Available online at Pelagia Research Library. Advances in Applied Science Research, 2016, 7(4):

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1 Available online at Pelagia Reseach Libay Advances in Applied Science Reseach, 016, 7(4: 35-4 ISSN: CODEN (USA: AASRFC Impact o Chemical Reaction on MHD Fee Convection Heat and Mass Tanse om Vetical Suaces in Poous Media Consideing Themal Diusion and Diusion Themo Eects D. Sinivas Reddy Depatment o Mathematics, Vadhaman College o Engineeing, Shamshabad, Hydeabad , India ABSTRACT Many eseaches have been studied the inluence o the heat and mass tanse by natual convection in poous medium due to its many applications, such as gowth o moden technologies o nuclea waste oganization, heating o gound wate in hot dike complexes in volcanic egions, etc. In this pape, a two dimensional steady magneto hydodynamics (MHD ee convection low o heat and mass tanse om a vetical suace in poous media subjected to a chemical eaction has been analyzed numeically consideing themal-diusion and diusionthemo eects. Hee, we tansomed the govening non linea patial dieential equations into system o odinay dieential equations by a simila tansomation to discuss the numeical solution o govening equations using implicit inite dieence method. We displayed the dimensionless tempeatue poiles, velocity poiles and the concentation poiles gaphically o the dieent values o the Lewis numbe, Soet numbe, Duou numbe, Magnetic numbe, chemical eaction paamete. Key wods: Poous medium, MHD, Duou eect, Soet eect and the Finite dieence method INTRODUCTION Seveal studies have been ound to analyze the inluence o the combined heat and mass tanse pocess by nomal convection in a themal and/o mass statiied poous medium, due to its boad applications, such as development o moden technologies o nuclea waste management, hot ditch complexes in volcanic egions o heating o gound wate, sepaation pocedue in chemical engineeing, etc. When heat and mass tanse occu simultaneously in a moving luid, the elation among the luxes and the diving potentials ae o moe complex natue. It has been ound that an enegy lux can be geneated by tempeatue gadients as well as composition gadients. The enegy lux caused by a composition gadient is called the diusionthemo eect. On the othe hand, mass luxes can also be ceated by tempeatue gadient and this is the themal diusion eect. These eects ae consideed as second-ode phenomena on the basis that they ae o lesse ode o magnitude than the eects descibed by Fouie s and Fick s laws but they may become impotant in aeas such as hydology o geosciences. The study o magneto hydodynamics(mhd lows have moe concentation due to its signiicant applications in thee divese ields, such as an cosmological, geophysical and engineeing poblems. Fee convection in electically conducting luids though an extenal magnetic ield has been a subject o consideable eseach concen o a huge numbe o scholas o a long time due to its divese applications in the aeas such as nuclea eactos, geothemal engineeing, luid metals and plasma lows among the othes. Fluid low contol unde the magnetic oce is also Pelagia Reseach Libay 35

2 D. Sinivas Reddy Adv. Appl. Sci. Res., 016, 7(4:35-4 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 luid-satuated poous medium has attacted signiicant attention in the pevious yeas, due to seveal impotant engineeing and geophysical applications. Seveal eseaches have discussed about Steady and tansient ee convection coupled heat and mass tanse by natual convection in a luid-satuated poous medium, due to vaious applications in engineeing. Nield and Bejan [ 1] and Ingham and Pop [, 3] had pesented a complete inomation in the above said ield. Muthy et al[4] have studied eect o doubly statiication on ee convection in Dacian poous medium. Lakshmi Naayana and Muthy [5] 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 the eect o MHD, (Viscous dissipation. Adian postelnicu [6] studied the heat and mass tanse popeties o natual convection about a vetical suace embedded in a satuated poous medium subjected to a chemical eaction by consideing the duou and soet eects. Madhusudhana Reddy and Pasada Rao[7] discussed the eect o adiation, diusion and chemical eaction on the mixed convective heat and mass tanse low o a viscous electically conducting luid though a poous medium. Shaiul Alam and Shiazul Hoque Mollah [8] have studied the eect o nth ode chemical eaction and heat geneation o absoption on MHD ee convective low with heat and mass tanse ove an inclined pemeable stetching sheet unde the inluence o Duou and Soet eects with vaiable wall concentation and tempeatue. Dulal Pal and Babulal Talukda [9] studied themal adiation and magnetohydodynamic on mixed convection low o a viscous incompessible electically conducting luid though a poous medium with vaiable pemeability. Saada and Shanke [10] have analyzed the chemical eaction eect on an unsteady MHD ee convection low though an ininite vetical plate by taking into consideation o vaiable suction. The objective o this pape is to investigate simultaneous heat and mass tanse by natual convection om a vetical suace in dacian poous medium unde the inluence o the magnetic ield including eects o chemical eaction, Soet and Duou. Notation we used: S Soet numbe ψ Steam unction T Tempeatue γ Dimensionless chemical eaction paamete x, y Catesian co-odinates along the nomal to the Ra suace espectively x Local Rayleigh numbe α m Themal diusivity u, v Dacian velocities in the x and y diections β T Coeicient o themal expansion D m Mass diusivity β Coeicient o concentation expansion Dimensionless steam unction C φ Dimensionless concentation K Dacy pemeability η Similaity vaiable k T υ Kinematic viscosity C Concentation C θ Dimensionless tempeatue p C s Concentation susceptibility ρ Density Themal diusion atio Speciic heat at constant pessue Pelagia Reseach Libay 36

3 D. Sinivas Reddy Adv. Appl. Sci. Res., 016, 7(4:35-4 Poblem Fomulation: Figue a: Flow model Conside the natual convection in a poous medium with a Newtonian luid on a vetical lat plate. The x-coodinate is measued along the suace and the y-coodinate nomal to it (see Figue a.the tempeatue o the ambient medium is T and the wall tempeatue ist 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. The ollowing assumptions ae used thoughout the pesent pape: (a the luid and the poous medium ae in local themodynamic equilibium; (b the low is steady-state, lamina and two-dimensional; (c the poous medium is homogeneous and isotopic; (d the chaacteistics o the luid and the poous medium ae constant; (e the Boussinesq appoximation is valid and the bounday laye appoximation is applicable; ( the concentation o dissolved A is small enough. In-line with the above assumptions, the govening equations descibing the consevation o mass, momentum, enegy and concentation can be witten as ollows u v + = 0 x y (1 gk u = T T + C C B u ( υ ρ σ [ βt ( βc ( ] ( 0 Dm kt y s p y T T T C u + v = α m + x y C C C C C DmkT T u + v = Dm + K x y T 1 C (4 y m y Whee all quantities ae deined in the list o symbols. The bounday conditions o the poblem ae 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 non dimensionalised using the ollowing quantities: (3 Pelagia Reseach Libay 37

4 D. Sinivas Reddy Adv. Appl. Sci. Res., 016, 7(4:35-4 ψ = α Ra 1/ ( η, = ( T T / ( T T, m x θ w y φ = ( C C / ( Cw C, η = 1/ Rax, (6 x Whee the steam unction ψ is deined in the usual way ψ ψ u =, v = (7 y x and Ra = gkβ ( T T x / ( υα is the local Rayleigh numbe. x w m The govening equations become / (1 + M = θ + Nφ (8 // / // θ θ + D φ = 0 (9 1 // / // φ + φ γφ + Sθ Le = 0 (10 Whee Le, D, S and N ae Lewis, Duou. Soet numbes and sustentation paamete espectively αm DmkT ( Cw C Le =, D =, S D C C α ( T T m s p m w 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 1 K x γ =. (11b αm Rax 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 Mathematical Solution: The System o non-linea odinay dieential equations (8 - (10 with bounday conditions (1 have solved numeically, by using Cank Nicolson implicit inite dieence method. A step size o η = 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 a vey good ageement between them. RESULTS AND DISCUSSION Numeical Computations wee caied out o non dimensional velocity poiles, tempeatue poilesθ and concentation poiles φ with dieent values o D, S, M, γ and N. The velocity poile D, S, M Pelagia Reseach Libay 38

5 D. Sinivas Reddy Adv. Appl. Sci. Res., 016, 7(4:35-4 pesented o ixed values o D, S, M in Fig. 1. We obseved that Non dimensional velocity inceases with the inceasing o N. Fig., explains velocity poile o dieent values o M, N, S and D. Fom Fig. we conclude that by inceasing o Lewis numbe the velocity poile inceases. Fom Fig. 3, we obseved that the velocity poile inceases with incease o the magnetic paamete. The tempeatue poiles o dieent paametes Lewis numbe Le and M have been discussed in Fig. 4 and it shows that the Tempeatue poile inceases by inceasing o Soet numbe S. Fom Fig. 5 we obseved that by incease o the magnetic paamete, the tempeatue poile deceases. The concentation poile plotted o dieent Lewis numbe in ig.6 and we can obseve that the concentation poile deceases with incease o Lewis numbe. Fom Fig. 7-8 we obseved that concentation poile inceases by inceasing o chemical eaction paamete o dieent values o Le, M, N, S and D. Concluding Remaks: In this pape, studied a two dimensional steady MHD ee convection low o heat and mass tanse om a vetical suace in poous media subjected to a chemical eaction and obtained numeical solution consideing themaldiusion and diusion-themo eects. Fo this poblem, velocity poiles, tempe poiles and concentation poiles ae studied with dieent paametes. Study eveals that, velocity poile inceases with the incease o divese paametes,tempeatue poile deceases by inceasing magnetic paamete and concentation poile inceases with the incease o chemical eaction. Figue 1: Velocity poile o dieent values o N with S = 0.001, D = 10.0, M = 0 Figue : Velocity poile o dieent Le with S = 0.001, D = 10.0, M = 0, N = 1 Pelagia Reseach Libay 39

6 D. Sinivas Reddy Adv. Appl. Sci. Res., 016, 7(4:35-4 Figue 3: Velocity poile o dieent Magnetic paametes with S = 0.001, D = 10.0 Figue 4: Tempeatue poiles o dieent soet paametes with D = 10.0, M = 0, N = 1 Figue 5: Tempeatue poiles o dieent magnetic paamete with S = 0.001, D = 10.0 Pelagia Reseach Libay 40

7 D. Sinivas Reddy Adv. Appl. Sci. Res., 016, 7(4:35-4 Figue 6: Concentation poiles o dieent Le with D = 10.0, M = 0, S = Figue 7: Concentation poile o dieent γ with D = 10.0, S = 0.001, Le = 1, N = 1, M = 0 Figue 8: Concentation poile o dieent γ with D = 0, S = 0, Le = 1. N = 1, M = 0 REFERENCES [1] Nield D A, Bejan A, Convection in Poous Media, Spinge, New Yok, 006. [] Ingham D, Pop I, Tanspot phenomena in poous media I, Pegamon, Oxod, Pelagia Reseach Libay 41

8 D. Sinivas Reddy Adv. Appl. Sci. Res., 016, 7(4:35-4 [3] Ingham D, Pop I, Tanspot phenomena in poous media II, Pegamon, Oxod, 00. [4] Muthy P V S N, Sinivasachaya D, Kishna P V S R, Tans.ASMEJ Heat Tanse,,004, 16, [5] Lakshmi Naayana P A, Muthy P V S N, Jounal o Poous Media, 007, 10(6, [6] Postelnicu A, Int. J.Heat Mass tanse, 004, 47, [7] Madhusudhana Reddy Y, Pasada Rao D R V, Intenational Jounal o Mathematical Achive, 01, 3(4, [8] Shaiul Alam Md, Shiazul Hoque Mollah Md, Asian Tansactions on Basic and Applied Sciences, 01, 0(5. [9] Dulal Pal, Babulal Talukda, Intenational Jounal o Nonlinea Science, 01, 14(1, [10] Saada K, Shanke B, Intenational Jounal o Moden Engineeing Reseach, 013, 3(, Pelagia Reseach Libay 4