INDUSTRIAL AND ENGINEERING CHEMISTRY
December, 1941
1495
Literature Cited
(2) Baylis, J. R.,J.A m . Water Works AS800., 12,213 (1924). (3) Beijerinck, M.W., Zentr. Biochem. Biophys., 16,277 (1914). (4) Harden, E. C.,U. S. Geol. Survey, Professional Paper 113, 46-7 (1919). (6) Hoover. C. P., J . Am. Water Works hsOC., 23,1282 (1931). (6) Hopkins, E.S., Eng. News-Record, 100,870 (1928). (7) Hopkins, E. S.,and McCall, G. B., IND.ENQ.CHIOM., 24, 106 (1932). (8) Hopkins, E.S.,and Whitmore, E.R., Ibid., 22,79 (1930). (9) Purcell, L.T.,J . Am. Water Works Assoc.. 31,1776 (1939). (10) Tillmans, J., Hirsch, P., and Haffner, F., Gus- a. Wasserfach, 70, 26 (1927); 71,481 (1928). (11) Weber, O.,Chem-Zto., 51, 794 (1927). (12) Weiser, H.B.,“Hydrous Oxides”, p. 294, New York, McGrawHill Book Co., 1926. (13) Weston, R. S.,and Griffin, A. E., J. New En& Water work8 Assoc., 47, 40 (1933).
(1) Am. SOC.of Agronomy, “Hunger Signs in Cropa”, Waahington, D.C.,1941.
PRESENTED before the Division of Water, Sewage, and Sanitation Chemistry at the 102nd Meeting of the Amerioan Chemical Society, Atlantic City, N. J.
tions of soluble manganese each autumn, and with the turnover this manganese will be diffused through the water, ridding them of the trouble until the warm weather period of the following year. If reservoirs are of sufficient depth to preclude a turnover, it is believed that the manganese will remain constant for a long time. This situation is believed to be true regardless of whether the reservoir has been previously stripped or not. The data in this paper depict conditions on the bottom of all deep reservoirs that have flooded large areas of vegetation, and show that purification processes for the removal of manganese are a necessity for all supplies obtaining water from this type of storage.
Mean Temperature Difference in
Multipass Exchangers Correction Factors with Shell Fluid Unmixed KARL A. GARDNE Mean temperature difference correction factors are presented for heat exchangers where the shell fluid is not mixed. These factors are shown to be higher than those for exchangers with the shell fluid completely mixed for the same temperature conditions. The limits within which the true correction factors may vary due to an unknown degree of mixing are thus defined by Nagle’s curves (lower limit) and the author’s (upper limit). Conditions are discussed under which it may be desirable to design a heat exchanger deliberately to prevent mixing of the shell fluid.
ORRECTION factors for the mean temperature difference in multipass heat exchangers may be derived on the basis of either of two assumptions: The shell fluid is completely mixed over any cross section; or the shell fluid does not mix at all. I n deriving his correction factors, Nagle (6) considered both assumptions and, concluding that the former was probably the more accurate, proceeded on that basis. Subsequent work of Underwood (Y), Fischer @), and Gardner (4) was predicated upon this same assumption.
C
The Griscom-Russell Company, New York, N. Y.
It is not my intention to question the validity of the assumption of perfect mixing in baffled heat exchanger shells. However, it appears that the possibilities of baffling systems deliberately designed to discourage mixing have received little attention, and i t seems desirable to investigate the results obtained under such conditions. Correction factors derived on this basis, when compared with Nagle’s, will show the limits within which the true correction factors may lie, and will provide a quantitative indication of the relative merits of the two types of flow. Equations for MTD in Multipass Exchangers with Shell Fluid Unmixed Equations are derived for correction factors by which the logarithmic temperature difference may be multiplied to obtain the true mean temperature difference in multipass heat exchangers with the shell fluid unmixed. The basic assump tions are: 1. The over-all heat transfer coefficient, U,is constant throughout the exchan er. 2. The rate of flow o? 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. 5. Heat 13sses are negligible. 6. There is equal heat transfer surface in each pass. 7. There is equal rate of flow of the shell fluid around each tube pass. 8. The equal streams of shell fluid are independent, and no heat is. transferred between them by mixing or any other mechanism.
1496
INDUSTRIAL A N D ENGINEERING CHEMISTRY (TI - T ; )
=
Vol. 33, No. 12
wc(t, - t l )
Equations 8 and 10, and 9 and 11 may be combined to yield
J
I LOCATION OF CROSS
SECTION
OF A 1-2 EXCHANQER WITH UNMIXED FIOURSI 1. DIAGRAM SHELL FLUID
17 I
I ?I I
1
The first six assumptions are common to both mixed and unmixed flow; the seventh and eighth apply only to unmixed flow. Figure 1 is a diagram of a 1-2 exchanger and the temperatures involved. A heat balance over the entire exchanger gives : WC(T1 - Ta) = wc(t2
-
W C ( T ~- T ~ ) U A
~
- tl)
t or~ Atm ; =
(1)
-
U A ( T ~ Ta)
(2)
FIGURE2. DIAGRAM OF A 1-2M EXCHANGER WITH UNMIXED SHELLFLUID
The correction factor is defined as
(TI- Tz) (3)
Solving Equations 12 and 13 for (TI- T ( ) and (TI- T ; ) , respectively, and adding the results,
Let
UA
In+ = RWC! --
(3
In(i%J
F-
Then
( R - 1) In
@
or when R = 1,
P = (1 - PP ) In +
(74
Equations 7 and 7A are valid for all exchangers; is not an independent variable, however, but a function of P, R, and the nature of flow within the exchanger. - , so it is now necessary to determine this function for the cage in hand. Heat balances over the upper and lower halves of the exchanger give :
+
Courtesy, The Griscom-Russell Cornpanu
K-FINAIR COOLER WITH A SINGLE SHELL PASSIN CROSS FLOW AND SIX TUB^ PASSES
The helical fins tend to produce unmixed
flow of
the air through the shell.
.
I N D U S TRS A L A N D T'N G I N E E R IN=-C H E MIS T R Y
December, 1941"
or since Ti +..Tl= n,,
1497
1.0
.6
Unfortunately Equation 15 cannot be solved explicitly for +; however, for a given value of R, can be assumed, P calculated from Equation 15,and the corresponding value of F obtained from Equation 7 (or 7A). A similar series of calculations for the condition where the shell fluid enters at the opposite end results in the same final equation. An exchanger with 2 M tube passes where the shell fluid is unmixed may be considered as M 1-2 exchangers connected in parallel on the shell side and in series on the tube side (Figure 2). The following equations may then be written:
+
Rz = MR In +1 =
1
F .7
1-2 EXCHANGER WITH A UNMIXED SHELL FLUID
3
1.0
.S
(16)
In +
.6,
F.
(17)
.7
.6
.S
where subscript 1 refers to values for a 1-2 exchanger. It is obvious from Equation 18 that Pi must be the same for all the component exchangers; 1. e.,
FIGURE 3. ( A ) CORRECTION FACTORS FOR 1-2 EXCHANQER~ WITH UNMIXED SHELL FLUID,AND ( B ) COMPARISON WITH 1-2 E X C ~ L ~ ~ ~ Q E B S WITH MIXEDSHELLFLUID
By subtracting each term from 1, (20)
(21)
terms of P and R for any even number of tube paases per shell Dass. 6 e n the number of tube passes is very large (M --* a) the correction factors should approach those for cross flow where one fluid is mixed and the other (whose inlet and outlet temperatures are TI and TI)is unmixed. In thisl cme Equation 22 becomes indeterminate, but by applioation of the customary rules, it mayibe reduced to 1
'ii(4-
Therefore, from Equations 18 and 21
2MR
- 1)
P a l - e (23) which agrees with the equation derived by Smith (6) for this case. Equation 23 can be solved for C#J and the result substituted into Equation 7 to give
-+
Equation 22 is the general relationship sought; with it and Equation 7 (or 7A) correction factors can be calculated in
Rq&)
F - (1
- R)ln [l + R l n (1 - P ) ]
Courtesy. The Uriucom-R~serllCompany
TUBING FOR EXCHANQERS SHOWN ON PAGBIS 1496 AND 1499 Above, helical fins; below, longitudinal fins
(24)
Vol. 33, No. 12
INDUSTRIAL AND ENGINEERING CHEMISTRY
1498
GROSS FLOW
.P
.3
.4
.5
.s
.7
10 .
.Q
.8
FIGURE 4. ( A ) CORRECTION FACTORS FOR CROSS FLOW EXCHANGERS WITH ONE FLUIDUNMIXED, AND ( B ) COMPARISON WITH CROSS FLOW EXCHANGERS WITH BOTHFLUIDS MIXED
Discussion of Curves Correction factors have been calculated for 1-2 exchangers from Equations 7 and 15; the results are shown in Table I and Figure 3A. The comparison with 1-2 exchangers where the shell fluid is mixed is shown in Figure 3B. Calculations have also been made from Equations 7 and 22 for 1 4 exchangers; the factors obtained differ so slightly from those for 1-2 axchangers that a separate chart is unnecessary. A similar result was observed by Nagle (6) for 1-2, 1-4, and 1-6 exchangers with mixed shell fluid. A few points for 1 4 exchangers are shown in Figure 3B to indicate the degree of approximation.
F
k
-
Figure 4A gives correction factors for cross flow with one fluid mixed and the other unmixed; the comparison with cross flow where both fluids are mixed is given in Figure 4B. These factors were computed from Equation 24, assuming that the fluid whose temperature change is (Ti - Ts)is unmixed. Bowman, Mueller, and Nagle (g) give an equivalent chart based on Smith’s curves (6) where, however, the fluid whose temperature change is ( t z - tl) is considered unmixed. Examination of Figures 3B and 4B reveals that the correction factors for unmixed flow are consistently better than those for mixed flow, especially a t low values of R. The factors for 1-4 exchangers differ only slightly from those for 1-2 exchangers in the useful region of the chart (roughly, above F = 0.8); even with infinite tube passes (cross flow), the curves in the upper region of the chart very nearly coincide. The same condition is true for exchangers with mixed shell fluid, as pointed out by Bowman (1). These results could have been predicted qualitatively by consideration of the fact that, in a multipass exchanger with the shell fluid mixed, it is possible to heat the tube fluid to a temperature higher than the outlet temperature of the shell fluid, but in doing so, a portion of the surface in one or more of the passes may transmit heat in the wrong direction; indeed this effect may reach a stage where the addition of more surface actually results in a lower heat transmission, as pointed out by Smith (6) andGardner (4). Obviously such a condition cannot exist when the shell fluid is unmixed, and this is the physical explanation for the improved correction factors obtained above. Figure 4B is drawn showing the full range of correction factors to illustrate the reverse curvature of the lines of constant R obtained for cross flow with both fluids mixed (and, to a lesser extent, for multipass exchangers with mixed shell fluid) and its absence for cross flow with one fluid unmixed. The results for single-pass shells may be extended to exchangers with any number of shell passes by the method described by Bowman (1) for exchangers with mixed shell fluid. Concise descriptions of this method may also be found in
FOR FIGURES 3 AND 5 TABLEI. COORDINATES
0.2 R
3
0.4 R
0.975 0.95 0.90 0.85 0.80 0.70 0.60 0.50 0.00
0.600 0.711 0.826 0.885 0.922 0.962 0.982 0.988 1.000
0.460 0.560 0.675 0.741 0.785 0.850 0.895 0.931 1.000
0.975 0.95 0.90 0.85 0.80 0.70 0.60 0.50 0.00
0.828 0.907 0.964 0.984 0.991 0.997 0.999 1.00
0.677 0.775 0.870 0.915 0.940 0.969 0.983 0.992 1.000
1,000
-
P
0.6 R = 0.8 R = 1.0 R = 1.5
R = 2.0 R
Figure 3, Single-Pees Shell, Two-Pass Tubes 0.230 0.196 0.325 0.282 0.380 0.292 0.246 0.404 0.361 0.475 0.355 0.296 0.447 0.502 0.577 0.393 0.328 0.558 0.498 0.636 0.345 0.532 0.419 0.596 0.678 0.371 0.578 0.452 0.652 0.741 0.384 0.610 0.473 0.784 0.688 0.485 0.393 0.632 0.818 0.716 0.500 0.400 0.667 0.769 0.910 Figure 5. Two-Pass Shell, Four-Pass Tubes 0.440 0.355 0.585 0.500 0.422 0.531 0.681 0.593 0.487 0.617 0.777 0.689 0.519 0.664 0.737 0.825 0.695 0.541 0.770 0.857 0.567 0.732 0.814 0.900 0.582 0,758 0.843 0.927 0.590 0.775 0.862 0.945 0.600 0.801 0.899 0.984
0.297 0.347 0.399 0.423 0.437 0.454 0.462 0.467 0.470
3.0 R
-
7
4.0
0.149 0.189 0.226 0.245 0 257 0,271 0.278 0.283 0.286
0.127 0.156 0.183 0.196 0.201 0.209 0.216 0.219 0.222
0.224 0.264 0,292 0.305 0.311 0.517 0.322 0.324 0.325
0.183 0.210 0.229 0.235 0.238 0.242 0.245 0.246 0.246
December, 1941
but the shell-side heat transfer coefficient will usually be improved by mixing. Figure 6 shows cross sections of (a) a n exchanger with a vertical lane between tube passes, ( b ) one with a horizontal lane, and (c) the same exchanger with a longitudinal baffle in the lane. When the tube passes and baffle openings are arranged as in Figure 6a, the assumption of perfect mixing is fully justified provided there are many transverse baffles; furthermore, the lane being normal to the flow of the shell fluid, there is no tendency to by-pass through it. The arrangement in Figure 6b will tend to give unmixed flow although there will probably be some mixing as the fluid from
1.0
S
6
F .?
6 UNMIXED SHELL FLUID
'
5 0
.I
.2
1499
INDUSTRIAL AND ENGINEERING CHEMISTRY
.3
.4
,b
+
-
.6
P*It,-tJ ( Te ti)
.?
.8
.8
1.0
FIGURE 5. CORRECTION FACTORS FOR 2-4 EXCHANCIEIRS WITH SHELLFLUIDU N M I ~ D
other papers (2, 4 ) ; Figure 5 and Table I for 2 4 exchangers were computed on this basis.
Conclusions and Examples I n the absence of other reasons to the contrary, it is advantageous t o design a baaing arrangement that will prevent mixing of the shell fluid between tube passes; from a temperature difference standpoint it is immaterial whether the shell fluid surrounding individual tube passes is mixed or not,
a
b
COMPLETE MIXING
C NO MIXING
PARTIAL MIXING
8
INDICATES FLOW OF TUBE FLUID INTO PLANE OF PAGE
0
INDICATES FLOW OF TUBE FLUID OUT FROM PLANE OF PAGE
'gZzG~
8
~
~
~
the upper and lower halves of the shell makes the turns around the transverse baffles. This arrangement is objectionable because the horizontal lane offers a path of lower resistance through which some of the shell fluid can by-pass the heating surface. The addition of a longitudinal baffle of sufficient thickness to block off the opening through the bundle will also ensure unmixed flow of the shell fluid, as in Figure 6c. If the additional cost of such a longitudinal baffle (or equivalent) does not exceed the saving in surface brought about by using correction factors for unmixed instead off mixed Bow, the arrangement shown in FiguYe 6c is to be preferred to that in 6a. A simple example will illustrate this point. EXAMPLE.It is required to heat the shell fluid from 100' to 220' F. by cooling the tube fluid from 400' t o 200' in a multipass exchanger, thereby exchanging 3,000,000 B. t. u. per hour. The heat transfer coefficient is found to be 50 in a 1-2 exchanger utilizing all the allowable pressure drop. Here R = 0.6 and P = 0.667, and the logarithmic temperature differenceis 136' F.; the correction factors are 0.815 and 0.700 for unmixed and mixed flow, respectively, and the corresponding surface requirements are: 3'0007000 = 641 sq. ft. 60 (136 X 0.815)
Courteay, The Uri.3aorn-Ruasell Company
STACK OF G-FIN POLYSECTIONS These exohangers have two shell pa8ses and two tube prtmes ertoh, but they are frequently furnished with single-pass shells where unmixed flow occurs. The tubes are equipped with longitudinal fins.
3,000,000 60 (136 X 0.700)
(for unmixed flow) f330sq' ft' (for mixed flow)
The cost of adding 89 square feet more heating surface (16.6 per cent) and increasing the shell size to accommodate it is to
~
INDUSTRIAL AND KNGIWBERING CHEMISTRY
1500
be compared with the cost of suitable baffling to prevent mixing; for this particular case unmixed flow is preferable. It is not contended that conditions such as those in the foregoing example will be encountered frequently, but for such cases there is a definite application for exchangers designed to prevent mising on the shell side.
Acknowledgment Acknowledgment is made t o Joseph Price of The GriscomRussell Company for permission to publish this paper. The writer also wishes to thank G. IT.Sedat of the sitme company for his assistance in checking the mathematical derivations.
Nomenclature A C
$
In
=
total heat transfer surface
p
= -t s
p1
- Tn
Ti
=E--
shell inlei temperature
=
= shell outlet temperature from parallel flow section = shell outlet temperature from counterflow section = shell outlet temperature (average)
.
t., 0,. . .t. = intermediate temperature between
first and second, second and third,. . . . (2M - 1)th and 2 Mth tube passes tl = tube inlet temperature tz = tube outlet temperature U = heat transfer coefficient W = weight rate of flow of ehell-side fluid 20 = weight rate of flow of tube-side fluid At, = mean temperature difference . Atlog = logarithmic temperature difference =:
dimensionless parameter
1- p =
(1 ~
- PR)
-- -
UA RWC
1
F(R
1)
= e
Literature Cited
logarithm to the base e half the number of tube passes per shell pass
= =
TI T: 2'; TZ
+
= specific heat of shell-side fluid = specific heat of tube-side fluid = mean temperature difference correction factor
M
wc
Vol. 33, No. 12
- tl - tl b - tl = -t= TI
Id - f b etc., in exchangers with more than 2 Ti -ttl Tt - f b ' tube passes per shell pass
(1) Bowman, R. A., IND. ENG.CHEM..28,541 (1936). (2) Bowman, K. A., Mucller, A. C., and Nagle, W. M., Trans. Am. So,.. Jlech. Engrs.,62, 283 (1940) (3) Fischor, F. I
