THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING DESIGN OF FINNED TUBE CONDENSERS Edwin H. Ygung Dennis J. Ward October, 1956 IP-184

DESIGN OF FINNED TUBE CONDENSERS Finned tubes can usually be used to advantage in shell and tube condensing applications where the fin side condensing coefficient is lower than the tube side coefficient. Such a situation normally exists in most organic condensers. This is illustrated by the universal use of finned tubes for condensing refrigerants in the refrigeration industry. How to determine if finned tubes can be used to advantage in petroleum and petrochemical applications is the subject of this second article in a series on finned tube heat transfer.(l) An understanding of how finned tubes perform in condensing applications is necessary in order to determine whether finned tubes should be specified. A number of papers have been published reporting the performance of finned tubes in condensing applications.(28) Nusselt (9) has presented theoretical relationships for condensing of saturated vapors on horizontal and vertical surfaces. These relationships may be applied to the horizontal tube portion (root section) and vertical fin section of a finned tube for predicting the mean coefficient on a horizontal finned tube. Nusselt's relationship for a 1

single horizontal bare tube is: 1 kf3 Pf2 gc 4 hr = 0.725 f. (1) LPf D A tcf J Nusselt's relationship for condensing on a vertical surface is: 3k Pf2 g A1 4 hf = 0.943 (2) Pf L A tcf Equations 1 and 2 were derived by Nusselt by taking into consideration the forces acting on the condensate film, using the assumption that the flow of the condensate is laminar. Beatty(10) developed a relationship involving an equivalent diameter for finned tubes which incorporates Equation 1 for the horizontal portion (root section) of the finned tube and Equation 2 for the vertical finned portion of the tube. The equivalent diameter for condensation is obtained in the following manner: Let q h' A At = hr Ar A t + hf Af 0 A t (3) total o eq cf r r cf Where the subscripts r and f refer to the root and finned portion of the tube respectively and Aeq is given by Equation 13 of Reference 1. Substitution of Equation 1 and Equation 2 into Equation 3 and solving for ho assuming that A tcf in Equation 1 is the same as A t f in Equation 2 gives: kf3 Pf2 c A 1.0 Af\ h = 0.725 r p f- + ] - ( (4) LPf A tceq Dr4 Aeq L 2

A comparison of Equation 4 with Equation 1 indicates that the equivalent diameter of a finned tube is: 1 1 XV Ar 0 Af + 1.30 (5) DeV Ae D r A L4 eq r eq where area of one side of one fin L =,..........(6) diameter of fin Therefore the temperature difference form of Beatty's equation for a single finned tube may be written as: kf3 p2 g Pf 4 h = 0.725 (7) f tcf eq In deriving Equation 7 Beatty assumed that the temperature drop across the condensate film on the fin was the same as the temperature drop across the condensate film on the root portion of the tube. A more precise relationship can be derived without making this assumption by allowing for the effect of the fin efficiency on the A tcf over the fin portion of the tube. The resulting relationship for a single tube is: 1 3 kf3 pf2 gc A' 1.A0 04 A 0 I ho = 0.725 +ft i 1 ) (8) 1Lf A tcf Nke Dr4 Aeq L4 where A t = the temperature drop across the condensate cf film in the root portion of the tube. A comparison of Equation 8 with Equation 4 indicates that the fin efficiency 0 appears in Equation 8 to the 3/4 power and in Equation 4 3

to the first power. The A tcf of Equation 4 is given by: A t C (At)m (9) c f whereas the A t f of Equation 8 is given by: /A t (-) (At)m (10) r where U = the overall coefficient for the root portion of the r finned tube, h = given by Equation 1. In normal organic condensing applications involving low finned tubes the fin efficiency is close to 100% and the use of Equation 8 is not warrented. If low conductivity metal finned tubes are used with relatively high condensing coefficients the use of Equation 8 is recommended since it is more exact and gives slightly higher condensing coefficients. For a fin efficiency of 100% Equation 4 and Equation 8 give identical results. The use of Equation 4 is illustrated in the design of a condenser. An alternative design procedure is to replace the temperature difference, A tcf by a tube loading term, W1. The necessary relationship is derived in the following manner. Raising both sides of Equation 7 to the fourth power: k 3 p 2 gc \ (h')4 = (0.725)4 ( kf ) (11) af A tcf DIq

but by Equation 3 = Aeq cf and q = W1 L (12) where W = pounds per hour per foot of tube length, L = length of tubing (feet), A = latent heat of vaporization, btu per lb. Substituting Equation 12 into Equation 5 and then substituting the result into Equation 11 and simplifying gives: 3 2 k of gc Aeq ho = o.65( -f Pf —) _e (13) Pf Deq W1 / which is the tube loading form of Beatty's equation for finned tubes. Relationships 7, 8, and 13 are for single tubes. As indicated earlier these relationships assume laminar flow of the condensate film. The theoretical correction factor for the influence of 1 the number of tubes in a vertical row is given by Nusselt as (1)4 for Equation 7 and 8 and (1) for Equation 13 where N is the average numN ber of tubes in a vertical row. The work of Katz,(5'7) Geist,(5 Short and Brown,(11) and Young and Wohlenberg(l2) indicates that the correction factor of Nusselt is too severe and that a turbulence correction factor probably should be included in Nusselt's equations when applied to multitube units. The correction for turbulence Cn as given by Katz, 5

Young, and Balekjian(7) can be included in Equation 7 and Equation 13 as follows: C rkf3 Pf2 gc 1 4 ho = 0.725 (7a) N/.)f A tcf Dq 4 1 C kf3 Pf2 g A eq h= 0.65(f --- (13a) 0 \Ns /lf D w Figure 1 presents a plot of 1 for condensation of Acetone, N4/ N-Butane and Freon 12, on six finned tubes in a vertical row from the /t a s(5 4 data of Katz and Geist.(5) Figure 2 presents C) from the same data \N) for use in the tube loading method. Examination of these figures indicates that the Nusselt correction factor 1 4 is counterbalanced N by a Cn value which for all practicable purposes results in a correction factor virtually independent of the number of tubes in a vertical row. This conclusion based on the data of Katz and Geist substantiates the findings of Short and Brown.(^l) A Cn plot based on the data of Young and Wohlenberg(12) for five bare tubes in a vertical row is compared with the Cn curve for Freon 12 condensing on six finned tubes in a vertical row in Figure 3. This figure indicates that the Cn correction for bare tubes is the same as for finned tubes. The data of Young and Wohlenberg(12) supports the conclusion that the correction(Cn\for plain tubes is also virtually independent of the number of tubes in a vertical 6

row. It is recommended that an average value of Cn be used in 1 Equation 7a for tube bundles having an average number of tubes in a vertical row less than 10. The recommended values are tabulated in Table 1 for three condensing vapors. Table 1 Average values: 4 Fluid (C C ANeo0 09N3 Acetone 0.97 0.96 N-Butane 0.94.92 Freon-12 0.80.74 For bundles having more than 10 tubes (average) in a vertical row, it is recommended that the lines in Figures 1 and 2 be extrapolated. The average number of tubes in a vertical row for condensers having more than twenty tubes and single pass arrangements on the shell side can be estimated by the use of Equations 14 and 15: N = 0.815 (X)52 for square pitch arrangement (14) N = 0.40 (x)05 for triangular pitch arrangement (15) where X = total number of tubes. 7

Economic Considerations,,, Industrial bare tube heat exchanger cost curves can be used for estimating finned tube heat exchanger costs by subtracting the cost of the bare tubing and adding the cost of the finned tubing. For example the current cost of a 3/4 inch - 19 fin per inch admiralty tube (.065 inch root wall) costs 20% more per pound or 40% more per foot than a 3/4 inch - 16 gage bare tube. Donohue,(13) Zimmerman and Lavine,(l4) and Kern(l5) have presented recent cost data on heat exchangers. Example Design of a Debutanizer Overhead Condenser Comparable designs of bare tube and finned tube condensers to condense 150,000 pounds per hour of a 20-80 mole per cent mixture of C3 and N-C4 at 160 psia are desired. Treated cooling tower water is available at 800F and is not to exceed an outlet temperature of 120~F. Three-fourths inch O.D. admiralty tubes are to be used in the designs. Table 2 Tube Specifications Description Bare Tube Finned Tube 3/4" - 18 BWG Admiralty 19 ft/inch - 3/4" O.D. - 18 BWG Admiralty _.,, l I II 0.D. inches 0.750 0.737 I.D., inches 0.652 0.541 Droot inches. 0.641 X = wall thickness, inches 0.049 0.050 8

Table 2 continued: Description Ao, sq. ft./ft. Ao Ai Acs' ft.2/tube Bare Tube 0.1963 1.15 Finned Tube 0.438 3.18 0.00232 0.001605 A. Preliminary Calculations. 1. From the equilibrium values for these compounds(16) the dew point and bubble point of the mixture were computed as: Dew point = 166~F Bubble point = 154~F 2. Assuming a linear temperature drop of the condensate with Q and the maximum temperature rise of the water, the mean temperature difference is: AT lm (154-80) -(166-120) 74 LIn = 58.8~F 3. The change'c3 Qc3 LHc4 4 in enthalpy of the pure components is:(17) = 126 Btu/lb. = 126 Btu/lb x 24,750 lb/hr = 3,120,000 Btu/hr. = 137 Btu/lb. 9

Qc4 = 137 Btu/lb x 125,250 lb/hr = 17,180,000 But/hr for a total heat duty of Q = QC + TQc = 3.12x106 + 17.18x106 = 20.3x106 Btu/hr 4. The water flow sate is: 20,300,000 W = --- = 507,500 lb/hr or (assuming p = 62 lb/ft3) 2.27 cu. ft./sec. B. Design of Finned Tube Condenser. Assume a 27 inch ID a 15/16" triangular I "I... shell containing 656 tubes ten feet long on pitch each with two passes on the tube side. 1. Water flc 3w rate and film resistant 656 Aflow = - 0.001605 2 2.27 Vt = = 4.31 0.526 ce = 0.526 ft.2 ft/sec. Vt0.8 t = (4.31)o.8 = 3.23 for water(19) 150 (l+O.Ollt ) Vt0'8 hi = -- diO.2 10

150 (2.1) (3.23) 1150 Btu/hr - ~F - sq. ft. (0.541)0~'.2 A 3.18 o. = -- = 0.00276 Aihi 1150 2. Metal Resistances. The fin resistance from Table 2 of the first article(l) is: rf = 0.00011 The root metal resistance is: X A (0.050) (o.438) rm = = = 0.00019 K Am (12) (65) (0.152) 3. Fouling Resistances. The outside fouling resistance (from TEMA(8)) is: r = 0.0005 The fouling resistance on the inside of the tube is: Ao ri = 0.001 x 3.18 = 0.00318 Ai 4. Condensing Coefficient. The condensing coefficient is computed from equation (7a): h'= 0.725(-)l -f - ( ) ( ) ()) 1 kf3 Pf2 g, 1 The Nusselt condensing physical property group, L ef 11

for the condensing mixture is given in Table 3. The / Cn value obtained from Table 1 or Figure 1 for condensing butane is 0.94. The equivalent condensing diameter, defined by Equation 5 is calculated for various outside resistances. The computed values for the finned tube given in Table 2 are tabulated below (see Table 4). Table 3 Nusselt Physical Property Group of Propane - Butane Mixture 1 Temperature f k g2 OFf _ 100 170 120 171 140 171.5 160 172 12

Table 4 Equivalent Diameter of Tube D/ 1 \ \ eq 1 + r] [ 0 4oo 400 8oo 1200 Average Value 3.53 3.52 3.49 3.48 3.50 Examination of Table 4 indicates that the equivalent condensing diameter of this tube is essentially constant with an average value of 3.50. Substituting in Equation 7a: 1 ho = 0.725 (0.94) (35.50) (3.4) (171.5) 1 )4 h'~~~~~~~~~~ } The c 1 = 1390 x / 1 4 ~ondensing coefficient is first ho = 600 1 - 1 = 0.00011 + 0.0005 + assumed to be: 0.00019 Uo ho Substituting 1 Uo Uo + 0.00318 + 0.00276 r for ho = 600: = 1 + 0.00674 = ( 600 = 119.0 = 0.00674.0oo841 13

The temperature drop across the condensate fill 9 is: Uo 119 Atc = - (Tm) = (58.8) = 11.65~F ho 600. Checking the assumed ho of 600: 1 1 4 ho = 1390 x ( - = 756 \11.65 which does not check. Second Trial Assume ho = 800, then 1 1 -= - + 0.00674 = 125 UO 800 125 At = - x 58.8 = 9.19~F 800 or ho = 1390 ( ) = 800 9.19 which checks exactly the assumed value of 800. Therefore m by Equation Uo The required Q A = UATm = 125 and Tm = 58.8~F area is 20,300,000 2,= 500,- 0 = 2670 sq. ft. (external) (125) (58.8) 14

The provided area is A = 656 x 10 x 0.438 = 2870 sq. ft. external for an excess of 110 XS = --- = 4 2760 A plain tube condenser, designed using the same procedure, required a 35 inch shell containing 988 tubes ten feet long with six passes on the tube side. A summary and comparison of the plain and finned units are given in Table 5. 15

Table 5 Summary of Requirements Item Plain Tube Finned Tube --., Heat duty, btu/hr Atm, ~F Twater in' OF Twater out, F Water velocity, ft/sec Outside fouling factor Inside fouling factor Uo, btu/hr-~F-sq. ft. (outside) U1, btu/hr-~F-ft. of length Shell diameter, inches No. of tubes Tube length, ft. No. tube passes Required area, sq. ft. Area provided, sq. ft. %XS area, % Exchanger cost Savings, dollars Savings, % 20,300,000 58.8 80 120 5.94.0005.001 20,300,000 58.8 80 120 4.31.0005.001 183 125 35.9 54.7 35 988 10 6 1890 1941 2.7 $12,000 _ — _ 27 656 10 2 2760 2870 4.0 $10,400 1600 13.3 16

Acknowledgment Permission by Wolverine Tube Division of Calumet and Hecla, Inc. to publish this paper is appreciated. 17

References 1. Young, E. H. and Ward, D. J., Petroleum Refiner. 2. Katz, D. L., Hope, Hope, Datsko and Robinson, Refrigeration Engineering, pp. 53, 211, 315, 1947. 3. Katz, D. L. and Robinson, D. B., Heating and Ventilating, 44, 1947. 4. Beatty, K. 0., Jr. and Katz, D. L., Chem. Eng. Prog., January, 1948. 5. Katz, D. L. and Geist, Trans. ASME, 70, November, 1948. 6. Ames, G. W. and Newell, R. G., Chemistry in Canada, January, 1954. 7. Katz, D. L., Young, E. H. and Balekjian,G., Petroleum Refiner, Vol. 33, November, 1954. 8. Katz, D. L. and Gillespie, D. C., NGAA, Proceedings of 27th Annual Convention. 9. Nusselt, W. Z., Ver. Deut. Ing., 60, 1916. 10. Beatty, K. 0., Ph.D Thesis, University of Michigan, 1946. 11. Short, B. E. and Brown, H. E., Institution of Mechanical Engineers and A.S.M.E. Proceedings of the General Discussion on Heat Transfer, Section I, pp. 27-31, London (1951). 12. Young, F. L., and Wohlenberg, W. J., Trans. ASME, Vol. 64, pp. 787-794, 1942. 13. Donohue, D. A., Petroleum Processing, March, 1956. 14. Zimmerman, 0. T. and Lavine, I., Petroleum Refiner, Aug., 1956. 15. Kern, D. Q. and Associates, Petroleum Refiner, August, 1956. 16. N.G.S.M.A. Data Book, 1951. 17. Maxwell, J. B., Data Book on Hydrocarbons, D. Van Nostrand Co., 1951. 18. "Standards of the Tubular Exchanger Manufacturers Association", 2nd Ed., TEMA, New York, 1949. 19. McAdams, W. H., Heat Transmission, Third Ed., McGraw-Hill Book Company, 1954. 18

I.I ir i I I I I -- I I I I I I I I I 1.2 1.1 1.0.90.80 Acetone N- butane Freon 12.70 II I I _ I I I I W.J %.. I I I 2 3 4 N, NUMBER OF TUBES IN A VERTICAL 5 ROW 6 7 8 9 10 1. Correction for number of finned tubes in a vertical row for use with temperature difference method.

I: A 2- - 1 l l l l l l 1u. I.I! I I I I I I I I - 1.0.90 N-butane l n.80 ZItz.70 0 2.70 Freon 12 I.. I I 1 I 1 I I I I I 2 3 4 5 N, NUMBER OF TUBES IN A VERTICAL ROW 6 7 8 9 10 2. Correction for number of finned tubes in a vertical row for tube loading method.

1.5 1.4 1.3 1.2 1.1 1.0 Z.90. 0C.80.70 I LEGEND o Plain tube data of Young and Wohlenburg(12) X Finned tube data of Katz and Geist (5) L re I I I I I I II.501 I 2 3 4 5 6 N, NUMBER OF TUBES IN VERTICAL ROW 3. Comparison of Cn values for Freon 12 condensing for plain and finned tubes. 7 8 9 10

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