RECORRELATION OF DATA FOR CONVECTIVE HEAT-TRANSFER BETWEEN CASES AND SINGLE CYLINDERS WITH LARGE TEMPERATURE DIFFERENCES W. Ji N.. -.L: [woJ-{ MILAS S.Wo CHURCHILL To Be Presented in Louisville, Kentucky at the March.22, 1955 Meeting of the AIChE IP-110 Prepaper for Limitied Distribution by the College of Engineering Industry Program February, 1955

Ef Al23 —a~YW&YW)

ABSTRACT The data and correlations for the rate of convective heat transfer for gases flowing across a cylinder have been re-examined. The data obtained at large temperature differences between the gas and solid were found to demonstrate the dynamice dissimilarities indicated by dimensional analysis for heating and cooling. Recalculation of the original data of the early investigators indicated that the correlation presented by McAdams did not adequately represent the data. Despite theoretical dissimilarities, the reliable data for both heating and cooling were correlated satisfactorily by plotting hD/kf versus DVo/vf, where h is the heat transfer coefficient, D is the cylinder diameter, Vo is the free stream velocity, and kf and vf are the thermal conductivity and kinematic viscosity, respectively, of the gas, evaluated at the arithmetic average of the free-stream and surface temperatures. Investigations of the effect of surface temperature variation, free-stream turbulence and the free-stream velocity profile were also reviewed. ii

INTRODUCTION This paper presents the results of a comprehensive re-examination of the data nd correlations for convective heat transfer for gases flowing across a cylinder, particularly with respect to large temperature differences between the gas and surface. Attention has been confined to the mean coefficient over the entire circumference. Data obtained for convective heat transfer with small temperature differences between the fluid and surface have been successfully correlated in terms of the dimensionless groups, DVp/4, and hD/k, and cp.i/k or combinations of these groups, For large temperature differences, the physical properties of the fluid, p, cp, k, and p may vary considerably from the bulk of the fluid to the transfer surface and the above dimensionless groups are not uniquely defined. Jakob (1) has pointed out that for complete dynamic similarity between two nonisothermal systems, the same ratios must exist between the significant physical properties at geometrically equivalent points. A lack of similarity will necessarily exist for any two fluids whose properties do not vary identically with temperature. A liquid and a gas fail in this respect, but two gases provide provide reasonable similarity. For all gases the variation of p., k and p with temperature is quite similar and can be approximated by the expression l/e e b Nusselt (2) accordingly suggested that the dimensionless ratio (T5/To) be included in correlations for convection with gases, ioe., hD (DTp cpa Ts\ \' k' T) (2) For this expression hD/k, DVp/p., and cpp./k can be evaluated at Ts, To, orx-at integrated mean properties, since the properties can be converted from one temperature to another or to integrated mean values by multiplying by some power to Ts/Too. Even for gases a similarity deficiency can be expected for heating and cooling under otherwise similar conditions, i~e., a single expressionrl of the type of Equation (2) cannot completely define both the heat1

ing and cooling process. Deissler (3) has derived expressions for convective heat transfer inside tubes with large temperature differences, using experimental velocity distributions and assuming an analogy between heat and momentum transfer, The bounding temperatures enter the expression in a complex manner, and different equations were obtained for heating and cooling. HEowever, the calculated results for both heating and cooling are reasonably well represented when hD/kf is plotted versus DVoPf/ufo Comparable deviations have not been accomplished for convection outside tubes. For Reynolds numbers below perhaps 1,0, the flow pattern around a circular cylinder approaches potential flowo At higher values of the Reynolds number the boundary layer separates from the surface at about 800 from the forward stagnation point: and standing eddies are formed along the rear half of the cylinder. Beginning at a Reynolds number of about 1000, these eddies are shed periodically, and a turbulent wake is formed behind the cylinder. At very high Reynold numbers the boundary layer itself becomes turbulent and the point of separation shifts toward the rear of the cylinder. These phenomena and the Reynold numbers at which they occur are affected by the surface roughness of the cylinder, by the history and boundaries of the main stream, and by the presence of temperature gradientso No rigorous expression or general correlatioarn has been formulated for isothermal flow around a body when a turbulent wake is formed. The further inclusion of a temperature gradient to yield heat-transfer information is therefore beyond present methods of analysis. EXPERIMENTAL DATA Convective heat transfer for a gas stream flowing across a circular cylinder has been extensively investigated but most of the work has been confined to air and to near-ambient temperatures. However, BenkeU4) Kennelly and coworkers(5 6), King(7), and Hilpert(8) measured heat transfer to ambient air from wires with surface temperatures extending to 916~F, 1076 F, 1840 F, and 1915 F, respectively, At the other extreme, with cooled tubes, Reiher(9) and Vornehm(l0) measured heat transfer from air at 489~F and 390 F, respectively, and Churchill and Brier(11) reported data for nitrogen up to 1800~0F Kilham(2) measured the total energy transfer from flames at about 3700TF to tubes at 2100 to 2800oF, but interpretation of his data in terms of convection coefficients does not appear justifiable, 2

SURFACE TEMPERATURE7 TURBULENCE, A-ND VEOCITY PROFILE Surface temperature distribution,,the free-stream turbulence, and the free-stream velocity profile are generally thought to influence the heat-transfer rate. However, Krujilin(13) and Giedt(l4) observed no difference in heat-transfer coefficie ts when the surface temperature distribution was varied. Comings, et all15) made a comprehensive study of the effect of turbulence on convective heattransfero They found that the heat-transfer rate increased 25 percent when the intensity of turbulence was increased from 1 to 7 percent at Reynolds number of 5800. At lower Reynolds numbers *the effect of turbulence was less. Early experimenters did not recognize the importance of turbulence, and interpretation of their results in this context is impossible. Recent investigations have intentionally been carried out at levels of turbulence below the threshold level found by Comingso Most experimenters have made an attempt to obtain a uniform velocity profile, and no specific data have been reported for different profileso However, Comings concluded that the high rates of heat transfer reported by McAdams for tube banks and by Goukhman and Reilher may have been due to high local velocities rather than true turbulence, CORRELATION The data for both heating and cooling with moderate temperature. differences have been successfully combined in a general correlation by several authors, the errors associated with nonsimilarity apparently being small with respect to the experimental errorso As noted previously, the dimensionless groups generally used for correlation of heat transfer are not uniquely defined in nonisothermal systems, Various authors have accordingly evaluated the physical properties at the surface temperature, Ts., the free-stream gas temperature, To, the arithmetic-mean temperature, Tf = (Ts+T0)/2, the logarithmic-mean temperature, Tim = (To-Ts)/An(Tm/Ts), the geometric-mean temperature, Tgm = rToTs, and weighted arithmetic-mean temperature, Tp = Ts + P(To-Ts) Additionally, mean properties have been use eogo a = (j4o+,s)/2 and Tm = 1/(To-Ts) 0o'dTdo Although the properties may vary considerable between To and Ts,Snumerical d.ifferences between the various mean properties and properties at the various mean temperatures are small and in most cases less than the experimental error in the heat-transfer rate data, In some instances individual physical properties have been evaluated at different temperatures, e g.. DVOPoYLf and DVoDf/4f.o For isothermal systems the Reynolds number is conveniently written as DG/. o For nonisothermal systems the choice of DG/U, as made by McAdams(l6) and others, implies 5

that the density is eva luated at the free-stream temperature, T0, and thus sacrifices the above freedom in the evaluation of properties. Since there is no decisive theoretical justification for any of these forms, the choice should be based on the success and conrvenience of t;he corre.lation. The data for flow of liquids across cylinders indicate that hD/k is proportional to (cptl/k)~03. This same effect is generally assumed for gases. Since cp~l/k is almost identical for air and nitrogen and since (cpk./k)0.3 varies only 0o2 percent between 0 F and 1800 D), it has not generally been included explicitly in the correlations. For the data taken at high temperature differences, discrepancies have been observed which are attributable to nonsimilarityo Fishenden and Saunders(17) noted that the greatest deviations from their correlation occurred for data taken at the highest temperature difference. Hilpert(8) correlated his data and that of King(7) for high temperature differences by using integrated mean properties and including t e temperature ratio TO/Ts as an additional parameter. Churchill- and Brier 11 used bulk gas properties and also found it necessary to include the temperature ratio as a parameter. The effect of the temperature ratio is not the same in the two correlations, however, McAdams(16) included almost all the data in a plot of hD/kf versus DG/If. However, examination of the original sources of the data indicates that the results of several investigators may have been misinterpreted. The data of Hilpert(8) and Vornehm(10) were originally expressed in terms of hD/km and DVo/V the data of Ben'ke(4) in terms of hD/kf and DVo/Yf and the data of Reiher(k) in terms of hD/km and DV~Pm/im, where V~ is a modified free-stream velocity to take into account vwall effects, These data have apparently been. plotted directly by McAdams as hD/kf an.d DG/4f, although. DG/bf differs as miach as 60 percent from DV,/vm' DVO/Vf,9 and DVopm/llm. Jakob(l) and Eckert(18) also appear to have misinterpreted Hilpet's data, but less seriously, reporting, respectively, that arithmetic-mean properties and properties at the aritbmetic-mean temperature were used. The high-temp;erature difference data of Hiltpert, Vornehm, Reiher, King, Benke, and Churchill and Brier are replot;ted in Figure 1 in the form suggested by McAdams, along with -t.e low-temperature difference data of Hilpert, Hu hes(l9), Gibson(20) FPalt;z and Starr(2l), Griffiths and Awbery(22), Goukhmman(25), and Small(243+). For all hilgh-temperature experiments hD/kf and DG/uf were calculated from the original data. Except in the cases of Benke and Vornehnm, where only derived results were available, the raw experimental data of these authors were used along with thermal conductivity values given by Stops(25) and viscosity given by Tribus and Boelter(26). Wherever possible, the high-temperature data were checked for internal consistency with respect to velocity and diameter and for experimental conditions, The data of Kermelly, Wright, and van Hylevelt(5) were rejected

because of the anomalous results reported for the various wire diameters, The data of Kennelly and Sanborn(6) were not included in the correlation because the e;xceptionally high heat-transfer rates reported are apparently due to their unique experimental technique. Representative values only were selected from the very extensive data of King(7). To distinguish the high- for the low-temperature difference data on Figure 1, open symbols are used- for data taken at temperature differences of less than 1500F and solid symbols for data taken at temperature differences greater than 300~F4 It is apparent that the coordinates of Figure 1, while satisfactory for correlation of low-temperature difference data, are inadequate to correlated that taken at high-temperature differences. It may also be noted that all high-temperature cooling data lie above, MAdams' line, while heating data fall below. Numerous other forms were tested. None completely eliminated the effect of temperature difference, As expected, inclusion of the temperature ratio as a parameter permitted individual correlation of the heating and cooling data but failed to yield a general correlation. The most satisfactory correlation was obtained when hD/kf was plotted versus DVo/vf, as shown in Figure 2, Data are included for gas temperatures ranging from 600F to 18o00oF and for surface temperatures from 70TF to 19130F. A curve recommended for design was arbitrarily drawn through the data, The deviations from the curve appear to be random with respect to high- and low-temperature differences and to heating and cooling, and are undoubtedly due in part to undefined variations in the fred-stream turbulence level and velocity profile as well as to experimental errors. CONCLUSIONS The dynamic dissimilarity indicated by dimensional analysis for the heating and cooling of gas in flow across a single cylinder is apparent in the data taken at large temperature differences. The fluid dynamics of flow across a cylinder is insufficiently defined to yield a theoretical expression for the dissimilarities. The correlation given by McAdams(l6) where hD/kf is plotted against DG/pf, does not adequately represent the high temperature difference data. As suggested by Nusselt('2,the data for heating only and for cooling only can be correlated by the introduetion of the ratio of the free-stream temperature to the surface temperature as a parameter. Despite theoretical objections, a satisfactory working correlation for both heating and cooling is obtained when hD/kf is plotted versus DVo/Vf. The correlation presented in Figure 2 is probably representative for low intensities of turbulence in the free stream and the results of Comings etoal(l5) can be used to correct for high levels of turbulence, 5

1000 COMPARISION OF HIGH TEMPERATURE DIFFERENCE DATA WITH LOW TEMPERATURE DIFFERENCE CORRELATION 500 CONVECTIVE HEAT TRANSFER FOR GAS FLOWING ACROSS A SINGLE CYLINDER 200-_ ____/ I OC~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C 100 I I I I I I III I I I I 50 ____D~ - ~ High Temperature Difference Data ~~~~~~~~~~~~~~~~~~~~~hD ~~~~~~~~~~~~~~Heating Gas kf k~~~~~~~~~~~~~~~~~~~~~ * Hilpert i ~, I King 20'~.~ ~ d~" + Benke ~2C0~~ -Cooling Gas ~ Reiher ~ Churchill & Brier ~ Vornehm I O Low Temperature Difference Data Heating Gas Hughes ~5r ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ / re;, o____ ~~- -—,,i, Gibson Paltz a Starr ___ ~ ~I~~~ __ ____rl~~~~~~~~~~~~ - I 1 I IIGouhkman et al. _ _ _ _ _ _ _ _ _ _ _ _ S m all ~~~~~~~~~McAdamrn~ ~ ~~s'. ~ ~~ Line I~~~~~~~~~~~~~ -I~~~~~ 1Griffiths & Awbery Mc~......~..dams~"~"~~~~~~ ~ Hiipert -1 - - - Subscripts: X<,v~~~~~~~~~~~~~~~~~~~~~~~~ ~f property evaluated at arithmetic mean temperature between surface and gas. +~'~~~~~~~~~~~~~~~~~~~~~~~~~~~ o free stream value. I~~~~~~1 — 2%"jI I 2 5 10 20 50 100 200 500 1000 2000 5000 I 0,000 20,000 50,000 100,000 300,000 D V0p0o,uf FIGURE I

1000 GENERAL CORRELATION FOR LOW AND HIGH TEMPERATURE DIFFERENCE DATA CONVECTIVE HEAT TRANSFER FOR GAS FLOWING ACROSS A SINGLE CYLINDER 2 - 100 50 - __ __ hD High Temperature Difference Data kf ____ __ - __ __ - - Heating Gas kf~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Hilpert + King Benke 201 1 I I I I r 1, rl 1 I I I I I I I I I I I I I l~l~A~51~ I 1 1 7 rl I II Cooling Gas Reiher * Churchill & Brier * Varnehm IC Low Temperature Difference Data Heating Gas Hughes _ Gibson o Paltz & Starr _ Gouhkman et al. Small o Griffiths a Awbery Hilpert ________2____ ____ Subscripts: fproperty evaluated at arithmetic mean temperature between surface and gas. a free stream value. 2 5 10 20 50 100 200 500 1000 2000 5000 10,000 50,000 100,000 300O D V0 FIGURE 2

NOMENCIATURE C - Heat capacity, Btu/(hr)(pound mass) D - Cylinder diameter, ft G - Mass velocity, (pound mass)/(hr)(s., ft) h - He'at transfer coefficient, Btu/(hr)(OOF)(sq. ft.) k - Thermal conductivity, Btu/(hr)(OF)(ft) in - Natural logarithm of T - Absolute temperature, 0R V - Velocity, ft/hr p - Density, (pounds masS)/(cu. ft.) V - Kinematic viscosity, (sq. ft.)/(hr),: - Function of - Viscosity, (pound mass)/(hr)(ft) Subscripts a - Arithmetric average property f - Arithmetric.average temperature gm - Geometric mean lm- Logarithmic mean m - Intergrated mean o - Free stream p - Constant pressure sa- Surface

BIBLIOGRAPHY le Jakob M,, Heat Transferl Vol. I, John Wiley and Sons, New York (1949) 2. Nusselt, W., Gesund..-Ing, 38, 477 (1915). 3. Deissler, R. G., Nat. Advisory Comm. Aeronaut. Tech. Note 2242 (Dec. 1950); Ibid., Techn. Note 2410 (July 1951); and Trans. Am. Soc. Mecho Engrs., 73, 101 (1951). 4. Benke, R., o Arch. GWArmewirt. 19, 287 (1938). 5. Kennelly, A. E.o, Wright, C. A., and van Bylevelt, J. S., Trans. Am. Inst. Elect, Engrs., 28, 363 (1909) 6. Kennelly, A. E., and Sanborn, H. S., Proc. Am. Phil, Soc. 53, 55 (1914). 7. King, L. V.,, Trans. Roy, Soc., (London), A214, 373 (1914). 8, Hilpert, R,, Forsch. Gebiete Ingenieurw., 4, 215 (1933). 9, Reiher, H,, Forschungsarb. Gebiete Ingenieurw., No. 269, 1 (1925). 10, Vornehm, L. reported by Ulsamer, J.o Forsch, Gebiete Ingenieurw., 35 94 (1932), 11. Churchill, S'. W. and Bier, J. C., preprinted for St. Louis Meeting of A.IaCh.E. (Dec. 1953). 12. Kilham, J. K,. Third Symposium on Combustion Flame and Explosion Phenomena, Paper No. 98, p. 733, Williams and Wilkins, Baltimore (1949). 135 Krujilin G.,, and Schwab, B.,, Tech. Phys. U.S.S.R.,.2 312 (1935). 14. Giedt, W, H,, Trans. Am. Soc. Mech. Engrs. 71, 375 (1949). 15. Comings, E, W, Clapp, J. T, and Taylor, J. F., Ind. Eng. Chem., 40, 1076 (1948). 16, McAdams, W, H., Heat Transmission, 3rd Edition, McGraw-Hill, New York, 1954,

17. Fisehenden, M,, and Saunders, 0.,: The Calculation of Heat Transxnissiona His Majesty's Stationery Office, London (1932). 18. Eckert, E, R. G., Introduction to the Transfer of Heat and Mass, McGrawHill Book Co, New York (1950). 19. Hughes:, J. A., London Phil. Mag., 1, 118 (1916). 20. Gibson, A. H., London Phil. Mag., t 324 (1924). 21. Paltz, W. J. and Starr, C. E., Thesis in Chem. Eng'g., M.I.T., 1931. 22. Griffiths, E., and Awbery, J. H., Proc. Inst. Mech.]Engrs. 125, 319 ( 933). 23. Gouhkman A. Joukovsky, V., and Loiziansky, L., Techn. Phys. U.S.S.R., 1, 221 (1934). 24. Small, J., London Phil, Mag., 19, 251 (1935). 25. Stops, D. W., Nature, 164, (1949). 26. Tribus, MNo, and Boelter, L. Mo K.y Nat. Advisory Comm, Aeronaut., Wartime Report, R W9 (Qct. 1942). QL

UNIVERSITY OF MICHIGAN 3Ii 9015 02656 7399