THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING FUNDAMENTALS OF FINNED TUBE HEAT TRANSFER Edwin H. King Dennis J. Ward The research from which this paper resulted was carried out through Engineering Research Institute Project 1592, sponsored by the Wolverine Tube Division of Calumet and Hecla, Incorporated, Detroit, Michigan, August, 1956 IP-172

Fig. 1I. Ty*picall finned tube bundle. Fig. 2. Cross section of a 19 fin/inch low finned tube.

Fundamentals of Finned Tube Heat Transfer By EDWIN H. YOUNG And DENNIS J. WARD Department of Chemical and Metallurgical Engineering The University of Michigan Ann Arbor, Michigan Integral finned tubes can be used to advantage in certain shell and tube heat transfer applications. Finned tube bundles of the type shown in Fig. 1 are becoming widely used in shell and tube heat exchangers. As indicated in Fig. 2 the tube has a plain end for expanding into tube sheets of standard heat exchangers. Heat exchangers for certain applications can be designed so as to take advantage of the extended surface and many existing units can be retubed for the same purpose. The problem of how to determine when finned tubes can be used to advantage is the subject of this series of articles on the use of finned tubes in shell and tube applications. This first article is concerned with the fundamental heat tranfer relationships for determining the required finned tube heat transfer area for shell and tube units. Later articles are concerned with the determination of shell side (fin) coefficients and design of condensers, coolers, and partial condensers. Economic considerations play an important part in determining whether or-not finned tube units should be used. A special Petroleum Processing Report by D. A. Donohue, published in the March, 1956 issue of Petroleum Processing(1) presented an excellent

2 survey of the different types of exchangers, the major characteristics of each, some of the highlights of their design, and some economic considerations. In recent years a number of investigators have published the results of research work on finned tubes.(36-16) The determination of the applicability of finned tubes for a particular service involves the use of ordinary heat transfer considerations plus an understanding of the effect of the fin on the performance of the tube. The basic heat transfer relationship prescribed by the Standards of Tabular Exchanger Manufacturers Association (2) for bare tube surfaces can be modified for extended surface tubes. Bare Tubes Equation 1 gives the fundamental relationship for determining the required outside heat transfer tube surface area, A = (1) UoAt in which: A = required external surface area, sq. ft., = total heat to be transferred, BTU per hr., At = the proper mean temperature driving force, and U = overall service coefficient of heat transfer, 0 BTU per (hr.)(~F.)(sq. ft. of outside surface area). The overall coefficient of heat transfer U for bare tubes is further de~fined asr:~o fined as:

1 -- U = (2) 1 (A (A \ i -+r +r ( +ri -- + 0o o m in which: h = shell side coefficient of fluid medium on outside of o tubing, r = fouling resistance on the outside of the tubing, rw= resistance of the tube wall (root wall in the case of finned tubes) = X/k, ri = fouling resistance on the inside of tubing, hi = film coefficient of the fluid medium on inside of tubing, Am = mean heat transfer area of metal wall, and (Ao\ = the ratio of outside tube surface to inside tube surface. VAi Finned Tubes Equation 1 can also be used for determining the required external surface area for finned tube bundles. Equation 2 which defines the overall coefficient of heat transfer for bare tubes may also be used for finned tubes if the outside coefficient, ho, and outside fouling resistance, rO, are modified so as to include the efficiency of the finned surface. The inside film coefficient, hi, for finned tubes is determined in the same manner as for ordinary bare tubes TEMA fouling factors(1) are

4 used for the inside surface fouling. The resistance of the root wall of a finned tube can be computed in the usual manner since the resistance of the fin is taken care of separately. The remaining terms ro and ho involve the performance of the fin. Fouling tests reported by Katz, et al.,(3) indicate that the fouling factors on the outside of finned tubes are about the same as for bare tubes and that solution cleaning of fouled finned tubes can restore heat transfer to its initial condition. Additional finned tube fouling (4) (5) data have been published by Armstrong and by Ames and Newell. The usual normal fouling factors of TEMA can therefore be used with integral helical finned tube surfaces. The determination of the fin side coefficient has been the subject (6-16) of a number of papers. Gardner has presented fin efficiency curves (17) for several forms of straight fins, annular fins, and spines. Dusinberre has shown that the fin efficiency curves of Gardner can be approximated in simple algebraic form (in the high efficiency region). The fin extends out into the fluid stream and as a result has a "skin" temperature that lies somewhere between the root wall outer surface temperature and the fin side fluid "bulk stream" temperature. The fin efficiency 0 is defined as: 0 = q(actual at Q) tbs - tmf (if at b) tbs tmr

5 where q = actual heat transferred through the fin, q(t = the heat that would have been transferred if the fin surface temperature was at the root wall (or base of fin) temperature, Q = (temperature of bulk stream - temperature of metal fin), (tbs - tf) Qb = (temperature of bulk stream - temperature of metal root), and = (tbs - t). Dusinberre's relationship for a circumferential fin having a rectangular cross section is: 1 0= (4) m2 X r3 where 2 m = H V + +~ > Y (dimensionless) where H = fin height = (Do-Dr)/2, h' = actual fin side coefficient, 0 r = TEMA fouling resistance on outside of fin tubing, thermal conductivity of fin metal, K = thermal conductivity of fin metal, m

6 Y = fin thickness, D = diameter over the fin, and 0 D = root diameter. r The factors that affect the skin temperature of the fin and the fin efficiency are: 1. Fin material (thermal conductivity of fin metal). 2. Fin thickness (if the thickness of a fin is increased, the fin temperature tends to approach the tube root wall temperature rather than the bulk stream temperature). 3. Fin height (if the height of the fin is increased, the fin tends to approach the bulk stream temperature with a loss in temperature driving force). 4. Film coefficient (as the film coefficient rises the fin temperature approaches the bulk stream temperature). 5. Shape of fin. 6. Fouling on the fin side. The fin efficiency is simply a correction factor which must be included in the design to account for the fact that the temperature drop across the fin film coefficient is different than that across the root surface coefficient. This can be illustrated in the following manner. Let Qt = Qf + Qr (5)

7 where Qt = total heat transferred, t Qf = heat transferred thru fin, and Q = heat transferred thru prime metal (root metal). Considering a clean outside surface, f = h' Ai (tbs - t) (6) f o fin mf and Q = h' Arot (tb - tmr (7) r o root bs mr where tb = temperature of bulk stream, t = temperature of fin metal surface, mf t = temperature of root metal surface, and mr h' = actual fin side coefficient. 0 But (t - tf = 0 (tbs - t ) (8) bs bf s mr therefore Q = h' A n (tbs - ) (9) f o fin bs mr and QA -C t ) + h A (t =t ho Aroot bs tr +o fin tbs tmr) (10) An examination of Equation 10 indicates that a choice exists in the application of the fin efficiency 0. The fin efficiency can be considered

8 to either (a) reduce the temperature difference (tbs - tmr), (b) reduce the fin area, Afin, or (c) reduce the fin side coefficient, h'. All three interpretations are used and are of course equally correct. Another alternate design procedure consists of combining the fin efficiency 0 of the fin with the 100 percent efficiency of the root by taking a weighted average based on the relative fin and root areas and reducing the effective film coefficient, ho, by this amount after having factored out the ho in Equation 10. (19) This method is used by Skiba. Two other alternate methods currently in use are presented below; one is based on the application of fin efficiency to the fin area to give an equivalent area and the second is based on a modi(20) fication of the fin resistance method of Carrier and Anderson. Equivalent Area Method EquationlO can be factored to give the following relationship: t = h (Aroot + 0 Afin) (tbs - tmr) (11) or Qt = h Aeq (tbs - tmr) (12) where Aeq = Ar + Afin (13) The equivalent area, A, and actual fin side coefficient, h' are related to the total outside area, Ao, and the design coefficient ho as follows: ho = ho eq 0 0 0 eq (14)

9 The design coefficient ho can be determined by solving Equation 1 for ho to give: = h( Ae) (15) 0 ho - "o(A The fouling resistance ro in Equation 2 is defined as: r = r(AeO) (16) where r' = TEMA Fouling Resistance. The value of ho from Equation 15 and the 0 value of ro from Equation 16 can be substituted directly into Equation 2 and the overall coefficient U0 determined. Substitution of UO into Equation 1 gives the required external finned tube heat transfer area A. An illustrao tion of this procedure is presented following a discussion of the Fin Resistance Method. Fin Resistance Method Carrier and Anderson presented the fin resistance method for handling fin efficiency for the case of non-fouled fins in 1944.(20) The resistance of a non-fouled fin is defined as: f \ho where rf = resistance of the fin. A relationship including fin efficiency which relates ho and h' 0 is given by Equation 15 in which the definition of Aeq is given by Equation 13. If Equation 13 is substituted into Equation 15 and Equation 15 is sub

10 stituted into Equation 17 for h the following relationship of Carrier and Anderson is obtained, rf = (18) rf = h' A'r Af It can be shown that if the fin surface is fouled Equation 17 must be modified to include the fouling resistance as follows: J1 -+ [o o rt 0 (19) where r' = TEMA outside fouling resistance. 0 To use the fin resistance method in design applications, Equation 2 must be rewritten in the following form: U +' + Ai i A = 1 + r' + rf + rw )+ ri(20) Uo 0 o in which ho = actual film coefficient of fluid medium on outside of finned 0 tubing, r' = TEMA fouling resistance on outside of finned tubing, 0 rf = resistance of the fin (see Equation 18), r = resistance of root wall, w ri = fouling resistance on the inside of the tube, hi = film coefficient of fluid medium on the inside of tubing, and (-~) = ratio of outside tube surface to inside tube surface. AW~i

11 Equation 4 can be substituted into Equation 19 to give: f [hT r (21) rf = t r0 =_Ar 1 _m2 For given values of ho and r' the fin resistance rf can be directly determined for a particular finned tube by use of Equation 21. Table 1 gives the dimensions of a 3/4 inch, 19 fins per inch admiralty tube. The fin resistance rf corresponding to various values of + r] are given in Table 2. Table 1 3/4 Inch, 19 Fin-per-inch Admiralty Tube Dimensions A sq. ft. per ft. length 0.438 A/Ai 3.18 Afin sq. ft. per ft. length, (0.8 Ao) 0.350 Aroot sq. ft. per ft. length, (0.2 Ao) 0.088 N, number of fins per inch 19 H, fin height, inches 0.048 Y, fin thickness, inches 0.015 x, root wall thickness, inches 0.050 k, thermal conductivity 65 Do, outside diameter of fins, inches 0.737 Dry root diameter, inches 0.641 Diameter of plain end, inches 0.750 Wall thickness of plain end, inches 0.068

12 Table 2 Fin Resistance of 3/4 Inch 19 Fin-Per-Inch Admiralty Tube 1 rf + r' 0.00011280 10.00011277 20.00011274 50.00011264 100.00011248 200.00011217 500.00011123 1000.00010971 2000.00010679 An examination of Table 2 indicates that the fin resistance of this tube is relatively constant over the usual range of fin side coefficients encountered in finned tube applications. A fin resistance curve can be prepared for any particular tube. An illustration of the use of the fin resistance method of design is presented under Example Calculations. Example Calculations An illustration of the use of the equivalent area and fin resistance methods for the determination of Uo for a distillate cooler using 3/4 inch - 19 fin-per-inch admiralty tubes are presented as follows:

13 Table 3 Film Coefficients h' distillate film coefficient 200 0O hi, water film coefficient 1000 r' fin side fouling resistance 0.001 ri, tube side fouling resistance 0.001 Solution: A. By The Equivalent Area Method This method involves the use of Equations 2, 13, 15, and 16 and the determination of fin efficiency. Equation 4 indicates that the fin efficiency is a function of: m =H' (Do) m= H 2 and V + r KY Substituting the tube dimensions and the outside coefficient and fouling (from. Table 1) the expressions are evaluated as: m H ( 0.256 K.MY 12.012 mf- + H V 1o-. + 0.001 (65) V 0 1o 0 y\00 /'\ 1 12 / and = 0.77 1.072.r.641 The determination of fin side film coefficients is the subject of succeeding articles in this series.

14 Substituting in Equation 4, 1 ( = = 0.97' + (0256)2 (1.072) 3 The equivalent outside area is calculated using Equation 12: Aeq = Ar + 0f Af = 0.o88 + (0.975 (0.35) therefore Ae = 0.429 sq. ft./ft. Substituting Aeq, Ao, and ho into Equation 15 gives: h, = 200 (.429\ = 196 U.\0.438/ Substituting the outside fouling factor, Aeq r =.001 o.438 o 0.429 and Ao.00102 into Equation 16 gives: The tube wall resistance is given by: (A~\ X AO (0.050)(0.438) w VA K A (12)(65)(0.153) overall coefficient, Uo, is obtained by using Equation 2, The U = o 1 1 + 0.00102 + 0.000184 + 0.001(.318) + 5.18 = 79 Btu/hr-~F-sq ft (outside surface) 0.0127

15 B. By The Fin Resistance Method The overall coefficient is computed using Equation 20. The fin resistance is given in Table 2. For the value of: - - = 1 — 1- == 166.7 [ + r' + 0.001 h' ~ 200 the fin resistance from Table 2 is: rf = 0.000112 Substituting into Equation 20 gives: = -1 + 0.001 + 0.000112 + 0.000184 + 0.001(3.18) + 3.18 U 200 1000 = 0.0127 therefore U = 1 = 79 Btu/hr-~F-sq ft (outside surface) o 0.0127 Discussion It is a matter of personal choice as to which method is to be preferred. It is apparent that the corrections for fin efficiency are very slight in the above example. With fin efficiencies of 97 percent or higher the error introduced in assuming 100 percent efficiency will give satisfactory designs for most design purposes. The alternate methods give identical results and indicate how fin efficiency can be used in designing finned tube heat exchangers. Acknowledgment Permission by Wolverine Tube Division of Calumet and Hecla, Inc. to publish this paper is appreciated.

16 References 1. Donohue, D. A., Petroleum Processing, March, 1956. 2. "Standards of the Tubular Exchanger Manufacturers Association", 2nd Ed., TEMA, New York, 1949. 3. Katz, D. L., Knudsen, J. G., Balekjian, G., and Grover, S. S., Petroleum Refiner, Vol. 33, No. 11, 1954. 4. Armstrong, R. M., Trans. ASME, 67 (1945Y. 5. Ames, G. W., and Newell, R. G., Chemistry in Canada, January, 1954. 6. Kays, W. M., and London, A. L., Trans. ASME, 72 (1950). 7. Katz, D. L., Young, E. H., Williams, R. B., and Balekjian, G., Petroleum Refiner, Vol. 33, No. 11 (1954). 8. Beatty, K. 0., Jr., and Katz, D. L., Chem. Eng. Prog., 44 (1948). 9. Myers, J. E., and Katz, D. L., Chem. Eng. Prog. Symposium Series No. 5, Heat Transfer, Vol. 49, 1953. 10. Williams, R. B., and Katz, D. L., Trans. ASME, 74 (1952). 11. Jameson, S. L., Trans. ASME, 67 (1945). 12. London, A. L., Kays, W. M., and Johnson, D. W., Trans. ASME, 74'(1952). 13. Schmidt, T. E., Institute of Mechanical Engineering and ASME, Proc. of the General Discussion on Heat Transfer, Section II, London, 1951. 14. Katz, D. L., and Geist, J. M., Trans. ASME, 70 (1948). 15. Zieman, W. E., and Katz, D. L., Petroleum Refiner, Vol. 26, No. 8, 1947. 16. Katz, D. L., Young, E. H., and Balekjian, G., Petroleum Refiner, 33 (1954) 17. Gardner, K. A., Trans. ASME, 67 (1945). 18. Dusinberre, G.'M., Mech. Eng., Vol. 78, No. 6, p. 570, 1956. 19. Skiba, E. J., Chem. Eng., Vol. 61, No. 12, 1954. 20. Carrier, W. H., and Anderson, S. W., Heating, Piping, and Air Conditioning May, 1944.