SIt1ILARITY SOLUTIONS FOR CONVECTION OF GROUNDWATER ADJACENT TO HORIZONTAL IMPERMEABLE SURFACES WITH AXISYMMETRIC TEMPERATURE DISTRIBUTION

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1 .. :.... St1LARTY SOLTONS FOR CONVECTON OF GRONDWATER ADJACENT TO HORZONTAL MPERMEABLE SRFACES WTH AXSYMMETRC TEMPERATRE DSTRBTON. TECHNCAL REPORT No. 14 April Prepared nder. NATONAL SCENCE FONDATON Research Grant No. G and ENERGY RESEARCH AND DEVELOPMENT ADMNSTRATON Research Grant No. E(04-3)-1093 By Ping Cheng and W. C. Chil Department of Mechanical Engineering niversity of Hawaii Honoll Hawaii. ----DSCLAMER sored bv anof the nited States (iovefnment. This book was prepared as an accont of work span thereof nor any of their employees makes any Neither the nited StiltS nment nor any age7" liabil;tv or responsibility for the accracy warranty express or npiled. or assmes any ega odct at" process disclosed. Of comp'etness. or seflness of 8vf'infort; =th:- Rence herein to any specific represents that its se wold not nnge pnv8 name trademark manfactrer. or otherwise does commercial prodct process. o Sl!rvl bv trade recommendation. or f8\fofing by the nited not necessarily constitte or mplv ts endorment. d 0 inions of athors expressed herein do not States Government or any agency thereof. The Vews an P thefeof necessarily state or reflect those of the nited States Government or any agency...

2 NOMENCLATRE constant defined by Eq. (6a) specific heat of the convective flid dimensionless stream fnction defined by Eq. (16) acceleration de to gravity local heat transfer coefficient average heat transfer coefficient defined permeability of the poros medim thermal condctivity of the poros medim local Nsselt nmber N r = hr/k average Nsselt nmber N = hr/k pressre over-all heat transfer rate local heat transfer rate coordinate in the radial direction radis of the impermeable srface modified Rayleigh nmber Ra = p=gktw-t=r/a modified local Rayleigh nmber Ra r temperatre velocity component in r-direction velocity component in z-direction coordinate in the vertical direction eqivalent thermal'diffsivity coefficient of thermal expansion momentm bondary layer thickness thermal bondary layer thickness = p=gktw-tlr/a i

3 r. L L L 11 dimensionless similarity variable.defined by Eq. (15) 11m vale of 11 at the edge of momentm bondary layer 11 r vale of 11 at the edge of thermal bondary layer e dimensionless temperatre defined by Eq. (17). constant defined by Eq. (6a) viscosity of convective flid. p density of convective "flid stream fnction V dimensionless stream fnction V =/R(Rar}1/3 Sbscript condition at infinity w condition at the wall ii

4 L ABSTRACT The axisynmetric boancy-indced grondwater flow adjacent to horizontal impermeable srfaces with wall temperatre being a power fnction of radis is discssed in this paper. With the bondary layer simplifications the governing non-linear partial differential eqations can be transformed into a copled pair of no.n-linear ordinary differential eqations with two-point bondary conditions that canoe integrated nmerically by established techniqes. Simple algebraic expressions for bondary layer thickness and heat transfer rate are obtained. Applications to free convective flow in a liqid-dominated geothermal system at high Rayleigh nmber are discssed.. iii

5 i L. ntrodction The prediction of hecl.t transfer rate from a tieatedor cooled horizontal srface to srronding grondwater in a sbsrface environment has import applications to the assessment of geothennal resorces and the design ofa geothermal power plant. The problem can be idealized as a horizontal flate plate embedded in a satrtedporos medim of infinite extent. t has been recognized for some time that convection in a satrated poros medim and in an incompressible flid have mch in common. -Ths in analogos to the classical free convection problems at high Grashof nmbers treated by Stewartson[l] and Gill [2] it can be conjectred that convective heat transfer in a poros medim at high Rayleigh nmbers takes place in a thin layer adjacent to the heated and cooled srfaces. ndeed recent finite difference soltions by Cheng Veng and La [3] for free convection in a liqid-dominated goethermal reservoir shows that bondary layer behavior becomes increasingly prononced in the flow field near the heated or cooled srfaces as the Rayleigh nmber of the reservoir is increased. The bondary layer approximations have been employed earlier by Wooding [4] McNabb [5] Vih [6] and recently by Cheng and Minkowycz [7] to obtain analytical soltions to a nmber of free convection problems in a satrated poros medim at high Rayleigh nmbers. n this paper the axisymmetric boyancy indced flow in a satrated poros medim above a heated or below a cooled horizontal impermeable srface with prescribed wall temperatre being a power fnction of radis is stdied. Bondary layer approximations simi1.ar to the previos work [4-7] are invoked and similarity soltions are obtained.- -" ';- '-: Nmerical integrat1on'ofthe reslting two-point bondary vale problem is carried ot and expressions for heat transfer rate and bondary layer thickness are derived. Comptations for heat transfer rate and the size of the hot water zone above a heated bedrock with -1-

6 ij l ' ' f 1 J! 1 km in radis maintained at different wall temperatre distribtions are carried ot.. Analysis Consider the asymmetric boyancy flow in a satrated poros medim above a heated horizontal impermeable srface or below a cooled srface with wall temperatre being a fnction of radis. The coordinate system is shown in Fig. 1 where rand z are cylindrical coordinates in horizontal and vertical directions with positive z axis pointing toward the poros medim. f we assme that (i) the convective flid and the poros medim are everywhere in local thermodynamic eqilibrim (ii) the temperatre of the flid is everywhere below boiling point (iii) properties of the flid and the poros medim are consant eqations are given by and (iv) the Bossinesq approximation is employed the governing 1... (r) (rw) = ar az 0 = _ K par ' (l) (2) ' (3) p = p [1 - a{t-t}] co co (4) (5) where the 11+" sign in Eq. (3) refers to the case of a heated impermeable srface facing pward (Fig. lb) while the "_" sign refers to the case of a...1 / -2- r''''

7 r i i.l l.l cooled impermeable srface facing downward (Fi. la). n Eqs. (1) to (5)'.and ware the velocity components in the horizontal and vertical directions respectively; p p and a are the density viscosity and the thermal expansion coefficient of the convecting flid respectively; K is the permeability of the poros medim; a = k/(pooc)fis the eqivalent thermal diffsivity with k denoting the therma condctivity of the satrated poros medim and (PooC)f the prodct of density and specific heat of the convecting flid; T p and 9 are respectively the temperatre pressre and the gravitational acceleration. The sbscript " 00 " refers to the condition at infinity. The bondary conditions for the problem are z = 0 T =T ±Ar').. Woo" T = T 00' w= 0 = 0 (7ab) i where A>O and the "+" and "_" signs in Eq. (6a) are for a heated impermeable srface facing pward and for a cooled impermeable srface facing downward respectively. Eq. (6a) shows that the prescribed wall temperatre is a power fnction of radis from the origin. The continity eqation is atomatically satisfied by introdcing the stream fnction Was =l r az and w=_l. r ar (8) Eliminating p from Eqs. (2) and (3) by cross differentiation the reslting eqation in terms Wis 3.(1 1t). 3r r ar (9) -3-

8 .. 1. Eqation (4) in terms ofw can be rewritten as (10) j i The appropriate bondary conditions for Eqs. (9) and (10) are z = 0 T = T.+ Ar>". w ClO ' (11 ) :! t J i i. Similarity Soltion T = T ClO ' 2.t = 0 az.. f we assme that the bondary layer behavior exists Eqs. (9) and (10) can be approximated by (12) (13) 1 (14 ) J To seek similarity soltions to Eqs. (13) and (14) with bondary conditions (11) and (12) we now introdce the following dimensionless variables Kp gsa] 1/3 =. ClO zr(>..-2)/3= (Ra )1/3 n [ pa r r ' K gas] 1/3. W.= a [ P ClO r(4+>..)/3 fen) = a(ra )1/3rf(n) pa r (15) (16) (17)..J.J where Ra = p KgSTw-T r/pa is the modified local Rayleigh nmber. r ClO ClO -4-

9 ! 1.J i. n terms of new variables it can be shown that the ve10citycomponehts are given by K gaaj 2/3. = [. Pa r(2a-1 )/3 f' (n) [ KP gaal1/3 (A-2)/3.. w = -a. a J [{A-2)nf' + (4+A)f] (18) (19) 1 i J t! j J and the governing eqations (13) and (14) and hei.r conditions are and a(o) = 1 f(o) = 0 a(oo) = 0 f' (00) = 0 appropriate bondary (20) (21 ) (22ab) (23ab)...J V. Reslts and Discssion The bondary layer approximations sed in obtaining Eqs. (20) and (21) are '. i t..l valid if (i) z» r and (ii) w«. From Eq. (15) it follows that z/r = 0(Ra;1/3). Frthermor the ratio of Eqs. (19) and Eq. (18) gives w/ = 0(Ra;1/3). Ths the two conditions are satisfied if Ra r is large. Conseqent-. 1y the bondary layer approximations are not expected to be valid near r = O. To find the range of Afor which the problem is physically realistic we follow the approach discssed by Gebhart [8] by examining Eq. (18) and the expressions for the bondary layer thickness which can be obtained from Eq. (15) to give and -5- <5 T nt -= r (Ra )1/3 ' r (24ab)

10 ... i 1.;!. t j!i j i (25) i.l. t! W where the vales of [-6'(n)] is plotted in Fig. 4. is given by Eq. The local srface heat flx (25) with n = 0 which shows that the local srface heat flx is constant for A = 1/2 and increases with respect to r for other vales of A. the definition of the 1oca1 Nse1t nmber N r = h; = k(tqt) with'h denoting. '. w 00 the local heat transfer coefficient Eq. (25) can be rewritten as N r (Ra )1/3 = [-6'(0)] r (26) From where the vale of [-6'(0)] for selected A is tablated in Table 1. The overall srface heat transfer rate for a circlar srface with a radis R can be compted from i where n m and nt denote vales of n where /r.or 6 has a vale of Since the wall temperatre differs from that of the srronding flid at r =0 both a.!ld the bondary layer thickness mst be increasing or at least constant with respect to r [8]. t follows from Eqs. (18) and (24) that these conditions are satisfied if 1/2 A< 2 Nmerical integration for the two-point bondary vale problem Eqs. (20) (23) can be obtained sing the Rnge-Ktta method by first converting to an initial vale problem with a systematic gessing of the vales of fco) and 6'(0) by the shooting techniqe. Nmerical reslts for selected A in the range of 1/2 A 2 are presented in Figs. 2 throgh 7. The vales of n m and nt in Eqs. (24) can be obtained from Figs. 2 and 3. These vales are tablated in Table 1 which shows that the momentm and thermal bondary layer thickness are of the same order of magnitde. With the aid of Eqs. (15) and (17) the local srface heat flx is given by -6-

11 (27) which can be integrated after the sbstittion of Eq. (25) to give (28) The average Nsse1t nmber is defined by N = hr/k where the average heat transfer coefficient h depends on the choice of the temperatre difference between the wall and the temperatre of the flid away from the wall. Consider the temperatre differene based on the mean temperatre difference defined by W Also from the definition of the average heat transfer coefficient. we have (29) (30) Eqating Eq. (28) with Eq. (30) and from the definition of the average Nsse1t nmber defined earlier we have! (31 )! where Ra = Tw-TmlpwgBKR/a. To obtain the indced pressre along the wall. we eqate Eqs. (2) and (18) and integrate along the line z=o to get..[k 9BA] 2/3. per 0) = -. Pm f' (0)r(2).+1 )/3 2{).+1) a ' (32) -7-

12 i f! f 1 ) 1 j i j..... which shows that the indced pressre is increasingly more negative with 'r. t is noted that althogh + o otside the momentm bondary layer the. vertical velocity component otside the bondary layer in general has a nonzero vale given by [ ] 1/3 = -cx. p009a8 r(x-2.)/3 (4+X) f(oo) 00 a3 which is obtained from Eq. (19) with the aid of Eq. (23b). Eqation (33) shows that wis 00 negative for the range of A considered since f(oo) is a non-zero " negative vale as shown in Fig. 5. The variation of the dimensionless vertical velocity is lotted (33) in Fig. 6 where it is shown that its magnitde increases from zero to a finite vale as n is increased. To plot the convective pattern of grondwater we note from Eqs. (15) and (16) that ' = ' (Rar)1/3;x] where ' =w/r(rar)1/3 is the dimensionless stream fnction. A representative plot of streamlines is presented in Fig. 7 where the dimensionless coordinates. are and (Rar) 1/3. To gain some insights of the magnitdes of varios physical qantities in a geothermal application consider an pward facing heated impermeable srface of radis 1 km with wall temperatre increasing from 288 Kat r=o to 573 Kat r=l km. in 'Fig. 8. The wall temperatre distribtion for different vales of Xis sketched For n4merical comptations the following physical properties are sed: (3 = 2.8xlO4/K Poo = O.92x10 6 gim 3 C = 4.2X10 Jo1e/Kg-K and k = 2.4 Watt/m-K. The vale of permeability reported in the literatre differs greatly ranging from 10-14m2 given by McNabb [5] for the rock formation at Wairakei New Zealand to abot m 2 given by Soroos [9] for the island of Hawaii. The vale of is a strong fnction of temperatre varying from -8-

13 ! 0.54 xlo- 3 Newton-sec/m 2 at 288 K to x 10-:- 3 Newton-sec/m 2 at 573 K. f the vales of K= 10-12m2 and = 0.54 x 10-3 Newton-sec/m 2 are sed the bondary layer thickness at r=l km and the total heat transfer rate for selected A are tablated in Table 2. Since at a fixed location the prescribed wall temperatre decreases as A is increased (Fig. 8) conseqently both the bondary layer thickness and the total heat transfer rate decrease as A is increased (Table 2). f the vales of = 0.042xl0-3 Newton-sec/m 2 and K= 10-10m 2 are sed the bondary layer thickness wold be considerably thinner with an assciated increase in heat transfer rate.! W V. Conclding Remarks Althogh the foregoing analysis is discssed in terms of the convection of grondwater adjacent to horizontal impermeable srfaces the reslts are applicable to a wide range of geophysical and engineering problems whenever they can be idealized as a horizontal flat plate embedded in a satrated poros medim. The analysis is based on the bondary layer approximations which are valid for large Rayleigh nmbers. The simple algebraic expressions for total heat transfer rate and bondary layer thickness (or the hot water zone) are sefl Jor a qick estimate of geothermal resorces. Acknowledgment. This stdy is part of the Hawaii Geothermal Project fnded in part by the RANN program of the National Science Fondation of the nited States (Grant No. G-383l9) the Energy Research and Development Administration of the nited States (Grant No. E(04-3)-1093) and by the State and Conty of Hawaii. -9-

14 REFERENCES t' ' ' i' ' 1 j i' i f J 1. Stewartson K. lion the Free Convection from a Horizontal P1ate1 ZAMP v (1958). 2. Gill W.N. Zeh D.W. and Casal LD. Free Convection on a Horizonta1 P1ate1 ZAMP v (1965). 3. Cheng P. Yeng K.C. and La K.H. Nmerica1 Soltions for Steady Free Convection in sland Geothermal Reservoirs." To appear in Proceedings of 1975 nternational Seminaron Ftre Energy Prodction -- Heat and Mass Transfer Problems. 4. Wooding R.A..Convection in a Satrated Poros Medim at Large Rayleigh Nmber or Pec1et Nmber1 J. Fli Mechanics v (1963). 5. McNabb A. lion Convection in a Satrated Poro's Medim Proceedings of the 2nd Astralian Conference on Hydralics and Flid Mechanics C161-C171 (1965). 6. Yih C.S. Dynamics of Nonhomogeneos Flids Macmillan Pblishing Co. nc. New York (1965). 7. Cheng P. and Minkowycz W.J. Similarity Soltions for Free Convection Abot a Dike" Hawaii Geothermal Project Technical Report No. 10 October Gebhart B. Heat Transfer McGraw-Hill Book Company (1971). 9. Soroos R.L. Determination of Hydralic Condctivity of Some Oah Aqifers with Step-Draw Down Test Data1 M.S. Thesis Department of Geology and Geophysics niversity of Hawaii May i -10-

15 TABLE 1. Vales of -8 1 (0) nt and Tlm for selected vales o A i A -8 1 (0) nt n ".... m " t f i TABLE 2. Vales of 0T' om at r=lkm and Qfor selected vales of A A (ot)r=lkm (om)r=lkm Q m m 25.8 MW m m 24.2 MW m m' 22.9 MW m m 21.9 MW J J -11-

16 ! " i!!. '. LST OF FGRES 1. Coordinate System 2. Vales of 6 Verss n for Selected Vales of A 3. Dimensionless Velocity Distribtion Verss n for Selected Vales of A 4. Vales of -6 1 Verss n for Selected Vales of A 5. f Verss n for Selected Vales of A 6. Dimensionless Vertical Veiocity Verss n for Selected Vales of A 7. Streamlines Distribtion for a Heated mpermeable Srface Facing pward 8. Sketch of the Wall Temperatre Distribtion for Selected Vales of A! -12-

17 o Tw =Teo - Ar>" r : ".. - z z!.. '. '. o Fig. 1 Coordinate System -13-

18 !' L!. t : i.j A Fig. 2.Va1es of e Verss n for Selected Vales of A -14-

19 f 1 1.5r : j jj..j x J 0 L----1 L_ Fig. 3 '1J Dimensionless Velocity Distribtion Verss n for Selected Vales of A -15-

20 ..! i f i' - Q) r 1 LJ Fig. 4 Vales of -e l Verss n for Selected Vales of A -16-:-

21 \. 1.0' =:========:::=:J i J 1 i ij... t r i r ) i.j..! i A' !.l 0.2 L 1 j Oe-..L '- --.L- ---L. ----" o : Fig. 5 f Verss n for Selected Vales of -17-

22 ;! W LJ ! i W! ' Lj. f L.j 1... i! 1 ill i' 1 O----""" L----L L f' Fig. 6 Oimension1ss Vertical Velocity Verss n for Selected Va1es of 3 'TJ

23 .. J ; ; i 1.0 t-----r---r-----r--r-----_ X=1.5 :: o J:. R Fig. 7 Streamlines Distribtion for a Heated mpermeable Srface Facing pward l. -19-

24 1! j w J i j J " 1!' j. o r Fig. 8 Sketch of the Wall Temperatre Distribtion for Selected Vales of A 1.J -20-