Unsteady flow in a porous medium between parallel plates in the presence of uniform suction and injection with heat transfer

Transcription

1 Waer- Hydroogy Unseady fow in a porous medium beween parae paes in he presence of uniform sucion and injecion wih hea ransfer Absrac H. A. Aia, W. A. E-Meged, W. Abbas, M. A. M. Abdeen 3, * Received: January, Revised: June 3, Acceped: January 4 he unseady fow in porous medium of a viscous incompressibe fuid bounded by wo parae porous paes is sudied wih hea ransfer. A uniform and consan pressure gradien is appied in he axia direcion whereas a uniform sucion and injecion are appied in he direcion norma o he paes. he wo paes are ep a consan and differen emperaures and he viscous dissipaion is no ignored in he energy equaion. he effec of he porosiy of he medium and he uniform sucion and injecion veociy on boh he veociy and emperaure disribuions are invesigaed. Keywords: Unseady fow, Viscous incompressibe fuid, Hea ransfer, Porous medium, Numerica souion.. Inroducion he fow of a viscous eecricay conducing fuid beween wo parae paes has imporan appicaions as in magneohydrodynamic (MHD) power generaors, MHD pumps, acceeraors, aerodynamics heaing, eecrosaic precipiaion, poymer echnoog peroeum indusr purificaion of moen meas from non-meaic incusions and fuid dropes-sprays []. he fow beween parae paes of a Newonian fuid wih hea ransfer has been examined by many researchers in he hydrodynamic case considering consan physica properies [-6]. he exension of he probem o he MHD case has araced he aenion of many auhors [7-]. In his paper, he ransien fow wih hea ransfer hrough a porous medium of an incompressibe viscous fuid beween wo infinie horizona porous paes is invesigaed. A consan pressure gradien is appied in he axia direcion and a uniform sucion and injecion is imposed in he direcion norma o he paes. he fow hrough a porous medium deas wih he anaysis in which he differenia equaion governing he fuid moion is based on he Darcy s aw which accouns for he drag exered by he porous medium [3-5]. * orresponding auhor: mosafa_a_m_abdeen@homai.com Deparmen of Engineering Mahemaics and Physics, Facuy of Engineering, Fayoum Universi E-Fayoum-6345, Egyp Basic and Appied Science Deparmen, oege of Engineering and echnoog Arab Academy for Science, echnoog and Mariime ranspor, airo, Egyp 3 Deparmen of Engineering Mahemaics and Physics, Facuy of Engineering, airo Universi Giza, Egyp he wo paes are mainained a wo differen bu consan emperaures. his configuraion is a good approximaion of some pracica siuaions such as hea exchangers, fow meers, and pipes ha connec sysem componens. he cooing of hese devices can be achieved by uiizing a porous surface hrough which a cooan, eiher a iquid or gas, is forced. herefore, he resus obained here are imporan for he design of he wa and he cooing arrangemens of hese devices.he inear paria differenia equaions of moion are soved anayicay using he mehod of Lapace ransform o obain he veociy disribuion as a funcion of space and ime. he inhomogeneous energy equaion is soved numericay considering he viscous dissipaion using he mehod of finie differences and by appying he ran-nicoson impici mehod. he effec of he porosiy of he medium and he sucion and injecion veociy on boh he veociy and emperaure disribuions are repored. he ransien souions proved ha he seady sae souion is approached as he asympoic deveopmen of a ime-dependen process and presened some ineresing resus.. Descripion of he Probem he wo parae horizona paes are ocaed a he y=±h panes and exend from x=- o and z=- o as shown in Fig.. he ower and upper paes are ep a he wo consan emperaures and, respecive where >. he fuid fows beween he wo paes in a porous medium where he Darcy's mode is assumed [3-5]. he moion is driven by a consan pressure gradien dp/dx in he x-direcion, and a uniform sucion from above and injecion from beow which are appied a = wih veociy v o.due o he infinie Inernaiona Journa of ivi Engineering, Vo., No. 3, ransacion A: ivi Engineering, Sepember 4

2 dimensions in he x and z-direcions a quaniies apar from he pressure gradien dp/dx which is assumed consan, are independen of he x and z-coordinaes. he veociy vecor of he fuid is given as v( ) u( ) i v j o wih he iniia and boundary condiions u= a, and u= a y=±h for >. he emperaure () a any poin in he fuid saisfies boh he iniia and boundary condiions = a, = a y=+h, and = a y=-h for >. he fuid fow is governed by he momenum equaion [6] u u dp u vo u y dx y K Fig. he geomery of he probem () NuU = ( ˆ / ŷ ) ŷ = is he Nusse number a he upper pae, Equaions ()-() are wrien as (he "has" wi be dropped for convenience) u u dp u S Mu, y dx y u S Ec( y Pr y y (3) (4) where and are, respecive he densiy and he coefficien of viscosiy and K is he Darcy permeabiiy [3-5]. o find he emperaure disribuion inside he fuid we use he energy equaion [7] u c cvo ( () y y y where c and are, respecive he specific hea capaciy and he herma conduciviy of he fuid. he second erm on he righ side represens he viscous dissipaion. Inroducing he foowing non-dimensiona quaniies x y z hu ˆ Ph ˆ ˆ ˆ ˆ ˆ x, y, z, u, P,, h h h h S voh /, is he sucion parameer, Pr c / is he Prand number, / Ec ch ( ) is he Ecer number, M h / K is he porosiy parameer, NuL = ( ˆ / ŷ ) ŷ =- is he Nusse number a he ower pae, he iniia and boundary condiions for he veociy become u,, u, y, (5) and he iniia and boundary condiions for he emperaure are given by :, :, y,, y. (6) 3. Anayica Souion of he Equaions of Moion Equaion (3) is a inear inhomogeneous paria differenia equaion which is soved anayicay using he Lapace ransform (L) mehod, under he iniia and boundary condiions given by Eq. (5) o give he veociy fied as funcions of space and ime. aing he L of Eq. (3) yieds d U ( du ( S K( U ( dy dy s where U(=L(u()), is he consan vaue of dp/dx and K( M s. he souion of Eq. (7) wih y as (7) 78 H. A. Aia, W. A. E-Meged, W. Abbas, M. A. M. Abdeen

3 an independen variabe is given as U ( Ks sinh( S / )sinh( y) cosh( S / )cosh( y) exp( Sy / ) sinh( ) cosh( ) where S / 4 K. Using he compex inversion formua and he residue heorem [6], he inverse ransform of U( is deermined as u( ) M sinh( S / )sinh( y) cosh( S / )cosh( y) exp( Sy / ) sinh( ) cosh( ) n exp( Dn ) ( ) [( n ) sinh( S / )sin( ( n ) y) ( Dn M ) Dn n exp( Sy / ) exp( Dn) ( n.5) cosh( S / ) cosh( ( n.5) y)] (8) ( Dn M ) Dn where, Dn [ ( n ) S / 4 M ], Dn [ ( n.5) S / 4 M ], S / 4 M. Equaion (8) shows ha u is direcy proporiona o he pressure gradien, ha is u/ is independen of. he expression for he veociy u is o be evauaed for various vaues of he parameers M and S. 4. Numerica Souion of he Energy Equaion he anayica souion of Eq. (8) deermines he veociy fied for various vaues of he parameers M and S. he vaues of he veoci when subsiued in he righside of he inhomogeneous energy equaion (4), mae i oo difficu o sove anayicay. herefore, he energy equaion is o be soved numericay using he ran- Nicoson impici mehod [8] wih he iniia and boundary condiions given by Eq. (6). he finie difference equaions are wrien a he mid-poin of he compuaiona ce and he differen erms are repaced by heir secondorder cenra difference approximaions in he y-direcion. he diffusion erm is repaced by he average of he cenra differences a wo successive ime eves. he viscous dissipaion erm is evauaed using he veociy componens and heir derivaives in he y-direcion which are obained from he exac souion. We inroduce he variabes v u / y and H / y o reduce Eq. (4) from he second order o he firs order. he compuaiona domain is divided ino meshes each of dimension and y in ime and space, respecivey. he finie difference represenaion for he energy equaion is given by Fina he boc ri-diagona sysem is soved using homas' agorihm [8]. Equaion (9) is rewrien in he foowing forms b b b3h b4h b5 () Where b m (where m =,,,5) are he coefficiens of he difference equaions () ha is corresponding o Eq. (9), and are counries equaions o (i, j+) and (i+, j+), respecivey. he posiion y=- is designaed by =. We wrie he generaized homas-agorihm as in he foowing seps [8]. he unnowns are wrien as H ˆ () H H H Hˆ () where, he variabes, ˆ, H and Ĥ are represening homas' coefficiens. he equaions reaing he emperaure o is derivaives H are given by y H H (3) Subsiuing from Eqs. () and () ino (3) and afer manipuaions we ge, (( y / ) H y / ) H ( y / ) Hˆ ˆ (4) From which we can ge ( y / ) H y /, ˆ ( y / ) Hˆ ˆ (5) Subsiuing from Eqs. (3) and (5) ino Eq. () we can obain he wo coefficiens H and Ĥ. Unie he veociy u, he emperaure disribuion depends on. A cacuaions are carried ou for Pr= and Ec=.. 5. Resus and Discussion Fig. shows he ime progression of he veociy and emperaure profies up i he seady sae and for M= and S=. I is cear from Fig. a ha he veociy chars are asymmeric abou he y= pane because of he sucion. he veociy componen u reaches he seady sae faser han which is expeced as u acs as he source of emperaure. i, j i, j i, j i, j Hi, j Hi, j Hi, j Hi, j ( ) S( ) 4 ( Hi, j Hi, j ) ( Hi, j Hi, j ) vi, j vi, j vi, j vi, j Ec ( Pr y 4 (9) Inernaiona Journa of ivi Engineering, Vo., No. 3, ransacion A: ivi Engineering, Sepember 4 79

4 u y.5 =.5 = = y Fig. ime deveopmen of he profie of: u; and (M= and S=) Fig. 3 indicaes he effec of he porosiy parameer M on he ime progression of he veociy u and emperaure a he cenre of he channe and for S=. I is cear from Fig. 3a ha increasing he parameer M decreases u and is seady sae ime as a resu of increasing he resisive porosiy force on u. Fig. 3b shows ha increasing M decreases and is seady sae ime as increasing M decreases u which, in urn, decreases he viscous dissipaion which decreases. u 3 =.5 = = 3 4 M= M= M= Fig. 4 indicaes he effec of he sucion parameer on he ime progression of he veociy u and emperaure a he cenre of he channe for M=. In Fig. 4a, i is observed ha increasing he sucion decreases he veociy u a he cener and is seady sae ime due o he convecion of fuid from regions in he ower haf o he cener, which has higher fuid speed. In Fig. 4b, he emperaure a he cener is infuenced more by he convecion erm, which pushes he fuid from he cod ower haf owards he cenre. u S= S= S= S= S= S= Fig. 4 Effec of S on he ime variaion of: u a y=; a y=. (M=) abe presens he variaion of he seady sae Nusse number a he upper pae and he ower pae for various vaues of and for Pr, Ec.. I is cear ha increasing he magniude of he pressure gradien decreases whie increases. abe presens he variaion of he seady sae Nusse number a he upper pae and he ower pae for various vaues of Pr and for 5,Ec.. I is shown ha increasing Pr decreases and increases he magniude of. abe 3 presens he variaion of he seady sae Nusse number a he upper pae Nu U and he ower pae for various vaues of Ec and for 5,Pr. I is shown ha increasing Ec increases whie decreases. M= M= M=3 Fig. 3 Effec of M on he ime variaion of: u a y=; a y=. (S=) 8 H. A. Aia, W. A. E-Meged, W. Abbas, M. A. M. Abdeen

5 abe Variaion of he seady sae Nusse number a he upper pae Nu U and he ower pae Nu L for various vaues of (, abe Variaion of he seady sae Nusse number a he upper pae Nu U and he ower pae Nu L for various vaues of Pr ( 5, abe 3 Variaion of he seady sae Nusse number a he upper pae Nu U and he ower pae Nu L for various vaues of 5, oncusions he unseady fow hrough a porous medium beween parae paes of a viscous incompressibe fuid has been sudied in he presence of uniform sucion and injecion. he effec of he porosiy and he sucion and injecion veociy on he veociy and emperaure disribuions is invesigaed. I is found ha boh he porosiy and sucion or injecion veociy has a mared effec on decreasing boh he veociy and emperaure disribuions. References [] Harmann J, Lazarus F. Experimena invesigaions on he fow of mercury in a homogeneous magneic fied, K. Dan. Videns. Ses. Ma. Fys. Medd, 937, No. 7, Vo. 5, pp. 45. [] Orhan A, Mee A. Laminar forced convecion wih viscous dissipaion in a couee poiseuie fow beween parae paes, Appied Energ 6, Vo. 83, pp [3] udor B, Ioana B. An unified numerica approach of seady convecion beween wo parae paes origina, Appied Mahemaics and ompuaion, 9, Vo. 5, pp [4] Apher RA. Hea ransfer in magneohydrodynamic fow beween parae paes, Inernaiona Journa of Hea and Mass ransfer, 96, Vo. 3, pp [5] Nigam SD, Singh SN. Hea ransfer by aminar fow beween parae paes under he acion of ransverse magneic fied, he Quarery Journa of Mechanics and Appied Mahemaics, 96, Vo. 3, pp [6] Aia HA. Ha effec on ouee fow wih hea ransfer of a dusy conducing fuid in he presence of uniform sucion and injecion, African Journa of Mahemaica Physics, 5, No., Vo., pp [7] Aia HA. Effec of porosiy on unseady ouee fow wih hea ransfer in he presence of uniform sucion and injecion, Kragujevac Journa of Science, 9, Vo. 3, pp. -6. [8] Eri S, Vajraveu K, Rober A, Van G, Pop I. Anayica souion for he unseady MHD fow of a viscous fuid beween moving parae paes, ommunicaions in Noninear Science and Numerica Simuaion,, Vo. 6, pp [9] Aia HA, Abdeen MAM. Unseady MHD fow and hea ransfer beween parae porous paes wih exponenia decaying pressure gradien, Kragujevac Journa of Science,, Vo. 34, pp. 5-. [] Joseph DD, Nied DA, Papanicoaou G. Noninear equaion governing fow in a sauraed porous media, Waer Resources Research, 98, No. 4, Vo. 8, pp [] Kesavaiah D.h, e a. Effecs of Radiaion and free convecion currens on unseady couee fow beween wo verica parae paes wih consan hea fux and hea source hrough porous medium, Inernaiona Journa of Engineering Research, 3, No., Vo., pp 3-8. [] Das SS. Effec of consan sucion and injecion on MHD hree dimensiona couee fow and hea ransfer hrough a porous medium, Journa of Nava Archiecure and Marine Engineering, 9, No., Vo. 6. [3] Narahari M. Effecs of herma radiaion and free convecion currens on he unseady couee fow beween wo verica parae paes wih consan hea fux a one boundar Wseas ransacions on Hea and Mass ransfer,, No., Vo. 5. [4] Ingham DB, Pop I. ranspor phenomena in porous media, Pergamon, Oxford,. [5] Khaed ARA, Vafai K. he roe of porous media in modeing fow and hea ransfer in bioogica issues, Inernaiona Journa of Hea and Mass ransfer, 3, Vo. 46, pp [6] Schiching H. Boundary Layer heor McGraw-Hi, New Yor, 986. [7] Kaac S, e a. Handboo of Singe-Phase onvecive Hea ransfer, John Wie New Yor, 987. [8] Miche AR, Griffihs DF. he Finie Difference Mehod in Paria Differenia Equaions, John Wie New Yor, 98. Inernaiona Journa of ivi Engineering, Vo., No. 3, ransacion A: ivi Engineering, Sepember 4 8