Finite Element Study of Soret and Radiation Effects on Mass Transfer Flow through a Highly Porous Medium with Heat Generation and Chemical Reaction

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1 Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume 1, Number 1 (17), pp Research Inda Publcatons Fnte Element Stud of Soret and Radaton Effects on Mass Transfer Flow through a Hghl Porous Medum wth Heat Generaton and Chemcal Reacton B. Shanar Goud and M.N. Raa Shear Department of Mathematcs, JNTUH College of Engneerng Kuatpall, Hderabad- 585, TS, Inda. Department of Mathematcs, JNTUH College of Engneerng Nachupall, Karmnagar -5551, TS, Inda. Abstract The problem of Soret and Radaton effects on mass transfer flow through a hghl porous medum wth heat generaton and chemcal reacton has been analzed numercall. Exact solutons of the governng equatons are solved b Galern fnte element technque dependng on the phscal parameters ncludng the Prandtl number (Pr), Thermal Grasof number (Gr), mass Grashof number (Gc), the Schmdt number (Sc), the Soret number (So), chemcal reacton parameter (Kr) and radaton parameter (R), The effects of phscal parameters are dscussed wth the help graphs. Kewords: Soret effects, Radaton effect, Chemcal reacton, FEM, Heat Generaton, MHD. INTRODUCTION Free convecton flow s an mportant factor n several practcal applcatons that nclude coolng of electronc components, n desgns related to thermal nsulaton, materal processng, and geothermal sstems etc. Magnetohdrodnamcs has attracted attenton of a large number of scholars due to ts varous applcatons. N.G Kafoussas and E.W Wllams [1] nvestgated thermal-dffuson and dffuson -

2 5 B. Shanar Goud and M.N. Raa Shear thermo effects on mxed free forced convectve and mass transfer boundar laer flow wth temperature dependent vscost. Ahmed M. Salem and Mohamed Abd El-Azz [] have studed the effect of Hall currents and chemcal reacton on hdromagnetc flow of a stretchng vertcal surface wth nternal heat generaton/absorpton. Magd A.Ezzat et.al [3] have studed free convecton effects on a vscoelastc boundar laer wth one relaxaton tme through a porous medum. Mohamed AbdEl-Azzz [] has studed Thermo - dffuson and dffuson effects on combned heat and mass transfer b hdromagentc three- dmensonal free convecton over a permeable stretchng surface wth radaton. A. Rapt et.al [5] studed effect of thermal radaton on MHD flow. J.H Mern and I.Pop [6] nvestgated the forced convecton flow of a unform stream over a flat surface wth a convectve surface boundar condton. Orhan Adn and Ahmet aa [7] observed mxed convecton of a vscous dsspatng flud about a vertcal plate. Fnte element stud of radatve free convecton flow over a lnearl movng permeable vertcal surface n the presence of magnetc feld was studed b S.Rawat and S. Kapoor [8]. M.S Alam et.al [9] carred out a research on Dufour and Soret effects on stead free convecton and mass transfer flow past a sem nfnte vertcal plate n a porous medum. S. Shutee [1] presented thermal radaton and Buoanc effects on heat and mass transfer over a sem nfnte stretchng surface wth sucton and blowng. K Varavelu et.al [11] reported unstead convectve boundar laer flow of a vscous flud at a vertcal surface wth varable flud propertes. M.Turmazogulu and I.Pop [1] presented the Soret and heat source effects on the unstead radatve MHD free convecton flow from an mpulsvel started nfnte vertcal plate. P.A Lashm Naraana and P.Sbanda [13] consdered the nfluence of the Soret effect and double dsperson on MHD mxed convecton along a vertcal plate n non Darc porous medum. Effects of chemcal reacton and radaton on MHD free convecton flow of Kuvshnsh flud through a vertcal porous plate wth heat source have been studed b P.Mohan Krshna et.al [1]. G Palan et.al [15] have analzed the effect of vscous dsspaton on an MHD free convectve flow past a sem nfnte vertcal cone wth a varable surface heat flux. G.Seth et. al [16] presented the effects of hall current, radaton and rotaton on natural convecton heat and mass transfer flow past a movng vertcal plate. MHD flow and heat transfer of a vscous flud over a radall stretchng power - law sheet wth sucton/ necton n a porous medum has been studed b M.Khan et.al [17]. S.Mohammed Ibrahm and K.Suneetha [18] presented heat source and chemcal effects on MHD convecton flow embedded n porous medum wth Soret, vscous and Joule s dsspaton. A.G Va Kumar and S. Va Kumar Varma [19] studed thermal radaton and mass transfer effects on MHD flow past an mpulsvel started exponentall accelerated vertcal plate wth varable temperature and mass dffuson. G.S. Seth et.al [] nvestgated MHD natural convecton flow wth radatve heat

3 Fnte Element Stud of Soret and Radaton Effects on Mass Transfer Flow. 55 transfer past an mpulsvel movng vertcal plate wth ramped temperature n the presence of hall current and thermal dffuson. In ths paper the unstead MHD free convecton and mass transfer flow past a vertcal porous plate has been nvestgated analtcall b usng Galern fnte element technque. The effects of the flow parameters on the temperature, concentraton and veloct have been studed graphcall. MATHEMATICAL ANALYSIS A two dmensonal flow of an ncompressble and electrcall tang vscous flud along an nfnte vertcal plate that s embedded n a porous medum n the presence of thermal radaton, heat generaton, and chemcal reacton s consdered. It s assumed that there s radaton onl n flud. The x - axs s taen along the nfnte plate and - axs perpendcular to t and all the varables are functons of and t. Under these condtons and assumng varaton of denst n the bod force term (Boussnesq s approxmaton), the problem can be governed b the followng set of equatons: Equaton of contnut: v --- (1) Momentum equaton: u u 1 p v v t x K v v g ( T T ) g ( C C ) u --- () Energ equaton: Dffuson equaton: T T T q Q v u t c c c c r v ( T T ) p p p p C C C D T t T m T v D K ( ) r C C m Where the Rosseland approxmaton s used, whch leads to --- (3) --- () q r 3K s e T --- (5)

4 56 B. Shanar Goud and M.N. Raa Shear Wth the approprate ntal and boundar condtons are gven b n t n t p w w w w u U, T T ( T T ) e, C C ( C C ) e at u U, T T, C C at --- (6) Assumng that the temperature dfference wthn the flow s such that expanded n Talor s seres and expandng T about T ma be T, the free stream temperature and neglectng hgher order terms we get 3 T T T T --- (7) From the contnut equaton (1), t s clear that the sucton veloct normal to the plate s ether a constant or a functon of tme. Hence the sucton veloct normal to the plate s taen as v v --- (8) where v s scale of sucton veloct whch s a nonzero postve constant. The negatve sgn ndcates that sucton s towards the plate. Outsde the boundar laer, () gves 1 p v U x K Introducng the followng non dmensonal quanttes, --- (9) U u U t C u p t T T C C,,,,,,, p U U Tw T Cw C g T T g C C vq K v Gr, Gc, Sc, Q, n, K, U v U v C v v v w w v vn D p Kv K v Cp Kr, R,Pr, S v r e 3 st D K T T m t w v, Ec m w P w T v C C c T T --- (1) In the vew of above equaton, the basc flow feld equatons can be expressed n the followng form: t K u 1 u u Gr GcC 1 u --- (11) u d Q Ec t C C 1 C KrC So t Sc --- (1) --- (13) Where d 3R 3R Pr and Gr, Gc, Pr, Sc, Kr, R, Q,and K are the thermal Grashof number, Solutal Grashof number, Prandtl number, Schmdt number, chemcal reacton

5 Fnte Element Stud of Soret and Radaton Effects on Mass Transfer Flow. 57 parameter,radaton parameter, heat generaton parameter, and permeablt of the porous medum respectvel. nt nt u U, 1 e, C 1 e at p u 1,, C as --- (1) SOLUTION OF THE PROBLEM B applng the Galern element method wth Cran Nolson dscretzaton, tang h.1,.1 and r for equaton (11) over the two noded lnear h element () e, ( ) s --- (15) ( e ) ( e ) ( e ) ( e )T u u u ( e ) Nu R 1d t 1 Here R1 Gr GcC N,N K Integratng the frst term n equaton (15), b parts, one obtans ( e ) ( e )T ( e ) ( e ) ( e )T u u ( e )T u ( e ) ( e )T d d N u d R1 d t --- (16) Snce the dervatve u s not specfed at ether ends of the element () e, ( ) we neglectng the frst term n equaton (16) to obtan ( e )T ( e ) ( e ) u ( e )T u ( e ) ( e )T d d N u d R1 d t --- (17) Fnte element model ma be obtaned from equaton (17) b substtutng fnte element approxmaton over the two noded lnear elements () e, ( ) of the form: ( e ) ( e ) ( e ) u N ( e ) ( e ) Here, u u T --- (18) Whereu, u are the veloct components at ) and, ( th and are the bass functons defned as follows. th nodes of the tpcal element (e)

6 58 B. Shanar Goud and M.N. Raa Shear obtaned:,. Substtutng equaton (18) nto (17), the followng s ( e) ( e) ( e) u l u 1 u u Nl Rl 1 1 ( e) l 1 1 u u u u Where denote the dfferentaton wth respect to tme, ( e ) l s the length of the element. Assemblng the element equatons b nter-element connectvt for two consecutve elements 1 and 1.we get u1 1 1 u1 1 1 u1 1 1 u N R1 1 1 u ( e) u e u u l l u u u 1 1 u (19) On equatng row correspondng to the node to zero, the followng dfference schemes wth l ( e) h s obtaned: N u u u u u u1 u u1 u u u R h h () Applng the trapezodal rule and from the equaton (18), followng sstem of equatons n Cran Ncholson method are obtaned: r N u 6r 3rh N u 1R r rh N u r N u r rh N u r rh N u Au A u A u A u A u A u R --- (1) n 1 n 1 n 1 n n n Where A 6r 3 rh N, A 81r N, A 6r 3 rh N, A 6r 3rh N 1 3 A 8 1r N, A 6r 3 rh N, R 1R 1 (( Gr) ( Gm) C ) Smlarl applng the Galern fnte element method for equaton (1) (13) the followng equatons are obtaned: B B B B B B R --- () C C C C C C C C C C C C --- (3)

7 Fnte Element Stud of Soret and Radaton Effects on Mass Transfer Flow. 59 Where B1 6rd 3 rh Q, B 8 1rd 1Q, B3 6rd 3 rh Q, B 6rd 3 rh Q, B5 8 1rd 1Q, B6 6rd 3 rh Q, R 1ScS C Sc 6r 3 rh. Sc ScKr, C 8Sc 1r ScKr, C Sc 6r 3 rh. Sc ScKr, 1 3 u C Sc 6r 3 rhsc ScKr, C5 8Sc 1r ScKr, C6 Sc 6r 3 rh. Sc ScKr, R 1Ec Here r and hare, mesh szes along -drecton and t drecton respectvel. h Index, refers to the space and tme. In the equatons (1), () and (3), tang 1(1) n usng ntal and boundar condtons (1), the followng sstem of equatons are obtaned: A X B, 1(1)3 --- () Where A s are matrces of order n and X, B s column matrces havng n - components. The solutons of above sstem of equatons are obtaned b Thomas algorthm for veloct, temperature, concentraton. For varous parameters the results are computed and presented graphcall. The sn frcton, Nusselt number and Sherwood number are mportant phscal parameters for ths tpe of boundar later flow. Wth nown values of veloct, temperature and concentraton felds, the Sn-frcton at the plate s gven b non-dmensonal form The rate of heat transfer coeffcent can be obtaned n terms the Nusselt number n non-dmensonal, gven b N u T The rate of mass transfer coeffcent can be obtaned terms of the Sherwood number n non-dmensonal form, gven b S b. C. u. RESULTS AND DISCUSSION We have analzed the effects of the varous parameters such as Prandtl number (Pr), thermal Grashof number(gr), mass Grashof number(gc),schmdt number(sc), radaton parameter (R), permeablt of the porous medum (K), chemcal reacton

8 6 B. Shanar Goud and M.N. Raa Shear parameter (Kr), heat generaton parameter (Q), Ecert number (Ec) and are presented graphcall. The nfluence of the mass Grashof number on the veloct s presented n fgure 1. It s observed that the veloct ncreases as the mass Grashof number ncreases. The effect of thermal Grashof number on the veloct profles s shown n fgure. As the value of Gr ncreases, the veloct ncreases. Fgure 3 dsplas the effect the permeablt of the porous medum on veloct profles. It s observed that the permeablt of the porous medum s ncreases the veloct ncreases. Fgures (a) and (b) show the nfluence of the radaton parameter on the veloct and temperature profles. It s observed that the veloct and temperature decrease wth ncreasng radaton parameter. The effects of the Prandtl number on veloct and temperature profles are presented n fgures 5(a) and 5(b). The numercal results show that the effect of ncreasng value of Prandtl number results n decreasng veloct and temperature. Fgures 6(a) and 6(b) show the effects of Schmdt number on veloct and concentraton profles respectvel. From these fgures t s observed that an ncrease n Prandtl number decreases both veloct and concentraton profles. Fgures 7(a) and 7(b) llustrate the veloct and concentraton profles for dfferent values of the chemcal reacton parameter; t s observed that an ncrease n the Kr values results n ncreasng veloct and decreasng n concentraton. The effect of the Soret number on the veloct and concentraton profles s depcted n fgures 8(a) and 8(b). It s observed that veloct and concentraton ncreases wth ncrease n Soret number. The effect of the heat generaton parameter Q on veloct and temperature are shown n fgures 9(a) and 9(b). It s notced that an ncrease n the heat generaton parameter Q results n an ncrease n the veloct and temperature. CONCLUSION In ths artcle a mathematcal pattern has been presented for the Soret and radaton effects on mass transfer flow through a hghl porous medum wth heat generaton and chemcal reacton. The non-dmensonal governng equatons are solved b Galern fnte element method. The conclusons of the model are as follows: The veloct ncreases wth an ncrease n thermal Grashof number (Gr), Solutal Grashof number (Gc), chemcal reacton parameter (Kr). The veloct and temperature decreases wth an ncrease n Prandtl number (Pr), radaton parameter (R), The veloct and concentraton decreases wth an ncrease n Schmdt number (Sc).

9 Fnte Element Stud of Soret and Radaton Effects on Mass Transfer Flow. 61 The veloct and temperature ncreases wth an ncrease n heat generaton parameter. An ncrease n the Soret number (So) extends to an ncrease n veloct and concentraton. An ncrease n the chemcal reacton parameter (Kr) nduces to decrease n the veloct.

10 6 B. Shanar Goud and M.N. Raa Shear

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