tubular heat exchangers is by inhgration of wcdt =
.I
U(1
- t)dA.
The
,~
tubular hest exchanger is a common piece of equip .4&h cbemical plantst Whenever possible, the ex~ W : , m . d e & p e & t ohsve the two fluid streams countercurrent at evety $obit. The adpsatagss of counter cdrrent dMtion a& '&ll knows; ,j& under eertsin cirnunstanew the exchanger has to be m o a & into a more compact constmc+iion at the exponee of&-icliycountamurent operation. When thls isdone, =inadtipass heatexchsngers, the no more striotifcombcurrent. Alone, oertain paths they flow co t, .and along others, parallel. It is up to the desi e '&e choice between the two bjpsa of exchangem. However, for the purpose of this &le, it is ju@$ to state $hcpoblem .. 88 if the exchanger w w to he s t r i c t & c o u n t e r c ~ L In hept exchanger design the most important quantity to b8'calculatedis the heit branefea area. It can he &own thd for adiabatic (wellinaulstsd) heat exchangers, the.hes't transferwea is d c ~ t e d . f mthe n wuation &fd U(T 6)dA '(1)
w h b I heat -fer t
-- "pi&) -
(0
'
T
area, sq. ft.
p e w of cold fluid,B. u.f .F.constant )
eat at
c
.tsmperetUre of aold flmid, .* F.
tampdratumof bot add
u =-pver-~ll ..caaaciertt of
F.
heat tran$er, B. t.u./(hr.). (sq. ft.) (' F.) w = rata of flow of cold fluid, 1b.fir. Rearranging for,iotsgantiOn, for stsady flow conditions Equation 1 is written;
~
-
-
'
'
Equation 2 is integrated beat by the graphic method.
When f is plotted ageinst c/U(T - t), the pres under the
c w e b a t w e a n ~ l i m i tkandk e is equaltod/w (Figure 1). TJm relationbetweenc/CJ(T f)andt isnotreprwnted by a straight h6 on ordinarygraph paper, and eyen though t4a form of the e m may -8 that it is on log-log or semilog
-
I N D U S T R I A L A N D E N 0 I,NE E R E N G C H E M I S T R Y
August, 1943
scale paper, it is not. Therefore, it is necessary to calculate the ordinates of several points. At any given cross section where the temDerature of the cold fluid t and that of the hot fluid is T, i h e over-all COefficient is calculated from the following formula: U = 1
1
2
1 D I + ~ +&' ha' DZ outside diameter of tubes 1
L D 1
& + ha; + 5 ' E
(3)
inside diameter of tubes log mean diameter, or for D I / D , 5 2, arithmetic mean diameter film coefficient of outer fluid, B. t. u./(hr.) (sq. ft.) ( " F.) coefficient Of inner fluid* B. t. u./(hr*) ('q. ft.) ( " F.)
fouling factor for tube outer surface fouling factor for tube inner surface thickness of tube metal wall, ft. heat conductivity of tube metal, B. t. u./(hr.) (ft.) (" F.) The over-all coefficient, as calculated from Equation 3, is for a square foot of tube outside area, and for this reason the heat transfer area, A , in Equation 2 is based on the tube outside area. It is evident that a long time is needed to complete the calculations necessary for the integration of Equation 2. For this reason other simpler methods have been suggested in the literature for calculating the heat transfer area. Naturally these simpler methods are based upon one or more assumptions. The desire to simplify calculations is always justified, provided the assumptions made do not lead to serious errors; if they do, the resultant final error and its sign must be known so that the necessary correction can be made after the completion of the calculations. The purpose of this article is to compare these methods as to simplicity and accuracy. It is somewhat supplementary to the paper on the same subject by Friend and Lobo (4),but the present article is justified because: (1) Friend and Lobo compared the simpler methods with one another on the basis of the implied assumption that one of them (listed as method 2 in the present article) gave the correct answer; the comparison should have been made with the answer obtained by integration of Equation 2 . ( 2 ) For reasons not stated Friend and Lobo used the Sieder-Tate (7) graphic correlation for calculating the film coefficients. The Sieder-Tate correlation is based upon plotting DG/p against (hD/k)/(cp/k) 1 1 3 (pLa/pLru) 0.14 on log-log scale. The physical properties of the fluid,
841
heat conductivity, B. tt. u. /WJ (ftJ ( " F.) P = viscosity, lb./(hr.) (ft.) c = sp. heat at constant pressure, B. t. u./(lb.) ( O F . ) in the Reynolds, Prandtl, and Nusselt groups are taken a t the main-stream temperature. I n palpLw,pa is the viscosity of the fluid a t the main-stream temperature, and pw a t the temperature of the tube wall. For strictly turbulent flow (in the sense of heat transfer)-i. e., for Reynolds number greater than 8000-a single straight line is obtained which can be represented by the equation (6): hD - = IC
0.027 (DT CT )0 '8 ( K CP) 1 ' 8
(E)'
(4)
For Fkynolds numbers less than 8000, the ratio of length to diameter, LID, enters the picture as an important factor. Previous to Sieder and Tate (1936), for strictly turbulent flow inside pipes, the Dittus-Boelter (3) equation (1930),
F'_ --
0.0225
(?)"'"(?)'-
(5)
was considered the best general correlation. Sherwood and Petrie (6) also found that their data could best be correlated by Equation 5 with a coefficient equal to 0.024. The physical properties of the fluid are taken a t the main-stream temperature. Equation 5 represents a straight line on log-log scale when D G / p is plotted against (hD/k)/(cp/k)' n, cchere n = 0 6 for heating and n = 0.7 for cooling. It is evident that the Sieder-Tate correlation offers an advantage in the sense that heating and cooling data are brought into agreement. On the other hand, to calculate the ratio pLa/pCm, it is necessary to know the tube wall temperature. The tube wall temperature can be calculated from the data already given in the problem, but more time will be required to calculate the film coefficients from the Sieder-Tate correlation than from that of Dittus-Boelter. The time needed to calculate tube wall temperatures can be shortened by estimating the over-all and film coefficients as Friend and Lobo have done. An experienced designer can roughly estimate these coefficients. I n any event, there should be some reason other than bringing heating and cooling data into agreement to justify the introduction of the factor (pLa/pw)0.14. If the two correlations give the same answer, it is natural that the simpler of the two, the DittusBoelter equation, should be preferred. On the other hand, if the answers differ, the question to be settled is: Which is the more accurate? Before answering the question, it is desirable to review the various equations, proposed on account of their simplicity, for calculating the heat transfer area.
Heat Exchangers A r e of Myriad Design, Depending upon the Task to Be Performed Courtesy, The Lurnmus Company
INDUSTRIAL AND ENGINEERING CHEMISTRY
842
If U and c were constant, Equation 2 combined with q = wc(th- t , ) would reduce t o the simple form, A = q/UA,
where q
= Am =
( 6)
heat transfer capacity of exchanger, B. t. u./hr. log mean over-all temperature difference, or for A h / A c 5 2, arithmetic mean. Subscripts h and c refer to hot and cold ends, respectively, of the
exchanger.
The conditions stated above are obtained only when the fluids undergo small changes in temperature, such as when they pass through the exchanger a t very rapid rates. Such rates are not practieal. However, in its general form, Equation 6 is still useful provided the change in U is taken into account in one way or another. This is done in methods 1, 2, and 3. COMPARISON OF METHODS
METHOD1. Using an over-all coefficient U,which is calculated using film coefficients a t the arithmetic mean mainstream temperatures of the fluids-i. e., a t To = ( T A T J / 2 for the hot fluid and t, = ( t h t,)/2 for the cold fluid. The equation is:
where R = AJAh
c ( U h - Uo)/Ue subscripts c, h = cold and hot ends of exchanger, respectively
Using Ts and t, in calculating the over-all coefficient Us, the equation for method 2 is A = q/U.&,
A = q/Lra Am
(7)
METHOD2. The arithmetic mean temperatures, T , and t,, are appreciably higher than the integrated mean tempera-
tures. For instance, in problem 1 referred to below, T , = 400" F. and t, = 95" F., while the integrated mean temperatures are 356" and 88" F., respectively. The film coefficients calculated a t the arithmetic mean temperatures are higher than when calculated a t the integrated mean temperatures; correspondingly, the over-all coefficient is higher in the former than in the latter case. I n order to determine the integrated mean temperatures, it is necessary to know the temperatures of the fluids a t various points along the length of the exchanger. Such data are available only after the calculations for the integration of Equation 2 are completed. Therefore the suggestion of using the integrated mean temperatures is of no practical value. However, temperatures T , and t,, approximately equal t o the integrated mean temperatures (compare T, = 344" F. with integrated mean temperature of 356" F., and t, = 88" F. with integrated mean temperature of 88" F.), can be calculated from T , = F(Th - T,) T,and t, = F ( t h - t,) t,. The factor F is obtained from a chart originally prepared by Colburn ( 2 ) on the basis of the assumption that the relation between U and T , or t , is linear. When the chart is not available, F is calculated using
(9)
METHOD3. Assuming that the over-all coefficient and temperature relation is linear, and using the arithmetic average of the Coefficients a t the hot and cold ends of the exchanger A = 2 q / ( u 4~ uc)A m
00)
METHOD4. Making the same assumption as in method 3 and, in addition, assuming c constant and the exchanger adiabatic, Colburn (2) integrated Equation 2 and, combining it with p = wc(th to), derived the equation
-
2.303 log ( UA&/U,A h )
+
+
Vol. 35. No. 8
A
a
[
UhAo
- UcAh
3
COMPARISON.Three problems were selected. The first is for cooling oil and is the same as that used by Friend and Lobo ( 4 ) . The second problem is for cooling aniline: An exchanger is t o be designed for 21,000 pounds of aniline per hour which is to be cooled with clean cooling water. The exchanger is a double-pipe arrangement; for the inner a standard 2inch pipe and for the outer a standard 3-inch pipe are used. It is assumed that a coefficient of 1000 will take care of the
+
+
-
4
Figure 1. Graphic Integration of Equation 4
log R
AREAFOR COOLING OIL, BASEDON OUTSIDE P I P E ARE.4, TABLE I. VALUES OF HEATTRANSFER CALCULATED BY DIFFERENT METHODSAND FOR DIFFERENT OVER-ALLTEMPERATURE DIFFERENCES OF THE HOT AND COLDENDS OF THE EXCHANGER
-600
Deviation Reaiilts . ~.~~ ~ . from of Friend and Correot, ~
Temperature O F .->WaterIn, TI Out, TZ In, ti Out, t z 600 200 70 120
--Oil-
100
70
120
-0ver-All Tz
Ac,
-
tl
130
30
Ti
AAh,
- tn
480
480
(11)
Method
No. 1 2 3 4 5 1 2 3 4 5
Equation
No. 7
9 10 11 2 7 9 10 11 2
Area, Sq. Ft. 186.7 198.5 195.0 209.0 191.5 385.5 480.0 422.0 531.5 474.0
~~
-
%
2.5 3.7 1.8 9.1 0.0
-18.7 1.3 -11.0 12.1 0.0
Lobo
173.9 187.0 1ili:o
... .. .. .. ... ... ...
.
August, 1943
INDUSTRIAL AND ENGINEERING CHEMISTRY
a43
Courtesy, The Lummus Company
Conservation of Heat Energy through the Use of H e s t Exchangers Is an integral Part of Chemical Engineering Design. Bank of H e a t Exchangers in a Recently Built Defense Plant.
resistances of the pipe metal and of any deposit collecting thereon. The third problem is for heating oil of the same physical properties as that in probleni 1. Heating is to be in a double-pipe heat exchanger. The oil flows inside a standard 2-inch pipe a t the rate of 3 feet per second a t the cold end. Saturated steam condenses on the outside. The steam-side temperature is assumed to be constant a t 227' F. along the entire length of the outer jacket. The film coefficient for the steam side is assumed to be 2150. The terminal operating conditions are listed in Tables I, 11, and 111. The temperature of the oil or aniline was kept the same at the inlet and varied a t the outlet. The temperature of the cooling water was varied a t both ends. This approach offers an opportunity for comparing each method with the integration method for different over-all temperature differences a t the hot and cold ends of the exchanger. The film coefficients were calculated from the DittusBoelter Equation 5 with n = 0.7 for cooling and n = 0.6 for heating. The tables show the calculated results. Those calculated by Friend and Lobo (4)are included as the last column of Table I. DISCUSSION
Of the five methods listed in the tables, the fifth is the correct method for calculating the heat transfer area of countercurrent tubular heat exchangers. The calculations are, how-
Shown A b o v e I s a
ever, lengthy. Of the remaining four methods, No. l is the simplest. It requires about half as much time as the others to complete the calculations. However, when the larger over-all temperature difference is a t the hot end (i. e., when Ah> A,) it gives a low answer, and there is the possibility that the deviation from the correct answer will be appreciable and greater than the accuracy required would permit (Tables I and 11). The accuracy of the method improves as the over-all temperature a t the cold end, Ac, is increased. There is no way of predicting offhand the deviation to be expected for a given combination of operating conditions specified in the problem. As Friend and Lobo (4) pointed out from practical experience, in most commercial design problems a deviation of -10 to -15 per cent may be expected from method 1. Therefore, using this method and adding about 11 to 18 per cent to the answer may prove satisfactory. This will be the case, for instance, when, as in problem 1 on the cooling of oil, the exchanger is to be assembled in sections of specified length. For greater accuracy and for reliability in accuracy, method 2 appears to be the most satisfactory as long as Ah>Ac. The deviation from the correct value is small in all cases listed in Tables I and 11, and will possibly be so in other cases. , I n other words, method 2 gives a result within the accuracy required in such problems. When the larger over-all temperature difference is a t the cold end (i. e., A,>Ah) the accuracy of method 1is as good as that of 2 and is better when Ah is de-
844
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 35, No. 8
pared (1) with those predicted from the Dittus-Boelter equation. This comparison showed that Nusselt values predicted from the Dittus-Boelter equation varied from -20 to as much as +60 per cent. -Temperature, C.-Deviation -Aniline-Water-0ver-All AEquafrom The comparison covered a range of experiAh, Method tion Area, Correct, In, Out, Out, In, mental Nusselt values from about 15 to Tz - tr TI- ts No. No. Sa. Ft. % TI TI tr k ._ -18.6 500. To be more specific, for Nusselt 125 25 60 1 20 65 6 7 298.0 2 9 370.0 1.1 values from 15 to 50, all predicted values 3 10 289 0 -21.0 4 11 360.0 -1.6 are high by as much as 15 per cent; from 5 2 366.0 0.0 50 t o 100, all predicted values are high by 17 125 25 65 8 60 1 7 199.2 -10.5 as much as 60 per cent; from 100 to 180, 2 9 231.0 3.8 3 10 192.5 -13.5 the predicted values gradually shift to 4 11 223.0 0.1 lower values and change sign; and from 5 2 222.6 0.0 180 to 500, all predicted values are low 27 65 60 1 125 35 8 7 144.6 -6.5 2 9 156.0 1.0 by as much as -20 per cent. In other 3 4 11 10 156.7 140.8 - 81. 9. 4 words, the plot of experimental against 5 2 154.5 0.0 p r e d i c t e d N u s s e l t values showed a 60 1 7 109.7 -4.6 “bumped” portion. Whether this nature 37, 125 45 8 65 2 3 10 9 1116.0 07.8 - 6 1.1 .0 of the plot is significant is not nom known. 4 11 114.5 -0.2 For the present, the important fact is 5 2 114.7 0.0 that film coefficients predicted from the 52 125 60 8 65 60 1 7 74.5 -2.2 Dittus-Boelter equation are lower than 2 9 76.6 0.5 3 10 74.0 -2.9 those predicted from the Sieder-Tate 4 11 76.1 -0.1 correlation; yet as shown above, they 5 2 76.2 0.0 may be high by as much as 60 per cent 75 52 125 60 8 50 1 7 83 0 -1.2 2 9 85 0 1.1 when compared with experimental values. 3 10 82.6 -1.8 It is evident that there is a great need 4 11 84.5 0.5 5 2 84.1 0.0 for examining anew all available data L for the purpose of recommending a correlation which is simple enough to be acceptable and still compares favorably creased (Table 111). It is interesting t o observe the deviation with the analytical equation. Until then it is left to the insigns (Table 111) and to compare them with those of the predividual designer to make the choice between the Dittusceding cases (Tables I and 11). When &>Ah, method 1 apBoelter equation and the Sieder-Tate correlation, when the pears to be the best of the simpler methods. This observation Reynolds number is greater than 8000. For a Reynolds numis supported by the statement from Friend and Lobo (4) that ber less than 8000, the use of the Dittus-Boelter equation is ((when the larger temperature difference is a t the cold end, out of the question; therefore the Sieder-Tate graphic cormethod 1 gives more nearly correct results”. relation which extends t o include the transition and the visTable I shows that the author’s results are higher than cous flow regions should be used, those of Friend and Lobo. This is due to the fact that the former values are based upon film coefficients calculated TABLE111. VALUESOF HEATTRAKSFER AREAFOR HEATING OIL,BASEDON OUTSIDE from the Dittus-Boelter Equation 5, PlPE AREA, CALCULATED B Y DIFFEREXT hIETHODS BXD FOR DIFFERENT OVER-ALL the latter from the Sieder-Tate correTEMPERATURE DIFFERENCES OF THE HOTAND COLD ENDSOF THE EXCHANGER lation (7). The film coefficients cal--Temperature, F.Deviation culated from the Dittus-Boelter equa-Steam----Oil-0ver-All AEquafrom In, Out, In, Out, Ac, Ah Method tion Area, Correct, tion are lower than those from the TI Tl ti t2 !Pa - t i TI-‘ tz No. No. Sq. Ft. % Sieder-Tate correlation. For instance, 227 227 80 200 147 27 1 7 107.0 3.1 2 9 101.8 -1.9 the film coefficient of the cooling oil 3 10 106.0 2.1 in problem 1 a t 400” F. is 326 when 4 11 102.2 -1.5 calculated from the Dittus-Boelter 5 2 103.8 0.0 227 227 80 220 147 7 1 7 186.0 1.1 equation, and is 378 (according to 2 9 174.6 -5.1 3 10 187.2 1.7 Friend and Lobo) when calculated 4 11 171.5 -6.8 from the Sieder-Tate correlation. It is 5 2 184.0 0.0 difficult to explain this difference as both equations are based upon experimental data, Boelter, Martinelli, and LITERATURE CITED Jonassen (1) and others before them recognized the need Of an approach to heat transfer by conduction(1) Boelter, L. M. K., Martinelli, R. C., and Jonassen, F., Trans. convection in the turbulent region. Using the analogy Am. Soc. Meoh. E n g ~ s .63, , 447 (1941). of heat and momentum transfer, they analytically correlated (2) Colburn, A. p., cHEXM.. 25, 873 (1933). the Nusselt. Prandtl. and Revnolds moduli. While their (3) Dittus. F. W., and Boelter, L. M.K., Univ. Calif. Pub. Eng., 2, 443 (1930). correlation is too complicated-to be of practical value to (4) Friend, Leo, and Lobo, W. E., IND. ENG.CHEY.,31,597 (1939). designers, it serves an excellent purpose as a reference or (5) McAdams, w. H., “Heat Transmission”, 2nd ed., p. 168, New standard equation. Thus, experimental Yusselt values, taken York, McGraw-Hill Book Co., 1942. with those predicted from from the literature, were (6) Sherwood, T. K,, and Petrie, J, M,, IsD.ENG.CaBM,,24, 736 the analytical equation and were found to agree within *20 (1932). (7) Sieder, E. N., and Tate, G. E., Ioid., 28,1429 (1936). per cent. The same experimental Nussek values were com-
TABLE11. VALUESOF HEATTRANSFER AREA FOR COOLING ANILINE, BASEDON OUTSIDE P I P E AREA, CALCULATED B Y DIFFERENT METHODS AND FOR DIFFERENT OVER-ALLTEMPERATURE DIFFERENCES OF THE HOTAND COLDENDSOF THE EXCHANGER ACl
O
