MHD And Thermal Radiation Effects On Linearly Accelerated Isothermal Vertical Plate With Chemical Reaction Of First Order

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1 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December 1 MHD And hermal Radiaion Effecs On Linearly Acceleraed Isohermal Verical Plae Wih hemical Reacion Of Firs Order Geeha.E 1 Mhcmarasamy.R Assisan Professor Deparmen of Mahemaics Sri handrasekharendrasarasahivisamahavidyalaya niversiy Enahr Kanchipram India 1. Research Scholar Research and Developmen enremanonmaniam Sndaranar niversiy irnelveli Professor Deparmen of Applied Mahemaics Sri Venkaesara ollege of Engineering Irngakoai India. Absrac: heoreical solion of hermal radiaion and chemical reacion effecs on nsaedy flo pas an niformly acceleraed infinie isohermal verical plae ih niform mass diffsion nder he acion of ransversely applied magneic field has been presened. he plae emperare is raised o and he concenraion level near he plae is also raised o. he dimensionless governing eqaions are solved sing Laplace-ransform echniqe. he velociy emperare and concenraion fields are sdied for differen physical parameers like magneic field parameer radiaion parameer chemical reacion papermeer hermal Grashof nmber mass Grashof nmber Schmid nmber Prandl nmber and ime. I is observed ha he velociy increases ih increasing vales of hermal Grashof nmber or mass Grashof nmber. B he rend is js reversed ih respec o he chemical reacion parameer. I is also observed ha he velociy increases ih decreasing magneic field parameer or radiaion parameer. Keyords:Acceleraed isohermal radiaion verical plae hea and mass ransfer magneic field chemical reacion. MHD plays an imporan role in agriclre perolem indsries geophysics and in asrophysics. Imporan applicaions in he sdy of geological formaions in exploraion and hermal recovery of oiland in he assessmen of aqifersgeohermal reservoirs and ndergrond nclear ase sorage sies. MHD flo has applicaion in merology solar physics and in moion of earhs core. Also i has applicaions in he field of sellar and planeary magneospheres aeronaics chemical engineering and elecronics. In he field of poer generaion MHD is receiving considerable aenion de o he possibiliies i offers for mch higher hermal efficiencies in poer plans. Radiaive hea and mass ransfer play an imporan role in manfacring indsries for he design of fins seel rolling nclear poer plans gas rbines and varios proplsion device for aircraf missiles saellies and space vehicles are examples of sch engineering applicaions. England and Emery[1] have sdied he hermal radiaion effecs of a opically hin gray gas bonded by a saionary verical plae. Radiaion effec on mixed convecion along a isohermal verical plae ere sdied by Hossain and akhar[]. he governing eqaions ere solved analyically. Das e al[] have analyzed radiaion effecs on flo pas an implsively sared infinie isohermal verical plae. hemical reacions can be codified as eiher heerogeneos or homogeneos processes. his depends on heher hey occr a an inerface or as a single phase volme reacion. In ell-mixed sysems he reacion is heerogeneos if i akes place a an inerface and homogeneos if i akes place in solion. In mos cases of chemical reacions he reacion rae depends on he concenraion of he species iself. A reacion is said o be of firs order if he rae of reacion is direcly proporional o he concenraion iself. hambre and Yong [] have analyzed a firs order chemical reacion in he neighborhood of a horizonal plae. Das e al [5] have sdied he effec of homogeneos firs order chemical reacion on opyrigh o IJIRSE.ijirse.com 715 I.INRODION

2 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December 1 he flo pas an implsively sared infinie verical plae ih niform hea flx and mass ransfer. Again mass ransfer effecs on moving isohermal verical plae in he presence of chemical reacion sdied by Das e al [6]. he dimensionless governing eqaions ere solved by he sal Laplace-ransform echniqe. Gpa e al [7] sdied free convecion on flo pas an linearly acceleraed verical plae in he presence of viscos dissipaive hea sing perrbaion mehod. Kafosias and Rapis[8] exended he above problem o inclde mass ransfer effecs sbjeced o variable scion or injecion. Free convecion effecs on flo pas an acceleraed verical plae ih variable scion and niform hea flx in he presence of magneic field as sdied by Rapis e al[9]. MHD effecs on flo pas an infinie verical plae for boh he classes of implse as ell as acceleraed moion of he plae as sdied by Rapis and Singh[1]. Mass ransfer effecs on flo pas an niformly acceleraed verical plae as sdied by Sondalgekar[11]. Again mass ransfer effecs on flo pas an acceleraed verical plae ih niform hea flx as analyzed by Singh and Singh[1]. Basan Kmar Jha and Ravindra Prasad[1] analyzed mass ransfer effecs on he flo pas an acceleraed infinie verical plae ih hea sorces. Hence i is proposed o sdy hermal radiaion and hydromagneic effecs on nseady flo pas an niformly acceleraed infinie isohermal verical plae ih hea and mass ransfer in he presence of chemical reacion of firs order. he dimensionless governing eqaions are solved sing he Laplace-ransform echniqe. he solions are in erms of exponenial and complemenary error fncion. Sch a sdy fond sefl in chemical proceesing magneic conrol of molen iron flo in he seel indsry liqid meal cooling in nclear reacors and magneic sppression of molen semi-condcing maerials. II.MAHEMAIAL ANALYSIS he magneohydrodynamic flo of a viscos incompressible flid pas an niformly acceleraed isohermal verical infinie plae in he presence of hermal radiaion and chemical reacion of firs order has been considered. Here he nseady flo of a viscos incompressible flid hich is iniially a res and srronds an infinie verical plae ih emperare and concenraion. he x-axis is aken along he plae in he verically pard direcion and he. A ime y-axis is aken normal o he plae. A ime he plae and flid are a he same emperare he plae is acceleraed ih a velociy ' in is on plane agains graviaional field and he emperare from he plae is raised o and he concenraion levels near he plae are also raised o.i is assmed ha he effec of viscos dissipaion is negligible in he energy eqaion and here is a firs order chemical reacion beeen he diffsing species and he flid. A ransverse magneic field of niform srengh B is assmed o be applied normal o he plae. he flid considered here is a gray absorbing-emiing radiaion b a non-scaering medim. hen nder sal Bossinesq's approximaion he nseady flo is governed by he folloing eqaions: * B g ( ) g ( ) y p qr k y y () D y (1) K ( ' ) l () opyrigh o IJIRSE.ijirse.com 7151

3 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December 1 opyrigh o IJIRSE.ijirse.com 715 Wih he folloing iniial and bondary condiions: y as y a y all for > : () On inrodcing he folloing non-dimensional qaniies: v y Y v ) ( D Sc K K gv Gr l (5). Pr ) ( 16 k B M gv Gc k a R P he local radian for he case of an opically hin gray gas is expressed by * a y q r (6) I is assme ha he emperare differences ihin he flo are sfficienly small sch ha may be expressed as a linear fncion of he emperare. his is accomplished by expanding in a aylor series abo and neglecing higher-order erms hs (7) By sing eqaions (5) and (6) eqaion () redces o ) ( 16 * a y k p (8) in eqaions (1) o () leads o M Y Gc Gr (9) Pr 1 R Y Pr (1)

4 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December 1 1 K (11) Sc Y he iniial and bondary condiions in non-dimensional qaniies are > : 1 1 for all a as Y Y (1) Y III.SOLION PROEDRE he solions are in erms of exponenial and complemenary error fncion. he relaion connecing error fncion and is complemenary error fncion is as follos: erfc(x) 1- erfc(x)(1) he dimensionless governing eqaions (9) o (11) sbjec o he iniial and bondary condiions (1) are solved by he sal Laplace-ransform echniqe and he solions are derived as follos: 1 exp( R ) erfc Pr a exp( R ) erfc Pr a 1 exp( KSc ) erfc( Sc K ) exp( KSc ) erfc( Sc K ) (15) (1) opyrigh o IJIRSE.ijirse.com 715

5 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December 1 exp( M ) erfc( M ) d e exp( M ) erfc( M ) M exp( ( M c) ) erfc( ( M c) ) e exp( c) exp( ( M c) ) erfc( ( M c) ) exp( ( M b) ) erfc( ( M b) ) d exp( b) exp( ( M b) ) erfc( ( M b) ) exp( Pr( a b) ) erfc( Pr ( a b) ) d exp( b) exp( Pr( a b) ) erfc( Pr ( a b) ) exp( Sc( K c) ) erfc( Sc ( K c) ) e exp( c) exp( Sc( K c) ) erfc( Sc ( K c) ) e exp( exp( KSc ) erfc( M ) erfc( M ) exp( Sc K ) exp( M ) erfc( M ) KSc ) erfc( Sc K ) d exp R. erfc Pr a exp R. erfc Pr a (16) Where R M R M KSc Gr Gc a b c d e and Y Pr Pr1 Sc 1 b(1 Pr) c(1 Sc). IV. RESLS AND DISSSION For physical ndersanding of he problem nmerical compaions are carried o for differen physical parameers Gr Gc Sc K R M and pon he nare of he flo and ranspor. he vale of he Schmid nmber Sc is aken o be.6 hich corresponds o aer-vapor. he vale of Prandl nmber Prischosen sch ha hey represen air (Pr.71).he nmerical vales of he velociy emperare and concenraion are comped for he above menioned physical parameers. he concenraion profiles for differen vales of he chemical reacion parmeer (K. 5) Sc.6 and. are shon in Fig. 1. he effec of chemical reacion parameer play an imporan role in concenraion field. From Fig.1 I is observed ha for he larger vales of chemical reacion parameer (K 5) he concenraion profile rapidly decreases compare o he vale of K..Fig. illsraes he effec of he concenraion profiles for differen vales of he Schmid nmber ( Sc ) K. and..he profiles have he common feare ha he concenraion decreases in a monoone fashion from he srface o a zero vale far aay in he free sream. I shoed ha he beer improvemen of he concenraion profile for he smaller vale of Schmid nmber Sc.16 hich corresponds o aer-vapor compare o he larger vale of Schmid nmber ( Sc.6 Sc.1 ). he emperare profiles are calclaed for differen vales of hermal radiaion parameer( R. 5 ) a ime. and hese are shon in Fig.. he effec of hermal radiaion parameer is imporan in emperare profiles. I is opyrigh o IJIRSE.ijirse.com 715

6 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December 1 noiced ha he emperare of he plae eradically increases for he smaller vales of he hermal rediaion parameer R. R in he presence of air ( Pr.71 ) and i is beer han he vale of R 5. his rend shos ha here is a fall in plae emperare dr o higher hermal radiaion. he velociy profiles for differen vales of ime (...6) K Gr Gc 5 R 5 and M are sdied and presened in Fig.. I is noiced ha he rend shos ha for higher vales of ime.6. he velociy of he plae rapidly increases compare o he vale of..fig. 5. demonsraes he effecs of he hermal radiaion parameer on he velociy hen ( R 8 1 )M K5 Gr Gc5 and. in he presence of hermal radiaion. I is conclded ha he velociy profile of he plae repidly decreasing for he larger vales of hermal radiaion parameer R 1 and R 8 compare o he vale of hermal radiaion parameer R. Fig. 6. demonsraes he effec on he velociy fields for differen vales of he chemical reacion parameer ( K 51 ) Gr Gc 5 R 5 M and.. I is demonsraed ha for he larger vale of he chemical reacion parameer K 1 and K 5 he velociy profile eradically decreasing hile comparing o he smaller vale of K. Here his rend shos ha here is a fall in velociy de o increasing vales of he chemical reacion parameer. Fig.7. illsraes he effecs of he magneic field parameer on he velociy hen ( M 5) R7 K1 Gr Gc5 and.. Here here is a beer improvemen for he smaller vale of magneic field parameer M in he velociy of he plae compare o he larger vales of M and M 5. his shos ha he increase in he magneic field parameer leads o a fall in he velociy. his agrees ih he expecaions since he magneic field exers a rearding force on he free convecive flo. V.ONLSION he heoreical solion of hermal radiaion and hydromagneic flo pas an niformly acceleraed infinie isohermal verical plae in he presence of chemical reacion of firs order. he dimensionless governing eqaions are solved by he sal Laplace-ransform echniqe. he effec of differen parameers like hermal Grashof nmber mass Grashof nmber chemical reacion parameer radiaion parameer magneic field parameer and are sdied graphically. he conclsions of he sdy are as follos: (i) he concenarion of he plae increases ih decreasing vales of he chemical reacion parameer or Schmid nmber. ' p D Gc Gr g (ii) (iii) (iv) he plae emperare decreases ih increasing vales of he hermal radiaion parameer. he velociy increases ih increasing vales of he magneic field parameer or chemical reacion parameer or hermal radiaion parameer. B he rend is js reversed ih respec o he ime. species concenraion in he flid dimensionless concenraion all concenraion concenraion far aay from he plae specific hea a consan pressre mass diffsion coefficien mass Grashof nmber hermal Grashof nmber accelerraed de o graviy VI.NOMENLARE GREEK SYMBOLS opyrigh o IJIRSE.ijirse.com 7155

7 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December 1 k hermal condciviy K chemical reacion parameer M magneic field parameer Pr Prandl nmber R hermal radiaion parameer Sc Schmid nmber emperare of he flid near he plae emperare of he plae emperare of he flid far aay from he plae ' ime dimensionless ime velociy of he flid in he x-direcion x y Y * erfc velociy of he plae dimensionless velociy spaial coordinae along he plae coordinae axis normal o he plae dimensionless coordinae axis normal o he plae volmeric coefficien of hermal expansion volmeric coefficien of expansion ih concenraion cofficien of viscosiy kinemaic viscosiy densiy of he flid dimensionless skin-fricion kg. dimmensionless emperare similariy parameer complemenary error fncion REFERENES [1] EnglandW.G. and EmeryA.F. hermal radiaion effecs on he laminar free convecion bondary layer of an absorbing gas Jornal of Hea ransfer Vol.91 pp [] HossainM.A. and akharh.s. Radiaion effec on mixed convecion along a verical plae ih niform srface emperare Hea and Mass ransfer Vol.1 pp [] Das.N. DekaR.K. and SondalgekarV.M. Radiaion effecs on flo pas an implsively sared verical infinie plae Jornal of heoreical MechanicsVol.1 pp [] hambrep.l. and YongJ.D. On he diffsion of a chemically reacive species in a laminar bondary layer flo he Physics of Flids Vol.1 pp [5] Das.N.DekaR.K. and SondalgekarV.M. Effecs of mass ransfer on flo pas an implsively sared infinie verical plae ih consan hea flx and chemical reacion Forschng im Ingenieresen Vol.6 pp [6] Das.N. and DekaR.K. and SondalgekarV.M. Effecs of mass ransfer on flo pas an implsively sared infinie verical plae ih chemical reacion he Bllein of GMA.Vol.5 pp [7]Gpa A.S. Pop. I. and SondalgekarV.M. Free convecion effecs on he flo pas an acceleraed verical plae in an incompressible dissipaive opyrigh o IJIRSE.ijirse.com 7156

8 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December 1 flid Rev. Rom. Sci. echn.-mec. Apl. Vol. pp [8] KafosiasN.G. and Rapis A. Mass ransfer and free convecion effecs on he flo pas an acceleraed verical infinie plae ih variable scion or injecion Rev. Rom. Sci. echn.-mec. Apl. Vol.6 pp [9] RapisA.zivanidisG.J. and Peridikis.P. Hydromagneic free convecion flo pas an acceleraed verical infinie plae ih variable scion and hea flx Leers in hea and mass ransfer Vol.8 pp [1] RapisA. and SinghA.K. MHD free convecion flo pas an acceleraed verical plae-leers in Hea and Mass ransfer Vol. 8 pp [11] SondalgekarV.M. Effecs of mass ransfer on flo pas a niformlyacceleraed verical plae Leers in hea and mass ransfer Vol.9 pp [1] SinghA.K.SinghJ. Mass ransfer effecs on he flo pas an acceleraed verical plae ih consan hea flx Asrophysics and Space sciencevol.97 pp [1]Basanh Kmar Jha and Ravindra Prasad Free convecion and mass ransfer effecs on he flo pas an acceleraed verical plae ih hea sorces Mechanics Research ommnicaionsvol.17 pp c K Fig. 1.oncenraion profiles for differen vales of K opyrigh o IJIRSE.ijirse.com 7157

9 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December Sc Fig..oncenraion profiles for differen vales of Sc opyrigh o IJIRSE.ijirse.com 7158

10 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December Fig..velociy profiles for differen vales of opyrigh o IJIRSE.ijirse.com 7159

11 ISSN: Inernaional Jornal of Innovaive Research in Science Engineering and echnology (An ISO 97: 7 erified Organizaion) Vol. Isse 1 December K Fig. 6. Velociy profiles for differen vales of K opyrigh o IJIRSE.ijirse.com 716

E. GEETHA Department of Mathematics Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya University Enathur, Kanchipuram , INDIA

E. GEETHA Department of Mathematics Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya University Enathur, Kanchipuram , INDIA In. J. of Applied Mechanics and Engineering, 13, vol.18, No.3, pp.77-737 DOI: 1.478/ijame-13-44 CHEMICAL REACTION EFFECTS ON MHD FLOW PAST A LINEARLY ACCELERATED VERTICAL PLATE WITH VARIABLE TEMPERATURE