Effect of Melting and Thermo-Diffusion on Natural Convection Heat Mass Transfer in a Non-Newtonian Fluid Saturated Non-Darcy Porous Medium

Transcription

1 The Ope Trasport Pheomea Joural, 29,, Ope Access Effect of Meltig ad Thermo-Diffusio o Natural Covectio Heat Mass Trasfer i a No-Newtoia Fluid Saturated No-Darcy Porous Medium R.R. Kairi ad P.V.S.N. Murthy * Departmet of Mathematics, Idia Istitute of Techology Kharagpur-72 32, Idia Abstract: This paper ivestigates the ifluece of meltig ad thermo-diffusio effect o atural covectio heat ad mass trasfer from vertical flat plate i a o-newtoia fluid saturated o-darcy porous medium. The wall ad the ambiet medium are maitaied at costat but differet levels of temperature ad cocetratio such that the heat ad mass trasfer occurs from the wall to the medium. The Ostwald-de Waele power law model is used to characterize the o-newtoia fluid behavior. A similarity solutio for the trasformed goverig equatios is obtaied, computatio is carried out for various values of the o-dimesioal physical parameters. The variatio i velocity, temperature, cocetratio, heat ad mass trasfer coefficiets with the power law idex, iertia parameter, meltig parameter, thermodiffusio parameter, buoyacy ratio ad wis umber is discussed for a wide rage of values of these parameters. Keywords: Natural covectio, thermo-diffusio, meltig, o-newtoia fluids, o-darcy porous medium. INTRODUCTION The study of covective heat ad mass trasfer accompaied by meltig effect i porous media has received a much attetio i the recet years because of its importat applicatios i castig, weldig ad magma solidificatio, permafrost meltig ad thawig of froze groud etc. Epstei ad Chao [] studied the meltig heat trasfer from a flat plate i a steady lamiar case, while Kazmierczak et al. [2, 3] aalyzed meltig from a vertical flat plate embedded i a porous medium i both free ad forced covectio processes. The heat trasfer at the meltig surface i the lamiar boudary layer by usig Karma-Pohlhause method is discussed by Pozvokov et al. [4]. Bakier [5] studied the meltig effect o mixed covectio from a vertical plate of arbitrary wall temperature both i aidig ad opposig flows i a fluid saturated porous medium while Gorla et al. [6] cosidered similar study with uiform wall temperature coditios. Tashtoush [7] studied the magetic ad buoyacy effects to ivestigate the flow, temperature profiles ad heat trasfer characteristics for meltig effect associated with uiform wall temperature based o o- Darcy flow model. The meltig pheomea o usteady ad steady mixed covectio heat trasfer from a vertical plate i a liquid saturated porous medium with aidig ad opposig exteral flows has bee studied by Cheg ad Li [8, 9]. Later they exteded their work to the usteady mass trasfer [] by cosiderig double diffusio processes. No-Newtoia power law fluids are so widespread i idustrial processes ad i the eviromet that it would be o exaggeratio to affirm that Newtoia shear flows are the exceptio rather tha the rule. Poulikakos ad Spatz [] aalyzed the meltig pheomea o free covectio from a vertical frot i a o-newtoia fluid saturated porous *Address correspodece to this author at the Departmet of Mathematics, Idia Istitute of Techology Kharagpur-72 32, Idia; pvsm@maths.iitkgp.eret.i matrix. Natural covectio of a o-newtoia fluid about a vertical wall ad that aroud horizotal cylider ad sphere i a porous medium was preseted by Che ad Che [2] ad Che ad Che [3], respectively. Nakayama ad Koyama [4] aalyzed the more geeral case of free covectio over a o-isothermal body of arbitrary shape embedded i a porous medium. Rastogi ad Poulikakos [5] examied the problem of double diffusive covectio from a vertical plate i a porous medium saturated with a o- Newtoia power law fluid. Sheoy [6] preseted may iterestig applicatios of o-newtoia power law fluids with yield stress o covective heat trasport i fluid saturated porous media cosiderig geothermal ad oil reservoir egieerig applicatios. A uified similarity trasformatio for Darcy ad o-darcy forced, free ad mixed covectio heat trasfer i o-newtoia ielastic fluid-saturated porous media is studied by Nakayama ad Sheoy [7]. Later, Sheoy [8] aalyzed o-darcy atural, forced ad mixed covectio heat trasfer i o-newtoia power-law fluid saturated porous media. The diffusio of eergy caused by a compositio gradiet is called the Dufour or diffusio-thermo effect. O the other had, diffusio of mass due to temperature gradiet is called Soret or thermo-diffusio effects. I liquid mixtures the Dufour terms is ideed very small ad thus the Dufour effect will be egligible i compariso to the Soret effect. Hece oe may igore the Dufour terms whe dealig with liquids. Alam ad Rahma [9] studied the Dufour ad Soret effects o steady MHD free covective heat ad mass trasfer flow past a semi-ifiite vertical porous plate embedded i a porous media cosiderig Soret Dufour effects. A aalytical study of liear ad o-liear double diffusive covectio i couple stress liquids with Soret effect reported by Malashetty et al. [2]. Postelicu [2] studied umerically the ifluece of chemical reactio o heat ad mass trasfer by atural covectio from vertical surfaces i porous media cosiderig Soret ad Dufour effects. Thermal diffusio ad MHD effects o combied, free-forced /9 29 Betham Ope

2 8 The Ope Trasport Pheomea Joural, 29, Volume Kairi ad Murthy covectio ad mass trasfer of a viscous fluid flow through a porous medium with heat geeratio is examied by Abdel-Rahma [22]. The liear stability aalysis of Soretdrive thermo-solutal covectio i a shallow horizotal layer of a porous medium subjected to iclied thermal ad solutal gradiets of fiite magitude is ivestigated theoretically by Narayaa et al. [23]. The mai purpose of the preset ivestigatio is to illustrate the effect of meltig ad thermal-diffusio o atural covectio heat ad mass trasfer i a o- Newtoia fluid saturated o-darcy porous medium, with a power law model for o-newtoia fluid, usig the similarity solutio techique. Fig. (). Schematic of the iterface (vertical lie) separatig the solid ad liquid saturated porous medium. Mathematical Formulatios Cosider the free covectio heat ad mass trasfer from a vertical plate embedded i a o-darcy porous medium saturated with a o-newtoia fluid. It is assumed that the plate costitutes the iterface betwee the liquid phase ad the solid phase durig meltig iside the porous matrix. The co-ordiate system ad flow model are show i the Fig. (). The x -coordiate is take alog the plate, the y -coordiate is measured ormal to the plate, while the origi of the referece system is take at the leadig edge of the plate. The fluid flow is moderate ad the permeability of the medium is assumed to be low so that the Forchheimer flow model is applicable ad the boudary-drag effect is eglected. The plate is at a costat temperature T m at which the material of the porous matrix melts. The liquid phase temperature is T (> T m ) ad the temperature of the solid far from the iterface is T (< T m ). The cocetratio at the wall is C w ad the surroudig porous medium is maitaied at costat cocetratio C. The flow is steady, lamiar ad two dimesioal. With the usual boudary layer ad liear Boussiesq approximatios, the goverig equatios, amely the equatio of cotiuity, the o-darcy flow model (i.e. the model give by Sheoy [8]), the eergy equatio ad the cocetratio equatio for the isotropic ad homogeeous porous medium may be writte as u x + v y = () u y + bk* μ * u 2 y = gk* μ * T T y + C C y (2) u T x + v T y = 2 T y 2 (3) u C x + v C y = D 2 C y + D 2 T 2 y. (4) 2 I the above equatios, u ad v are the Darcia velocity compoets alog the x ad y directios, T ad C are the temperature ad cocetratio, respectively, is the power law idex. is the referece desity, g is the acceleratio due to gravity, = k /( c f ) is the equivalet thermal diffusivity, k is the effective thermal coductivity of the saturated porous medium, D is the effective solutal diffusivities, respectively, D quatifies the cotributio to the mass flux due to temperature gradiet, T ad C are the thermal expasio coefficiet ad cocetratio expasio coefficiet, respectively. Also, b is the empirical costat associated with the Forchheimer porous iertia term ad μ * is the cosistecy idex for power law fluid. Followig Christopher ad Middlema [24] ad Dharmadhikari ad Kale [25], the modified permeability of the flow k* of the o-newtoia power law fluid is defied as k* = 2c t where K = 25 2 c t = K d 2 5( ) 2 ad ad for =, c t = ( 3) (+) Christopher ad Middlema [24] Dharmadhikariad Kale [25] The boudary coditios ecessary to complete the problem formulatio are writte as: y = : k T y = L + c s (T m T ) v, T = T m, C = C w (5) y : u, T T, C C. (6) where L ad c s are latet heat of the solid ad the specific heat capacity of the solid phase, respectively.

3 Meltig ad Thermo-Diffusio o Natural Covectio Heat Mass Trasfer The Ope Trasport Pheomea Joural, 29, Volume 9 The boudary coditio (5) at the iterface states that the temperature of the plate is equal to the meltig temperature of the material saturatig the porous matrix ad the other coditio meas that the heat coducted to the meltig surface is equal to the sum of heat of meltig ad the sesible heat required to raise the temperature of the solid, T to its meltig temperature T m. It is importat to ote that the aforemetioed equatio is cosistet with a coordiate system fixed to the meltig surface, so that the iterior of the solid appears to move towards the (statioary) meltig surface with costat velocity equal to the meltig velocity v(x,). Accordig to the preset formulatio, trasiet effects i the solid have bee eglected. This assumptio is valid as log as meltig solid is large compared to its thermal boudary layer thickess []. The cotiuity equatio is automatically satisfied by defiig a stream fuctio (x, y) such that u = y ad v =. We itroduce the followig similarity x trasformatio: = y x Ra, () = Ra x x f (), () = T T m, T T m () = C C w C C w where Ra x = x k * g T (T T m ). μ * The above trasformatio reduces the system of partial differetial equatios ito the followig system of o-liear ordiary differetial equatios: f ( + 2Gr * f ) f = ( + N ) (7) + 2 f = (8) + 2 f + S = (9) r The boudary coditios are =, f + 2M =, =, = (), f,,. () I the above, is the power law idex. The power law fluids with < are called pseudoplastics, while those with > are termed as dilatats. M = c (T T ) f m is the L + c s (T m T ) meltig parameter, S r = D (T T m ) (C w C ) / is the Soret umber, Gr* = b k * 2 2 [g (T T m )] 2 is the modified μ * 2 Grashof umber. The buoyacy ratio is N = c w T w ad = D is the wis umber. The parameter N > represets the aidig buoyacy ad N < represets the opposig buoyacy. The o-dimesioal heat ad mass trasfer coefficiets are defied as Ra x = () (2) Ra x = (). (3) RESULTS AND DISCUSSION The resultig ordiary differetial equatios (7)-(9) alog with the boudary coditios ()-() are solved usig matlab BVP solver bvp4c which is a fiite differece code that implemets the 3-stage Lobatto IIIa formula. The itegratio legth varies with the parameter values ad it has bee suitably chose each time such that the boudary coditios at the outer edge of the boudary layer are satisfied. The results obtaied here are accurate up to the 4 th decimal place. I order to assess the accuracy of the solutio, the preset results for the Nusselt umber are compared with those obtaied by Poulikakos ad Spatz [] with N = ad S r =, the tred shows that the results are i good agreemet. The followig values are cosidered for the parameters:.5, M, N 3, S r, 5 ad. Gr*.. The variatio of heat ad mass trasfer coefficiets are show for some selected values of the parameters through figures. Aidig Buoyacy Figs. (2-4) show the o-dimesioal velocity f /, temperature ad cocetratio profiles for various values of power law idex ad meltig parameter M for fixed values of N, Gr*, ad S r. It ca be see from Fig. (2) that the values of the slip velocity, f / (), decreases as the power law idex icreases. As the power law idex icreases, the mometum, temperature ad cocetratio boudary layer thickesses icreases. Icreasig the meltig parameter M icreases the velocity iside the boudary layer ad, therefore, it decreases the temperature ad cocetratio distributios. The variatio of Nusselt ad Sherwood umber versus the power law idex parameter for differet values of M ad S r is show i Figs. (5, 6). As the power law idex ad the meltig parameter M icreases, the Nusselt ad Sherwood umber decreases. The reaso for this is that icreasig the power law idex ad the meltig parameter M icreases thermal ad cocetratio boudary layer thickess (see Figs. 3, 4) which results i a reductio i

4 The Ope Trasport Pheomea Joural, 29, Volume Kairi ad Murthy f' N = 2, Gr* =., =, S r =. whole rage of. The variatio of mass trasfer coefficiet agaist for differet values of ad M is illustrated i the Fig. (8). It is observed that as icreases the reductio i the mass trasfer coefficiet due to icrease i the value of the meltig parameter M icreases. It is also oticed that this reductio is more for dilatats whe compared to pseudoplastic fluids N = 2, Gr* =., =, S r = Fig. (2). Variatio of velocity profiles with similarity variable for aidig buoyacy (N > ). θ.8.6 N = 2, Gr* =., =, S r = Fig. (3). Variatio of temperature profiles with similarity variable for aidig buoyacy (N > ). temperature ad cocetratio gradiet at the surface of the plate. Also it ca be oted that as the Soret parameter S r icreases the heat trasfer coefficiet icreases but the mass trasfer coefficiet reduces for all values of. This is because, either a decrease i cocetratio differece or a icreases i temperature differece leads to a icrease i the value of the Soret parameter S r. Hece icreasig the parameter S r icreases the o-dimesioal heat trasfer coefficiet ad decreases the mass trasfer coefficiet. The effect of Soret parameter o the heat ad mass trasfer coefficiet reduces with icreasig M for all values of. Fig. (7) depicts the effect of o the heat trasfer coefficiet for varyig ad M. It is clearly see from this figure that as icreases the reductio due to icreasig M i heat trasfer coefficiet reduces for both pseudoplastic ad dilatat fluids. Also, it is more for the dilatats i the φ Fig. (4). Variatio of cocetratio profiles with similarity variable for aidig buoyacy (N > ) Fig. (5). Variatio of heat trasfer coefficiet agaist for varyig M ad S r. Opposig Buoyacy N =, Gr* =., = The variatio of o-dimesioal velocity f /, temperature ad cocetratio agaist the similarity variable for differet values of ad M is show i Figs. (9-). It is observed from these figures that slip velocity, f / (), icreases with ad the mometum, temperature ad cocetratio boudary layer thickess decreases with, therefore, heat ad mass trasfer rate icreases with the power law idex. With the icrease i the value of the meltig parameter M, a idetical behavior is observed, as i =- =-

5 Meltig ad Thermo-Diffusio o Natural Covectio Heat Mass Trasfer The Ope Trasport Pheomea Joural, 29, Volume case of aidig buoyacy, o the velocity, temperature ad cocetratio fields observed that as icreases the reductio due to icremet i the meltig parameter o the Nusselt ad Sherwood umber icreases. Also it ca be oted that the reductio is more proouced for dilatat fluids i the porous medium N =., Gr*=., = =- = N =, S r =-, Gr* Fig. (6). Variatio of mass trasfer coefficiet agaist for varyig M ad S r. 5 Fig. (8). Variatio of mass trasfer coefficiet agaist for varyig ad M N =, Gr*, S r =- f'.3 N =-, Gr*, =, S r Fig. (7). Variatio of heat trasfer coefficiet agaist for varyig ad M. The variatios of the Nusselt ad Sherwood umber as a fuctio of power law idex is show for three differet values of M ad S r i Figs. (2, 3) respectively with fixed values of the other parameters. From these figures it is evidet that icreasig the meltig parameter M reduces the heat ad mass trasfer coefficiet. Also it plays a vital role i reducig the heat ad mass trasfer coefficiet for dilatats i the medium compared with that for pseudoplastics. It ca be oted from these figures that icrease i S r reduces the heat ad mass trasfer coefficiet for all values of the power law idex. The effect of S r o the heat ad mass trasfer coefficiet reduces as the meltig parameter icreases. Figs. (4, 5) illustrate the variatio of the Nusselt ad Sherwood umbers agaist for differet values of ad M, respectively with fixed value of other parameters. It is. 2 4 Fig. (9). Variatio of velocity profiles with similarity variable for opposig buoyacy (N < ). CONCLUSIONS I this study the meltig pheomea with thermodiffusio effect o free covectio from a vertical flat plate with costat wall temperature ad cocetratio embedded i a o-darcy porous medium saturated with o- Newtoia fluid is aalyzed. Both the aidig ad opposig buoyacies are cosidered for aalysis, ad the heat ad mass trasfer coefficiets are obtaied for various values of flow ifluecig parameters. The results are aalyzed thoroughly for differet cases of <, =, ad >. It is oted that the velocity, temperature ad cocetratio profiles as well as the heat ad mass trasfer coefficiets are sigificatly affected by the meltig pheomea ad thermal-diffusio i the medium. The major coclusio is 6 8 2

6 2 The Ope Trasport Pheomea Joural, 29, Volume Kairi ad Murthy θ N = -, Gr*, =, S r.3 N =-, Gr* =., = =- = Fig. (). Variatio of temperature profiles with similarity variable for opposig buoyacy (N < ). φ N = -, Gr*, =, S r Fig. (). Variatio of cocetratio pr variable for opposig buoyacy (N < ). iles with similarity Fig. (3). Variatio of mass trasfer coefficiet agaist for varyig M ad S r Fig. (4). Variatio of heat trasfer coefficiet agaist for varyig ad M N =-.3, Gr*, S r =, M =.5, M N =-, Gr* =., = =- =-.8.6 N =-.3, Gr*, S r =, M =.5, M Fig. (2). Variatio of heat trasfer coefficiet agaist for varyig M ad S r Fig. (5). Variatio of mass trasfer coefficiet agaist for varyig ad M.

7 Meltig ad Thermo-Diffusio o Natural Covectio Heat Mass Trasfer The Ope Trasport Pheomea Joural, 29, Volume 3 that the heat ad mass trasfer coefficiet are reduces with icreasig value of the meltig parameter M for all (>, =, < ). The effect of Soret parameter o the heat ad mass trasfer coefficiet reduces with icreasig value of the meltig parameter i the whole rage of. Fially, the effect meltig o the heat ad mass trasfer coefficiet is more sigificat for dilatat fluids i the medium, compared to pseudoplastics. NOMENCLATURE b = Coefficiet i the Forchheimer term c f = Specific heat capacity of the covective fluid [J / kg K] c s = Specific heat capacity of the solid phase [J / kg K] d D D f = Pore diameter [m] = Solutal diffusivity [m 2 /s] = Soret coefficiet = Porosity of the saturated porous medium = Dimesioless stream fuctio g = Acceleratio due to gravity [ m/s2 ] k * = Itrisic permeability of the porous medium for flow of power law fluid [ m 2 ] K = Permeability of the porous medium [ m 2 ] Gr* = No-Darcia (iertia) parameter or Grashof umber based o permeability for power law fluid = Power law idex N = Buoyacy ratio Nu = Nusselt umber Sh = Sherwood umber S r L T C = Thermal-diffusio or Soret parameter = Latet heat of the solid [J /kg] = wis umber = Temperature [K] = Cocetratio x, y = Axial ad ormal co-ordiates [m] u,v = Velocity compoets i x ad y directios [m/s] Greek Symbols μ * = Fluid cosistecy of the ielastic o-newtoia power-law fluid [ kg/(ms) ] = Referece desity at some poit [kg/m 3 ] = Thermal diffusivity [m 2 /s] T = Coefficiet of thermal expasio [ /K ] c = Coefficiet of solutal expasio [ /K ] w w = Dimesioless stream fuctio = Similarity variable = Dimesioless temperature = Dimesioless cocetratio = T T m = C w C Subscripts w, = Coditios o the wall ad at the ambiet medium ACKNOWLEDGEMENTS The authors are thakful to the reviewers for their costructive criticism which helped us i improvig this article. Both the authors would like to ackowledge sicerely the fiacial assistace received from the CSIR, Idia through the research project grat umbers respectively, 9/8(596)/26-EMR- ad 25(55)/7/EMR-II. REFERENCES [] M. Epstei, ad D. H. Cho, Meltig heat trasfer i steady lamiar flow over a flat plate, ASME Joural of Heat Trasfer, vol. 98, pp , 976. [2] M. Kazmierczak, D. Poulikakos, ad I. Pop, Meltig from a flat plate embedded i a porous medium i the presece of steady covectio, Numerical Heat Trasfer, vol., pp , 986. [3] M. Kazmierczak, D. Poulikakos, ad D. Sadowski, Meltig of a vertical plate i porous medium cotrolled by forced covectio of a dissimilar fluid, Iteratioal Commuicatio i Heat Mass Trasfer, vol. 4, pp , 987. [4] F. M. Pozvokov, E. F. Shurgalskii, ad L. S. Akselrod, Heat trasfer at a meltig flat surface uder coditios of forced covectio ad lamiar boudary layer, Iteratioal Joural of Heat ad Mass Trasfer, vol. 3, pp , 97. [5] A. Y. Bakier, Aidig ad opposig mixed covectio flow i meltig from a vertical flat plate embedded i a porous medium, Trasport i Porous Media, vol. 29, pp , 997. [6] R. S. R. Gorla, M. A. Masour, I. A. Hassaie, ad A. Y. Bakier, Mixed covectio effect o meltig from a vertical plate i a porous medium, Trasport i Porous Media, vol. 36, pp , 999. [7] B. Tashtoush, Magetic ad buoyacy effects o meltig from a vertical plate embedded i a saturated porous media, Eergy Coversio ad Maagemet, vol. 46, pp , 25. [8] W. T. Cheg, ad C. H. Li, Trasiet mixed covective heat trasfer with meltig effect from the vertical plate i a liquid saturated porous medium, Iteratioal Joural of Egieerig Sciece, vol. 44, pp , 26. [9] W. T. Cheg, ad C. H. Li, Meltig effect o mixed covective heat trasfer with aidig, ad opposig exteral flows from the vertical plate i a liquid-saturated porous medium, Iteratioal Joural of Heat ad Mass Trasfer, vol. 5, pp , 27. [] W. T. Cheg, ad C. H. Li, Usteady mass trasfer i mixed covective heat flow from a vertical plate embedded i a liquid saturated porous medium with meltig effect, Iteratioal Commuicatio i Heat Mass Trasfer, vol. 35, pp , 28. [] D. Poulikakos, ad T. L. Spatz, No-Newtoia atural covectio at a meltig frot i a permeable solid matrix, Iteratioal Commuicatio i Heat Mass Trasfer, vol. 5, pp , 988. [2] H. T. Che, ad C. K. Che, Natural covectio of o- Newtoia fluids alog a vertical plate embedded i a porous medium, ASME Joural of Heat Trasfer, vol., pp , 988.

8 4 The Ope Trasport Pheomea Joural, 29, Volume Kairi ad Murthy [3] H. T. Che, ad C. K. Che, Natural covectio of a o- Newtoia fluid about a horizotal cylider ad sphere i a porous medium, Iteratioal Commuicatios i Heat ad Mass Trasfer, vol.5, pp , 988. [4] A. Nakayama, ad H. Koyama, Buoyacy iduced flow of o- Newtoia fluids over a o-isothermal body of arbitrary shape i a fluid-saturated porous medium, Applied Scietific Research, vol. 48, pp. 55-7, 99. [5] S. K. Rastogi, ad D. Poulikakos, Double-diffusio from a vertical surface i a porous regio saturated with a o- Newtoia fluid, Iteratioal Joural of Heat ad Mass Trasfer, vol. 38, pp ,995. [6] A. V. Sheoy, No-Newtoia fluid heat trasfer i porous media, Advaces i Heat Trasfer vol. 24, New York: Academic Press, 994, pp. -9. [7] A. Nakayama, ad A. V. Sheoy, A uified similarity trasformatio for Darcy ad o-darcy forced, free ad mixed covectio heat trasfer i o-newtoia ielastic fluid-saturated porous media, The Chemical Egieerig Joural, vol. 5, pp , 992. [8] A. V. Sheoy, Darcy-Forchheimer atural forced ad mixed covectio heat trasfer i o-newtoia power-law fluidsaturated porous media, Trasport i Porous Media, vol., pp , 993. [9] M. S. Alam, ad M. M. Rahma, Dufour ad Soret effects o MHD free covectio heat ad mass trasfer flow past a vertical flat plate embedded i a porous medium, Joural of Naval Architecture ad Marie Egieerig, vol. 2, pp , 25. [2] M. S. Malashetty, S. N. Gaikwad, ad M. Swamy, A aalytical study of liear ad o-liear double diffusive covectio with Soret effect i couple stress liquids, Iteratioal joural of thermal sciece, vol. 45, pp , 26. [2] A. Postelicu, The ifluece of chemical reactio o heat ad mass trasfer by atural covectio from vertical surfaces i porous media cosiderig Soret Dufour effects, Heat Mass Trasfer, vol. 43, pp , 27. [22] G. M. Abdel-Rahma, Thermal diffusio ad MHD effects o combied, free-forced covectio ad mass trasfer of a viscous fluid flow through a porous medium with heat geeratio, Chemical Egieerig Techology, vol. 3, pp , 28. [23] P. A. L. Narayaa, P. V. S. N. Murthy, ad R. S. R. Gorla, Soretdrive thermo-solutal covectio iduced by iclied thermal ad solutal gradiets i a shallow horizotal layer of a porous medium, Joural of Fluid Mechaics, vol. 62, pp. -9, 28. [24] R. H. Christopher, ad S. Middlema, Power law fluid flow through a packed tube, Idusrial ad Egieerig Chemistry Fudametals, vol. 4, pp , 965. [25] R. V. Dharmadhikari, ad D. D. Kale, The flow of o-newtoia fluids through porous media, Chemical Egieerig Sciece, vol. 4, pp , 985. Received: May 8, 29 Revised: Jue 27, 29 Accepted: Jue 29, 29 Kairi ad Murthy; Licesee Betham Ope. This is a ope access article licesed uder the terms of the Creative Commos Attributio No-Commercial Licese ( 3./) which permits urestricted, o-commercial use, distributio ad reproductio i ay medium, provided the work is properly cited.