CENTRAL RESEAECH LIBRALY it i o DOCUMENT COLLECTION VAR 3 445k D3I49L3Y9 4 ORNL 1701 Engineering FORCED CONVECTION HEAT TRAIilSFER BETWEEN PARALLEL PLATES AND IN ANNULI WITH VOLUME HEAT SOURCES WITHIN THE FLUIDS H. F. Poppendiek L. D. Palmer CENTRAL RESEARCH LIBRARY DOCUMENT COLLECTION LIBRARY LOAN COPY DO NOT TRANSFER TO ANOTHER PERSON If you wish someone else to see this document, send in name with document and the library will arrange a loan, OAK RIDGE NATIONAL LABORATORY . OPERATED BY CARBIDE AND CARBON CHEMICALS COMPANY A DIVISION OF UNION CARBIDE AND CARBON CORPORATION POST OFFICE BOX P OAK RIDGE, TENNESSEE ORNL-1701 Copy No. 4/ ;7 N Contract No. W-TL05, eng 26 Reactor Experimental Engineering Division FORCED CONVECTION HEAT TRANSFER BETWEEN PARALIEL PLATES AND IN ANNULI WITH VOLUME HEAT SOURCES WITHIN THE FIUIDS by H. F. Poppendiek L. D. Palmer DATE ISSUED: MaY 11 1954 OAK RIDGE NATIONAL LABORATORY Operated by CARBIDE AND CARBON CHEMICALS COMPANY A Division of Union Carbide and Carbon Corporation Post Office Box P Ogk Ridge, Tennessee MARTIN MARIETTA ENERGY SYSTEMS L| RIES TR 3 Y456 D349L39 y - AN R et e s e S T 1. 2. 3. 45, 60 T-11. 12. 13& 1k, 15. 16. 17. 18. 19. 200 21. 220 23. 2k, 25, 6. 270 28. 29. 30. 31. 32. 330 3k, 36. Lo, Lh, 118, 119. 120-369. INTERNAL DISTRIBUTION C. E. Center Bilology Library Health Physics Library Central Research Library Reactor Experimental Engineering Library Laboratory Records Department Laboratory Records, ORNL R.C. C. E, Larson L. B. Emlet (K-25) J. P, Murray (Y-12) A. M. Weinberg E. H. Taylor E. D. Shipley C. E. Winters F. C. VonderLage R. C. Briant J. A. Swartout 8. C. Lind F. L. Culler A. H. Snell A. Hollaender M. T. Kelley W. J. Fretague G. Ho Clewett K. Z. Morgan T. A. Lincoln A. 5. Householder C. S. Harrill D. 8. Billington D. W. Cardwell E. M. King R. N, Lyon Jd. A. Lane A. 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Manson) Given distribution as shown in TID-A500 under Engineering Category DISTRIBUTION PAGE TO BE REMOVED IF REPORT IS GIVEN PUBLIC DISTRIBUTION TABILE OF CONTENTS INTRODUCTION . csveceveccenscrssonccs . IAMINAR FLOW ANALYSIS .. eeeeccosccscconssosncccananse searesssssrae ceees TURBULENT FLOW ANALYSIS.....4.s tesccsasccsasesernnsrenae cocnns cereesnn DISCUSSION..... Ceescescsesssesseteesesseerat et oes st ne toessesssrases APPENDIX l.cecceccecsccnsesnsscsseossasasonsse P PAGE 1h SUMMARY This paper concerns itself with forced convection heat transfer between parallel plates which are infinite in extent and ducting fluids containing uniform volume heat sources; also heat is transferred uni- formly to or from the fluids through the parallel plates. Dimensionless differences betweén the plate wall temperature and the mixed-mean fluid temperature are evaluated in terms of several dimensionless moduli. These analyses pertain to the laminar and turbulent flow regimes and liquid metals as well as ordinasry fluids. The solutions may also be used to estimate heat transfer in annulus systems whose inmer to outer radius ratios do not differ significantly from umity. NOMENCLATURE letters cross sectional heat transfer area, ft2 £luid thermel diffusivity, £t2/hr parameter in equation (o), ft/hr fluid heat capacity, Btu/lb °F parameter in equa%ion (r), dimensionless gravitational force per unit mass, :E‘t/hr2 heat transfer conductance, Btu/hr £ OF £luid thermsl conductivity, Btu/hr £t (°F/ft) fluid pressure, 1bs/ft2 heat transfer rate, Btu/hr radisl distance from centerline of parallel plate system, ft radial position at which the reference tempersture tq is stipulated, ft half the distance between the two parallel plates, ft fluid temperature st position n, °F a reference temperature at radius ry, OF mixed-mean fluid temperature, Op fluid temperature at plate walls, Op fluid temperature at the parallel plate system center, Op fluid velocity at n, ft/hr mean fluid velocity, ft /hr ny, Nu Re -8 - volume hegt source, Btu/hr ft2 axial distance, ft radial distance from parallel plate walls, ft fluid weight density, lbs/ft” eddy diffusivity, £t2/hr friction factor defined in equation (i) dimensionless absolute viscosity of fluid, 1b hr/ft2 fluid kinematic viscosity, fte/hr fluid mass density, lbs h:a:'e/.f"l:}+ £luid shear stress at position n, lbs/ft? f£1luid shear stress at parallel plate wells, los/ft® Dimensionless Moduli Y/ To Y1./To h bro/k, Nusselt Modulus ¥ v¢p/k, Prandtl Modulus up bro/ P -9 - INTRODUCTION The mathematical heat transfer analyses to be presented here for a parallel plates system are accomplished much in the same manner as were those for a pipe system presented previously in reference 1. The present analyses as well as those given in reference 1 can be used to determine the tempera- ture structure in flowing flulds that possess internal sources of heat gener- ation. Such volume heat sources may result from nuclear or chemical reactions or may be generated electrically. The ideslized volume-heat-source system considered in this paper is defined by the following postulates: 1. Thermal and hydrodynemic patterns have been established (parallel plates of infinite extent). Uniform volume heat sources exist within the fluids. Physical properties are not functions of temperature. : Heat is transferred uniformly to or from the fluid at the plate walls. In the case of turbulent flow the generalized turbulent velocity profile defimnes the hydro- dynamic structure. In the case of turbulent flow there exists an analogy between heat and momentum transfer. - 10 - LAMINAR FLOW ANALYSIS The differential equation deseribing heat transfer in the parallel plates system for the case of laminaer flow is o | 3 ey ot L. 0% W .é.um[l __.)]_a_x_a_é.;é.+ (1) To Yep Where, Uy, mean fluid velocity t, temperature X, axial distance T, radial distance a, thermal diffusivity W, wniform volume heat source Y, fluid weight density Cps fluid heat capacity One boundary condition is represented by the uniform wall-heat -flux which may be positive, negative or zero, d _ _/aq\ _ 4 Ot (. _ . a%(r—rq)—(fi)o- k-—a-—l-;-(r“"’ro) (2) where %&.is the radial heat. flux and (%%) is the wall heat flux. The second boundary condition is, td,‘h reference te;perature, such as a wall or center- line temperature, t(r = rq) = t4 (3) Note, the mixed-mean fluid temperature may also be specified as the reference temperature. -11 - Downstream from the entrance region where the thermal pattern (tempera- ture gradients) of the system has become established, the axial temperature gradient, .ig...;% » 1s uniform and equal to the mixed-mean axial fluid temperg - ture gra.dientl, ...g.;.fi - The latter gradient can be obtained by making the following heat rate balance. The heat generated in a lattice whose volume is 2r, dx (the width of the lattice being unity) plus the heat transferred into or out of the lattice at the plate walls must all be lost from the Ei d.x adlc - 2 Y — Hence, in the established flow region the axial temperature gradient is w-31 (dq dt _ btm _ To (dA>o (5) 0x 0JXx Un ¥ cp Upon substituting equation (5) into equation (1), the following total differ- ential equation results: (- @) © 1. DNote, that the mixed-mean fluid temperature at any given axial position is defined as, -ro -r / t u dr ° ty = 2 = X tu dr Tq Upre g [ o I SR e e e e i N R T -12 - vhere F' = 1 - _l_.(gg) - Equation (6) can be solved upon making two Wro integrations. The first integration plus boundary equation (2) yields, dt=E '—52!_ F! 3 at k{(e 1)1--.2.;;21-} (7) A second integration gives the desired temperature solution, =2 [ ) 5@ ) ® where the reference temperature is, to, the wall temperature. The tenpera- ture solution in terms of the centerline temperature rather than the wall temperature is given by t -t = 2 o, b E,_?E*[GE - 'el‘)(%) -5 (&) ] (9) where t4 1s the centerline temperature. Equation (9) is graphed in Figure 1 for several values of the function F'. The difference between the plate wall temperature and mixed-mean ~luid temperature is defined by To \/ér u(ty - t)ar . (10) to -ty = nes b Upon substituting the laminar velocity profile relationfiand equation (8] into equation (10) there results, Yo - tm _ 1TF - 14 (11) Wr,© 35 k -13 - UNCLASSIFIED ORNL-LR-DWG-176 0.75 | T | l 7‘ 0.70 0.60 / // ) | 0.50 / // 0.40 S // 0.30} / // - 0.20 / / . | — ~ 0.10 - F= e fn"fq_ / //‘, . 2 / / ‘ \LV'EQ' "] F'=3/4 o mm——— “ \ \ \ \‘\ \ F'=‘|/2 -0.40 \ \ . \ ‘-\ -0.20 ™ Fi=0 -0.30 AN ~0.40 \\\ -0.50 I ! | | \ 0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 4 r To Fig. 4. Dimensioniess Radial Temperature Distributions in a Parallel Plotes System for Laminar Flow (Equation 9) -1k - TURBULENT FLOW ANALYSIS Fluid flow in pipes and channels (parallel plates systems) under turbulent flow conditions has been characterized in terms of a laminar sublayer contiguous to the wall, a buffer layer, and a turbulent core by Nikursdse, von Karman, and others. This structure has been presented in a general fashion by the well known generalized velocity profile which was shown together with the experimental data of Nikuradse, Reichardt, and Laufer in reference 1. Table 1 gives some of the specific hydrodynamic relations for the various flow layers in a parallel plates system; a discussion of some of the details of this table can be found in Appendix 1. The differential equation describing heat transfer in a parallel plates ‘system.for the case of turbulent flow is u(r) $% = 5= [(a re ) g;] ' 3% (12) where, u(r), the turbulent velocity profile (given by the generalized velocity profile) € > the eddy diffusivity? given in Table 1 Upon substituting equation (5) into equation (12) for the established theraml region, the following total differential equation resulis, _ 1 (4q u(r)[ To (dA)o] _ W _.@_[(“e)éfi-] (13) ar dr 2, It is postulated that the heat and momentum transfer eddy diffusivities are equal as proposed by Reynolds and successfully used by von Karman, Martinelli and othens. TABLE I HYDRODYNAMIC RELATIONS FOR THE VARIOUS FLOW LAIERS BETWEEN PARALLEL PLATES REGION GENERALIZED VELOCITY DISTRIBUTION SHEAR STRESS STRESS EQUATION EDDY DIFFUSIVITY Laminar Sublayer T < + _'o . o y<5 u _ p y T:To T=p‘D%E ""%“—"—“O or o< i i2led To v Y To Re” - P Buffer ILayer , . 5 — o no o Re=100,000 N ° \\ \\ AN \\\ \i\ ~ _ A\\ N . 4—Re=1,000,000 & 0 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 n Fig. 4. Dimensionless Radial Temperature Distributions Within a Fluid Flowing Between Parallel Plates with Insulated Plates for Several Reynolds Moduli and Pr=0.04 _a_ The difference between the plate wall temperature and the mixed-mean fluid temperature was obtained by evaluating the integral ()« @) 1 E,-!-, : '}”\u o : i PN L ‘ " '« The dimensionless temperature difference, to -~ tm , is graphed as a function Wr02 of Reynolds and Prandtl moduli in Figure 5. k The superposition of solutions of the boundary value problems (14) and (15) yields the more general boundary value problem defined by equations (13), (2) and (3). In the superposition process, all temperatures are expressed as tempefature increments above datum temperatures. The radial temperature distribution above the wall temperature, centerline temperature, or mixed- mean fluid temperature for the composite boundary value problem defined by (13), (2), and (3) is obtained by adding the radial temperature distributions above the wall tempefatures, centerline temperatures, or mixed-mean fluid temperatures, respectively of boundary value problems (14) and (15). Also, the rise in mixed-mean fluid temperature, at some point in the esfablished flow region of the parallel plates system, above its value at the entrance for the problem defined by (13), (2) and (3) is obtained by adding the corresponding temperature rises for problems (14) and (15). The solution of boundary value problem (15) expressed in terms of Nusselt, Reynolds, and Prandtl moduli as developed by Martinelli is presented in Appendix 3. -22 - UNCLASSIFIED CRNL —~ LR—DWG. 180 1 Y - ~ O {Pr (oo 1 0_6 | [ Ll ] | l I 1)1 \ | l v 102 2 4 6 8103 2 Fig. 5. Dimensionless Differences Between the Wall Functions of Reynolds and Prandtl 4 Moduli 6 for 8404 2 Re Parallel q 6 8105 2 4 6 Plate System (Walls Insulated). and Mixed —Mean Fluid Temperatures as ...’2‘3.. DISCUSSION The forced convection analyses presented here pertain to the parallel plates system. These analyses may also be used to estimate heat transfer in annulus systems where the inner to outer wall radius ratio does not differ significantly from unity; under such circumstences, the annulus satisfactorily approximates a parallel plates system. The present report is the second one in a Planned series which are to explore the experimental as well as theoretical aspects of volume-heat-source forced convection. Two specific research activities have almost been com- Pleted and are to be reported in the near future. One activity involves an experimental study of volume-heat-source forced convection in a pipe system in the laminar and turbulent flow regimes; comparisons are made with the previously developed theory. Another activity is a mathematical study of volume -heat -source forced convection in the lsminar regime including a temperature dependent fluid viscosity. - ol - APPENDIX 1 HYDRODYNAMIC RELATIONS FOR TURBULENT FLOW IN A SMOOTH PARALIEL PLATES SYSTEM The hydrodynamic relations given in Table I characterize turbulent flow in a smooth channel (parallel plates system). The manner in which this table was developed is illustrated below for the buffer layer. The turbulent shear stress equation is expressed as =(0+€)g~% (a) Dl.q In the buffer layer, the shear stress is very closely equal to the wall shear stress, 1., and the velocity distribution is given by, ut = -3.05 + 5.00 ln y* (b) Upon differentiating equation (b) it can be shown that 5o ~x P (c) J — &l Upon substituting equation (c) and the wall shear stress in equation (2) and solving for the eddy diffusivity, there results, T _Jz;g; (a) 3 -1 < — 64 ™ i - 25 - Two relations describing the pressure drop and wall shear stress in the parallel plates system are, LD _ ELZ 'u?m ax " Ir, Zg (e) and AD where the quantity, hro, is sometimes called the equivalent duct diameter and, § , is the friction factor which is uniquely related to the Reynolds modulus. Upon substituting equation (e) into equation (f) there results, JTQ=J§% (g) The Reynolds modulus for the parallel plates system (based on the equivalent diameter) and the friction factor relation are expressed as, b roup Re = ——— e = 2 (1) 3 and _ 023 5 6 . 5= 5 for 5 x 10