EFFECT OF NON-LINEAR DENSITY VARIATION ON CONVECTIVE HEAT AND MASS TRANSFER WITH THIRD ORDER BOUNDARY CONDITIONS

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1 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, 015 EFFE OF NON-LINEAR DENSIY VARIAION ON ONVEIVE HEA AND MASS RANSFER WIH HIRD ORDER BOUNDARY ONDIIONS K.Sree Ranga Vani 1 and D.R.V. Prasada Rao 1 Department o Mathematics, Sri Satha Sai Institte o Higher Learning, Anantapram,A.P. Department o Mathematics, Sri Krishnadevaraa Universit, Anantapram, A.P., India ABSRA In this paper, e analze the combined inlence o thermal radiation, Soret, Dor eects, heat sorces on convective heat and mass transer lo o a viscos,electricall condcting, chemicall reacting lid past a vertical plate ith a convective srace bondar condction. he governing eqations are transormed b emploing similarit transormations and the resltant non-dimensional eqations are solved nmericall sing Rnge-Ktta method along ith Shooting techniqe. he eects o varios parameters velocit, temperatre, concentration,sin riction, Nsselt nmber and Sherood nmber are ehibited in igres,tables and analze in detail. Keords :hermal Radiation, Radiation Absorption, Soret Eect and Dor eects, Dissipation NOMENLAURE: M Magnetic parameter= e H o U g, Gr hermal Grasho nmber= } U N Boanc ratio= Dm D Dor parameter = Dm, Sr Soret nmber= s p,,ec Ecert Nmber= m p U Pr Prandtl nmber= p h, Bi onvective heat transer parameter= } U Sc Schmidt nmber=, Kr hemical reaction parameter= r D m U 11

2 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, 015 U Re Renolds nmber= Q, Q Heat sorce parameter = 0 U p Q 1 Radiation absorption parameter= r U, v velocit components along and -aes respectivel lid temperatre, K thermal condctivit, g acceleration de to gravit, ree stream temperatre, dimensional stream nction, Gree smbols 3 R N 1 hermal radiation parameter= 4 lid concentration Dm mass disivit ree steam concentration K permeabilit o the poros medim p speciic heat at constant pressre hemical reaction parameter= r, Kinematic viscosit U electrical condctivit, e magnetic permeabilit, thermal and concentration epansions dimensionless temperatre, dimensionless concentration Similarit variable, Stream nction, plate srace concentration eponent 1.Introdction: Hdro magnetic bondar laers ith heat and mass transer over a lat srace are ond in a man engineering and geophsical applications sch as geothermal reservoirs, thermal inslation, cooling o nclear reactors. Man processes in engineering areas occr at high temperatre and a noledge o radiation heat transer becomes ver important or the design o the pertinent eqipment. Nclear poer plants, gas trbines and the varios proplsion devices or aircrat, missiles, satellites and space vehicles are eamples o sch engineering areas. Bestman [5] eamined the natral convection bondar laer ith sction and mass transer in a poros medim. His reslts conirmed the hpothesis that sction stabilizes the bondar laer and aords the most eicient method in bondar laer control et non. Abdl Sattar and Hamid Kalim [1] investigated the nstead ree convection interaction ith thermal radiation in a bondar laer lo past a vertical poros plate. 113

3 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, 015 Mainde [15,16,17] eamined the transient ree convection interaction ith thermal radiation o an absorbing-emitting lid along moving vertical permeable plate. Raptis [0] analzed the thermal radiation and ree convection lo throgh a poros medim b sing pertrbation techniqe. With regard to thermal radiation heat transer los in poros media, hamha [6] stdied the solar radiation eects on poros media spported b a vertical plate. Implsive los ith thermal radiation eects and in poros media are important in chemical engineering sstems, aerodnamic bloing processes and geophsical energ modeling. Sch los are transient and thereore temporal velocit and temperatre gradients have to be inclded in the analsis. Raptis and Singh [19] stdied nmericall the natral convection bondar laer lo past an implsivel started vertical plate in a Darcian poros medim. he thermal radiation eects on heat transer in magneto aerodnamic bondar laers has also received some attention, oing to astronatical re-entr, plasma los in astrophsics, the planetar magneto-bondar laer and MHD proplsion sstems. Abd-El-Nab et al [] have presented a inite dierence soltion o radiation eects on MHD nstead ree convection lo over a vertical poros plate. Shatei et al [1] have analzed the hermal Radiation and Boanc Eects on Heat and Mass ranser over a Semi-Ininite stretching Srace ith Sction and Bloing. Dlal Pal et al [7] have analzed nstead magneto hdrodnamic convective heat and mass transer in a bondar laer slip lo past a vertical permeable plate ith thermal radiation and chemical reaction. Rajesh et al [18] have considered the radiation eects on MHD lo throgh a poros medim ith variable temperatre or variable mass dision. he std o heat generation or absorption eects in moving lids is important in vie o several phsical problems sch as lids ndergoing eothermic or endothermic chemical reactions. he volmetric heat generation has been assmed to be constant or a nction o space variable. For eample, a hpothetical core disrptive accident in a liqid metal ast breeder reactor LMFBR cold reslt in the setting o ragmented el debris on horizontal sraces belo the core. he poros debris cold be satrated sodim coolant and heat generation ill reslt rom the radioactive deca o the el particlate. Vajravel and Hadjinicolao [] stdied the heat transer characteristics in the laminar bondar laer o a viscos lid over a stretching sheet ith viscos dissipation or rictional heating and internal heat generation. Hossain et al [1] stdied the problem o natral convection lo along a vertical av srace ith niorm srace temperatre in the 114

4 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, 015 presence o heat generation or absorption. Alam et al [3] stdied the problem o ree convection heat and mass transer lo past an inclined semi-ininite heated srace o an electricall condcting and stead viscos incompressible lid in the presence o a magnetic ield and heat generation. Had et al [11] stdied the problem o ree convection lo along a vertical av srace embedded in electricall condcting lid satrated poros media in the presence o internal heat generation or absorption eect. In all these stdies Soret/Dor eects are assmed to be negligible. Sch eects are signiicant hen the densit dierences eist in the lo regime. For eample, hen species are introdced at a srace in lid domain, ith dierentloer densit than srronding lid, both Soret and Dor eects can be signiicant. Also, hen heat and mass transer occr simltaneosl in a moving lid, the relation beteen the les and the driving potentials are o more intricate natre. It has been ond that an energ l can be generated not onl b temperatre gradients bt b composition gradients as ell.he energ l cased b a composition gradient is called Dor or dision thermo eect. On the other hand mass les can also be generated b temperatre gradients and this is called Soret or thermo-dision eect. he thermal disionsoret eect,or instance it has been tilized or isotope separation, and in mitre beteen gases ith ver lie moleclar eighth,h e and o medim eightn,air,the dision thermo Doreect as to be o a considerable magnitde sch that it can not be ignored. In vie o the importance o these above mentioned eects, Drsnaa and Wore [8] stdied dision thermo and thermal-dision eects in transient and stead natral convection rom a vertical srace. Kaossias and Williams [13] presented the same eects on mied ree-orced convective and mass transer bondar laer lo ith temperatre dependent viscosit. Angel et al [4] investigated the Dor and Soret eects on ree convection bondar laer lo over a vertical srace embedded in a poros medim. Gnanesar Redd et al [10] reported a Soret and Dor eects on stead MHD ree convection lo past a semi-ininite moving vertical plate in a poros medim ith viscos dissipation. Recentl Gangadhar [9] and Madhsdhan Rao [14] have investigated soret and dor eect on convective heat and mass transer past a vertical plate ith convective srace. In this paper e analze the combined inlence o thermal radiation, Soret, Dor eects on convective heat and mass transer lo o viscos, electricall condcting, chemicall reacting lid past a vertical plate ith a convective srace bondar condction. 115

5 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, Similarit soltion o hdro-magnetic heat and mass transer over a vertical plate. he governing eqations are transormed b emploing similarit transormations and the resltant non-dimensional eqations are solved nmericall sing Rnge-Ktta method along ith Shooting techniqe. he eects o varios parameters velocit, temperatre, concentration, sin riction, Nsselt nmber and Sherood nmber are ehibited in igres and tables and analze in detail.. Mathematical Analsis: Let s consider a stead, laminar, hdro magnetic convective heat and mass transer lo o a viscos electricall condcting chemicall reacting lid past a vertical plate. A niorm magnetic ield o strength H 0 is applied normal to the plate. Since the magnetic Renolds is ver small or most o the lid sed in indstrial applications, e neglect the indced magnetic ield in comparison to the applied magnetic ield. he densit variation and the eects o boanc are taen into accont in the momentm eqation Bossinesq s approimation. In addition, there is no applied electric ield and all o the Hall eects and Jole heating are neglected. We choose a rectanglar coordinate sstem O,,z ith -ais ling along the plate and the -ais normal to the all. he plate at =0 is nder a convective srace bondar condition hile the concentration on the all is constant o. he eqations governing the lo, heat and mass transer nder bondar laer approimations are Eqation o ontinit 0 v 1 Momentm eqation 1 g g g U B v v o Energ eqation 1 1 Q q D Q v R p p s m o 3 Dision eqation D D v r m m m 4

6 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, 015 Where,v are the velocit components along the -and -ais respectivel. and are the lid temperatre and concentration he bondar conditions at the plate srace and ar aa rom the plate are,0 v,0 0, h,0,0 A, U,,,, 5 In vie o the continit eqation.1 e deine the Stream nction as, v 6 A similarit soltion o eqations 1-6 is obtained b deining an independent variable and a dependent variable in terms o the stream nction are U, U 7 We introdce the non-dimensional temperatre and concentration as, 8 Where is the temperatre o the hot lid at the let srace o the plate. Sbstitting the eqations 6-8 in the eqations 1-5,e obtain 4 1 3N 1 1 M D 1 G N Pr Pr D Pr Ec D Sc ScSr rsc , 0 0, 0 Bi 0 1, 0 1, 1 1, 0 13 here prime denotes dierentiation.r.to.it is noteorth that the local parameters Bi, M, G and N in eqations 9-13 are nction o. Hoever, in order to have a similarit soltion all the parameters Bi, M, F, N, Sr. D, Ec mst be constant and e thereore assme h c 0.5 Where a,b,c and d are constants., a, b, c d

7 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, 015 he phsical qantities o interest in this problem are sin riction parameter 0.5 Re 0, the plate srace temperatre 0, Nsselt Nmber.5 N Re 0 0 and the Sherood Nmber Sh Re can be calclated. For local similarit case, integration over the entire plate is necessar to obtain the total sin riction, total heat and mass transer rate. 3. Soltion o the problem: We investigate the eect o non-linear densit temperatre variation on convective heat & mass transer lo o a viscos electricall condcting chemicall reacting lid past a vertical plate in the presence o dissipation, radiation absorption. he set o copled nonlinear governing bondar laer eqations - together ith the bondar conditions - are solved nmericall b emploing the Rnge-Ktta orth order method along ith shooting techniqe. First o all, higher order non-linear dierential eqations - are conventional in to simltaneos linear dierential eqations o irst order and the are rther transormed into initial vale problem b appling the shooting techniqe. he resltant initial vale problem is solved b sing Rnge-Ktta orth order techniqe. he step size n = 0.05 is stressed to obtain the nmerical soltion ith decimal place accrac as the criterion o convergence. From the process o nmerical comptation, the sin-riction oeicient, the Nsselt nmber and the Sherood nmber, hich are respectivel proportional to 11 0-ϴ 1 0 are also sorted ot and their nmerical vales o presented in a tablar orm. 4. Reslts and discssion: he governing eqations9-11 sbject to the bondar conditions1-13 are integrated as described in section.3.nmerical reslts are reported in tables.1.he Prandtl nmber Pr as taen to be Pr=0.7 hich corresponds to air, the vale o Schmidt nmbersc ere chosen to Sc=,,1.3,.01,representing dising chemical species o most common interest in air lie,h,h O,NH 3 and Propl Benzene respectivel. Attention is ocsed on positive vale o boanc parameters that is, Grasho nmber G>0hich corresponds to the cooling problem. he boanc ratio N>0 hen the boanc orces are in the same direction and N<0, hen the are in opposite directions. In order to benchmar or nmerical reslts, e have compared the plate srace temperatre 0 and the local heat transer rate at the plate srace 0 in the absence o both magnetic ield and boanc 118

8 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, 015 orces or varios vales o Bi ith those o Gangadhar [9] and ond them in good agreement as shon in table.1. he velocit component 1 is shon or dierent vales o,q 1, Ec, Sr, D. 1 redces in both the degenerating and generating chemical reaction casesig.1. An increase in the radiation absorption parameter Q 1 leads to a depreciation in 1 ig.4. Higher the dissipative heat larger 1 ig.7. Increase Sror decreasing D reslts in a depreciation in 1 at η=0ig. 10. he non-dimensional temperatre ϴ is shon or dierent parametric vales. We ollo the convention that the non-dimensional temperatre is positive/negative according as the actal temperatre is greater/lesser than. From igs.,5 &8 e ind that the actal temperatre redces ith increase in chemical reaction parameter and radiation absorption parameter Q 1 and enhances dissipative heat Ec. Increasing the Soret parameter Sror decreasing D leads to a depreciation in the actal temperatre ig.11.he concentration dissipation is shon or dierent parametric vales. he concentration is positive or all parametric vales. From igs.3 & 6 e notice that the actal concentration redces ith increase in and redces ith Q 1. Higher the dissipative heat Ec lesser the actal concentration ig.9. Increasing Soret parameter Sr or decreasing D reslts in a depreciation in ig.1. With reerence to the chemical reaction parameter, We ind that τ & Sh enhances and N redces in both degenerating and generating chemical reaction cases. An increase in Ec enhances N & Sh and redces τ at ῃ=0. An increase in the radiation absorption parameter Q 1 enhances τ, N & Sh at ῃ=0. Increasing soret parameter Sror decreasing Dor parameter D reslts in an enhancement in τ, N, Sh. table omparison: omparison ith Gangadhar013 or sin riction,nsselt Nmber N and Sherood Nmber or G=, N=1,Sc=,D=0.3,Ec=0.001,Sr=, Pr=0.71, Bi=

9 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, 015 Gangadhar 013 Present WorN 1 =0,Q=0,D -1 =0 M N Sh N Sh M =, Sc=1.3, N=1, Q=0.5, =0.01, Q 1=0.5, Ec=0.1, Sr=0.3, D=, Bi=0.5 = 0.5,1.5,.5 0. M =, Sc=1.3, N=1, Q=0.5, =0.01, Q 1=0.5, Ec=0.1, Sr=0.3, D=, Bi= = 0.5,1.5, Fig.1 : Variation o ith M =, Sc=1.3, N=1, Q=0.5, =0.01, Q 1=0.5, Ec=0.1, Sr=0.3, D=, Bi= Fig. : Variation o emperatre ith M =, Sc=1.3, N=1, Q=0.5, =0.5, =0.01, Ec=0.1, Sr=0.3, D=, Bi=0.5 = 0.5,1.5,.5 Q 1 = 0.5,1.5,.5,3.5, Fig.3 : Variation o oncentration ith Fig.4 : Variation o ith Q Q 1 = 0.5,1.5,.5,3.5, M =, Sc=1.3, N=1, Q=0.5, =0.5, =0.01, Ec=0.1, Sr=0.3, D=, Bi=0.5 Q 1 = 0.5, 1.5,.5,3.5,5 0. M =, Sc=1.3, N=1, Q=0.5, =0.5, =0.01, Ec=0.1, Sr=0.3, D=, Bi= Fig.5 : Variation o emperatre ith Q Fig.6 : Variation o oncentration ith Q 10

10 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, 015 Ec = 0.1, 0.3, 0.7, Ec = 0.1, 0.3, 0.7, M =, Sc=1.3, N=1, Q=0.5, =0.5, =0.01, Q 1=0.5, Sr=0.3, D=, Bi= M =, Sc=1.3, N=1, Q=0.5, =0.5, =0.01, Q 1=0.5, Sr=0.3, D=, Bi= Fig.7 : Variation o ith Ec Fig.8 : Variation o emperatre ith Ec Sr= 0.3,1,1.5, D=,,,0.3 M =, Sc=1.3, N=1, Q=0.5, K=0.5, =0.01, Q 1=0.5, Sr=0.3, D=, Bi= Ec = 0.1, 0.3, 0.7, M =, Sc=1.3, N=1, Q=0.5, =0.5, =0.01, Q 1=0.5, Bi=0.5, Ec= Fig.9 : Variation o oncentration ith Ec Fig. 10 : Variation o ¹ith Sr and D 1.5 M =, Sc=1.3, N=1, Q=0.5, =0.5, =0.01, Q 1=0.5, Bi=0.5,Ec=0.1 Sr= 0.3,1,1.5, D=,,, Sr= 0.3,1,1.5, D=,,, Fig. 11 : Variation o ith Sr and D 0. M =, Sc=1.3, N=1, Q=0.5, =0.5, =0.01, Q 1=0.5,Bi=0.5, Ec= Fig. 1 : Variation o ɸ ith Sr and D 11

11 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, Ec o0 N0 Sh Q Sr/D / / 1/ / able.1 omparision ith Madhsdhan Rao01 or sin riction,nsselt Nmber N and Sherood Nmber or G=0.1, N=1,Sc=,D=0.3,Ec=0.01,Sr=, Pr=0.71, Bi=0.1, D -1 =0.5, K r =0.5 Madhsdhana Rao01 Present WorN 1 =0,Q 0,D -1 0 M N Sh N Sh onclsions : he velocit component 1 and N redces and τ & Sh enhances in both the degenerating and generating chemical reaction cases. he actal temperatre and the actal concentration redces ith increase in chemical reaction parameter. An increase in the radiation absorption parameter Q 1 leads to a depreciation in 1, the actal temperatre and the actal concentration and an enhancement in τ, N & Sh at ῃ=0. Higher the dissipative heat larger 1, the actal temperatre, N & Sh and lesser the actal concentration and τ at ῃ=0. Increase Sr or decreasing D reslts in a depreciation in 1, the actal temperatre and the actal concentration and an enhancement in τ, N, Sh. 1

12 Mathematical heor and Modeling ISSN Paper ISSN 5-05 Online Vol.5, No.1, Reerences: 1. Abdl Sattar.M.D, Hamid Kalim.M.D 1996 : Unstead ree-convection interaction ith thermal radiation in a bondar laer lo past a vertical poros plate, J Math Phs Sci, V.30, pp: Abd El-Nab M.A, Elsaed M.E, Elbarabar and Nader Y.A.004 : Finite dierence soltion o radiation eects on MHD ree convection lo over a vertical poros plate, Appl. Maths omp. Vol.151 pp: Alam,M.S.,Rahman M.M. and Samad M.A006: Dor and Soret Eects on Unstead MHD Free onvection and Mass ranser Flo Past a Vertical Poros Plate in a Poros Medim, Nonlinear Analsis:Modelling and ontrol,113,pp: Anghel,M.,H.S., and Pop I 000: Dor and Soret eects on ree convection bondar laer over a vertical srace embedded in a poros medim, J.Heat and Mass ranser,v.43,pp: Bestman. A.R 1990 : Natral convection bondar laer ith sction and mass transer in a poros medim, Int J Energ Res;V.14,pp: hamha A.J 1997 : Solar Radiation Assisted natral convection in a niorm poros medim spported b a vertical heat plate, ASME Jornal o heat transer, V.19, pp Dlal Pal and Bablal aldar 010 : Pertrbation analsis o nstead magneto hdro dnamic convective heat and mass transer in a bondar laer slip lo past a vertical permeable plate ith thermal radiation and chemical reaction, ommnications in Nonlinear Science and Nmerical Simlation, Volme 15, Isse 7, Jl 010, P Drsnaa, Z. and Wore, W. M. 199: Dision-thermo and thermal dision eects in transient and stead natral convection rom a vertical srace, Int. J. Heat Mass ranser,v.35,pp: Gangadhar,K 013 : Soret and Dor Eects on Hdro Magnetic Heat and Mass ranser over a vertical plate ith a convective Srace Bondar ondition and hemical Reaction, J. Applied Flid Mechanics,V.6,pp: Gnanesar Redd. M., and Bhasar Redd. N., 010: Soret and Dor eects on stead MHD ree convection lo past a semi-ininite moving vertical plate in a poros medim ith viscos dissipation, Int. J. Appl. Math and Meth., 61, pp: Had F.M., Mohamed R.A., Mahd A.006 : MHD ree convective lo along a vertical av srace ith heat generation or absorption eect, Int. omm. Heat Mass transer, Hossain M.A., Molla M.M., Yaa L.S.004 : Natral convective lo along a vertical av srace temperatre in the presence o heat generation/ absorption, Int. J. hermal Science, V.43, pp: Kaossias,N.G.and Williams,E.M.,1995: hermal Dision and Dision-hermo eects on ree convective and mass transer bondar laer lo ith temperatre dependant viscosit, Int.J.Eng.Science,V.33,pp: Madhsdhan Rao 01: Soret and Dor eects on hdro-magnetic heat and mass transer over a vertical plate in a poros medim ith a convective srace bondar condition and chemical reaction, Int. J.Engg.Research and applications 4,pp: Mainde, OD.,010: Similarit soltion o hdro magnetic heat and mass transer over a vertical plate ith a convective srace bondar condition, Int.J.Ph.Sci.,56,pp: Mainde OD 005: Free convection lo ith thermal radiation and mass transer past moving vertical poros plate, Int omm Heat Mass ranser,v.3,pp: Mainde O.D. and Ogl, A.,008: he eect o thermal radiation on the heat and mass transer lo o a variable viscosit lid past a vertical poros plate permeated b a transverse magnetic ield,hem.eng.ommn.,1951,pp: Rajesh. V and Varma.S.V.K 010 : Radiation eects on MHD lo throgh a poros medim ith variable temperatre or variable mass dision, Int.J. o Appl.Math and Mech.61,pp: Raptis.A.A and Singh.A.K 1985: Free convection lo past an implsivel started vertical plate in a poros medim b inite dierence method, Astrophsics space science J, Vol. 11, pp: Raptis. A.A 1998 : Radiation and ree convection lo throgh a poros medim, Int. commn. Heat mass transer, Vol. 5, pp : Shatei.S 008: hermal Radiation and Boanc Eects on Heat and Mass ranser over a Semi-Ininite stretching srace ith Sction and Bloing, jornal o Applied mathematics,v.008,article id ,1pages.. Vajravel.K, Hadjinicolao.A1993 : Heat transer in a viscos lid over a stretching sheet ith viscos dissipation and internal heat generation, Int. omm. Heat Mass transer 0, pp: