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ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR CORRELATION OF HEAT TRANSFER AND PRESSURE DROP FOR AIR FLOWING ACROSS BANKS OF FINNED TUBES By DONALD L. KATZ Professor of Chemical Engineering EDWIN H. YOUNG Assistant Professor of Chemical Engineering RICHARD B. WILLIAMS GAREN BALEKJIAN ROBERT P. WILLIAMSON Graduate Students Project M592 WOLVERINE TUBE DIVISION CALUMET AND HE3CLA CONSOLIDATED COPPER COMPANY August, 1953

ABSTRACT The lack of a generalized correlation for the outside convection coefficient of gases in crossflow through banks of finned tubes necessitated a study of the available data. The purpose of this work was to provide a procedure for the design of finned-tube banks. Such units have been used either to cool process water or to heat or cool gases. The data for air are correlated by using the tube characteristics and arrangement of the tubes in the units to relate the outside heat-transfer coefficient and the pressure drop with the rate of flow of air. The heat-transfer and pressure-drop correlations for air are generalized for any gas by using the physical properties of the fluid. Two sample calculations are included to illustrate the use of the proposed correlations. iii

TABLE OF CONTENTS Page ABSTRACT iii LIST OF FIGURES v LIST OF TABLES v GENERAL DISCUSSION1 DATA AVAILABLE 2 HEAT-TRANSFER. CORRELATION FOR THE OUTSIDE FILM COEFFICIENT 3 CORRELATION OF HEAT TRANSFER FROM AIR TO BANKS OF FINNED TUBES 5 GENERALIZED HEAT-TRANSFER CORRELATION7 CORRELATION OF PRESSURE-DROP DATA7 BOND RESISTANCE IN BIMETAL FINNED TUBES 9 THE EFFECT OF TUBE SPACING9 REFERENCES 11 APPENDIX 13 Other Correlations Considered 13 Sample Design Calculations 14 Cooling Water with Air15 Cooling Hydrogen with Water 20 iv

LIST OF FIGURES Fig. 1 Nomenclature for Dimensions of Finned Tube and Tube Spacing Figs. 2, 35 4, 5, and 6. Outside Convection Coefficient as a Function of the Maximum Air Velocity Fig. 7 Range of Heat-Transfer Data Fig. 8 Variation of Pressure Drop with the Number of Fins per Inch Figs. 9, 10, and 11. Pressure Drop as a Function of the Maximum Air Velocity Fig. 12 Heat-Transfer Correlation for Air Fig. 13 Comparison of the Heat-Transfer Correlation with Nontriangular and other Tube Banks Fig. 14 Generalized Heat-Transfer Correlation Fig. 15 Pressure-Drop Correlation for Air Fig. 16 Generalized Pressure-Drop Correlation Fig. 17 Heat-Transfer Correlation of T.E. Schmidt Applied to Air Fig. 18 Heat-Transfer Correlation for Air as a Function of S, N, and Dr Fig. 19 Empirical Pressure-Drop Correlation for Air Fig. 20 Generalized Pressure-Drop Correlation LIST OF TABLES Table I Dimensions of Tubes and Units for Available Data Table II Dimensions of Tubes and Units Table III Variables in Heat Transfer and Pressure Drop v

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CORRELATION OF HEAT TRANSFER AND PRESSURE DROP FOR AIR FLOWING ACROSS BANKS OF FINNED TUBES Finned tubes are often used for transferring heat from a fluid in-. side the tubes to air flowing across a bank of the finned tubes. The purpose of these heat exchangers is either to cool the fluids inside or to heat the air outside 1the tubes. The size of such units varies from bus heaters and automobile radiators up to large industrial cooling units. The latter, which might be 24 feet square, are used to replace cooling towers in process plants. At the time this work was started, no general correlation of data for any group of tubes was available in the literature. For purposes of commercial design, individual companies often obtain experimental data on particular tubes and tube arrangements for specific installations. The recent paper by Schmidt7 presents an equation of the outside heat-transfer coefficient for banks of finned tubes in terms of the coefficient for plain tubes, the fin height, and the number of fins per inch. It is the purpose of this report to correlate available heat-transfer and pressure-drop data for air flowing across banks of finned tubes with helical fins, The data are on tubes of the type manufactured by the Wolverine Tube Division of the Calumet and Hecla Consolidated Copper Company. These tubes are constructed of copper, extruded by the Trufin process, or made of bimetallic construction with a liner metal which is different from that of the fin metal. Experimental data on 30 different units for air flowing outside the tubes have been provided by 11 organizations. In some cases complete test reports were provided, while in other cases only performance curves were made available. Banks of plain tubes have been studied rather thoroughly by Grimison. Data on specific banks of finned tubes have been reported by Katz, Beatty, and Foust4 and recently by Kays and London.'6 These papers do not provide a correlation to allow the prediction of heat-transfer coefficients or pressure drop for finned tubes of a given dimension or for the effect of tube spacing and bundle arrangement in tube banks. Jameson3 has reported heat-transfer and pressure-drop data obtained from extended surfaces made of fins soldered to ---— 1 - 1-I

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN bare tubeso Gunter and Shaw2 have proposed a generalized pressure-drop correlation based on experimental data obtained from units using various types of extended surfaces, including helical fins. This report proposes correlations for evaluating heat-transfer coefficients and pressure drops between air on the outside surface of finned tubes and water or steam on the inside of the tubes. Within the limit of the available dimensions of the tubes and arrangements in the 30 tests reported the various variables of fin spacing, fin height, and tube arrangement are investigated. In order that these data might be applied to fluids other than air in transferring heat to or from the outside of finned tubes, the correlation has been placed on a dimensionless basis. Such a correlation requires more careful data and includes the properties of the fluid at the mean bulk temperature of the fluid passing across the tube banks. In addition to the correlations of heat-transfer coefficients and pressure drops, sample problems illustrating the use of the correlations for the design of a water cooler and of a hydrogen cooler are presented Data Available: Data are available on 30 different units which employed a total of 27 different types of finned tubes. Most of the tube-bank units consisted of tubes on equilateral, triangular pitch. Some tube-banks had nonequilateral triangular tube arrangements and 3 units had square pitch layouts. The frontal face area varied from 0.605 sq ft to 480 sq ft. The 30 units referred to above represent data on banks of finned tubes of monometallic (copper) or bimetallic construction. The dimensions of the various tubes on which the data are available and their arrangement in the tube banks are summarized in Tables I and II. Figure 1 illustrates the application and indicates the nature of the dimensions used. The available heat-transfer data are plotted as outside coefficients of heat transfer between the air and the outside surface of the tube as a function of the velocity of the air at the minimum cross-sectional area in Figures 2 through 6. This velocity is computed on the basis of air at 70~F and 50 per cent relative humidity. Wherever possible, actual data are shown. However, in some cases only performance curves were available (necessitating calculation of the individual heat-transfer coefficients). The range of the experimental data on both the heat-transfer coefficients and the velocity of the air are indicated on Figure 7. This plot summarizes the smooth curves of Figures 2 through 6. The lines in Figure 7 terminate at the reported minimum and maximum air velocities. The reported pressure-drop data are plottec versus the maximum air velocity on Figures 9^ 10, and l1. ~~~___~~_____________ 2

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The effect of the number of vertical rows of tubes normal to the direction of flow on heat transfer and pressure drop should be mentioned. For plain tubes the added turbulence caused by the air passing the initial rows of tubes causes an increase in heat transfer in the first two or three rows. Schmidt7 studied the effect of the number of rows in finned-tube air heaters in which the fan was located downstream from the tubes HEe reported heat-transfer coefficients as a function of the mass velocity raised to some power and he found that this power varied slightly with the number of rows. The constant of proportionality also varied with the number of rows of finned tubes. He states that for units having blowers upstream from the tubes the effect of the number of rows is less than when the fan is downstream. Jameson3 studied units containing wrapped-on finned tubes with a blower located upstream so that there was a considerable calming section prior to the finnedtube unit. This is nearly equivalent to a downstream fan. Jameson also found some slight effect of the number of rows on the heat-transfer coefficient. The data studied in the present report show no effect of the number of rows of tubes in the unit. For a bank of finned tubes with an upstream blower there is little increase in heat transfer from the first to the second or later rows. A considerable portion of the heat-transfer data were reported as overall coefficients of heat transfer between air on the outside of the tube and either water or steam inside the tubes. The outside film coefficients were evaluated from these overall coefficients. The inside film resistance and the metal resistance were subtracted from the overall resistance to heattransfer in order to compute the outside film resistance, which is the reciprocal of the outside film coefficient. In the case of condensing steam the heat-transfer coefficient on the inside of the tube was estimated. A coefficient such as 1500 Btu/hr-~F-sq ft was assumed. For some of the data, a series of the reported fluid velocities on the inside of the tube were plotted against the overall resistance and extrapolated to an infinite velocity. This results in an overall resistance having no film resistance on the inside of the tubes (Wilson plot). It should be appreciated that these computations result in outside film coefficients which are higher numerically than the overall coefficient by 5 to 20 per cent. Thus, an overall coefficient of 15 Btu/hr-OF-sq ft might give a computed outside coefficient of 17 Btu/hr-~F-sq ft'. Therefore, it follows that even though the conversion from overall coefficient to an outside film coefficient may include these approximations, the film coefficient error due to this is relatively small.. Heat-Transfer Correlation For The Outside Film Coefficient: A correlation of the heat-transfer data involves description of the bank of finned tubes, properties of the flowing air, and the conditions of flow l_______________,__,___,__,_____ 3 ______....._..._-__.....___.__.,

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN such as temperature, pressure, and velocity. For convection between solid surfaces and a fluid, studies have shown that Equation 1 is normally suitable for correlating the data. Nu = cRein Pr, (1) where Nu = Nusselt number, Re = Reynolds number; Pr = Prandtl number; and c, m, and o = constants. A definition of these dimensionless groups, used in Equation 1, is given in Equation 2: hD = fG mcAf (2) k k where h = film coefficient; D = diameter for single tubes; k = thermal conductivity of fluid; Cp = specific heat of fluid; = viscosity of fluid; and G = mass rate of fluid flow. The heat-transfer coefficient between the air and the outside surface of the tube is included in the Nusselt number as the convection coefficiait The condition of flow of the fluid across the tube bank is indicated by the Reynolds number and includes an equivalent diameter for the tube bank, mass velocity of the air, and viscosity of the air. The thermal properties of the fluid passing through the tube bank are included in the Prandtl number as viscosity, specific heat, and thermal conductivity. For tube banks containing plain tubes, the significant dimension which is used as D is either the diameter of the tube or the spacing between the tubes. Actually, it has become necessary to use different correlations depending on the spacing between the tubes. For finned tubes the problem is even more complex in that it is desirable to use a single dimension to represent the mechanical structure of the tubes end their arrangement in the tube bank. It must be admitted that any single dimension which is used is likely to be relatively empirical. To prove the validity of such a dimension, the only test _____________________l. ______________________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN which can be applied is that the use of this dimension in the general equation will bring test data on different tube banks into a single correlating relationship. It should be noted that different tube banks give different coefficients of heat-transfer at a given velocity, as indicated in Figures 2-6. Since there are numerous variables in a tube bank such as number of fins per unit length, the fin diameter, the root diameter, and the pitch of the tube, the grouping of the variables to find an equivalent diameter must be rather arbitrary. After considerable study, the selected equivalent diameter is given by Equation 5, in which the equivalent diameter is for a tube bank composed of a specific kind of finned tube and is a function of the number of fins per inch, the root diameter, the fin diameter, and the spacing between t.e tubes based on an equilateral triangular arrangement. In case the longitudinal and transverse spacing of the tubes were not the same, an average of the two spacings was.,used. De (3) (12) Kr where N = number of fins per inch; Do = fin diameter, in.; Dr = root diameter, in.; and S = tube pitch for an equilateral triangular arrangement, in. Correlation Of Heat Transfer From Air To Banks Of Finned Tubes: In all the reported experimental data the air was being heated from approximately room temperature up to a value around 3000F. Under these conditions the Prandtl number of air, i.e,, the viscosity of the air,' the thermal conductivity of the air, and the heat capacity of the air do not vary appreciably, so that Equation 2 can be simplified to Equation 4 by introducing a constant average Prandtl number. It will be noted that in Figures 2-6, the'. maximum linear velocity was used in preference to the mass velocity; in other words, Vf' was used instead of Gm -word max max. ho = f (D, Gm), (4) The exponents "r" and "m" as obtained from the correlation are -0.5 and 0.56 respectively. This results in Equation 5: ho = f (~e-0.5, G,6) (5) ---------------------- 5 ________________________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A further simplification of Equation 5 is possible by replacing the mass velocity by the maximum linear velocity based on air at a standard pressure, temperature, and humidity. Under standard conditions of 70~F, 1-atm. pressure, and 50% relative humidity, air has a density of 0.074 lb/cu ft. It should be noted that the maximum velocity is computed on the basis of the minimum free-flow area in the transverse direction. The data for individual units are plotted in Figures 2 through 6 as ho versus Vmax. In Figure 12, the data for all units, except units 11 through 18 and unit 22, are plotted as ho(S/Dr)(NDo)O-5 versus Vmax. The single line drawn through the points is represented by Equation 6. It is apparent that a group of the original data does not fit the curve. No adequate explanation has been found for the discrepancies between the results of the various groups of tube banks. It is believed tha ththe points which lie below the average curve indicate the presence of some resistance, either on the inside of the tube or in the compound tube wall, that is not properly accounted for. The curve on Figure 12 is based on tube banks having triangular staggered pitches. 0.56 ~ho 1.9 r).(6 ) (SN Do) 0.5) max (6) Equation 6 represents a correlation of the bulk of the heat-transfer data for air flowing across banks of finned tubes in units having triangular tube arrangements. The outside coefficient is a function of the tube dimensions, the tube spacing, and the maximum velocity through the minimum free transverse area. When it is necessary to determine the outside coefficient for a unit having similar tubes and spacing to the units represented on Figures 2 through 6 it is advisable to use the curves on these figures. However, when it is desired to find the effect on the heat-transfer coefficient of changing the tube dimensions or the tube spacing, Equation 6 is recommended. Equation 6 is also recommended when determining coefficients for tubes and spacings not covered by the original test data which are included in this report. An equally good correlation of the data was obtained by plotting the group ho (S/Dr) (1NDo)0-5 versus the face velocity. Table II has been included to facilitate the conversion of maximum velocity to face velocity. The seccnd column of Table II gives the ratio of the free-flow area to the face area for each unit studied. The product of this ratio and Vmax gives the corresponding face velocity. The third.column of Table II gives the ratio of outside to inside surface area for each unit and one can then convert the heat-transfer coefficient based on outside surface to one based on inside surface if it is desired. The meager data available for tube arrangements other than triangular staggered pitch are presented in Figure 13, together with triangular arrangements not indicated in Figure 12. Equation 6 is also plotted on this figure. 6 ______________________

- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Generalized Heat-Transfer Correlation: The data on air have been computed in the form of a generalized dimensionless correlation of heat transfer, employing a modification of Equation 2. Most of the reported data were in the form of inlet and outlet bulk air temperatures, so that the properties of the air could be employed in the correlation. In the case of such available information, it was possible to compute the Prandtl number, the Nusselt number, and the Reynolds number using the equivalent diameter given in Equation 3. Accordingly, these calculated groups have resulted in the correlation indicated on Figure 14. It should be noted that those points which did not fit the curve on Figure 12, likewise, fall in a similar position on Figure 14. The data were not sufficient to check the validity of the 1/3 power assumed for'the Prandtl number. A viscosity ratio was not used in the correlation because the fluid was gaseous and the effect of temperature on the viscosity was relatively small. Figure 14 may now be used for predicting heat transfer coefficients for any fluid for which the properties are known at the mean bulk temperature in the tube bank. Correlation of Pressure-Drop Data: As in the case of heat-transfer, the pressure-drop data are correlated in a simplified form in which the properties of the air are inherent in the correlation and in a generalized form in which the properties of the fluid must be known to obtain the pressure drop. The individual pressure-drop data as plotted in Figures 9, 10, and 11 show the variation of the pressure-drop in inches of water per row with the maximum air velocity. Figure 8 indicates that pressure-drop varies as the 1/2 power of the number of fins per inch for constant maximum air velocity and physical characteristics of the tube bank. From theoretical considerations, the pressure drop due to fluid flow past solid surfaces is given by Equation 7: AP = f(Re) 2L, (7) p 2gc D where AP = pressure drop; P = fluid density; V = fluid velocity; L = length of flow path; ----------------------- 7 -----------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN D = diameter for a single tube; f(Re) = friction factor; and gc = gravitational constant. For plain-tube banks, D is usually taken as the spacing between the tubes. The function of the Reynolds number is usually referred to as the "friction factor" and varies inversely as the 0.2 power of the Reynolds number, in which D is the length dimension used in Equation 7. In the following correlation the equivalent diameter of tne finned tube bank as defined by Equation 3 is substituted for D in Equation 7. Within the range of temperatures studied, the density and viscosity of air are assumed to be constant, whereby Equation 8 is obtained as the simplified form of Equation 70 IPF Dt vu AP FDe Vmax. (8) De where AP = pressure drop, inches water/row; and F = constant including the physical properties of the fluid. This equation is the basis for correlating the pressure-drop data in terms of APDeas a function of the maximum air velocity as indicated in Figure 15. A parameter, including the root diameter of the tube, the diameter over the fins, and the number of fins per inch as K = NDo/Dr 2 is also used to complete the pressure-drop correlation for air. To obtain a generalized pressure-drop correlation for gases, Equation 7 must be modified to include the properties of the fluid in addition to the tube-bank characteristics. Thus Equation 7 may be written as follows: AP e = f(Re) (9) max Using the properties of air at the average temperature in the tube bank available from 24 units, the "friction factor" function of the Reynolds number (f) was calculated and plotted against the Reynolds number in Figure 16. This generalized correlation includes also the parameter K, defined previously. Figure 16 indicates a break in the curves at a Reynolds number of 300,000,,....... 8

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN possibly due to the transition from laminar to turbulent flow. Therefore, for a given set of conditions in any finned-tube bank, the pressure-drop can be evaluated from this figure and the properties of the flowing fluid. Bond Resistance In Bimetal Finned Tubes: In the construction of Bimetal finned tubes the liner and the root metal are pressed against one another in the fabrication process. This tends to give a tight metal-to-metal bond. From heat-transfer tests on banks of finned tubes, it has been observed that the tightness of this Bimetal bond varies during the manufacturing process. The effect of the bond resistance in Bimetal finned tubing is thought to vary up to about 10 percent of the total resistance to heat transfer for low heat fluxes such as for air. This resistance is inherent in the data taken with copper-aluminum Bimetal finned tubing and was not considered in the correlation of data for this report. Therefore, the heat-transfer coefficients reported for Bimetal tubing are already adjusted for the bond resistance which is inherent in the tubing. Bond resistance is a factor which must be considered in the design of heating units made of Bimetal tubing and in some instances it may be fairly large. Work is in progress on this problem and it is considered that at hii heat fluxes, bond resistance should not be neglected. However, at low heat fluxes such as are encountered in the heating of air, no correction for bond resistance is necessary with the correlations in this report. The Effect Of Tube Spacing: Equation 6 indicates that the heat-transfer coefficient for air is inversely proportional to the tube spacing if all other variables remain constant. This is shown also by the test data for units 12 and 13. These units are identical except that the tube spacings are 0.875 and 1.0 inch respectively Figure 3 shows that unit 13 has a coefficient about 12 percent below that of unit 12. If the test data are plotted as ho(S/Dr)(NDo)0O5 versus Vma the points for these two units may be adequately represented by a single curve. The quantitative effect of tube spacing may be shown by the following calculations for a Bimetal finned tube with Do = 2.00 inches Ao = 3.61 sq ft/ft Dr = 1.070 inches Ao/Ai = 14.8 N = 9 fins/inch K = 17.75 L. _______________- 9 -----------------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN For S = 2.00 inches (fins touching), triangular arrangement De = 5.26 ft Free-flow area/sq ft face area = 0.385 sq ft Outside surface/sq ft face area = 21.6 sq ft S/Dr (mNDo)~'5 7.94 For a face velocity of 500 ft/min, Vma = 500/0.385 = 1300 ft/min From Figure 12 ho (S/Dr) (IDo)0'5 = 105.6 Thus ho = 105.6/7.94 = 153, Btu/hr sq ft ~F Heat transfer/sq ft face area = (13.3) (21.6) = 288 Bt hr) (~F) (sq ft face) per row of tubes, From Figure 15 AP = 0.16 in water/row For S = 2.25 inches, triangular spacing De = 6.69 ft Free-flow area/sq ft face area = 0.452 sq ft Outside surface/sq ft face area = 19.2 sq ft S/Dr (MND)0'5 = 8.95 For a face velocity of 500 ft/min, Va = 500/0.452 = 1106 ft/min From Figure 12, ho (S/Dr) (1Do)0-5 = 96.5 Thus ho = 96.5/8.93 = 10.8 Btu/hr ~F sq ft Heat transfer/sq ft face area = (10.8) (19.2)= 207 Btu/hr "F sq ft face area per row of tubes. From Figure 15, AP = 0.10 in water/row For each row of tubes, increasing-the tube spacing from 2.00 to 2.25 inches decreases the rate of heat transfer by 28 percent and decreases the power requirements by 37 percent. The tube bank with the larger tube spacing will require more rows of tubes to transfer the same amount of heat as the bank with the smaller tube spacing. From the standpoint of heat transfer alone, the tubes should be as closely spaced as possible. _.........___________________________ 10 _____ _.:...'-.:,..,,,-, —:-'- -

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN REFM;ENCES 1. Grimison, E.D., "Correlation and. Utilization of New Data on Flow Resistance and Heat Transfer for Cross Flow of Gases over Tube Banks", Trans. A.S.M.E.,, 583-94 (1937). 2. Gunter, A.Y., and Shaw, W.A., "A General Correlation of Friction Factors for Various Types of Surface in Cross Flow", Trans. A.S..ME., 6_D, 643-60 (1945). 3. Jameson, S.L., "Tube Spacing in Finned-Tube Banks", Trans. A.S.M.E., 63, 633-42 (1945). 4. Katz, D.L., Beatty, K.O., Jr., and Foust, A.S., "Heat Transfer through Tubes with Integral Spiral Fins", Trans. A.S.M.E., 6Z, 665-74 (1945). 5. Kays, W.M., and London, A.,., "Heat Transfer and Flow Friction Characteristics of Some Compact Heat.Exchanger Surfaces, Part I-Test System and Procedure, Part II-Design Data for Thirteen Surfaces", Trans. A.S.M.E., 72, 1075 and 1087 (1950)...__________________________.1 _________________1_

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 6. London, A.L., Kays, W.M., and Johnson, D.W., "Heat Transfer and Flow Friction Characteristics of some Compact Heat Exchanger Surfaces, Part III-Design Data for Five Surfaces", Trans. A.S.M.E., 74, 1167-78 (1952). 7. Schmidt, T.E. l "Heat Transmission and Pressure Drop in Baniks of Finned Tubes and in Laminated Coolers", Institute of Mechanical Engineering and A.S.M.E., Proc. of the General Discussion on Heat Transfer, Section II, 186, London, (1951),

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX OTHER CORRELATIONS CONSIDERED Heat-Transfer Correlation: Figure 17 shows the grouping of heat-transfer data for air on the basis of the recently proposed Schmidt correlation7. The outside coefficient for banks of finned tubes is related to the tube dimensions and the corresponding outside coefficient for a bank of plain tubes by the following equation: hf = 1-0.18 [( r 063 (10) hp [ 2 where hf = outside convection coefficient for finnedtube bank; hp = outside convection coefficient for plaintube bank; Do = diameter over the fins, inch; Dr = root diameter, inch; and N = fins/inch. A similar distribution of data for air is obtained by the simple empirical correlation shown in Figure 18. The diameter over the fins (Do) is eliminated because of the constant ratio of Do/Dr that was observed within the limits of the available data. The equation fitting the data best on Figure 18 is: | ho = --- A V.0.56. S \ / max (11) These correlations may be useful in.some applications with air. 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - Pressure-Drop Correlation Empirical correlations for the pressure-drop data are shown in Figures 19 and 20 using a different set of dimensions to describe the finned tubes. The correlation for air is satisfactory and could be used advantageously as an alternate to Figure 15. The grouping of data in Figure 20 is not favorable for the generalized pressure-drop correlation, but is shown to indicate how the data spread when these variables are used. SAMPLE DESIGN CALCULATIONS Two sample calculations are given to illustrate the use of the recommended heat-transfer and pressure-drop correlations for air as well as for other gases for the design of finned tube heaters and coolers. The first example is of a heat exchanger for cooling water with air. The second example illustrates similar calculations for a hydrogen cooler using the generalized correlations. Example I. It is required to cool 400,000 lbs/hr of water from 125~F to 115~F with air in cross flow being heated from 70~F to 1100F on the outside of a bank of finned tubes. "~ -r — ~~v\>.l XWATER I N ATER IN H --- WIDTH - WATER COOLER L1______________________ lI.................-,-.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Sample Design Calculations Cooling water with air: T1 = 70~F T2 = 110~F t2 = ll5~F t1 = 125~F Heat Load = 4,000,000 Btu/hr Heat Exchanger — 2 tube passes - 1 shell pass. Tube Specifications — Wolverine Trufin Copper tube Catalog No. 0901 Do = 1.030 in. Di = 0.384 in. or 0.032 ft. Tube wall thickness = 0.038 in. y = mean fin thickness = 0.019 in. N = 9 fins/in. AO/Ai = 10.8 Dr = 0.384 + (2) (0.038) = 0.460 in. A = (3.4 (o.84) = 0.1005 sq. ft/ft i = ~(12) Ao = (0.1005)(10.8) = 1.086 sq. ft/ft 1st Approximation -- Assume Uo = 10: q = UoAATLM = 45 - 15 27.F At, total heat-transfer area = (4 10 = 14,650 sq. ft. (10)(27.5) Lt, total tube length = 1 0 = 13,500 ft 1.086 Assume tube size and arrangement: Length of each tube = 20 ft/tube, and 34 tubes/row n, number of rows = 13-500 19.85 or 20 staggered rows (20)(34) on equilateral triangular pitch. Calculation of V air. (minimum channel-area velocity) ~m ----~- max C air = 0.25 Btu/lb~F at avg air condition.................... 15

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN ATir = T2 - T = 110 - 70 = 400F o, 4 ooo oo? Wir = Q =. 400,0000 lbs/hr air CpAT (0.25)(40) Specific volume of Std. air (50% R.E. and 70~F) = 13.5 l volume = (400,000)(1535) = 90,000 cu ft/min (60) With a clearance of 0.125 in. between fin edges, (equilateral triangular pitch of 1.155 in.): Width of Unit normal to flow (1.030 + 0.125)(34) 3.27 ft (12) Unit height = 20 ft Total face area = (20)(3.27) = 65.5 sq ft Projected fin area per tube = af1ins = (fin thickness)(Do - Dr)(N)(tube length) (12) 144 3i~ -= (o.019)(1.050 - 0.460)(9)(20)(12) B~~~~~~~~! ~(144) = 0.163 sq ft/tube Projected tube area excluding fins = aroot (20)(0460) = 0.767 sq ft/tube (12) Total projected area per tube = at = 0.163 + 0.767 = 0.930 sq ft/tube Total face area of tube metal (projected area per row) atF = (0.930)(54) = 31.6 sq ft. aFF, free face area = total area - obstructing area = 65.5 - 31.6 = 33.9 sq ft V0,000 = 2650 ft/min 33.9 This linear air velocity is excessively high. To reduce the air velocity to a reasonable value of about 1/2 of this velocity, revision of the assumed tube bundle arrangement is necessary S = 0.125 + 1.050 = 1.155 in..____ ________________________ 16 ____________________ _.. i-,...

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 90 tubes/row, 15-ft tubes Number of rows, n = (13,500) = 10 90 x 15 Width of unit = (90)(1.155) = 8.66 ft (12) Height of unit = 15 ft Total face area = (15 x 8.66) = 130 sq ft FF = 130 - (O.93)(15)(90) = 67.3 sq ft Vm 90,000 1340 ft/min = 1.34 x 103 ft/min Vmax' 67-3 From Figure 12, (ho S/Dr) (NDo)05 = 108 = ho De'05 De~.5 = (S/Dr)(NDo)0'5 = (1.155/0.460)(9 x 1.030)0-5 =7.65 (Note unit of De in Figure 12, in.) ho = 108/7.65 = 14.1 Btu/hr-~F-sq ft outside area 0 Metal Resistance: r = Lu Ao/kCu Aavg where: r = metal resistance; m L = thickness of copper; Cu A = outside area of tube; 0 kc = thermal conductivity of copper = 220; and A = average heattransfer area of tube. avg de, tube equivalent diameter to calculate Aavg and LCu, is: de = Dr + (Do - Dr)(N)(y) de 0.460 + (1.00 - 0.460)(9)(0.019) 0.5575 in. = 0.0465 ft L o0.0465 - 0.032 2 o.007 ft LCu 2 DI,, log-mean diameter = 0.5575 -0.384 = o.466 in. =0.0389 ft ln 0.5575 0.584 -------------- ~~~~~17 _

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Aavg = (3.14)(DLM) = (3.14)(0.0389) 0.122 sq ft/ft length of tube rm = metal resistance = (0.0073)(1.086) (220)(0.122) 0.000295 ~F-hr-sq ft/Btu Calculation of the Inside Water-Film Resistance: Q wwater Cp AT Twater = C T ATwater = (tI - t2) = 125 - 115 = 10~F Cp = 1.0 Btu/lb- F Wwater 4, 00 000 = 400i000 lbs/hr (10)(1) ai, total internal cross-sectional area per pass, = (5.14) (71 )(1) (total no. tubes per pass) = (3.4)(0.584)2 (5)(90).64 sq ft (4) (44).yv __400,000 Vwater = water velocity (5600)(62.4)(0.364) 4.90 ft/sec Using Equation 9c, p. 185 in McAdams, 2nd edition, hi = 150 (1 + 0.Olt)(V')0-8/(Di)02 hwater = (150)(1 + 0.011 x 120) (4'90)08 (0.3584)0.2 1510 Btu/hr - "F-sq ft inside area, where average water temperature = 120~F rwater = film resistance = Ao/hbter Ai= 10.8/1510 =.0075 fi = fouling factor (water) =.001 Fhr-f (TEMA) fiA/Ai =..001(10.8) =.0108 Btu ---------— 18

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - Calculation of the actual overall coefficient, Uo: 1/Uo = rair + rmetal + fi + rwater = 0.071 + 0.0003 + 0.0108 + 0.0072 = 0.0893 U0 = 11.2 Btu/hr-~F-sq ft outside, as opposed to an assumed value of 10.0. From P. 289, Trans. A.S.M.E. (1940), correction to LMTD = 0.992 (from Figure 10) Actual LMTD = (0.992)(27.3) = 27.1~F Required heat-transfer area = 4,000,000 13,200 sq ft (11.2)(27.1) Total required tube length = 1,200 = 12,150 ft 1.086 Actual tube length used = (90)(15)(10) = 13,500 ft Excess heat-transfer area = (13,500 - 12150) (100) = 11.1% 12,150 Pressure-Drop Calculation: Using the Figure 15 correlation, where De = (S/Dr)2(ND) = (7.65)2 4.9 ft, 12 12 E = NDo/DrO^2 (9)(1.030) 10.8 (0.460)0. - From Figure 15, at Vmax = 1340 ft/min APDe = 0.440,;.AP = 0.440/4.9 = 0.0898 in. water row Total pressure drop = (0.0898.)(10) = 0.898 in. water or 0.0325 lb/sq in. 19

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Example II. It is required to cool 28,600 lb/hr of hydrogen from 150'F to 100~F with water being heated from 90~F to 105~F inside the tubes of a finned-tube cooler. WATER WATER IN OUT 1J-J WI~i A............. __.._ _-_ I -:z: I I - -A - + -- -- LENGTH -- HYDROGEN COOLER Cooling Hydrogen with Water: T = 150~F T = 100~F t~ = 105~F ti = 90~F Heat Load = 5,000,000 Btu/hr Heat Exchanger -- 2 tube passes - 1 shell pass Tube Specifications -- Wolverine Trufin Bimetal tube Catalog No. 1106 Do = 1.272 in. D. = o.504 in...____..__._.... -.. _______...... 20 -..

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Tube wall thickness, total = 0.043 in. copper liner thickness = 0.028 in. y = mean fin thickness = 0.015 in. N = 11.0 fins/in. AO/Ai = 16.2 Dr = 0.504 + (2)(0.043) = 0.590 in. Ai = (3.14)(0.504) 0.132 sq ft/ft (12) Ao = (0.132)(16.2) = 2.14 sq ft/ft Assume Uo = 10: q = Uo AATiM 4:aD. 23'3~F In 45 -10 Approximate total heat-transfer area A = 5,000,000 21,500 sq ft UO^TM (10) (23.3) Total tube length = 21,500 = 10,040 ft 2.14 Assume bank of tubes 10 lows deep.(10,040) 1004 ft/row (10) (with 12-ft tubes, 84 tubes/row) Cp, specific heat of hydrogen = 5.5 Btu/lb-~F w, hydrogen = 5000,000 = 28,600 lbs/hre (50)(5.5) Density of hydrogen = (density of Std. air) (Mol wt H2) (Mol wt air) (0074)(2.02) 0.00515 lb/cu ft (29) Volume of hydrogen = 28,600 = 92,500 cu ft/min (0.00515)(60) S = pitch = 1.75 in.; tube clearance = 0.428 in. Height of unit = (84)(175) = 12.25 ft (12) Length of unit = 12 ft 1 -- __ --- —---- ~~~~21

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Total face area = (12)(12.25) = 147 sq ft afins = (0.015)(1.272 - 0.590)(11)(12)(12) afins (144) = 0.1127 sq ft/tube aoot (12)(0.590) = 0.590 sq ft/tube aroot = (12) atF, total projected area (0.1127 + 0.590)(84) = 59.0 sq ft aFF = 147 - 59 = 88 sq ft Vmax = e = 1050 ft/min G, = 283600 = 325 lbs/hr-sq ft 88 De = (S/Dr)2 (NDo/12) = (1.75/0.590)2(11 x 272) (12) = 10.28 ft At t of hydrogen = 125 F / = 0*0227 lb/ft-hr k = 0.115 Btu-ft/hr-~F-sq ft Cp = 3.5 Btu/lb-~F Re = Gma De// = (35)(128) = 147,000 (0.0227) From Figure 14, he /(Cp)0'335 = 1,225 k / (k) and (Cp//k)0333 = (3.5 x 0.0227/0.115)0'3 = o.884 Therefore h, (1225)(0.1)( 5 (10.28) = 12.1 Btu/hr-~F-sq ft outside r, hydrogen = 1/12.1 = 0.0826 TEMA fouling factor = 0,001 rf = (0.001)(16.2) = 0.0162....... --- —- 22,

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Metal Resistance: rm = LAAo/kALAavg Al + LCuAo/kCuAavg Cu where: rm = metal resistance, L = thickness of metal, A ~ = outside area of tube, k = thermal conductivity, and Ag = average heat transfer area of tube. - 0.0585 2 00f48 LA1 = 0585 48 0.0053 ft 0.o585 - o,048 DLM log-mean diameter =.n 0.585.0528 ft 0.048 avg Al = (3.14)(0.0528) = 0.166 sq ft/ft 0.048 - 0.042 LCu = o.8 = 0.003 ft o.o48 - 0.042 DLM of copper liner = n 0.48 = 0.0451 ft. 0.042 Aavg Cu = (3.14)(0.0451) = 0.142 sq ft/ft rm = rAl + rCu r (0.0053)(2.14) + (0.003)(2.14) -.000831 rm = (llo) (0.166) (220)(0.142) Calculation of the Water-Film Resistance. w - = 5, o. = 333,000 lbs/hr C AT (l)(15) ai, total internal cross-sectional water flow area/pass (3.14)(0.504)2(5)(84).581 sq ft (4)(144) Vwater = ( 3300):0.58l) = 2.56 ft/sec Vwater. (3600) (62.4) (0.581) From - Equation 9c in McAdams, p. 183, at tavg = 97.5~F I______________________ 23 _____23________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN hi = 150(1.0 + 0.011(97-5)) (2.56)0.8 (.504)0.2 = 754 Btu/hr-~F-sq ft inside Calculation of the actual UO: l/Uo = 0.0826 + 0,000831 + 0.0162 + 0.0215 = 0.1211 correction to IMTD for cross flow is negligible. UO = 1/0.1211 = 8.25 Btu/hr-~F-sq ft outside Required heat-transfer area = 5('000)(000X = 26,100 sq ft (8.25)(25.2) Required tube length 2612,200 ft 2.14 Actual tube length used = (12)(84)(10) = 10,080 ft Add two vertical rows,. 12 rows deep. 2nd Approximation: 4 water passes and one hydrogen pass unit 84 horizontal tubes in one vertical row 12 vertical rows deep No modification in tube size or pitch The hydrogen gas-film coefficient remains unchanged Revised Water-Side Coefficient: ai. total internal cross-sectional water-flow area (3.14)(0.504)2(3)(84) 0.5 8q ft (4)(144) V water 33o,0 4.25 ft/sec wter (3600) (62.4) (0.35) hi = 150(1.0 + (0.011)(97.5)) I(52) 08 = 1150 Btu/hr-~F(0. 504).2 nside 24

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 16.2 = 0.01435 water 113050 Overall Coefficient: l/Uo = 0.0826 + 0.000851 + 0.0162 + 0.01435 = 0.114 Uo = 1/0.114 = 8.76 Btu/hr-~F-sq ft outside Required heat-transfer area = 5^000,000 = 24,500 sq ft (8.76)(23.3) Required tube length = 24 = 11,440 ft Actual tube length used = (12)(84)(12) = 12,100 ft Excess heat-transfer area = (12,100 - 11i440) (100) = 5.76% 11,440 Pressure-Drop Calculation: De = 10.28 ft NDo/Dr02 = (1)(1.272) = 15.5 (0.590)0.2 Re = 147,000 V = 1050 ft/min max p = 0.00515 lb/cu ft 2 = (1050)2 = 1.1 x 106 sq ft/sq min max From Figure 16, APDe 7.8 x 10-6 p max Therefore AP = (7.8 x 10-6)(0.00515)(1.l x 106) 10.28 = 0.0043 in. water/row Total pressure drop = (12)(0.0043) = 0.0515 in. water = 0.00186 lb/sq in. 25 -..t.i

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TABLE III VARIABLES IN HEAT TRANSFER AND PRESSURE DROP Do -- diameter over the fins, in. Dr -- root diameter, in. N -- number of fins per in. S -- center-to-center distance for tubes in an equilateral staggered bank of tubes. De -- equivalent diameter of the bank of finned tubes, (S/Dr)2(NDo/12),ft. n -- number of tube rows normal to direction of flow. Vmax — velocity of gas measured at 70~F, 1 atmosphere, and 50 per cent humidity, through the minimum free-flow area. C -- heat capacity of outside fluid at the average fluid temperature, Btu/lb- ~F. k -- thermal conductivity of outside fluid at the average fluid temperature, Btu-ft/hr-~F-sq ft. - -- viscosity of outside fluid at the average fluid temperature, lbs/hr-ft. ho -- outside convection coefficient, Btu/hr-~F-sq ft outside. G -- mass-flow rate of outside fluid through the minimum free-flow area, lbs/hr-sq ft free area. Re -- Reynolds number, DeGmax//. Pr -- Prandtl group, Cp7/k. Nu -- Nusselt group, hoDe/k. AP -- pressure drop, in. water/row of tubes. p -- density of fluid at the average fluid temperature, lb/cu ft. 0.2 K -- pressure-drop parameter, NDo/Dr f -- friction factor or function of the Reynolds number, Pe p48ax 48