ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR NUMERICAL ANALYSIS OF THE TEMPERATTRE DISTRIBUTION IN THE OF TWO HIGH-FIN ALL-ALUMINUM TUBES ROOT Report No. 41 Edwin H. Yout4 Chemical aid Metallurgical Engineering Assistant Professor of Luis 0. Gonzalez Dennis J. Ward James R. Wall David Ing Research Ass istants Project 1592 CALUMET AND HECLA, INC. WOLVERINE TUBE DIVISION DETROIT, MICHIGAN February 1956

- ENGINEERING RESEARCH INSTITUTE i UNIVERSITY OF MICHIGAN - TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv OBJECTIVE iv INTRODUCTION 1 THEORETICAL CONSIDERATIONS 2 METHOD OF ANALYSIS 3 DESCRIPTION OF SYSTEMS ANALYZED 7 ANALYTICAL RESULTS 8 DISCUSSION OF RESULTS 16 CONCLUSIONS 17 APPENDIX A Sample Calculations 18 APPENDIX B Calculation of the Effective Heat Transfer Coefficient for Analysis I 19 REFERENCES 21 J - ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN LIST OF ILLUSTRATIONS Table Page I. Dimensions and Heat Transfer Conditions Assumed for Analyses 7 II. Computed Temperature Distribution in the Fin and Root Walls, Using Grid Presented in Fig. 1 13 III. Computed Temperature Distribution in the Fin and Root Walls, Using Grid Presented in Fig. 2 14 IV. Fin Efficiencies Obtained for Assumed Conditions 15 V. Influence of Root-Wall Temperature Distribution on Effective Heat Transfer Coefficient 16 Fig. 1. Coarse grid used in numerical solution of temperature distribution in the root of a finned tube. 4 2. Fine grid used in numerical solution of temperature distribution in the root of a finned tube. 5 3. Scale drawing of fin and root profile considered for the 0.020-inch wall tube. 9 4. Resulting system of isotherms and adiabat "a - b" for tube A in analysis I (0.020-inch root wall, h'o = 10). 10 5. Resulting system of isotherms and adiabat "a - b" for tube A in analysis II (0.020-inch root wall, ho = 50). 11 6. Resulting system of isotherms and adiabat "a - b" for tube B in analysis III (O.010-inch root wall, h'o = 10). 12 iii

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ABSTRACT The results of this investigation indicate that the effect of the root-wall temperature distribution on the fin efficiency of 9-fin-per-inch aluminum tubes is small for commercial tubes in normal applications. This reduces the fin efficiency by 4 to 59 for a tube with a 0.020-inch root wall and by 11.5% for one with a 0.010-inch root wall, OBJECTIVE The purpose of this investigation is to determine the effect of the longitudinal root-wall temperature distribution on the fin efficiency of thin-walled, 9-fin-per-inch, all-aluminum tubes in normal applications. iv

-- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN INTRODUCTION Finned tubes can be used in many different applications, such as condensing, boiling, cooling, and various combinations of the above. In using the finned tube for transferring heat in a desired application, the heat is transferred under the influence' of a "temperature-difference" driving force. This "temperature difference" is distributed across the outside film resistance, the outside fouling resistance, the metal resistance, the inside fouling resistance, and the inside film resistance. The heat is transferred through the metal by conduction under the influence of the temperature drop across the metal. The performance of a finned tube can be greatly influenced by the distribution of the heat flux within the metal wall. In order to handle high heat transfer rates through a finned tube, it is essential that a large portion of the energy be conducted through the fin itself to the base of the fin and then be dissipated to the film at the inside of the tube. The relative thickness of the fin and root-wall metal is an important factor in the channeling of heat flux to the inner Surface of a tube from the surface of the fin. It is apparent that if for a given fin thickness the root-wall thickness is continually decreased, the heat-flux pattern in the root wall will approach a direction perpendicular to the axis of the tube. Consequently, as the wall thickness decreases, a smaller portion of the heat flux will channel through the fin, resulting in a decrease in the rate of heat transfer. Therefore, a small ratio of root-wall thickness to fin thickness could result in reduced efficiency of the extended surface for a particular set of conditions. Such a situation will be referred to as "crowding of the heat at the base of the fin." The critical ratio of root-wall to fin thickness is the value below which "crowding" becomes a significant factor in the performance of the tube. This critical value depends on the individual heat transfer coefficients and the thermal conductivity of the metal tube. This investigation was undertaken in order to determine whether the ratio of root-wall to fin thickness used ina standard one-half-inch fin height, all-aluminum tube having a 0.020-inch root wall is sufficiently above the critical value to prevent occurrence of "crowding of the heat flux at the base of the fin" in ordinary applications of this tube. 1

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN In heat transfer theory it can be shown that an adiabatic line or line of constant heat flux is in every point perpendicular to the isotherm passing through that point.1-3 In other words, once a system of isotherms is known, the heat-flux pattern can be obtained by drawing a system of orthogonal trajectories. Consequently, a fundamental part of this investigation was the determination of the temperature distribution within the root wall and fin. From this distribution a system of isotherms could be obtained. Once the system of isotherms had been determined, the heat-flow pattern could be obtained simply by drawing a system of lines perpendicular to the isothermal lines. This method results in a graphical presentation of the temperature distribution and heat-flux pattern throughout a cross section of the finned tube. A quantitative measure of the degree to which the root-wall thickness affects the heat transfer to the fin can be made from fin-efficiency determinations. In the paper on fin efficiency by Mr. Karl Gardner,8 assumptions were made in order to obtain solutions of the equations derived for predicting fin efficiency. Listed among the assumptions was the following: "The temperature of the base of the fin is uniform." With the exception of the above assumption, the conditions used in the analyses presented in this report are the same as those used by Mr. Gardner. Therefore, a comparison of the fin efficiencies as obtained by Mr. Gardner with those obtained in these analyses is a test of the above assumption. This comparison gives a direct indication of the degree to which the root-wall temperature distribution affects the efficiency of the fin. THEORETICAL CONSIDERATIONS The process of heat transfer from the inside fluid to the inside surface of the tube wall is described by the equation q = hi (Twi - Ti) Ai ~ (1) Similarly, the heat transfer from the external surface of the root wall and fin to the outside fluid is described by the equation q = h o (To - T) A^, (2) where q = rate of heat transferred, Btu/hr, A = inside surface area, ft, -' 2

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Ao = outside surface area, ft2, hi = inside film coefficient, Btu/hr/ft2/~F, Twi = temperature of the inside wall, ~F, Two = temperature of the outside wall, OF, Ti = mean temperature of the inside fluid, ho = outside film coefficient, and To = mean temperature of the outside fluid. For constant inside and outside coefficients and temperatures with a homogeneous metel wall, the conduction of heat through the metal wall involves a process of two-dimensional heat conduction. If the discussion is restricted to the steady-state condition, it can be said that the temperature at any point in the wall is a function of the coordinates of that point, i.e., T = T (x,y) (5) By establishing a heat balance around a differential element of the wall, the following differential equation can be obtained:1-3 62T e2T a t + P = o. (4) This is known x2s Laplaces equation for two-dimensional heat flow. The inteThis is known as Laplace's equation for two-dimensional heat flow. The intztegration of this equation would give the function T = T(x,y) and, therefore, the temperature distribution in the wall. However, the analytical determination of a particular solution for this equation involves the introduction of both initial and boundary conditions. The complexity: and uncertainty of boundary conditions makes it extremely difficult, in practice, to attempt a direct analytical solution of this equation. A practical method consists of approximating the solution of Laplaces equation by substituting the differentials by the technique of "finite differences."2I4 5 This substitution leads to a numerical analysis solution. The method is briefly summarized below. METHOD OF ANALYSIS Several approximate methods have been proposed for the solution of the Laplace equation.2-7 The underlying principle in most of the methods is the substitution of differentials by finite differences. In order to apply the method of finite differences, a cross section of the tube wall is subdivided into grids as shown in Figs. 1 and 2. Under 3

t — - 0.10. II" 4 0.125" o 24 m z C0 Z m m z m 7I n z m l, o23 Fig. 1l Coarse grid used in numerical solution of temperature distribution in the root of a finned tube. Drawing not to scale. z LL Uo22 0 T. o21 -Jl II 16 0.0025" 11 1 4 1 1 _, I 3f 2 *7 012.17 0.005 3 8.8 I.13.18' 0.005" W. 1 _ _ _ _ _ _ _ _ _ _ L _ _ _ _ _ _ _ _ _ I' z m 0 -n (Ij z 4 le 4 *9 *14 *19 0.005 " 5 It10! 5 20 0.0025" l 0.005"II 0. 005-ow I ~- 0.010 - 0.020" 0.020 ]

- ENGINEERING RESEARCH INSTITUTE Fig. 2. Fine grid used in numerical solu of temperature distribution in the root a finned tube. Drawing not to scale. UNIVERSITY OF MICHIGAN - 59 I IL 58 > 0 5o 57 1 -F cJ 0 i 56 1 6 11 16 21 26 31 36 41 54 55 * 0 0.0025" -J -J -t3 0 0 IL 0 * * * * * * ** * * 0.005" 2 7 12 17 22 27 32 37 42 52 53 3 8 13 18 23 28 33 38 43 50 51 * 0 0 0 0 0 0 ~ 0 0 0 0.005" 4 9 14 19 24 29 1 34 39 44 48 49 5" ~ * * * * * * * ~0.005" 0.0025" 5 10 15 20 25 30 35 40 45 46 47 0.005' EACH 5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - steady-state heat transfer conditions there will be no accumulation of thermal energy in any one grid. Therefore, the rate of heat transfer to the grid under consideration is equal to the rate of heat transfer from the grid. The grids are numbered in order and a temperature Ti is assigned to each one, where the subscript "i" is the identification number corresponding to that particular grid. A heat balance can be written around each grid. Since the temperatures of the outside and inside fluids are known, a set of "n" linear equations can be obtained, involving "n" unknown temperatures. Theoretically this system of equations could be solved simultaneously. However, if accurate results are desired, the grids should be very small. Consequently, the number of equations would become very large and the exact solution would become extremely tedious and time consuming. Professor Southwell5-7 introduced the idea of relaxation, which gave practicality to the solution of a large number of simultaneous equations. Essentially the method requires a good estimatdon of the temperatures in the various grids. The residuals in the grids are evaluated from heat-balance equations. A systematic procedure is followed for relaxing the various residuals. The final set of temperatures which reduces all the residuals very close to zero gives a satisfactory solution. For the solution of this particular problem, the section under consideration was divided into 22 blocks for the first approximation. The resulting system of 22 equations was solved by successive elimination. After the first approximate temperature distribution had been established for the 22-grid system, a new grid of 57 blocks was adopted. Taking as a basis the previous approximate temperatures, this new system of 57 equations was solved by relaxation and the solution was considered complete when all the residuals fell between + 0.05 and -0.05 (Btu/hr). The restriction of the residuals to fall within + 0.05 (Btu/hr) results in an average maximum error for the computed temperature of 0.006~F. Once the temperatures at the various points were known, a system of isotherms was drawn in by resorting to linear interpolation between any two temperatures whenever necessary. For the tube having a root-wall thickness of 0.01 inch, a similar analysis was carried out. The first coarse grid consisted of 14 blocks and the resulting system of equations was solved simultaneously. For the final solution a network of 32 grids was used. Since no useful purpose would result from reproducing all the above equations in this report, they are not reproduced here. The equations and their solutions are being maintained in project file for future reference. 6

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - DESCRIPTION OF SYSTEMS ANALYZED Longitudinal sections of the tubes were considered, as shown in Figs. 1 and 2. By consideration of symmetry it is permissible to restrict the analysis to a section between the axis of symmetry of the fin and the midplane between adjacent fins. The analysis was conducted for two different root-wall thicknesses. Table I summarizes the tube dimensions and heat transfer conditions assumed for the analyses. TABLE I DIMENSIONS AND HEAT TRANSFER CONDITIONS ASSUMED FOR ANALYSES Tube: A A B Analysis: I II III Heat Transfer Conditions h', Btu/hr/ft2/~F 10 50 10 k1, Btu/hr/ft2/OF 120 120 120 hi, Btu/hr/ft2/OF 1000 1000 1000 Inside fluid temperature, ~F 240 240 240 Outside fluid temperature, ~F 100 100 100 Dimensions Fin height, in. 0.50 0050 0o50 Fin thickness, in. 0.02 0.02 0.02 Fin spacing (center to center), in. 0.110 0.110 0.110 Root-wall thickness 0.02 0.02 0.01 The heat transfer conditions assumed for analyses I and III (Table I) correspond to a typical application of high-finned tubes (such as for condensing steam inside the tubes by air on the outside of the tube in forced convection). The 0.020-inch root wall was selected because a tube having approximately this root-wall thickness gave unusually low heat transfer performance for a monometallic tube. This tube had been tested in a steam-condensing apparatus in which the air coefficient was of the same order of magnitude as assumed in the first analysis (Table I). The second analysis was made in order 7 I

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN to determine if crowding of heat flux would occur under more severe heat transfer conditions than were assumed for the first analysis. The third analysis applies to a 0.010-inch root wall. This analysis was made to determine the effect of a thinner root wall on crowding of the heat flux at the base of the fin. Other assumptions used in the solution of the problem were: (a) The thermal conductivity and heat capacity of the metal is independent of temperature. (b) The convective heat transfer coefficients for the inside and outside of the tube remain constant over their entire surfaces. (c) No heat sources or sinks are present within the metal. A scale drawing of the fin and root profile considered for the tube with'a 0.02-inch root wall is shown by solid lines in Fig. 35 ANALYTICAL RESULTS The results of the three analyses are graphically presented in Figs. 4, 5, and 6. These figures show the lines of constant temperature (isotherms) and lines of constant heat flux (adiabats). In all these figures line a-b is the heat-flux line of greatest interest. This line indicates that the heat entering the inner root-wall surface to the right of point "b" channels into the fin. The heat entering the inner root-wall surface to the left of point "b" flows through the root wall only, The position of point "b" therefore indicates the relative portion of the heat that is transferred through the fin and through the outer root wall. Fin efficiencies were obtained by graphical integration, using the fin and outer root surface temperatures given in Table II and Table III. The fin efficiencies obtained are presented in Table IV. Also included in Table IV are the fin efficiencies obtained from the curves presented by Gardner.9 The air-side coefficients actually act on the surfaces with which they are in contact. The fin efficiency is a measure of the reduction in temperature drop across this film encountered along the fin due to the conduction of heat through the fin metal radially toward the root. This means that either a variable temperature difference must in some way be applied, or the coefficient reduced, on the area to obtain an "equivalent area." A widely used method is that of combining the fin efficiency with the outside area to give 8

-- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Fig. 3. Scale drawing of fin and root profile considered for the 0.020-inch wall tube. I I -I I I I I I I I I LL. I o I I I I 0I el I I I I I I L I I I I I -J I3: 0 0 Mr I I I I I Li A.A L —-[ _ _ 2 _ _ m _ 9

--- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - 10

I I Fig. 5. Resulting system of isotherms and adiabat "a - b" for tube A in analysis II (0.020-inch root wall, h' = 50). 21 Z m z CI) z m rri nn rm Ijr -I rmr nn rn z~ tA rri I H p P LL UIL I x -J -J 0 0 or LL. 0 6 I I k I 6 "I 13 N N I t~ 7\ 1\ \12 17 3 8 " 13 1 49 14\ 17 LiZZ_^ _ \ \o 15 \ \ — \o \ c z 70 "n -, I > z 1

H r -- Fig. 6. Resulting system of isotherms and adiabat "a - b" for tube B in analysis III (0.010-inch | root wall, ho = 10). z c.j -- L 7 o r UL. 30~~~~~~ 1 {;~~~~~0 I I 6 I1I 16 21 26 31 36 41 \54\ 55 55 / 54/' —-2 i7 12 \17 S 227X 32 37 42 5 35S3 /5/52 5 3 8 13 18 23 28 33 38 43 50 51 51 50 < ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.o ~ I= C I I 6 i, 16 21 26 31 36 41 54 55 55 54 3 8 13 18 23 28 33 38 43 50 51 51 50 -I In Z'I" In n rl I, n mi n I n n'fi 7'ff m~

-- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN -- TABLE II COMPUTED TEMPERATURE DISTRIBUTION IN THE FIN AND ROOT WALLS, USING GRID PRESENTED IN FIG. 1 Tube: A A B Pos ition Analysis: I II III 1 228.181 228.003 227.739 2 228.189 228.031 227.755 3 228.332 228.063 227.788 4 228.341 228.101 5 228.369 228.143 6 228.212 227.989 227.707 7 228.229 228.017 227.733 8 228.267 228.049 227.767 9 22833 228.090 --- 10 228.340 228.128 -- 11 228.140 227.887 227.607 12 228.152 227.914 227.623 13 228.176 227.967 227.657 14 228.210 228.007 15 228.247 228.056 16 227.734 227.072 227.095 17 227.831 2273.75 227.184 18 227.907 227.577 227.243 19 227.967 227.754 20 228.015 227.804 21 222.352 216.656 224.720 22 219.302 202.153 222.456 23 216.900 192.975 199.152 24 21.3624 187.825 187.745 13

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE III COMPUTED TEMPERATURE DISTRIBUTION IN THE FIN AND ROOT WALLS, USING GRID PRESENTED IN FIG. 2 1 2 4 5 6 7 8 9 10 1.1: 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 50 228.181 228.189 228.552 2283.41 228.569 228.178 228.186 228.303 228.317 228.555 228.173 228.185 228.270 228 292 228.330 228.165 228.175 228.235 228.262 228.303 228.149 228.160 228.195 228.230 228.268 228.125 228.157 228.150 228.195 228.227 227.759 227.755 227.788 227.722 227 748 227.778 227.696 227.718 227.758 227.667 227.682 227. 747 227.627 227.642 227.685 227.585 227.600 227.634 31 52 55 54 55 36 57 58 39 40 41 42 45 44 45 46 47 48 49 50 51 52 55 54 55 56 57 58 59 228.080 228.095 228.102 228.154 228.181 228.011 228.050 228.042 228.096 228.125 227.885 227.925 227.965 228.016 228.050 227.865 227.585 227.730 227.250 227.650 227.160 227. 520 227.920 22'7.265 227.405 223.425 220.005 216.95 215.620 227,525 227,543 227.566 227,455 227 445 227o471 227.275 227.297 227.336 227.00 226.53 226.79 226.07 226.60 225.80 216.548 202 113 192.545 187.765

r- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - TABLE IV FIN EFFICIENCIES OBTAINED FOR ASSUMED CONDITIONS Tube: A A B Analysis: I II III Fin efficiency from graphical integration of temperature distribution Fin efficiency from Gardner curve, Fig. 3A, Reference 9 Grid I Grid II 92.1% 92.0o 85.0% 85.2%... 85.0 o 89 % 96 % 96 % an equivalent area: Aeq = Ar + EfAf where Aeq Ar Af Af = equivalent area, = root area, = fin efficiency, and = fin area. The relationship between the two areas Ao and Aeq is given by hoAo = h o Aeq where hto = actual outside film coefficient and ho = coefficient compensated for fin efficiency. This equation can be used for computing ho from the areas and fin efficiency. This can be done, using either the Gardner fin efficiency or the analytical efficiencies given in Table IV. For comparison purposes the ho coefficients, using both efficiencies, are tabulated in Table V. A typical calculation is given in Appendix B. Table V shows that the effect of the root-wall thickness is relatively small for the 0.02-inch tube but becomes a more significant factor for the tube with the 0.01-inch root wall (4% aS compared to 11%). 15

-- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN -- TABLE V INFLUENCE OF ROOT-WALL TEMPERATURE DISTRIBUTION ON EFFECTIVE HEAT TRANSFER COEFFICIENT Analysis h0 Ef' (cEf)Gardner h0 (ho)Gardner Percent -l % (oe)Gardner h Difference I 10 92.0 96.0 9.25 9.65 4,32 II 50 85.0 89 43.4 44.9 3.46 III 10 85.0 96 8.68 9.65 11.2 DISCUSSION OF RESULTS The root-wall thicknesses employed in these investigations were thinner than normally encountered. in industrial finned tubes. The heat transfer conditions assumed here were of the same order of magnitude as would be encountered in normal operation using forced air outside high-finned tubes (with condensing steam or water flowing inside the tubes). Since, as shown in Table IV, the effect of increasing the root-wall thickness is to decrease its influence on fin efficiency for normal applications of industrial tubing, the effect of the root-wall temperature distribution would be somewhat less than indicated in this investigation. The gation can be ing all other effect of some of the variables on the results of this investideduced qualitatively. The influence of these variables, keepconditions constant, would be expected to be as follows: 1. Number of Fins per Inch (Fin Spacing).-The effect of reducing the fin spacing (increasing the number of fins per inch) would be to increase the influence of the root-wall temperature distribution, resulting in lower fin efficiencies. 2. Fin Height.-An increase in fin height would increase the influence of the root-wall temperature distribution on fin efficiency, while a decrease in fin height would decrease this influence. 3. Root-Wall Thickness.-The effect of a change of root-wall thickness is shown in Table IV. A comparison of analyses I and III shows that the 16

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN -- thicker the root wall, the closer the fin efficiency to the predicted value given by Gardner's curves. CONCLUSIONS The influence of the temperature distribution in the root wall of a high-finned tube on fin efficiency is quite small for a 0.02-inch root wall under the conditions of analysis b"ut becomes more significant for thinner root walls. Since the root-wall thickness of the present commercially manufactured tubing is greater than 0.02 inch, the effect of the root wall on the fin efficiency and the corresponding effective heat transfer coefficient can be considered to be small for air-cooling applications with high-finned tubing. 17

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - APPENDIX A Sample Calculations Following is a sample of the calculations involved heat balance around an elementary block. Consider block 6 of Fig. 2. Heat flows to block 6 7, and 11 by conduction, while heat is lost by convection to Therefore, calling q the net rate of heat flow to the block, in writing a from blocks I, the outside. q = kAy (T - T6) + kAx (T7 - T6) + kAy (T1 - T6) + hAx (100 Te) Ax Ay Ax If steady state is considered, q = 0 Substituting values from Table I and Fig. 2 120x0.002 025 120x0.005 120x0.0025 0 = 0.00o (T1-T) + 0.00' (T7-T6) + o0oo5 (TI1-T6) + 10x0.005 (100-T6) 12 or, after simplification, 4.oooo69 T6 - 0.0069 - T, - T, - 2T7 = 0 18

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX B Calculation of the Effective Heat Transfer Coefficient for Analysis I The effective outside heat transfer coefficient is defined as ho = hto eq, Ao where h'o = outside film coefficient, Aeq = equivalent outside area, = Ar + efAf Ao = total outside area, Ar = outside root area, Af = outside fin area, and ef = fin efficiency. The root area for a tube of the following dimensions is 0.28 ft2/ft. Tube Dimensions Outside diameter Root diameter Fin spacing Fin thickness 2.00 inches 1.04 inches 0.11 inch 0. 02 inch The total outside area is 5.59 ft2/fti therefore, Af = Ao - Ar = 3.59 - 0.28 = 3.31 ft2/ft. The fin efficiency obtained in analysis I was 92%; therefore,,(h) (0 100.92 x 3 31 (ho) =10.. 3.59 + 0.28 - --- = 9.25. The fin efficiency from Gardner's curves was 96.0%; therefore, 19

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN (ho Gardner = 10 (.96 x 3.31 + 0.28 3.59 / 9.65 The percent discrepancy is = 965 5 x 100l = 4.32% 9.25 J 20

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - REFERENCES 1. Jakob, M. Heat Transfer, Vol. I. New York: John Wiley and Sons, Inc., 1949. 2. Dusinberre, G. M. Numerical Analysis of Heat Flow. New York: McGraw-Hill Book Co., 1949. 3. Schneider, P. J. Conduction Heat Transfer. Cambridge, Mass.: AddisonWesley Co., 1955. 4. Milne, W. E. Numerical Solution of Differential Equations. New York: John Wiley and Sons, Inc., 1953. 5. Douglas, J., and Peaceman, D. W. tNumerical Solution of Two-Dimensional Heat-Flow Problems," A.I.Ch.E. Journal, Vol. I., pp. 505-512 (1955). 6. Southwell, R. V. Relaxation Methods in Engineering Science. New York: Oxford University Press, 1940. 7. Emmons, H. W. "The Numerical Solution of Heat-Conduction Problems," Trans. A.S.M.E., Vol. 65, pp. 607-615 (1943). 8. Gardner, K. A. "Efficiency of Extended Surface," Trans. A.S.M.E., Vol.67, pp. 621-632 (1945). 9. Ibid., p. 624, Fig. 3A. 21