PROCEEDINGS SECOND WORKSHOP GEOTHERMAL RESERVOIR ENGINEERING December 1-3,

Transcription

1 SGP-TR-2-34 PROCEEDNGS SECOND WORKSHOP GEOTHERMAL RESERVOR ENGNEERNG December -3, 'Conducted under Grant No. NSF-AER supported by the RANN program of the National Science Foundation and Contract No. E PA-5 from the Geothermal Energy Division, Energy Research and Development Administration.

2 BUOYANCY NDUCED BOUNDARY LAYER FLOWS N GEOTHERMAL RESERVORS Ping Cheng* Department of Petroleum Engineering Stanford University, Stanford, CA Most of the theoretical study on heat and mass transfer in geothermal reservoirs has been based on numerical method. Recently at the 975 NSF Workshop on Geothermal Reservoir Engineering, Cheng () presented a number of analytical solutions based on boundary layer approximations which are valid for porous media at high Rayleigh numbers. According to various estimates the Rayleigh number for the Wairakei geothermal field in New Zealand is in the range of -5, which is typical for a viable geothermal field consisting of a highly permeable formation and a heat source at sufficiently high temperature. The basic assumption of the boundary layer theory is that heat convective heat transfer takes place in a thin porous layer adjacent to heated or cooled surfaces. ndeed, numerical solutions suggest that temperature and velocity boundary layers do exist in porous media at high Rayleigh numbers (2). t is worth mentioning that the large velocity gradient existing near the heated or cooled surfaces is not due to viscosity but is induced by the buoyancy effects. The present paper is a summary of the work that we have done on the analytical solutions of heat and mass transfer in a porous medium based on the boundary layer approximations since the 975 ~orkshop. Similarity Solutions to Boundary Layer Equations Free Convection about a Vertical Heat Flux mpermeable Surface with Uniform The solution for free convection about a vertical impermeable surface with wallatemperature being.a power function of distance, i.e., T = T + Ax for x> is given by Cheng and Minkowycz (3). W The constant heat flux solution can be obtained by a simple transformation of variables and by setting A = /3 in Ref. 3. The ~Visiting Professor. On sabbatical leave from Department of Mechanical Engineering~ University of Hawaii, Honolulu, Hawaii '

3 expressions for local Nusselt number, the mean Nusselt number, and thermal boundary layer thickness are Nu [~a"; ] 3 = x x.7723, Nu / [Ra'" ] 3 = L L.29, <5 T = 4. 8 [Ra,'; ] 3", x x hx ox with Nux - -k = k (T~-T-:r' NUL Rat - PmgSKqL2/~ak, and TW-T oo () (2 ) (3) ql K(TW-T ) L CXl -t (TW-T oo ) dx, where the density of the fluid at infinity; g the gravitational acceleration;~ and B the viscosity and the thermal expansion coefficient of the fluid; K the permeabil ity of the porous medium; L the length of the plate; q the surface heat flux; a and K the equivalent thermal diffusivity and the thermal conductivity of the saturated porous medium. The equivalence of Eqs. () through (3) and the corresponding expressions given by Cheng & Minkowycz (3) was shown recently by Cheng (4). Free Convection about a Horizontal Heat Flux mpermeable Surface with Uniform The constant heat flux solution for a horizontal impermeable surface can be obtained by a simple transformation of variables and by setting A = /2 in the solution given by Cheng and Chang (5). The expressions for local Nusselt number, the mean Nusselt number, and thermal boundary layer thickness for the present problem are Nu [Ra*4 =.8588, x x (4) NUr! [RatJ"4 =.288, (5 ) (6) -237-

4 Mixed Convection from a Vertical sothermal mpermeable Surface The problem of mixed convection from a vertical impermeable surface with a step increase in wall temperature (i.e., T W = Too + A for x~o), embedded in a porous medium is considered by Cheng (6). The expressions for local Nusselt number, average Nusselt number, and thermal boundary layer thickness are Nu x --- [Re Pr]l/2 x [-6' ()] ( 7) 2[-8' ()] (8) x [Re Pr]/2 x (9 ) U x v with Re = and Pr = where U is the velocity outside x v a the boundary layer. The values of NUx/[RexPr]~ and [RexPr]~6T/x as a function of Gr Re are shown in Figs. and 2, where the corresponding value~ fo~ free convection about a vertical isothermal plate as given by Cheng and Minkowycz (3) can be rewritten as NU/[Re.Pr] '2 =.444 x x () [Grx/Re:J 2 () Mixed Convection from a Horizontal Surface Heat Flux mpermeable Surface with Uniform The expressions for local Nusselt number, average Nusselt number, and thermal boundary layer thickness are given by (4,7) Nu x = [Re Pr]2 x l [- (Q)]' (2) -238-

5 3 = 2[ -<pcott' (3 ) (4), t The val~es of Nu /[Re Prl~ and [Re PrJ ~/x as a function of Ra*/(Re Pr) are giv~n inxref. 4, wher~ the asymtotes for free conve~tionxabout a horizontal plate with uniform heat flux are given by Eqs. (4) and (6), which can be rewritten as Nu /[Re PrJ x x "2 = O " [ Ra ~': / (Re pro) ], x x (5) pr]28t/x=4./[ra~/(repr)2]4". [Re x x x (6) The Effect of U ro on Heat Transfer Rate and the Size of Hot-Water Zone To gain some feeling on the order of magnitude of various physical quantities in a geothermal reservoir, consider a vertical impermeable surface at 25 C is embedded in an aquifer at 5 C. f there is a pressure gradient in the aquifer such that groundwater is flowing vertically upward, the values of heat transfer rate and the size of the hot water zone can be determined from Figs. and 2. The results of the computations for U oo varying from. cm/hr to cm/hr are plotted in Figs. 3 and 4 where it is shown that the total heat transfer rate for a vertical surface, km by km, increases from 2 MW to MW, whi e the boundary layer thickness at x = km decreases from 3 m to 2 m. Val idity of Boundary Layer Approximations The validity of the boundary layer approximations can be accessed by a comparison of results obtained by similarity solutions to that of numerical solutions of exact partial differential equations, or to experimental data. For free convection' in a porous medium between parallel vertical plates separated by.a distance H, the correlation equation given by Bories and Combarnous (8) is -239-

6 (7) -24- where L is the length of the plate, NU = H h~ and RaH3= pgbk (T'p(T ) HhlCX. Eq. (7) issval id for Ra L H from 2 to and for H/l between.5 and.5. On the other hand, the heat transfer rate as obtained from boundary layer approximations for an isothermal vertical plate (3) is ( 8) which can be rewritten as Nu ' = O ( Ra H ) 5 (!i) 5 H L, (9) Eqs. (7) and (9) for H/L =.5 and.5 are plotted in Fig. 5 for comparison. t is shown that they are in good agreement, especially at high Rayleigh numbers where the boundary layer approximations are val id. Concluding Remarks As in the classical convective heat transfer theory, boundary layer approximations in porous layer flows can result in analytical solutions. Mathematically, the approximations are the first-order terms of an asymptotic expansion which is valid for high Rayleigh numbers. Comparison with experimental data and numerical solutions show that the approximations are also accurate at moderate values of Rayleigh numbers. For problems with low Rayleigh numbers where boundary layer is thick, higher-order approximations must be used (9). References. Cheng, P., "Numerical & Analytical Study on Heat and Mass Transfer in Volcanic sland Geothermal Reservoirs." Proceedings of 975 NSF Workshop on Geothermal Reservoir Engineering,pp (976). 2. Cheng, P., Yeung, K.C., & Lau, K.H., 'Numerical Solutions for Steady Free Convection in sland Geothermal Reservoirs." To appear in the Proceedings of 975 nternational Seminar on Future Energy Production--Heat & Mass Transfer Problems. Hemisphere Publishing Corporation.

7 3. Cheng, P. and Minkowycz, W.J., "Free Convection about a Vertical Flat Plate Embedded in a Porous Medium with Appl ication to Heat Transfer from a Dike." Accepted for publication by J. Geophysical Research. 4. Cheng, P., "Constant Surface Heat Flux Solutions for Porous Layer Flows." Submitted for publication to Letters in Heat & Mass Transfer. 5. Cheng, P. and Chang, -Dee, "Buoyancy nduced Flows in a Saturated Porous Medium Adjacent to mpermeable Horizontal Surfaces." nt. J. Heat Mass Transfer,.!., 267 (976). 6. Cheng, P., "Combined Free & Forced Boundary Layer Flows about nclined Surfaces in a Porous Medium. ' Accepted for publ ication by nt. J. Heat Mass Transfer. 7. Cheng, P., "Similarity Solutions for Mixed Convection from Horizontal mpermeable Surfaces in Saturated Porous Media." Accepted for publication by nt. J. Heat Mass Transfer. 8. Bories, S.A. and Combarnous, M.A., "Natural Convection in a Sloping Porous Layer," J.F.M., 57, (2973). 9. Cheng, P. and Chang, -Dee, "Higher-Order Approximations for Convective Heat Transfer in Porous Layers" (in progress). -24-

8 FGURE. HEAT TRANSFER RESULTS FOR MXED CONVECTON FROM AN SOTHERMAL VERTCAL PLATE., , Free Convection Asymptote Forced Convection Asymptote N.l: N..5['---'-'-' O.ll- -J...-_ L-...o.-L-L...L..L. --t_--j.._l.-j.-j..-.-oa-l L-_.-l--J--J-...L-...L-.oo.-J-J Grx/Rex.

9 )( "- - to r - >< C\...,.---, \- N J::- W (l) :: ~,.-, , t~ _.~.-.-._ r, 2 f-. ~.5 f-.4.3 r L.2 _ Free Convection Asymptote _._.- Forced Convection Asymptote.. --._ l.-j-.!...j- J-_L.--l--...L-..J---J-.l..l.l..l -._--... _J.-J-l.-ll--..l...l.!~./ Gr x / Rex Fig. 2 Boundary Layer Thickness for Mixed Convection From An sothermal Vertical Plate

10 FGURE 3. THE EFFECT OF U oo ON TOTAL HEAT TRANSFER RATE FOR MXED CONVECTON FROM AN SOTHERMAL VERTCAL HEATED SURFACE N A GEOTHERMAL RESERVOR. 5~ K = - 2 m 2 3; f!. = 6 9 m 3 co 2... f3 =.8 x- 4 C 4t Q.) ::: 3 2r '+- W +- \- () N.j::-.j::- fl =.27g/sec-m a = 6.3 x- 7 m 2 /sec T - Tco = 2 C (f) c Area = xlooonl \- - r- f Free convection asymptote /' ~ - 4-., Forced convection asymptote ~ Q.) 5 ~ 4 L '" +- r- 3 -.,.'/ ~ L-_--L~ l_l_lj_l.j...l L_J_l._J L_.L.J_L.l L l l_..l._...l..._j"..j_.j Ua:> crn/hr

11 E... E or- -- Ṉ l:""..--. \n f- ro -- = -2 2 K - m 5f P oo - 6 g / m 3 4 f3 -.8 x -4/ o C 3 - fl -.27 g/sec-m a - 2 Tw - ToO = 2 C ~ r- "' ~ ~ 5 = -~ 4[ 3, Free convection asymptote Forced convection asymptote xlO- 7 m 2 /sec '---_---L._--l.---J---'-...l.-..J.-----J-l L._.l-..J---'-~ l.._... J.... '...4_" ' ' '_' U co ern /h r Fig. 4 The Effect of U en on the Boundary Layer Thickness (Size of Hot Water Zone) for Mixed Convection from a Vertical sothermal Heated Surface in a Geothermal Reservoir

12 FGURE 5. COMPARSON OF THEORY AND EXPERMENT FOR VERTCAL POROUS LAYERS. roor , --- Theoretical (Cheng 8 Minkowycz 975) Experimental (Bories a Combarnous 973) 4-3r 2 N J: (J'\ NU H r,v :[ /./;;~~~ 3 4~ r /' /' /' ~ 2~,?-',/',// ~ -l-_..l--j--l-j--,l.,,!-.lor...l.t...! -.-_-,---,--,-, 2,,