THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING HEAT AND MASS TRANSFER IN CLOSED, VERTICAL, CYLINDERS WITH SMALL INTERNAL HEAT GENERATION AS APPLIED TO HOMOGENEOUS NUCLEAR REACTORS By Elayne M. Brower May, 1958 IP-290

ACKNOWLEDGEMENTS The author wishes to express her sincere appreciation to Dr, F. G, Hammitt for his encouragement and aid in the use of the digital computer program and to Dr. H. A. Ohlgren for his aid and advice in the initial phase of the problem, The financial support of the Atomic Power Development Associates over a part of this program is also gratefully acknowledged. ii

TABLE OF CONTENTS Page ACKNOWLEDQGEMENTS....,.... OO.................o O..... ii P T..e...... * e.ooeee o ee eeee oeeee a *ee oeeoeeeea ii LIST OF TABLES............. o.. o0 oo...........,o*......... iv LIST OF FIGURES 0 o... a...0............................. O O. O0 V NOMENC LATUE 5.... o...............,...... o..................... vi INTROENCLATIONo..O..ooooo.ooooooo *oo.*. *. O~**.....*. *....* 1vi INTROPUCTIONOOO O O OO - o O o O O -O 0 o O O 1 REVIEW OF LITERATURERE........o.. o...........ooooo.ooo o.o 3 ANALYTICAL APPROACH..........o........ o....OO..... o o........ 4 RESULTS....... o 00......... o o.. o........... o. o 12 CONCLUSIONS..............,.............................. 39 APPENDIX.......6...................0.... oo,........0,..... o 40 REFERENCES I..ON......... * oooo....o............ooo. eoo.o o.ooe 348 iit

LIST OF TABLES Table Page I Su.wSary of Results, Constant Wall Temperature 46 II Summary of Results, Variable Wall Temperature 47 iv

LIST OF FIGURES Figure Page 1 Sketch of Cylindrical Test Section Assumed for Analysis.......... O 0 o..........o O..O o............... 6 2 Velocity and Temperature Profiles Assumed in Analysis. Oa.o...a. O***.~.O.~.S......e 7 3 Non-Dimensional Heat Source Strength Vs. NonDimensional Maxmum Temperature Differential.o a.... 13 4 Non-Dimensional Boundary Layer Thickness Vs. Axial Position, Constant Wall Temperature..........o 15 5 Non-Dimensional Heat Source Strength Vs. Maximum Non-Dimensional Boudar yer. 0 0 0 0..... o... 1.6 6(a-c) Non-Dimensional Boundary Layer Thickness Vs. Axial Position, Variable Wall Temperature. o o............. 17-19 7(a-d) Non-Dimensional Centerline Temperature Distributioe Vs, Axial Positiono...... a *... o... o o...* * 20-25 8(a-c) Non-Dimensional Radial Temperature Differential Vs. Axial Position- Variable Wall Temperature. o..,0 oo.. 25-27 9(a-d) Normalized Wall Conduction Vs. Axial Position....... 29-33 lO(a-b) Non-Dimensional Boundary Layer and Core Velocity Vs. Axial Position.,..... e o. 3. 57 ~. ~ e e 0O. e.. eo o o S oe,$ oe.V "

NOMENCLATURE cv, Specific heat of fluid g Acceleration due to gravity k Thermal conductivity la Length and radius of tube Nua Nusselt Number based on radius, ha/k 9~ D t'Dimensional volumetric heat source qv Non-dimensional volumetric heat scrce, Qv a6 a g p V K Cv Ra Rayleigh Number based.on radius and maximum temperaRaa ture differential, a g a3 (Twalin - Tfluidax) VK R r Dimensional and non-dimensional radial coordinates T Temperature t Non-dimensional temperature differential between wall and centerline at any axial position, ( aa4 A T; y K 1 subscript o refers to the top of the tube tr Nondimensional temperature differential between wall and any radial point r tw Nondimensional temperature differential along wall between bottom and any axial position; two is between bottom and top of tube, ioe. maximum tw vi

NOMENCLATURE (CONT' D) t Non-dimensional temperature differential between fluid at centerline and wall at bottom, t + tw, t = to + t Ra a a maximum non-dimensional temperature differential in systemr Ulu Dimensional and non-dimensional axial velocity X,x Dimensional and non-dimensional axial coordinates a Coefficient of volumetric expansion pffiB~ Non-dimensional core thickness; 1-3 is the nondimensional boundary layer thickness y, 7,t F(), Non-dimensional functions defined in text G (), H(p, ) F~K ~Thermal diffusivity, k p cv v Kinematic viscosity p Density of fluid vii

INTRODUCTION At present the interest in the field of homogenous nuclear reactors both aqueous and liquid metal, is increasingo However, from the standpoint of a complete powerplant based on- such a reactor it appears that considerable data would be desirable, and in fact necessary, on the heat transfer characteristics of fissionable fuels in closed vessels. The most useful consideration at the present seems to be a system where the heat is transferred either from a fissioable fuel inside tubes to a coolant surrounding them or from the fuel in the surrounding areas to the coolant in the tubes, In either case, considerable data on the heat transfer characteristics of the heat generating fluid is desirableO This investigation was undertaken in an effort to promote some further unranerstaing beyond the original investigation on this subject by HamittlI2, into the natural convection heat and mass transfer characteristics in closed vertical vessels with internal heat generation. The work is based on the procedure developed by Hammitt and extends the analytical solution to a lower range of the parameter q, i.e. a in-diaensional parameter composed of the tolumetric heat source, the length to diameter ratio, and various fluid properties. This extention is of particular interest since it appears that the small passage diameters under consideration for various power reactor concepts result in qv values below those previously studied, even with the very large voltr ic heat fluxes anticipatedo Although still restricted to flids with a Prandtl Number of approximately one (aqueous solutions) this -1

-2represents a first effort in extending the previous study (References 1 and 2) in the direction of liquid metal fuels. For such fluids the non-dimensional heat source term, qv, is an order of magnitude lower than that of an aqueous fluid with the same volumetric heat flux and core dimensions. The present study relates to qv values of the order that might be anticipated in certain liquid metal fuel reactor designs,

REVIEW OF LITERATURE The work done up to the present in the field of natural convection has covered many geometrieso However, until the inception of the homogenous nuclear reactor little interest was taken in closed vessels nd more especially in closed vessels containing a fluid with internal heat generation. The approach of Lighthill3 considers a tubular vessel of moderate length to diameter ratio, with one end closed and the other end open to an infinite reservoir, and a constant wall temperature. In this analytical investigation no internal heat source and no variable wall distribution are considered. Ostrach4 extended one phase of the work by Lighthill considering a linear axially-varying wall temperature. An analytical study of a vertical closed tube with internal heat source was made by Murgatroyd5. In this case the tube considered had a very large length to diameter ratio so that only fully developed flow, i.e. no change in conditions due to axial position, was considered. The end effects not considered here are thought to be of importance in the reactor concept and hence this solution could not be applied directly. Experimental investigations were carried out by Martin6 in an attempt to validate the analytical solution of Lighthill. In general, the types of flow regimes were confirmed, but the experimental heat transfer data exceeded the predicted by a factor of approximately 2. Haas and Langsdon7 carried on some short experimental work for the Atomic Energy Commission which compared very well with the experimental results of Hammitt2 -35

ANALYTICAL APPROACH The method presented in Reference 2, and upon which this work is based, considers a completely closed tubular vessel with the heat generated in an arbitrary axial distribution of heat source strength and removed through the walls under an arbitrary temperature distribution, Since the heat source term in a power reactor is considerably higher than conventional sources it was assumed that a modified boundary layer solution applied. Lighthill5 pointed out that internal flow could be expected to fall within one of three regimes depending on a parameter which is proportional to the product of the Rayleigh Number and the radius to length ratio of the tube. These regimes are: 1) Similarity Regime - if the parameter is small, then the temperature and velocity profiles are fully developed and their shapes (not their magnitudes) do not vary with axial position~ 2) Boundary-Layer Regime - if the parameter is very large, the boundary layer does not have sufficient extent to grow into the central portion of the tube and thus occupies only a negligible portion of the radial extent of the tube. 3) Intermediate Regime - those cases in which the boundary layer fills a substantial portion of the tube radially but the "fully developed" regime is not attained. The boundary layer regime, type 2, of Lighthill was extended in References 1 and 2 to include an internal heat source, both ends closed, and an arbitrary wall temperature distribution, This effectively means that the absolute tube dimensions are large, the heat source is strong, -4

-5the length to radius ratio of the tube is small, or the thermal diffusivity and kinematic viscosity of the fluid is small. These factors tend to limit the analysis to aqueous solutions, however, it is felt that some of the trends indicated would apply to liquid metals, especially considering the low values of the overall non-dimensional heat source term qva The analytical solution which was developed appears suitable as qy increases without limit since the boundary layer becomes increasingly thin. However, as qv decreases, the boundary layer fills as increasing portion of the cross-sectional area of the cylinder so that at a sufficiently low qy, presumably depending o the wall temperature distribution, the solution loses significanceo It is the purpose of this paper to explore the lower range of qv, below that considered in Reference 2, and to define the limits for this type of solution, i.e. boundary layer type. To analyse this problem the following assumptions were made: 1. Boundary layer approximations apply, i.e. partial derivae tives of quantities normal to the wall are large compared with those parallel to the wall. 2. Inertia forces of fluid are small compared to shear forces. 5. Boundary layer thickness of velocity and temperatures profiles is the same, ieo Prandtl Number near unity. Integral equations for the conservation of mass, momentum, and energy were written for each radial disc or element of the cylinder (Figure i). Velocity and temperature profiles as shown in Figure 2 were

I a | *^ —" -' I /-OVERALL TUBE tS F- IRADIAL DISC o1i / X CONTROL VOLUME ~~II Jlr, Y Il _j I i I t - I\R l/ I I |. s Q-0, UNIFORM HEAT _I Fie;ure 1. Sketch of Cylindrical Test Section Assumed for Analysis, w HEAT NDUCTION SOURCE F~~~~~~~~~Igr 1SkchoCyiriaTetetionAsmd o nlss

-7d<~O TEMPERATURE dr t =0 —O I/' r /> I —rO — /,.. < I. X y —--— Y -- ^____ ~ ~ I VELOCITY I coI CORE BOUNDARY LAYER =/-f ---------- Figure 2. Velocity and Temperature Profiles Assumed in Anaylsis.

-8assund and iade to satisfy the physical conditions at the wall, centerline, and the boundary layer - core interface for the quantities and their first derivatives.o Hauitt miodified these equations to include a vowie heat soruje and an arbitrary wall temperature distributionb Under these conditions the integral equations for the conservation of mass, momentum and energy are: Jf- Lu dcL O (1) )ridft. + + 0(2) 0 (21) Ji ^.() (^ - f /L tL t((3)n~= ^ LC,, t^ =n /Bjt\ - U(5) ax 4> 1 /Jn- 1 > The velocity and temperature profiles following Lighthill are assumed to have the form: ^ - yr 0< (< tto c(3 (/ t(x) 1p<^<~

-9Substitution of (4) ana (5) into (L) and (2) gives the terms () and 6(tp) as in both the work of Lighthill and Hammitt. The difference between the two analyses evident at this point is that the temper* ature term, t, is a function of axial distance, x, in Hammitt's case, while it is a aonstant in Lighthillls work, The evaluation of the integral energy equation (siown in detail in the Appendix) takes into account the internal heat source term and the variable wall temperature distribution. As can be seen from this derivation the term involving the wall temperature drops out and the equation left is in t only, the temperature between the centerline and the wall as a function of axial position. t ttx, {X (c3] FL (6) Since the equations for the conservation of maIs, momentum, and energy have all been used in the development of equation (6) and since there are still two independent variables, t and x, another equation must be developed to permit a solution. This relation (Reference 1) is that between the axial position and the temperature along the centerline of the vessel. cJt) o (7) This is based on the assumption of negligible heat transfer between the aseending care the decending boundary layer. Patting equation (6) and (7) into a more usable form we have

tam)F(2 - i t r -J AX (8) - O X).r =,, t(x~- *~.,) (9) The temperature term in equation (9) is actually ti by the nomenclature of Figure 1, where t = t+tw, i eo the temperature from the cylinder centerline to the wall plus the axial variation along the wall from Twallmin. Hence, equation (9) can be rewritten as t(x) + tW()] =. ( -F ) (o10) The calculation of these equations was simplified by programming them into an IBM - 650 high speed digital computer. For this purpose equations (8) and (10) were reduced to the approximate difference equations +to to R FN- ae+ t~v-v + = (t1) (t tw)^ (t. tw)-/ - - f ) (w) E(tr,),,_,'GN ~ G,,.,) For a consistent solution A-M0O The cylindrical tube was divided into a series of radial discs (Figure 1) and the independent variables could be read into the machine for each dtacp

-11Thus, it was possible to consider arbitrary non-dimensional heat source and non-dimensional wall temperature distributions

RESULTS The results for a heat flux range, qv, from 2 x 109 to 1 x 105 are reported by Hanmmitt2o The work contained herein covers the q. range from 1 x 104 to a lower limit as defined by the breakdown of the equations and the boundary layer philosophy, ioeo the Case where the boundary layer completely fills the tube, At that point a new analysis need be considered and these equations no longer applyo This occurs approximately at a q of 5 x 102o The results in this report are presented in a similar manner to those in Reference 2 since the subject matter is so closely connectedo One of the most significant results is the relation between the non-dimensional temperature, t^, and qv (Figure 3)0 There is a very slight curvature to these curves over the range 109 to l02y This is so small, however, that it cannot be detected over the range 10 to 10 alone or over the range 105 to 109 alone9 Since this curvature is so slight it can be represented by a straight line to a good approxiMation and the empirical relation tv ^k developed in Reference 2 is valid. Considering the larger qv range, from 109 to 102, more realistic values of the constants k and n can now be determined0 These are listed below: tto /to k n 1.0 0o921 1o,22 3~0 0Q314 1.24 10o0 0o.0791 1o24 20~0 o 0357 125 40o.0 o 0172 1.25 -12

IJ4 sf~ 9 Fj s8 CD / m j 5 0 t9=pI3 10 20 40 ct rd- 3. 3 CD) 1 m2 / / / / /S. / ^-LIMIT AS BOUNDARY LAYER y}0' / / / ^I~ COMPLETELY FILLS TUBE 7 CQ * 6 0 (DC(D 5 p33 P^ e 3 4 5 e 7 689 103$ 4 5 6 7 89 1042 3 4 5 t^0 0

l14These slopes are very close to those found in Reference 2 and the change in the intercept is only slightly greater. Therefore, it appears that the values of k and n of Reference 2 can be used to a good approximation for the wall temperature and heat source distributions not covered in this report. The boundary layer thickness, Figure 4, increases more rapidly at the lower heat source strengths. Whereas for qv = 1 x 104 the boundary layer fills 55 percent of the tube for the constant wall temperature case, it fills 88 percent of the tube at qv = 1 x 105, and for v = 1 x 102 the boundary layer has crossed the tube centerline and, if such could be tolerated, fills 142 percent of the tube. This point can easily be seen by the rapidly decreasing slope of a plot of qv vsa the maximum boundary layer (Figure 5)0 From this curve, as mentioned previously, it can be seen that the boundary layer just fills the tube at a heat source strength of about 5 x 102, Figures 6a, b, and c show that the increase in boundary layer thickness due to linear variation of the wall temperature is also larger at the lower qv's6 In all eases, the lowering of qv causes a progression up the tube of the point of maximum boundary layer thicknesso At = 1 x 106, (1 -)max occurred at x/l = 085 (Reference 2), while at a = 1 x 102, (1 P )max had progressed to x/l = Oo.40, again conaidering the constant wall temperature case as typical. The axial temperature distributions are shown in Figure 7 as the ratio tt /t4o, i.e. the non.-dimensional temperature difference from the centerline at any axial position to the wall at the bottom as related to the maximum value, from the centerline at the top of the cylinder to the wall at the bottom.

- 15 - 0 0 6 Z, I I. 0 0 If,50Dl D ~ 0x z 0! a \ LO 0: < z z C)O OO p t O 0 m Z D L. - -O 0 ( 1-)'SS3N>IIHI U3AV1 AHVCGNnoe O1VNOiSN3JN1G-NON Figure 4. Non-Dimensional Boundary Layer Thickness Vs. Axial Position, Constant Wall Temperature. Position, Constant Wall Temperature.

- 16 - 9 8 - 7 6 5 4 3 2 9 8 7 6 4 3 2 * \. ~~~~~I0p 9 8 7 6 5 4 3 2 I 0.3 0.4 0.6 0.8 1.0 1.2 1.4 MAXIMUM NON-DIMENSIONAL BOUNDARY LAYER Figure 5. Non-Dimensional Heat Source Strength Vs. Maximum Non-Dimensional Boundary Layer Thickness, Constant Wall Temperature.

O0-8 ~3 ~~~~~~~O6 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~t co 20 coo~ 2 /// 40 w 0-4 I 0-2 Qz 2 0 co o 01 0-2 0-3 0-4 0-5 0-6 0-7 0-8 09 10 NON-DIMENSIONAL AXIAL POSITION, x/9 Figure 6a. Non-Dimensicaial Boundary Layer Thickness Vs. Axial Position, Linear Variable Wall Temperature, qv = 1 x 104.

1* 0 < > f U al o: 0 -6 NON-DIMENSIONAL AXIAL POSITION, X/t Variable Wall Te.erature _. = 1 x 10. zI z b.II"z-J C 20 z 0: 400 0-1 012 - 0-3 - 0-5 0-6 0-7 0-8 09 0 NON -DIMENSIONAL AXIAL POSITION,x/I Figure 6b on-Dimensioal. Boundary yer hickness Vs* Axial Psition, inear Variable Wall Temperature, qv = 1 x 03.

U) w 1.2 z 1.0 to 0 o 0.26 0.7 -- z 20 4 a m 40 — _J z NON-DIMENSIONAL AXIAL POSITION, x/e Figure 6c on- ensonal Bundary Layer Thi e Vs Axial ositi, near Variable Wall Tempratu, x 1 W z 0 z 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.89 1. NON-DIMENSIONAL AXIAL POSITION, x/i Figure 6c. INon-Dimensional Boundary Layer Thlekness:Vs. Axial Position, Linear Variable Wall Temperature, qv = I x 103.

Io0 0.8 qI xiPOS 4Posit W l k. 0.6- Xle tcL 0-4 0-2 0 0-1 0-2 0-3 004 0'5 0-6 0-7 0-8 0-9 I0 NON- DIMENSIONAL AXIAL POSITION,x/g Figure 7a, Non-Dimensional Centerline Temperature Distribution Va. Axial Position, Constant Wall Temperature.

1.0 06- 20140, o,, 10 20 0. 1 0.2 0.3 0.4 5 0.6 0.7 0.8 0.9 1.0 NONDIMENSIONAL AXIAL POSITIN0.5 0.6 07 0.8 FigurONDIMENSIONAL AXIAL POSITIONemperature, q = 1 x Figure 7b. Linear Variable Wall Temperature, qv = 1 x 104.

1.0 0.8 t 0.6 t tih.o 20 o 0.4 _,40 0.2 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 7c. Linear Variable Wall Temperature, q = 1 x 10.

1.0 0.8 tt 0.o ] l l''^^'^^Jto 0.4 40 0.2 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/c Figure 7d. Linear Variable Wall Temperature, qv = 1 x 12.

-24At the higher values of qv (Figure 29a, Reference 2) there was a small increase in tl /tto with decreasing qv, approximately 27% over the qv range 2 x 109 to 1 x 105, or roughly four decades, at x/1 = 0,7o At the lower qv range, 1 x 104 to 1 x 1i, o only two decades, the similar increase in t, /t, is of the order of 50% or approximately an increase of a factor of fourb Th variable wall temperature runs produced the same effect, iLeo a pronounced increase in the spread of te /t o for wall temperature variations from 5 to 40 at the lower qvs compared to the high qv range0 The spread of t /to for t /to from 3 to 40 at a q. = 1 x 10 is about 26% while at qv = 1 x 102 the spread has increased to about 45%o All the trends indicated in Reference 2 at the high qv range are corrborated by this supplementary data in the low qv rangeo The radial temperature distributions, t/to y show little difference for the qv range presented in this report (Figure 8) from the higher qv range In all cases, the curves for t. /to of 10, 20, snd 40 are close to horizontal. The curve for tto/to = 3 shows best the tendency for a rapid decrease in temperature at the top and bottomY with the intermedinte section nearly constant. For the constant wall temperature case t = tt, since t,, in the general relation t + t = t, equals zero, Therefore, the plots of tt /tt, previously discussed, are identical to these for the special case of constant wall temperature. The wall heat flux is presented vs0 axial position in the tiTbe in terms of the function normalized to the value at the axial midpoint

0.8 0.6 t 0.4, t3 0.2 0 0.1 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9 1.0 NON-DIMENSIONAL AXIAL POSITION, X& Figure 8a. Non-Dimensional Radial Temperature Differential Vs. Axial Position, Linear Variable Wall Temperature, qv = 1 x 10.

0.8 0.6 1~ k, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ro 0.4'c o........10 2 to 40 0.2 0 0.1 0.2 03 O0. 0.5 0.6 0.7 0.8 09 1 0 0.1 ^^^^^le1 NON- DIMENSIONAL AXIAL POSITION, X/ Figure 8b. Non-D-nioral Radial Texperature Differential Vs. Axial position, Lnear Variable Wall TeWperature, qy= XiOj.

0.6 0.4 Jt t L to 10 20 \ -------------— \ ^ ^ 40 t 0.2 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, X1 Figure 8c. Non-Dimensional Radial Temperature Differential Vs. Axial Position, Linear Variable Wall Temperature, qv = 1 x 102.

of the tube, x/l = 0o50o Figure 9a shows the results for a constant wall temperature Again, as with the other parameters discussed, the rapidity with which the normalized wall conduction approaches the limiting condition of low qv is apparent0 The limiting condition, in this case, is a horizontal line at lo-, which according to the mathematics of the problem would represent a boundary layer completely filling the tube radially and axiallyy The effects of low qv and consequently a large boundary layer on the wall conduction can be derived from Equation (ll)o The wall conduction term t/- t-3/' reduces to a constant, t, for constant wall temperatures as approaches zero and the boundary layer, 1- 3, approaches 1*OO The constant, t, is very small because the heat source is small. The non-dimensional function, G( ), increases rapidly as 3 decreases, from 10-8 at (3 = 0O99 to l.1 at ( = 00 This causes the e a n.^ xf — term in Equation (12) to become very small and thus the square root to approach 1.Oo Thus, taN to,- or since the wall temperature is constant, tN' tNlo When normalized to the axial middle f the test section or the point where the boundary layer initially approaches the centerline, the length of tube over which the wall conduction is constant will appear as a line at lOo The axial pEortio of the tube over which this occurs increases with decreasing qvo The limit axially would occur when the wall conduction became constant over the entire length of the tube. It should be noted th the plots shown are not for the limiting case of fully developed boundary layer but rather show'the results of negativ boundary layer thickness for the low v and hence are not physically significant, Nevertheless, the results approach those expected for the fully developed boundary layer0

6 OaM4.9jadmall tTU1M W.u34uoo fuO4TgTOd TPT)V OsA xUOT.OnPUOo TTVIA PgzTt-;mUoN ~6 omngT';i I/x'NOISOd 1VIXV 1VNOISNJIOla —NON 0 1 8*0 9g0 Vaoo?~0 0 OJ ____ 81 ____ 9*0 ___ fO ___ 2.0 ____ 0 I'I z 0 r" m * c~3 ~~N 9 O.i 9 64 0.110 0 z C C) -6 30 x I ~~~z 01 x Iz 01 xl Ib 9 0I OZ

-50The variable wall temperature case presented in Figures 9b, c, an d d is s"ewhat more complicated Now, especially for large t. /t0 ratis the proressin tward a constant wall oandwutton term is even me rapid, The significance of a large ratio of to /t is that the wall teoperature differential is increasingly larger than the radial treperature differential anad. hence, becomes the dominent factor, Although the wall temperature doas not enter Equation (11) directly, it does enter the second equation, (12). Therefore, it indirectly effects the wall conduction tern through the radial tmpyernture, t. This is apparent by a omparison of Figures 9b and d The fluid velocities attained in both the ore and the boundary layer are shown in Figure iO. The boundary layer vlity ealuen lated is the imaxi value based the assumed profile (Figure 2) and ccur:s relatively near the tube wallo In Reference 2 it was noted that the velocity was always less than the bqncary layer velocity aMd that the ratio of budary layer/cre velocity decreased with decreasing qVo This same trend continues into the lower qv range, h.owerve: at a qv of 1 x 10,4 with a constant wall temperature, where the baundary layer and core each Ocupy roughly 50% of the tube radiallythe velocities are approximately equal (Figure 10b)0 As the qv become even lwler, and the core oceupies a smaller part of the tube, the core velacity bec-esO greater than the boundary layer velocityo The ratio of boundary layer velocity to core velocity while approximately 1 at qv = 1 x 104 is 0.39 at.v = 1 x 105 and Ool2 at qv = x 102 for the constant wall temperature case.

NORMALIZED WALL CONDUCTION *~,,,* 4 cm 8 O 0 I's > -0 0 pz (c 0

- 32 - 10 s -~~~~3 ~ 3 1.1 9 7 6 5 4.2.I to 3.8 -J.? Z.3 0 0.2 0.4 0.6 0.8 1.0 NON-DIMENSIONAL AXIAL POSITION,x/e Figure 9c. Linear Variable Wall Temperature, qv = 1 x 103.

33 - 8 7 6 5 4 3 1.2k 2.to z 06 08 1.0 o 20 o0 0 40 z 0 j.9 <.8.7 N.3.2 0 0.2 0.4 0.6 0.8 1.0 NON-DIMENSIONAL AXIAL POSITION,x/l Figure 9d. Linear Variable Wall Temperature, qv = 1 x 102.

qI=1x104 - x 044 100 UCORE a q,__ I "x10~___ I 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/g Figure 10a. on-Dimensonal Boundary Layer and Core Velocity Vs. Axial Position, Constant Wall Temperature.

12 |UCORE ----------- B.L MAX. to 10 2 03 04 0 0 0 to 10 20 w o ~~0 06 4 0.2 0.3 0.4 05 0.6 0.7 08 0.9 1.0 NON-DIMENSIONAL AXIAL POSITION, x/2 Figure lOb. Linear Variable Wall Temperature qv = 1 x 102.

if~-3 U "CORE to.BMAX. 50 20 40 -- o4o ~~~04~~~~~~~~~~~ NON- DIMENSIONAL AXIAL POSITION, x/e Figure lOc. Linear Variable Wall Temperature, qv = 1 x 103.

10 to 0 810 \ 20 20 40 rJ 40 I) 0 z 4~ 0 0.1 0.2 0.3 0-4 ITION X/90.9 1. NON-DIMENSIONAL AXIAL POSITION x/e Figure 10d. Linear Variable Wall Temperature qv =x

-38The variation of UBLmax and ucore with qv and P noted here can be derived directly from the basic velocity profile assumption This i shown in the Appendix. The velocity profile shown in Figure 2 and expressed by Equation (4) yields the following relation.ship between UBLm and ucore across the tube. Boundary Layer UBsma/ucore 0o25 2o798 0Q50 lo.62 0.75 0.518 lo00 0.280 Thus, when the boundary layer fills the tube, the velocity in the core is 306 times the maximum boundary layer velocity. The general trend is the same for the variable wall temperature caseo Howevers for any one heat source strength the boundary layer and core velocities are more nearly equal at the larger temperature ratios, t, /to, This is simply a function of the fact that the boundary layer thickness increases less rapidly as t 0 /to increases. The absolute value of the non-dimensional velocities is much larger at heat source strengths of 1 x 106 and higher (Referene 2) than in the range considered here4 Also, the ratio of boundary layer to core velocity has a much greater rate of change at the higher qv condition than at the very low qv This is inherently a function of the basic velocity assumptions, manifested through the G( (3), = G( () t, term in the velocity profile.

CONCLUSIONS The general trends and empirical relations found in Reference 2 for a range of high heat source strength can be accurately extended to the lower. rage for the cases eonsideredo It may nw be assumed that this will hold true for wall temperature distributions and heat soarce distributiont other than those considered in this reporto The rate at which the boundary layer increases is more rapid once it has occupied approximately half of the tube. As an effect of this then, the rates of change of temperature velocity and wall cona duction are all increasedc At a non-dimensional heat source strength of about 5 x 12 the bundary layer completely fills the tubea At this point the equations and assuptions on which this analysis is based no longer apply and a new ystem must be constructed.o 359~

APPENDIX Derivation of Energy Relation for Radial Disc for Variable Wall Temperature Condition From Reference 1, an energy balance for a disc (Figure 1) is, f2UTRdI - 1v (S 9R (~ eX Substitute X = xl R = ra T = TWALL tt, tw) U =,u Then _t a T U 4 - (t Ta) 2 t, LU/^A- o<T ( +t + qQ (A-) e & j rC7 MIN CA L C404-L) m4om

But a) ( WALL M,)iK^ ^o a (TwA ^M) = ~ Therefore -rL ~ / ^ +t $ I OC L -ax j(trL Sty) V JrL = Am (ts~tW~v, - 2 v (Ax) tw does not vary with respeet to r, but does vary with respect to x0 tA (A7) u /L O/ n- a tw /i O/ /L d6 j-/ut2.a/ +JtavJA = L tf X/i/LJ + X o aX a X #SatLr-dr +ftJOLd^ +UL (A8) a^ ^^^ ^^^

-42Substituting the assumed velocity and temperature profiles r - y O^^/L. L( = I tA < -- ( A - (A9) Q j-ntAAJd ~ t' Y 2 i+{f/JtA E-/fA T __aXJ __L __ L / [( For constant wall temperature tw = 0 tA = t (does not vary with x) But for variable wall temperature tw = f(x) f(x) is some function of x/l tA = f(x)

-43As derived by Lighthill, Part I of Equation (A9) becomes _- ptA jp ]e (A10) where F(3)- (/_-)3 (3) (5+ _/3_2 + /_/P ft p3) 30240 (3 + 4 f J+3 ) Part II o Equation (A9) becomes whch,equals, (- Sty) AX) (A12) where H(.3,)= — ~(f7 —^ (A15) + ^ 20 6 6' /, 30/ Fro References 1 and 2 Y tA G() where ( \ (3 ) \3 3) (3+ ja (3)

-44Thus, Equation (A12) which is Part II of Equation (A9) equals _h~ & G(?) NOS) 4tA4 (A14) And Equation (A5) becomes J [ Z 9) S tw ]= - _ + (A15) Substituting & - = 3 J +/ ) (Reference 1) in Equation (A15) and solving, the second ter on the left side drops out So that (A5) is dxi - X ) =_ (A16) where tA can be replaced by t(x) indicating variation with x. Derivation of Maximum Boundary Layer Velocity The maximum velocity in the boundary layer will occur where 6a = 0 (Figure 2). Thus, differentiating the velocity equation for the boundary layer L s ) ) ( ^ - ___ -& 6 3 ) (A17) Setting this equal to zero, the radial distance at which the velocity is maximm is ~^4 -~063/ (A18)

-45Since the core velocity = t, the ratio of uBLmx/ 6 is the ratio of the maximum boundary layer velocity to the core velocity. This is found for any g by substituting ruBLmax and ( into the velocity equation for the boundary layer. US L MAX / O (A-)]9)

IA7R3t- r - J5uy < e7 f j/i1 op FZdL E_________4_A7" /. ~E T ~o - -J/AY/. y' o,-J' |'.~ \,,,,,'- -_ _, _ __,e_,,_,_- i,,.,/ o..fp/..,/,.,,<.' o. 3oo/ o II B. 4O ~. o0. 4P.y~. - 3 O..// / e. vCo___ o_____ 0-. y m._ o. o. t/ ____o_____._ d./.____ / -~._07______ _________.1 J _ ____'o._ _ 0. o.4____ o.__ o._ _ -o. e7_ 10 ___3_ o-s -../________ oo 0./.0 8 _ _____________o.f> 3.40_o 3.__'/ 3./.^ 6 I______.__ -— _ _3S7_ _o o 4.o4_ _ __/00 G O_______. _ o /3so..._____.d ______ a.. o. o/.___ o. 0 /._/__I_ o. /______ o.o_______. o.Co y /-./_ o.o7 3_____ ___ __________ ____ __ ______________ o>.,f' ~ ^. /) /. ^! o ~, i..,,,L /. ______ 0-___________ _^ 0. --- a. o _ _.>J _ < _ 0 o. A,63 "T a.;,......,,o0.o2 o -O --- 6 —-0- _ /_ 5 ^ 4 ^ ^ —----— ^ / f f O: —----— _____~_~3_i_0_4__fOZ.. _O-.4/, -54kw.-.L&.z -. ______ _ I__0 2^_____ L/XX//?./ x/. ___,, __ ________ 7^ _____l./, x/o 3./s- x /dx/___ I x. —. /.'> /' b o. f __ _ /z A'3^ /-Sf.A/to /. l0 /e O A/c; /* O>6

T-A 73LE - 54(/A~fjIfA Y a'= -R..E-J47Z ~~/V/P0~A7 A/4/e4~~~ vdRI473,-i6- Wodz 7; AW fe.4 y-4/,R - ~r iuN 4 6~LI a g/ ________________ 0 ~~~~~~~A /000- 0 A Z. A oo A000 000 -0 A_ 00/ L$ 0 6/43 Omi 8~ II II 110.~03 1 o.7o76S ~ p ~ Zzz.L41 I p~ ~~ 63 d4t o.~~~ o p.#;7 b.b 0.60 P -~. O 0470 o/)-7 -o. 1v4 0 ii ~ ~ ~ ~ ~ 7.I II2 034( s'#6 ___ -o/s'2'Z o6.o./U 0.i7.~ l.3 )- a~~._. ff /./ aokw4 0- f 0) 2-4 J ~0 /0..0.6 0 x _ _ _ _ - b 0x0. o x _ _ _ /.o4Ix.3 O7 ~ S X _ _ _ 0.6. 6xo.r K. K/a. 32-x,?/o/.le6K/O.?5l.ex,0)..Z- I/O _____ ____ 4. b ~ ~ ~ ~ 2cx A EI/D II. Il rO.Lt~~~~~~~~~~~~~~~J-DY X/O1444- A P-to I ttoP oY/d/0X/Y /~~I b ~ I. L /dLdr/~~~bI/q 319 r/o~/'OR d jr /o/l I Z0y O". /q X/0'f/w X 0X I(. z. (/Oirl / A I 03 x/b ~~~~~c~~t tI x Y 16' 4 -114"10" 4 lo olt~ /lLex d I-/3Y~0.///o )- ~r/ I x/?bU2e dS., Id 432-x/O X le 2462- 11 AO F6 l le &- 1503 /40, Ouo j6 - i 7cw II ii II 9. 40~~~~~~~~~~~~~~~~~4 -.3 WA X. ~ 6 OV. -4 A.4*v 24 b. 30w:230y ~~~'/b J Y/O~~~~~~~~~~~IX0 /X/O II O d.4 I Q3 X 200 3 x d Z f L - 5 r /40 o Zf A4 d4

REFERENCES 1. Hammitt, F*G., "Modified Boundary Layer Type Solution for Free Convection Flow in Vertical Closed Tube with Arbitrarily Distributed Internal Heat Source and Wall Temperature," ASE Paper No. 57-81* 2. Hammitt, F*G., "Heat and Mass Transfer in Closed, Vertical^ Cylindrical Vessels with Internal Heat Sources for Ho ogeneous Nuclear Reactors," Doctoral Thesis, Nuclear Engineering, University of Michigan, Feb. 1958. 35* Lighthill M.J. "Theoretical Coniderations on Free Convection in Tubes," Quarterly Journal of Mechanics and Applied Mathematics, Vol. 6, 1953, pp. 398-439. 4. Ostrach, Simon, and Thornton, P.R. "On the Stagnation of NaturalConvection Flows in Closed-End Tubes," ASME Paper No. 57-SA-20 5. Murgatryd, W., "Thermal Convection in a Long Cell Containig a Heat Generating Fluid," Atomic Energy Research Establishment, ED/R 1559, Harwell, England, 1954. 6. Martin, B.W., "Free Convection in an Open Thermsyphen, with Special Reference to Turbulent Flow," Proc Royal Society, Series A, Vo1. 230, 1955, pp. 502-530. 7. Haas, P.A., and Langsdon, J.K., "HRP-CP Heat Removal fro a Proposed Hydroclone Underflw Pot Geometry for a Volume Heat Source,' CF-55-10-7, Oak Ridge National Laboratory, October 5, 1955. -48

UNIVERSITY OF MICHIGAN 3 9015 02523 0692