ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR FORCED CQNVECTION FROM NONISOTHERMAL SURFACES By JOHN KLEIN MYRON TRIBUS Project M992-B WRIGHT ATR DEVELOPMENT CENTER, Uo S. AIR FORCE CONTRACT AF 18(600)-51, E O No.. 462-BR-1 August, 1952

The convection of heat from surfaces at nonuniform temperatures is reviewed Solutions for systems previously analyzed by others are collected and. compared A few new solutions are proposed. A method. of treating heat fluxes with variable wall temperature by using the solutions for constant wall temperature is describedo ii

SYIBOLS AND NOMENCLATURE A = k/pCpgc b = (1/6) Pr f" (0) B = arbitrary constant c = (/4) (1 + m) C = coefficients in Graetz and. Poppendiek solutions Cp C = unit heat capacity at constant pressure, BTU/lb-~F d = diameter of tube, ft f = Hartree velocity function (Ref. 19) g = integrating kernel, ~F/(BTU/hr-ft) gc= 32.2 ft/sec2 h = integrating kernel, BTU/hr-ft2-~ F k = thermal conductivity, BTU/hr-ft2-(~F/ft) ke ='eddy conductivity", BTU/hr-ft2-(OF/ft) m = exponent for "wedge"t flows p exponent in Poppendiek solution Pr= (3600k Cpgc/R), Prandtl modulus q = heat flux, BTU/I1r-ft2 Rex = local Reynolds modulus, S = diameter or width of channel or slot, ft T = temperature ~F T = free-stream temperature, OF Ts = slot temperature, OF Tw = wall temperature, 6F T o = initial stream temperature^, ~F u = velocity parallel to surface, ft/hr v = velocity perpendicular to surface, ft/hr W = fluid weight rate, lbs/hr x = distance along surface, ft y = distance from surface, ft z = dummy variable, ft a = exponent in Graetz solution = cR/2WCp ftP3n = coefficient in Poppendiek solution = dummy variable, ft (,unheated starting length) 9 - dimensionless temperature field, OF p = fluid density, slugs/ft3 = fluid viscosity, lb sec/ft2 f = wall shear stress, lb/ft2 iii

L -- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN FORCED CONVECTION FROM NONISOTHERMAL SURFACES INTRODUCTION Rubesin "2 has outlined the problem of heat transfer from surfaces whose temperature varies in the direction of fluid flow. The analysis of heat flow for such cases shows that the heat fluxes which would be predicted when the surface temperature variation is not taken into account differ markedly from the case where the varying temperature effect is included. In some cases the actual heat flux is in the opposite direction to the prediction based on an isothermal surface. In this paper the method of'ubesin will be reviewed and some new applications demonstrated. The reaider who wishes to consider this problem further should also consult referiences 10 and 13, where numerical and graphical methods of analysis are givebn. OTHER PAPERS ON THIS SUBJECT Heat transfer from nonisothermal surfaces has in recent years attracted the attention of many workers. A number of exact solutions to the differential equations for conservation of momentum, mass, and energy have been given,- as well as several suitable approximate solutions. One graphical method has been cited, Table I is a tabulation of analytical solutions of the above type known to be available at the present time. Experimental data to test the various analytical solutions have been almost completely lacking. The data of Scesa12 were taken for the express purpose of testing the predictions and, as shown by Scesa12 and Rubesin2, the agreement is satisfactory. Sherwood and Maisell8 give data 1

TABLE I Author Boundary-Layer Character Flow Conditions Prescribed Properties of the Fluid Surface-Temperature Description Reference Rubesin (approximation) Laminar Flat plate, zero pressure gradient Constant ~Step function 1 ChapmanRubesin (exact) Laminar Flat plate, zero pressure gradient /*.p = constant Polynomial in x 4 Pr = 0.72 Levy (exact Laminar Wedge flows = cx ul = cx Constant 0.7< Pr < 20 Polynomial in x 5 Lighthill (approximation) Laminar Arbitrary, skin friction is taken as parameter Constant Arbitrary, solutions given in integral form 7 Graetz (exact) Lipkis (exact) Seban (approximate) Leveque Laminar Laminar Laminar Laminar Flow in a tube, parabolic velocity distribution Flow in a tube parabolic velocity distribution Arbitrary Flat plate, velocity profile in boundary layer taken as linear Constant Constant Constant Constant step function Temperature a linear function of x Arbitrary Step function 6 8 10 9 Rubesin (approximation) Turbulent Flat plate, zero pressure gradient Constant Arbitrary, power function variation worked out as example 2 Sage, et al, (electrical analog) Turbulent Flat duct Constant Step function 11 Tribus (Graphical or numerical) Turbulent Round or flat duct Flat plate Entrance to a tube Eckert Laminar Constant Constant Constant (liquid metals) Arbitrary Step function Step function 11 15 Poppendiek Turbulent Poppendiek Turbulent Flat plate Constant (liquid metals) Step function 25 Yih-Cermak Laminar Round or flat duct Constant Arbitrary, opposite walls of duct not necessarily at same temperature 26

1 - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN for the case of mass transfer. Yih and Cermak2 have presented a report which considers the laminar flow of a fluid in a pipe or duct with nonuniform wall temperature. Unfortunately, the report has not been widely circulated and the present author did not learn of its existence until after the completion of this paper. THE GENERAL METHOD Consider, for example, the form of the energy equation in two dimensions as used in boundary-layer calculations, upC ~+ vpCT = VC {(k + ) e ) (1) p ax Zy ay subject to the following restrictions: 1. The fluid properties p, CpJ k, are independent of temperature. 2. The velocity field (uy v) is known and independent of temperature 3. The "eddy" conductivity" is not a function of temperature. The effect of the last term may -be taken into account by adding a particular solution of- the- above equation to the solution of the following equation. aT T T (la.) uCp + pCpp c T A (ke + k) (la) The particular solution of most interest is known as the "adiabatic wall temperature" and is the temperature assumed by the wall when the heat flux from the wall to the fluid is everywhere zero. When a solution to Eq la is found, that is, a relation between q(x) and Tw(x), there must be added to Tw(x) the adiabatic wall temperature Tad(x). The important property of Eq la is its linearity with respect to temperature. I 2

.- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Consider now a function 0(5, x, y) which satisfies the above equation and a set of boundary conditions as follows: 9(f, x, O) 9(S Ix, 0)= oe(, x,.O) = 0(, x, o) = o(9, x, 0) = o(9, x, 0O) Gy(, x, ) =,(, y ) 1 0 0 0 TO TO y> 0 x < f 1 0 0 0 if the flow is a "boundary-layer" type if the flow is of the "conduit" type with y = r a line of symmetry for the flow and temperature to construct the temperature field stepwise fashion by considering 0 The above function may be used surface temperature varies in a when the the sum: T -Tn = i2,3, n = 1,2,3,^.. ) - Tw(n-l )] (n x, y) (2) Tw(O-) = T. We may represent the above summation by an integral taken in the Stieltjes sense (see Appendix A). x T - T = J (f, x, y) dT (f ), T-= o (3) T (O-) = T The above integral is the formal solution to the energy equation. It remains now to determine the form of the function 9(5, x, y) for Some particular systems of practical interest. In every case the function 0($ x, y) is the solution to the energy equation in the presence of a "step function" in temperature. A number of "step-function" solutions exist. For example the Graetz6 and Leveque9 solutions are of this type. The solutions by Rubesinl12 are of such a nature. The analysis by Lighthill7 is in the form above. In general, it is not the temperature field but the heat flux at the wall which is sought. Thus we are usually interested not in (I, x, y) but in its derivative at y = 0, ie., q(x) = [-k aTy.(x) L. 7x Jy = 0 X -k 9y(g, x, O) dTw(f). = o (4) 5

lm ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN As Rubesin has pointed out2, -k times the normal gradient of 9 at y = 0 is the usual definition of the local unit thermal conductance for the case where the wall temperature is a step function. I Defining we have: h(J, x) = q(x) =:= O -key -(,, 0), h(, x) dTw (f). (5) Table II gives a summary of the function h(g, x) currently known for various systems. FINDING THE WALL TEMPERATURE WHEN THE HEAT FLUX IS PRESCRIBED The previous discussion has considered only cases where the wall temperature is prescribed and the heat flux to be found. In many cases of practical interest the heat flux is given and te ttemperature of the wall is to be found. The kernels must be modified for use with this second type of problem. Most of the kernels of Table II are of the form. h(<, x) - f(x) (x^->5 a xl I i i with f x q(x) = f O (7) (7) h(,3 x) dT(J) I Now multiply both sides by (xc z- )-1 d(x') and integrate from x = 0 to x = z. z q(x)j (Xs- zS)a-ld(xs) x=0 - (x( z ) d(x) x=0 x ( (x adTw()=0. f =0 (8) Margenau and Murphyl7 discuss the above type of double integral; since x varies from 0 to z andy varies from 0 to x for every x, the result is' equivalent to varyingf from 0 to z and x fromf to z for every value off 4

TABLE 3 INTEGRATING KERNELS FOR NON-ISOTHERMAL CONVECTION q(x) =-(c, Tx) d T(C) T(x)-Do, (C, x)q(f)dC AUTHOR REF. SYSTEM METHOD OF SOLUTION' h(. x) g(,x) Laminar flow over a flat plate. Velocity and temperature profiles I R1UBESIN I Zero pressure gradient. Fluid postuted linear in y. Thermal /3 2_(3 3 6! -p R2 properties constant. boundary layer thickness proportional x 6Pr Re -< to momentum thickness.(Integral) Laminar flow over a flat plate. odity and tempertue profiesposht 3 | ECKERT |15 Zero pressure gradient. Fluid lledasoacubiciny. Integralmethod Po.l R e 1 63! properties constant. of solution. e [ ] eWl3)(2/3) k P R) Laminor flow over a flat plate. Velocity profile taken as u-cy, not 3 Shear at the wall postulated dependent upon x. Term vaT/8y.k2 (p/9)[(du/dy) x — pr'13( f //t LEVEUE 9 constant. Fluid properties thereby drops out of energy equation 3(1/3)! du/dy x- 3k(/3)/ constant. and resulting equation is solved. Laminar flow over a surface with Velocity profile taken as ufy rX-__ - ITT i LIGHTHILL 7 known variation in surface shear where r(x)- wall shear. k, I3 f LIGHTHILL |7 stress. Constant fluid properties Resulting differential equations ( /3)!r p/, [ (z)d solved. "Wedge-flows. Fluid properties Velocity near wall taken as linear in y. ( 4m b Re. E -( L BORND 16 constant. (Velocity over wedge Resulting differential equation solved Pr (4/3)f 3 -2 2 _ _given by u c=x) for two coses: (a) Step function in b = f"(o) c=-3(+m) (3la+m) k b Re c temnaur Step function inheat fu (o) is a dnnsioless veocity grdent tabulated as a function of m in reference 19. Laminar flow over a surface. Velocity profile taken as r MODIFIED Fluid properties constant.'u-ty. dr/dx small enough _.Apr / 2) Jdp LEVEQUE dT/dx small. to make vaT/8y negligible. W).! L J ) r(x)=woll shear. Turbulent flow over a flat Velocity and temperature /3 3s l RUBIESIN |2 |plate. Fluid properties con- profiles taken as followinq 0o02z88k i. E- 946 |3 (28/195) e -0r3 88 R x~x40~ 40j stont. I/7 power low. Integral method. 3x r Rex x J(/9)3o.oes k Turbulent flow over a flat elocity profile token as 17 power 9 |nr| SEBAN 12 plate. Constont fluid prop- of y. Temperature profile taken as o. x, Re9 ( 9 0s Pr _I-o x - erties. linear in y near the wall, V7 power x(/(9).o9k * I0} of y outside the laminar sub-layer Experimental measurements on Empirical equation'best-tit' to MAISEL AND a mass transfer apparatus. data. (Translated to air heat aCsk 08r e0 SHERWOOD transfer system by present x ex R -x J author.) Laminar flow in a tube. Para- Differential equation solved by 4k e-a^i(X-C) i I 2 3 X GRAETZ 6 bolic velocity distribution separation of variables. Only d i= a 7317 4435106 See Eq. 26 Constant fluid properties. first three eigen values known. k c 0.749 0539 0 119 9 -2wcp' ________________________ Turbulent flow of a liquid metal Thermal conductivity of meol kesPrI2+ ) C ir POPPENDIEK 24 in a tube. Velocity profile postulated large enouqh to L2( j established. Constant fluid render eddy heat transport )neg-! (pf n.2 n es (p___ __)! n/' ——...3 Oonstants cfi and properties. ligible. Velocity profile foll.' s. An tabulated in reference Ps. POPPENDIEK Turbulent flow of a liquid metal Thermal conductivity of metd poal s- i - IT: MODIFICATION 25 in a tube. Velocity profile tulated large enough to render k r (- 2 (P3) (P-C) M MODIFICATION 2 established. Constant fluid eddy heat transport negligible. (/ p+3) 2 SP (X ( (P k \ (P)I - OF LEVEQUE properties. Velocity profile follows. u-B(? P+2 *L I ( P -2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Therefore, z 0 q(x) f(x) (x"- z ) - d(x 0 dTw ()f (x - z (x - d(xS) (9) Let P) = x - Hence f~.-~ zrq~.r(x )' This latter integral value 1 (xd.- d(xd 1 (10) is an Eulerian integral of the firet kind and has the (-z) ( )'. -.! (a - 1); Tl(x) - T = g q()(x-d 1 i. T fo i(n-a)r' (a - 1)e The above integral may be written in the form (11) (12) T,.(x) - T4 J q( ) g(., x) d (i3) Table II gives values of g(, x) for some of the systems for which h(, x) is known. In the cases of the Lighthill solution and the Leveque modification (see Table II), when (x)Y = Cxn, as often occurs, X -n e t^^ dz = / - n 2 -n (:X U -n 7 ) (14) and X dz 1 Cz-n C(n + 1) (n + n + n + 1 (x >} (15) Therefore, these solutions are of the form: h(,x.) = I,.1/6 1/53 / 1/3 -n/2 kC / Pr /9/ x (1/5)' (1 - n/2) -1/ (-n/2 (-n/2 -1/3 (x(Lighthil (Lighthill) (16) 6

-- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN and h(~, x) _ kCl/3prl/3(n + 1)l/3(9p/p2)/3 1/......._..(./ (x+. n+l)1 3)/5Y (17) (Leveque Modification) Most,of the kernels'of Table II have been published before, Two, however, are new and their derivation is presented in later sections of this paper. (The generalization of the Graetz solution was given in reference 26 ) USE OF THE POPPENDIE A ND GRAETZ SOLUTIONS WEE,HEAT FLUX IS GIVEN The integrating kernels used in the Graetz or Poppendiek solutions are of the form: h(f, x) = h(x - ~) (18) and the heat flux is given by: q(x) f h(x - ) o IdT(d) d * (19) Define now: u(z) =f e-zt h(t)dt v(z) o b0 e-7 t dT dt dt o = -T(o) +.zr eZtT(2t)dt. (20) Then it follows21 u(z) v(z) 0 Oc = J e-z q(t) dt o (21) Now if q(x) and h(x) are known, the temperature may be found by the FourierMellin integral *Yih and Cermak consider the case where the temperature outside a poorly insulated pipe is specified026 7

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN C + ieo T(x) 1 -eZ [(v(z). T(o)] dz 2ni C - i z (22) or T(x) -T(o) = T () f e 2-^c _ it - v(z) dz (23) For the integrating kernels in question h(t) = Cn e-ant' n (24) Therefore u(z) = (An + z)n and the formal solution for T(x) is (25) T(x) - T(o) C +,i o Ie-Zt(t)at 1. ez X. 2tc i - i z. Cn n An + z (26) The difficulty in the evaluation of tle above integral tude of zeroes of lies in the infini ZiCn (An + z)-1 (27) An approximate solution may be obtained by taking only a few terms of the series. (In the case of the Graetz solution only the first three eigenvalues are known.) By way of example, the Graetz kernel is h(x) 4= 4 [0.749 e-7'317x + 0o: 9e4435x + 0.179e'lO6x + o..; hence, approximately8 henceo approximately Az An +Z = 0.749 + 0.359 + 0.179 7.317P + z 44,353 + z 106p + z (30o.91 + z)(95.035 + z) (29) (7.317P + z)(44,35P + z)(lo06 + z) 8

- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN and if q(x) is a step function, q.(x) =q + i c T(x) - T(O) = Lqd ( 3173 2ii4k cTh i 2; (3i The integral is evaluated to give: xz-O the temperature is: x7 0 + z)(4435 + z(lO + z))(z6 + z) ).915 + z)(9503f + z) (3o) (31) T(x) - T(O) = qd 4R -30.915x3 -95. 53xB [Il.7Px - 3.05 + 3.27e + 2.51e The above equation is in error at the origin ( as in the Gra-etz soluLtion, since only the first three terms of the infinite series are taken. However, the error disappears quickly since the exponentials decay rapidly. FLOW IN A ROUND PITPE WITH PARABOLIC VELOCITY DISTRIBUTION ESTABLISHED (Graetz Solution Modification26) The usual form for the Graetz solution is6: amvq(x) = 4k (to _ tw) d i i=l -Ciep Cie (32) where the Ci and ai are given as: 1 1 i = C. = 0749'i = 753136 2 0 539 44,61 3 0 179 106 and P = t.:Jk 2WC In the Gretz solution hence the generalzatio is simple In the Graetz solution, = 0; hence the generalization is simple~ h(g, x) 4k i i C-e p(.x - ) C eeoiB+i (33) 9

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN C.nd therefore, if' the pipe has nonuniform wvll temperature 4(x) 4k I d i Cie ii3 x 0 eeyiix drt'( 5 ) (34) Note that the conductance h(, x) is based on the inlet temperature of the fluid rather than on the mixed mean temperature at each axial position. Certain wall temperature variations yield solutions readily, since the integrals are already known, For example, if the wall temperature variation is of the form; T'x = Axn'l, q(x) (n + 1)A Ci i al. (-x) [1 - _ x -— ~i _x nlr - 1) cS6x2 I -'L upBX (-5) (35) if the wall temperature is of the form' T(x) = A sin wx, q(x) = Aw 1 C (caif cos wx + w sin wx \i. ap2 + w2 ai2 + cef2 + w+ / 1 (36) Yih and Cermak consider the case where the temperature is prescribed on the outside of an insulated pipe or plane duct2'. MODIFICATION OF THE LEVEQLUE SOLUTION Consider the velocity profile u = Y(x)y// and sufficiently slowly with x that v T<< u _T; i.e., if u cY dx let r'(x vary -= x )Y then by continuity y v = JJu dy =r'(x) y2 oTe x eu The energy equation is written. d-Z should be small.'Jx UyxT a=2T u = a d 2 or x) jT or - =x a d2T y. d y2 (37) 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - Define x dS =/a dx/1x) or S = (_adx (58) Then: dT _ 1 82T S y aH A solution to the above is: T-T= C1 (eca5io T - Tw a3 T - T 1d satisfying the boundary conditions T = Tw as y = o T- T as y - o9 The heat flux at the surface y = o Y y =: kCl (Tw - T The constant C1 is the reciprocal of _3 e dca = 0.89297 0 W 7 0 w = y(95)-1/3 X > 0 x > o or x- o09 y > o. is given by,)(95)-1/3. (39) (40) which satisfies the requirement that T -- T, as y -- c. The heat flux is given by: q(s) = (Tw - T, ) k S-1/3 91/5 (0.89297) or (41) q(, X) = (Tw - To/) kl/3 -1/3 913 (0.89297) 7t) (42) 11

-- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN - The unit thermal conductance is thus h(f, x) = k~ -I/3a-1/3 91/5 (0.89297) xix~ dx t. -1/ 7~xxt) (4.) For the case where the wall temperature varies with x, we have then q(x) 1/ (-/3a.8/ 91/3 (0.89297) =, f(x)t 1/3 o ) r(x) d.T(f) (44) ka pCp The Leveque solution is useful as an approximation of the Graetz solution very close to the descontinuity in temperature. For example, near x = g = 0 a large number of terms should be taken to get accuracy in either the Graetz or Poppendiek series solutions. The Poppendiek generalization of the Leveque solution may be used in the thermal entrance region of a liquid metal system. APPLICATION TO TEE PROBLEM OF DISTRIBUTED HEAT- SOURCES The integrating kernels in the right-hand column of Table II permit the investigation of several problems of practical interesto For example, consider the flat plate in laminar or turbulent flow with a heat source, q, which is uniform between points x = a and x = b and zero everywhere elseo Then for x > b we have: T(x) - Too S=b - bg(F, x) q(J) d 5= a a = of g(f, x) do - p g(f, x) do 0 0 (45) ('46) Figs. 1 and 2 show the results-of the integration for the flat plate in laminar and turbulent flow. 12

.7.6 X 4- -I T(X)- TC -1/3 -1/2 f ~ 2Pr Rex30 3(1/3) (2/3)!0.304k q a f =. TOO U|.4 001, I 17,7.77z _za2~ b Xx - -. l HEAT FLUX ~ qo x<a, x> b a<x<b I X/X FIGURE I. EFFECT OF A HEAT SOURCE ON A DOWNSTREAM TEMPERATURE. LAMINAR BOUNDARY LAYER.

1.0.9.8 _< X'6.- *7 +-.7.6.5.4.99 AT x_.c -1/3 -0.8 T(x)To 32 Pr Rex qox ) 00 39 ( 32/39)!(7/39)!0.0288k af ) f U l l x, x f —— o --— ~- FLU x<a b <x HEAT FLUX:= 0 X < b<X f~ ___ _q o Kx<b f~~~~~~~~~~~~~i):o~~~~~~~~~~~~~~~~~~A&aA A& I 1.5 2 3 4 5 6 7 8 9 O1 FIGURE 2. X/x EFFECT OF A HEAT SOURCE ON A DOWNSTREAM TEMPERATURE. TURBULENT BOUNDARY LAYER.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN As an example of the application of the above data, Figs. 3 and 4 show a comparison of temperature distributions when a uniform heat source is used and when a discontinuous heat source dissipating the same power is used. The discontinuous source is taken as twice the average power forna<x <(n + l)a when n is even and zero when n is odd, (a = constant)o Another interesting case is that of the line source of heat, such as is approximated by a fine wire. For this case, the interval a b in the above integrals shrinks to zero, but the source intensity increases to keep qdg = Q, constant. Therefore: T(x) - To = g( x) Q(n) n (47) for n > x, g(n, x) = 0. Figs. 3 and 4 show the temperature distributions which result when the limiting case of the line source is approached. COMPARISONS WITH EXPERIMENT The integrating kernels proposed by Seban12 and Rubesin2 have been favorably compared to the data of Scesa 12,20 In these cases step functions of temperature were imposed on a flat plate with a turbulent boundary layer parallel to an air stream~ Maisel and Sherwood18 experimented with mass transfer and, as indicated in Table II, obtained results which were close to the prediction of Seban12. The data of Spielman and Jakob22 for evaporation of water are also in good agreement with Seban's equation. Flow In a Tube The doctoral thesis of Kroll23 yields data on heat transfer in a pipe with laminar flow and a variable wall temperature. Fig. 5 shows typical variations in q(x) and Tw(x) measured by Kroll for three runs. Fig. 6 shows the calculated value of q(x) using the data on Tw(x) and graphically integrating Eq 48. 153

I~ " — 24 a.:2.5 7 <- I \ Ir~l@l I L - / 2.'' l (coSE (2 w m / 5 vr) > > / \-\V (STRIPS) CSE (b) uj x KO f/ } ] (WIRES) CASE (c) —-..2.3.4.5.6.7.8.9 11. x/L FIG. 3. EFFECT OF NON UNIFORM HEAT FLUX ON TEMPERATURE OF FLAT PLATE IN LAMINAR FLOW. Effect of changing the distribution of a given heat flux to a length L of laminar boundary layer. In case (a), a constant heat flux is used. In case (b) the heat intensity is twice that of case (c), but the heat is applied to half the area, alternating with regions of zero heat flux. Case (c) considers line sources of heat such as fine wires at x/L = 0 0,2,, o.6, and 0.8. In all cases qavL = ) q(x)dx. U

U) LLJ On cr + 5 h_ 5 < ao n 2 4 0 CJ r I yI I V)' -- o 2 z FIG. 4. FLAT x/L EFFECT OF NON UNIFORM HEAT PLATE IN TURBULENT FLOW. FLUX ON TEMPERATURE OF The effect of a nonuniform heat source on surface temperature for a turbulent boundary-layer. (See Fig. ) for symbolso)

[ -- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN,q(x) = 4 L Cie-ai x d 1,Y2,.I O- -1i n "d T ( 9) (48) The points shown in Fig. 6 are measurements by Kroll. Fig. 7 shows the computed local'Nusselt modulus compared to the measured values. The computed values are taken from Eq 49: hD 4 x k - Tw - Tg /2 w g 1,2^5 Cie - eaiPB1 s= dT,(3). w (49) The measured values are taken from: bD =.(x), (50) k (TW -Tg) k where q(x) and (Tw- T ) were obtained by Kroll. w g The agreement in runs 46 and 22 is satisfactory,. The deviations in run 54 are attributable to the changed fluid properties caused by the large wall-temperature change. The better agreement in Fig. 7 is due to the fact that the "local" thermal conductivity.was used in Eq 50 in reducing the experimental points to the nondimensional Nusselt modulus, thus in a measure compensating for the changed fluid properties, BOUJ.ARY AIY PS PRODUCQBED B JETS OF AIR DISCEHAGED PARATIE. TO TEE SUTRFACE'Wieghardt27 has presented data for the temperature distribution downstream from a slot of hot air, as in-Fig, 8,* The results far downstream of the slot were represented by the empirical relation: - The following comparison was suggested by EOIRG. Eckert. 14

(A) DISTANCE FROM TUBE INLET, INCHES 1000 - R RUN 54 600 400 - RUN 46 RUN 22 200... 0 2 3 4 5 6, (B) DISTANCE FROM TUBE INLET, INCHES FIG. 5. (A) VEAT FLUX. (B) TEMPERATURE OF TUBE WALL. AT TUBE ENTRANCE, RUN 46, Red=489, RUN 22, Red= 1,390, RUN 54, Red= 2,850.

I — Uf.:OOI __ __ -r 800 -J F 66 00_COPS__ _ A__I _\ __,z 200 __ 0 0 I 2 3 4 5(-6.8 AT)6 DISTANCE FROM INLET, INCHES FIG. 6a. COMPARISON BETWEEN ANALYSIS AND EXPERIMENT. THE POINTS ARE FROM REFERENCE 23, THE SOLID LINE IS THE INTEGRATION OF EQUATION 47 AND THE DASHED LINE IS THE PREDICTION BASED UPON "ISOTHERMAL WALL" EQUATIONS. 7

20C It^ IOC n 160 i1 r I 120 x D 100 i8 LLJ I 80 -— f — - i. — -- ~___ ~___ _____RUN 22___ 0. )0 0 0 __.___. —_... I L I L L i: i 400 0 2 3 4 5 6 7 DISTANCE FROM INLET, INCHES FIG. 6b. COMPARISON BETWEEN ANALYSIS AND EXPERIMENT. THE POINTS ARE FROM REFERENCE 23, THE SOLID LINE IS THE INTEGRATION OF'EQUATION 47 AND THE DASHED LINE IS THE PREDICTION BASED UPON "ISOTHERMAL WALL" EQUATIONS.

9000 pu. 8000'r I I 7000 L__ __ / I - 6000 - 3000 u. 5000, 4000 __ ^_ _^" ^^ _ __ - - 3000.. 0 I 2 3 4 5 6 7 DISTANCE FROM INLET, INCHES FIG. c. COMPARISON BETWEEN ANALYSIS AND EXPERIMENT.'THE POINTS ARE FROM REFERENCE 23, THE SOLID LINE IS THE INTEGRATION OF EQUATION 47 AND THE DASHED LINE IS THE PREDICTION BASED UPON "ISOTHERMAL WALL" EQUATIONS. f

0 RUN 54 X RUN 46 - EQUATION 48 D 3 X =\: I,.J X46 FIG. 7. MEASURED AND COMPUTED NUSSELT MODULUS. I __________________ 0I 2 3 4 5 - 0 I 2 3 4 5 DISTANCE FROM TUBE INLET, INCHES 6 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN T -T0_ 21.88 (51). Ts - T S Up Except for the distLurbance to the boundary later caused by the jet, the situation studied by Wieghardt is similar to the case of heat source of strength UspsSCp (Ts - TA) placed at the origin of the plate. Using the integrating kernel on line VII of Table II and setting q(f) d= Q=U ssp, S Cps (Ts - T), = 0 we have: T (x) - To -1/3 -o.8 S (Ts -T) 8 2 r.. ReP R Sc,.q (Ts. 195 (32/39)' (0.0288k) (52) A minor rearrangement yields 0.2 / \-o.8 Tw(x) - TG= 28 Pr Prs (ks/k Res x U p^(5) - T - 32/39) (7/39) (0.0288k) sp where Re = UsPsS s For the range of temperatures used by Wieghardt Pr = Pr = 0.72, (/4s) (ks/k) = 1. (54) Thus Tw(x) - T- _ 46 0.2!_p__ -0.8 Tw-,) ~ 4.62 Res Xsps (55) Ts - T- s Us / 55 The data given by Wieghardt did not include the actual temperature of the air during the tests, but if the laboratory air temperature is guessed to be 20~C, then the "slot" Reynolds modulus, Re, may be cons pulted. The tests thus appear to have been run. at values 3760 e Res 12,630. Eq 55 thus becones t24 Tw(x) - Toto_ 0'5 (56) Ts - TCo C30oUoS s./ Part of the discrepancy between Eqs 56 and 51 may be ascribed to the fact that the issuing jet has a unifrom temperature profile, whereas a turbulent boundary layer heated from one side has a profile similar to the 15

U FIGURE 8. DISCHARGE SLOT USED BY WIEGHARDT IN TESTS OF REFERENCE 26.

L - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1/7-power law. For the same enthalpy in the boundary layer, the wall temperature will be higher in the case of a heat source than for a jet. FRICTIONAL HEATING As mentioned in the beginning of this paper, the effects of viscous heating may be taken into account by adding the "adiabatic wall temperature" to the solutions obtained when viscous heating effects have been put equal to zero. In the case of flow in the absence of a pressure gradient the adjustment is most simply accomplished by replacing the free-stream temperature by the adiabatic wall temperature. When the adiabatic wall temperature varies, as it does over a wedge or where there is a pressure gradient, the effect of frictional heating may be introduced by writing Tw(x) - Tad (x) in place of Tw(x) wherever it appears in the equations. For example, a nonisothermal surface in frictional flow has heat flux related to wall temperature by. x q(x) = h(, x) d [T(T ) - Tad()] (57) f = o If the fluid properties vary significantly, the above result is not valid. i

ACKNOWLEDGEMENTS The author would like to express his appreciation to Professor Ruel V. Churchill for reviewing the paper and suggesting some of the methods of analysis. Professor James Neal Addoms of Massachusetts Institute of Technology read the manuscript and offered several helpful criticisms which were adopted. The author wishes to give specific thanks to both Professor Addoms and Dr. Co L. Kroll for permission to use data from an unpublished thesis. 17

APPENDIX A The Stielties Integral The more commonly known Riemann integral, which for most engineers is pictured as the area under a curve of the integrand plotted against the variable of integration, does not yield a value if the integrand is not "well behaved". The Stieltjes-type integral is so defined that for "well behaved" functions the result of the integration is the same as ordinary or Riemann integration. The Stieltjes integral does have an advantage, however, in providing (by definition) an integral in some cases where the Riemann integral is ambiguouso It is a shorthand notation for expressing a sum plus an integral. A discussion of Stieltjes integrals is contained in Reference 20. In this appendix we give only the interpretation necessary to the discussions of this paper Consider, for example the integral, I, I = ff(x) dg( dx dx So long as f(x) and dg(x)/dx are well behaved, no difficulty arises in finding the Riemann integral, I. Now let f(x) = X and suppose g(x) is given by the graph below I g(x) J f x A discontinuity in g(x) occurs at9. Everywhere except at x =, dy(x)/dx Co. If one attempts to form the Riemann integral for this case, the immediate vicinity of does not yield an unambiguous "area under the curve" of f(x) Cdg(x)/dx. plotted versus x. 18

The question is resolved as follows. Everywhere the integral is evaluated as an ordinary or Riemann integral. At x = -we note f(x) = f(') and consider ~ $ + f.(x) -Xo) dx dx, 6+ ( = f () f' X) dx )' dx 9 _ - f(5) dg SI f(g) g(.) -6(?, Thus the Stieltjes integral is seei to be the. ordinary Riemann integral plus. contributions which occur whenever g(x) has a discontinuityo Therefore, we interpret the Stieltjes integral as: Xh(Sx) d Ty) = 3 h5,x) d, 5 3-~ S — o (Stieltjes) (-Riemann). +L h(gi x) all jumps in T (at 5 i) T ( f) - T(i.] i.e., a Riemann integral plus a summation. 19

REFERENCES 1. Rubesin, Morris William, "An Analytic Investigation of the Heat Transfer Between a Fluid and a Flat Plate Parallel to the Direction of Flow Having a Stepwise Discontinuous Surface Temperature", MS Thesis, University of California, Berkeley, 1945o 2. Rubesins, Morris Wo. "The Effect of an Arbitrary Surface Temperature Variation Along a Flat Plate on The Convective Heat Transfer in an Incompressible Turbulent Boundary Layer", NACA TN 2345, April, 1951. 3. Johnson, H.Ao, Rubesin, M.oWo Sauer, F.M., Slack, E.G., and.PossnerIL., "Bibliography of Aerodynamic Heating and Related Subjects", AAF Techo Rept. 56339 September 10,1947 4. Chapman, D., and Rubesin, MW.,. "Temperature and Velocity Profiles in the Compressible Laminar Boundary Layer with Arbitrary Distribution of Surface Temperature", Jour. Aero. Sci. 547, September, 1949. 5. Levy, Solomon, "Heat Transfer to Constant Property Laminar Boundary Layer Flows with Power Function Free Stream Velocity and Wall Temperature Variation", Jour. Aero. Sci., 19,.No.5, May, 1952. 6. Jakob, M., Heat Transfer, Volo I, Wiley, 1949, page 451. 7. Lighthill, M.J., "Contributions to the Theory of Heat Transfer Through a Laminar Boundary Layer", Proc. Roy0 Soc. Londo Ser. A. 202, 359-377 (1950). 8. Lipkis, Robert, "Heat Transfer to an Incompressible Fluid in Laminar Motion", Engineering 299 Interim-Report, University of California, Los Angeles, (Unpublished), June, 1949. 9. Boelter, L.M.Ko, Cherry, V.H., Johnson, H.A., and Martinelli' R.Ci "Heat Transfer Notes" University of California Press, 1946, page x-38. 10, Seban, R.A., "Calculation Method for Two Dimensional Laminar Boundary Layers With Arbitrary Free Stream Velocity Variation and Arbitrary Wall Temperature Variation", Institute of Engineering Research, No. 12, Series 2, University of California, Berkeley, May 10, 1950. 20

11. Schlinger, W.G., Berry, V.Jo, Mason, J.Lo, and Sage, B.H. "Prediction of Temperature Gradients in Turbulent Streams'", General Discussion on Heat Transfer, London Conference, September 11-13, 1951. Proceedings to be published by I.M.E. and A.S.M.E. (Section II). 12, Scessa, Steve, "Experimental Investigation of Convective Heat Transfer to Air from a Flat Plate with a Stepwise Discontinuous Surface Temperature", MS Thesis, University of California, Berkley, 1951. 13. Tribus, M., Comments published with paper cited as Reference 11 above. 14. Jakob, M., and Dow, WoM., "Heat Transfer from a Cylindrical Surface to Air in Parallel Flow with and without Unheated Starting Sections", Trans. ASMEo 68, 1946. 15. Eckert, E.^REG., Introduction to the Transfer of Heat and.is McGraw Hill, 1950, page 88. 16. Bond, R., "Heat Transfer to a Laminar Boundary Layer with Nonuniform Free Stream Velocity and Nonuniform Wall Temperature", University of California, March 1 1.950. 17. Margenau, H. and: Murphy, GoM. The Mathematics of Physics and Chemistry, D.Van Nostrand, 19435 page 507. 18. Maisel, DO.S, and Sherwood, T.K., "Evaporation of Liquids into Turbulent Gas Streams", Chem.Eng Prg. 131, March 1950.,9o Hartree, D.RE, " On an Equation Occuring in Falkner and Skauls Approximate Treatment of the Equation of the Boundary Layer", Proc. Camb. Yhil. Soc. 335, 223-239, 1937. 20. Widder, David V^, Advanced Calculus Prentice-Hall, Inc. New York,19k7 21. Pipes, Louis A,, Applied Mathematics for Egineers and Pysicists McGraw-Hill, 1946, page 555. 22. Spielman, M., and Jakob, M,"Local Coefficients of Mass Transfer by Evaporation of Water into an Air Jet", A.S.oME. Paper No.52-SA-1 23. Kroll, C.L., "Heat Transfer and Pressure Drop for Air Flowing in Small Tubes" Thesis, Chemical Engineering Dept. Massachusetts Institute of Technology. 1951 24. _Poppendiek, e.F.F, "Forced Convection Heat Transfer in Thermal Entrance Regions - Part I", Oak Ridge National Laboratory, TennesseeORNL 913 Series A Physics March 1951. 21

25, Poppendiek, H.F. and Palmer, L.,D..tForced Convection Heat Transfer in Thermal Entrance Begions - Part II", Oak Ridge National Laboratory, Tennessee, ORNL 914 Metallurgy and Ceramics. March 1951. 26. Yih, C.S. and CermakJ.E., "Laminar Heat Convection in Pipes and Ducts", Civil' Engineering Dept. Colorado Agriculture and Mechanical College, For Colollins, Colorado. September, 1951. (ONE Contract No. N90 nr 82401, RE 065 - 071/1-19-49) 27. Wieghardt, K,, "Hot Air Discharge for De-Icing", AAF Translation F-TS 919RE December 1946, Central Air Documents. ATI No. 24 536 22