EFFECT OF RADIATION ON NATURAL CONVECTION FLOW FROM A POROUS VERTICAL PLATE IN PRESENCE OF HEAT GENERATION

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1 EFFECT OF RADIATION ON NATURAL CONVECTION FLOW FROM A POROUS VERTICAL PLATE IN PRESENCE OF HEAT GENERATION Amen Ferdousi 1*, M. Mostfizur Rhmn, Mohmmd Slek Prvez 3, M. A. Alim 4 1 Fculty of EEE, Estern University, Dhk, Bngldesh Deprtment of CSE nd CIS, Dffodil Interntionl University, Dhk, Bngldesh 3 Deprtment of Nturl Sciences, Dffodil Interntionl University, Dhk, Bngldesh 4 Dprtment of Mthemtics, Bngldesh University of Engineering nd Technology, Dhk *Corresponding Author: men@esternuni.edu.d ABSTRACT The effects of Rdition on Nturl Convection Flow from Porous Verticl Plte in Presence of Het Genertion hve een presented here. The governing oundry lyer equtions re first trnsformed into non-dimensionl form nd the resulting nonliner system of prtil differentil equtions re then solved numericlly using finite difference method together with Keller-Box scheme. The numericl results of the surfce sher stress in terms of skin friction coefficient nd the rte of het trnsfer in terms of locl Nusselt numer, velocity s well s temperture profiles re shown grphiclly nd tulr form for selection of prmeters set of consisting of het genertion prmeter Q, rdition effect Rd, Prndtl numer Pr. Keywords: Rdition effect, Porous plte, Het genertion, Nturl convection. 1. INTRODUCTION The study of het genertion or sorption in moving fluids is importnt in prolems deling with chemicl rections nd those concerned with dissociting fluids. Possile het genertion effects my lter the temperture distriution; consequently the prticle deposition rte in nucler rectors, electronic chips nd semiconductor wters. is constnt, wheret w >T. Here T is the mient temperture of the fluid, T is the temperture of the fluid in the oundry lyer, g is the ccelertion due to grvity, the fluid is ssumed to e grey emitting nd soring, ut non scttering medium. In the present work following ssumptions re mde: Vritions in fluid properties re limited only to those density vritions which ffect the uoyncy terms. The effect of rdition on free convection hs een drwn forth not only for its fundmentl spects ut lso for its significnce in the contexts of spce technology nd processes involving high temperture. In the presence of het genertion, nturl convection oundry lyer flow from porous verticl plteof stedy two dimensionl viscous incompressile fluid nd the rdited het trnsfer hs een investigted. In this nlysis considertion hd een given to grey gses tht emit nd sor ut do not sctter therml rdition. Over the work it is ssumed tht the surfce temperture of the porous verticl plte T w, The rditive het flux in the x-direction is considered negligile in comprison with tht in the y direction, where the physicl coordintes (u, v) re velocity components long the (x, y) xes. Vjrvelu nd Hdjinicolou[1] studied the het trnsfer in viscous fluid over stretching sheet with viscous dissiption nd internl het genertion. In this study, they considered tht the 3 volumetric rte of het genertion q m [ W / m ] should e:

2 Q q m ( T T ) for T T for T T where Q is the het genertion constnt. The ove reltion explined is vlid s n pproximtion of the stte of some exothermic process nd hving T s the onset temperture. When the inlet temperture is not less thn T they used Q (T- T ). Merkin[] studied free convection with lowing nd suction. Lin nd Yu[3] studied free convection on horizontl plte with lowing nd suction. Hossin et l[4] studied the effect of rdition on free convection flow with vrile viscosity from porous verticl plte. Hossin et [5]l studied flow of viscous incompressile fluid with temperture dependent viscosity nd therml conductivity pst permele wedge with vrile het flux. Hossin nd Tkhr[6] studied rdition effect on mixed convection long verticl plte with uniform surfce temperture. Moll et l.[7] studied nturl convection flow long verticl wvy surfce with uniform surfce temperture in presence of het genertion/sorption. Akhter[8] studied the effect of rditions on free convection flow on sphere with isotherml surfce nd uniform het flux. Ali[9] studied the effect of rdition on free convection flow on sphere with het genertion. Mkinde nd Moitsheki [1] studied on nonperturtive techniques for therml rdition effect on nturl convection pst verticl plte emedded in sturted porous medium. Mkinde nd Ogulu[11] studied the effect of therml rdition on the het nd mss trnsfer flow of vrile viscosity fluid pst verticl porous plte permeted y trnsverse mgnetic field. Ogulu nd Mkinde [1] studied unstedy hydromgnetic free convection flow of dissiptive nd rditing fluid pst verticl plte with constnt het flux. Hossin et l. [13] studied the effect of rdition on free convection flow from porous verticl plte. They [4] nlyzed full numericl solution nd found, n increse in Rdition prmeter cuses to thin the oundry lyer nd n increse in surfce temperture prmeter cuses to thicken the oundry lyer. The presence of suction ensures tht its ultimte fte if verticlly incresed is lyer of constnt thickness. None of the forementioned studies, considered the het genertion effects on lminr oundry lyer flow of the fluids long porous plte with rdition het loss. The present study dels with effects of rdition on nturl convection flow from porous verticl plte in presence of het genertion. The results will e otined for different vlues of relevnt physicl prmeters nd will e shown in grphs s well s in tles. The governing prtil differentil equtions re reduced to loclly non-similr prtil differentil forms y dopting some pproprite trnsformtions. The trnsformed oundry lyer equtions re solved numericlly using implicit finite difference scheme together with the Keller ox technique[14]. Here, we hve focused our ttention on the evolution of the surfce sher stress in terms of locl skin friction nd the rte of het trnsfer in terms of locl Nusselt numer, velocity profiles s well s temperture profiles for selected vlues of prmeters consisting of het genertion prmeter Q, Prndtl numer Pr nd the rdition prmeter. In order to check the ccurcy of our numericl results the present results re compred with[13].. FORMULATION OF THE PROBLEM We hve investigted the effects of rdition on nturl convection flow from porous plte in presence of het genertion. The fluid is ssumed to e grey, emitting nd soring ut non scttering medium. Over the work it is ssumed tht the surfce temperture of the porous verticl plte, T w, is constnt, where T w > T. The physicl configurtion considered is s shown in Fig.1. The conservtion equtions for the flow chrcterized with stedy, lminr nd two dimensionl oundry lyers; under the usul Boussinesq pproximtion, the continuity, momentum nd energy equtions cn e written s: u v x y u u u ( u v ) g ( T T ) u x y y T T T c ( u v ) k qr p x y y y (1) () (3) 1

3 With the oundry conditions x, y, u, T T. y, x, u, vv, T Tw y, x, u, T T (4) where is the density, is the strength of mgnetic field, is the electricl conduction, k is the therml conductivity, is the coefficient of therml expnsion, is the reference kinemtic viscosity = /, is the viscosity of the fluid, C p is the specific het due to constnt pressure nd q r is the rditive het flux in the y direction. 1 Vy 4x 4, V gt 3 3 V g T f TT 4 T T, w T 4 3 w, 1 T w 1 T w T w w, Rd T T T T k( s ) (6) Where, is the non-dimensionl temperture function, w is the surfce temperture prmeter nd is the rdition prmeter. Sustituting (6) into Equtions (1,, 3) leds to the following non-dimensionl equtions f f 3ff f f f f f v f pr 3 Rd w f f f (7) (8) Where Pr=C p /k is the Prndtl numer is the het genertion prmeter nd M= / is the mgneto hydrodynmic prmeter. Figure1.The coordinte system nd the physicl model The oundry conditions (4) ecome f, f, 1 t f, s (9) In order to reduce the complexity of the prolem nd to provide mens of comprison with future studies tht will employ more detil representtion for the rditive het flux; we will consider the opticlly thick rdition limit. Thus rdition het flux term is simplified y the Rosselnd diffusion pproximtion [13] nd is given y 4 T 4 q r (5) 3 y r s In Eqution (5) r is the Rosselnd men sorption co-efficient, s is the scttering coefficient nd is the Stephn-Boltzmn constnt. Now introduce the following non-dimensionl vriles: The solution of equtions (6), (8) enle us to clculte the non dimensionl velocity components u,v from the following expressions u u f (, ) Vg ( Tw T ) (1) 1 f v (3 f f ) V In prcticl pplictions, the physicl quntities of principle interest re the shering stress w nd the rte of het trnsfer in terms of the skin-friction coefficients C fx nd Nusselt numer Nu x respectively, which cn e written s V Nux ( qc qr ), C fx (11) VT gt

4 where u T w nd qc k y y q c is the conduction het flux. (1) Using the Equtions (6) nd the oundry condition (9) into (11 nd 1), we get C f x, fx Nu Rdw x, x 3 (13) Velocity profiles.5.4 Q = 17.9 Q = 15. Q = 1. Q = 5. Q =. Pr = 1, =, The vlues of the velocity nd temperture distriution re clculted respectively from the following reltions: u f (, ), x, y 3. METHOD OF SOLUTION Solutions of the locl non similr prtil differentil eqution (7) to (8) sujected to the oundry condition (9) re otined y using implicite finite difference method with Keller-Box Scheme[14], which hs een descried in detils y Ceeci[15]. 4. RESULTS AND DISCUSSION In this exertion the effects of rdition on nturl convection flow on porous verticl plte in presence of het genertion is investigted. Numericl vlues of locl rte of het trnsfer re clculted in terms of Nusselt numer Nu x for the surfce of the porous verticl plte from lower stgntion point to upper stgntion point,for different vlues of the forementioned prmeters nd these re shown in tulr form in Tle:1 nd Tle: nd grphiclly in Fiqure 6-9. The effect for different vlues of het genertion prmeter Q on locl skin friction coefficient C fx nd the locl Nusselt numer Nu x, s well s velocity nd temperture profiles re displyed in Fig. nd 6.The im of these figures re to disply how the profiles vry in, the selected streetwise coordinte. 3 Temperture profiles Q = 17.9 Q = 15. Q = 1. Q = 5. Q =. Pr = 1, = Figure. () Velocity nd () temperture profiles for different vlues of het genertion prmeter Q with others fixed prmeters. Figures ()-() disply results for the velocity nd temperture profiles, for different vlues of het genertion prmeter Q with Prndtl numer Pr = 1., rdition prmeter = nd surfce temperture prmeter w = 1.1. It hs een seen from Figures () nd () tht s the het genertion prmeter Q increses, the velocity nd the temperture profiles increse. The chnges of velocity profiles in the direction revels the typicl velocity profile for nturl convection oundry lyer flow, i.e., the velocity is zero t the oundry wll then the velocity increses to the pek vlue s increses nd finlly the velocity pproches to zero (the symptotic vlue). The mximum vlues of velocity re recorded to e 59, 874, 6866 nd for Q=., 5., 1., 15. respectively which occur t the sme point =.8353 nd for Q=17.9, the mximum vlues of velocity re recorded to e Here, it is oserved tht t =.97931,

5 the velocity increses y 16.8%s the het genertion prmeter Q chnges from. to 15.. The chnges of temperture profiles in the direction lso shows the typicl temperture profile for nturl convection oundry lyer flow tht is the vlue of temperture profile is 1. (one) t the oundry wll then the temperture profile decreses grdully long direction for the vlue Q less then 1. to the symptotic vlue. But for Q 1. the temperture profile increses (t = temperture is.416 for Q = 17.9) nd gin it decreses grdully long direction to the symptotic vlue. Velocity profiles = = = =.5 =. Pr = 1, Q =., Velocity profiles Temperture profiles Pr = 1, =, Q =. Pr = 1, =, Q =. Temperture profiles = = = =.5 =. Pr = 1, Q =., Figure 3. () Velocity nd () temperture profiles for different vlues of rdition prmeter with others fixed prmeters Figure 4. () Velocity nd () temperture profiles for different vlues of het flux prmeter θw with others fixed prmeters The effect for different vlues of rdition prmeter the velocity nd temperture profiles in cse of Prndtl numer Pr = 1., het genertion prmeter Q =. nd surfce temperture prmeter w = 1.1 re shown in Figures 3()-3(). Here, s the rdition prmeter increses, the velocity profile increses nd the temperture profile increses slightly such tht there exists locl mximum of the velocity within the oundry lyer, ut velocity increses ner the surfce of the verticl porous plte nd then temperture decreses nd finlly pproches to zero. 4

6 Velocity profiles Pr = 1.15 Pr = 1.1 Pr = 1. Pr =.9 Pr =.8 Q =., =, Skin friction.4 Q = 17.9 Q = 15. Q = 1. Q = 5. Q =. Pr = 1, =, Temperture profiles Figure 5. () Velocity nd () temperture profiles for different vlues of prndtl numer Pr with others fixed prmeters The effect of different vlues of surfce temperture prmeter w, the velocity nd temperture profiles while Prndtl numer Pr = 1., het genertion prmeter Q =. nd rdition prmeter = re shown in Figures 4()-4(). Here, s surfce temperture prmeter w increses, the velocity profile increses nd the temperture profile increses such tht there exists locl mximum of the velocity within the oundry lyer, ut velocity increses ner the surfce of the verticl porous plte nd then temperture decreses nd finlly pproches to zero. However, in Figures 5()-5(), it is shown tht when the Prndtl numer Pr increses with w = 1.1, = nd Q =., oth the velocity nd temperture profiles decrese. Pr = 1.15 Pr = 1.1 Pr = 1. Pr =.9 Pr =.8 Q =., =. Rte of het trnsfer Q = 17.9 Q = 15. Q = 1. Q = 5. Q =. Pr = 1, =, Figure 6. () Skin friction nd () rte of het trnsfer for different vlues of het genertion prmeter Q with others fixed prmeters. Figures 6()-6() show tht skin friction coefficient C fx increses nd het trnsfer coefficient Nu decrese respectively for incresing vlues of het genertion prmeter Q. in cse of Prndtl numer Pr = 1., rdition prmeter = nd surfce temperture prmeter w = 1.1. The vlues of skin friction coefficient C fx nd Nusselt numer Nu x re recorded to e 818, 769, 6844, 67, 537 nd.6579,.6661, ,.46974, for Q =17.9, 15., 1., 5. nd. respectively which occur t the sme point = 3. Here, it is oserved tht t = 3, the skin friction increses y 18.5% nd Nusselt numer Nu decreses y % s the het genertion prmeter Q chnges from 17.9 to.. The effect of different vlues of rdition prmeter on the skin friction coefficient nd the locl rte of het trnsfer while Prndtl numer Pr = 1., het genertion prmeter Q =. nd surfce temperture prmeter w = 1.1 re shown 5

7 in the figures 7()-7(). Here, s the rdition prmeter increses, the skin friction coefficient nd het trnsfer coefficient increse. Skin friction Rte of het trnsfer 3 1 = = = =.5 =. Pr = 1, Q =., Figure 7.() Skin friction nd () rte of het trnsfer for different vlues of rdition prmeter Rd with others fixed prmeters Pr = 1, Q =., = = = =.5 =. Rte of het trnsfer Figure 8. () Skin friction nd () rte of het trnsfer for different vlues of het flux prmeter θ w with others fixed prmeters From Figures 8()-8(), it cn lso esily e seen tht n increse in the surfce temperture prmeter w leds to increse in the locl skin friction coefficient C fx nd the locl rte of het trnsfernu x while Prndtl numer Pr = 1., het genertion prmeter Q =. nd rdition prmeter =. It is lso oserved tht t ny position of, the skin friction coefficient C fx increses nd the locl Nusselt numer Nu x increse s w increses from. to 3.. This phenomenon cn esily e understood from the fct tht when the surfce temperture prmeter w increses, the temperture of the fluid rises nd the thickness of the velocity oundry lyer grow, i.e., the therml oundry lyer ecome thinner thn the velocity oundry lyer. Pr = 1, =, Q =. Skin friction Pr = 1, =, Q =. The vrition of the locl skin friction coefficient C fx nd locl rte of het trnsfernu x for different vlues of Prndtl numer Pr for w = 1.1, Rd = nd Q =. re shown in Figures 9()-9(). We cn oserve from these figures tht s the Prndtl numer Pr increses, the skin friction coefficientdecresesnd rte of het trnsfer increse... 6

8 Skin friction Rte of het trnsfer 6 4 Pr = 1.15 Pr = 1.1 Pr = 1. Pr =.9 Pr =.8 Q =., =.4.4 Figure 9. () Skin friction nd () rte of het trnsfer for different vlues of Prndtl numer prmeter Pr with others fixed prmeters. Pr = 1.15 Pr = 1.1 Pr = 1. Pr =.9 Pr =.8 Q =., = Tle:1 Skin friction coefficient nd rte of het trnsfer ginst x for different vlues of het genertion prmeter Q with other controlling prmeterspr = 1., =, w =1.1. Q=. Q= C fx Nu x C fx Q=15. Q=17.9 C fx Nu x C fx Nu x COMPARISON OF THE RESULTS Numericl results of skin friction nd rte of het trnsfer re clculted from eqution (13) for the surfce of the porous plte from lower stgntion point to upper stgntion point t =.1 to =3. Numericl vlues of C fx ndnu x re depicted in Tle.1. Here in the elow tle the vlues of skin friction coefficient C fx nd Nusselt numer Nu x re recorded to e 818, 769, 67, 537 nd.6579,.6661,.46974, for Q=17.9, 15., 1., 5. nd. respectively which occur t the sme point = 3. Here, it is oserved tht t = 3, the skin friction increses y 18.5% nd Nusselt numer Nu x decreses y % s the het genertion prmeter Q chnges from 17.9 to w= 1.1 Hossin Present C fx Nu x C fx Nu x w =.5 Hossin Present C fx Nu x C fx Nu x

9 In order to verify the ccurcy of the present work, the vlues of Nusselt numer nd skin friction for Q =, =.5.Pr = 1. nd vrious surfce temperture =1.1, =.5 t different w position of re compred with Hossin et l. [13] s presented in Tle. The results re found to e in excellent greement. 6. CONCLUSION The effect of rdition on nturl convection flow on porous verticl plte in presence of het genertion hs een investigted for different vlues of relevnt physicl prmeters including Prndtl numer Pr, nd surfce temperture prmeter w. Significnt effects of het genertion prmeter Q on velocity nd temperture profiles s well s on skin friction nd the rte of het trnsfer hve een found in this investigtion ut the effect of het genertion prmeter Q on rte of het trnsfer is more significnt. An increse in the vlues of het genertion prmeter Q leds to increse oth the velocity nd the temperture profiles, the locl skin friction coefficient C fx increses t different position of nd the locl rte of het trnsfer Nu x decreses t different position of for < nd decrese symptoticlly when Pr=1.. The increse in the vlues of rdition prmeter leds to increse in the velocity profile, the temperture profile, the locl skin friction coefficient C fx nd the locl rte of het trnsfer Nu x. All the velocity profile, temperture profile, the locl skin friction coefficient C fx nd the locl rte of het trnsfer Nu x increses significntly when the vlues of surfce temperture prmeter w increse. The increse in Prndtl numer Pr leds to decrese in ll the velocity profile, the temperture profile, the locl skin friction coefficient C fx ut the locl rte of het trnsfer Nu x increse. Nomencltures r C f C p Rosselnd men sorption co-efficient Locl skin friction coefficient Specific het t constnt pressure w 8 f g k Nu x Pr Q q w q q c r Dimensionless strem function Accelertion due to grvity Therml conductivity Locl Nusselt numer Prndtl numer Het genertion prmeter Het flux t the surfce Conduction het flux Rdition het flux Rdition prmeter T Temperture of the fluid in theoundry lyer T Temperture of the mient fluid T w Temperture t the surfce ( uv, ) Dimensionless velocity components long V (x, y) the (x,y) xes Wll suction velocity Axis in the direction long nd norml to the surfce respectively Greek symols Equl to 4 R 3 d Coefficient of therml expnsion Equl to 1 T Equl to Tw T Similrity vrile w Dimensionless temperture function w Surfce temperture prmeter Viscosity of the fluid Kinemtic viscosity Similrity vrile s f w Suscripts w Density of the fluid Stephmn-Boltzmn constnt Scttering co-efficient AsoluteViscosity t the film temperture Coefficient of skin friction Shering stress Non-dimensionl strem function wll conditions Amient temperture

10 References [1] K. Vjrvelu nd A. Hdjinicolou, Het trnsfer in viscous fluid over stretching sheet with viscous dissiption nd internl het genertion, Int. Commun. Het Mss Trnsf., vol., no. 3, pp , My [] J.. Merkin, Free convection with lowing nd suction, Int. J. Het Mss Trnsf., vol. 15, no. 5, pp , My 197. [3] H.-T. Lin nd W.-S. Yu, Free Convection on Horizontl Plte With Blowing nd Suction, J. Het Trnsf., vol. 11, no. 3, pp , Aug [4] M. A. Hossin, K. Khnfer, nd K. Vfi, The effect of rdition on free convection flow of fluid with vrile viscosity from porous verticl plte, Int. J. Therm. Sci., vol. 4, no., pp , Fe. 1. [5] M. A. Hossin, M. S. Munir, nd D. A. S. Rees, Flow of viscous incompressile fluid with temperture dependent viscosity nd therml conductivity pst permele wedge with uniform surfce het flux, Int. J. Therm. Sci., vol. 39, no. 6, pp , Jun.. [6] M. A. Hossin nd H. S. Tkhr, Rdition effect on mixed convection long verticl plte with uniform surfce temperture, Het Mss Trnsf., vol. 31, no. 4, pp , Apr [7] M. M. Moll, M. A. Hossin, nd L. Shin Yo, Nturl convection flow long verticl wvy surfce with uniform surfce temperture in presence of het genertion/sorption, Int. J. Therm. Sci., vol. 43, no., pp , Fe. 4. [8] T. Akhter, Effect of Rdition on Nturl Convection Flow on Sphere with Isotherml surfce nd uniform Het Flux, Bngldesh University of Engineering nd Technology (BUET), Dhk, Bngldesh, 7. [9] M. M. Ali, Numericl Study of Rdition on Nturl Convection Flow on Sphere with Het Genertion, Bngldesh University of Engineering nd Technology (BUET), Dhk, Bngldesh, 7. [1] O. D. Mkinde nd R. J. Moitsheki, On Nonperturtive Techniques for Therml Rdition Effect on Nturl Convection pst Verticl Plte Emedded in Sturted Porous Medium, Mth. Prol. Eng., vol. 8, Oct. 8. [11] A. Ogulu nd O. D. Mkinde, Unstedy Hydromgnetic Free Convection Flow of Dissiptive nd Rditing Fluid Pst Verticl Plte with Constnt Het Flux, Chem. Eng. Commun., vol. 196, no. 4, pp , 8. [1] O. D. Mkinde nd A. Ogulu, The Effect of Therml Rdition on the Het nd Mss Trnsfer Flow of Vrile Viscosity Fluid Pst Verticl Porous Plte Permeted y Trnsverse Mgnetic Field, Chem. Eng. Commun., vol. 195, no. 1, pp , 8. [13] M. A. Hossin, M. A. Alim, nd D. A. S. Rees, The effect of rdition on free convection from porous verticl plte, Int. J. Het Mss Trnsf., vol. 4, no. 1, pp , Jn [14] H. B. Keller, Numericl Methods in Boundry- Lyer Theory, Annu. Rev. Fluid Mech., vol. 1, no. 1, pp , [15] T. Ceeci nd P. Brdshw, Physicl nd Computtionl Aspects of Convective Het Trnsfer. Springer,