Mean Temperature Difference in Unbalanced-Pass Exchangers KARL A. GARDNER The Griscom-Russell Company, New York, N. Y .
The mean temperature difference in multipass heat exchangers with unequal tube passes may be calculated from the logarithmic mean temperature difference and the correction factors presented in this paper. The derivation of equations for exchangers with single-pass shells and any even number of tube passes is given in detail, and the method for extension to any number of shell passes is described. The equations of previous investigators appear as special cases of the author’s general equations. The results are based on the usual as-
sumptions, and are presented in terms of the customary parameters plus an additional parameter involving the ratio of unbalance of the surfaces and rates in the counterflow and parallel-flow tube passes. A discussion of the advantages and limitations of unbalanced-pass heat exchangers is included and examples are given. An additional use of the data as an approximate correction to mean temperature difference for variation of heat transfer rate in conventional multipass exchangers is suggested.
Courtesy, The Qriscom-Russell Cornpanu
CUTAWAY VIEW OF
A
SIMPLE UNBALANCED PASSHEATEXCHANGER WITH TWOUNEQUAL TUBE PASSES AND A SINGLE SHELL PASS
N SELECTING heat transfer equipment for specified conditions of temperature and flow, the design engineer is continually confronted with the problem of how best to utilize the available temperature difference between the fluids in an exchanger of practical design. It is well known that the ideal solution to this problem from a Purely thermal standpoint is to arrange the heat transfer surface in such a way that the two fluids enter a t opposite ends of the exchanger and flow countercurrently, under which conditions the familiar logarithmic mean temperature difference is obtained:
I
Atm = FAtl,,
(2)
For various structural and economic reasons it is not usually practical to attempt to obtain countercurrent flow in a shell and tube exchanger, so the heat transfer industry has adopted the multipass tube exchanger as the mostsatisfactory compromise between thermal efficiency and practical limitations. The aim is, of course, to arrange the surface in such a way that the highest over-all heat transfer coefficient compatible with the allowable pressure losses is obtained, and at the same time to keep the mean temperature difference factor as ,,lose to unity as possible. I n countercurrent flow, given sufficient (T* t e ) - (Te ti) Atlog (1) surface, it is possible to heat the cold fluid to a temperature differing from the inlet temperature of the hot fluid by only an infinitesimal quantity, whereas in parallel (or cocurrent) This formula is generally accepted as the criterion t o which the flow it is possible only to approach the outlet temperature of mean temperature differences for other arrangements of the hot stream. Therefore it is obvious that in a multipass flow are compared, since it is customary to express the true tube exchanger, where some of the passes are necessarily in mean temperature difference as the product of a correction parallel flow, certain conditions of crossed temperatures factor times the logarithmic mean temperature difference: might be postulated which the exchanger could meet only 1215
-
1216
INDUSTRIAL AND ENGINEERING CHEMISTRY
with a greatly reduced mean temperature difference or which, possibly, it could not meet a t all, even with infinite surface. Such conditions make themselves known through correction factors rapidly nearing zero as the limiting temperature is appoached.
6. Heat losses are negligible.
!:
~
any cross section.
Vol. 33, No. 10
z:;,,
~
Over
With the aid of the correction factor curves presented in these papers it is possible to determine the minimum number of shell passes necessary to make an exchanger operate with a reasonably complete utilization of the ideal temperature difference. Unfortunately, cases frequently occur where the number of shell passes found in this manner is quite large and, since it is W generally impractical to design an exchanger 2 IY lwith more than two shell passes per shell, a a W several shells must be used in series, a procetr a dure which usually results in a higher cost per 3 Isquare foot of heating surface without B corresponding reduction in the amount of surface t, required. In the majority of cases this condition must be accepted as an inevitable consequence of the multipass design; therefore, LOCATION OF CROSS SECTION any method which makes it possible to install the heating surface in one shell instead of two, RELATIONS IN MULTIPASS HEATEXCHANGER FIGURE 1. TEMPERATURE or in two shells instead of three, etc., is a welcome solution to one of the designer's problems. One such method, applicable in many cases, is the use of Various methods have been adopted or suggested for obtaining a closer approximation to true counterflow (higher corunbalanced-pass tube exchangers. Examples showing the advantages of this method and a discussion of its limitations are rection factors) in exchangers which still retain the structural given. advantages of multipass design. The writer here proposes to investigate another such method which has hitherto reEquations for M T D in Unbalanced-Pass ceived no attention in the literature-the use of unbalancedExchangers pass tube exchangers (i. e., shell and tube exchangers having more heat transfer surface in the counterflow tube passes than The writer has derived equations for the mean temperature in the parallel flow passes). difference in single-pass shell exchangers with any even number of tube passes, the surface in the counterflow passes having Methods of Overcoming Low Correction Factors any desired ratio to that in the parallel-flow passes. The basic assumptions are the same as those listed above with the exNagle (4) was the first to derive and publish mean temception that assumptions 1 and 6 are modified as follows: perature difference correction factors for one- and two-pass shell exchangers with two, four, or six tube passes per shell; la. The over-all heat transfer coefficient, U , is constant Underwood (6) derived the exact equations for the curves throughout the exchanger. Nagle had obtained by graphical methods, and Bowman ( I ) b. Coefficient U , for the counterflow tube passes is constant and is the same for each counterflow ass. extended these results to exchangers with any number of c. Coefficient U p for the parallel-flow tuEe passes is conshell passes and two, four, or six tube passes per shell pass. stant and is the same for each parallel-flow pass. Bowman also showed that the correction factors for ex6a. There is equal heat transfer surface in each shell pass. changers with an infinite number of tube passes per shell pass b. There is equal surface in each counterflow tube pass. do not differ appreciably from those with two tube passes per c. There is equal surface in each parallel-flow tube pass. shell pass. One result of these investigations was to show the extent to which low correction factors could be improved Figure 1 is a diagram of a heat exchanger with 2M tube by using a series of multipass exchangers with counterflow passes ( M counterflow and M parallel flow) showing the through the series, and it is this method which is almost extemperatures involved, The basic equation is obtained from clusively used a t present. Fischer (9), reasoning that exa heat balance over the right end of the exchanger: changers with three tube passes per shell pass (two counterflow and one parallel flow) should have a better mean temperature difference than those with an even number of tube passes, derived correction factors on this basis and was able to show Also, by definition, some improvement. However, exchangers with any odd number of tube passes have the same structural disadvantages as true counterflow exchangers, so Fischer's correction factors for two-six and four-twelve exchangers are the most useful. Differentiation of Equation 3 gives: Bowman, Mueller, and Nagle ( 2 ) summarized and coordinated all the known work on mean temperature difference up to 1940; they listed the basic assumptions made in deriving the various correction factors as follows: From heat balances on differential lengths of each tube pass, 1. The over-all heat transfer coefficient, U , is constant throu hout the exchanger. 2. %he rate of flow of each fluid is constant. 3. The specific heat of each fluid is constant. 4. There is no condensation of vapor or boiling of liquid in part of the exchanger.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Cctober, 1941
1217
C o u r t e s y , The Griscom-Russell Company
HEAT EXCHANGER WITH
SINGLE-PASS SHELL AND
FOUR-PASS TUBES,SHOWING
AND
HIDAD COVQR
SEALING PLATE,
RINGGASKET,
etc., for the M parallel-flow passes. Also,
etc., for the M counterflow passes. By adding Equations 6A, 6B, etc., and 7A, 7B, etc., --WCdT =
M(Ac
+ l ) T - + + . .. + AB) [ M -( KK(tlI + + . . . W ) ) ](8) (tI
tlI1
tIV
(TI
- Til
(14)
K = U,A,/U,A,
where
+ +
tIV . . . P))is taken from Equation 3 and substituted into Equation 8 which, after rearrangement, becomes --Mwc(Ao +
An expression for (t"
Ud,
dT dA
=
( K + 1)(MT t(M-9)
b
where
then At,,, =
R
- wc
wc
-
- t1
-
tIII
- ...
- K2 (T - T2) (9)
-- tl
T2
ta
The over-all heat transfer coefficient may be defined arbitrarily as
An expression for dx/x is obtained by adding Equations BA, 6B, etc., and dividing the result by Equation 9:
!E= 5
R(K
( M z + 1)dz + l ) ( M z + 1 ) - KZ - Z ( M B+ 1)
(15)
INDUSTRIAL AND ENGINEERING CHEMISTRY
1218
The mean temperature difference correction factor is obtained from Equations 17 and 1, noting that the latter may also be written:
P =
where
Vol. 33, No. 10
Generality of Equations and Limiting Cases The expression for the correction factor derived above is quite general and includes most of the familiar equations as special cases. Some of these are listed below. CASE 1. M = 1, K = 03, true counterflow: F = 1. CASE2. M = 1, K = 0, true parallel flow:
(H) -
/ 1
P\
CASE3. M = 1, K = 1, single-pass shell, two-pass tube exchanger with balanced tube passes:
When R = 1, Equations 20 and 21 become indeterminate; for this case,
F -
M(1
-
P ) In
- Z l ( 1 - JW
2
+ q)
1
(23)
This is the equation obtained by Underwood (6) and rewritten in terms of P and R.
In order to use these equations, it is necessary to find an expression for al. This may be obtained from Equations 6A, 6B, etc., and 7A, 7B, etc., as PR
21
=
M(1
- PR)
-P
[-M+
/M -- 11 +I
(24)
From Equations 2 and 4 it can be seen that
This may be substituted into Equations 24 and 21 (or 23) and the latter solved by trial and error for F. However, the writer finds that a trial-and-error solution can be avoided entirely by retaining q5 as a parameter. Equation 21 may be written: CASE4. M = 2, K = 1, single-pass shell, four-pass tube exchanger with balanced tube passes:
Therefore, from Equations 24 and 27, P -
R
+
+
2
,
+v/&r z(gn,.+r
+1 -
1)
1
+ 1 m
+1
1 - P (28)
-g ( m )
F = (R
Underwood's equation for this case is presented in a form requiring trial-and-error solution but may be resolved into Equations 29 and 33. CASE5. M = 0 3 , single-pass shell, infinite-pass tube exchanger or, as pointed out by Bowman ( I ) , cross flow with both fluids completely mixed:
1 - P 1 - PR F = ( R - 1) In 4
In
or when R
=
1,
F =
P
(1
- P ) In 4
- 1) In 4
(30)
By assuming successive values of q5, P can be found from Equation 28 and substituted into Equation 29 (or 30) to determine F directly. A similar calculation on an exchanger with the shell fluid entering a t the opposite end results in the same final equations.
1
P = (+R
-
1)
+ L -~1 (4
- 1)
In
(34)
+
This is identical with the equation derived by Smith (6) for mixed cross flow when expressed in terms of the parameters P , R , and 4.
October, 1941
INDUSTRIAL AND ENGINEERING CHEMISTRY
1219
Simplification of Equations for Two-Pass Tubes When M = 1, a direct solution is possible without the use of parameter Cp since Equation 24 reduces to 21
When R
=
=
~
PR 1 - PR
1, 2P
F =
(1
(35)
- P ) V'X?jTfIn
+P
- P ) V'K-
[ (1 ( l - P ) v'K
-+ 1 1
(37)
-P
The writer has calculated correction factors from Equations 36 and 37 for K = 0 , 2 , and 3; those for K = 0 (parallel flow) have little practical utility and were used only to furnish a fourth point for the cross plots described below. It was found that a set of nearly straight lines is obtained when P is plotted against K for constant values of F for any given value of R; a sample plot for R = 1 is shown in Figure 2. Eight such curves were constructed for other values of R from the writer's data for IC = 0 , 2 , and 3 and from published data for K = 1; i t was thus found possible to interpolate on these curves for any value of K between 0 and 3 with satisfactory accuracy. Curves of the correction factors for K = 1.5, 2.0, 2.5, and 3.0 are shown in Figure 3; values for constructing these curves are given in Table I. Figure 3 (bottom graph) also compares the factors for various values of K. Calculations were also made from Equations 28 and 29 to obtain correction factors for single-pass shell-four-pass tube exchangers with K = 2 and 3. The results are given in Figure 4 and Table 11; Figure 4 also shows the comparison between the factors for various values of K .
Discussion of Curves The correction factors for four-pass tubes are not so good as those for two-pass tubes; even with K = 00, no better results are obtained than with two-pass tubes and K = 3. They are, however, slightly better than those for balnncedtube passes. As the number of tube passes is increased, the correction factors continue to approach those for cross flow with both fluids completely mixed (Equation 34), and the advantage of unbalanced tube passes over balanced passes becomes negligible. It is only because the factors for 1C = l, M = 1 are so close to those for M = 03 that increasing the number of tube passes in balanced-pass exchangers has so little effect. In this connection, Figure 5 shows the full range of correction factors for mixed cross flow, with special reference to the maximum values of P attained by the curves. This effect was pointed out by Smith (6) for this type of flow. It is not, however, confined to cross-flow exchangers, as may be verified by differentiating Equation 28 with respect to Cp and setting the result equal t o 0, thus obtaining P = (t.$J+(T-
t,)
FIQUI~E 3. CORRECTION FACTORS FOR ONE-TWO EXCHANQERS UNBALANCED PASSES,AND ( b o t k m graph) COMPARISON When M '= 1, Equation 38 has the solutions (b = 0 and 4 5 m ; of these, (b = 00 represents a maximum with F = 0, OF CORRECTION FACTORS
WITH
FOR FIGURE 3 (Single-Pass Shell-Two-Pass Tubes) TABLE I. COORDINATES
K 1.5
2.0
3.0
R=0.2 R = 0 . 4 0.595 0.446 0.702 0.557 0.793 0.660 0.838 0.715 0.865 0.750 0.893 0.792 0.907 0.815 0.913 0.826 0.917 0.833
0,975 0.95 0.9 0.85 0.8 0.7 0.6 0.5
0.625 0.483 0.412 0 731 0.585 0 500 0'818 0.691 0:598 0.860 0 743 0.652 0.884 0:777 0.687 0.910 0.817 0 , 7 2 7 0.923 0 838 0 , 7 5 1 0.928 0:847 0.762 0.930 0.853 0.775
0 , 3 5 7 0.320 0 438 0 400 0:5RO 0:478 0.580 0.522 0.613 0.552 0.654 0.589 0.676 0,610 0.689 0 , 6 2 2 0,702 0,634
0.975 0.95 0.9 0.85 0.8 0.7 0.6 0.5 0
0.647 0.750 0.837 0.876 0,900 0,922 0.932 0.936 0.938
0.510 0.615 0.711 0.765 0.796 0.833 0.855 0.864 0.869
0.429 0.522 0.621 0.673 0.707 0.748 0.770 0.781 0.793
0.375 0.460
0.975 0.95 0.9 0.85 0.8
0.667 0 770 01852 0 889 0'911 0:934 0 944 0'945 0:946
0 532 0'635 0:730 0 783 01814 0.848 0.865 0.874
0.447 0 543 0:640 0.692 0.726 0.766 0.789 0.800 0.810
0.390 0 482 0:56S 0.620
0.7
0.6 0.5 0
0.880
R=0.6 0.390 0.474 0.573 0.627 0.660 0.700 0.724 0.738 0.752
P R = 0 . 8 R = 1 R=1.5 R = 2 R = 3 R = 4 0.335 0.412 0.506 0.555 0.586 0.629 0.653 0.665 0.878
F 0.975 0.95 0.9 0.85 0.8 0.7 0.6 0.5 0
0
2.5
Vol. 33, No. 10
INDUSTRIAL AND ENGINEERING CHEMISTRY
1220
0.256 0.323 0.382 0.416 0.440 0.467 0.483 0.491 0.500
0.215 0.171 0.271 0.209 0.320 0.242 0.349 0.261 0.366 0.273 0.388 0.285 0,398 0.291 0,400 0.293 0,407 0.293
it is not sufficient to calculate a few points in the upper region of the chart and one point a t F = 0, and to draw the best line through them. Such a procedure may be permissible for exchangers with two tube passes per shell pass, but for all other designs, points should be calculated throughout the full range of the chart.
Equations for Exchangers with Any Number of She11 Passes
0.144
0 171
01195 0.208 0.215 0.221 0.225 0.227 0.227
I n a series of identical exchangers, it can be seen from Equations 4 and 17 that for any arbitrary exchanger,
0.550
0.601 0.632 0.673 0.695 0.707 0.720
0,650
0.690 0.713 0.725 0.737
Or
0.355 0 437 0:511 0 558 0:590 0.626 0 646 0:656 0.667
0 280 0'349 0:409 0 443 01466 0.493 0.509 0.516 0.523
0 243
0'295 01341 0 366 0'382 0:403 0.414 0 419 0:423
0 190 01226
0.256 0.272 0.283 0.295 0.200 0.300 0.301
1.0
which means that there is no backward curvature of the correction factor lines for exchangers with two Equation tube passes per shell pass. When M = 38 becomes
9
8
F 7
and all the curves exhibit a maximum as shown in Figure 5 . For intermediate values of M maxima also occur a t correction factors greater than 0. With fixed inlet temperatures, constant rates of flow on both streams, and constant heat transfer rate, this backward curvature is to be interpreted as meaning that the quantity of heat transferred increases with increasing surface up to a certain point, beyond which the addition of more surface actually decreases the amount of heat which can be exchanged. This becomes apparent upon considering the fact that P is directly proportional to the heat transferred and that qi is proportional to the exponential function of the surface (Equation 25). The seeming paradox is explained by Smith ( 5 ) as follows: "The physical explanation of this maximum heat transmission with surface of finite area is that when the outlet temperatures pass one another there is a region of the surface in which the fluid with the high inlet temperature is colder than the fluid with the low inlet temperature, and therefore heat is transmitted in the wrong direction in this region, and the region of reversed heat transmission becomes relatively greater as the surface is increased." Since this effect occurs a t correction factors less than those tolerated in practice, it is chiefly of academic interest; FIGURE 4. CORRECTION FACTORS FOR ONE-FOUREXCHANGERS WITH however, in conUNBALANCED PASSES, AND (bottom s t r u ~ t i n g correcgraph) ~ 0 n i p A R I s o NOF c O ~ E C T I O N tion factor curves, FACTORS
6
0 153 0'180 01202 0.213 0 220 0:228 0 231 01231 0.232
+
U(A WCN , A,)
where N
=
= f(P,
R, K , iM)
(41)
number of shell passes in series
All the quantities in Equation 41 except
P are constant by definition and independe n t of w h i c h s h e l l p a s s i s c o n s i d e r e d ,
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
October, 1941
1221
in which case P must be the same for all shell passes; furthermore, if P is constant, then F must also be the same for each shell pass and for the exchanger as a whole. This was noted by Nagle (4) and Bowman (1) and proved by Fischer (3) for the case where K = 1. Bowman has shown that for such a condition the value of P for any of the shell passes may be determined from the value, PN,for the entire exchanger by means of the relations:
-P d .
P =
I
')*"
H
whenR f 1 (42)
R-(E
Thus, the mean temperature difference correction factor for an exchanger with N shell passes is found by substituting the P value from Equation 42 (or 43) into any of the equations for single-pass shells. This factor is the same, whether the shell fluid enters at the stationary head end or the floating head end of the shell, provided K is the same for each shell pass.
Examples The examples given in Table I11 illustrate the mean temperature difference for various designs of multipass heat exchangers; example 1 is taken from the paper by Bowman, Mueller, and Nagle (2) with the addition of some unbalanced-pass designs. The following example (No. 3) carried out in some detail will indicate the use of the Courtesy, The Griscom-Russell Company equations and curves: It is required to VACUUMCONDENSER FOR HEAVY PETROLEUM VAPORS exchange 1,000,000 B. t. u. per hour in The two upper tube bundles have six passes through the tubes; the lower bundle a shell and tube exchanger by coolhas four tube passes. ing the shell fluid from 200" to 143" F. and heating the tube fluid from 100" to 157" F.; removable tube bundles are specified and packed so, too, is a single-pass shell-single-pass tube exchanger beglands are not acceptable. For these conditions P = 0.57, cause of the structural restrictions. The remaining possibiliR = 1, and Ahos = 43" F. ties are: ( a ) two single-pass shell-two-pass tube exchangers The usual correct>ionfactor curves show that a single-pass in series, (6) a two-pass shell-two-pass tube exchanger with a shell exchanger (with a factor of 0.50) is out of the question; removable longitudinal baffle, and (c) a single-pass shelltwo-Dass tube exchanger with unbalanced passes. Each of these cases will have to be c o n s i d e r e d t o d e t e r m i n e w h i c h is m o s t TABLE 11. COORDINATES FOR FIQIJRE 4 (Single-Pass Shell-Four-Pass Tubes) economical, P CASE 1. It is found possible to obtain a K F R-0.2 R = 0 . 4 R = 0 . 6 R-0.8 R 1 R-1.5 R = 2 R = 3 R 4 heat transfer rate of 87, utilieing all the avail0.575 0.450 0.379 0.330 0.295 0.236 0.197 0.154 0.128 2.0 0.976 able pressure loss. Since the counterflow mean 0.95 0.685 0.552 0.470 0.410 0.370 0.292 0.244 0.188 0.160 temperature difference is 43" F. in each case, 0.9 0.786 0.655 0.567 0.497 0.448 0.357 0.299 0.227 0.185 0.85 0.836 0.709 0.619 0.547 0.493 0.394 0.329 0.247 0.199 the surface required is 0.8 0.862 0.744 0.6h4 0.679 0 . 5 5 2 0.418 0.347 0.259 0.207
-
3.0
I
0.7 0.6 0.5
0.893 0.907 0.915
0.784 0.806 0.817
0.695 0.717 0.728
0.619 0.641 0.651
0.976 0.96 0.9 0.85 0.8 0.7 0.6 0.5
0.590 0.710 0.805 0.849 0.877 0.905 0 919 0.927
0.458 0.569 0.677 0.728 0 760 0.803 0.824 0.835
0.393 0.483 0.580 0.635 0.669 0.711 0.735 0.746
0.342 0.425 0.613 0.662 0.695 0.634 0.664
-
0.559 0.579 0.589
0.446 0.461 0.467
0.370 0.379 0.384
0.274 0.280 0.282
0.215 0.219 0.221
0.307 0.884 0.460 0.505 0.534 0.571 Q 692 0.668 0.602
0.247 0.306 0.369 0 406 0.430 0.457 0.470 0.477
0.208 0.258
0.162 0.199 0.235 0 251 0.263 0.277 0.283 0.285
0.135 0.163 0 I91 0.202 0.209 0.217 0.221 0.222
0.810
0 337 0.356 0 379 0.389 0.390
where the correction factor, 0.923, is taken from Nagle's curves (4). This surface is ins t d e d in two 12-inch shells with tubes 8 feet long.
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1222
Vol. 33, No. 10
Courtesy, The Grisoom-Russell Company
TUBEBUNDLEOF
A
TWO-PASSSHELLAND TWO-PASSTUBE EXCHANGER WITH SEALING REMOVABLE BAFFLE
CASE2. In this case true counterflow is obtained and a rate of 89 is possible. The surface is then -Oo0
Oo0
43 X 89
- 261 sq. ft.
in a 12-inch shell with 16-foot tubes and a removable longitudinal baffle. CASE3. It is decided to use 75 per cent of the surface in the counterflow pass and 25 per cent in the parallel-flow pass. The corresponding rates are found to be 7 2 and 94, respectively; the over-all rate, from Equation 10, is 77.5. The value of K is
A
SELF-
between the tube passes does not throw too heavy a load on the tube joints of the smaller pass.
Use of Curves as Correction for Variation of Rate in Conventional Exchangers Although the expressions for correction factors presented here have been derived primarily for use with heat exchangers having tube passes of unequal heat transfer surface, they may also be used as an approximate correction to mean temperature difference for the variation of heat transfer rate through single-pass shell-two-pass tube exchangers with equal tube
(75 X 72)/(25 X 94) = 2 . 3
and from Figure 2, F = 0.79. The surface requirement is then 1,000,000 (43 X 0.79) X 77.5 = 376 sq’ ft’
and a 13-inch shell with 16-foot tubes is necessary. An actual comparison of costs shows that, despite the smaller surfaces, the two-pass shell-two-pass tube eschanger (case 2) and the two-single-pass shell-two-pass tube exchangers (case 1) are more expensive than the unbalancedpass unit (case 3) by 14 and 23 per cent, respectively. The arrangement in two shells will usually be found more costly because of the duplication of parts which contribute nothing to the transfer of heat (stationary heads, shells, shell covera, etc.) ; however, single shells with removable baffles, regardless of the considerable additional expense of the baffle, may often be operable under conditions which are impossible for practical unbalanced-pass designs. The preceding example also indicates some of the limitations of exchangers with unequal tube passes. The pressure loss through the tubes is necessarily higher than it would be for the same exchanger with equal tube passes, and the over-all heat transfer coefficient is less. It is also to be noted that the gain in counterflow surface is counteracted to a certain extent by the reduced rate, so that an unbalance ratio of 3 in surface has a net effect of 2.3; this effect varies with the relative contribution of the tube-side film coefficient to the over-all coefficient. The unbalanced design is further limited to such temperature ranges that the differential thermal expansion
FIGURE5. CORRECTION FACTORS FOR MIXEDCROSS-FLOW EXCHAKCERS, SHOWING MAXIMA REACHED BY CURVES
*
October, 1941
