The Molecular Diffusion of Heat and Mass from Two Spheres

Transcription

1 Iteratial Jural f Mder Studies i Mechaical Egieerig (IJMSME) Vlume 4, Issue 1, 018, PP 4-8 ISSN (Olie) DOI: The Mlecular Diffusi f Heat ad Mass frm Tw Spheres Abdullah Abbas Kedush Dept. f Nuclear Egieerig Techlgy, Augusta Techical Cllege, Augusta, Ga, USA *Crrespdig Authr: Abdullah Abbas Kedush, Dept. f Nuclear Egieerig Techlgy, Augusta Techical Cllege, Augusta, Ga, USA Abstract: A aalytical sluti t the heat cducti equati applied t tw spheres was btaied. Bispherical crdiates system were used. The derived sluti applies t bth fluid ad slid spheres. The sluti revealed a decrease i the heat cducti betwee the tw spheres whe they apprach e ather. The sluti cmpared well with the available theretical mdels Keywrds: Heat Cducti, Tw Spheres, Heat Trasfer, Temperature, Aalytical Sluti 1. INTRODUCTION Heat ad mass trasfer frm spheres immersed i fluids ccur i may egieerig idustries, techlgies ad scietific applicatis. Amg these are: dryig, adsrpti, extracti, fixed ad fluidized beds, clud physics, aersl physics, cmbusti f fuel drplets ad clig f spherical uraium fuel elemets i certai types f uclear reactrs. Althugh a idividual spherical particle r bubble is rare t be fud i practical systems, yet mst f the ifrmati f the scietific literature deals with studies f the idividual sphere r bubble. The heat Cducti frm a sigle sphere t the surrudig mass f fluid is give as fllws (Leal 199) ( Nu) s (1) The abve equati applies als t the case f mass trasfer simply by replacig Nu by the Sherwd umber Sh. Mrris ad Reed ( 1974) derived a sluti fr the cductive heat trasfer frm tw tuchig spheres. They fud that the Nusselt umber f a sphere was reduced by the presece f a tuchig sphere accrdig t the fllwig Nu l () The prblem f tw, three ad multi-sphere r multi-bubble system are w beig addressed by a umber f research wrkers e.g. Ruzicka ( 000) ad Ramachadra et al. ( 1989) wh slved the frced cvective heat trasfer t a liear array f three spheres by usig the fiite elemet methd. I geeral, a icreased rate f cvective heat r mass trasfer with sphere separati was btaied. The mlecular heat cducti ad mass diffusi f the tw spheres are gvered by the fllwig pheme: the diffusi f heat r mass frm e sphere mderates the gradiet f temperature ad ccetrati arud the ther sphere ad thus Nu r Sh are decreased with the separati distace betwee spheres. This pheme is ecutered i lw Reylds umber flw (e.g. Che ad Pfeiffer (1970), Amizadeh et al. (1974) ad Ramachadra ad Kleistreuer ( 1985). The fllwig cclusis were deduced frm the literature review: a. Acfusi exists as t the icrease r decrease f heat trasfer with sphere spacig. (see fr example Mulhlad et al. (1988). b. The majrity f the theretical research was cfied t the umerical sluti f the cservati equatis, except fr the wrk f Kedush (007). c. Tw-sphere iteractis were t treated experimetally, except fr the wrks f Wag ad Liu Iteratial Jural f Mder Studies i Mechaical Egieerig (IJMSME) Page 4

2 The Mlecular Diffusi f Heat ad Mass frm Tw Spheres (199) f the tw slid spheres, Kk (1989), ad Saada (005) f the tw bubbles. The ideal case f the preset paper serves the purpse f mdelig practical situatis f cductive flw f heat i a system f tw spheres. The tw-sphere aalysis may be csidered as the lwest rder effect f thermal iteracti f multi-spheres i a swarm. The preset presetati may be csidered as a cmplemetary t that f Kedush (007).. THEORETICAL ANALYSIS Assume tw spheres f equal radii a separated by a certai distace L as shw i Fig. 1. Yvavich (1978) develped a methd f dealig with heat cducti i cmplicated gemetries. The methd depeds chsig a rthgal curviliear crdiate system which is mst apprpriate fr the prblem at had, ad slvig Laplace s equati i that crdiate system. Accrdigly, the bispherical crdiate system was chse here fr the preset prblem. The heat cducti equati, i bispherical crdiates, is give as fllws (cshη csθ) 3 c siθ [siθ η ( 1 csh η csθ η ) + θ ( siθ csh η csθ θ )] = 0 (3) Fig1. The tw spheres i bispherical crdiates. It will be suppsed that heat is geerated withi the tw spheres at a rate such that their surface are maitaied at a uifrm temperature T a. The budary cditis, which specify the temperature ad heat flw f the system becme the fllwig T = T a at η = η (4) η θ = 0 at η = 0 (5) = 0 at θ = 0 π (6) Laplace s Eq. (3) is slved by separati f variables. A sluti f sufficiet geerality fr ur purpse is the fllwig T = (csh η csθ) 1/ [Be (+0.5)η + De (+0.5)η ]P (csθ) (7) where B ad D are arbitrary cstats ad p (cs ) is the assciated Legedre fucti f the first kid. Applyig the budary cditis makes the cmplete sluti as fllws Iteratial Jural f Mder Studies i Mechaical Egieerig (IJMSME) Page 5

3 The Mlecular Diffusi f Heat ad Mass frm Tw Spheres T = T a [(csh η csθ)] 0.5 e (+0.5)η csh(+0.5)η =0 csh(+0.5)η P (csθ) (8) Assume that the mass f the statiary fluid rud the tw spheres has a cstat thermal cductivity k. The heat flux frm the spheres surfaces S is give by the fllwig T (9) 1/ q kg ds S where g1 c / (csh cs ) ad c si ds ( g g ) d d 1/ 3 (csh 0 cs ) The metric cefficiets f the crdiates are the fllwig g g (11) 1 Ad c si g (csh cs ) 3 The fllwig dimesis are defied as fllws: a c / sih ad L c / tah. The temperature gradiet is btaied frm Eq. (8) ad substituted it Eq. (9) t get the fllwig equati ( 1) sih q 4 akt (13) sih( 1) a 0 Equatig the abve equati t q h(4 a ) T, gives the fllwig equati ( 1) sih Nu (14) sih( 1) 0 Or the fllwig Nu 0 ( 1) sih csh 1 sih( L a L a 1 1) csh Equati (15) reduces t the sigle sphere sluti (that is, Nu= f Eq.(1) up icreasig the value f ɳ r L. This icrease teds t make the tw spheres a ifiite distace apart. O the ther had, whe ɳ =0. (the case f almst tuchig spheres), we btai frm Eq. (15) the sluti f Mrris ad Reed [] (that is, Nu=l f Eq. ()). Dividig Eq. (15) by the sigle sphere, we get the fllwig equati s 0 Nu / ( Nu) ( 1) sih / sih( 1) (16) 3. DISCUSSION The abve aalyses f the tw spheres apply t bth fluid ad slid spheres. Figure shws that whe the ceters f the spheres are ly 4 radii apart the Nusselt umber is apprximately 1.6 ad whe the ceters f the spheres are 100 radii apart the Nusselt umber is apprximately 1% less tha the limitig value f. It ca be deduced frm Fig. that as the umber f spheres is icreased the Nusselt umber becmes prgressively smaller util a cditi is reached i which a particular sphere is surruded by multiple spheres f the same surface temperature. This meas that a pit will be reached where the Nusselt umber appraches zer. Figure shws a Iteratial Jural f Mder Studies i Mechaical Egieerig (IJMSME) Page 6 (10) (1) (15)

4 The Mlecular Diffusi f Heat ad Mass frm Tw Spheres cmparis betwee the preset sluti ad that f Mrris ad Reed ( 1974). Bth slutis agree the decrease f Nu umber with separati but the preset sluti has the fllwig merits (i) It is i a clsed frm. This wuld eable researchers t utilize the preset sluti tgether with the earlier sluti f the authr (Kedush (007)) t btai the cmplete sluti f heat trasfer t tw bubbles vertically abve e ather, as fllws L 1 ( 1) sih csh a 1 1/ 1/ A B 0 1 L (17) Nu H Pe G Pe sih( 1)csh a where 3 H [1 ( a / L) ] ad 6 a G 1 L Subscripts A ad B refers t tp ad bttm bubbles respectively. (ii) The preset results are clser t thse f Russell (1911) tha Mrris ad Reed ( 1974). Russell (1911) calculated electric charges tw spherical cdesers whse surfaces were at the same ptetial. His results were preseted i apprximate frmulas ad tables. Fig. Cmparis betwee the preset sluti (------) ad that f Mrris ad Reed ( 1974) ( ) 4. CONCLUSION The preset paper revealed ad prved aalytically the fllwig pheme: A decrease i the heat cducti ad mass diffusi betwee tw slid r fluid spheres ccurs whe they apprach e ather. This is ctrary t the case f heat cvecti Kedush (007). I geeral, the preset aalytical results cmpared well with the available theretical mdels. ACKNOWLEDGMENT The authr wishes t dedicate this wrk t the livig memry f the late Prfessr Hugh C. Simps f the Uiversity f Strathclyde (UK), frmer editr ad fuder f the Iteratial Jural f Multiphase Flw. The authr had ejyed his brilliat lectures ad had gd frtue f wrkig uder his supervisi. Iteratial Jural f Mder Studies i Mechaical Egieerig (IJMSME) Page 7

5 The Mlecular Diffusi f Heat ad Mass frm Tw Spheres NOMENCLATURE ɑ = radius f the sphere h = cvective heat trasfer cefficiet k = thermal cductivity f the fluid L = the distace betwee ceters f the tw spheres q = the heat flux the sphere surface T = temperature Ta = abslute temperature f the sphere surface REFERENCES Amizadeh, K., Al-Taha, T.R., Crish, A.R.h., Klasky, M.S. ad Pfeiffer, R., Mass trasprt arud tw spheres at lw Reylds Number, It. J. Heat Mass trasfer, vl. 17, pp , Che, W.C. ad Pfeiffer, R., 1970, Mass trasfer rates with first rder hmgeeus chemical reacti arud tw spheres, Chem. Eg. Prg. Symp. Ser., N.105, 66, pp , Kedush, A.A., Ehacemet f cvective heat ad mass trasfer frm tw bubbles at high Reylds umber, ASME J. Heat Trasf., vl.19, pp , 007. Kk, J.B.W., Dyamics f gas bubbles mvig thrugh liquid, Ph.D. Thesis, Uiversity f Twet, The Netherlads, L.G. Leal, Lamiar Flw ad Cvective Trasprt Prcesses, Butterwrth-Heiema: Bst 199. Mrris, F.A. Jr. ad Reed, L.D., Lw Kudse umber heat trasfer frm tw spheres i ctact, ASME J. Heat Trasfer, v l. 96, pp , Mulhlad. J.A., Srivastava, R.K. ad Wedt, J.O.L., Ifluece f drplet spacig drag cefficiet i evapratig, mdisperse streams, AIAA Jural, vl. 6, pp , Rama chadra, R. S. ad Kleistreuer, C., Mass trasfer with chemical reacti i tw iteractig drps, Chem. Eg. Cmmu., vl. 36, pp , Rama chadra, R.S. ad Kleistreuer, C.,Yag, T.-Y., 1989, Frced cvecti heat trasfer f iteractig spheres, Numerical Heat Trasfer, vl. 15, pp , Russel, A., The electric stress at which iisati begis i air, Prc. Phys. Sc., vl. 3, pp , Ruzicka, M.C., O bubbles risig i lie, It. J. Multiphase Flw, vl.6, pp , 000. Saada, T., Iteracti ad calescece f bubbles i quiescet liquid, Ph.D. Thesis, Kyushu Uiversity, Japa, 005. Wag, B.-X. ad Liu, T., Research hydrdyamics ad heat trasfer fr fluid flw arud heatig spheres i tadem, It. J. Heat Mass Trasfer, vl. 35, pp ,199. Yvavich, M. M., Advaced Heat Cducti, Hemisphere: Washigt, Citati: Abdullah Abbas Kedush et.al. (018) The Mlecular Diffusi f Heat ad Mass frm Tw Spheres, Iteratial Jural f Mder Studies i Mechaical Egieerig, 4(1), pp.4-8. DOI: rg/ / Cpyright: 018 Abdullah Abbas Kedush, This is a pe-access article distributed uder the terms f the Creative Cmms Attributi Licese, which permits urestricted use, distributi, ad reprducti i ay medium, prvided the rigial authr ad surce are credited. Iteratial Jural f Mder Studies i Mechaical Egieerig (IJMSME) Page 8