THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEC-E OF ENGINEERINGFREE CONUVECTION HEAT TRANSFER AND LTi.Ii) FLOW IN CLOSED VESSELS WITH INTERNAL HEAT SOURCE F. G, Hamrnitt E. M. Brower Paul T, Chu November, 1959 IP —399

ACKNOWLEDGEMENTS Tie authors would like to acknowledge the fiinanzcial assista~ice of The lUniversity of Michigan Research TInstitite. the Chrjyser Corporation and Atomic Power Development Associates in the work upoan which this report is based. Also the assistance of the following studeints at this Uoiversity is acknowledged: Ao E. Brownm R. P. Hanson ald Jo Lo S-umiers.

TABLE OF CONTENTS Page I. INTRODUCTION............................................... o. 1 IIo EXPERIMENTAL PROGRAM.................................. 1 A. Description of Facilities.....*..................... 1 B. Experimental Results..........o.......... o....o... 5 1. General Applicability......................... 5 2. Heat Source Strength vs. Temperature Differentials..... 6 35 Axial Temperature Distribution....................... o o *.. 12 4. Radial Temperature Gradient and Axial Symmetry........ 12 5o Velocity and Boundary Layer Thickness Observations..... 22 6. Laminar-Turbulent Transition and Non-Steady State Effects......... a..................................... 24 7, Wall Heat Flux Distribution..,.................. 29 III. ANALYTICAL PROGRAM.........e...... *........................5 32 A. General Approach e.............. o 32 B. Limitations of Method.,........,....o........... 36 1. Axial Symmetry and Laminar Flow..................... 36 2. Heat Source Strength and Wall Temperature Distribution. 37 C. Coverage of Computer Solutions of Boundary Layer Solution.. 41 Do Extension to Low qv - Fully Developed Region.*....*......... 41 E. Extension to Non-Steady State.. o o o...................... 44 F. Results of Computer Calculations * *.... o..*..*..... 44 1o General Scope o..a.... o... oo o o *a o. ee 0 0 - o a o <0e. 44 2. Heat Source vs. Temperature Differentials.....o....... 44 5. Velocity...o.... *.......o... 0........... 0 48 4. Boundary Layer Thicknesso.............................. 53 5 - Wall Heat Flux.................. o *............. 4 56 Go Comparison with Experimental Observations..o.....*...o. 63 IVo CONCLUSIONS................................. *........... * *...... 64 B. BIBLIOGRAPHY..... o o.................. o....................... * 65 iii

LIST OF TABLES Table Page I Laminar - Turbulent Transition Data *....*... 26 II Constants of q = ktp n. 45 iv

LIST OF FIGURES Figure 1. Photograph of Water Cooled Facility........................... 2 2. Layout Drawing of Water Cooled Facility..................0. 5. Non-Dimensional Overall Temperature vs. Overall/Radial Temperature Ratio. Experimental Data (High qv Range)........ 7 4o Non-Dimensional Overall Temperature vs. Overall/Radial Temperature Ratio, Experimental Data (Low qv Range)............ 8 5. Non-Dimensional Heat Source, qv I vs. Non-Dimensional Overall Temperature Differential, Experimental Data.. o..........*.... 10 6. Non-Dimensional Heat Source, qv ^ vs. Nusselt Number.......o 11 7. Non-Dimensional Centerline Temperature vs. Non-Dimensional Axial Position, Experimental Data, Compare to Linear and Parabolic Wall Temperature Distribution, q.v *' 109......o.... 13 8. Non-Dimensional Centerline Temperature vs. Non-Dimensional Axial Position, Experimental Data, Compare to Linear and Parabolic Wall Temperature Distribution, qv" 10 lO....,...,.. 14 9. Non-Dimensional Centerline Temperature vs, Non-Dimensional Axial Positiont, Experimental. Data, Compare to Linear and Parabolic Wall Temperature Distribution, 1 x 105 < qv ix lx60. 15 10. Non-Dimensional Temperature Differential Centerline to Wall vs Axial Position..*o. o * o a. o r... *.o...o... v.......o 16 11. Non-Dimensional Radial Temperature Distribution vs. Radial Distance, qv > 1 x 10 o 0 o o aao o* * o. ao e o o > 0 Q oa 4 * 0: e e oa o 17 12. Non-Dimensional Radial Temperature Distribution vs. Radial Distance, 2 x 105 < qv 1 x 10 8T4o.......................... 18 13. Non-Dimensional Radial Temperature Distribution vs. Radial Distance, 5 x 10 < qv < 1 x 105**........,............ o 19 14. Non-Dimensional Radial Temperature Distribution vs. Radial Distance, 1 x 10 - q~v <- 4 x 10..,........................... 20 Distance, 1 x 10 L qv <- 4 x!0.X 20 15o Non-Dimensional Radial Temperature Distribution vs. Radial Distance. e..........o.....o.................. 21 v

LIST OF FIGURES (CONT'D) Figure Page 16. Non-Dimensional Boundary Layer and Core Velocity vs. NonDimensional Heat Source, Comparison of Experimental and Calculated Data..,......................... o 23 17. Non-Dimensional Boundary Layer Thickness vs. Non-Dimensional Axial Position o o oe o o e. o.. o 25 18. Normalized Wall Conduction vs. Overall/Radial Temperature Ratio, experimental Datao ^ 3. - ee o ea o.... o......e...... 50 19. Normalized Wall Conduction vs. Non-Dimensional Axial Position, Linear and Parabolic Wall Temperature Distribution Comparison, Calculated and Experimental Data Comparison, Uniform Heat Source - 4* o o a.... e oe....o o.. e,. oee..,... o...ao... o.... 51 20. Test Section Nomenclature Schematic...o.........,............ 34 21. Non-Dimensional Heat Source vs. Overall Temperature Differential, Experimental and Calculated Data, Positive Wall Temperature Differential o8 O. oO OOOO.e 38 22. tbr-Dimensionlal Heat Source vs. Overall Temperature Differential, Calculated Data, Negative Wall Temperature Differential..*..o 539 23. Various Wall Temperature Distribution.,.................... 42 24. Non-Dimensional Wall to Centerline Temperature Differential vs. Axial Position, Fully Developed Flow RegimeConstant Wall Temperature..o o o o o.ee, oG4oeo o o o o o o 47 25. Non-Dimensional Core and Boundary Layer Velocity vs. NonDimensional Axial Position, Constant Wall Temperature, Comparison of Sine and Uniform qv Distribution................,... 49 26. Non-Dimensional Core and Boundary Layer Velocity vs. NonDimensional Variable Linear Distribution of Overall/Radial Temperature, Uniform qv o................~..oo o 50 27. Non-Dimensional Core and Boundary Layer Velocity vs. Axial t.Position, Constant Wall Temperatureo............ 51 28, Non-Dimensional Core and Boundary Layer Velocity vs. Axial Position, Linear Variable Wall Temperature.,.........,,...o... 52 vi

LIST OF FIGURES (CONT D) Figure 29. Non-Dimensio:nal, Bourdary Layer Thickness VS. Non-Dimensional Axial Position, Comparison of Various Axial, Heat Source D istributione.o.......o...o.o.o.oooo.c...o...o.oo o.ooo4.........e...o.oo.o 54 350 Non.-Dimensional Boundary Layer Thickness vs. Non-Dimensional Axial Position,'uhiform Heat Source Distribtion..oo o.o. o.o..... 55 31. Non-Dimensional Boundary Layer Thickn.ess vso Axial Position, Constar.t Wall Temperaturea oo 0.o..aoo..oaoaQ.o.oo4aoo.o.o..o.on 57 52. Normalized Wall Coduction.. vs. Non-Diensional Axial Position, Con.stant Wall Temperature Uniform. Heat; Source Distributiono.... 58 335 Normalized Wall Conduction vs. Non-Dimensional Axial Position, Constant Wall Temperature, Variable AxiLa.l. Heat Source Distributionr0<. <000 * — o00 0 ~ 0 6 11 0 * e~<~~~~ o 0 0~|0* ~0 0 * 0 0~<a o ~ la 0 59 534 Normalized Wall Condu.ction vs. Non. Dimensio:al Axial Position, Lin.ear Wall Temperatu.re Distributio:n, Tnif.orm Heat Source........ 60 35. Normalized Wall Conductio-n vs. Non-Dimension al Axial Position, Linear Variable Wall Temperature Uriform heat Source,.. o O O o 0. 61 36. Normalized Wall Conduction vs. Axial Positio:.r^ Constant Wall Temperature) iTf:orm Heat Source, Fully Developed Flow. 0. oo0*00 62 vii

NOMENCLATURE T Temperature U Dimensional velocity in axial position a2 u Non-Dimensional velocity in axial direction = Ej U xx Dizmensional and non-dimensional coordinate in axial direction Rr Dimensional and non-dimensional coordinate in radial direction 9, ^a Length and radius of tube p Density cv Specific heat k Thermal conductivity K Thermal diffusivity,^ = k/pcv v Kinematic viscosity As Area of tube wall ga4AT t Non-dimensional temperature V —; without subscript, nondimensional temperature differential, wall to fluid, at any given axial position. Subscript o applies to differential between wall and centerline at top of tube. Subscript S applies to differential between centerline at any given axial position and wall at bottom. Subscript to applies to differential between centerline at top and wall at bottom. Subscript R applies to differential from wall fluid at any radius, r, and at any axial position. tw Non-dimensional temperature differential between top and bottom of wall viii

NOMENCLATURE (Cont'd) t = to+tw Maximum non-dimensional temperature differential in the teo o system = Ra. a tp Non-dimensional temperature differential between fluid at centerline and wall at bottom Raa Rayleigh Number based on radius and maximum temperature differential (wallin- Tfluidmax) 6 qv Non-dimensional volumetric heat source = Qva ha PVK2c~ Nu Nusselt Number based on radius =-ig Acceleration of gravity Coefficient of volumetric expansion Non-dimensional core thickness 1-P is the non-dimensional boundary layer thickness and Ad is the radius at which the boundary layer is terminated. See Figure 20.'5.9, 7 " F(P),G(P) Non-dimensional functions defined in text h Film coefficient for heat transfer ix

I. INTRODUCTION The phenomenon of heat transfer and fluid flow under conditions of natural convection in a closed vessel wherein heat is generated as an internal heat source in the fluid and removed through an external coolant is of importance in the field of nuclear reactors and certain chemical processes. However, very little quantitative data is to be found in the literature regarding this phenomenon which will allow the design engineer to evaluate realistically the temperature differentials, fluid velocities, and wall heat flux distributions which are to be expected under a set of given physical conditions. It is the purpose of the research work herein reported to attempt to remedy this situation and to provide such engineering data. Some of this work has been reported in previous research papers by the authors and co-workers at the University of Michigan.1 2 ),k 5 6 This report summarizes the previous papers and includes additional work performed subsequently. Both experimental and analytical investigations are included. These cover a very large range of internal heat source strength, length to diametjer ratio of the vessel, wall temperature and heat source distribution. They are limited to cases of axial symmetry in cylindrical vessels and to fluids of Prandtl Number near unity (i. e.: aqueous or gaseous fluids in general). II. EXPERIMENTAL PROGRAM A. Description of Facilities The experimental qata is derived from several pieces of apparatus which are similar in basic respects though differing somewhat in detail. In all cases heat is generated by ohmic resistance in an aqueous solution contained within a vertical, cylindrical~ glass vessel; and removed to a coolant through the vessel walls. Heat loss through the vessel ends is essentially negligible. Temperature is measured within the fluid by a glassencased thermocouple probe from which a small bead is exposed to the fluid. Velocities are estimated by timing the motion of dye or particles within the fluid. The major portion of the data results from the water-cooled facility shown in Figures 1 and 2 and its subsequent modifications. Basically a cylindrical glass test section is held between carbon-steel electrodes at either end. It is surrounded by a larger plexiglass tube, forming an annulus -1

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COOLANT FLOW- -COOLANT FLOW OUT IN I' N Es^^^^ <\\K\\Ks r n X 1111 l g SJll _______________/ ^ CHANNEL ITEST SECTIONI z %|~~~| ITHERMOCOUPLE'^W -~ -I~ ~ PROBE _ Y (1111 1 111l~liiir!;' I'1 i ELECTRIC LECTRIC LEAD Figure 2. Layout Drawing of Water Cooled Facility.

-4for cooling water. The arrangement is such that test sections of various diameters and lengths can be used. In the tests conducted the length has been varied from 22-1/4 to 6-1/4 inches and the inside diameter from 3-7/8 to 7/8 inches, giving a range of length to diameter ratio from about 25 to about 1.6. An eccentric vertical glass tube, guided by bushings in the electrodes, carries the thermocouple bead, measuring local fluid temperatures. It is capable of axial motion and rotation so that it may be positioned at any desired distance from the wall (except near the tube centerline) and at any desired axial position. Hence if axial symmetry is assumed it is possible to obtain a nearly complete temperature survey. The diameter of the eccentric thermocouple tube was approximately 20% of the test section inside diameter. The independent variables which may be controlled with this facility are heat input rate, test section dimensions, and magnitude and direction of coolant flow. Power input could be measured most accurately electrically, but a check through a heat balance on the coolant was also used. Two other slightly different facilities were used to obtain some of the test data. The first of these was identical in all major aspects to that described above except that the coolant was air in free convection rather than water in forced or free circulation. This arrangement was utilized initially because of its extreme simplicity. The second variant arrangement differed from the others is that the vertical eccentric thermocouple was replaced by a multiplicity of glass thermocouple rods inserted through the sides of the vessel and capable of motion in the radial direction. This arrangement was used to investigate the validity of the assumption of axial symmetry which had been implicit in the other experimental and analytical work. Forced convection water cooling was used with this design and it was attempted to induce asymmetry of the internal flow by non-symmetrical cooling. A final source of data was provided by several tests in a facility somewhat similar to that of Figures 1 and 2 (a thermocouple glass rod was extended from the top to reach any desired position by pivoting from the vessel top) which was used in tests at Oak Ridge described in Reference 4.

-5B4 Experimental Results 1. General Applicability The experimental data cover several categories which are listed below. All are presented in terms of non-dimensional parameters which are defined under Nomenclature. These are so constituted that the results are theoretically independent of length to diameter ratio, a) Heat source strength vs. temperature differentials for various wall temperature distributions magnitude and local direction. b) Velocity vso heat source strength. ) Radial and axial temperature gradients as a function of heat source strength. d) Wall heat flux distribution as a function of axial position and heat source strength. e) Condition of turbulence as a function of heat source strength and temperature differentials* The parameters of major importance are the non-dimensional heat source strength, qv $ and the non-dimensional temperature differentials, t. The overall heat transfer results of Figures 5, 6, 16^ 21 and 22 are presented in terms of these two. The ratiQo (maximum fluid to wall temperature differential) / (axial wall temperature differential), is used as the parameter to define a family of curves. An examination of qcv discloses that it is proportional to the volumetric heat source strength / the vessel radius to the sixth power, and inversely to the vessel length.* Consequently a large volumetric heat source in a small-bore vessel is the equaivalent of a much smaller volumetric heat source in a larger diameter vessel. The non-dimensional temperature differential is actually a modified Rayleigh Nimbero It involves a grouping of dimensional and physical constants, which is thevt of;the Ra-yleigh Number based on radius, multiplied by the radius to length ratio. In this report, the temperature differential used is defined by the subscript. The derivation on the various constants is is xplained furthber in the section under analytical results. It is also a fnctio.n of' various of the physical properties of the fluid.

-62. Heat Source Strength vs. Temperature DifferentialsThe experimental data in all cases has been taken under conditions of apparent axial symmetry, but with the temperature of the wall adjacent to the test fluid falling from a maximum at the top of the test section to a minimum at the bottom. It is difficult to avoid a wall temperature gradient in this direction with the equipment used. For such a closed-end internal heat source cell, the wall heat flux (with uniform internal heat source) is a maximum at the top and minimum at the bottom. Hence the temperature differential across the wall decreases toward the bottom. When the outside wall temperature was maintained constant or even made to increase slightly toward the bottom by downward-flowing coolant, the effect of the wall temperature differutial was sufficient to give an inner wall gradient decreasing toward the bottom. For a given physical configuration and heat source strength, the axial wall temperature gradient can be controlled by varying the coolant flow rate and direction. For most of the experiments, runs at a constant qv } but varying coolant rate and direction were made so that points for a given qv were attained for different axial wall temperature gradients. In some cases it was possible to obtain the same q-v with test sections of different length to diameter ratio. It was found that variation of this ratio within the limits of the experiments did not affect the results in terms of the non-dimensional parameters. qv depends not only on actual volumetric heat source and test section dimensions but also on the fluid physical properties which are functions of temperature. These were defined in relation to an average fluid temperature taken to be that measured on the vessel centerline at a point about 9/16 of the vessel length from the bottom. This point was selected asapproximately representative of the volume-averaged temperature. In some cases qv was not held entirely constant over a series of runs. In these cases it was prorated to the closest of the various fixed qv values which had been used by assuming the validity of the relation between qv and non-dimensional temperature differential derived analytically. The error so introduced should not be large since the prorated range is small. The experimental points resulting from the above procedure are shown in Figures 3 and 4. The overall non-dimensional temperature differential* is plotted against the ratio of overall differential to *Based on the differential between the fluid at the vessel centerline at the top and the wall at the bottom; i.e., the maximum differential existing.

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-86 0 q V _ _ _ _ _ 10 y = 7.5 x 10e i30.......0 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 tc / to Figure 4. Non-Dimensional Overall Temperature vs. Overall/Radial Temperature Ratio, Experimental Data (Low q, Range).

-9centerline-wall differential at the tcp* at the various qv levels investigated. The qv values range from approximately 1010 to 103* The data of Figures 3 and 4 have been cross-plotted from the smoothed curves at values of the temperature ratio described above ranging from 1.0 to 20. Figure 5 is the result-ant plot of qv vs. ts. On logarithmic coordinates, this is a family of substantially straight lines with a slope ranging from approximately 1.3 for low qv to 1.5 for higher qv and low temperature ratioso Since a slope of about lo24 is predicted by the laminar flow analysis^ it, is believed that these larger slopes are a result of more pronounced turbulent mixing as q is increased. A discontinuity in the curves is shown, at qv of about 10 o. It is believed that this is the result of rapidly increasing turbulent mixing in this area. However, since unfortunate.ly there was no overlapping of data between the configuration producing the lower set of points and that producing the _upper, adjacent to the disco.atin.uity, it may be simply the result of experimental error resultuing fJrom slightly different measuring techniques. As the temperature ratio is increased-? t'e overall temperature differential for a given qv increases. This trend and the approximate magnitude of the shift agrees with the laminar flow arnalysis. Curves derived from the analysis for the corresponding conditions are shown in Figure 5 as dotted lines. It is noted that the experimental curves show a smaller temperature differential for a given heat source strength than the analysis predicts. Presumably this improved heat transfer is a result of the turbulent mixing which occurs to some extent in all the experiments but is not considered in the analysis. The improvement factor varies from about 1.2 for low qv to about 1.5 for high qvo The calculated curves of Figure 5 are all based on a uniform wall temperature gradienrti whereas the experimental. gradients were parabolic in nature (Figures 7p8,9 and 10). Calculations for a gradient typical of fthe experiments have been compared with those assuming uniform gradient. It wras found that the effect on the overall temperature differential for a given q. was small. It was previously showns that Nusselt's Number based on the radius is simply qv/2t where the temperature differential used in the calculation of the film coefficient is that correspondirg to the non-dimensional temperature differential. Figure 6 is a plot of the experimental data showing Nussel.t s Number versus qv and is simply a cross-plot of the infortion of Figure 5*For constant wall temperature, this ratio is 1.0. **This corresponds to -proportionately greater wall temperature gradient.

-10 - 10_11_._ _-_ -0= I 4 8 1630 2 3 45679 -to 481630 Io O II I I II 1111,,. 01.1 1.5 810 C9 50 10? (I) z zI Io~~~~~~~~' 6 NON-DIMENSIONAL TEMPERATURE DIFFERENTIAL, to Figure 5. Non-Dimensional Heat Source, qv, vs. Non-Dimensional Overall Temperature Differential, Experimental Data.

-11 - 70 60 1.1 1.5 9 III I I I |II 11/ I 2 34 567810 50 z wJ I I 140 H — 3 400 z (* wI u) U) Z30 20 0 1o' 105 o10 107 o10 10o 10' NON-DIMENSIONAL HEAT SOURCE, qv Figure 6. Non-Dimensional Heat Source, qv, vs. Nusselt Number.

-1235 Axial. Temperature Distribut.ion: Figures 7, 8, 9, and 10, show the experimental. axial wall and fluid centerline temperature distributions for the various qv ranges. These curves are affected by the form and magnitude of the coolant flowo However. in general the wall temperature gradient is not uniform. but increases rapidly tooward the bottom. Also the differential between fluid cen.t:erlirne ar.d wall at a given axiaL position is generally fairly constant in' the upper part of the vessel and decreases rapidly toward the bottomo For similar wal.l temperature distributior this trend is predicted by the analysis, 4. Radial Temperature Gradient and Axial Symmetry For the range of qv studied, the analysis predicts a boundary layer type of flow pattern wherein the outer portion of fluid adjacent to -the wall will descend a.d'the inner portion, or core, ascend, The radial extent of the outer descending boundary layer will be small and at high qv values (corresponding to most of the experimental range), but it will increase as qv is reduced, A substantial upward velocity along the centerline is then required for the closed vessel. Since, over most of the range, the descending layer occupies only a small portionr of the total area, the descending bou-dary layer velocity, will be greater than the ascending core velocit'y. Wi.th such a boundary layer flow pattern it is to be expected on theoretical grounds5' 6 that the radial temperature gradien.t will have approximately the same radial extent as the corresponding velocity gradient.9 and that the temperature w.ll be approximately constaint across the core at a given axial position. These expected patterns have been observed. Lgures 11, 1.2, 135,:14 a:nd 15 show representative radial temperature profiles for widely spaced q., values. As qv is reduced the radial extent of the temperature gradient is increased as anticipated. The values agree approximately with. the theoretical expectations. With the exception of Figure 15, the above data were take.. with the eccentrLic vertical thermocouple probe previously described. He:-.,ce poi.nts at varying distances from the wall are also taken at varying angul.a:r. posit.ions. Since the points taken with the eccentribe probe form a regular profile, there is a strong presumption that approximate axial symmetry must; exist in these caseso It is the authors' opinion, based on their observations of the flow, that there is no significant variation from axial symmetry in these tests and that the effect of the probe is small. Its

-131.0 4 o.Z z 0. 0.4 -0. 0.0 C, 2o 9 \\ 0. 8 ONX 7 At Z IJ:: Z i0 --— 0 C1 U. 0 1.0 Compare to Linear and Parabolic Wall Temperature wi, q 0.4 w.I V 0 0o.2 04.0 NON-DIMENSIONAL AX0IAL POSITION, X/. Distribution, 10' Distribution., qv X O~ log

-14 - 1.0 Gx 3. 0 to~~~~~~~~~ t 0o6 / 0t.2 0.4x 0.60..0\I~ 0.6C e to L r Paraboc Wl T at 0.41 ~ O =v~-'0 1 to 0.2 CALCULATIONS FOR LINEAR TEM DROP\ EXPERIMENTAL CALCULATIONS FOR PARABOLIC WALL TEMP DROP 0 0.2 0.4 0.6 0.8 1.0 NON-DIMENSIONAL AXIAL POSITION,x/I Figure 8. Non-Dimensional Centerline Temperature vs. NonDimensional Axial Position, Experimental Data, Compare to Linear and Parabolic Wall Temperature Distribution, qv = 10

51.0 w 0 w w 0 0.6 z 0 U) z 0.4 I I 0O I X IOW< e rI D z _ 0.2 z 0 0.2 0.4 0.6 0.8 1.0 NON - DIMENSIONAL AXIAL DISTANCE Figure 9. Non-Dimensional Centerline Temperature vs. Non-Dimensional Axial Position, Experimental Data, Compare to Linear and Paabolic Wall Temperature Distribution, 1 x105 < q 1 X 10

1.O ---- EXPERIMENTAL DATA -\ --- CALCULATED DATA t t IO7<q< IdO (AVERAGE VALUES) 0.8 0.4 PARABOLIC DISTRIBUTION 0 0.1 0.2 03 Q4 0.5 0.6 0.7 NON-DIMENSIONAL AXIAL POSITION, x/I oo Figure 10. Non-Dimensional Temperature Differential Centerline to Wall vs. Axi Position. 0 0.1 0.2 0.3 54 0.5 0.6 0.7 0.8 0.9 1.0 NON-DIMENSIONAL AXIAL POSITION, x/I Figure 10. Non-Dimensional Temperature Differential Centerline to Wall vs. Axial Position.

OT x T < Ab AGOTe;sTc T'ITP'E'SA uoTqnqTJtsTa J anq'eJadcI T'-TPBE TuoTsuaGTa.-uo *'IT aTnT i o/J-I'33NV.tSIG "IVtlV "IVNOISN3WIO-NON 0* 01 g' b' ~ Z' 1' 0 601 x I:b t6'11 D/l -- -- 9 A -----------— 6-:' 0/| D -------------- z':l/x —------------- / e' —,, —— ~' —-M.' o O'. r r" 0| it9~~~9' / IV' U I t ___ _xiol______ __ r -L:9' I~~I' = 3~~~~~/x~~0-J~ 0~ 0 -L,

NORMALIZED NON-DIMENSIONAL RADIAL TEMPERATURE DISTRIBUTION, ATr ~Jo o o Fl-. >0> — ~ rn| x:llix l... *0<> <! tO0\ 0P3~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~ - 0 iP_ _ _-I A|x _Ix 0 -— P p 0 (D Hz rU CA 03 Z Po (CD IA ~o CD P —x 0J. 0 x H 0 _ 0 C — 0 *co 0 0 ~ -o? o~~~

-1991.0 V —' — z A A cr Hv 1.0 0. = * 0.45 1.0 _ 13. e R - - A w 5 V < 01 0 z v AIx^s z A V x /X V ro Radial Distance, 5 x 10 - qv -1 x 10.

-20 - 1.0 IU Q~~X 0 /: vx 0 x 05 0 05.0.45V v\ z X- / x D w 0 I-a^~Rda Dsne1x0<v x1r/a _j 1.0-*0 \X z z z x w /X X V N = 0.72 Z x* A I 1, I X V A A A 1.0 0.5 0 0.5 1.0 r NON-DIMENSIONAL RADIAL DISTANCE, -- Figure 14. Non-Dimensional Radial Temperature Distribution vs. Radial Distance, 1 x 10 -qv r 4 x I10.

-211.0Ef - 0.9 ___ ——'"" —-__; 0.8 0.7 0.6 - w 0.6.- - - A__^ --— ^ —0 — _ u 0.3 q 1.48 x 10* ~0.1 --— x -= 0.1 0 - L 0</ —— t —-- x 0.5 A x=0.9: E0 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 POSITION -L to Figure 15. Non-Dimensional Radial Temperature Distribution vs. Radial Distance.

-22contribution to shear area is obviously small and exists in a region of low velocity. Also its share of cross-sectional area is extremely small. The data of Figure 15 was taken from a facility wherein the temperature was measured by radial probes at various angles of rotation at a given axial position. Thus it was possible to measure the radial temperature profile at several angles of rotation and to check the existence of axial symmetry. In all cases it was found to exist within experimental accuracy although attempts were made to induce non-symmetry. This experiment was performed to induce a concentration of the downward velocity on one side of the vessel an.d upward on the other as reported for parallel flat plates.9 However, such a result was not obtained. Fu.ll details are given in Reference 7o An examination of the radial profile curves discloses that in many cases the temperature reaches a maximum at the juncture between the boundary layer and the core and then decreases at greater radii. It is believed that this is not simply experimental, error since similar curves have been shown by other investigators. In an over-simplified manner the phenomenon can be explained by the fact that the fluid in this region is retarded in its motion and kept away from the cooling surfaces for a longer time than fluid traveling upward in the central portion of the vessel. However, heat is generated uniformly throughout the fluid, and conduction is almost negligible as a heat transfer mechanism throughout the body of the fluid. 5. Velocity and Boundary Layer Thickness Observation Velocity observations were made visually. In some cases dye was injected at known locations, using an injector attached to the thermocouple probe, and its motion timed. It was possible to obtain a semi-quantitative indication of the velocity direction, magnitude, and state of turbulence at any point within the test section. In other cases the motion of particles within the fluid was timed, giving an approximate indication of the mean upward and downward velocity magnitudes. It was found, as expected, that the interface between upward and downward flow was radially at approximately the same position as the inner extreme of the radial temperature gradient. The experimentally observed maximum boundary layer and core velocities are plotted as a function of qv in Figure 16. These are compared with the values derived from the laminar flow analysis. The observed bounadary layer velocity is less than that expected and the core velocity is more.

28 26 24 22 20 ------ EXPERIMENTAL --- - -- -- _ — - - 18 s _- — ~ —----- CALCULATED (Porobolic Distribution)I - [-.-. C 14 zo 6 4 - 510 _ 10 5 1~ 5 - Figure Le Ce Velcit vs. N2 Calculated Data.x 6 R av —o 10~ 5 lot 5 Io* 5 Iog 5 Io' ~ o10" NON- DIMENSIONAL HEAT SOURCE, qw Figure 16. Non-Dimensional Boundary Layer and Core Velocity vs. NonDimensional Heat Source, Comparison of Experimental and Calculated Data.

-24Presumably, the retardation of the boundary layer velocity is a result of the turbulent mixing. The boundary layer thickness from the radial temperature profile measurements is plotted as a function of qv in Figure 17. It is noted that the thickness decreases with increasing qvu A curve derived from the analysis is shown for comparison. Fair agreement exists. 6. Laminar-Turbulent Transition and Non-Steady State Effects Observation of the flow discloses that truly laminar conditions did not apply to the entire vessel in any run except perhaps those tests conducted at the lowest qv. The lack of laminar flow (perhaps small eddy is a better description) was evidenced in the dye tests and also by the fact that the local temperature readings fluctuated continually. Typically, the period of fluctuation was of the order of 15 to 30 seconds. However, the magnitude of the period may be due more to the characteristics of the precision potentiometer used than to the flow. Somewhat similar observations have been made in the past by other investigators.10 There is no abrupt change from laminar to turbulent flow in this configuration. Laminar conditions break down first in regions of high shear at the interface between upward and downward flow and, in general, in the upper rather than the lower portion of the vessel. Since the lower portion is relatively stagnent, laminar conditions persist here longest. Within the range of observations made, it appeared that in no case was the flow either entirely laminar or entirely turbulent. Rather laminar conditions were restricted to a larger or smaller portion of the vessel. including always the extreme lower portion. The various experimental runs are listed in Table I in order of ascending Rayleigh Number based on the temperature differential from fluid centerline to wall across the top of the vessel radius. The division between substantially laminar conditions and substantially turbulent (ioe., where there is turbulent in perhaps 1/2 the vessel volume) occurs for a Rayleigh Number value so defined of about 4 X 10(. This parameter appears more successful than either length Rayleigh Number or qv in delineating the degree of turbulence~ This is perhaps to be expected on theoretical grounds. However, considerably more basic research is required. Prediction is complicated by the closed loop nature of the flow leading to reappearance of general turbulence.

-25 -.12- v E l — a —. —.-10.10, xIo<q <3xl0 ____,,.09 to Sn.X08: w.07.06 1 0.05 3 to.04.03 El EXPERIMENTAL 4J 02 __ -- _-PARABOLIC WALL TEMPERATURE _j.02 CALCULATION >..01 cr_ 0 2 - 0 —----------- | 1 3 x 310q<DIMSIO x/10 o.10 v z.09.08.07 - Z.06 Z.05 L-.04 ________.02.01 to 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/I Figure 17. Non-Dimensional Boundary Layer Thickness vs. NonDimensional Axial Position.

TABLE I LAMINAR -TURBULENT TRANSITION DATA Raa Ra v Remarks 1.22 x 10 2.5 x 109 2.0 x 107 All laminar 1.28 x 107 4.44 x 1010 54 x 106 All laminar 1,4 x lo All00 1.59 x 107 1.32 x 1010 10 x 107 All laminar 1.72 x 107 6.66 x 108 6o x 108 Slight turbulence 2.00 x 107 7,79 x 108 20 x 108 No comment 2.06 x 107 4.17 x 109 4.2 x 107 All laminar 2,4 x 107 2,17 x 1010 1,6 x 107 All laminar 2.89 x 1071.0 x 10111.7 x 107 No comment 3.00 x 107 5.05 x 10102.0 x 108 No comment 3505 x 107 5,11 x 1010 20 x 108 No comment 3.28 x 107 65 x x 10 All laminar 7~59 x 107 10 7 3-39 x 107 3.5 x 1010 350 x 10 All laminar 3556 x i07 20 x x 020 10 3561 x 107 1.39 x 109 2.0 x 109 3578 x 107 6.11 x 1011 2,0 x 108 Some turbulence 4.31 x 107 1.67 x 109 2.0 x 109 No comment 4.55 x 107 172 x x 109 20 x 109 Some turbulence 4.89 x 107 1.83 x 109 2.0 x 109 Some turbulence 5.4 x 107 1.89 x 10121 1 x 107 Some turbulence 5o50 x 107 1.11 x 1010 10 x 108 No turbulence 5.55 x 107 2,17 x 109 2.0 x 109 Some turbulence

-27TABLE I (.CONT'D) LAMINAR -TURBULENT TRANSITION DATA Raa Rae qv Remarks 6.11 x 107 6.66 x 101 5.6 x 107 No turbulence 6.66 x 107 1,17 x 108 Some turbulence 7.235 x 1071.17 x 101 2.0 x 108 No comment 7.23 x 107 1.22 x 1011 60 x 108 Very turbulent 7.23 x 107 1.22 x 1011 6. x 108 No comment 7.23 x 107 1.22 x 1011 6.o x 108 No comment 7.23 x 107 61.28 x 1011 No comment 8.34 x 107 2.89 x 1011 3.1 x 107 Some turbulence 8.9 x 10x 1016.x 10 No comment 1.11.5 x 1019 1.11 x 108 4.11 x 109 6.o x 109 Very turbuleat 1.111.89 8 6.89 x 1016. No coment 1.17 x 108 4.44 x 109 6.o x 109 Very trbulent 1-17-V x 10turb ulent 1.22 x 108 4.5 x 109 6.o x 109 Very turbulent 1.22 x 108 4.56 x 100 6. x 109 Very turbulent 1.50 x 108 2.56 x o10 6.0 x 10 No comment 1.50 x 108- 5.55 x 109 60 x 109 No comment 1.55 x 108 2.61 x 101 2.0 x 109 No comment 1.78 x 10.0 x 1011 2.0 x 109 Some turbulence 1.78 x 108 3.0 x 1011 2,0 x 109 No comment 1.89 x 108 3.16 x 1011 2.0 x 109 No comment

-28TABLE I (CONT'D) LAMINAR-TURBULENT TRANSITION DATA Raa Ray qv Remarks 2.44 x 108 4.11 x 1011 2.0 x 109 No comment 2.78 x 108 4.7 x 1011 2.0 x 109 Very turbulent 5.28 x 108 1.22 x 1010 2.0 x 1010 Very turbulent 8.34 x 108 4.39 x 1011 1.1 x 1010 No comment 1.05 x 109 5.55 x 101 1.4 x 110 No comment 1.17 x 109 6.11 x 1011 1.9 x 1010 No comment

-29The ability to predict the degree of turbulence in a given configuration is of great importance to the design engineer. A comparison of the experimental (essentially turbulent) results and the predictions of the laminar analysis show that there may be an improvement in Nusselt's Number by a factor of about two from turbulence. 7. Wall Heat Flux Distribution It is predicted analytically that, for a uniformly distributed internal heat source, the wall heat fllx will be a maxim7um at the top of the vessel and a minimum at the bottom for all of the axial wall temperature distributions investigated. This trend was confirmed ex:perimertally. However, the data is only approximate. Wall heat fluxc values are inferred from the temperature differential across the test section wall. This differential was measured in the air-cooled tests as well as in the water-cooled tests. The axial variation of the coolant temperature of water cooled apparatus is much less than that of the inside wll surface so that coolint temp'erature can be considered almost constant. An *rror in estimating its gradient between the measured end points has little effect on the calculation of the temperature differential across the test, section wall. Also the film coefficient on the outside of the wall is high because of forced convection, so that the temperature difference between wall surface and mixed mean coolant temperature at a given axial position is small. The resul.ting data, normalized to the wall heat flux at the axial midpoin-t is presented in Figure 18 as a function of temperature differential ratio, (overall)/(fluid centerline to wall at top), This agress qualitatively with the analytical predictions (Figure 34). The experimental data covers only a range of points over the central 50% of the test section length. However, the analysis slhows that the deviation of wall heat flux from the mean value is much greater at the ends. Reliable measurements near the ends were not possible. tVle test section wall thickness was not sufficiently small to substantially eliminate conduction, and there was uncertainty as to the quantity of heat flow from the vessel ends. Even though this were small compared to the total flow, it cqould affect the local temperatures quite significantly.

-303 0 WATER, MAX LENGTH, 2x 10 <S qv 2 x 10 X WATER, MIN LENGTH, 2 x 108 qv< 2 x 10'~ 2 -— l 0 AEC, qv 10' + AIR, qv 07 +I. t+ O+ 1 + x/1=.25 ~~o + +.8.6.5.3 z 0 0 1.0 x —-- -- /.50 ___..6 - w N..3 0 o.2 _.,.0 _- 4 -----.8.6.4.2 0 10 20 30 40 50 60 to / to Figure 18. Normalized Wall Conduction vs. Overall/Radial Temperature Ratio Experimental Data.

~*aGonogS q-H uLojTu fl'UOSTFJ'dUIo'esCJ TiuaniG-UTdx, pue paTn'IOO'IIOSTJSdIxoo UOTnqT.fXsT( Ga.nqjaduIGi TTM DOTTIoqs'ed pus J.Xau'q'uoiqTsod T'STxV Tl:UOTSUGTTla-UOM *SA uoTonpuoo TTi'e pzTTlo'aON'6T @nS-F I / x'NOIIISOd t1VIXV -1VNOISN3WIY-NON 0'1 6'0 8'0 0 tL' 90 O'0 ~'0 IO'0 0.'0'O 0 z o rn.. ll = i 9'0 r /- //i - o.oI o* - I /..I...' C./- 1-~' 01 80 0 L-0^1 0y - C) l-lN3W183dX 3 - I —— 0 - l -01 z Ii_............. ~....;\ ~ //,,~____ o ____':: d 9I, C) l 9 1 3 -( Ob~0 " -IV3NI1) ~ ~-I-0 01"10aVEIVd > 0I 1V.IlN3IWI3dX 3. 03.V"inolng'V3. ----------------------------— Iz

III. ANALYTICAL PROGRAM A. General Approach The general method utilized was patterned after that of Lighthill in his analysis of a vessel with constant temperature walls and open at one end (closed at the other) to an infinite reservoir which is maintained at a temperature different from that- of the vessel walls. Lighthill considered three possible flow regimes which covered the full range from infinitje to small non-dimensional temperature differentials between inrfinite reservoir and wall. For the largest differential the postulated flow regi me is of boundary layer type sLmilar to that, encountered with a flat plate in an infinite fluid~ This has been extended somewhat into the range of lower differentials by considering the velocity in the core of the vessel which is of necessity of the opposite direction to that in the boundary layero It was upon this particular approach that the analysis of the case of natural convection with internal heat source was basedo The details of the basic analysis used here have been given in 12 previous papers, o 2 However, the general outlines will be repeated for convenience The vessel is considered to be composed of a multiplicity of discs normal to the centerline and of infinitesiimal. thickcness. Boaudary layer assumptions are made so that partial derivatives of the quantities parallel to the wall are neglected compared with. those normal to the wall. Axial symmetry and steady-state are assumedo In.tegral relations for the conservation of mass, momentum, and energy; considering each disc as a control volume are writte.n ar.d non-dimensionalized by diivision: with the proper terms The conservation of momentum relation is further simplified by dropping the inertial terms with respect to those related to viscous shearo It was stated'by Lighthill and has been verified by the present work, that the assumption is reasonable if Prandtl Ni uber is in +th.e order of unity or greater; but is not defensible for fluids of -very low Prandtl Number such as liquid metals. The only difference up to this point between the present analysis and that of Lighth.ill is that a volumetric heat source term is added to the energy conservation relation. Non-dimensionalization. of this relation results in a non-dimensional heat source, qvThe general characteristics of the flow pattern are prescribed both on physical and theoretical groundso The exact analysis of Pobhhausen and the experimental work of Schmidt and Beckman aa.d of Saunders 1 are

-33available for a. vertical flat plate in an infinite fluid. It has been shown that the flow is of the boundary layer type and that velocity and. temperature profiles extend approximately equal distances into the fluid if Prandtl Number is near unityo For the flat-plate geometry the solution has bee, evaluated out by numerical procedures for fluids of any Prandtl Number. For those cases where the temperature differential is sufficient to cause the boundary layer thickness to be small, the circular tube differs to no sig.ificart extent from the flat plate. Furthermore, the predictions of Lighthill. for hi.s particular case have been largely verified experimez.tally by Marvtlinx iL Considering all the above, it seems reasonable to assume that the flow pattern in the closed vessel will also be of boundary layer type, if the interal heat source is sufficiently strong. There will thus be a descending boudaary layer along the wall and an ascending core along the vessel centerlineo The temperature and velocity profiles will extend approx-imately equal distaaces into the fluid provided the Prandtl Number is near unity. Thus approximately unity Prandtl Number is required for two reasons. Following Lighthill s procedure, velocity and temperature profiles were assumed such tChata at a given axial position the temperature increases parbolically from wall temperature to core temperature at the in.er extreme of the boundary layer. It; is assumed uniform across the core. A cubic relatsion is used for the velocity so that it vari.es from zero at the wall9 through a maximum dow:cr.ard bo'undary layer velocity, and then upward to equality with the upward core velocity. T.his is assumed uziform at a given axial positio. The point of tangency of boun-dary layer velocity and temperature profiles wATth th.e corresponding core values is defined as the irner extreme of the boundary layer (Figure 20). The assumed relations are reproauced below: - 7 for 0 < B < r U' -B 2 - 71 - (LB) 1 + 6(r-l)' j for B < r < 1 (l) t(x) for 0 < B < r t(x)!l-(=-)2 for B < r <1 (2)

-34CORE ~r -- BOUNDARY LAYER x ASSUMED VELOCITY t~O PROFILE ^t \\\ ///X/ /- ASSUMED TEMPERATURE PROFILE t Figure 20, Test Section Nomenclature Schematic.

-35 - The assumed velocity profile is identical with that of Lighthill's. Because of the presence of the parameters y and j which are functions of boundary layer thickness, the magnitude and radial extension of the profile is left as a function of axial position. The temperature profile is also identical to Lighthill's except that the core temperature is considered to be a function of axial position rather than a constant as in Lighthill's case where there was no internal heat source. Then core thickness, is a function of axial position / These profiles and their first derivatives obviously satisfy the conditionrs at the inner extent of the boundary layer and also at the wall, The first derivatives at these points are used to give wall heat flux and and wall shear in the energy and momentum relations. They are substituted into the integral conservation relations. A further relation describing core temperature is required. This was not necessary in Lighthillts case where it was assumed constant. For the present analysis it was assumed that heat conduction would be negligible within the core and that the core temperature would be the result of the heat generation rate within the core and the time for which a portion of the fluid was exposed. Thus the axial gradient becomes proportional to the volumetric heat source strength and inversely proportional to the core velocity. The core temperat.ure relations result in two integral-differential equations, one representative of the core axial gradient and the other of the conservation of energyo1 These relations apply individually to the radial discs into which the overall vessel is considered to be divided. The overall boundary conditions are: 1.) Heat generation for the entire vessel must equal to the wall coliai tJ>zc:oca tc l e ves. sel as a whole, i.e6: net energy convection for the en.;.tirie vessel is zero~ 2) Mixed mean, boundary layer and core temperatures must be equal. at either extreme of thLe vessel; i. e,: fluid leaving the boundary layer at the bottom. is identical to that entering the core at that point. A similar stateme:.rt( applies at the top. 3) BouLndary layer thickness at the top is zero since this is the point where the boundary layer starts.

-36As explained in previous papers,l2 the two independent integraldifferential equations are written as difference equations, considering the vessel as a series of radial discs of small height, and the boundary conditions satisfied by an interative process using a high-speed digital computers Since it is assumed that the vessel is composed of a finite number of discs, it is possible to consider different values of heat source strength and wall temperature for each disc so that the problem can: be solved for arbitrary axial variation of wall temperature and heat source strength. However, no radial variations are possible. One of these possible variations is the Lighthill case. This is obtained by assuming that all the heat is generated in a thin radial disc at the bottom. UTfder these conditions the core temperature is constant as required. Bo Limitations of Method When programmed for a high-speed digital computer, the method described above has a wide applicability since it is suited for completely arbitrary specification of wall temperature and heat source distribution. However, there are certain mathematical limits which have been explored. In some cases, before these are reached, some of the assumptions substantially lose their validity, so that a real limit of usefulness is imposed more quickly. 1. Axial Symmetry and Laminar Flow In all cases the method is limited to axial symmetry and laminar flowo The latter is important since the experimental observations in the closed cell geometry indicate that a completely laminar flow may exist only with extremely low heat source strengths. For the case where the boundary layer hypothesis becomes questionable Lighthill considers the possiblility of turbulence in his paper but makes no analysis for the present range of interesto However, it is indicated that the existence of turbulence considerably increases the heat transfer. Murgatroyd16 makes an analysis based on turbulent flow in a long cell containing a heat generating fluid. This is of great interest in itself but does not match the physical configuration considered in the present analysis, because Murgatroyd's cell is considered to be of infinite length to diameter ratio so that there is no transport of fluid between core and boundary layer except through the turbulent mixing (i.e.: no end effects).

-572o Heat Source Strength and Wall Temperature Distribution There are also limits upon the wall temperature distribution ap.the non-dimensional heat source strength, qv. Since this involves the physical properties of the fluid, the dimensions of the vessel, and the volumetric heat source strenrgth, the limitations are upon the combination of these rather th.an upon either length to diameter ratio or volumetric heat source considered aloneo Except through turbulence there is no apparent limit in the direction of high qua si.nce the solution becomes asymptotic to that of a flat plate in an infinite fluid as qv is increased and the boundary layer becomes increasingly thin. In the direction of low qv there is a definite limitation when the boundary layer at some axial position occupies the entire crosssection of the tube. When this occurs no further boundary-layer development along with axial distance is possible and the limit of the solution has been reached (the prediction of a negative radius for the rising core results). The attainment of this limit depends upon the wall temperature distribution. It is shown by the envelope line of Figure 21 that this limit occurs for positive*- wall temperature gradients in most cases at a qv between 102 and 103. For negative* wall temperature gradients (Figure 22), it occurs at higher and higher qv as the gradient becomes more negative. This range of values corresponds to very small volumetric heat sources easily attained in the laboratory, if the test section diameter is of the order of at. inch. To a typical nuclear power reactors, this diameter is only about 1/1.6 of an inch. This lower limit on qv is explored in detail in previous papers. 3517 The ratio of non-dimensional temperature differentials. tr /to can also be con:sidered a limiting parameter in combination with q-v as shown in Figu'res 21. and 22O A unity value for this ratio corresponds to a uniform wall temperature. Values greater than uznity (positive wall temperature) imply a wall temperature gradient falling toward the bott;om of the vessel and values below unity, indicate as opposite temperature gradiento *Positive wall temperature gradient refers to cases where the wall tem.perature decreases toward the bottom. ~*Negative to cases where it increases.

-3810 - 10 to 9 L __ _ —-- EXPERIMENTAL DATA /,// 10 -- LINEAR DISTRIBUTION o / / / 0 PARABOLIC DISTRIBUTION t // / - ~-CALCULATED -_3 3//, / 7- DISTRIBUTIONS \ /// v; \,, // V~o —. — /./////, /. to / PARABOLIC (3) /// / 7t~~~~~~~~~~~ ~~/, t o t# -3 / to //.I,/ t. 10b. 10 25 ~/~/ ~/., / to' t, 0* =/' 50.'0/ /I. // // //// o // / / /^' 10 3 4 -' 7 - 8 10o o' o10 10 0o o0 10o t 0 Figure 21. Non-Dimensional Heat Source vs. Overall Temperature Differential, Experimental and Calculated Data, Positive Wall Temperature Differential.

-391.0 0.63 1 2 3 4 5 6 78910 to. T052 9 I I I S //// 10 I O / o //// 0og /. NEGATIVE TEMP. DIFFERENTIAL 4 / 10 w z / 0 10 105 06 107 8 NON-DIMENSIONAL OVERALL TEMPERATURE DIFFERENTIAL, to Figure 22. Non-Dimensional Heat Source vs. Overall Temperature Differential, Calculated Data, Negative Wall Temperature Differential.

-40 - With the experimental equipment on hand it proved impossible to impose such a gradient upon the test section inner wall. However, with a sufficiently small cooling stream directed downward in conjunction with a sufficiently strong internal heat source, it seems that such a gradient would be inevitable. In this case the boundary layer temperature would increase as it descended so that an apparently unstable flow pattern would result. Perhaps it would separate into various discreet circulating cells arranged axially rather than the single cell which has so far been observed. The analysis yielded results over a rather limited range of these inverted wall gradients (i.e.: t./to < 1), as shoTwn by the envelope curve in. Figure 22, This curve is the*adivision between cases where the core thickness is positive as physically required, and cases where the calculation shows negative core thickness. Its position is a function of both qv and temperature ratio at all points. However, the dependence upon temperature ratio becomes much greater for those ratios less than unity. Over the range explored (up to qv = 2 x 109) the lowerst ratio for which a physically possible an:swer could be obtained was about 0.5. This limiting ratio increases as qv decreases, aLd becomes unity at qv of about 5 x 10 2 The minimum overall temperature differential tE, for a given qv, occurs at constant wall temperature where to /to is 1o The calculation shows that. either a positive or a negative wall temperature gradient results in a larger overall temperature differential. There is no apparent mathematical limit in the direction increasing positive wall temperature gradients, and calculations have been made for tr /to values as high as 60. The effect of increasing the gradient is a dimu.n.taio of the boundary layer thickness for a given qrv so that at high values of the ratio (about 40), thelimiting qv is reduced from about 2.5 x 102 to about 102 Although from thte viewpoint of the computer program, physically meaningful results can be obtained up to the envelope shown, the validity of some of the assumptions upon which the solution was based becomes quetionable as the limit is approhed. The assumptions of temperature and velocity gradients concentrated in the boundzry layer and uniform across the core region becomes doubltfuli when the core occupies only a small portion of the vessel around the centerlinee Because of these assumptions, for example, as the envelope is approached the analysis shows a core upward velocity considerably greater than the boundary layer downward velocity. However, this does not seem physically reasonable or in agreement with the observation.

-41An.-other liTitati,:on upon the validi.ty of the solution at high wall tenmerature ratio is the assurptiLon that heat conduction in the axial direction is:negligible. Order of mrn;itude calculations show that this is reasonable i. most cases si-ce the gradient in the axial direction is sma l compared with the radial gradleJ-nt near the wall where conduction is importa.t. However for very large wall. tempera.ure gradients h.lis is not the case. Also t.his assumption is least defelsible for vessels with sma.ll len:.gth. to; d.iam.neter ratio s ince the end effects is rmore pronounced Fc..r example,, t. i, t..e l.ength/diameerg d = 10_'boun.dary layer thickness /radius = C0 2 and t /O = 20 te approximate ratio be;tween radial gradieal in the boun.da y. l.aer and axi.al wall gradient woul; d be 2 x.O x 1o x 2 5 0.2 20( In thi`s case the assrump-tion of negligibl.e axial con:.duction may be questionabl'eo'.uti-,J.er reduction of length to di aeter ratio makes.t more so. At tthis time no inor.ration is available o: the ef'ect of this assmption.r on'the overall res -1t So Co Coverage of/ Computer Solutions of Boun;dar y Layer Solution he co.mpute':r soiutions cover the ran'ge of gv from 2 x L09 to about 102 (lower lir`it of applicab iity of solution) for t, /t ranging from about o 5 (lower limi.t of. applic.abitL. ) to aboutt 60o In general the Lnvestigation has been concenx.ttrataed upon a unLforn, heat source distribution.and lin.ear wall terrperat-ure gradiert. How^ever, to afford compariso..s over lilmted r-anges9 compuations have been made for a half-cosine wall temperature distribution (Figu.e 235), since this reseazrbles closely the expe-ri.mental distributions^ and. also for varcious heat source distributions These are desc:ri.fbed 2in detail in a later sectl ion.o Do Ext;ension to Low qv (Fully-Developed Region.) lhe t3wer limit of the boun-dary layer solution previously described is reached w'-he:. at some point within the,vessel2 the boundary layer has grown to the ext:e:,-nt. that it occupies the en{;tire tubel I:. general, this occurs somewhere in;the mid.-ortiocn of the t~ube, s:..nce the boundary layer thick.:ess decreases in the ex'.:reme lower portion f If qv val.ues below the envelope shown inr Figures 2! 22 are considered, a soluti.on w.l1. result wherein the boundary layer grows from zero thickness at the top to a thick ess equal to the radius at, some po.int within the vessel. At this poiInt the core thickness is necessarily zero andaccordin..g to t-hiEs solu;ions becomes negative at points

-421.0 i i 0.8 UJ \ Iw O \ \ \ \ | \ \\ | 0. 0.6 IJ z \ \ \ 0.2 -- - EXPERIMENTAL 3= to --- LINEAR DISTRIBUTION... Cosine DISTRIBUTION \ 0 0.2 0.4 0.6 0.8 1.0 AXIAL POSITION, x/I Figure 23. Various Wall Temperature Distribution.

-43 lower along the axis. Thus the solution is physically meaningless below the point at which the boundary layer thickness becomes equal to the radius of the vesselo For such a case a combined solution may be used. The original boundary layer solution is used for t-hat portion of the vessel where the boundary layer is growing with axial distance. Below that point where the boundary layer fills the entire cross-section, it is considered to be of uniform thickness equal to the radius so thatt the flow becomes "fully developed" ia the sense of pipe flow away from the entry section. A procedure similar to that of the original analysis can be used for this portion except that the boundary layer thickness is no longer a function of position but is constant. Details of the method are given in a previous paper. A difference equation relating the non-dimensional temperature differentials, heat source and axial position can be derived which can be solved directly without iterative procedures. This "fullydeveloped solution" is then used for the lower portion of the vessel. It is easily sh.own- that i.n this portion, starting from the bottom of the tube, the non-dimen.sional temperature difference between wall and fluid centerline rapidly approaches qv/4, a value that also applies to "rod flow"; i.eo: radial transport of heat by conduction only. Therefore, at the poi.nt. of joining of the solutionsg it is necessary that the boundary layer portion also g ives a centerline to wall differential of this amount. A suitable soiution. for the boundary layer portion can be obtained by iteratio.:r to obtain this-. relation between core thickness and temperature differential. The solution for each portion individually assures that the conservation Cof rmass, momentum, and energy are observed for planes normal to the axis at ay axial position within that portion. Hence the combined soulution makes this assurance for-the entire vessel, and also assures continuity of temperatures, wall heat flux, and velocities at the point of joiin.g of the solutions. The a.alysis for the "fully-developed" portion of the vessel is applicable both for con.stant and variable wall temperature, provided only that the axial temperature gradients are not large compared with the radial.3 Regardless of the wall temperature differential between wall and centerline at any axial position in the fully developed portion is approximately qv/4 as predicted by an analysis based simply on conduction theory.

-44E. Extension to Non-Steady State It is believed that the steady-state analytical results so far achieved can be extended to the prediction of non-steady state heating and cooling rates of closed tanks by assuming that the flow regimes in such processes are a succession of quasi-steady states. Order of magnitude calculations of the time intervals involved.indicate that this is not a:n unreasonable assumption. No numerical results of this type are as yet available~ Fo Results of Computer Calculations 1. General Scope For a given qv and non-dimensional wall temperatures, tw specified at the centerline of each of the discs of the vessel considered, all other quantr.tities are uniquely determined within the assumptions of the analysiso In addition to the overall heat transfer results, the computer program provides detailed information on local temperature, velocity, and heat flux. 2< Heat Source vso Temperature Differentials The overall non-dimensiornal heat source vs. non-dimensional temperature differential results are shown in Figures 21. 22 for the boundary solutiom qLv is the ordinate and tgo is the abscissa. This latter is measured from the fluid at the top of the vessel centerline to the wall at the bottom, and represents the maximum temperature differential existing in the fluid. The curve parameter, tro/to represents the proportionate stren.,gth of the wall temperature gradient in the positive direction. Within the range of the solution, the curves appear as virtually parallel straight lines on the logarithmic coordinates. Because of this fact it is possible to formulate relations as below to represent the curves. = k t The best values for these constants over the entire range investigated for

-45the curves of uniform heat source distribution and linear wall temperature gradients are given below (from Reference 17). TABLE II CONSTANTS FOR q = k t~ for 102 < < 1010 o tS /to k n 110 0.921 1.22 350 0.314 1' 24 10.0 0.0791 1.24 200 0.03557 1.25 4o.o 0.0172 1.25 The exponent, 1.25) appears In the exact solution for the natural convection flow along a flat plate in an infinite fluid. The temperature differential corresponding to a given heat source st;rength increases as the proportionate.wall temperature gradient increases. However, as shown in Figure 18, the temperature differential at a given axial position between fluid certerline and wall actually decreases. This car. be explained by the fact that the boundary layer thickness decreases as the proportionate wall temperature gradient increases anzd hence smaller fluid to waal temperature differentials are required across the reduced boundary layer to remove a given quantity of. heat. As qv is reduced beyond the range of applicability of the boundary'layer solution, the presentation of overall heat transfer resul.%s in terms of ratio of temperature gradient becomel less meaningful. Under this condition, the centerline to wall temperature differential becomes virtually a constart over most of the vessel length, increasinrg somewhat near the top and decreasing near the bottom, Its value over most of the vessel is simply qv/4 and the overall differential, t~o is controlled almost entirely by the wall temperature distribution, The increase at top and decrease at bottom becomes extremely local as qv is reduced substantially

-46below the boundary-layer solution range. This is shown in Figure 24 where non-dimensional temperature differential, wall to centerline, is plotted against axial position for various qv at constant wall temperature. The general character of the results for a positive wall temperature gradient and the fluid temperature follow the wall temperature in the same manner. Returning to the boundary-layer solution (Figures 21 and 22), curves were run in addition. to those previously described to determine the results of various non-uniform heat source distributions and of wall temperature distributions other than uniform gradients. a. Heat Source Distributions The heat source distributions listed below were investigated, In all cases the heat source was uniform for planes normal to the axis at a given axial position. All these computations are limited to the range of the boundary layer solution and constant wall temperature. 1) Constant volumetric heat source gradient from maximum at top to zero at bottom, 2) Constant volumetric heat source gradient from maximum at bottom to zero at top, 3) Sine distribution - zero at ends and maximum in center ReSults are shown in Figure 21. The constant gradient distributions are signified by anl inverted and an upright triangle respectively. These are listed below in the order of decreasing to necessary to remove a given heat quantity. a) Constant gradient, maximum at bottom b) Sine distribution, c) Uniform distribution, d) Constant gradient, maximum at top. The relatior of the constant gradients to the uniform distribution can be explained as follows: It is advantageous to add the major portion of heat near the bottom since the core temperature is increased by virtue of the heat addition. Hence, the mean temperature differential between fluid centerline and wall is incrxeased and heat transfer is facilitated. The maximum difference in tyo due to the variation of heat source distribution is a factor of about 53

Cr w 1.I4 ^q^ 2z - ~ q =10_ __ _ _ _ _ w 0V —-------- z z ------- 0.8 qV I 0 w 00 I 0.6 —U j' 0.4 0 0 0 I 0 a. NO -IESOA AXA OITO 0 Co0.2 z 0 cn 0.002.2 00.4 005 00.... z 0 z Figure 24. Non-Dimensional Wall to Centerline Temperature Differential vs. Axial Position, Fully Developed Flow Regime, Constant Wall Temperature.

-48Another heat source distribution was investigated. It was assumed that all the heat is added into a very thin radial plane (normal to the axis) at the bottom. This condition is analogous to the case of a vessel closed at the top and open at the bottom to an infinite reservoir maintained at a temperature greater than the wall temperature, and the volumetric heat source is zero. Since for constant wall temperature, this is the case investigated by Lighthill, it is possible to compare these results with his exact solution and also to investigate the effect of variable wall temperature. It was found that the results obtained for constant wall temperature from the computer program were identical to those given by Lighthill. These are shown by the curve, L _ in Figure 21. To compare with the results in the lighthill's paper, it is necessary to compute Nusselt Number from Nua = qv/2t *. In principle it is possible to use the program to compute results for the "Lighthill Case" with any desired wall temperature distribution within the bounds of applicability of the boundary layer theory. However, time has not been available to obtain these results within the present project. b. Wall Temperature Distribution The majority of the calculations consider either constant wall temperature or constant wall temperature gradient. However, the experimental results were taken with more or less parabolic distributions where the wall temperature is fairly constant near the top of the vessel and falls rapidly toward the bottom. Consequently a temperature distribution of this sort was run (half-cosine) to afford a comparison with the linear distribution of the same tn /to. As is shown in Figure 21, the overall heat transfer behavior is qui'e insensitive to the wall temperature distribution. It was also found that the local parameters were little affected. 3. Velocity Figures 25 through 28 show the maximum boundary layer and non-dimensional core velocities resulting from the analysis over the full range of qv, including both boundary-layer and fully-developed regions. Figure 25 for constant wall temperature applied to qv from 2 x 109 to 106 for various heat source distributions. Their effect upon the magnitude *t6A is identical to Lighthill's t, in this case.

-492000........ UCORE /, _ _ __ UB/. q Lx 1800 - - MAX. // / / / \ \v / //'f"UNIFORM q 1600 / / / D A 1400 SINE \ \ 1200 - f -- -^\ // \ \ 1000 z / \ \UNIFOR 400 SIE\ o 4x \ \ ~, / V NFRMCw\ \IO 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/1 Figure 25. Non-Dimensional Core and Boundary Layer Velocity vs. NonDimensional Axial Position, Constant Wall Temperature, Comparison of Sine and Uniform q,\ Distribution.

I00 ko i UCORE ^*- ^ """ ~ ~'~'~-"~~-~~^ B.L. MAX 60 40 / I I0 ~,< 40 -— ^^^ ^ ^ —----- --- V —-~ —-— "^-^ ---- -^^ -- ^ --- 4 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/-t Figure 26. Non-Dimensional Core and Boundary Layer Velocity vs. NonDimensional Variable Linear Distribution of Overall/Radial Temperature, Uniform qv.

qV:I X x4 q — -- =IX104 101 —-~^^ --- I --—' ---,I UCORE I0 - NON-DIMENSIONAL..... UXIL POII /MAX. " I Ii/ //' qv= jxIo Z 4,qv: I xl. qv=Z Ix I 0 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/I Figure 27. Non-Dimensional Core and Boundary Layer Velocity vs. Axial Position, Constant Wall Temperature.

10 _______ —------ ^o. |. —^=1 -------— t~~ —-'__UCORE t~~:~~ ~ ~ ~~~~~~~~~~~ I t~-3, UB.LMAx 8 10 I — 3 -------- -— 0 - 4^^0 ^ ^^-^////^ 20 40 0 z 4 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9. NON- DIMENSIONAL AXIAL POSITION, x/I Figure 28. Non-Dimensional Core and Boundary Layer Velocity vs. Axial Position, Linear Variable Wall Temperature.

-53and distribution of velocity is not great. However, the velocity magnitude decreases by about the 0.5 power of qv over this range and the ratio between maximum boundary layer and core velocity decreases by about the 0.1 power. In this qv range, the boundary layer velocity is a factor of 3 to 4 greater than the core velocity, somewhat greater than the experimental factor (Figure 12). This factor decreases and reverses itself as qv decreases. Figure 26 shows the effect of wall temperature gradient upon the velocities at a given qv. It is noted that the effect of increasing wall gradient is a reduction of velocities. Figure 27 shows boundary layer and core velocities at constant wall temperature in a lower qv range (102 to 10 ). As qv is continually decreased, the results show that the core velocity eventually becomes greater than the boundary layer velocity (the factor is about 5 at qv= 102). Figure 28 shows the effect of wall temperature gradient for low qv' Again, increasing gradient reduces velocity. The qv range covered by Figures 27 and 28 is the lower end of applicability of the boundary layer solution and also includes the fully developed regime (qv = 102 curve). Anticipated velocity profiles for infinitely long tubes with both laminar and turbulent flow are given in the literature. These cannot be presented in the framework of the present analysis because infinite length implies qv = 0 4. Boundary Layer Thickness In the range of application of the boundary layer solution, the boundary layer is assumed to start from zero thickness at the top of the vessel. The analysis shows a very rapid growth near the top followed by a leveling off to a thickness which is maintained approximately throughout most of the length. The mathematics of the solution is such that the boundary layer thickness falls very steeply near the bottom also to zero. However, the predicted behavior at the vessel ends is probably not physically significant. The flow model employed is one-dimensional, and cannot accurately predict the flow pattern in the vessel ends where cross-flow must predominate. Figures 29 and 30 show typical boundary layer thicknesses plotted against axial position in the high qv range. The effect of variable wall temperature and variable heat source distribution is also shown.

0.3T Cl w | o6 UNIFORM qV I1~~~~~~~~~~~~~ I / 0.2 j SINE O" 0.1 0. 0. 0.7. Co ^P tUNIFORM qv vL <01 C:4 z SINE 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 29. Non-Dimensional Boundary Layer Thickness vs. Non-Dimensional Axial Position, Comparison of Various Axial Heat Source Distribution.

.3 tqo qvI =I x z o'20.0 0-. " 0 40.0 ca z 0 z _____ _ 0.1.2.3.4.5.6..8.9 1.0x Figure 50. Non-Dimensional Boundary Layer Thickness vs. Non-Dimensional Axial Position, Uniform Heat Source Distribution.

-56 - Figure 51 shows similar results for low qv above and below the boundary-layer solution level of Figure 21. The curve for = 10 is plotted as computed from the machine program showing negative core thicknesses over a major portion of the vessel. Hence it is not physically meaningfull. Figure 16 shows the comparison between analysis and observation. 5. Wall Heat Flux In many physical applications the distribution of wall heat flux is of great importance since it controls the temperature differential across the vessel wall. This is limiting in some nuclear power reactor designs because of the corresponding thermal stresses. Also, knowledge of its distribution is necessary to the solution of a problem. Suppose, for example, the heat source strength and distribution and also the coolant inlet conditions and flow rate are known. In this rather typical case the wall heat flux is a coupling parameter between, the heat balance equations and consequent temperature rise of the coolant, and, via the wall differential, is the internal wall temperature distribution. A trial and error solution could be made whereby the heat flux distribution is assumed, the coolant temperature distribution can be calculated, and then the internal wall temperature distribution is obtained. If, per data of the sort herein given. the assumed heat flux distribution is consistent with the calculated internal wall temperature distribution and the known heat source distribution, the problem would be solved. In all cases the analysis shows an increase of wall heat flux toward the top of the vessel and a decrease toward the bottom, even for those cases where the heat source is predominently in the lower portion (as the "Lighthill case"). The results in the high qv range are plotted in Figures 32, 335 and 54 showing the effect of variation of heat source strength, distribution, and wall temperature distribution. The sharpness of the rise at the top and the decrease at the bottom is increased by a reduction of qv or an increase in proportionate wall temperature gradient. Similar results for the low qv range (above and below the limit of the boundary layer solution) are shown in Figures 35 and 36. The trends. mentioned above persist. For very low qv, the wall heat flux is virtually constant over most of the tube (for uniform heat source distribution) but shows a sharp peak at the top and drop at the bottom. However, these deviations from the mean are extremely local, covering only a few percent of the length for qv - 102. Such local peaks may not be of physical significance. They would induce axial heat flow in the wall which would reduce the otherwise predicted wall differential in this region,

.. 1.4 V). (0 I I.-1 1.0 -TUBE CENTERLINE. 4 03 >z 0 0.62 0.3 0-4 0-5 0.6 0- 7 — 8 0.9 10 o 0.4 —..% 0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 NON-DIMENSIONAL AXIAL POSITION, X/. Figure 51. Non-Dimensional Boundary Layer Thickness vs. Axial Position, Constant Wall Temperature.

-5820 I0 w tc 8 z 6 T 2 — 5 F-D 4 I3 2 "J I5 q4v1lxI0 -'- ^ qv ~ 4x107 o q0u~ x 10 L\ __-qv' I x icP41 N 0.8 lq'v4x O \ q -Ix 0.2 ------- ----- 0.. 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 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 / Figure 32. Normalized Wall Conduction vs. Non-Dimensional Axial Position, Constant Wall Temperature, Uniform Heat Source Distribution.

-5920...... | -— D qv DISTRIBUTION n- 10 < -- -7 qv DISTRIBUTION w 8,, 8 & i rSINE qv(lx106) DISTRIBUTION aU 6 c \-SINEq (4x107) DISTRIBUTION 6 O. S V:3 \ —SINE q(2x 109) DISTRIBUTION cn 5 z. | I N E qVV ( 2 DISTRIBUTION10 ) \_ o 0.8 0- 1.0 0. 2LN qv Ixt0DISTRIBUTI ON / SINe Source q D ISTRIBUTION-tUTION7 C z 0.6 0.5 SINE q (2 xl0) DISTRIBUTION N 0 0. 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 5533. Normalized Wall Conduction vs. Non-Dimensional Axial Position, Constant Wall Temperature, Variable Axial Heat Source Distribution.

-60 - 20 CALCULATED RESULTS (AVERAGED) IxlO6 q, 2x10 I 0 w 8 cr. W 6 0 4 3 Z ~ I.0 10 —------ to 3.0 2 010.0 0. I....... w 0.8 —- PARABOLIC N ___ N - o: 3.0 S 0.6 to o 05 z \ 0.4 0.3 0.2 -.-. —--- 0.1.2.3.4.5.6.7.8.9 1.0 NON-DIMENSIONAL AXIAL POSITION, x / I Figure 34. Normalized Wall Conduction vs. Non-Dimensional Axial Position, Linear Wall Temperature Distribution, Uniform Heat Source.

-6130 20 10 9 8 7 6 5 4- t —0 = — 0 to 0 40 z 0 2 40 —1.0,-i I 1.0 -.8.2.4.6..7.6.5.4.3.2 0.2.4.6.8 1.0 NON- DIMENSIONAL AXIAL POSITION, X/t Figure 35. Normalized Wall Conduction vs. Non-Dimensional Axial Position, Linear Variable Wall Temperature Uniform Heat Source.

-62\ I U 4 0: =10 Z I —. z V Vl02 Z crq0 AXIAL POSITION, x/l Temperature, Uniform Heat Source, Fully Develoed Flow. w D q=10 \ 0 0.02 0.04 0.06 0.7 0.8 0.9 1.0 AXIAL POSITION, x/I Figure 56. Normalized Wall Conduction vs. Axial Position, Constant Wall Temperature, Uniform Heat Source, Fully Developed Flow.

-63Since the temperature differential between fluid centerline and wall is at a maximum at the top and the boundary layer thickness is at a minimum, the radial temperature gradient at this point is increased. Hence, the greatly increased heat flux at the top for a uniform internal heat source distribution is expected. Figures 35 and 36 consider the only uniform qv distribution in lower range. However, heat transfer for this range of qv where the flow pattern is fully developed over most of the vessel is substantially as predicted by conduction. Hence the wall heat flux distribution would closely follow the internal heat source distribution, with the exception of the extremely local variations at the ends which were previously discussed. G. Comparison with Experimental Observations Order of magnitude agreement between the predictions of the laminar flow analysis and the experimental non-laminar observations exists with respect to velocity, boundary layer thickness, and wall heat flux distributions. These comparisons are illustrated in the previous figures. The most important point of comparison (Figure 21) is between the overall temperature differentials for a given non-dimensional heat source and wall temperature distribution. The experimental temperature differentials are in all cases less than those derived from the laminar analysis. It is presumed that this is primarily a result of turbulent mixing in the actual case. Aside from the effect of turbulence, the assumption of a single mean temperature as applying to the entire vessel seems a likely cause of significant discrepancy. Because of the large temperature dependence of viscosity, the temperature differences were such that a typical viscosity variation from minimum to maximum within the test section might be as much as 20%. The factor between experimental and analytical, non-dimensional temperature differentials is approximately 0.5 for the high q. range and 0.9 for the low range. It is felt that if this type of analysis is used for a different configuration, heat source strength, or wall temperature

-64distribution, these approximate factors can be applied to give at least an engineering estimate of the temperature differentials to be expected. Also, the detailed comparisons given for wall heat flux distributions velocities, and boundary layer thicknesses can be used as a guide for the desired quantities. IV, CONCLUSIONS The experiments described provide a wide range of data for the engineering evaluation of a physical system consisting of a vertical cylindrical vessel, closed at the ends, with internal heat source. Coolant and heat source distribution must be axially symmetrical. The laminar flow analysis allows a further extrapolation of these data to conditions other than those covered by the experiments. In addition it should allow an extension into the problem of unsteady-state heating and cooling of closed vessels.

BIBLIOGRAPHY 1. Hammitt.,. F. G. ASME Paper No, 58-SA-30. 2. Hammitt, F. G. ASME Paper No. 58-A-212. 35 Hammitt, F. G. and Brower, Eo M. University of Michigan Research Institute Report, IP-340. 4, Brower, E. M. Master Thesis, Nuclear Engineering Dept., University of Michigan, 1958. 5. Yip, So Master Thesis, Nuclear Engineering Dept., University of Michigan, 1959. 6. Galbraith, D. Master Thesis, Nuclear Engineering Dept., University of Michigan, 1959, 7. Quarterly Journal of Mechanics and Applied Mathematics, 6, (1953), ppo 398-439. 8. Hartuett, J. P., Welgh, W. E, ASME Paper' 56-A 113. 9. Siegel, R., Norris, R. H. ASME Paper No. 56-SA-5. 10. Poppendick, H. F., Palmer, L. D. Nuclear Science and Engineering, 3, No. 1, (Jan, 1958). 11. Mohr, D. Master Thesis, Nuclear Engineering Dept., University of Michigan, 1958. 12. Schmidt, E. and W. Beckman, Tech. Mech. Thermodynamics, 1, (1930), PP 34-39 and 391-406. 13. Saunders, 0. A. Proc Roy. Soc. A-157 (1936), pp 278-91. 14. Murgatroyd, W. ED/R 1559. Atomic Energy Research Establishmento Harwell Berks, England, 1954. 155* Martin, B. W., Cohen, H. Brito Journal Applied Physics, 5, (March, 1954) pp. 91-95. -65