NATURAL CONVECTION IN A RECTANGULAR CAVITY WITH INTERNAL HEAT GENERATION Willy Smith Frederick G. Hammitt* The University of Michigan Department of Nuclear Engineering Laboratory for Fluid Flow and Heat Transfer Phenomena Internal Report 05031-2-I April 1965. o* Research Associate Nuclear Engineering Department, The University of Michigan. ** Professor, Nuclear Engineering Department, The University of Michigan.

AKNOWLEDGMENTS Financial support for the work described in this report was provided by the Michigan Memorial Phoenix Project and the National Science Foundation. ~sM<t~

-i1. Introduction Contributions to the field of natural convection have been relatively limited in recent years, due primarily to the inherent complexity of the applicable non-linear differential equations and the scantiness of important applications. Hovever, high-speed digital computers have made practical the finding of accurate enough numerical solutions, while important applications have come into being both in nuclear reactors and space technology, thus reviving the interest in the subject. The present work considers a closed rectangular cavity with internal heat generation, cooled along a pair of vertical side walls, and simulating the channels in an internally-cooled homogeneous nuclear reactor. Although the experiment used wiater, the theoretical analysis considers fluids of arbitrary Prandtl number. A formal relationship between the applicable non-dimensional parameters, following the one-fourth power law typical of natural convection, was obtained. Good agreement between experimental and theoretical reaults was found. 2. Theoretical Analysis The conventional fluid fl-ow equations of continuity, NavierStokes, and conservation of energy, properly modified to include a heat source term, provide the starting point. The following assumptions are made: i) the fluid is quasi-incompressible, i.e., the density is constant except for its temperature dependence in the body force per unit mass term, where: F = - LeA(T-To).

-2ii) internal heat dissipation by viscous forces is negligible compared to heat source input. iii) viscosity, thermal conductivity, and specific heat/are constaht and evaluated for a conveniently defined mean temperature. Then^: div V = 0 -= - g3(T-To) - grad p +72V DT _721:.+ K Dt -KT+ Q The x-axis is vertical, and the geometry of the cavity such that one of the horizontal dimensions, z, can be assumed infinite. The equations are normalized by simultaneously making: x = xb, y = y'a, v = v' t = ta2 u = u, T-To=T'(Tm- To) T'T, Kb('.~2 a = pt pVJKh2 k/NT p = p, ~_ pVKh2 P Q =, q' = q, kAT Nu a4 a where the primes indicate non-dimensional variables, givingl: bu v= +- 0 bx by lDu_ = RaT-_ + (a)2 b2u 2u a-Ut'B^x - + y-? 1Dv b -2 b 2 b b2 (1) DT ba22T _ b2T

-3where the primes have been dropped for convenience. Further simplifications are achieved by applying the boundary layer approximation, i.e., assuming the gradients of velocity and temperature in the vertical direction x are small compared to the same gradients in the horizontal direction y. The approximation is improved when ()2 < 1. The second momentum equation (1) then vanishes, and the remaining equations become: bu Ev bx by 1 Du a RaT - + 2u (2) n S Rt - dx (2) DT = 2T + q Nu Following a procedure similar to Lighthillts although the normalization method is somewhat different, the pressure can be eliminated between eq.(2) and the same equation written at the,all (y= 1), where the boundary conditions are:v = u = 0 (non-slip), and: T a Tw(x), Two being a knowmn all temperature distribution. Then, for the steady-state case: bu bv ~+ o;"0 (3) u &+ < v bu =; Ra(T- T2) + zy2 (4 C( Ix byb Iy2j^ u + v- = - + q Nu (5) bx by by2 These expressions are still complex. A previously successful 2-5 method is to integrate them across the cavity. Assuming synmmetry about the x-axis, i.e., velocity and temperature are even functions, one has: f u dy = 0

-416 =a Ra (1 C bu b2i L 1 u2dy = Ra (T- T)dy + - y21 C- bx o o Iby y 2 -If uT dy t] + q N expressing respectively conservation of mass, momentum and energy for the fluid filling each cross section of the vessel. Following Lighthill2 again, the second derivative is eliminated by writing eq.(4) at the center line of the cavity (y 0O). 1 bu A term of the form - u((ux) appears, which can be neglected if F'randtlts number is large, or in any case by the boundary layer approximation of small gradients in the x direction, and certainly line when the vertical velocity u at the cavity center/is small. The integrated form of the equations is then: 1 j u dy 0 B u 2,ady = Rai (T-Tc)dy + ( (6) x uT dy= () qNu To this point, the analysis has departed from the classical work of Lighthill2 onlyin a different choice of non-dimensional parameters, the inclusion of a non-dimensional heat source term, and the rectangular geometry. A modified Squires method is used now, whereby temperature and velocity profiles consistent with the boundary conditions are assumed and substituted into eqs$(6), Experimentally and intuitively there is a downward moving boundary layer adjacent to the cooling walls, and an upward moving core. The present experimental data indicate that by and large the core temperature is a function only of vertical

-5position, i.e., Tc = Tc(x), and in fact, the core is defined here as that region over which the transverse temperature gradient is zero. As substantiated by the experimental data, the temperature profile has a maximum within the boundary layer, and then decreases monotonically toward the wall, as shown in figure 1. A suitable expression for T(x,y) can be shown to bel T(x,y) T + T (y-l)(y2 + Gy + H) (7) w n (l-d)3 for d y l, and where: G = G(d) = n(l-d) - 2d H = H(d) = n(l-3d+2d2) + d2 n being a function of d to be determined. For Oy Cd, the temperature is corntant; thus: T(x,y) = T(x) 2 5 The present work, then, differs from prior studies25 by the use of this maximum in the temperature profile. Experimental indication of such a maximum had been observed previously by 3 one of the authors in cylindrical geometry, but was not incorporated in that analysis. The maximum can be suppressed by making n unity in eq.(7). The profile then reverts to a shape 2-5 similar to that used in earlier analyses, but is not exactly the same, being now a third degree polynomial: T(x,y) = Tw(x) + - - (y-l)(y2+ (1-3d)y+ (1-3d+3d2)] wy (l-d)3 The same general behavior has been previously obtained by a 2-5 second degree polynomial5 T(x,y) Tc(x) + (T- T) (lc w ~~~~~ 1-cl~~~m

-n&i The velocity is assumed constant and positive over the interval O y d, implying equal extent for thermal and frictional boundary layers.Physical arguments justify this assumption, although it contradicts an order of magnitude boundary layer analysis. It has been used successfully in preceeding 2-5 6 work, and is also supported by numerical solutions6 for the flat plate case. The present data indicate the validity of the above assumption within experimental error. Moreover, the data show that a maximum also exists in the velocity function, as shown in figure 2 (dotted lines). However, the analysis assumes the simpler velocity profile1 (full line): 0 y d: u(x,y) = r(x) l-d dTye l: u(x,y) = rel a( r)n [ l+s(y-l)]} The longer a portion of the fluid remains in a certain location, the more its temperature will be increased because of the internal heat source. It is then reasonable to assume that the location of the temperature maximum and the zero of the velocity function coincide, The relation between n and d is then found to be: n = 19d+ 29- 3 V3 (1-d) (5d+ll) 16(d+ 2) By substitution of u(x,y) into the first of equations (6), it. expression for s(d) is found: s(d) = -4(d+2 (d-1)d But now there are three unknown functions: d(x), T (x), and r(x), with only two equations available. A third equation is obtained3 by integrating the energy equation (5) over the

-7wall r —- - Tc Ti; — ^ eq.(3.29), | |" — eq. (3.29.2) Tw d m 1 (1586) Fig.l. Assumed Temperature Profiles u, wall ---------— H, -~ r \ eq.(332) i dy Fig\ 2 / A u e V l(15o7 ) Fig.2, Assumed Velocity Profile

-4central core only (-d y d), and assuming that no fluid is exchanged between core and boundary layer. Then: b J uT dy = q Nu d(x) ) Substituting the velocity and temperature profiles into the second and third of equations (6), and into (8), one obtains after considerable amount of algebra: i d dr2M(d)) = a Ra(T- T)P(d) + rR(d) (9) d (r(Tw-Tc)N(d)] ) (Tw-Tc)S(d) + q Nu (10) dTc q Nu (11) cdx r where the expressions for the coefficients depend 6n the temperature profile assumed (see Table I;. These equations must satisfy the following boundary conditions at top and bottom of the cavity: Tc(l) = 1, T (-1) = 0 r(~l) = 0 (12) d(tl) = 0 The first follows from the definition of the non-dimensional temperatures. The second is the non-slip condition, while the third states that the boundary layer thickness must become zero at top and bottomI'3'4. This is due to the requirement that at either place the mixed-mean core and boundary layer temperatures must be equal. The heat distribution q is arbitrary, the strength of the source being proportional to the Nusselt number. However, for the

-9Temperature profile characteristics Has maximum No maximum No maximum n " 1 2nd. degree n / 1 3rd. degree polynomial polynomial M(d) d2 + 1d +4 35 1 - d.................... N(d) |37d+!l-56n(2d+l- | 2 ( + 2d) 420 n 28 15 p (ld)n- l 1) 1- d l'd 12n 4 3 R(d) 6(1+ d) __, _ _(1- d)2 S(d) 21+ 2n - 2 n(l-d) 1-d 1-d Table I. Coefficients of the basic equations corresponding to the different temperature profiles.

-l0present work, uniform heat distribution was assumed (q - 1). The wall temperature distribution can also be selected arbitrarily, with the only restriction that TW(-l) = 0, which results from the definition of non-dimensional temperature differential. Only linear variations of Tw(x) were considered: Tw(x) -(x+ 1) (13) where tn>l. The case m co. corresponds to constant wall temperature. 3. Numerical Solution The set of equations (9),(10) and (11) is too complex for analytical treatment, but can be successfully attacked by numerical methods. In the present cases, satisfactory solutions were found using a fourth order Runge-Kutta integration procedure in a 7090 IBM digital computer. An additional difficulty is that the nunerical integration can be started only if a well defined point is knovn. In the case under analysis, the points x = +1 where the values of the functions are given by the boundary conditions (12) are also singular points. This difficulty was circumvented by replacing the differential equations by a set of algebraic equations which will provide an approximate solution valid only in the neighborhood of x= 1, the accuracy of this asymptotic solution increasing as x= is approached. The resulting equations are still complicated enough to require iteration methods and a computer 1 For given Prandtl number and specified wall temperature distribution,a value of Ra was selected and a reasonable corresponding value of Nu assumed on the basis of experimental or

-11previous numerical results. Starting from the top of the cavity, the asymptotic solution, valid only about x= 1, is used to determine the initial values in terms of the input parameters Ra, Nu, and m. The Runge-Kutta method is then applied, and progresses toward the bottom of the cavity (x=-1). The increments selected for the independent variable x are very small at the beginning of:.the calculation, as well as when the bottom is approached, and somewhat larger in the middle region. The exactness of the solution is increased with decreasing values of Ax, but by the same token the computing time is increased, so that a suitable comprOmise must be made. lWhen the bottom of the cavity is approached, if the boundary cnditions (12) are not satisfied within a preset value, the Nusselt number is modified accordingly, and the computation repeated starting again from the top. Two types of solutions were analysed: a) Large Prandtl number: Considerable simplification is achieved since the:'left-hand side of eq.(9) can be altogether neglected. This equation provides then an explicit expression for r(x), which when substituted into (10) and(ll),and &lso using (13), gives a system of two differential equations in two unknowns: dFi- a Ra(Tw- TC)2 P(d)N(d)) (Tw- Tc)S(d) + Nu a c R(d) d(T, -Tc) 1 + b Nu R(d) dx 2m a Ra(Tw- Tc)P(d) These equations were numerically solved following the procedure described above, for values of m ranging from 1.5 to 1000. The results, in terms of the non-dimensional parameters, are presented in figure 9.

-12b) Arbitrary Prandtl number: The solution for arbitrary values of QT presents greater difficulties. The Runge-Kutta method requires not only knowledge of an initial point, but also that the equations.can be cast into the form: d' = fl(x,d,Tc) r' = f2(x,d,Tc) T = f3(x,d,Tc) where the derivatives with respect to x are expressed as functions of only the dependent variables. This was accompl!ished after some manipulation. The input parameters for the theoretical solution are the Prandtl number O-, characterizing the fluid, and m, fixing the slope of the imposed linear wall temperature distribution. For each pair of values T, m, many pairs of corresponding values of the non-dimensional parameters Nu and.a Ra are calculated, b and in each computation, values of the functions d(x), r(x), and Tc(x) across the cavity were obtained. Values of 0 considered range from 0.01 to 10. The results are presented in two ways: i) in terms of the parameter m, for fixed Prandtl number; and ii) for different values of r and given wall tbmperat.tre distribution (i.e., fixed m). For the first case, figures 3, 4, and 5, the curves refer to large 0 > ( 10), and correspond to a constant value of Ra. Irhen the wall temperature has large slope (m= 1.5), the thickness of the boundary layer is quite uniform far from the ends of the cavity, and the core velocity r(x) also remains fairly constant. As the constant wall temperature case is approached (m- o ), the changes in boundary layer thickness become greater, with

0.95 I -.f M=2 c) C,) Lu1 C/) 3 u w l 0r 0.9 0 MA I = 10 o 0 I \ ~~~~~~~~~~~m s100 M 1000..0 -0.8 -0.6.00.4 -0.2 o Q. NON- DIMENSIONAL COORDINATE x Fig.3. Theoretical core t-, 9 ORNTE Fig3 Theoretical Core thickness for variable wall temperature distribution and Constant Rayleigh number.

00 > B 591000 o >.608 40 0 z 20 o 1m589 z -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 08 1.0 NON - DIMENSIONAL COORDINATE, x Fig.4. Theoretical core velocity for variable wall temperature distribution and constant Rayleigh number.

' 1.0:Q: ~c 0.8 1L0.6 8 0.4 0 m=-0.8 -0.6 -0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 LU 0.2 z o 1590 0NON-DIMENSIONAL COORD, -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 NON-DIMENSIONAL COORDINATE, x Fig,5. Theoretical core temperature for different wall temperature distribution and constant Ra.

-16increasing thicknesses appearing toward the bottom of the cavity. Core velocity varies in a similar manner with m, while cbre temperature is not so sensitive to changes in wall temperature distribution. In the second set of curves, figures 6, 7, and 8, corresponding to constant wall temperature and fixed Rayleigh' number, the boundary layer thickness is rather constant for large r, although it tends to increase toward the bottom. As r decreases, the trend reverses and the thicker parts of the boundary layer appear close to the top. Core temperature shows little change for different values of o. The following relationship among the non-dimensional parameters was found: 1/4 Nu = C(m,Q)(| Ra) (14) This expression not only verifies that the one-fourth power law, known to be associated with other cases of natural convection, is also valid for the present problem, but also allows easy comparison between the present results and those of other investigators, as shown in fig.9. Theoretical results by Chu5 indicate that Lighthill's2 "large" value of.' is about 10, and for this value thhir solutions coincide. Chu's solutions cover the range 0.02< T a 10, and apply to cylindrical geometry similar to Lighthill's, but there is no internal heat generation. Previous work by Hammitt3,4,7 was performed in closed cylindrical geometry with internal heat generation. The * Cylinder with reservoir on top.

I -.- l U s I ~ ~ - — I7 -^ I 0.95 0.90 oc =1 0.85'0 Cl) 1.J c) co z 0.80 C-) x uJ 0.75 0 C-) 0.70 1591 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 NON-DIMENSIONAL COORDINATE, x Fig.60 Theoretical core thickness as obtained from the general solution for ib.e CT, constant Ra and constant wall temperature.

100 ~ —0' = I0 0.-J 6 3:6- / —': I -.0 - 0 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 NON-DIMENSIONAL COORDINATE, x z 0 I Fig. 7. Theoretical core velocity as obtained from the general solution for variable a-, constant Ra and constant wall temperature.

0.8 0. LU Ir=0.01 O. -'O */ 1593 Z 0.4 1.0 -0.8 - -0.4 -0.2 0 2 0.4 0.6 1.0 -1.0 -0.8 -0 -0.4 -0.2 0 0.2 0.4 0.6 OB 1.0 NON-DIMENSIONAL COORDINATE, x Fig.8. Theoretical core temperature as obtained from the general solution for variable CT, constant Ra and constant wall temperature.

Nu 100 u1TuLS SOWTI M sLUT- {#D. - um 100'-> o CYLOfOCAL GEOMETRTY LIH 1 ~^ CNh onLUTIO IOfn 0 - \ UMwmj L CAS FO HM Mt FOR WATER./_ a Ra0 Fig.9. Summary of theoretical and experimental data. i~~~~~~~~~~~~~~~~~~~~~~~~~~~;~

-21theoretical solution for large o agrees well with the solution found in the present work a9lso for large C. For a given a Ra value, both solutions show lower Nu than those found by Chu5. Since the Chu and Lighthill solutions are in agreement, and Hamr.itt reproduced Lighthill's results in a closed geometry by assuming the heat source to be an infinitely thin disc at the bottom of the cavity, it appears that the reduction in Nu is due to the inclusion of the volumetric heat source. Solutions for closed cylindrical and rectangular geometries are almost coincident, indicating relative independence orn geometry. Present theoreticalresults for large and arbitrary Prandtl number are also shown in fig.99 The solutions are for constant wall temperature. For r>10, the lines are almost coincident, althpugh the solution for Co = 10 is still below that obtained for "large" Prandtl number. This discrepancy results from the use of different temperature profiles: for solution the large T /, a simplified temperature profile was used, while for the arbitrary Prandtl number solution a temperature profile with a maximum was assumed, in accordance with the experimental observations. Thus, significant changes in the results have been brought about by the adoption of this more realistic temperature profile. Values of the coefficient C(m, <) appearing in equation (14) have been calculated for different oc and various wall temperature distributions, as sumr~arized in Table II.

-220.01 0.1 i 1 6.58 10 o 1.42 l.1305 1.44.1338 1.47.1404 1.48 *1417 1.50 _____.1073 C.1445 1.o52.1490 1.55.1539 1.58.1593 1. 62 9.1654 1.63.1665 1.7$.1344 1.86.1411 1.95.1482 2.00.1537.2072 2.02.1539 2.10.1585 2.20.1636 2.25.1669 3.50.2693 5.00.2286 10.2521.3215 100.2732.3344' $1000.1208.1820.2585 o.2815.2863.3449 C(m, o1. * correspond to the solution for "large" Prandtl number.

4. Experimental Results A schematic and a photograph of the experimental facility designed to verify the theoretical treatment are shown respectively in figures 10 and 11. Essentially, it consists of a rectangular cavity (24" high, 8" wide, 36" long) closed at top and bottom and at the smaller vertical sides by plexiglas plates. The larger vertical.walls are 1/4" thick brass plates. An A.C. voltage applied across these plates results in heat generation in the cavity due to ohmic resistance. Heat is removed from the cavity through the same brass plates, which are also the interior walls of two vertical cooling tanks.ThE; fluid used was tap water, which provided sufficient electric resistivity for the purpose. An L-shaped probe electrically insulated from the fluidc except at the tip where a small thermocouple bead is located, is inserted through the plexiglas top. It can be moved verti. cally and/or rotated to reach points from the wall to beyond the center line and from top to bottom. The position of the tip of the probe, in terms of non-dimensional distance, can be estimated with an error not larger than ~ 0.01. A complete temperature mapping of the cavity at power levels ranging from 2.2 x 10-3 to 4.3 x 10-2 watts/cm3 was obtained. All temperature measurements, more than 2000 in number, were very consistent and reproducible, and were obtained with a precision of the order of i 0o05~F. Fig.lZ shows typical experimental results corresponding to Q - 1.02 x 10-2 watts/cm3. The temperature profiles were found to have in all cases the shape assumed for the theoretical

PLEXIGLAS TOP VARIAC T 1 L A POTENTIOMETER'~~~~~~~~-~~~- -~"\~~~~~~ * — 220 l1~ l1. l__._\.+ | L TRANSFORMER BRASS PLATES o MAIN SWITCH -T 480 VOLTS LINE PLEXIGLAS BOTTOM T -THERMOCOUPLES IROTAMETER COOLING WATER _= —.___=~_ 1240 Fig.10. Schematic of rectangular geometry natural convection facility.

PLEXIGLAS TOP VARIAC l I 11 I I I A ET H 1 C OIN \TRANSFORMER BRASS PLATES " Schematic of rectangular geometrMAIN SWITCH 480 VOLTS LINE PLEXIGLAS BOTTOM T - THERMOCOUPLES IROTAMETER COOLING WATER 1240 Fig.10. Schematic of rectangular geometry natural convection facility.

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-26mV~F -75 X 0.916 1 I WALL 1.20 o I \I X 0.666 \ 1.10 X 0.33 0' LUIO / jry~ \ r. oX -0.333 I.,1.00 - 7 0.95 \ 0.85 C.X " O - I 1595 NON-DIMENSIONAL DISTANCE, y Fig.12. Experimental temperature prffiles for a Dower level of 1,02 x 10 watts/cm3.

-27analysis, showing a distinct maximum within the boundary layer region. Curves Tc(x) for each power level were obtained, and are presented in fig.13, where the range of variation at a given position has been indicated with a short vertical line, and a single curve drawn across those lines. Non-dimensional core temperature at all power levels are practically coincident, and compare well with the theoretical curve, thus supporting the underlying theory. Semi-quantitative velocity measurements were performed by injecting a tracer into the cavity using a similar L-shaped probe terminated in a hypodermic needle. Although these measurements are not so precise as the temperature results, they do compare well with results obtained by a aompletely separate experimental method (explained below) and with the values resaltin from the theoretical analysis. Fig.lJ shows a typical velocity profile obtained by the injection method. Similar plots corresponding to other power levels also show unquestionably the existence of a positive maximum within the boundary layer and adjacent to the core, wiich has not been previously reported in the literatures All the theoretical analysesl'5 have been performed using a velocity profiles without that maximum. The second method of study of the velocity field consisted of taking photographs at equally set time intervals, afte: mall amounts of dye had been injected into the flow. From these photographs, velocities were estimated and found to compare well with the values: already obtained by direct timing of the

Ui.8 w ~50.~~ ~CORE: THEORETICAL —---- lp w.6 EXPERIMENTAL $^ BOTTOM' / s 4 44 i /^^^~~~~~~~~~~~~~~~~~WALL KWu 252 0 I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,B'2/^:::a,^ -~W 5 z~~!^ -V- W-4 950 0. _ 1596 -1.0 -.8 -.6 -.4 -.2 0.2.4.6.8 1.0 NON-DIMENSIONAL COORDINATE, X Fig.13. Experimental core and wall non-dimensional temperatures at different power levels.

-29/"- WALL 2-~~~~~~~~~~ 3820 0o 8 -44 | NON-DIMENSIONA COORDINATE, y -.J -Il -6 -I 8 -18-20 x x- 0 for w - 3.37 x 10 watts/cn3 -22 -24 1597

-30dye motion. The photographs provide a permanent record of the fluid motions, and show how these are affected by power level. The sequence of photographs in Figo.15 were taken at a powrer level of 6.6 x 10-3ratts/cm3, and show that the flow is esentially laminar throughout. In the boundary layer region, some very slow eddying is observed near the bottom of the cavity, while during the same time interval the small amounts of dye injected in the core region have moted upwards without losing their identity. As the power level is increased, some slow eddying appears in the core near the top of the cavity, as shown clearly in fig.16, corresponding to a power level of 1.96 x 10 i2 wtts/ cm3, and as also detected by small fluctuations of large period (about 20 sec.) observed in connection with the temperature measurements. A:nother effect of the increase in the value of the heat source is to sharpen the maximum in the velocity profile. Values of ", Nu and a Ra were obtained at each power level. b Since T is temperature dependent, and the average temperature within the cavity varies with poower level, o' was found to range from 5 to 8, with an average value of 6.58. The other input parameter of the problem, i.e., Rwll temperature distribution, is also a function of power level. However, for all power levels, Tw(x) shows a linear variation for the middle part of the cavity (see Fig.10), in agreement with the theoretical assumptions. The corresponding values of m range from 1.75,2.25 to y. Consequently, all experimental conditions are approximated if the theoretical input parameters are selected to be C = 6.58 and m = 2. A computer calculation with these values was performed, and the agreement between experimental and theoretical results is excellent (see Fig.9). This good correlation is ( I, s7JRa = 1.15 x 108, Nu = 13.3 (*.) 0 = 6.22, Ra = 4.33 x 106, Nu = 16.4

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-33 - unexpected, since previous work3 in a closed cylindrical cavity with internal heat generation, as well as a study5 in a cylinder with an infinite reservoir on top and no heat source, showed discrepancies between theory and experiment. These discrepancies, small when the fluid was water3'5 but large for the case of mercury5, were attributed to turbulence, which existing in the actual experiment was not taken into consideration by the theory. An experimental check of Lighthillts geometry by Martin and Cohen8 did show good agreement for flow presumably in the laminar regime. In the present work, the observed turbulence was very slight. Thus, the good agreement tends to verify that the previous differences were due mainly to unaccounted turbulent effects. 5. Conclusions Natural convection flow in a closed cavity has been sttudied both theoretically and experimentally. The theoretical analysis follows the general lines of Lighthills procedure, but differs from previous work15 by considering arbitrary values 6f the Prandtl number,as well as by the following aspects: internal heat generation in rectangular geometry, and temperature profiles with a maximum. Extensive computer calculations verified the existence of a relation (14) among the non-dimensional parameters that follows the one-fourth power law. The experimental program produced data in good agrement with the theoretical analysis. Tihe temperature mappind.ndicatesthat a maximum in the temperature profile exists as assumed, while the velocity results reveal that also the velocity profile has a positive maximum, heretofore not reported. The flow was

-34m found to be esentially laminar up to a Ra M 4.0 x 108, although for values of A Ra larger than 2.9 x 10 some slow eddying appears in the upper part of the rising central core and in the lower part of the boundary layer. **I**

BIBLIOGRAPHY 1. Willy Smith, "Natural Convection in a Rectangular Cavity", Ph.D. Thesis, Nuclear Engineering Dept., The University of Michigan, 1964. 2. M. J. Lighthill, "Theoretical Considerations on Free Convection in Tubes", Quart, lournal Mech. and Applied Math., Vol.VI, Pt.4, 1953, pp.398-439. 3. F. G. Hammitt, "Free Convection Heat Transfer in a Closed Cylindrical Tube with Internal Heat Generation" Ph.D. Thesis, Nuclear Eng. Dept., The University of Michigan,1957. Also available as Industry Program Report IP-259, The University of Michigan. 4. F.G. Hammitt, E, M. Browre and P. T. Chu, "Free Convection Heat Transfer and Fluid Flow in Closed Vessels with Internal Heat Source", Industry Program Report IP-399 The University cf Michigan. Also available as NP-8780, UC-34, O.T.S., Dept. of Commerce, Washington, D.C. 5. P. T. Chu, "Natural Convection Inside a Circular Cavity", Ph.D. Thesis, Dept. of Mech. Engineering, The University of Michigan, 1961. 6. Simon Ostrach, "An Analysis of Laminar Free-Convection Flow and Heat Transfer About a Flat Plate Parallel to the Direction of the Generating Body Force", NACA TN 2635, February 1952. 7. F. G. Hammitt, " Modified Boundary Layer Type Solution for Free Convection Flow in Vertical Closed Tube with Arbitrarily Distributed Internal H at Source"? ASME Paper No.58-SA-3O 1958; abstractea Mech. Engineering 3, No.8, Aug. 195k, 8. B. W. Martin and H. Cohen " Heat Transfer by Free Convection in an Open ThermosypAon Tube", J. of App. Phys. (London), j, March 1954. ws/65