THE TUNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF TIE COLLEGE OF ENGINEERING NATURAL.'CONVECTION. FLOW. IN LIQUTD-MEAL MOBILb-FUEL NUCLEAR REACTORS Frederick G. Hammitt Associate Professor Departments of Mechanical and Nuclear Engineering Elayne Mo Brower Research Associate University of Michigan Research Institute Dec-ember, 1958 IP-340

TABLE OF CONTENTS Page LIST OF FIGURES............................................. i INTRODUCTION................................................ 1 LIMITING CASES.........00 000.000 O o 7 Ao Bounrdary Layer Solutions - Aqueous Fluids....................................... 7 B. Infinite Length/)iameter Case - Aqueous or Liquid Metal oO...........................O 14 Co Effect of Liquid Metals as Compared with Aqueous Fluids.................o.o... 16 APPLICATION OF RESULTS TO SMALL-BORE LIGAMENT WITH LIQUID METALS........................................ 17 A. Temperature Differertials - Centerline to Wall...............o..... o........O.o 17 B. Wall Heat Flux...................................... 18 C. Velocity.............................................. 20 CONCLUSIONS................................................. O O O 22 APPENDIX.................................................... 23 NOMENCLATURE................................................ 28 BIBLIOGRAPHY. e............. 0..................... 30 ii

LIST OF FIGURES Figure Page la Reactor with Coolant Flow passing through Tubes in the Core...... O... 4.............o 2 lb Reactor with Fuel Contained in Separate Closed Tubes.................... o o 2 lc Reactor with Fuel Contained in Separate Tubes Connected at Top and Bottom by Headers os.-. 3 ld Thermal Reactor with External Heat Exchanger..... 3 2 Test Section Nomenclature Schematic............. 8 3 Normalized Wall Conduction vs. Non-Dimensional Axial Position, Constantt Wall Temperature, Uniform Heat Source Distribution..... o. 10 4 Nor Non-Dimensional Heat Source vs. Overall Temperature Differential, Experimental and Calculated Data..................,....,.... o 11 5 Temperature vs. Axial Position, Constant Wall Temperature................... 13 6 Normalized Wall Conduction for Constant Wall Temperature and Uniform Heat Generation - Fully Developed Flow ooo..0.0.0.......0...o. 19 7 Non-Dimensional Boundary Layer and Core Velocity vs. Axial Position, Constant Wall Temperature................... o.ooo...ooo o....... 21 iiiL

INTRODUCTION Various advanced power reactor concepts envision a "mobilefuel" consisting of a molten alloy, slurry, or solution containing the fissionable material. In fast reactor designs such as LAMPRE-II under development at Los Alamos, the mobile fuel is contained within the core vessel and heat is removed by pumping a coolant through the core, utilizing the core as a conventional closed heat exchanger. It is conceivable that the fuel side of the core be a single, connected vessel with the coolant in separate tubes passing through the core vessel; or that the fuel be contained in separate sealed tubes or ligaments with the coolant on the "shell side". (Figure 1-a and l-b). A variant is a ligament design with the ligaments connected at one or both ends into headers. (Figure l-c). Economic considerations limiting the inactive fissionable material inventory, and requiring maximum utilization of the active inventory result in very small cores, high heat fluxes, and the impossibility of a circulating system. For these reasons, the passage of the coolant through the core rather than the use of an. external exchanger seems mandatory for the fast power reactors. Thermal power! reactor concepts such as the LMFR also involve a mobile fuel consisting of fissionable material carried in solution in a molten metal. In this case the core is sufficiently large and the critical loading sufficiently low that a circulating system with external heat exchanger seems most desirable. (Figure l-d). -1

COOLANT COOLANT / - O~' LIQUID FUEL OO I- -0 FUEL REACTOR CORE PRESSURE VESSEL REACTOR WITH COOLANT FLOW PASSING THRU REACTOR WITH FUEL CONTAINED TUBES IN THE CORE IN SEPARATE CLOSED TUBES FIGURE Ia FIGURE lb

HEAT EXCHANGER FUEL REACTOR COOLANT KHk>Y"""'"~~~7 Ir~~~l~~~nY~~~ VESSEL te fo ~~~~~~~~CORE~ SECONDARY COOLANT OR -e 00 HEAT ENGINE WORKING FLUID.0 ~~~~~SHIELD Ek BLANKET PUMP PRIMARY COOLANT REACTOR WITH FUEL CONTAINED IN SEPARATE TUBES CONNECTED AT THERMAL REACTOR WITH TOP AND BOTTOM BY HEADERS EXTERNAL HEAT EXCHANGER FIGURE Ic FIGURE Id

In all of these systems natural convective forces influence to a significant extent the velocity and temperature profiles and the heat flux distributions. This is particularly so in the internally cooled cores typical of the fast designs, since no forced convection is superimposed upon the fuel movements The externally cooled, LMFRtype design exhibits a combined forced-natural convection within the core where the heat is generated and within the fuel cooling in general. This has been treated in Reference 1. The present paper is concerned primarily with the internally-cooled arrangement, where the liquid fuel is within a closed vessel and is motivated by the internal heat generation due to fission and heat loss across the tube walls due: to -the forced.-c6n3vetion' -oolant. (Figures l-a, b, or c). Limiting the discussion to the internally cooled cores, there are two types of natural convection of interest: 1) The flow and heat transfer within a single, vertical, smallbore ligament (Figure lb), in which heat is generated by fission within the fuel and removed to the coolant through the ligament wall; 2) The overall flow and heat transfer within a larger vessel consisting of vertical, parallel, single-ligament passages which are either connected through headers at the ends or continuously connected (Figures l-a and c). The flow pattern pertaining to the second configuration is a superposition of the pattern found for a closed-loop with hot and cold leg (Figure Nld) and that found for a single ligament (item 1 above). The closed loop can be simply approximated if knowledge of the effective friction factor and density change with temperature is available

(Reference 2). This type of behavior would be encountered if the rate of heat generation did not match the coolant flow in a similar manner at all radii (from core centerline). The case of the single, sealed ligament is more difficult to evaluate and will be the primary subject of this paper. In typical cases involving liquid metals the passage diameters may be very small (perhaps of the order of 50 mils). Although the volumetric heat source is very great, it will often be found that the temperature differences existing between passage centerline and wall are approximately those predicted on the basis of pure conduction; ie, considering the fuel to be a solid rod. Nevertheless, knowledge of the actual temperatures and velocities is important for the following reasons: 1) Actual motion of the fuel (velocities of the order of 100 feet/hour seem typical) even for sealed ligament. Larger values for parallel ligaments may allow mass transport of container material between hot and cold regions. 2) A significant perturbation of the axial distribution of wall heat flux seems probable. Since this is the limiting factor in many designs, realistic knowledge of its magnitude is mandatory. 3) Disposition of the fission gases is affected since the macroscopic velocities are sufficient to overcome the effects of static diffusivity, to affect a hold-up of gas since the velocities are of the magnitude of the rising velocity of small bubbles, and perhaps to affect the bubble growth on ligament walls. 4) If somewhat larger passage diameters are involved, there is a significant effect upon the temperature distributions. The parameter

-6delineating the effectiveness of-the natuioal convection involves the sixth power of the diameter. Thus small increases in diameter rapidly become of importance.

LIMITING CASES No analytical or experimental data is presently available for the case of interest; ie, a liquid metal, contained within a vertical, sealed ligament of arbitrary length to diameter ratio within which heat is generated (steady-state being maintained by the removal of this heat through the ligament walls to a coolant), and the walls maintained at arbitrary axial temperature distribution. However, certain limiting cases have been explored both experimentally arnd analytically and serve to provide a guide for the understanding of the more general case. A. Boundary Layer Solutions -Aqueous Fluids Analytical predictions and experimental corroboration for the temperature and velocity profiles for the identical case except that fluids of Prandtl Number of unity or greater were ccasidered, (liquid metal Prandtl Numbers range from perhaps 0.001 to 0.01*) have been detailed in previous papers by one of the present authors (References 3 and 4). The effect of the low ratio of momentum to thermal diffusivities of the liquid metals in general would be to reduce the temperature differentials and velocities. The flow pattern and temperature profiles, found both experimentally and analytically for the aqueous fluids, are illustrated schematically in Figure 2 (taken from Reference 4). In general there is a rising core of fluid along the centerline which has approximately constant It was shown in Reference 5 that the assumptions are not justified for Prandtl Number below about 0750

-8r - - -I _....r. F. tRS..t~~.o ~CORE t% BMAX t PRC FILE two Figure 2. Test Section Nomenclature Schematic.

temperature and velocity at a givenr axial, positione Along the ligament walls there is falling boundary layer wherein the temperature and velocity both vary rapidly as a function of radiuso The temperature difference between wall and cen.tertine is a maximumn a the top and falls to approximately zero at the bottom~ The boundary layer thickness generally grows as the distance from the top increases (behavior typical of any boundary layer) to a maximum near the bottom before, theoretically becoming abruptly zero at the bottom. As a result of these temperature and boundary layer thickness variations, the wall heat flux is a maximum at the top even though the rate of heat generation for the vessel and the wall temperature is constanto Typical wall heat flux distributions are shown in. Figure 3 (taken from Reference 4>) It was founld that the overall temperature differences could best be presented in a plot of non-dimentional volumetric heat source versus non-dimensional temperature differential~ Such a plot, taken from Reference 4 is included (Figure 4) for convenience~ The various curves apply to different wall temperature distribu-tions (all linear in the axial direction and unliform around the circumference at each axial position) and axial heat source distributions (ro variation of heat source with radius was considered)> Because of presentation in nondimensional parameters, the results are applicable to any length to diameter ratio. The meaning of the symbols is given. in the Nomenclature and on Figure 2o It is noted from Figure 4 that the -lines terminate, depending upon wall temperature distribution, at a qv between 102 and 103. This termination signifies the lower imit; of applucabilty of the boundary layer solution used. At this qv vleue tihe boun-dary layer, at some point

'olq.nqTl:jsT-G a: nos ARoH mJxOjTufl'afWnJ;UamaL rTTM 4usPUMoD'u~OT4SO6d TvPxY T1UOTSU;a.cI-UON eAs uo';O- npu T PuoD pazTTe lXIoN P ZnflTx 3/x'NOILISOd VIlXt -VNOISN3W laI-NON 0'1 6 9' L' 9' 9' A'd ~' z 0 2'o 9'0 +_A___1XI-^ \\.I I\ \ 11 1 OlXZAb.9.,Ol 9l~s b \ i'' T( ] t solxzshb I I I'l o J_.... il _ \ -— 1o I o OI X= A-b -- XIOXb rAb =A 08 rA 0'' / 0 -010'0 rI o'Og........ 0'01~~~ _ _ _ _ _ _ _ _ _ z~~~~~~~~~

-1_190 EXPERIMENTAL DATA --- LINEAR DISTRIBUTION 7 t 0 PARABOLIC DISTRIBUTION ALCULATED DATA / // HEAT SOURCE DSTRIBUTNS> 6 __ _s__ 10 _ _ _ _ _ _ _ _ _ _ _ _ 10_ _ _ _ _ _ _ 1c 0'/ / // // /.////4 to' I08..................10 Figure k. Non-Dimensional Heat Source vs. Overall Temperature Differential, Experimental and. Calculated Data.

412 along the axis, has grown to the extent of filling the entire vessel (as here used, the boundary layer includes some upward velocity'region, see Figure 2, so that continuity is not disregarded)~ The solution was based upon the requirements of conservation of mass, energy, and momentum, and then the substitution of assumed temperature and velocity profiles of the type shown in Figure 2, with bounrdary layer thickness a function of axial position. Once the boundary layer has grown to the extent of filling the entire vessel, it is obvious that the assumed profiles, allowing a core of uniform temperature and velocity, no longer apply, and that the boundary layer thickness can no longer be a function of axial position Calculation of typical cases show that the region of interest for liquidmetal mobile-fueled fast reactors is usually in this region of qv, ie, below 102 However, the significance of such a qv is not known since the analysis is limited to higher Prandtl Numberso If a completely similar analysis is followed except that the boundary layer thickness is fixed equal to the vessel radius, a difference equation relating the non-dimensional temperature differentials, heat source, and axial position can be derived~ The steps in addition to the analysis of Reference 2 are shown in the appendix. A typical plot of non-dimensional temperature differEnce between centerlirne and wall as a function of axial position is shown in Figure 5 for various heat source strengths at constant temperature. If a variable wall temperature is assumed, the temperature differential in the center section will be unchanged and the top and bottom will be changed only slightly from the constant wall temperature case4 It is noted that the temperature differential starts at zero at the bottom and becomes assymptotic to qv/4

w cr I-. w Cw C6. qv 102 q~~~~~~~~~~~y 10~~~~~~~~~~~~~~~~~.1 4 I I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%V'I 0 01 0.2 0.3 0.4 05 0.6 0.7 0.8 0.9 1.0 Fvn Figure 5. Temperature vs. Axial Position, Constant Wall Temperature

approaching this value very rapidly for small qvo This analysis, like the original of Reference 3, is limited to Prandtl Number near unity~ This value of the non-dimensional temperature differential corresponds to the case of static conduction. For those cases where the non-dimensional heat source is sufficiently low so that the boundary fills the entire vessel at some point, it is necessary to use a combined solution. The original solution of reference 3 can be used for that portion of the vessel near the top where the boundary layer is growing. The "fully-developed" solution (boundary layer thickness equal to radius and constant) can then be used for the remainder of the vesselo Such a combination will assure that conservation of mass, momentum, and energy are observed for any radial plane through the vessel at any axial position. To assure consistency at the point of joining of the solutions it is necessary that the original boundary layer solution give a temperature differential of qv/4 at the axial position where the boundary layer thickness becomes equal to the vessel radius. As pointed out in Reference 3, the implementation of the boundary layer solution involves a numerical procedure with an IBM-650 program. It may be noted that the situation is somewhat analogous to that of pipe flow where the boundary layer grows in the entry section (analiogous to the ligament top) until it occupies the entire pipe. Henceforth. it is "fully-developed" and no longer grows with axial distance. B. Infinite LengtL / iameter Case Aqueous or Liquid Metal At the opposite extrleme from the boundary layer case explored in Referperce ~4, where effecteively the len-Pgth to diameter ratio was

sufficiently small that the temperature and velocity gradients were limited only to the vicinity of the wall and the boundary layer effects could not extend into the central portion before reaching the bottom of the tube, is the case of infinite length to diameter ratio. An examirnation of the parameter grouping which forms qv discloses that this corresponds to zero non-dimensional heat source. The infinite length case has been examined'n detail by Murgatroyd, Refer'ence 60 A short examination of the physical situation will disclose the significant features. If the flow is laminar and the tube infinitely long, it is obvious that transfer of heat normal to the axis can be accomplished only by conduction. There are no convective effects either from turbulent mixing or transport of ascending fluid along the centerline to descending fluid along the wall at the ends (since they are infinitely distant). This is the case of "rod flow" for which the Nusselt's Number based on diameter and mixed mean temperature is 8 (Reference 7). It is shown in the appendix to this paper that such a Nusselt's Number corresponds to a non. dimensional temperature differential between wall and centerline equal to qv/4 at a given axial position~ As previously mentioned the fully-developed solution was asymptotic to this value for low qvo We have thus an independent verification of this solution. It should be mentioned that the assumption of laminar flow is well justified for the fast reactor applications. As mentioned in Reference 4, the transition from a generally laminar condition to a generally turbulernt one appnears tno occur at a Rayleigh Number based on vessel radius of about 4 x 107, This roughly corresponded to qv of about 108 for the tests conducted. The corresponding Rayleigh Number

and qv for a typical fast reactor application are about 50 and 10. C. Effect of Liquid Metals as Compared with Aqueous Fluids No analysis or experiment applicable to the boundary layer regime has been made for fluids with low Prandtl Number in the applicable geometry. However, the infinite length analysis applies regardless of Prandtl -Number- It seems certain that the general type of flaw behavior observed and predicted for aqueous fluids with high qv would also be observed with liquid metals. However, the low Prandtl Number should result in the temperature differences for a given heat source being reduced and the temperature gradients extending further into the fluid. Since the temperature differences would be less, the coefficient of volumetric expansion is generally small, and the conductivity large, the Rayleigh Numbers would be much lower, the forces motivating natural convection circulation less, and the velocities probably reducedo Nevertheless, the asymptote reached as qv approaches zero is the same. On the basis of these qualitative considerations, it seems likely that the general type of behavior would be the same, but that the fully-developed flow condition would be reached at a somewhat higher qv since the temperature gradients should extend more deeply into the fluid.

APPLICATION OF RESULTS TO SMALL-BORE LIGAMENT WITH LIQUID METALS Utilizing the limiting results discussed in the previous section, it is possible to delineate the significant results to be expected from natural convection in a small-bore ligament filled with liquid metal fuel in a typical fast power reactor design. Ao Temperature Differentials - Centerline to Wallo The results for aqueous fluids within the range of the boundary layer solution (qv above about 103) are shown in detail in References 3, 4, 8 and 9o These are not directly applicable to liquid metals but the type of behavior for high qv is probably typical~ However, the cases of interest to the liquid metal fast reactor concepts which have come to the attention of the authors, show qv values within the fullydeveloped range Figure 5 shows the non-dimensional temperature differential (in units of qv/4) as a function of axial position and qv for constant wall temperature. The significant results are listed in Table I. It is noted that the temperature differential is equal to approximately qv/4 for most of the vessel length but shows an increase near the top and a decrease at the bottom. The degree of increase at the top is a function of qv but is a maximum of about 70% of qv/4 at qv = 10. This is also very local (occurring within 6% of the top) and ray not be of any real significance since small axial heat flow in the tube wall could relieve over-heating. As far as the central portion of the tube is conc-erned the results apply directly to any axial wall temperature di stributLonl -so l ong as the axial gradient is not so great that axial conduction of heat becomes significant compared with radial.

The results at the tube ends apply strictly only to constant wall temperature. However, they are typical of any distribution since the substitution of different temperature configurations produces differences only of degree. Also, the end results apply only to aqueous fluids, while the central portion is of general applicability. However, it is believed that fluids of low Prandtl Number would exhibit similar behavior, although the end effects would be somewhat reduced in magnitude and axial extent. B. Wall Heat Flux The axial wall flux distribution for uniform heat source, constant wall temperature, and aqueous fluids for the fully-developed condition is shown in Figure 6 and listed in Table I. Figure 3 showed similar results for the boundary layer solution at high qv but included the effects of wall temperature distributionl In both cases it is noted that there is a decrease of wall heat flux at the bottom, a generally constant region in the central portions and a sharp rise toward the top. These effects become very much more localized as qv is reduced into the fully-developed region of interest in the present application. As in the case of temperature differential, the central portion of the curves are applicable for any fluid and wall temperature distribution, again with the qualification, that the axial gradients cannot be so sharp that axial conduction becomes relatively large. The end portions apply strictly to aqueous fluids and constant wall temperature. However, a local rise of somewhat similaer magnitude is to be expected with liquid metals and.various wall temperature distributioas. As in the case of the temperature rise at the top, this increase of heat flux may not be physically significant because of its very local charactero Local over

-1910 8 6 o 0 LL9 Z6 qvKlO z z 0 4N- t-. 65~ I0 (f) 6 Z m 0 0.02 0.04 0.06 0,7 0.8 0.9 1.0 AXIAL POSITION X/* Figure 6. Normalized Wall Conduction for Constant Wall Temperature and Uniform Heat Generation - Fully Developed Flow

-20O heating encountered in. this respect may well be relieved by axial heat flow. Only experimental results could resolve the seriousness of the problem~n The problem is of potential importance because the thermal stresses due to the wall heat flux are the limiting factor in many designso C. Veloc ity Detailed plots of the velocities anticipated and also observed for aqueous fluids in the boundary layer regime are given in references 3, 4, 8, and 9o Reference 6 shows analytical predictions for laminar and turbulent flow in vessels of infinite length. Figure 7 shows the maximum velocity to be expected in the laminar fully~ developed regime as a function of axial position and qv. As previously mentioned, the physical cases of interest seem limited to laminar flow. Except for the extreme ends9 the values apply to any fluid and any wall temperature distributiono Also the end values should be correct in direction of shift and order of magnitude. Under the assumed profiles used in this solution (a quadratic curve was used) the maximum velocity is the ascending velocity along the centerline although in the high qv boundary layer cases it was the descending velocity adjacent to the wall. Whethetr the prediction of maximum velocity along the centerline is justified or is merely a result of the assumptions used is not knowno However, the order of magnitudes of velocities predicted is of prime importance and this is believed to be essentially correct.

qv= I 104 = —9 1010 I- = I x 10 I0 UCORE UB.LIx a6 6 0 u ~~~~~~~~~~~~~qv~l I0Xlo x 2 2 z4 qv= I x 10o..J 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 NON-DIMENSIONAL AXIAL POSITION, x/l Figure 7. Non-Dimensional Boundary Layer and Core Velocity vs. Axial Position, Constant Wall Temperature

CONCLUSIONS Although no directly applicable theoretical or experimental results are available for natural convection in single, small-bore vertical liquid-metal filled ligaments as encountered in mobile-fuel fast power reactors, it is possible to delineate the general nature of behavior expected with respect to temperature, velocity, and wall heat flux profiles from limiting analyses and experiments which are available. These data are illustrated in the paper, and it is shown that natural convection may be of importance from the viewpoints of perturbation of wall heat flux (a limiting design condition), and the motivation of velocities important as a possible mechanism for mass transport and also as an influence on the disposition of fission gas created within the fluid. -22

APPENDIX Long Tube (Fully-Developed Flow) Analys In order that the solutions will be consistent at the transition points, the same arbitrary velocity and temperature profiles are assumed as in Reference 8. However, it is assumed that the boundary layer thickness is no longer a function of x but is constarnt and equal t o the tube radiuso From Reference 8 t = t [1 - (2 - 2 t E l(1 _) ] (1) TU.=- yyl_-() {z + ~(= - 1)}] ((2) but 0 = 0, therefore t = t (1 r2) (3) u = - y [1 - r2(l + 6{r - 1})] (4) Substituting equation (4) into the continuity relation (Reference 8) f r u dr = 0 = - yr r dr - r dr + bf r dr (5) from which 6 = -5. Equation (4) can then be written as u =- y(5r3 -6r2+ 1) (6) -23 -

-24 - and at r = 0, d = - y(15 r2 12r) = (7) Now, substituting the velocity and temperature profiles, Equations (3) and (4) into the momentum equation (Reference 8) J' r t dr + z (t)r=O (7) = ('rd+2() -0 (8) we get y = r and (9) the velocity, Equation (6) becomes u = t (5r3 - 6r + 1) (10) 12 Repeating the same procedure with the integrated energy equation (Reference 8) ax S r u t dr =( r= (11) we have lr u t dr = 1 (ad =at-2t (13) and Equation (11) reduces to ~- 7 -; (t2) + 2t + v = 0 (14) axial convection wall heat conduct ion source From Equation (14) it can be seen that t approaches qv/4 as a limit

-25 - as the tube length becomes infinite (8/8x - O). The axial convection term is a function of the axial temperature and the wall conduction term is a function of the wall to centerline differential temperature. Hence -1 v t 2+2t. O- (15) 2 where tf = t + tw Therefore (t + tw)N (t + tw)N-1 + 672t + 168qv = o (17) and tN -i= (twN 168) + wN 1 + 168sx)2 + (tN + t )2 - 336Axt + 168sqjx - t 2 (18) wN-1 N wN N -1 where Ax and t are positive numbers~ Natural Convection Laminar Flow in Infinate Length Tube The heat flow in the axial direction must be negligible compared with flow in the radial direction for the case of pure conduction, and the heat lost through the wall in a given axial section must equal the heat generated in that section. Thenrsv Tr2 Then T = _v- +- C1 in r + C2 (Reference 2, page 662) T MT Q'T (R r 2)

-26T -T~ Qv (R2 r2) vk Tmean T dv = Lo Tr (R - r)2t r dr V LicR2 Qv R2 T hb R(T - Tw) Qwall Qv RaL = RQv h!= -wall (-TE - TW) 2iRL{(T - TW) 2 (T- Tw) = 2h T TW R but previously T, = TW = R2 so T~ Tw 2h (T= - Tw h R (T - TW) -4k R 2k 2k and h, = for pure conduction T + TW mean- 2 McAdams, Reference 7, page 233, shows that for an infinately long tube with laminar (rod-like flow) and uniform heat flux the Nusselt's

_27l Number based on the diameter and the mean temperature is 8 away: from the entrance section. NuTan h D = 8 or NuT h a 4 Nu mean k Tean - Thus, based on the radius and the temperature difference from wall to centerline NUaTj TW 2 In the present analysis using the norn-dimensional heat source term, clt~~~~~v'9 ~ Nua.Tg T w = by simple algebraic substitution for qv and t in terms of the physical quantities. And when t = qv/4 NUaT, - TW = v - c — 2 27

NOMCLATURE a Radius of test section cv Specific heat g Acceleration of gravity h:Film coefficiemnt for heat transfer k Thermal conductivity Length of test section Nua Nusselt's Number based on radius; ha k Pr Prandtl Number; 6 Q.Qa ag qv Nonadimensional volumetric heat source Vol umetric- heat source -- energy per unit volume Raa Rayleigh Number based on radius and maximum temperature differential;, ga3(ATMax. to Wall ) BPv. Rayleigh Number based on length and maximum temperature differential; cxgi (Tax. to wall) R,r Dimensional and non-dimensional coordinates in radial direction. T Temperature $t Non-dimensional temperature differential ga without subscript non-dimensional temperature differential wall and fluid at any given axial positibon6 Subscript o applies to top of tube centerline~ Subscript. applies to centerline~ Nia = qv/2t. U Velocity in axial direction u rNonmdimensional velocity in axial direction = a2U/K e v Volume 28

NOMENCLATURE (CONT'D) X,x Dimensional and non-dimensional coordinates in axial direction a Coefficient of volumetric expansion 3 MNon-dimensional core thickness~ 1 - p is non-dimensional boundary layer thicknesso K Thermal diffusivity = k/pcv lo Kinematic viscosity p Density -29 -

BIBLIOGRAPHY 1. Poppendiek, H.Fo, and Palmer, L.D.,'!Heat Transfer in Heterogeneous Circulating-Fuel Reactors, Nuclear Science & Engineering, Vol. 3, Noo 1, January 1958. 2. Glasstone, S.,'Principles of Nuclear Reactor Engineering," Van Nostrand Coo Inc, New York, 1955 3. Hammitt, F.oGo., "Modified Boundary Layer Type Solution for Free Convection Flow in Vertical Closed Tube with Arbitrary Distributed Internal Heat Source and Wall Temperature,-" ASME paper number 58-SA-30, Semi-Annual Meeting, Detroit, Michigan, June 15-19, 19580 4. Hammitt, F.G., "Natural Convection Heat Transfer in Closed Vessels with Internal Heat Sources - Analytical and Experimental Study," to be presented at ASME Annual Meeting, New York, Novo 30 - Dec. 5, 1958. 50 Mohr, Dale, "Evaluation of the Prandtl Number Error in the Modified Boundary Layer Type Solution for Free Convection Flow in Vertical Closed Tube with Arbitrarily Distributed Internal Heat Source and Wall Temperature," Master's Thesis in Nuclear Engineering, University of Michigan, Ann Arbor, Michigan, June, 1958. 6. Murgatroyd, W., "Thermal Convection in a Long Cell Containing a Heat Generating Fluid," AERE ED/R 1559, Harwell, 1958.7. McAdams, W.H., "Heat Transmission," McGraw-Hill Book Company, New York, 1954, 8. Hammitt, F.G., "Heat and Mass Transfer in Closed, Vertical, Cylindrical Vessels with Internal Heat Sources for Homogeneous Nuclear Reactors," Ph.D. Thesis, University of Michigan, Ann Arbor, Michigan, February 1958, 9. Brower, E.M., "Heat and Mass Transfer in Closed Cylindrical Passage of Homogeneous Nuclear Reactors with Internal Heat Generation," Master's Thesis in Nuclear Engineering, University of Michigan, Ann Arbor, Michigan, June 1958. -30