652
From this equation, At = 1.44F. Then to establi,sh the fina1 values for G<, GO, hi and h.o. the surface temperatura should be taken as t,i = 291 - 1.44/2 = 296.3F and t,o = 291 + 1.44’2 = 291.7F. The fina1 values of the temperature differences will be:
With these temperature differences, the fina1 values of G; and Go become:
1.212 x 10-i 2.26 x lo-‘x 2.65 x 10~? (3.75 ~1m 3.39 x 160.31268.3)
= 46,500 lb’hr-sq ft Ga =
1.212 x 10-J
2.26 x 1O-9 x 6.35 x lo-‘” (2.5 + 2.76 x 268.3;1160.3:)
1 0.95:L = 6,000 lb/hr-sq ft.
The fina1 values for the surface coefficients are found us hi = 28.5~Btuihr-sq ft-“F and h.. = 13.4 Btu/hr-sq ft-“F.
Main Dimensions. The cross area of une tube S; = 0.7854 x 0.0672’ = 0.00333 sq ft. This giva the number of tubes required for a single Pass arrange- ment, by Eq. 17-25:
41
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80,000 z = 46,500 x 0.00333
= 516 tuba
as against 297 tubes in Example 1’7-2. This oecurs because of the much smaller rate of gas flow result- ing from the economie factors. The necessary length of the tubes is found from the tota1 heating surface Ai required for the heat flow Q:
SOLUTION: The Uver-al1 coefficient Ui iu found by Eq. 13-15, using the arithmetic-mean wea:
4 = 80,000 x 0.240 (195 - 65) = 2,500,OOO Btu/hr
1
ut= 1 0.0652 (0.0833 - 0.0652) 0.0652 28.5 +24.85 (0.0833 + 0.0652) T 0.0833 x 13.0
= 10.96 Bt,u/hr-sq ft-“F
and the required surface is found from the beat balance as :
and by Eq. 18-8: (& -k Xnij =
EXAMPLE 18-2. Find the optimum tota1 cast for Example 18-l.
Ai = 2,500,000
(290.3 - 130) x 28.5 = 547 sq ft
The inside surface of 1 ft of the tubes &’ = r x 0.0652 = 0.2048 8q ft and therefore, the length L of one t,ube must be:
L= 547
516 x 0.2048 = 5.2 ft
Either a G-ft or the smallest standard 8-ft length will probahly be used.
18-6. Optimm velo&/ in tzcbes. In many cases t,he power to be applied to the second Ruid on the Shell side is small compared to the power required far the Auid flowing through the tubes. In these cases, Iro is small compared to ki; Ato becomes very small eompared to At, and for DiGlIpi 2 10,000, Eq. 18-2 simplifies to:”
The procedure for finding G; is the same as for finding Gi by Eq. 18-2 and illustrated by Example 18-1.
The minimum tota1 cast, in dollars per Btu, ex- cluding the cast of the heat itself, is giva by the equation:‘,4
Eq’ “-’
In Eq. 18-8, X,, is the fixed cast, dollars per Btu trnnsferred :
x,; = Cd Ai Ca r! L’io Atm
Eq. 18-9
and X,< is the cast of power delivered to the fluid inuide the tubes, allowing far over-al1 efficiencies of motors, pumps, ete, also expressed in dalla-s per ,Btu transferred:
Eq. 18.10
Combining Eq. 18-8 and 18-9 giveu:
x, f x,; 1 & = 1 - (At,/At,,,,)n/(3 - mi)
Eq. 18.11
For turbulent flow, (GD/p) > 10,000, m. = 0.2 and n = 0.8.
Petro/Chem Engineer August 1966
1 1 - (160.3/4303 0.8/(3 -0.2) 1
= 2.88 x~O-~ $;Btu
or the sum of the fixed charges and cast of power is 2.88$ per l,OOO,OOO Btu transfewed.
EXAMPLE 18.3. Check the answer of Example 18-2 hy means of Eq. 18-B.
SOLUTION: Al1 values entering in Eq. 18-9 and 18-10 can be taken from Examples 18-1 and 18-2. Henee:
X”,< = 1.212 x 10;’ 10.96 x 430
= 2.5’7 x D8 $/Btu
x,,i = 2.26 x lo-” x 6.35 x 10-l” x 46,500’.*
10.96 x 430 = 0.35 x lo4 $jBtu
and X,,,, + X,wi = 2.57 + 0.35 = 2.92 x lo-” $/‘Btu. The discrepnncy of about 17& is due to errors in slide rule calculations. These computations show that, in this example, power constitutes only 12% of the heat transfw cast and 88% of the costs aie Iixad charges.
18-7. O@imum doeitl, outside tcbes. If the cast of power to push the fluid through the tubes is negligible, the outside thermal resistance is eon- trolling and DoGo/po lies between 1000 and 40,000; the optimum mass velocity Gor can be computed from the equation:”
G‘, 1 0.381 GO" = - CmKu (4.75 At ,,/, /Ah - 1)
where:
Eq. 18-12
Ro’ = KiDJDi .’ Eq. 18-13
and the minimum, or optimum cast of transferring heat, in~dollars per Btu, is:
Eq. 18.14
where X,,, the eost of power delivered to the floid on the Shell side, in dollars per Btu, is:
C,,,A; C,oK”G<, 3-m x,,= q -
Ui Atm Eq. 18-15
Similarly to Eq. 18-11, Eq. 18-14 can be pre- sented as:
43
-L + x,,, 1 X,0 = 1~ (At,/At,) (no/ (3 - vaa
Eq. 18.16
EXAMPLE 18.4. Find the optimum tota.1 cast for the heat exchanger of Example 18-1, if it ‘i8 used to preheat water instead of air, al1 other data being the same.
SOLUTION: By Eq. 18.13, K’O = 2.62 x lo-’ x 0.0652/0.0833 = 2.05 x 10-r. Also. bv definition. c 00 = C,iDJD,, oi C,a = 1.212 x’ lo-4 x 0.0652; 0.0833 = 0.950 x lo-‘. Thus, by Eq. 18-12:
2.26 x ,;x;O:I"I j:,,(&)- l]r
= 4780 Ih/hr-sq ft
The corresponding Reynolds number is:
Nm = 0.0833 x 4780 =7210
0.068
and the surface coefficient, from Eq. 17-10:
h 0.33 x 0.248 x 4780 0
= 1.25 x 7,210".* x 0.66%
~ 14.5 Btu/hr-sq ft-“F
The over-al1 heat transfer coeffieient lJu could be taken equa1 to ho. However, it is better to determine it from Eq. 13-14 using an average value for hi of 1200 Btu/hr-sq ft-“F, as giva in Table 15-2. Thus:
1 u. 0.0833 = + 0.0833 (0.0X33-0.0652)+ 1
0.0652 x 1200 (0.0833 + 0.0652) 24.85 14.5
‘= 14.2 Btu/hr-sq ft-“F
and, by Eq. 18-14:
1 0.950 x 104 0.752> 14.2 i 430
= 1.787 x 1O-8 $/Btu
or the sum ,of fixed charges and costi of power is 1.79$ pei 1.000,000 Btu transferred, or almo& 40% less than the cast of Example 18-2.
18.8. Heating or couling of gas. If a gas is heated by condensing vapor (usually steam) or cooled by water, the thermal resistance on the gas side is the controlling factor and it is necessary to decide whether the @,s should flow inside or outside the tubes. The factors and exponents in the foregoing equations may be assumed to have the following values: ?L = 0.8, m = 0.2, no = 0.6, and mo = 0.16. Then the ratio of the optimum sums of power and fixed charges becomes:
where (At,,,)i is the mean temperature differente if the controlling thermal resistance is inside t,he tuba and the outside resistance is negligible, and (At,), is the mean temperature differente if the
~controlling resiutance is outuide the tube and the inside resistance is negligible.
q = W’c’(tl’ - ti) = w”(tr” -- tl”)
Eq. 18-17 shows that the choice is determined by
Let U represent the over-al1 coefficient evaluated at the optimum velocity of the fluid that must be co&d in order to maintain a proper balance be- tween fixed charges and pumping costs for the warm fluid as expla.ined in Section 18-5. The cooling water costs Cw dollars per lb and is available in
44 August 1966 Petro/Chem Engineer
a combination of severa.1 factors. If the gas Flora either inside or outside the tuba of a certain type of exchanger, in which the hourly fixed charge per foot of tube is the same, the” C,o/G = WDo. If the gas at atmospheric pressure is heated with steam condensing under pressure, the steam should be inside the tubes and the Shell can be made of light sheet iron well insultited on the outside. On the other hand, if gas under pressure must~ be cooled with water, a ‘consid+rably less expensive type of construction - a trickle, or drip cooler - along the lines of Fig. 14-6, can be used. In a trickle cooler the gas will flow inside the tubes instead of water and the Shell must be designed to withstand the gas pressure.
18-9. Water far eoolers a,nd ccndensers. It fre- quently happens that the pressure in the water mains of a plant is sufficient to give the necessary rate of flow through the beat exchange apparatus and the cast of cooling water is directly propor- ei tional to the amount used. The optimum water rate will correspond to the lowest annua1 sum of the water costs and of the fixed charges on the beat exchange apparatus.
Suppose that W’ lb per hr of warn~ fluid enters the heat exehanger at ti and leaves at t2’ losing the beat Q = W’c’(tl ~ t?‘). This beat is absnrbnd by W” Ib per hr of cooling water which enters at t,” and leaves at tì”; for the specific beat of water c” = 1 may be used. The temperature t*” is de- termined by the heat balance:
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Petro/Chem Engineer August 1966 57, iurcw Informario” on ldYP,tkOd Prod”<lr lea lader h,“iW tmer 45
sufiicient nmnnnt. If the water is under sufficient, ~ressure, the highest practicable water velocity should be used. As before, let C,i designate the hourly fixed charges per square foot of the heat transfer surface Ai and 7 desi,gnate the number of hours per ycar of exchanger operation. Then the yemly fixed charges vii = A& T. The yearly water cast y,, = W” &,,j (tt” - t,“). At the uame time Q = UR,F Ati, where the factor F depends on the arrangemmt uf the shell and tube passes in t,he exchsnger as discussed in Section 14-2, and At, is the man logarithmic over-al1 temperature differente in cuunterflow~ and must be valculnted from Eq. 14-4.
By combining these relations, the minimum value of the sum of the yearly fixed charges, and water cast ‘va -Im y., iS found by differentiating the expres- Simon for y. + II<” with respect to the tempwkure differenee (t,’ - tlnI at the hotter end and setting the derivative equa1 to zwo. This results in:
Eq. 18-18
where At, = ti’ ~ tI” is the maximum temperature differente at the cold end and Ab,,, = t,’ - t2” is the optimum over-al1 differente at the hot end.
Since At,,, appears in severa1 places in Eq. 18.18, the latter can be salved only by trial and error. Homever, the calculations can be simplified by using charts in which the ratio Y = (t,’ ~ t%“j/ ih’ - ti”) = At,,/At, is plotted against values of X representing the right side of Eq. 18-18. X = UFCJC,i far various values of Z = (tl’ ~ t2’),’ (ti - h”). The caves in Fig. 18-3 serve for values of 2 from 0 to 4’ and the alignment chart of Fig. 18-4 can be used far higher values of Z, up to 100.”
IE, as in Example 18-5, the line connecting the X and Z values intersects the Y scale st a sharp
Figure 18-4. Chart for solution of Eq. 18.18 for higher values of 2.
46 August 1966 Petro/Chem Engineer
Pre-publication Announcement
Applied Heat Transfer
Design For the HPI
will be auailable shortly in
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Petro/Chem Engineer August 1966 ror rdler I”hrnBli.” on Adrarfirsd hd”<,$ sce Reodar serviw cen,u 47
anrle and which makes it dificult to read the cor- responding value of Y, it is advisable tti check the value of to” with Eq. 18-18. If the left side turns out to be smoller than the right side, the value of tz” must be slightly increased; this will decrease the value of Ati, and increase the left side of the equation. An accnrate value for ts” is necessary in order to obtain a correet value of At,,, and thereby 3 correct cooling area Ao,
When t,he temperature of the warm fluid is con- stant, as in a condenser, Eq. 18-18 reduce to:
62” - h” ~ In ($) Azz Eq. l*.lg
Ah
EXAMPLE 18.5. A counterflow exehanger must cool 8000 lb per hour of air from 210 to 80F; the water has a temperature of 75F and eosts $0.07 per 100 CU ft. The tota1 fixed charges aie $0.60 per sq ft per year; t,he cooler is used con- tinuously every day of the year; allow 10 days for holidays, cleaning and repairs. Assume the optimum value of Uu as 11.6 Btu/hr-sq ft-“F, ineluding a safety factor F, = 1.25. Determine: (a) the opti- mum water outlet temperature, ib! the correspond- ing water consumption per hour, and (c) the rrquired conling surface of the, exchanger.
SOLUTION: (a) the specific weight of water at 68F is 62.3 lb/cu ft, therefore:
0.07 cw = 100 x 62.3
~- = 1.124 x 1O-5 $/lb
The number of hours of service per year:
7 = (52 x 7 - 10) x 24 = 8496 hr.
Similarly to Eq. 18.4:
0.60 C,i = m = 7.06 x 10-j $/hr~-sq ft
and by definition, with F = 1 for counterflow:
-7 = “% = Cd
11.6 x 1.124 x lo-” = 1,845 7.06 x 10-j
tl’ - ti’ z = ~---- 210 - 80 = 26 ti - t,” - 80 - 75
and from Fig. 15-4 Y = 14 = (210 - tz”i/(80 - 75). This giva tz” = 14OF. A check of Eq. 18-18 gives :
210 - 80 + (80 - 75) - (210 - 140) 2
[, (210 - 140) - (80 - 75) 1
l
80 - 75 210 - 140 - l + In
210 - 140 ~~..,- 80 - 75 1
= 1.712
This is slightly less than X = 1.845., Increasing tz” to 141.4F giva for the left side, 1.841, which is satisfnctory.
ib) With e = 0.24 Btullb for the air, t,he heat that must be absorbed by the water is:
q = 8,000 x 0.24 (210 - 80) = 250,000 Btu’hr
and the amount of water used is:
250,000 m = (141.4 ~ 75) x 62.3
= 60.4 CU ft/hr
48
(cj The mean over-al1 temperature differente is, by Eq. 11.19:
&, (210 -
141.4:1 (80 - -
_ 75) = 210 -- 141.4
24.3F h
80 ~ 75
The necessary cooling ara is:
250,000
18-10. Optinmm tmperature differerm in steam Ireatws. When waste heat is recovered from exhaust steam by an exchanger, an increase of the heat trsnsfer surface will increase the amount of beat recovered but will also increase the fixed charges. Wence there is an optimum mean temperature differente for any particular case.
If the temperature of a fluid must be raised from t, to t2, and ti is higher than the saturation tem- perature t’ of lom-pressure exhaust steam, part of the heating, up to an intermediate temperature t, can be done by the exha& steam, and the rest of the heat must be furnished by more expensive high-pressure steam condensing at t”, where t” > t2, The problem is to find the optimum over-al1 a temperature differenee At in the first, low-pressure steam heater. Designate the value of low-pressure steam in dollars per Btu of latent heat by CH’; the value of the high-pressure steam by CH”; assume that (C,, + C,]),, is the optimum sum of fixed charges Ca and cast of power C, required to force the heated fluid through the heaters expressed as dol- lars per hoor per square foot. The tota1 hourly cast t,hat depends on the intermediate temperature t can be presented as:
where the area A = Ai + A2, is equal to qLJAt,; At,,% is the logarithmie mean over-all temperature differente sine II can be cwsidered eonstant and F in Eq. 14-5 is unity for any arrangement of surfaces because the steam condensa in both heaters at constant temperature. Combining there relations, differentiating, and setting d Ey/dt equala to zero gioes the following equation for minimum totd cast:
(C,, -l- C,) (t” -t’ rt,-t> (t”-t) = *~(Cl,,pCw,j
Eq. 18-21
Eq. 18-21 can be solved as a quadratie equation or can be used directlp far finding (t’ - t) and t” - t).
References ‘W. H. MeAdams, Hrat ‘I’nmsmisuion. op. cit.,
p. 432. w. H. McAdams, Heat Tm?wmission, op. cit.,
p. 435 “Ibid., p. 436 ‘The suhscript O? as outside the parentheses
(X.; + X,;,) o or in lljO QlcAdams, op. cit., p. 440. “R. D. Dougleass and D. P. Adams, Inà. Eng.
