Numerical study of MHD free convection heat and mass transfer from vertical surfaces in porous media considering Soret and Dufour effects
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1 Available online at Advances in Applied Science Reseach, 013, 4(1: ISSN: CODEN (USA: AASRFC Numeical study o MHD ee convection heat and mass tanse om vetical suaces in poous media consideing Soet and Duou eects G. Seedha Sama 1, D. Seenivasu, K. Govadhan 3, K. Ramakishna Pasad 4 1, Depatment o Mathematics, CVR College o Engineeing, R. R. Dist., A. P., India 3 Depatment o Mathematics, GITAM Univesity, Rudaam, Medak, Hydeabad, A. P., India 4 Depatment o Mathematics, Si Venkateswaa Univesity, Tiupathi, A. P., India ABSTRACT In the pesent appoach, a two dimensional steady MHD ee convection low o heat and mass tanse om a vetical suace in poous media has been analyzed numeically conside with Soet and Duou eects. The govening non linea patial dieential equations have been tansomed by a simila tansomation in to a system o odinay dieential equations, which ae solved numeically by using implicit inite dieence scheme. The dimensionless velocity, tempeatue and concentation poiles ae displayed gaphically showing the eects o the dieent values o the Lewis numbe, soet numbe and Magnetic numbe. Keywods: ee convection, poous medium, MHD, Duou, Soet and inite dieence method. Nomenclatue S T, y Soet numbe Tempeatue Catesian co-odinates along and nomal to the suace, espectively α m Themal diusivity β T Coeicient o themal epansion β C Coeicient o concentation epansion φ η υ θ ρ ψ Dimensionless concentation Similaity vaiable Kinematic viscosity Dimensionless tempeatue Density Steam unction Subscipts w Condition at wall Condition at ininity Supescipt Dieentiation with espect to g 300
2 G. Seedha Sama et al Adv. Appl. Sci. Res., 013, 4(1: INTRODUCTION Seveal studies have been ound to analyze the inluence o the combined heat and mass tanse pocess by natual convection in a themal and/o mass statiied poous medium, owing to its wide applications, such as development o advanced technologies o nuclea waste management, hot dike complees in volcanic egions o heating o gound wate, sepaation pocess in chemical engineeing, etc. Hee statiied poous medium means that the ambient concentation o dissolved constituent and/o ambient tempeatue is not uniom and vaies as a linea unction o vetical distance om the oigin. When heat and mass tanse occu simultaneously in a moving luid, the elation between the lues and the diving potentials ae o moe inticate natue. It has been ound that an enegy lu can be geneated not only by tempeatue gadients but by composition gadients as well. The enegy lu caused by a composition gadient is called the Duou o diusion-themo eect. On the othe hand, mass lues can also be ceated by tempeatue gadient and this is the Soet o themal diusion eect. These eects ae consideed as second-ode phenomena, on the basis that they ae o smalle ode o magnitude than the eects descibed by Fouie s and Fick s laws, but they may become signiicant in aeas such geosciences o hydology. The study o magneto hydodynamics(mhd lows have stimulated etensive attention due to its signiicant applications in thee dieent subject aeas, such as a astophysical, geophysical and engineeing poblems. Fee convection in electically conducting luids though an etenal magnetic ield has been a subject o consideable eseach inteest o a lage numbe o scholas o a long time due to its divese applications in the ields such as nuclea eactos, geothemal engineeing, liquid metals and plasma lows, among the othes. Fluid low contol unde magnetic oce is also applicable in magneto hydodynamics geneatos and a host o magnetic devices used in industies. Steady and tansient ee convection Coupled heat and mass tanse by natual convection in a luidsatuated poous medium has attacted consideable attention in the last yeas, due to many impotant engineeing and geophysical applications. Recent books by Nield and Bejan [ 1] and Ingham and Pop [, 3] pesent a compehensive account o the available inomation in the ield. The MHD ee-convection and Mass Tanse low with Hall cuent, viscous dissipation, Joule heating and Themal diusion is studied by Singh, A.K [4].N.Kishan et all [5] studied the MHD ee convection heat and mass tanse in a doubly statiied Dacy poous medium consideing soet and Duou eect with viscous dissipation.eect o doubly statiication on ee convection in Dacian poous medium have been studied by Muthy et al [6]. G.Vidyasaga et all [7] studied the Mass Tanse eects on adiative MHD low ove a non isothemal stetching sheet and embedded in a poous medium. P.Sinivasulu and N.Bhaska Reddy [8] studied the Themo-diusion and Diusion-themo eects on MHD bounday laye low past an eponential stetching sheet with themal adiation and viscous dissipation. Lakshmi Naayana and Muthy [9] investigated the eects o Soet and Duou on ee convection heat and mass tanse om a vetical suace in a doubly statiied Dacy poous medium. They have neglected eect o MHD, (Viscous dissipation. Patha et al[10] studied the eect o magnetic ield and double dispesion on ee convection heat and mass tanspot consideing the Soet and Duou eects in a non Dacy poous medium. V.Sihai babu and G.V.Ramana eddy [11] studied the Mass tanse eects on MHD low om a vetical suace with ohmic heating and viscous dissipation. Satya saga Saena and G.K.Dubey [1] studied the MHD ee convection heat and mass tanse low o viscoelastic luid embedded in a poous medium o vaiable pemeability with adiation eect and heatsouce in slip low egime. Satyasaga Saena and G.K.Dubey [13] studied the Unsteady MHD heat and mass tanse ee convection low o pola luids past a vetical moving poous plate in a poous medium with heat geneation and themal diusion. P.R Shama and Singh [14] studied the eect o vaiable themal conductivity and heat souce/sink on MHD low nea a stagnation point on a linealy stetching sheet. Recently, Adian postelnicu [15] studied the heat and mass tanse chaacteistics o natual convection about a vetical suace embedded in a satuated poous medium subjected to a chemical eaction by taking account the Duou and Soet eects. The objective o this pape is to study simultaneous heat and mass tanse by natual convection om a vetical suace embedded in a luid satuated Dacian poous medium unde the inluence o magnetic ield including Soet and Duou eect. 301
3 G. Seedha Sama et al Adv. Appl. Sci. Res., 013, 4(1: PROBLEM FORMULATION Fig. 1 Flow model and physical coodinate system Conside the natual convection in a poous medium satuated with a Newtonian luid on a vetical lat plate. The - coodinate is measued along the suace and the y-coodinate nomal to it (see Fig. 1.The tempeatue o the ambient medium is T w and the wall tempeatue is T w. The low along the vetical lat plate contains a species A slightly soluble in the luid B, the concentation at the plate suace is C w and the solubility o A in B a away om the plate is C. Seveal assumptions ae used thoughout the pesent pape: (a the luid and the poous medium ae in local themodynamic equilibium; (b the low is lamina, steady-state and two-dimensional; (c the poous medium is isotopic and homogeneous; (d the popeties o the luid and poous medium ae constant; (e the Boussinesq appoimation is valid and the bounday laye appoimation is applicable; ( the concentation o dissolved A is small enough. In-line with these assumptions, the govening equations descibing the consevation o mass, momentum, enegy and concentation can be witten as ollows u v + = 0 y (1 σ [ βt ( βc ( ] ( β0 gk u = T T + C C u ( υ ρ Dm kt y s p y T T T C u + v = α m + y C C DmkT y m y C C C T u + v = Dm + y T (3 (4 30
4 G. Seedha Sama et al Adv. Appl. Sci. Res., 013, 4(1: Whee all quantities ae deined in the list o symbols. The bounday conditions o the poblem ae y : u 0, T T, C C y = 0 : v = 0, T = Tw, C = Cw (5a y : u 0, T T, C C (5b WheeT w,t, C w and C have constant values. Equations 1,, 3, 4, 5 ae now nondimensionalized using the ollowing quantities: ψ = α Ra 1/ ( η, = ( T T / ( T T, m θ w y φ = ( C C / ( Cw C, η = 1/ Ra, (6 Whee the steam unction ψ is deined in the usual way ψ ψ u =, v = y and Ra = gkβ ( T T / ( υα is the local Rayleigh numbe. The govening equations become w m (7 / (1 + M = θ + Nφ (8 // / // θ θ + D φ = 0 (9 1 // / // S 0 Le φ + φ + θ = (10 Whee Le, D, S and N ae Lewis, Duou, Soet numbes and sustentation paamete, espectively αm DmkT ( Cw C Le =, D =, D C C α ( T T m s p m w S DmkT ( Tw T =, C C α ( C C s p m w βc( Cw C N = β ( T T T w (11a We notice that N is positive o themally assisting lows, negative o themally opposing lows and zeo o themal-diven lows. Futhe, in ode to get similaity solutions, the constant dimensionless chemical eaction paamete γ 1 K =. (11b αm Ra Was intoduced in Eq. 10.we notice to this end that pimes denote with espect toη. The tansomed bounday conditions ae (0 = 0, θ (0 = 1, φ(0 = 1 (1a θ 0, φ 0 as y (1b 303
5 G. Seedha Sama et al Adv. Appl. Sci. Res., 013, 4(1: The paametes o engineeing inteest o the pesent poblem ae the local Nusselt numbe and local Shewood numbe, which ae given by the epessions 1/ / = Nu / Ra θ (0, 1/ / Sh / Ra = φ (0. (13 MATHEMATICAL SOLUTION The set o non-linea odinay dieential equations (8 - (10 with bounday conditions (1 have been solved numeically, by using Cack Nickels implicit inite dieence method. A step size o = 0.01 was selected to be 5 satisactoy o a convegence citeia o 10 in all cases. The value o η was ound to each iteation loop by the statement η = η + η. In ode to see the eect o step size η we an the code o ou model with two dieent step sizes η = 0.01, η = and each case we ound vey good ageement between them. RESULTS AND DISSCUSSION Numeical Calculations wee caied out o dieent values o D, S, M and N.Fo the Pupose o discussing the eect o vaious paametes on the low behavio some numeical calculations have been caied out o non dimensional velocity poiles, tempeatue poilesθ, and concentation poilesφ. In ig1, the velocity poile D, S, M pesented o ied values o D, S, M.The non dimensional velocity inceases with the inceasing o N.In ig., we plotted velocity poile o dieent values o D, S, M, N.Fom Fig. we obseved that the velocity poile inceases with the inceasing o Lewis numbe Le.Fom ig.3 we obseved that the velocity poile inceases with the inceases magnetic paamete M. η Fig.1: Velocity poile o dieent values o N with S = 0.001, D = 10.0, M =
6 G. Seedha Sama et al Adv. Appl. Sci. Res., 013, 4(1: Fig.: Velocity poile o dieent Le with S = 0.001, D = 10.0, M = 0, N = 1. Fig.3: Velocity poile o dieent Magnetic paametes with S = 0.001, D =
7 G. Seedha Sama et al Adv. Appl. Sci. Res., 013, 4(1: Fig.4: Tempeatue poiles o dieent soet paametes with D = 10.0, M = 0, N = 1. Fig.5: Tempeatue poiles o dieent magnetic paametes wit S = 0.001, D =
8 G. Seedha Sama et al Adv. Appl. Sci. Res., 013, 4(1: Fig.6: Concentation poiles o dieent Le with D = 10.0, M = 0, S = Fom igues 4-5 show that the tempeatue poiles o dieent paametes Le and M. Fig.4 shows that the Tempeatue poile inceases with the incease o Soet numbe S. Fom ig 5 we obseved that the tempeatue poile inceases with incease o magnetic paamete M.In ig 6, the concentation poile plotted o dieent Lewis numbe Le. Fom ig 6 we obseved that the concentation poile deceases with the incease o Lewis numbe Le. REFERENCES [1] D.A.Nield and A.Bejan, Convection in PoousMedia. NewYok, Spinge- velag.006. [] inghamd.,popi, Tanspot phenamina in poous media, I pegaman, ood,1998. [3] ingam D.PopI, Tanspot phenomina in poous mediaii,pegman,ood,00. [4] Singh A.K. The MHD Fee-Convection and Mass Tanse Flow with Hall Cuent, Viscous Dissipation, Joule Heating and Themal Diusion is Studied. Indian Jounal o Pue and Applied Physics, 003, Vol.41, Pp [5] N.Kishan,M.Chenna Kishna eddy, K.Govadhan, Intenational Jounal o Applied Mechanics and Engineeing 009,Vol 14/3. [6]P.V.S.N, Muthy, D. Sinivasachaya, and P.V.S.S.R, Kishna P.V.S.S.S.R, Tans. ASME J. Heat Tanse, 004, 16, Pp [7] G.Vidyasaga,P.Bala Ankieddy,B.Ramana and V.Sugunamma, Adv.App.science Reseach, 01, 3(6: [8] P.Sinivasulu and N.Bhaska Reddy, Adv.App.science Reseach, 01, 3(6: [9] P.A. Lakshmi Naayana, andp.v.s.n, Muthy, Jounal o Poous Media, 007, 10(6, [10] M.K, Patha,P.V.S.N. Muthy, and G.P. Raja Sekha, ASME J. Heat Tanse, 006, 18, pp [11] V.Sihai babu and G.V.Ramana eddy, Adv.App.science Reseach, 011, (4: [1] Satya saga Saena and G.K.Dubey, Adv.App.science Reseach, 011, (5: [13] Satya saga Saena and G.K.Dubey, Adv.App.science Reseach, 011,(4: [14] P.R Shama and G. Singh, Jounal o Applied luid mechanics, 009, (1, pp [15] A. Postelnicu,. Int. J.Heat Mass tanse, 004, 47, Pp
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