Analysis of third-grade heat absorption hydromagnetic exothermic chemical reactive flow in a Darcy-forchheimer porous medium with convective cooling

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1 Anlyss of thrd-grde het sorpton hydromgnetc exothermc chemcl rectve flow n Drcy-forchhemer porous medum wth convectve coolng S.O. Slwu Deprtment of Mthemtcs, Fculty of Physcl Scences, Lndmrk Unversty, Omu-rn, Nger. e-ml:kunleslwu@gml.com Astrct The stu exmne oundry lyer non-newtonn flud, lmnr, vscous nd ncompressle het sorpton chemcl rectve flow wth symmetry convectve coolng n Drcy-forchhemer porous medum. The electrclly conductng flud flow s drven y therml uoyncy force nd xl pressure grdent long fxed chnnel. The convectve exchnge het wth the surroundng temperture t the wlls surfce follows Newtons lw of coolng. The solutons to the dmensonless nonlner equtons governng the flow re otned usng weghted resdul method (WRM). The computtonl ssessment of the nlytcl results n the oundry lyer s crred out nd the grphcl results for the momentum nd energy dstrutons re otned. The coeffcent of skn frcton nd Nusselt numer re lso showed nd dscussed ccordngly for some pertnent prmeters entrenched n the flow. From the result shows tht rse n Frnk-Kmenetsk prmeter needs to e gude ecuse t contrute sgnfcntly to the destructon of the system thermo-flud lso there s n ncrese n the flud ondng force tht mkes t to e more vscoelstc s the non-newtonn prmeter ncreses. Keywords: Hydromgnetc; Exothermc recton; Drcy-forchhemer; Het sorpton; Convectve coolng 1. Introducton A porous medum s chrcterzed y ts porosty nd other propertes of the medum lke tensle strength, tortuosty, electrcl conductvty, permelty tht cn e otned from ther respectve consttuents propertes (flud nd sold mtrx), ther pores structure nd med porosty re htully complex [1-]. Drcy descres the flow pst porous med tht forms the scentfc strtng pont of flud permelty used n erth scences, prtculrly n hydrogeology. Ths s phenomenologclly resultng from consttutve equton tht llustrtes the flud flow pst permele med. For lrge veloctes n porous medum, nertl specl effects cn ecome sgnfcnt. An nertl term s ntroduced to the Drcy s equton whch s referred to s Forchhemer term. Ths term s le to ccount for the nonlner ehvor of the pressure dfference versus velocty dt [,]. The de of porous med s used n severl prctcl res of scence nd engneerng such s fltrton, geomechncs, o-remedton, petroleum geology, mterl scence, ophyscs etc. Studes ssocted wth porous medum nd flows of n electrclly conductng flud hve fscnted mny reserchers ecuse of ther pplctons n numerous nturl nd technologcl processes. A revew of mgnetohydronmcs (MHD) studes s relted to technologcl felds ws estlshed y [,]. The sc nfluence of convectve het trnsfer n MHD ws exmned n [,9]. Whle, [11,1] reported on Joule hetng nd vscous nfluences on hydromgnetc flow wth het trnsfer. Specl weght ws gven to uoyncy forces on mgnetohydronmcs flow pst porous medum y [1,1]. The ove cted E-ISSN: - Volume 1, 1

2 work never tkes nto ccount the lkely effects of het sorpton. However, severl processes n engneerng hppen t hgh tempertures nd het sorpton long wth dfference n the flud vscosty n desgn of some equpment [1]. Also, Theoretcl stu of non-newtonn lquds hs n present tmes ttrcted the ttenton of mny reserchers due to ther ndustrl pplctons. Non-Newtonn fluds cnnot e presented wth sngle consttutve formulton, therefore the consttutve models for non-newtonn flud depends on the ctegores of fluds eng consdered whch cnnot e stsfctorly cptured y the lnerly vscoelstc clsscl model. Among the severl consttutve equtons nclude the clss of thrd order fluds tht cnnot e solve nlytclly even for the smple form of the flows nd therefore computtonl technque s nevtle. Brod studes concernng the non- Newtonn fluds hve een exmned y [1-1]. In [19], nvestgton of overll thermonmcs, unqueness nd stlty of the models for vrous knd of thrd grde flud eng n unusul nstnce nvolvng het lnce ws consdered. Anlyss reltng to energy nd speces trnsfer n vscoelstc thrd grde fluds hve een studed n []; nonetheless most of the reserch dd not exmne the thermonmcs spect n relton to comned effects of non-drcy, het sorpton, chemcl knetcs nd hydromgnetc of the flow system n chnnel wth convectve coolng on the surfces. However, the stu of rectve fluds s very sgnfcnt n understndng the het trnsfer ehvour of hydronmc lurcnts n engneerng systems. The gol of ths reserch stu s to exmne the thrd grde chemcl rectve hydromgnetc flud flow wth het sorpton n Drcyforchhemer porous med etween two fxed wll n the presence of unvryng mgnetc feld nd convectve coolng. The mthemtcl model s presented n secton. In secton, the weghted resdul technque s estlshed nd mplemented for the soluton process. In secton, oth the computtonl nd grphcl results re offered nd quntttvely explned sed on some exstng flud prmeters entrenched n the flow system.. Mthemtcl Formulton of the Model Consder convectve coolng exothermc chemcl recton, lmnr nd ncompressle thrd grde flud flow through non-drcy porous prllel horzontl chnnel medum s presented n Fgure 1. The non-newtonn model s employed to produce the vscoelstc effects. The flow s nduced y moleculr chemcl knetcs nd ssumed to e drven y oth the xl pressure grdent nd uoyncy force. The flow s ssumed to e long x -xs wth y -xs norml to the flow. The plte surfces re sujected to exchnge of het wth the ment temperture. To smplfy the model equtons, the Mxwell equtons of electromgnetsm s neglected y supposng tht the flud hs smll electrcl conductvty nd therefore tht constnt electromgnetc force s susequently produced. In ths cse, the densty vrton s pproxmted ccordng to the Boussnesq pproxmton. Followng [,1], nd gnorng the the flud rectve vscose consumpton, tme-dependent effects nd ssumed low mgnetc Reynolds numer. The momentum nd energy lnce equtons governng the flow re s follows: dp d u ν dx ν u u * * K K Fgure 1. Geometry of the flow σb u d u du α ρ gβ ( T T T ) =, (1) E-ISSN: - 1 Volume 1, 1

3 d T k du KT QCA e υl σb u ρ m = du ν α E RT Q ( T T ) () The oundry condtons mposed tkes the form: dt y = ; u =, k = h( T T ) () dt y = ; u =, k = h( T T ). * were u, P, T, T, K,, ν, ρ, l, β, g nd α re respectvely the flud xl velocty, flud pressure, flud temperture, ment temperture, Permelty of the porous medum, Forchhemer prmeter of the medum, knemtc vscosty, densty, chnnel wdth, expnsvty coeffcent, grvty nd mterl coeffcents. The terms k, Q, C, A, R, K, l, m, E, υ, Q nd h re the therml conductvty, het of recton, speces concentrton, constnt recton rte, constnt unversl gs, Boltzmnn s constnt, Plnck s numer, numercl exponent, ctvton energy, vrton frequency, het sorpton coeffcent nd het trnsfer coeffcent respectvely. Usng the dmensonless qunttes eqn. () on eqns. (1)-(), x u E( T T ) P x =, u =, θ =, P = ν RT ν RT Fs =, D =, =, * n * K K E dp αν gβrt G =, β =, Gr =, dx ν E ν e γ = QA E RT QEA Ce δ = RKT υl C RT QT RKe λ = E QAC E RT E RT m υl KT, y = KT υl m m y.,, B = h K Therefore, the governng equtons trnsforms to: d u d u du G H u β D u F u Grθ =, s d θ θ m 1 nθ δ (1 nθ ) e du du γ H u 1 β λθ =. () Wth oundry condtons s follows:, () () u =, u =, dθ = Bθ t y = 1 dθ = Bθ t y =. () where G, H, β, D, F s, Gr, δ, n, γ, λ nd B prmeter represent the pressure grdent, Hrtmnn numer, non-newtonn, Drcy, Forchhemer nert numer, therml Grshof numer, Frnk-kmenetsk prmeter, ctvton energy, vscous hetng, het E-ISSN: - Volume 1, 1

4 sorpton nd Bot numer respectvely.. Method of soluton The de of weghted resdul method see [] s to look for n pproxmte result, n the polynoml form to the dfferentl equton gven s D[ v( y) ] = f n the Aµ [ v] = γ µ on R. where [ v] domn R, () D represents dfferentl opertor reltng non-lner or lner sptl dervtves of the dependent vrles v, f s the functon of known poston, A µ [ v] denotes the pproxmte numer of oundry condtons wth R een the domn nd R the oundry. By ssumng n pproxmton to the soluton v ( y), n expresson of the form v( y) w( y, 1,,... n ). (9) whch depends on numer of prmeters 1,,... n nd s such tht for rtrry vlue ' s the oundry condtons re stsfed nd the resdul n the dfferentl equton ecome E( y, ) = L( w( y, )) f ( y). (1) The m s to mnmze the resdul E ( y, ) to zero n some verge sense over the domn. Tht s E y, W =, = 1,,,... n (11) Y ( ). where the numer of weght functons W s exctly the sme wth the numer of unknown constnts n w. Here, the weghted functons re chosen to e Drc delt functons. Tht s, W ( y) = δ ( y y ), such tht the error s zero t the chosen nodes y. Tht s, ntegrton of equton (11) wth W ( y) = δ ( y y ) results n E ( y, ) =. By pplyng WRM to equtons () to (), ssumng polynoml wth unknown coeffcents or prmeters to e determned lter, ths polynoml s clled the trl functon whch re defned s follow: n u( y) = y, θ ( y) = y. (1) = n = By mposng the oundry condtons () on the trl functons (1) s well s susttutng the trl functons nto equtons () nd () to otn the resdul: = G 9y y y u r y y y 1 y 1y y 1 9 y 1 y 9 y y H y y y y y y1... θ = 9y r 1 y y 9 y 1 9. y 1 y 9 y y δ n y y y y y y1 9 y y 1y (1)... (1) Mnmzng the resdul error to zero t some set of collocton ponts t regulr ntervl wthn the domn when Gr =., m =., B = 1, n =., β =., H = 1, λ =, γ =.1, G =., Fs =., δ =. ( ) k nd D =.. Tht s, y k = where N k = 1,,..., N 1 nd =, = 1, N = 1. The soluton re otned usng MAPLE softwre. Hence, the dmensonless momentum nd energy equtons ecomes 1 9 u =.y., y.1y.1y.y.y.9y.1y.y.y E-ISSN: - Volume 1, 1

5 θ =.y.1y. y.1y.1y.y.9y 1 (1).9y.y.9y.9 (1) The procedure for weghted resdul method s repeted for vryng vlues of the emedded prmeters. The other qunttes of engneerng nterest re the skn frcton (τ) nd the wll het trnsfer rte (Nu) defned s follows: 9 u θ flow velocty occurs. The energy profles for τ =, Nu = (1) y y vrton n the numercl exponent m s Tle 1: Comprsm of results when β =.1, presented n Fgure. The het rses wth respect to n ncrese n the vlues of m from. to G = 1, m = γ = n =.1, B = 1, 1.. The temperture mesure the verge rte of D = Fs = Gr = δ = H =, the knetc energy possessed y the flud y [] [] (ADM) Present prtcles nd the hgher the vrton of the perturton (WRM) prtcle, the more the flud temperture E-1. ncreses. Therefore, het s trnsported from the center lne tht ncreses the flud verge knetc energy whch leds to rse n the temperture felds E-1.. Results nd Dscusson To get cler nsght of the physcl results, t s necessry to crry out computtonl stu of the prolem for the momentum feld, energy feld, the skn frcton, nd the therml grdent numer. stu re compred well wth the present Weghted resdul method (WRM) n the specl cse of the prolem. The recton of vryng n vlues of the therml Grshof numer Gr on the momentum profle s shown n Fgure. It s notced tht rse n the vlues of reltve effect of the therml uoyncy force to the vscous hydronmc force n the oundry lyer cuses n ncrese n the velocty dstrutons. Ths s due to the fct tht the flud flow get wrmer s t moves through the chnnel wthn the oundry lyer nd therey decreses the flow resstnce forces tht resulted n n enhncement n the flud flow rte. For low uoyncy effects, the mxmum Tle 1 shows the comprsm of the present stu to specl cse of exstng stu. The exstng nlyss on the stu under dverse method of solutons re n good greement wth the present technque of soluton s presented n the tle. The numercl results otned usng perturton technque nd Adomn decomposton method (ADM) n the prevous Fgure. Effects of (G) on Velocty E-ISSN: - Volume 1, 1

6 Fgure. Effects of (m) on Temperture Fgure depcts the consequence of Bot numer B on the temperture felds. It s oserved from the therml oundry lyer condton () tht the hgher the Bot numer the greter the chnnel convectve coolng tht resulted n correspondng decrese n the surfce tempertures nd the ulk flud. The entre system temperture dmnsh wth rse n the prmeter vlues B s the lqud temperture contnully modfes to the sme temperture throughout the system. The reducton n the temperture ncreses the flud vscosty tht n turn retrds the flud momentum s the therml oundry lyer gets thnner. Fgure shows the effect of Hrtmnn numer H on the flow profles. It s seen tht the velocty felds reduces wth rse n the mgnetc feld prmeter H. The reducton n the profles s due to n nduced n the mgnetc feld n n electrclly conductng flud tht stmulte drg force known s Lorentz force whch ressts the flud moton s shown n the fgure. As result of the opposton to the flud moton due to Lorentz force, n ddtonl extr work s done tht chnges the therml energy. Fgure. Effects of (B) on Temperture Fgure. Effects of (H) on Velocty Fgure demostrtes the het profles for dverse vlues of the het sorpton prmeter λ. The result portry tht n ncrese n the vlues of λ cuses decrese n the energy dstrutons s expected. Ths s ecuse het s le to leve the system s the exothermc chemcl recton tkes plce wthn the chnnel therey reduces the therml oundry lyer thckness s result, more het dffused out of the system nd cuses decrese n the temperture profle. Fgure represents the nfluence of vrtons n pressure grdent G on flud momentum. An ncrese n the prmeter vlues G results n n ncrese n the flud velocty.e. mxmum velocty s cheved s the pressure grdent rses whch men tht the greter the pressure pply on the flud n the chnnel, the fster the vscoelstc lqud flow. E-ISSN: - Volume 1, 1

7 Fgure. Effects of (λ) on Temperture Fgure. Effects of (β ) on Velocty Fgsures 9 nd 1 llustrte the effects of the Drcy nd Forchhemer prmeters D nd Fs respectvely on the velocty profles. It s notced from the fgures tht the velocty profles decrese wth n ncrese n the vlues of D or Fs respectvely. Ths s due to the fct tht the porosty prmeters ntroduces lner or second order qudrtc drg nto the flud n the chnnel y cusng reducton n the flow velocty rte wthn the oundry lyer whch then reduces the velocty dstruton. Fgure. Effects of (G) on Velocty Fgure portrys the response of vscoelstc prmeter β on the velocty dstruton. From the fgure, t seen tht n ncrese n the non- Newtonn prmeter reduces the flud flow rte. Ths s due to n ncrese n the flud prtcle ondng force tht mkes the flud to e more vscoelstc. Therefore, the flud momentum dstruton n the system dmnshes. The descendng trend s due to the mlnce etween the convectve coolng nd nonlner het t the surfces s the vscoelstc prmeter ncreses. Fgure 9. Effects of (D) on Velocty E-ISSN: - Volume 1, 1

8 s mgnetc feld prmeter vlues H ncreses ut ncreses s t move fr wy from the oundry surfce whle n opposte effect s notced when the pressure grdent term G s enhnced. The skn frcton rses wthn the rnge y. nd reduces wthn the rnge. y 1 s the vlue of G ncreses. Fgure 1. Effects of (Fs) on Velocty Fgures 11 represents the recton of the temperture to vrtonl ncrese n the Frnk- Kmenetsk prmeter δ. The fgures show tht n ncrese n the recton prmeter enhnces energy rte n the chnnel. Tht s, the nternl het generton rses s the rectng regents s enhnces. The exothermc chemcl recton ncreses the het trnsfer rte from the comuston regon to the coolng surfce. Furthermore, het s trnsfer over the flud to melt the flud vscosty n other to rse the collson of prtcles; smlrly, extr het s generted y the ntercton of vscous flud prtcle tht n turn ncreses the profle. Fgure 1. Effects of (H) on Skn frcton Fgure 1. Effects of (G) on Skn frcton Fgure 11. Effects of (δ ) on Temperture The responses of the skn frcton to n ncrese n the prmeter vlues H nd G respectvely re llustrted n the Fgures 1 nd 1. It s oserved tht the skn frcton ntlly decreses The therml grdent effect ntlly ncreses nd lter decreses s t move dstnce wy from the wll s the Frnk-Kmenetsk prmeter δ rse s dsplyed n Fgure 1 s result of respectve ncrese nd reducton n the therml oundry lyers. Whle converse effect s experenced when the vlues of the het sorpton s ncreses. From the fgure, erly reducton n the effect s notced ut lter E-ISSN: - Volume 1, 1

9 ncreses s t keep dstnce from the oundry wll n the rng. y 1 s presented n Fgure 1 t y 1. Fg. 1. Effects of (δ ) on Nusselt numer Fgure 1. Effects of (λ) on Nusselt numer. Concluson The nfluences of het trnsfer on thrd grde exothermc chemcl rectve flud flow pst non-drcy porous medum wth het sorpton hve een studed. The formulted equtons for the flow re non-dmensonlsed nd solved usng weghted resdul method (WRM) to get the velocty nd temperture dstruton s well s the skn frcton nd Nusselt numer. Clculted results re represented grphclly to show the mportnt of some prmeters on the flow. It s oserved tht: () rse n Frnk-Kmenetsk prmeter needs to e gude ecuse t contrute sgnfcntly to the destructon of the system thermo-flud whle the Drcy nd Forchhemer prmeters resst the free flow of vscoelstc flud profle. () An ncrese n het sorpton nd Bot numer retrds temperture dstruton nd ncreses the flud ondng force tht result n slow movement of the non-newtonn lqud. () An ncrese n the vscoelstc of the flud hs sgnfcnt effects on the flow flud. Reference [1] T. Dutt, Frctl pore structure of sedmentry rocks: Smulton y llstc deposton, Journl of Geophyscl Reserch: Sold Erth, (), pp.1. [] M. K. Hed, H. S. Wong nd N. R., Buenfeld, Chrcterston of Hdley Grns y Confocl Mcroscopy, Cement &Concrete Reserch, Vol., No., (), pp [] S. Peng, Q. Hu, S. Dultz nd M. Zhng, Usng X-ry computed tomogrphy n port structure chrcterzton for Bere sndstone: Resoluton effect, Journl of Hydrology, (1), pp.-1. [] nd M. S. Dd, Rdtve het trnsfer of vrle vscosty nd therml conductvty effects on nclned mgnetc feld wth dsspton n non- Drcy medum, Journl of the Ngern Mthemtcl Socety, Vol., (1), pp.9-1. [] R. A. Kreem nd S. O.Slwu, Vrle vscosty nd therml conductvty effect of soret nd dufour on nclned mgnetc feld n non-drcy permele medum wth dsspton, Brtsh Journl of Mthemtcs & Computer Scence, Vol., No., (1), pp.1-1 [] R. Moreu, Mgnetohydronmcs, Dordrecht: Kluwer Acdemc Pulshers. (199). [] M. G. Red nd N. Sndee, computtonl modellng nd nlyss of het nd mss trnsfer n MHD flow pst the upper prt of prolod of revoluton, Eur. Phys. J. Plus, 1:, (1). [] O. D. Mknde nd T. Chnyok, Numercl nvestgton of trnsent het E-ISSN: - Volume 1, 1

10 trnsfer to hydromgnetc chnnel flow wth rdtve het nd convectve coolng, Commun Nonlner Sc Numer Smult, Vol. 1, (1), pp [9] M. A. Hossn, Vscous nd Joule hetng effects on MHD free convecton flow wth vrle plte temperture, Int J Het Mss Trnsfer, Vol., (199), pp.. [1] M. G. Red nd N. Sndeep, Free convectve het nd mss trnsfer of mgnetc o-convectve flow cused y rottng cone nd plte n the presence of nonlner therml rdton nd cross dffuson, Journl of Computtonl nd Appled Reserch n Mechncl Engneerng, Vol., (1), pp [11] nd E. O. Ftunm, Dssptve het trnsfer of mcropolr hydromgnetc vrle electrc conductvty flud pst nclned plte wth Joule hetng nd non-unform het generton, Asn Journl of Physcl nd Chemcl Scences, Vol., (1), pp.1-1. [1] O. D. Mknde nd P. Snd, Mgnetohydronmc mxed convectve flow nd het nd mss trnsfer pst vertcl plte n porous medum wth constnt wll sucton, Trns ASME J Het Trnsfer, Vol. 1, (), pp.-1. [1] M. G. Red, Het nd mss trnsfer on mgnetohydronmc perstltc flow n porous med wth prtl slp, Alexndr Engneerng Journl, Vol., (1), pp [1] V. Rvkumr, M. C. Rju nd G. S. S. Rju, Comned effects of het Asorpton nd MHD on convectve Rvln-Ercksen flow pst semnfnte vertcl porous plte wth vrle temperture nd sucton, An Shms Engneerng Journl, Vol., (1), pp.-. [1] S. Asghr, K. Hnf nd T. Hyt, Flow of thrd grde flud due to n Accelerted dsk, Interntonl Journl for Numercl Methods n Fluds, Vol., No., (1), pp.-9. [1] T. Hyt, E. Momont nd F. M. Mhomed, Perstltc MHD flow of thrd grde flud wth n endoscope nd vrle vscosty, Journl of Nonlner Mthemtcl Physcs, Vol. 1, No. 1, (), pp [1] O. D. Mknde, Therml stlty of rectve thrd grde flud n cylndrcl ppe: n explotton of Hermte-Pdé pproxmton technque, Appled Mthemtcs nd Computton, Vol. 19, (), pp.9-9. [1] K. R. Rjgopl, Boundry condtons for fluds of the dfferentl type: Nver stokes equtons nd relted non-lner prolems, Plenum Press, New York,, (199). [19] O. D. Mknde nd T. Chnyok, Numercl stu of unste hydromgnetc generlzed couette flow of rectve thrd-grde flud wth symmetrc convectve coolng, Computers nd Mthemtcs wth Applctons, Vol. 1, (11), pp [] T. Chnyok nd O. D. Mknde, Anlyss of trnsent Generlzed Couette flow of rectve vrle vscosty thrd-grde lqud wth symmetrc convectve coolng, Mthemtcl nd Computer Modellng Vol., (11), pp.1 1. [1] S. O. Adesny, J.A. Flde, S. Jngl, O. Anwr Be g, Irreverslty nlyss for rectve thrd-grde flud flow nd het trnsfer wth convectve wll coolng, Alexndr Engneerng Journl, Vol., (1), pp [] M. S. Dd, nd, Anlyss of het nd mss trnsfer of n nclned mgnetc feld pressuredrven flow pst permele plte, Appl. Appl. Mth., Vol. 1, No. 1, (1), pp.19-. E-ISSN: - 9 Volume 1, 1