CENTRAL RESEARCH LIBRARY 277 DOCUMENT COLLECTION UNCLASSIFIED LRI 3 445L D3IY9LAS 7 ORNL 1769 Engineerin y-f FREE CONVECTION [N FLUIDS HAVING A VOLUME HEAT SOURCE D. C. Hamilton H. F. Poppendiek R. F. Redmond L. D. Palmer OAK RIDGE NATIONAL LABORATORY OFPERATED BY CARBIDE AND CARBON CHEMICALS COMPANY A DIVISION OF UNION CARBIDE AND CARBON CORPORATION (=4 POST OFFICE BOX P OAK RIDGE. TENNESSEE UNCLASSIFIED ORNL-1769 Copy No .fléL Contract No W-ThO5, eng 26 Reactor Experimental Engineering Division FREE CONVECTION IN FLUIDS HAVING A VOLUME HEAT SOURCE (Theoretical Laminar Flow Analyses for Pipe and Parallel Plate Systems) by Hamilton Poppendiek Redmond Palmer Homu U= HOQ DATE ISSUED NOV 15 1954 OAK RIDGE RATIONAL IABORATORY Operated by CARBIDE AND CARBON CHEMICALS COMPANY A Division of Union Carbide and Carbon Corporation Post Office Box P Oak Ridge, Tennessee L 3 445k 0349L85 7 \ )-—J-\ ORNL 1769 Engineering INTERNAL DISTRIBUTION l C E Center 50 R E Aven 2 Biology Library 60 H W Hoffman 3 Health Physics Library 61 M C Edlund ¥ U-5 Central Research Library 62 R W Bussard ™% Reactor Experimental 63 S Visner Engineering Library 64 P R Kasten T7-11 Leboratory Records Department 65 N F Lansing 12 Laboratory Records, ORNL R C 66 P C Zmola 13 C E Larson 67 M W Rosenthal 14 L B Emlet {XK-25) 68 W D Powers 15 J P Murray (Y-12) 69-118, D. C, Hamilton 16 A M Weinberg 113 F E Lynch 17T W H Jordan 120-124 L D Palmer 18 8 J Cromer 125 H ¢ Claiborne 19 E J Murphy 126 L G Alexander 20 E H Taylor 127 D G Thomas 21 E D Shipley 128 M A Arnold 22 C E Winters 1290 T K Carlsmith 23 J A 1lane 130 W K Ergen 2L F C Vonderlesge 131 ¢ B Mills 25 Jd A Swartout 132 N D Greene 26 S C Lind 133 & M Adamson 27T F L Culler 134 E S Bettis 28 A H. Snell 135 E P, Blizard 29 A Hollaender 136 A D Callihan 30 M T Kelley 137 S I Cohen 31 C P Keim 138 ¢ A Cristy 32 R S Livingston 139 W R Grimes 33 J H Frye, Jr 140 W D Manly 34 G H Clewett 141 L A Mann 35 A S Householder 142 E R Mann 3 C 8 Harrill 143 W B McDeonald 37 D S Billington bk J L Meem 38 R N Lyon 145 W W Parkinson 39 R B Briggs 14 D F Salmon 10 A S Kitzes 147 H W Savage 41 O Sisman 148 ¢ ¢ Lawson L2 ¢ B Graham 149 R A Charpie k3 W R Gall 150 M Tobias Lh-53 H P Poppendiek 151 L E McTaggart 54 8 E Beall 152 C A Moore 5 J P Gill 153 L F Parsly 56 J 0 Bradfute 154 E C Miller 57 A P Fraas 155 M J Skinner 58 P N Haubenreich EXTERNAL DISTRIBUTION 156 R F Bacher, California Institute of Technology 157 Division of Research and Medicine, AEC, ORO 158 ORNI, Document Reference Library (Y-12 Plant) 159-168 R F Redmond, Battelle Memorial Institute 169 AF Plant Representative, Wood-Ridge (Attn S V Manson) 170 AF Plant Representative, E Hartford (Attn: W S Farmer) 171-413 Given distribution as shown in TID-4500 under Engineering category DISTRIBUTION PAGE TO BE REMOVED IF REPCRT IS GIVEN PUBLIC DISTRIBUTION TABLE OF CONTENTS SUMMARY INTRODUCTION NOMENCIATURE GENERAL DISCUSSION OF THE PROBLEM IDEAL SYSTEM I (PARALLEL PIATES) Velocaty Solution Temperature Solution TDEAL SYSTEM II (PARALLEL PIATES - APPROXIMATE) Velocity Solution Temperature Solution IDEAL SYSTEM III (CYLINDRICAL PIPE - APPROXIMATE) Velocity Solution Temperature Solution DISCUSSION REFERENCES PAGE 10 16 17 19 23 23 26 30 30 32 38 39 SUMMARY Theoretical laminar flow analyses are given for free convection in fluids having a uniform volume heat source and for both parallel plate and cylindrical Pipe geometries The solutions are intended to be valid in the central region (vertically) of channels having small diameters and large lengths, that 1s, the solutions do not apply to short systems or near the ends of long systems where the velocity and temperature profiles are not yet fully established In addi- tion, the solutions are restricted to systems in which the long axis 1s vertical and in which the walls are uniformly cooled by a forced flow coolant flowing vertically upward parallel to the long axis of the system Solutions are obtained for the parallel plate geometry by two different techniques called exact and approximate’ In the "exact method the differ- ential equations for velocity and for temperature, which are interdependent in free convection systems, are solved simultaneously, in the 'approximate" method the form of the velocity distribution 1s postulated and substituted in the temperature equation which i1s then integrated Solutions by the two methods agree well in the range where the basic postulates are believed to be valid The velocity and temperature structures are functions of two new dimensionless moduli herein designated as Ny and Nyt INTRODUCTION The purpose of this report i1s to provide a wider distribution for three analyses performed in 1951 than was accomplished by the very limited local distribution of References 1, 2, and 3 Originally these analyses were per- formed as the first step i1n a theoretical-experimental free convection research program At that time 1t was planned to withhold publication of these analyses as a report until the experimental data were available which proved their validity Subsequently, other problems have diverted attention from free convection experiments so that this research has become a part time activity (Reference 4) This reduced experimental program 15 less comprehensive than would be required to adequately prove or disprove the validity of the basic assumption of these analyses Therefore the reason for delaying this publa- cation 1s no longer valiad It 1s expected that the results of the more modest experimental program will be reported in the near future The basic postulates that apply to all three analyses are discussed in the next section, following that 1s the 'exact" solution (Ideal System I) for the parallel plate geometry Then an approximate solution (Ideal System II) for the parallel plate geometry 1s presented Finally, an "approximate' solution (Ideal System III) for the cylindrical pipe geometry 1s given which is the cylindrical equivalent of Ideal System II NOMENCLATURE¥* &, 8o constants A= éfi,-unlform vertical temperature gradient (6L.-1), also area (L?) Z B1(z) - function of z in Equation (%) (L-1 T"l) Bo(z) - function of z in Equation (6) Cq,Co; constants cp - constant pressure specific heat (FLM'l B’l) C - circumference of flow channel (L) d - separation of parallel plates or diameter of cylindrical pipe (L) Also used as differential operator Dy = %%, hydraulic diameter (L) f = (280 Dh) [ dpPr), friction factor o W2 dz where 9Pf 1 the pressure gradient due to friction dz g - gravitational acceleration (ET'Q) g, - dimensional constant (IMF~1 7=2) h - heat transfer coefficlent (FT~+ L1 9"1) h - height of system (L) k - thermal conductivity (FT™' @71) L - length of fluid circuit (L) *The last part of the definition of each symbol will indicate 1ts dimensions in the force (F), mass (M), length (L), time (T), temperature (@) system, when no dimensions are given the symbol i1s dimensionless m = x, spatial coordinate (L) M=D1 X1 ABgd" Ny = oy’ form of Grashof times Prandtl modulus | 2 Nyp = £ P8 g form of Grashof modulus k2 hd L Nu = = X~ 3(0) Nusselt Modulus P - pressure (FL'2) Pr = %—, Prandtl Modulus q - heat transfer rate (FLT~1) q" - heat transfer rate per unit area (FL"l T'l) q' - volume heat source term (FL"2 T"l) r - radial coordinate (L) r, - value of r at the interface between the two free convection streams (L) 1 ro = &, pipe radius (L) 2 R =X To wD Re = I/h , Reynolds modulus + X 8 =X - (}_c.o__é__:‘;), spatial coordinate (L) 5o =( "o___"l) (L) 2 S =5 So t - temperature (8) u - x component of velocity (LT™1) v - y component of velocity (LT~1) w - z component of velocity (LT-1) Wy - average velocity in the middle, hot, or upward flowing free convection stream (LT-1) W, - average velocity in the outer, cold, or downward flowing free comvection stream (ILT-1) W = wd Y N1 , velocity function Wy = EEE__ , mean velocity function Y11 X - spatial coordinate (L ) X, - value of x at the interface between the two free convection streams (L ) Xo = g., half separation of the parallel plates (L ) ¥Y,2, spatial coordinates (L ) Greek Symbols a = ?Pfif » molecular thermal diffusivity, (L? T'l) k B - volume coefficient of expansion (871) 0(X), @(R) - temperature excess above wall temperature at the same value of z (8) 8.(0) = 8(0) for conduction only (@) 8.(0) = : P P 0 43" for parallel plates 8k e.(0) ' 4 lindrical pipe = or C narica 1 c TSl v PP 1/4 » = (NI ol /1 - dynamic viscosity (l'.\fll."l T-l) 3 = £ | kinematic viscosity (LZ T71) P p - mass density (ML-B) ® = __6 _ , temperature function AE) A &, - mean buoyasnt temperature difference - 10 - GENERAL DISCUSSION OF THE PROBLEM Laminar flow free convection systems are described by three equations of motion (Navier Stokes equations) and the heat conduction equation for a moving system These four partial differential equations are i1nterdependent and com- prise a set one would hardly attempt to solve It 1s intended here to briefly discuss the basic postulates that permit simplification of these equations to the quite elementary ordinary differential equations that are solved in this report Although the parallel plate or cartesian geometry of Figure 1 1s used in thas discussion the comments are equally applicable to the cylindrical pipe geometry The free convection system to be studied i1s the fluid in the channel between the parallel plates (Figure 1) separated by a distance, d, and of height, h, which 1s very long compared to & Heat 1s generated uniformly throughout the fluid and the heat 1s removed uniformly at the walls Because of these factors and because of the vertical orientation of the z axis there will be three parallel free convection fluid streams, the warm stream in the center of the channel will flow up and the two cool streams near the walls will flow down Below some critical velocity these streams should be quite stable and, in Tact, should behave much as three laminar forced flow streams separated by Physical boundaries might behave This tendency toward stability of the flows suggests that the flow would be one long vertical cell, not & number of small cells or laminar eddies a few diameters in length In forced flow heat transfer systems 1n conduits the velocity and temperature distributions are observed to -11- UNCLASSIFIED ORNL-LR DWG 3558 —_— e N V " V Y % l / / / /) ; ’ / / 7/ | 2 / 4 U / { wix) | ’ X A v | ' LA A +X|fi /| L A | A 7 % "‘""""Xo —>'/ / / y , y < d - Y/ / / | K \——COOLANT CHANNELS —/ Fig 1 Configuration of Ideal System I (Parallel Plates) and the Accompanying Coolant Channels I - 12 - become fully established or reach a stable form some diameters beyond the entrance Beyond this entrance region the velocity and temperature distributions no longer change as one proceeds down the pipe The similarity of the flow in the free convection system and the forced flow system above suggests that beyond some entrance region, near the ends of the present system, the velocity and tempera- ture profiles may also become fully established These are the two basic postulates of the systems analysed in this report and are stated more incisively as follows Postulate 1 Postulate 2 w = f(x) %E.= A, where A 158 a positive constant and Z uniform for the entire system Other postulates that are necessary to describe the three i1deal systems to be analysed are Postulate 3 Postulate 4 Postulate 5 Postulate 6 Postulate 7 Postulate 8 The volume heat source term, q'"', is uniform throughout the system and constant with time The height to diameter ratio, h/d, is very large The flow 1s laminar and steady (1 e , constant with time) The flow 1s two dimensional (1 e , the y component of velocity, v, 18 zero) All fluid properties except density are constants The density 1s constant in the heat equation and 1is a linear function of temperature in the dynamic equation - 13 - As & consequence of Postulate 1 one can prove that the x component of velocity, u, and the transverse pressure gradient _gg.vanlsh and that %E 15 uniform with x Thus, two of the dynamic equations are eliminated and the third is greatly simplified to iz_w;.zgfl(gg-l-pi) (1) ax2 /1. dz & As a result of Postulate 2 one can prove that the heat flux at the wall is uniform and therefore known, that is, each element of width, 4, and height, dz, loses through its own bounding wall surface exactly the amount of heat generated within that element Thus, no net heat loss occurs 1n the z direction for such an element An additional consequence of Postulate 2 is that the use of the temperature function, &, eliminates z as & variable and the equations involve only one independent variable, x The heat conduction equation i1s then simplified to i29 & " Z-a" "% (2) By definition p(t) = p(t,) (l - B(t - 'bo)) (3) Employing the function, &, and Equation (3), Equation (1) becomes a8 5= - Fe + m) (1) Note that the function Bj(z) 1s independent of x - 14 - The heat conduction and dynamic equations that result from employing dimensionless functions 1in Equation (2) and (4) are dz‘b X = QNI W(X) -2 (5) dX’S 2y T)'ddx(x = - 3}5 o(X) + By(z) (6) Equations (5) and (6) together with the accompanying boundary conditions define the parallel plate system to be analysed The equivalent set for a cylindrical pipe 1s %—.dER_ (R i‘é‘i@) = - 61137 (R) + Ba(z) (7) %% (R %5)_) = Ny W(R) - 4 (8) The boundary conditions and auxiliary information that go with the differential equations to complete the boundary value problem are given here Due to the definition of the temperature function & ®(1) =0 (9) It 1s evident from inspection that both the velocity and temperature functions are symmetrical, thus w(x) (10p) W(R) (10c) W(-X) W(-R) and o(X) (11lp) ¢(R) (11c) o(-X) o(-R) The velocity at the walls i1s zero, thus w(1) = 0 (12) No net flow occurs, therefore 1 / W(X) a&X = © (13p) 0 1 / W(R) RAR = O (13c) O No net heat transfer occurs in the z direction so the heat generated at a given level must transfer to the walls at that level, thus __ng 1) - _L_?_gg = -2 (1%4) Equations (5) to (14), inclusive, define the systems to be solved - 16 - IDEAL SYSTEM I (PARALLEL PLATES) The geometry was previously described in Figure 1 and the differential equations and boundary conditions were adequately discussed in the previous section It 1s sufficient here to define the system mathematically and then to obtein the solution The differential equations to be solved are 498 _ oy w(x) - 2 (5) dX T - - 2 o(x) + Bola) (6) The boundary conditions to be employed are W(-X) = W(X) (1op) W(1) =0 (12) 1 / W(X) dX = 0 (13p) 0 __ldgj(co -0 (11p) o(1) =0 (9) - 17 - Velocity Solution Eliminating the temperature from Equations (5) and (6) one gets the velocity equation () 4 I yx) L (15) The general solution to (15) is W(X) = %;.(1 + a1 s1n AX sinh AX + &, cos AX cosh X + 83 s1n AX cosh AX + &) cos AX sinh AX) (16) 1/% where A= (g&) / (17) By successive application of boundary conditions (10p), (12), and (13p) one obtains az = ay =0 (18) a) = - (élnl cosh A + cos A sinh A -2 A cos A cosh f) (19) sinh A cosh A - sSi1n A cos A H (éln A cosh A - cos A s1nh A -2 A sin A sinh l) (20) &2 sinh A cosh A - sin A cos A Thus, the velocity solution, plotted in Figure 2, 1s given by N1 W(X) = 1 + a7 sin X sinh AX + ap cos AX cosh AX (16a) The Reynolds modulus for the central or hot stream 1is Xy L Rey, = __Xj__‘:’.g. = 2 Nyg W(X)ax (21) _18_ ggfiLASSF: F:)EDG L-LR-DW 0 0006 | l | 3559 4 EN. - _ ABgd 3 1 N, 00004L~ 10 av ] \ w _ wd N, vNp 0 0002 \\ / N -0 0004 N 7 0 02 0 4 06 08 10 X Fig 2 Dimensionless Velocity Function, W, for Ideal System I (Parallel Plates) - 19 - Because the values of Rep computed by the numerical integration of Figure 2 disagreed by less than five percent with the equation obtained in Ideal System 11, that equation will be employed to display these results N Rew = II “h 3460 + 0 786 Ny (57) The critical value of Reyp above which the flow 15 no longer laminar must be determined by experiment Experiments in Reference (5) indicated that the critical value of Reynolds modulus for non-isothermal flow varies in a very com- plex manner and is not the same as for the i1sothermal flow case Temperature Solution At least three methods may be used to obtain the temperature solution, the method employed here 18 to substitute the velocity from Equation {1l6éa) into the temperature Equation (5) and integrate using the boundary conditions (9) and X X o(X) =///// dx///// (2w w(X) -2) X aX (22) 1 o Putting W(X) from Equation (16a) in Equation (22) and performing the (11p) integrations one obtains o(X) = ._12_ (al(cos A cosh A - cos AX cosh \X) + A -an(sin A sinh A - sin AX sinh JLX)) {22a) and o(0) = Eé. aj(cos A cosh A - 1) - ap sin A sinh)\) (23) A - 20 - A Nusselt modulus may be defined as follows = 5807 - T (24) The dimensionless temperature function, ®(X) 1s shown in Figure 3 as a function of X and Ny The value of Ny = O corresponds to the case of pure con- duction The variation of Nusselt modulus with N1 1s given 1n Figure b Tt is interesting to note the similarity in shape of this curve with conventional Nusselt modulus versus Grashof times Prandtl moduli plots for systems having no volume heat source -21 UNCLASSIFIED ORNL LR DWG 3560 I | _ABgd4 I° av 8 (X) o < 8¢ (0) 5x 103 \ 104 \ — 04 5 x 104 \ _——-—-—_-\‘ 10° N 02 S | | | |\ 0 02 04 06 o8 10 X Fig 3 Dimensionless Temperature Function @, for |deal System I (Paralle! Plates) _22_ UNCLASSIFIED ORNL LR-DWG 356/ 20 AB gdé NI" B ay hd 4q Nus — - —— YT T 300 10 / Nu | Fig 4 Nusselt Modulus for Ideal System I (Parallel {0 10° Ny 3 0 10° 10 Plates) 5 - 23 - IDEAL SYSTEM II (PARALLEL PLATES - APPROXIMATE) This solution is an approximeate method for obtaining an answer to the problem described by Ideal System I If the two solutions agree satisfactorily the approximate" method offers the two advantages of presenting a less diffai- cult boundary value problem and of requiring less time to perform the numerical calculations The technique depends upon the jJudicious postulation of the form of the velocity distribution to be substituted into Equation (5) Velocity Solution Iet the real flow system of Ideal System I be replaced by a counter-current heat exchanger system such as that depicted in Faigure 5 To emphasize the method used, the X coordinate i1s replaced by the coordinate, M, in the hot upward flowing stream, and by the coordinate, S, i1n the cold, downward flowing stream One can think of these streams as separated by parallel plates inserted at + X, (or M = + 1) The velocity distribution in each region is given by the equations with Figure 5, this 1s the familar parabolic expression for established isothermal, laminar, forced flow between parallel plates Since there 1s no net flow X, Wy = 285 W (25) To satisfy static equilibrium at the interface, x;i, the shear stress must be the same, or dw(xl) _ dW(Xl) (26) dm ds -24_ N NN\ | _ | | { — | R\ N N NN 3 M=-1{ M-0 X-0 w 3 2 = X -7 (1-M°) for —1< M<1 where M X, 2X {1+X,) .3 (52-1) for -1 =S <1 where S= we 2 (1-X,) (1 -X) Fig 5 Coordinate System and Postulated Velocity Distribution for |deal System IT (Parallel Plates) - 25 - Equations (25) and (26) require that X, =¥2' - 1and w, = V2 w, (27) The pressure drop due to friction around the fluid circuit of length, L, must be equal to the pressure rise due to the difference 1n the average density of the two streams, that is 1 1 2 2 3 pdS - dM/ gL = + < 28 2./{/// ////, ° ¢ Dy 2Dy (%) - O hot cold The friction factor for established, isothermal, laminar flow between parallel plates will be used r=2 (29) The left member of Equation (28) may be expressed in terms of a mean buoyant temperature difference, Ady,, defined as follows 1 1 Ady =/ oM - %/ 3ds (30) 2 /1 Employing Equations (3), (27), (29), and (30) Equation (28) may be expressed as Ady = 96(7 + 5 V2') Wy (31) The velocity structure 1s now completely defined in terms of constants and A&, which must be obtained from the temperature solution - 26 - Temperature Solution The temperature solution will again be obtained by substituting the velocity solution into Equation (5) and performing the integrations (X —(—l 2Np W(X) - 2 (5) The forms of Equation (5) that will be used here in the hot and cold stream regions, respectively, are d2e (M) o W 2 = 2X,° Ny W (L) - x (5M) M2 1 YI %h (Wh) 1 dEQ(S (1 -X ) W (1 -X = 2m 2 ) Ny oWy (e S -x)® (55) ds 2 W Wc 2 The temperature in the cold stream is obtained by integrating Equation (58) 0 a0 as ad (1 X,)° SNTW ds ( ) el _I_h (1-52) + 1} as (328) 5 ! b 1-X4 Integrating, one gets 2 8(s) = X (1 -:_%".‘leh)(l - 8) + }%— (1 + 5—,%@ NIWR) (1 - S2) + \/"‘x Ni¥n (1L -5 ) (338) - 27 - From Equation (33S) the temperature at the interface between the two streams may be computed for use as the boundary temperature in Equation (32M) 2 o(-1) = 2X, - V2'X,° NyWy (34) For the hot stream @ de M M aM ad 2 3 2 aMm d(afi) = X, dM 'é-NIWh(l-M ) - 1) aM (32M) 0 1 0 Integrating, one gets ®{(1) o(M) = 2X, - ¥2'%,% NpWy, + X, (1 - g.NIWh)(l-ME) + Ny, (1 - M*) (33M) and 2(0) = 1 - (2 @u 1) My (35) Also, recall _ il Nu = 36T (24) From Equations (30), (33M), and (33S) the mean buoyant temperature difference 15 computed as Ady, = Eg‘ - (——-——-—-——-——5 V?O- h) NIWy, (36) Eliminating A&, between Equations (31) and (36) get Wh = L (37) - Wh(10 + 7 VD) + f% (5 - 2V2) Np - 28 - The Reynolds modulus of the hot stream may be obtained from Equation (37) Re, = 2X,NyqHy, = — 1L (372) b I"b ™ 3480 + 0 786 Wg The temperature for Ideal System II was computed from Equations (33M), (33S), and (37) for various values of Ny and plotted in Figure 6 for comparison with the results of Ideal System I The two solutions are in excellent agree- ment for values of Ny up to 10% Above this value the solutions diverge rapidly so that for NI equal to lO5 the approximate solution yields tempera- tures that are too high o8 06 04 02 o -29- UNCLASSIFIED ORNL-LR-DWG 3564 ___.103 103 IDEAL SYSTEM I —a=—= |DEAL SYSTEM II (APPROXIMATE ANALYSIS) I | l l \ 0 02 04 06 08 { X Fig 6 Comparison of Dimensionless Temperatures for |deal Systems I and IT (Parallel Plates) 0 - 30 - S IDEAL SYSTEM III (CYLINDRICAL PIPE - APPROXIMATE) The excellent agreement of Ideal Systems I and II for values of Ny up to lOh supports the validity of the approximate method Since the solution to be presented here i1s i1dentical with Ideal System II, except for geometry, the accompanying discussion will be reduced to & minimum The "exact' solution of Equations (7) and (8) is not difficult, 1t 1s an uncommon form of Bessel's equation The disadvantage of the exact' solution in this case is the labor involved in the numericel calculations of the solutions Velocity Solution The postulated velocity distribution given in Figure 7 was obtained in the same manner as was used in Ideal System II, 1n this case the two regions are dynamically characteristic of isothermal, laminar, established forced flow in a pipe (for the hot stream) and in & circular annulus (for the cold stream) Since there 1s no net flow Yh R,2 = we(l - 312) (38) To satisfy static equilibrium at the interface %(Rl-)= %(Rfi) (39) Equations (38) and (39) require that R, % " 0 316198 (40) { and wy = 2 16258 w, (41) -39~ UNCLASSIFIED Z ORNL -LR-DWG 3563 v | V / | U f o} A | V Y | Y ) : ; U | / / I \L , | ' | v o T | ( | | / fi | c— 1, —'————)-fi Y - 2 0 " 2(1-C4R°) for O £ R 4R, - X -2¢,(1-R%2+C,INR) for Ry <« R <1 w 2 3 i C1=Ri2, C3- -(1_R|2)(IHR|)-1 C,= (1+RE=C31™! C4=CptCy- 1) Fig 7 Coordinate System and Postulated Velocity Distribution for ideal System IIT (Cylindrical Pipes) - 32 - Again, the buoyant force must be equal to the pressure drop due to friction 2 2 fowy fow p B g 8.,(0) A, = ) + c hot DH DH cold R, 1 2 where Ad = ®RAR - "“ELET' ®RAR g:E (1’R1 ) 0 R, The friction factors are (fpwc2) _ 6]4-!& Wh cy cold Dh d2 From Equations (42), (44), and (45) get 1605 Wh At 32(cl + Ch) " €5 Temperature Solution (k2) (43) (k) (45) (46) The temperature solution 1s obtained by substituting the velocity equations from Figure T into Equation (7) and then performing the integrations a(r 42) = LA (R =) (NIwhwh 1) 4RAR (7) _33- For the cold stream region, R; Redmond, R F , Hamilton, D C , and Palmer, L D #Part II - Heat Transfer from a Fluid in Lsminar Flow to Two Parallel Plane Bounding Walls , ORNL CF 52-1-1, Jan 22, 1952 4% Hamilton, D C , and Lynch, F E Unpublished Preliminary Free Convection Experiments, 1952-195k 5 Hamilton, D C , Lyach, F E , and Palmer, L D The Nature of the Flow of Ordinary Fluids in & Thermal Convection Harp , ORNL-1624, Feb 23, 1954 #The first three references are exclusively theoretical analyses and the general title to each 1s Theoretical and Experimental Analyses of Natural Convection waithin Fluids in Which Heat 1s Being Generated