Study of Thermal Radiation and Ohmic Heating for Steady Magnetohydrodynamic Natural Convection Boundary Layer Flow in a Saturated Porous Regime

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1 International Jornal on Recent and Innovation Trends in Compting and Commnication ISSN: -869 Volme: Isse: Std of Thermal Radiation and Ohmic Heating for Stead Magnetohdrodnamic Natral Convection Bondar Laer Flow in a Satrated Poros Regime Sahin Ahmed Heat Transfer & Flid Mechanics Research, Department of Mathematics, Rajiv Gandhi Central Universit, Rono Hills, Itanagar, Arnachal Pradesh-79, India, asahinrg@gmail.com Abstract An analsis is performed to std the thermal radiation and Ohmic heating effects on copled heat and mass transfer b stead magnetohdrodnamic natral convective laminar bondar-laer flow of a viscos incompressible electricall condcting Newtonian flid past a vertical permeable srface embedded in a Darcian poros medim. The heat eqation incldes the terms involving the radiative heat flx, Ohmic dissipation, viscos dissipation and the internal absorption whereas the mass transfer eqation incldes the effects of chemicall reactive species of first-order. The non-linear copled differential eqations are solved analticall b pertrbation techniqe. The nmerical reslts are benchmarked with previosl pblished stdies and fond to be in excellent agreement. Finall, the effects of the pertinent parameters which are of phsical and engineering interest on the flow and heat transfer characteristics are presented graphicall and in tablated form. It is observed that the effect of heat absorption is to decrease the velocit and temperatre profiles in the bondar laer. Kewords- Laminar bondar laer flow; Darcian drag force; Ohmic heating; viscos dissipation; Natral Convection; Heat absorption. ***** I. INTRODUCTION The std of flid flow problems associated with heat transfer is of widespread interest in almost all the fields of engineering as well as in astrophsics, biolog, biomedicine, meteorolog, phsical chemistr, plasma phsics, geophsics, oceanograph and scores of other disciplines. Hdromagnetic flows and heat transfer in poros media have been considered extensivel in recent ears de to their occrrence in several engineering processes sch as compact heat exchangers, metallrg, casting, filtration of liqid metals, cooling of nclear reactors high speed aerodnamics and magnetic braking technologies with fsion control [-]. Merkin [6] investigated a mixed convective bondar-laer flow on a semi-infinite vertical flat plate, when the boanc forces aid in the development of the bondar laer or prevent it. Watanabe [7] presented the effects of the srface mass transfer on a mixed convective flow on a permeable vertical srface. With the combined effect of heat transfer man challenging flow problems have been stdied in magnetohdrodnamic convection flows with different sitable configrations. The effects of transversel applied magnetic field, on the flow of an electricall condcting flid past an implsivel started infinite isothermal vertical plate stdied b Sondalgekar et al. [8]. MHD effects on implsivel started vertical plate with variable temperatre in the presence of transverse magnetic field were considered b Sondalgekar et al. [9]. Free convection flows in a poros media with chemical reaction have wide applications in geothermal and oil reservoir engineering as well as in chemical reactors of poros strctre. Man transport processes exist in indstrial applications in which the simltaneos heat and mass transfer occr as a reslt of combined boanc effects of diffsion of chemical species. Moreover, considerable interest has been evinced in radiation interaction with convection and chemical reaction for heat and mass transfer in flids. This is de to the significant role of thermal radiation in the srface heat transfer when convection heat transfer is small, particlarl, in free convection problems involving absorbing emitting flids. Khair and Bejan [] stdied heat and mass on flows past an isothermal flat plate. Lin and W [] analzed combined heat and mass transfer b laminar natral convection from a vertical plate. Yin [] stdied nmericall the force convection effect on magnetohdrodnamics heat and mass transfer of a continosl moving permeable srface. Achara et al. [] have stdied heat and mass transfer over an accelerating srface with heat sorce in the presence of sction and blowing. Mthcmaraswam and Janakiraman [4] stdied MHD and radiation effects on moving isothermal vertical plate with variable mass diffsion. Hossain et al. [] investigated radiation effects on the free convection flow of an opticall incompressible flid along a niforml heated vertical infinite plate with a constant sction. Orhan and Kaa [6] examined MHD mixed convective heat transfer along a permeable vertical infinite plate in the presence of radiation and soltions are derived sing Kellar box scheme and accrate finite-difference scheme. Ahmed and Li [7] examined the effects of mass transfer on a mixed convection three dimensional heat transfer flow of a viscos incompressible flid past an infinite vertical poros plate in the presence of transverse periodic sction velocit. The problem of combined heat and mass transfer of an electricall condcting flid in MHD natral convection adjacent to a vertical srface is analzed b Chen [8] b taking into accont the effects of Ohmic heating and viscos dissipation bt neglected chemical reaction of the species. Chadhr et al. [9] have analzed the effect of radiation on heat transfer IJRITCC September 4, 796

2 International Jornal on Recent and Innovation Trends in Compting and Commnication ISSN: -869 Volme: Isse: in MHD mixed convection flow with simltaneos thermal and mass diffsion from an infinite vertical plate with viscos dissipation and Ohmic heating. The classical model introdced ρv b Cogle et al. [] is sed for the radiation effect as it has d d = μ d σb + μ K the merit of simplicit and enables s to introdce linear term ρgβ T T T + ρgβ C C C, () in temperatre in the analsis for opticall thin media. Ahmed and Zeco [] stdied the effect of the transverse magnetic field on a stead mixed convective heat and mass transfer flow of an incompressible viscos electricall condcting flid past an infinite vertical isothermal poros plate taking into accont ρc P v dt the indced magnetic field, viscos and magnetic dissipations d = α d T d d + μ d q of energ in presence of chemical reaction of first order and +σb Q T T, (4) heat generation/absorption, and the non-linear copled eqations are solved b network simlation techniqe. The thermal radiation and Darcian drag force MHD nstead v dc d = D d C d R C C, () thermal-convection flow past a semi-infinite vertical plate immersed in a semi-infinite satrated poros regime with The second and third terms on RHS of the momentm variable srface temperatre in the presence of transversal eqation () denote the thermal and concentration boanc niform magnetic field have been discssed b Ahmed el al. effects, respectivel. Also second and forth terms on the RHS []. A nmerical analsis of condction-radiation, porosit of energ eqation (4) represent the viscos dissipation and and chemical reaction on nstead hdromagnetic free Ohmic dissipation, respectivel. The third and fifth term on convection flow past an implsivel-started semi-infinite the RHS of eqation (4) denote the inclsion of the effect of vertical plate embedded in a poros medim in presence of thermal radiation and heat absorption effects, respectivel. thermal radiation is presented b Ahmed []. In this paper, it is proposed to std the effects of viscos dissipation and Ohmic dissipation on stead two dimensional magnetohdrodnamic natral convection heat and mass transfer flow of a Newtonian, electricall condcting and viscos incompressible radiative flid over a poros vertical plate embedded in a poros medim taking into the accont of combined effects of boanc force and first-order chemical reaction. The present std ma have sefl applications in several transport processes as well as in processing magnetic materials. The analtical reslts for some particlar cases are compared with those from [9] and are fond to be in excellent agreement. The governing eqations for this investigation are formlated and solved b sing pertrbation techniqe. II. MATHEMATICAL FORMULATION A two-dimensional laminar bondar laer flow of a viscos incompressible electricall condcting and heat absorbing flid past a semi-infinite vertical permeable plate embedded in a niform poros medim which is sbject to thermal and concentration boanc effects has been presented. As shown in Fig., x*-axis is along the plate and * is perpendiclar to the plate. The wall is maintained at a constant temperatre T w and concentration C w higher than the ambient temperatre T and concentration C, respectivel. Also, it is assmed that there exists a homogeneos chemical reaction of first-order with constant rate R between the diffsing species and the flid. Under these assmptions, the governing eqations of the Newtonian flow model of electricall condcting radiative and chemicall reacting flid throgh poros medim in presence of magnetic field with heat generation and viscos dissipative heat are dv d = v = v Constant, () Fig. : Flow model of the problem For the radiative heat flx sing the Cogle model [] is given q = 4 T T I, (6) where I e bλ = K λw T dλ, K λw is the absorption coefficient at the wall and e bλ is Planck s fnction. The appropriate bondar conditions for velocit, temperatre and concentration fields are = : =, T = T w, C = C w, (7) :, T T, C C, (8) Introdcing the following non-dimensional qantities: dp d = p is independent of, () IJRITCC September 4, 797

3 International Jornal on Recent and Innovation Trends in Compting and Commnication ISSN: -869 Volme: Isse: = v, =, M = σb ν θ, + Prθ P r F + ψ θ + Pr + PrM =, () ν v μv θ = T T, φ = C C, γ = Rν φ + Scφ Scγφ =, () T w T C w C v, where N = M + K. Gr = ρgβ T T w T μv, Gm = ρgβ C C w C μv, Sc = ν D, ψ = Q ν ρc P v, F = 4νI ρc P v, Pr = μc P α, Ec = v C P T w T, K = K v ν (9) On sing (6) and (9), the eqations () () redce to the following non-dimensional eqations: d d + d d M + K = Gr θ Gmφ, () d θ dθ d + Pr + Pr E d d d = Pr E + ψ θ PrEM, () d φ dφ + Sc Scγφ =, () d d The dimensionless form of the bondar conditions (7) and (8) are = : =, θ =, φ = () :, θ, φ (4) III. METHOD OF SOLUTION Eqations () () represent a set of partial differential eqations that cannot be solved in closed-form. However, these eqations can be solved analticall after redcing them to a set of ordinar differential eqations in dimensionless form. Ths we can represent the velocit, temperatre θ and concentration φ in terms of power of Eckert nmber Ec as in the flow of an incompressible flid Eckert nmber is alwas less than nit since the flow de to the Joles dissipation is sper imposed on the main flow. Hence, we can assme = + Ec + o Ec θ = θ + Ecθ + o Ec φ = φ + Ecφ + o Ec () Sbstitting () in eqations () () and eqating the coefficient of zeroth powers of Ec (i. e. O Ec ), we get the following set of eqations: + Ν = Grθ Gmφ, (6) θ + Prθ Pr (F + ψ)θ =, (7) = : : =, =, θ =, θ =, φ =, φ =,, θ, θ, φ, φ () () The soltion of velocit, temperatre and concentration fields have restricted p to O Ec and neglected the higher order of O Ec as the vale of Ec. The soltions of eqations (6) () with the help of bondar conditions () and () are obtained as follows: = A e A 4 e A + A 6 e A 4 e m, (4) = θ = e A, () φ = e m, (6) A 7 e A 4 B e A + B e A +B e A 4 B e A + B 4 e m B e B + B 6 e B, (7) θ = B 9e A B e A B 4 e A 4 + B e A B 6 e m + B 7 e B B 8 e B, (8) φ = (9) The phsical qantities of interest are the wall shear stress τ w is given b τ w = μ, () = Using (4), (7) and () in (), we get C fx = C fx = τ w ρv =. () A 6 m A 4 + A A A 4 Ec B 7A 4 B A + B A + B A 4 () B A + B 4 m B B + B 6 B The local srface heat flx is given b The Local Nsselt nmber q w = κ T = () N x = xq w /κ(t w T ) (4) Using (), (8) and () in (4), then the Local Nsselt nmber can be written as φ + Scφ Scγφ =, (8) The coefficients of first-order of Ec (i. e. O Ec ), we obtain + Ν = Grθ Gmφ, (9) N x Re x = θ = A + EcB 9 +Ec B A + B 4 A 4 B A () +B 6 m B B 7 + B B 8 where Re x = v x/ν is the local Renolds nmber. IJRITCC September 4, 798

4 International Jornal on Recent and Innovation Trends in Compting and Commnication ISSN: -869 Volme: Isse: IV. VALIDATION This is de to the fact that the thermal bondar laer absorbs Validation of the analsis has been performed b energ which cases the temperatre fall considerabl with comparing the present reslts with those available in the open increasing the vale of internal heat absorption parameter. The literatre [9] and a ver good agreement has been established, effect the reaction rate parameter ( on the species when K=, =., =.. In order to verif the accrac of the concentration profiles () for generative chemical reaction is present reslts, we have considered the analtical soltions shown in Fig. 7. It is noticed from the graphs that there is a obtained b Chadhar et al. [9] and compted these soltions decreasing effect on concentration distribtion with increasing for varios phsical parameters for skin-friction coefficient and the vale of the chemical reaction rate parameter in the local Nsselt nmber. bondar laer. Table : Comparison of present reslts with those of In Table, it has been observed that the skin friction Chadhar et al. [9] with different vales of F for C fx and coefficient decreases and local Nsselt nmber raises sharpl N x /Re x ; at Pr =.7, Sc =.78, M =., Gr =., Gm =., de to the increase of radiation parameter F. Ec =.. VI. CONCLUSIONS Chadhar et al. [9] Present reslts F C fx N x /Re x F C fx N V. RESULTS AND DISCUSSION To get a phsical insight into the problem the nmerical evalation of the analtical reslts reported in the previos section was performed and a set of reslts is reported graphicall in Figres -7 for the cases cooling Gr> of the plate i.e. free convection crrents conve heat awa from the plate into the bondar laer. Dring the nmerical calclations the phsical parameters are considered as Pr=.7 (diffsing air), Gr= (thermal boanc forces are dominant over the viscos hdrodnamic forces in the bondar laer), F=>(thermal radiation is dominant over the thermal condction), Ec=. (Enthalp difference is dominant over the kinetic energ). Figs. and illstrate the inflence of the heat absorption and porosit parameters and K, respectivel on the flow velocit. The effect is observed on velocit profile b increasing the vale of the heat absorption parameter, and the bondar laer thickness decreases with increase in the absorption parameter as shown in Fig., which is expected. The opposite trend is observed in Fig. for the case when the vale of the poros permeabilit is increased. As depicted in this figre, the effect of increasing the vale of poros permeabilit is to increase the vale of the velocit component in the bondar laer de to the fact that drag is redced b increasing the vale of the poros permeabilit on the flid flow which reslts in increased velocit. Fig. 4 depicts the effect of radiation on the flow velocit. We note from this figre that there is decrease in the vale of flow velocit with increase in radiation parameter F which shows the fact that increase in radiation parameter decrease the velocit in the bondar laer de to decrease in the bondar laer thickness. The effect of chemical reaction parameter is highlighted in Fig. which shows that the velocit decreases with increasing the rate of chemical reaction. Hence increase in the chemical reaction rate parameter leads to a fall in the momentm bondar laer. The trend of the velocit profile in this figre is same as shown in Fig. 4. The effect of absorption parameter () on flid temperatre () is presented in Fig. 6. A theoretical analsis of the stead magnetohdrodnamic flow and natral convection heat and mass transfer in a viscos, incompressible, electricall-condcting flid along a semi-infinite vertical plate immersed in a poros medim with thermal radiation has been condcted. The flow model has been setp for homogeneos chemical reaction of first-order in the presence of Ohmic heating and viscos dissipation. The nonlinear and copled governing eqations are solved analticall b pertrbation techniqe. Analtical soltions sing the method of complex variables have been derived. Above investigation reveals the following facts: It is seen that the velocit starts from minimm vale of zero at the srface and increases till it attains the peak vale and then starts decreasing ntil it reaches the minimm vale at the end of the bondar laer. Increasing heat absorption acts to decelerate the flow velocit in the bondar laer. Flow velocit is accelerated with increasing porosit IJRITCC September 4, parameter in the poros regime. It is seen that with an increase in heat absorption of the stead motion, the temperatres are decreased For the stead state case, there is a strong redction in the concentration distribtion for the effect of generative chemical reaction. NOMENCLATURE B Uniform magnetic field C* Species concentration (Kg. m ) C P Specific heat at constant pressre (J. kg. K) C Species concentration in the free stream (Kg.m ) C w Species concentration at the srface (Kg.m ) D Chemical moleclar diffsivit (m.s ) Ec Eckert nmber/dissipative heat F radiation parameter g Acceleration de to gravit (m.s ) Gr Thermal Grashof nmber Gm Mass Grashof nmber Chemical reaction parameter K porosit parameter M Hartmann nmber/magnetic parameter N x Local Nsselt nmber Pr Prandtl nmber q* Heat flx per nit area Re x Local Renolds nmber Sc Schmidt nmber T* Temperatre (K) T Flid temperatre at the srface (K) Flid temperatre in the free stream (K) T 799

5 International Jornal on Recent and Innovation Trends in Compting and Commnication ISSN: -869 Volme: Isse: Dimensionless velocit component in x-direction (m. [8] V. M. Sondalgekar, S. K. Gpta, and N. S. Birajdar, Effects of mass s transfer and free convection crrents on MHD Stokes problem for a ) vertical plate, Nclear Eng. Des., vol., pp. 9-46, 979. * dimensional velocit along x* direction [9] V. M. Sondalgekar, M. R. Patil, and M. D. Jahagirdar, MHD Stokes v* dimensional velocit along * direction v Dimensionless sction velocit (m. s problem for a vertical plate with variable temperatre, Nclear E ng. ) Des., vol. 64, pp. 9-4, 98. Greek smbols flid thermal diffsivit β T coeff. of volme expansion for heat transfer (K ), β C coeff. of volme expansion for mass transfer ( ) Chemical reaction parameter θ Dimensionless flid temperatre (K), κ Thermal condctivit (W. m. K ), μ Coefficient of viscosit (kg. m ) ν Kinematic viscosit (m.s ), σ Electrical condctivit (VA m ), τ w wall shearing stress (N. m ) φ Dimensionless species concentration (Kg.m ψ heat sorce parameter Sbscripts w conditions on the wall conditions at the free stream REFERENCES [] B. Q. Li, and R. He, Comptational modeling of electrodnamic and transport phenomena in materials processing sstems, Int. J. Applied Electromagnetics and Mechanics, vol. 4, pp ,. [] T. Ando, K. Ueno, S. Tanigchi and T. Takagi, Indction pmp for high-temperatre molten metals sing rotating twisted magnetic field: Thrst measrement experiment with solid condctors, IEEE Trans. Magnetics, vol. 8, pp , 998. [] V. Bojarevics, and K. Pericleos, Liqid metal indction heating modelling for cold crcible applications, Applied Mathematical Modelling, vol., pp , 998. [4] D. J. 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6 International Jornal on Recent and Innovation Trends in Compting and Commnication ISSN: -869 Volme: Isse: E Fig. : Velocit distribtion for heat absorption ( : Velocit distribtion for porosit (K.6 F Fig. 4: Velocit distribtion for radiation (F Fig. : Velocit distribtion for chemical reaction ( Fig. 6: Temperatre for heat absorption (Fig. 7: Temperatre for chemical reaction ( IJRITCC September 4, 8