THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING Stuart W. Churchill Professor of Chemical Engineering University of Michigan Richard E. Balzhier Instructor in Chemical Engineering University of Michigan October, 1958 IP-324

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LIST OF FIGURES Figure Page 1 Radial Heat Flux for Uniform Heat Flux at the Wall... 10 2 Radial Heat Flux Density for Uniform Heat Flux at the Wall.....r.. o..,,............................. 11 3 Radial Heat Flux for Laminar Flow and Constant Wall Temperature.................................. 12 4 Radial Heat Flux for Slug Flow and Constant Wall 5 Radial Heat Flux for Turbulent F]ow and Constant Wall Temperature......................... 14 6 Radial Heat Flux for Fully Developed Flow and Constant Wall Temperature.......4..* o......... 15

INTRODUCTION The shear stress is known to decrease linearly from the wall to the axis of a pipe. The radial heat flux density which is in some ways an analogous quantity varies much more complexly as will be shown. An energy balance over a differential ring within a fluid in fully developed flow in a tube can be written in terms of local, timemean variables as follows: 1 - rT _ uT (1) r ar = up cy z kz where q is the radial heat flux density in BTU/(hr)(sq ft). The other variables are defined at the end of the paper. For simplicity physical properties will be assumed constant and k will be assumed to be negligibly small in all subsequent derivations. The latter assumption has been shown by Schneider(l) to be reasonable for most circumstances. Integration of Equation (1) from the axis to any radius gives qr = pc u rdr (2) 0 and r 6T 2 r ~T 2 _w qwrwJr = uyd (3) f 2 fw U + d(r2) fu z d(r-) 0 0

where Q is the radial heat flux, in BTU/hr, and the subscript w designates the wall. The first integral form is more convenient near the center of the tube and the second form near the wall. The radial heat flux, the radial heat flux density and the above ratio can therefore be obtained by integration if only the local longitudinal temperature gradient and the local velocity are known as functions of radius. If the rate of radial transfer of energy by diffusion and eddy motion is expressed in terms of an effective radial conductivity kt = q/(i) the temperature distribution can be obtained by one integration rw.1 Tw T =,Jqdr (-a-w)rw- k w kt k Cw kt rw r r rw and the mixed mean temperature, Tm, and the Nusselt number by another: 1 1 1 Tw m (u )(Tm _ T)d (w)2 qwrw [- k r _)d( (r__] ) - TM w-ir)- k M (uqw kt rw r w 0 O r ~~~~rw ~(6) 2hrw 2rwqw 2 Nu (7) k (Tw - Tm)k 1 m U k rw The local Nusselt number can thus be determined from the radial heat flux and velocity distribution if the effective conductivity is also known as a function of radius.

Analytical solutions and detailed experimental data for heat transfer are primarily restricted to two boundary conditions: uniform wall temperature and uniform heat flux density. These two cases will be considered in detail. Uniform Heat Flux Density at the Wall Seban and Shimazaki(2) asserted that for a fully developed thermal boundary layer, h and (T - Tm)/(Tw - Tm) are invariant with length. For a uniform heat flux density at the wall, it follows that (Tw T ) is invariant, and that - and is also invariant. For this case az az Equation (3) reduces to r __& | (u ) d(LE)2 (8) um rw and the integral depends only on the velocity distribution. The results obtained for several important cases follow: 1. Slug Flow- u = 1 Urn QW rw' rw For k/kt 1= integration of Equation (7) then yields the well known result, Nu = 8. u2 2 Laminar Flow 2 - () Q 2 (L)l wL)] l' = 2 l (r )2] (1l) For ~ = 1, Equation (7) yields another well known result, Nu = 48/1X.

30 Turbulent flowa: u f Re, -—,L]O Um rw rw The integral in Equation (8) can be evaluated graphically using the generalized graphical correlations which have been developed for the turbulent velocity distribution(3). The results for Re of 4~0 x 103 and 2.35 x 106 in smooth pipe and Re of 2.5 x 106 in rough pipe (rw/e - 15) are illustrated in Figure lo The curves for slug and laminar flow are also includedo As could be anticipated the curves for turbulent flow lie between those for laminar and slug floW. Since these curves depend only on the velocity disc tribution they should be valid for Bny fluid. Although the heat flux ratio is a well behaved function for all of the illustrated cases, the heat flux density ratio as illustrated in Figure 2 actually goes through a maximum near the wall for laminar and turbulent flow. This maximum results from the competing effects of decreasing area and longitudin-al transfer which increase and decrease, respectively, the heat flux density'as the radius -decreaseso The integration for Nu has been carried out for smooth pipe by a number of investigators, all of whom assumed k/kt = 1 + a Pr (C) with a colistant Ca and used values of Ec/v obtained directly or indirectly from experimental velocity distribution. Empirical equations were used for E/v and analytical solutions were obtained by von Karman(4) and Deissler(5) who assumed q/qw constant, and by Martinelli(6) who assumed q/Jw = r/rw. A graphical rerpresentation for the velocity distribution was used by

-5(7) Lyon, who carried out the equivalent to the integrations in Equation (7) numerically for several values of Pr less than 0.1, and obtained a graphical correlation which can be represented approximately by the equation Nu = 7.0 + 0.25 (a Pr Re) (11) Equation (7) could be integrated rigorously for the entire range of conditions for which experimental velocity and a distributions are available but the results would not be expected to differ significantly from the above approximate results since the integrations are an effective smoothing operation. By an alternative procedure ~tilizing an analogue computer and not explicitly - involving the radial heat flux, Sleicher( obtained the first three coefficients, eigenvalues and eigenfunctions in series solution for the temperature field as well as for Nu. The ratio of a/Uair derived by Jenkins and experimental values of a e/v for air were utilized in the calculations. Sleicherts results could be utilized to integrate Equation (3) for a developing thermal boundary layer insofar as the series converges in three terms, However the radial heat flux in the inlet region will instead be illustrated for uniform wall temperature for which more extensive experimental data are available. Unifprm Wall Temperature For uniform wall temperature, invariance of (Tw - T)/(Tw -Tm) requires that a t = AP- (12)

Hence for a fully developed thermal boundary layer Equation (3) can be reduced to r J (Tw T) u d(r) Qw fw (13) (Tw- T) u d r2) 0 Thus the rad'al distribution of the l ongitudinal temperature gradient is required for a devel.oping thermal boundary layer but only the radial temperature distributon,;for a ful ly developed thermal boundary layer. Again several important cases will be considered in detail: I. Slug Flow with k/k.0 An analytical< solution for the temperature distribution was obtained by Graetz(9) as a function of r/rw and wc/kL only in the form of an infinite series, Since only the first several eigenvalues and eigenfunctions in the series have been computed the usefulness of the sol.ution is limited to values of wc/kL for which only the corresponding valups of the series contribute. Equation (3) was integrated graphically for several values of wc/kL in this range and Equation (3) for w/kL = 0 The results are shown in Figure 4. 2, Laminar Flow with k/kt = 1.0 A series solution analogous to that for slug flow was also accomplished for parabolic flow by Graetzo The results computed from. this solution are shown. in FigurLe 3, 3e Turbulent FlFow The temperature field in turbulent flow is a function of Re, Pr and e/rw. Experimental values for a developing temperature field have apparently been reported only by Abbrecht(l0) for air in smooth pipe at

-7Re of 151,000 and 65,oo000. The values of the radial, heat flux ratio computed from the temperature data at Re = 15,000 are shown in Figure 5, The data (8) of Sleicher for the radial temperature distribution for the same conditions but at L/rw of 62 were utilized in Equation (13) and the results are inclided in Figure 5, His theoretical results for L/rw - oo match the experimental curve for L/rw = 62 very closely. Seban and Shimazaki(2) integrated the equivalent of Equation (5) and (13) by reiteration assuming - 1I and using experimental values of e/v for air in smooth pipe, They reported values of Nu for a range of conditions but the temperature dist-ribution itself only for Re = 10,000 and Pr = 0,01. The radial heat flaux ratio calculated from. this distribution is compared with the values obtained from the Graetz solutions for wc/kL - O and with the values obtained from the theoretical results of Sleicher for L/rw - oo in Figure 6, Contrary to Figure 1 for uniform wall flux, the heat flux ratios for turbulent flow fall below the curve for slug flow since eddy transfer of heat enters Equation (2) indirectly via aT/az. In principle the series solution of Sleicher could be used to carry out the integrations for the radial heat flux for any L/rw, e/rw, Pr and Re, but such integrations are limited practically by the eigenvalues and eigenfunctions which he has computed. Plots of q/qw versus r/rw (not shown) reveal a maximum near the wall for all L/rw for laminar and turbulent flowo Conclus ions The radial heat flux provides a new basis for the interpretation and computation of heat transfer. The values of the heat flux ratio which

were computed for a variet~y of condAitions do rot yield to precise generalization but do permit interpolation with, reasonable confidence, The heat flux densit-, goes through, a maximumm n-ear the wal. for xmost conditions due to the competxng effecstus of iongitudina. taransfer and ehanging radius, but -the heat f.ux L'tself decreases monoton-icail7y from the wall to zero at the axis, The -'esults provide a more accurate basis for calculations witih eddy dicffusivities thlan any of the ideaUlzations previously utilized,

NOMENClATURE c'heat capacity, BTU/(lb )(~F) e height,of pipe roughness, feet h local heat transfer coefficientt' BTU/(hr-)(sq.fto )(.O~F) k thermal conductivity BTTU/(hr)(ft)(QF) L distance down pipe from start of heating, feeto Nu = 2h rw/k Pr =.c[/k Q = heat flux, BTU/(hr) q = heat flux denmsity, BTU/(hr) (sq. ft.) Re =2 rw PuF i r radial distance from axis, ft" T temperature,' F u = local velocity, ft/hr w = mass flow rate, lb/hr z - distance down pipe, feet = ratio of eddy diffusivities for heat and momentum transfer E eddy diffusivity for momentum transfer9 ft2/hr I= viscosity, lb/(ft)(hr) p = density, lb/(ft)3 Subscripts m mixed mean t total w =wall z= longitudinal -9

VELOCITY D STRIBUTION LAMINAR (PARABOLIC TURBULENT Re =2.5 x 106, w = 15 --- TURBULENT Re 4.x 103 SMOOTH TURBULENT Re= 2.3 5xI06 SMOOTH....... 0.8 o iS...LUG (FLAT) _++ /./ //'.06 /:.2. Figure 1. Radial Heat Flux for Uniform Heat Fluxat the Wall. Figure 1. Radial Heat Flux for Uniform Heat Flux at the Wall.

-1i T VELOCI Y DISTRIBUTION 1.0 ------ LAMINAR (PAR BOLIC) L- TURBULENT R = 4.0 x 10I SMOOTH | __.- - TUR.qULENT R-!e 2.35x1 6 SMOOT / /,' 4-+-+ SLUG (FLAT) 0.7__ __ _ /.. 0. __ 05/.,' o.? oi i_ / ~ 0 5.1 0.2 o.3 0.4 0.5 oi6 0.7 0.8 0.9 1.0 r Figure 2. IRadial Heat Flux Density for Uniform Heat Flux at //the WalL. the WCall.

-120. — 0 0.2 0.4 0.6 0.8 1.0 r rw Figure 3. Radial Heat Flux for Laminar Flow and Constant Wall Temperature.

-131.0 0.9 0.8 0.7 C iWc ______ I ~~~~kL 0.6,27 0.5 ~oi3/d~~~ o.4 O.0.2 I ~ ~ ~ ~ ~i;..-'; I 1,"i/ — ~~~~ 0... 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r Figure 4. Radial Heat Flux for Slug Flow and Constant Wall Te perature.

-1.41.0 0.9 Sleicher / /! I Abbrecht F f T F ~~~~~Wl 0.8emra [ " 0.6 0 i 0.2 0.3 0.4. 0 0.7 0.8./ 1. Figure 5. Radial Heat Flux -for Turbuent Flow and Constant Wall Temperature.

,.C 0.9 0.8 0.7 0.6/ 6 eL/r / o r/r W Figure 6a Radial Heat Flux for Ful~y Developed Flow and ConstantGraetz Laminar ture L~rw oo r/r w andContan Wil empratre

'REFERENCES 10 Schneider, P. Je, Trans o Amero Soc0 Mech_. Engrso 765 (1957)o 2 oSeban, Ro A, o and Shiimazaki, T. T., Transo Anero Soc Mecho hEngrs. 73, 803 (195'1) 35 Bakhmeteff, BO A,, The Mechanics of.Turbulent Flow, Princeton University Press, Princeton, I J. (19.41. 40 von Karman, T., Transo Amer. Soco MechO EngrsO 61, 705 (1939). 50 Deissler, R. G., Natl. Advo Comm. Aeronauto Tech Note 3145 (1954)o 60 Martinelli, R. C,, Tmans. Amero SOCo Mech. Engrs. 69, 947 (1947). 70 Lyont RO N,, Chem. Enmg Progg 47, 75 (1951). 8, Sleicher, C, A., Ph.D Thesis, University of Michigan, Ann Arbor (1955) and Trans. 0Amer0 So_0 Mech. Engrs.0 79, 789 (1957). 90 Jakob, Max, Heat Transfer, Vol. I9 John Wiley and Sons -New-York (1949). 10 Abbrecht, PO Ho, PhoD. Thesis, University of -Michigan, Ann Arbor (1956).

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