Multipass
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Lxchanger
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Calculations
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HOWARD TEN BROECK
06
Socony-Vacuum Oil Company, Brooklyn, N. Y.
05 04
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02 01
NDERWOOD (4), Nagle (3), and
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6 8 02 03 04 06 08 IO 15 20 3 4 01 Fischer ( I ) derived equations showing the correct average temperaUA/wc EFFICIENCIES OF 1-2 AND 2-4 EXCHAXGERS FIGURE1. THERMAL ture difference to be employed in multipass exchanger calculations. Underwood’s equations are too involved to be used directly for design purposes. Nagle prepared plots (2) in which F (the ratio of the true average termed the ‘‘thermal efficiency”). Nagle’s charts are for temperature difference to the logarithmic mean difference) 1-2 and 2-4 exchangers and can be used conveniently when is plotted against P (a dimensionless temperature ratio terminal temperatures are known or when they are set by the conditions of the problem. However, in cases where the terminal temperatures are to be determined-for examde. when an exTABLE I. VALUES OF UA/WC FOR 1-2 AND 2-4 EXCHANGERS changer of known area is be used for service for which it was not designed, use of R P 4 3 2 l’/z 1 3/4 =/1 8/8 1/4 Nagle’s or Fischer’s charts involve trial-and1-2 Exchangers error solutions. Fischer derived the correc0,1368 0.1265 0.1183 0.1147 0.1124 0.1098 0.1082 0.1075 0.1068 0.10 tion factors for 1-3, 2-6, 3-9, and 4-12 ex0,2630 . . . ... 0.15 ... ... ... ... ... ... changers, and presented the results of his work 0.3688 0.175 0.5720 0 : 3707 0 12961 0:2720 0:2527 0 :2443 0 :2366 0 12332 0 :2296 0.20 in the same form as did Nagle. 0.7449 ... 0.21 ... ... .... ... ... ... ... 0 ; 6785 ... 0.25 ... ... ... . . . The equation for heat transfer, including . . . . . . ... 0.27 1.0398 the correction factor F, is: ... 0 :6339 0 : 5139 0 : 4423 0:4i59 0:3931 0 : 3832 0 13738 0.30 ... ... . . . 1 ,0454 0.35 ... 0.40 ... ... 1 :0097 0 7239 0 : 6487 0 :5922 0: 5686 0 :5474 ... . . . 1.8173 0.45 ... ... ... ... 0.50 ... 1:2463 1:0022 0 ; 8608 o :8i49 0 : 7650 ... ... ... 0.55 . . . 1.8542 ... ... ... 0.60 ... ... 1:7i20 From this equation and from the definition ... 0.65 *.. ... ... . . . 2.8092 1 :2676 1 : i45s 1 : 0324 ... ... . . . 0.70 . . . ... ... 2 :0905 1 : 6635 1 :4757 of At,, ... ... 0.80 ... ... ... ... 3.4182 2.2730 ...
b
I-
0.85 Pmaz.
0.10 0.20
0.21 0.30 0.40 0.45 0.50 0.55 0.60 0.70 0.80 0.90 0.95 Pmax.
...
...
...
...
...
...
0.2192 0.2794 0.3817 0.4651 0.5865 0.6667 2-4 Exchangers 0,1356 0.1259 0.1177 0.1145 0.1112 0.1096 0.4802 0.3519 0.2897 0.2681 0.2506 0.2430 0.5611 1 i3i4 0:5750 0:4895 0:4320 0:4ii2 1.2376 0.8472 0,6794 0.6241 2.6260 ... 1 5986 1 0450 0:9iso ... 2.6580 ... . . . i 6704 1 3452 ... ... 3.3212 2.0619 ... ... . . . 4.8890
:
:
...
...
... ...
... ...
...
...
... ...
0.23
0.32
0.46
0.57
0.74
0.82
... . . . 3.2873 0.7633 0.8196 0.8771
1 - RP - ,
0.1081 0.1075 0.1070 0.2358 0.2324 0.2292
:
0 3932 0 : 3810 0:3722 0.5795 0.5602 0.5422
-
o:s220 0 7844 0 : 7506
:
1:i912 1 0776 1 :Oi60 1 , 6 3 2 0 1 ,4884 1.3758
2,4752 2.1502 1,9208 5.2520 3.6674 2.9954 , . 4.4206 ... 0.92 0.95 0.98
.
, 1041
-U=A
wc
ln(-)
P(l - R )
For the special case of R = I,
P UA _ F ( l - P) wc
(3)
These equations show that there is a relation involving UA/wc. . . R,. P.. and F . which is formally independent of any consideration of
INDUSTRIAL AND ENGINEERING CHEMISTRY
1042
the number of passes or type of flow. Therefore F can be eliminated between this equation and the plots of Nagle and Fischer, and we obtain a plot showing F as a function of UA/wc with R as a parameter. The same results can be obtained from Underwood’s equations, which can be simplified to the following f o r m :
VOL. 30, NO. 9
From these equations, values of UA/wc can be calculated for various values of R and P as accurately as desired. Figure 1 was prepared by means of Equations 4 and 5. The plots also show the threshold of crossed temperatures, or the conditions under which tz = TZ. The threshold is obtained when P = 1/(R 1). Table I gives the data used in preparing Figure 1. One shell pass, two tube passes: Table I1 summarizes the results for exchangers having one parallel- and two counterflow tube passes per shell.pass, and were prepared by means of Equations 2 and 3, using values of F from Fischer’s charts. Therefore, they carry the inaccuracies of chart readings but are sufficiently accurate for ordinary design purposes. Two shell passes, four tube passes: Use of the figure and tables of this article is limited-to cases where the heat R - d m ) P 2 2/(1 - P ) ( l - R P ) A -U= In 2 - (1 (5) transferred is sensible heat; i. e., they WC 2 - (1 R + d 1 ) P + 2 1/(1 - P ) ( l - R P ) 2/do not apply to condensers. Use of Figure 1 eliminates the necessity of calculaiing At,,, and of determining the value of F , with the resulting -elimination of trial-and-error solutions. In addition it should prove of value in determining the most economical TABLDZII. VALUESOF UA/wc FOR EXCHANGERS WITH ONE exchanger. PARALLELAND Two COUNTERFLOW TUBEPASSESPER SHELL
+ +
+
+
PAS5
P 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.85 0.90
0.138 0.262
PIUS.
0.22
... ...
..... . ... .... .. ... ... ... ... ...
0.10 0.135 0.20 0.467 0.30 0.40 ... 0.50 ... 0.55 0.60 ... 0.70 ... 0.75 0.80 ... 0.90 ... P ~ P x 0.24 .
... ... ...
0.10 0.15 0.20 0.24 0.30 0.40 0.45 0.50 0.60 0.70 0.80 0.90 0.95
0.135 0.251 0.472 1.08
Pmsx.
0.25
0.10 0.15 0.20
0.24 0.30 0.40 0.45 0.50 0.60 0.70 0.75 0.80 0.85 0.90 0.96 Prnax.
...
...
... .. .. ..
... ... ...
... 0.135 0.251 0.462 1.09
... ... ... ... .... .. ... ... ... ... ...
0.25
G/,
2
3
4
Acknowledgment 1
1-3 Exchangers with One Shell Pass 0.126 0.118 0.116 0.111 0:379 0.647
... ... ... ...
... ... ... ... ...
0.28
0.107
0:272
0:253
0 : 236
0:229
0:623 0.982
0:5i2
0:43s
0 : 384
0:3i1
o:iis 1.68 ...
o:ii7
0:5i9
0 546
... ...
1.19 2.66
0 863
... ...
... ... ...
0.47
0.60
...
...
... ... ...
...
I
.
.
0.38
...
... I
0.108
‘/4
0:29l
2-6 Exchangers with Two Shell Passes 0.111 0.118 0.116 0.125 0.285 0.261 0.250 0.353 0.571 0.487 0.429 1.003 1.125 0,837 0.680 ... ... 1.03 1.575 ... 2.52 ... ... 1.63 ... ... 3.03 ... ... ... ... 6.0 .
x/a
.
... ...
... .. .. .. ...
... ...
... ...
0.32
0.47
0.58
0.75
:
:
0.77
01763 1.045 1.47 2.17 2.87 5.46 0.90
0.108 0.236 0.388 0.575 0.819
0.107 0.229 0.371 0.541 0.746
1.34 1.99
... ... ...
:
:
1 iio 1.618
1 005 1.38
2:44 4.79 0.92
1.91 2.97
...
0.98
0.107
0:346
01285
0:26l
0.’250
0.‘236
0:229
1:022
0:566
0:482 0.827
0:429 0.667
0:388 0.575
0:3i1 0.541
...
1.46 3.69
o:Sii
...
...
... ... *..
... ...
1.02 1.55 2.56 5.97
0.33
0.48
0.62
0 746 1.005 1.35 1.87 2.82 3.86 0.98
:.,I.. 1.YY
...
... ...
...
... ...
...
... ...
1.13 1.62 2.29 3.83 6.10
0.82
0.97
4-12 Exchangers with Four Shell Passes 0.125. 0.118 0.116 0.111 0.108 0:346
:
1 000
.... .. ... ... ... ... ... .. .. ..
0:285
0.‘261
0:GO
0 482
1.13 1.82
... ... ...
:
0.811
...
1.46 3.15
...
...
...
... ... ... ...
... ... ... ... ...
0.33
0.49
0.64
:
0.107
0.’250
0.‘236
0:2i9
0:429 0.667
0:388 0.575
0:371 0.541
1:ooo
o:8ii
0:746 1.005 1.35
1.53 2.44 3.14 4.77 9.00
... ...
0.87
Nomenclature
&vA
1.12 1.56
...
:
2.24
1 848
3.63 5.34 0.98
2.78 3.74 0.99
...
...
heat transferred, B. t. u./hour over-all coefficient of heat transfer area of exchanger, square feet logarithmic mean temperature difference true average temperature difference = correction factor = At,,,/At, = inlet temperature of shell-side fluid = outlet temperature of shell-side fluid = inlet temperature of tube-side fluid = outlet temDerature of tube-side fluid =
= = = At, Atsv, =
F TI
Tz
3-9 Exchangers with Three Shell Passes 0.111 0.108 0.116 0.118 0.125
...
The author wishes to acknowledge the ’ valuable suggestions and criticisms made by G. E. Tate of the Foster Wheeler Corporation.
tl t2
p W w C C
=- 11
- 51
TI - ti = rate of shell-side fluid, pounds/hour = rate of tube-side fluid,. pounds/hour = specific heat of shell-side fluid = specific heat of tube-side fluid
Literature Cited (1) Fischer, F. K., IND. Elvo. CHEM.,30,377 (1938). (2) McAdams, W. H., “Heat Transmission,” p. 148, New York, McGraw-Hill Book Co. (3) Nagle, W.M., S.M. thesis in chem. eng., Mass. Inst. Tech., 1932. (4) Underwood, A.J. V., J.Inst. Petroleum Tech., 20, 146 (1934). R E C E I V ~ March D 8,1938.
