Mean Temperature Difference in an Array of Identical Exchangers KARL A. GARDNER The Griscom-Russell Company, New York, N. Y.
ARRAY OF THIRTY TUBEFLO SECTIONS ARRANQED FOR FIVEPARALLEL STREAMS ox BOTHFLUIDS, EACH FLOWINQ IN SERIES THROUGH SIXEXCHANG~RS
Heat exchangers whose mean temperature difference characteristics have been deter-
such a method for the important case of identical 1-2 multipass exchangers arranged in termining the correcmean mined are frequently used in an array series with counterflow through temperature difference a different tion factors in various types of which, in itself, the series, and other writers (3, heat exchangers. The basic deof exchanger. Since the inlet. and outlet 4) used this method to extend temperatures for the array are usually the their results for other types of signs thus far considered are: cross-flow exchangers with both only ones known, a method is required to exchangers. Bowman, Mueller, and Nagle (2), in a summary of fluids unmixed (', '); cross flow determine the characteristics of a cornwork done on mean temperature with one fluid mixed and the POnent exchanger from those Of the array difference up to 1940, reported other unmixed (6,8) ; cross flow in order to utilize available information to on several combinations of crosswith both fluids mixed (I, 4 , 8 ) ; 1-2, 1-4,and 1-6 multipass exthe fullest extent. A general relation is ' flowexchangers. given between the mean temperaturedifArrangements of exchangers changers with shell fluid mixed other than those cited above (6,') or unmixed l3 ference correction factor for an array of are common, however, and the pass exchangers with shell fluld mixed ( 3 ) ; 1-2 multipass exexchangers and that for a cornwriter proposes to clarify the changers with unequal tube ponent exchanger. A complete solution is situation somewhat by presentpasses, shell fluid mixed (4). given for three common flow arrangements. ing a general relationship beThe solution for any one type tween the correction factor for an individual exchanger and requires only a knowledge of the over-all correction factor for a n array of identical exthe inlet and outlet temperatures of both fluids. Frequently, however, such an individual exchanger is part changers. of an array which in itself constitutes a different type of exMathematical Development changer; in most cases the only temperatures known are the The following derivations are based on the assumption inlet and outlet temperatures of the array. If the fund of information summarized in the preceding paragraph is not that the component exchangers of the array are identical in all respects; i. e., the surfaces, heat transfer rates, specific to be wasted in such cases, a method is required for determining the temperature characteristics of an individual comheats, mass flow rates, and nature of construction and flow are the same for each. For these conditions, ponent exchanger from those of the array. Bowman (1) gave
M
ANY investigators have given methods for de-
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This equation becomes indeterminate if either both) = 1; the corresponding solutions are: where N = number of equal streams into which W is divided n = number of equal streams into which to is divided Subscript 2, y = conditions for a component exchanger One general relabionship appears immediately from Equation 2 : R,,,
=
R
(4)
(5)
Let
M
~
- p,
)];whonR
or R1 (or
= l , R 1 # l(8A)
1 - PiRl
F = F1
M'Pl(R
- 1)
; when R1 = 1, R # 1 (8B)
( 3)
where
Also
7'1
F = F1 iWR1(l NP P)( RIn1
R
= number of component exchangers; then
Combining Equations 4 and 5A gives
Since F = Atm/At,og and F,,, = AL,,/(A~L,~)~,~, Equation
6 may be written:
These equations give the general relationship between the correction factor for any component exchanger and that for the array considered as a single exchanger. I n order to use them, it is necessary to express R,and PI in terms of R and P; the former has already been done in Equation 3. It only remains to derive an expression for PI. Since such an expression will depend upon the flow arrangement within the array, i t will not be possible to continue with a general development; i t becomes necessary to consider specific cases of practical importance. CASE1. Although Bowman (1) gave a treatment for the series counterflow arrangement shown in Figure 1, it is repeated here for the sake of completeness. For this case, TZ = N = 1 and R1 = R; also
Since both PI and Rl are the same for each exchanger, any function involving them will also be the same. Therefore,
Multiplying all terms of Equation 10 together,
P1 =
where
p -
- tl T I - ti
t -t
When
R
=
1, Equation 12 becomes: p1
The mean temperature difference correction factor in any type of heat exchanger (assuming no heat transfer by radiation) may be completely expressed in terms of the two parameters, P,,, and RZJv. Equation. 5A might accordingly be written :
But (NUA/MTVC) is a constant by definition, and RZ,,is the same for all exchangers in the array; therefore, P,,, and F,,, must also be the same for each component exchanger. This being the case, there is no further necessity for differentiating between the various component exchangers and P,,,, R,,,, and Fs., may be replaced by P I ,R J ,and F1,where the change in subscript means merely that the reference is to any or each of the components rather than to a particular one. Equation 7 thus becomes:
=
M - P(M P - 1)
(12-4)
Substitution of Equation 12 or 12A into 8 or 8C gives F = F1 (13) where F1 is determined for the particular type of flow in a component exchanger using R1 = R and PI as calculated from Equation 12 or 12A. It is obvious that these expressions are entirely independent of the nature of flow within the component exchangers so they may be used for a series counterflow arrangement of identical exchangers of any type for which i t is possible to determine the mean temperature difference correction factor. CASE2. The arrangement shown in Figure 2 is often used when the pressure loss on the fluid whose flow rate is u: is severely restricted. For this case, N = 1, n = M , and
Proceeding as in case 1,
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September, 1942
When M is large, the correction factors for this case should approach those for cross flow with the fluid whose temperature change is (TI- T,) mixed and the other fluid unmixed. This may be verified by letting M + in Equations 19 and 19A and applying L’H6pital’s rule t o the resulting indeterminate expressions, noting t h a t F1 approaches unity as M increases, regardless of the type of flow. There results: EXCHANGERS CONNECTED I N SERIES ON FIGVRE1. ARRAYO F M IDENTICAL BOTHSTREAMS, WITH COUNTERFLOW THROUGH THE SERIES
F E -
1
+
- R ) In [l
-
- PR)]’
when R # 1 (20) -P F = (1 P ) 1n[l ln(1 P)I; when R = 1 (20A) Bowman, Mueller, and Nagle (.2) gave a curve for this case based on Smith’s equa-’ tions (8). CASE3. The arrangement shown in Figure 3 is simply t h a t of case 2 with the fluids interchanged. Here N = M and n = 1, Ri = M R (21)
+
-
EXCHANGERS CONNECTED IN SERIESON FIGURE 2. ARRAYOF M IDENTICAL THE HIGH-TEMPERATURE STREAM AND I N PARALLEL ON THE LOW-TEMPERATURE STREAM
In (1
pi = 1
-
- (1
If neither R nor RI
=
-
P)1’-)1
(25)
1,
le
FIGVRE 3. ARRAYOF 17.1 IDEXTICAL EXCHANGERS CONNECTED IN SERIESON THE LOW-TEMPERATURE STREIM,AND IN PARALLEL ON THE HIGH-TEMPERATURE STREAM
(1
- PiRJM =
TM+l
-
Ti - ti
= 1
- PR
If R1
=
1,
Since R1 = R / M Pi =
If neither R nor R1
If R
=
M [ I - (1 =
1,
and finally if Rl = 1,
1,
- PR)l’dM]
When R = 1, however, there is no need to distinguish between the two fluids, and Equation 19A holds for either case. The correction factors for cross flow with the fluid whose temperature change is (t, - tl) mixed and the other fluid unmixed are obtained by letting M + m . Equations and curves are given by Smith (8)and Gardner (4). Equations 19, 19A, 19B, 26, and 26A are chiefly valuable for cases where F is know to be unity-e. g., in arrays of true counterflow exchangers; however, for most cases i t is necessary t o calculate PIin order t o determine F I , so t h a t the result is obtained most directly from Equations 8, 8A, or 8B. The preceding three cases are those most commonly encountered in practice, and their solution is considerably simplified by the ability t o reduce them t o the form, .f(Pi,Ri) = [f(PjR)1”” as shown in Equations 11, 16, and 24. A few other uncommon cases-e. g., a group of identical exchangers connected
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GROUPOF N I N E EXCHANGERS ARRANGED FOR THREE C O U S T E R F L O R . STRE4MS CONUEC PED I N P 4 R I L L E : L
in series on both streams with parallel flow through the seriesmay also be reduced to this form. No such simple relation is found for other arrangements, and even the relatively simple rectangular array shown in Figure 4 involves the solution of the following quartic in P I , Fvliich is presented without proof:
Nore complicated arrays require the solution of equations of higher degree for which no formal solution for PI can be obtained. However, a solution for P can usually be written, as shomn above. Therefore, if any particular array is encountered frequently enough t o warrant it, a correction factor chart in terms of P and R can always be prepared by the following procedure : 1. Derive an expression for P in terms of P, and R analogous to Equation 27. 2. Let R have any desired value and calculate RI from Equation 3. 3. Assume a value of PI and calculate F1, P , and F. Repeat t o obtain sufficient points for a curve at constant R. 4. Repeat this process for other values of R.
Discussion and Examples
-I
T,
I
Many familiar types of heat exchangers may be resolved into combinations of identical components, although this fact may not in all cases be immediately apparent. Papers by Bowman ( I ) and Gardner (6) exemplify previous applications of this process under cases 1 and 3, respectively. A single numerical example will serve to illustrate the application of the equations under cases 1 and 2: Calculate the mean temperature difference for the arrangement shown in Figure 5 where each component is a 1-2 multipass shell and tube exchanger with mixed shell fluid. This obviously consists of tx-o identical exchangers connected as in case 2, each of which consists of two identical exchangers connected as in case 1. Let P, and R?represent the temperature characteristics for each group of two connected in series counterflon; then from Equation 18,
1
Te
FIGURE4. ARRAYO F FOURIDENTICAL EXCHANGERS CONNECTED IN SERIESON THE LOW-TEMPERATURE STREAM, AND I N T W O PARALLEL HIGH-TEMPERATURE STRE AXS
2
Po = 62 (1- 4 1 - P R ) and since P = (90 - 80)/(140 (140 - 100)/(90 - 80) = 4.0,
-
80) = 0.167 and R =
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G . W. Sedat of the same company for his assistance in checking the mathematics involved.
Nomenclature A C
total heat transfer surface = specific heat of shell or hot fluid' C = specific heat of tube or cold fluid' = over-all mean temDerature difference I I I I F 1 L-, = 1 A correction factorF Z f v , F1 = correction factor for component exte=90, BO" changer M = total nimber of component exchangers N = number of equal streams into which EXCHANGERS CONNECTED IN SERIES ON THE FIGURE 5. FOURIDENTICAL HIGH-TEMPERATURE STREAM, WITH Two PARALLEL LOW-TEMPERATURE shell or hot fluid is divided' n = number of e ual streams into which STREAMS EACHFLOWING IN SERIES COUNTERFLOW THROUGH TWO tube or fluid is divided' EXCHANGERS = -t s - 11 P &
=
-
Jb
,018
T.
- 0.667)
Pz = 2 2 (1 - 4 1
= 0.212
Also Rz = RI = R/2 = 2. Then from Equation 12, 1
Pi
-
+--
=
1 - Rz
1 - Pz 1 - PZRZ
=
dMz
-= 0.127
1-211-
0.788
I
(w)
for a component exchanger R, , R1 = = inlet temperature of shell or hot fluid for arrav or. Ti when used with subscript, for component exchanger' = outlet temperature of shell or hot fluid for array or. T. when used with subscript, for component exchanger' = inlet temperature of tube or cold fluid for array or, tl when used with subscript, for component exchanger' = outlet temperature of tube or cold fluid for array or, te when used with subscript, for component exchanger' U = heat transfer coefficient W = weight rate of flow of shell or hot fluid' W = weight rate of flow of tube or cold fluid' = subscript indicating arbitrary component exchanger z,y = mean temperature difference for array Atm = mean temperature difference for arbitrary component At,,, exchanger = logarithmic mean temperature difference for array or, Atlog when used with subscript, for component exchanger
_ _
U.3(0
a n d F , = 0.991 (from Underwood's equation for 1-2 exchangers). Therefore from Equation 8, =
- t.I,
(H:) for a component exchanger -
P,,,, Pi =
0.963
This same result could also have been obtained b y using
P2and Rz to determine the correction factor for a 2-4 multipass exchanger which is, of course, 0.991 as before. Then from Equation 19, using M = 2,
T h e difference is due t o the approximation involved in liniiting the number of significant figures t o three.
Literature Cited (1) Bowman, R.A.,IND.ENO.CHBM., 28, 541 (1936). (2) Bowman, R. A., Mueller, A. C . , and Nagle, W. M., Trans. Am. Soc. Mech. Engrs., 62,283 (1940). (3) Fisoher, F. K., IND.ENQ.CHDM.,30, 377 (1938). (4)Gardner, K.A.,Ibid., 33,1215 (1941). (5)Ibid., 33, 1495 (1941). (6) Nagle, W.M.,Ibid., 25,604 (1933). (7) Nusselt, W., Tech. Mech Thermodynamik. 1, 417 (1930). (8) Smith, D.M.,Engineering, 138, 479, 608 (1934). (9) Underwood, A. J. V., J.Inst. Petroleum Tech., 20, 145 (1934).
Acknowledgment is made to Joseph Price Of The GriscomRussell Company for permission to publish this paper, and to
1 Fluids and temperatures are designrtted as shell and tube or as hot and cold for the purpose of fixing ideas only. When dealing with shell and tube exohangers, that designation will probably be profitable.
