UNSTEADY COMBINED HEAT AND MASS TRANSFER FROM A MOVING VERTICAL PLATE IN A PARALLEL FREE STREAM

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1 Internatonal Journal of Energy & Technology.journal-enertech.eu ISSN 035-9X Internatonal Journal of Energy & Technology (9) (00) 3 UNSTEADY OMBINED HEAT AND MASS TRANSFER FROM A MOVING VERTIAL PLATE IN A PARALLEL FREE STREAM P. M. Patl* and Al J. hamkha** * Department of Mathematcs, J.S.S s Banashankar Arts, ommerce and S. K. Gubb Scence ollege, Vdyagr, Dharad , Inda. E-mal: pmpmath@gmal.com ** Manufacturng Engneerng Department, The Publc Authorty for Appled Educaton and Tranng, Shuekh, Kuat. E-mal: achamkha@yahoo.com ABSTRAT An unsteady med convecton flo over a movng vertcal plate n a parallel free stream s consdered to nvestgate the combned effects of buoyancy force, thermal and mass dffuson n presence of lnear chemcal reacton and heat generaton or absorpton. The unsteadness s caused by the tme dependent free stream velocty as ell as by the movng vertcal plate velocty. The governng boundary layer equatons are transformed nto a non-dmensonal form by a specal group of non-smlar transformatons. The resultng system of coupled nonlnear partal dfferental equatons s solved by the combnaton of a quas-lnearzaton technque and an mplct fnte dfference scheme. The obtaned results for specal case of the problem are compared th prevously reported ork and are found to be n ecellent agreement. Numercal computatons are performed for dfferent values of parameters and representatve velocty, temperature and concentraton profles are llustrated graphcally. In addton, representatve numercal results for the local skn-frcton coeffcent, local Nusselt number and the local Sherood number are presented and dscussed. KEYWORDS: Unsteady flo, med convecton, movng plate, parallel free stream, fnte dfference scheme, quas-lnearzaton.. INTRODUTION The flo, heat and mass transfer problem n the boundary layer nduced by a contnuously movng plate n a parallel free stream has more mportant practcal applcatons such as artfcal fbers, the aerodynamc etruson of plastc sheets, the coolng of an nfnte metallc plate n a coolng bath, the boundary layer along materal handlng conveyers, the boundary layer along a lqud flm n condensaton processes paper producton etc. The analyss of boundary layer flo nduced by a movng flat surface n an otherse ambent flud as frst ntated by Sakads [], ho analyzed t by both analytcal and numercal schemes. Remarkable results ere found beteen ths analyss and the analyss of the boundary layer on a statonary plate n a movng flud as consdered by Blasus []. The result found that the sknfrcton coeffcent s appromately about 30% hgher for the Sakads [] case compared to the Blasus case []. Tsou et al. [3] shoed epermentally that the flo and heat transfer problem from a contnuously movng surface s a physcally realzable one and studed ts basc characterstcs. Abdelhafez [4] eamned the boundary layer flo on movng flat plate n a parallel stream. He shoed that Blasus and Sakads problems are to specal case of hs problem. happd and Gunnerson [5] studed the lamnar boundary layer n to cases U U and U U separately and formulated to sets of boundary value problems. Afzal et al. [6] formulated a sngle set of boundary condtons by employng the composte reference veloctyu U U, here U s the movng plate velocty and U s the free stream velocty, nstead of consderng U and U separately as done by Abdelhafez [4], rrespectve of hether U U or U U. Ln and Haung [7] analyzed the case for horzontal sothermal plate movng n parallel or reversbly to a free stream here smlarty and non-smlarty equatons are used to obtan the flo and thermal felds. The detaled analyses of the problem of lamnar flud flo hch results from the combned motons of a boundng surface and free stream n the same drecton have been dscussed by Abraham and Sparro [8] and Sparro and Abraham [9] usng the relatve velocty model, hch uses the magntude of the relatve velocty n conjuncton th the drag formula for the case n hch only one of the partcpatng meda s n moton. Kare and Jalura [0, ] presented med convecton flo over a contnuous plate movng at a unform speed n materal processes, such as hot rollng, etruson and drang. Ingham [] nvestgated the estence of the sngular and non-unque solutons for the free convecton boundary layer flo near a contnuously movng vertcal plate th temperature nversely proportonal to the dstance along the plate. Al and Al- Yousef [3] dscussed the problem of lamnar med convecton flo adjacent to a unformly movng vertcal plate th sucton or njecton. The steady lamnar flo and heat transfer characterstcs of a contnuously movng vertcal sheet of etruded materal are studed close to and far donstream from the etruson slot by Al-Sanena [4]. Soundalgekar and Murty [5] studed the effects of poer la surface temperature varaton on the heat transfer from a

2 Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 contnuous movng surface th constant surface velocty. The effects of varable surface temperature and lnear surface stretchng ere nvestgated by Grubka and Bobba [6]. Smlarty solutons ere reported by Al [7] for the case of poer-la surface velocty and three dfferent thermal boundary condtons. Al [8] also etended hs ork for the stretchng surface subject to sucton or njecton. Moutsoglou and hen [9] consdered buoyancy effects on flo and heat transfer from an nclned contnuous sheet th ether unform all temperature or unform surface heat flu. More recently, ortell [0] etended the ork of Afzal et al. [6], by takng onto account vscous dsspaton effects n the energy balance. The effects of transpraton on the flo and heat transfer over a movng permeable surface n a parallel stream ere analyzed by Ishak et al. []. Ishak et al. [] eamned the boundary layer flo over a contnuously movng thn needle n a parallel stream. The development of the boundary layer on a fed or movng surface parallel to a unform free stream n presence of surface heat flu has been nvestgated by Ishak et al. [3]. Patl et al. [4] presented a numercal nvestgaton of a steady to-dmensonal med convecton flo along vertcal sem-nfnte poer-la stretchng sheet n a parallel free stream. Unsteady med convecton flos do not necessarly posses smlarty solutons n many practcal applcatons. Most of the unsteady med convecton boundary layer flo problems appearng n recent years demand detaled analyss takng non-smlarty nto account. The unsteadness and nonsmlarty n such flos may be due to the free stream velocty or due to the curvature of the body or due to the surface mass transfer or even possbly due to all these effects. Because of the mathematcal dffcultes nvolved n obtanng nonsmlar solutons for such problems, many research nvestgators have confned ther studes ether to steady nonsmlar flos or to unsteady sem-smlar or self-smlar flos [5-8]. Therefore, as a step toards the eventual development of studes on unsteady med convecton flos t s mportant as ell as useful to nvestgate the combned effects of thermal and mass dffuson on an unsteady med convecton flo along a sem-nfnte vertcal plate n presence of nternal heat generaton or absorpton and chemcal reacton of frst order. The am of the present numercal study s to nvestgate the smultaneous effects of nternal heat generaton or absorpton and chemcal reacton of frst order th thermal and mass dffuson on an unsteady med convecton flo along a sem-nfnte vertcal plate. The plate s supposed to move th parallel to the free stream velocty. The unsteadness s caused due to the tme dependent movng vertcal plate velocty as ell as free stream velocty. The coupled non-lnear partal dfferental equatons governng the flo have been solved numercally usng an mplct fnte dfference method n combnaton of quas-lnearzaton technque [8, 9]. Results are compared th some results reported by Tsou et al. [3], Soundalgekar and Murty [5], Al [8], Moutsoglou and hen [9] and Patl et al. [4] and are found to be n ecellent agreement. velocty U n the - drecton. The -as s taken along the plate n the vertcally upard drecton and the y- as s taken normal to t. Fgure shos the coordnate system and physcal model for the flo confguraton. The unsteadness n the flo feld s ntroduced by the free stream velocty U and vertcal movng plate velocty U hch are varyng th tme n the same drecton. The surface s mantaned th prescrbed all temperature T dfferent from the free stream temperature T suffcently aay from the surface. A heat source s placed thn the flo to allo possble heat generaton or absorpton effects. The concentraton of dffusng speces s assumed to be very small n comparson th other chemcal speces far from the surface, and s nfntely small. Hence the Soret and Dufour effects are neglected. The lnear homogeneous chemcal reactons are takng place n the flo. The buoyancy force arses due to the temperature dfference n the flud. All thermophyscal propertes of the flud n the flo model are assumed constant ecept the densty varatons causng a body force n the momentum equaton. The Boussnesq appromaton s nvoked for the flud propertes to relate densty changes, and to couple n ths ay the temperature feld to the flo feld [30]. Under these assumptons, the equatons of conservaton of mass, momentum, energy and concentraton governng med convecton boundary layer flo over a movng vertcal plate are gven by: u v 0, () y u t y t y u u u e u u v g T T T T T T T Q0 T T u v m t y y u v D k t y y The ntal condtons are: u 0,, y u, y v 0,, y v, y,, 0,,,, 0,, y, y T y T y, () p, (3), (4) (5) The physcal boundary condtons for the problem are gven by: ut,, yuw t U,0, vt,, y 0,,, W, t y W,, e,,,, t,, T t y T u t u t U T t T here,,,. (6) ; < 0 or > 0.. PROBLEM FORMULATION We consder the unsteady lamnar ncompressble vscous heat generatng or absorbng med convecton boundary layer flo along a sem-nfnte vertcal plate movng th

3 Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 3 Fg. Physcal model and coordnate system Applyng the follong transformatons: U U ; y ; U, y U f,, ; t ; u ;, v ; y T T,, TW T G ; ; W,, ; T T T T W W0 H ; W W0 U U U ; u U f ; / U v f, U f U f ; * Gr ; Gr N ; Re Re g TW 0 T Q0 Gr ; Q ; pu m * g W0 U Gr ; Re ; U k Pr ; ; Sc ; ; (7) U U D U W nto Eqs. () to (3), e fnd that Eq. () s dentcally satsfed, and Eqs. () and (3) reduce to f F F G N H d FF (8) d F F f F Pr f G G Pr QG G (9) Pr F G f G Sc f H H Sc H H, (0) Sc F H f H here f F d f; f0. It s orthy to note that f s 0 taken to be zero snce the plate s mpermeable. That s, there s no sucton or njecton. In Eq. (7), that represents on the flo feld has sgns; the plus sgn ndcates the buoyancyupard (or buoyancy asssted) flo, hle the negatve sgn stands for buoyancy-donard (or buoyancy opposed) flo. The reduced boundary condtons become: F,,0,,0 H,,0, F,,, G,, G,, 0, H,, 0, () for 0,. We have assumed that the flo s steady at tme 0 and becomes unsteady for 0 due to the tme dependent plate velocty U W tu,0 u e t U, here and free stream velocty ; < 0 or > 0. Hence, the ntal condton (.e. condtons at 0 ) are gven by the steady state equatons obtaned from (7) and (8) by substtutng, d d F G H 0 hen 0. The correspondng boundary condtons are obtaned from (). The man physcal quanttes of nterest are the skn / frcton coeffcent f Re, the local Nusselt / number Nu Re and local Sherood number / Sh Re hch represent the all shear stress, the heat transfer rate and mass transfer rate at the surface, respectvely. The local skn frcton coeffcent s defned as u y, U f Re F,,0 f F.e. Re,,0. () The local heat transfer rate n terms of Nusselt number can be epressed as T y Nu Re G,,0, T T f

4 Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 4.e. Nu Re,,0. (3) The local mass transfer rate n terms of Sherood number can be epressed as y Sh Re H,,0,.e. Sh Re H,,0 3. METHOD OF SOLUTION. (4) The non-lnear coupled partal dfferental Eqs. (8) - (0) under the boundary condtons () have been solved numercally usng an mplct fnte dfference scheme n combnaton th the quas-lnearzaton technque [5] and [6]. Quaslnearsaton technque can be veed as a generalzaton of the Neton-Raphson appromaton technque n functonal space. An teratve sequence of lnear equatons s carefully constructed to appromate the nonlnear Eqs. (8) - (0) under the boundary condtons () achevng quadratc convergence and monotoncty. Wth the help of quaslnearsaton technque, the nonlnear coupled partal dfferental Eqs. (8) - (0) th boundary condtons () are replaced by the follong sequence of lnear ordnary dfferental equatons. 3 F A F A F A F A4 F A5 G A6 H A7 3 G B G B G B G B G B F B 3 H H H H H F, (5), (6). (7) The coeffcent functon th teratve nde are knon and the functon th teratve nde (+) are to be determned. The correspondng boundary condtons are gven by F H F H G,0, G,, 0,,0,,,0,,,,,,, 0,, at 0, as (8) The coeffcents n equatons (5) - (7) are gven by A f f d ; A F ; d A3 F ; A4 ; A5 ; A6 N; d A7 F F ; d Pr B f f ; B PrQ; Pr B3 F ; B4 Pr ; Pr Pr B5 G ; B6 G F ; Sc f f ; Sc ; Sc 3 F ; 4 Sc ; Sc Sc 5 G ; 6 G F. (9) Snce the method s presented for ordnary dfferental equatons by Inouye and Tate [9] and also presented for partal dfferental equatons n a recent study by Sngh and Roy [30], ts detaled descrpton s not provded here. At each teraton step, the sequence of lnear partal dfferental equatons (5) (7) ere epressed n dfference form usng central dfference scheme n the - drecton and backard dfference scheme n and drectons. Thus n each step, the resultng equatons ere then reduced to a system of lnear algebrac equatons th a block tr-dagonal matr, hch s solved by Varga s algorthm [3]. To ensure the convergence of the numercal soluton to the accurate soluton, step szes, and are optmzed and taken as 0.005, 0.0 and 0.0, respectvely. The results presented here are ndependent of the step szes at least up to the fourth decmal place. A convergence crteron based on the relatve dfference beteen the current and prevous teraton values s employed. When the dfference reaches , the soluton s assumed to have converged and the teraton process s termnated. 4. RESULTS AND DISUSSION The computatons have been carred out for varous Pr 0.7 Pr , valuesof, , N0.5 N.5, , Sc0. Sc.57, and Q.0 Q.0, has. The edge of the boundary layer been taken beteen 6.0 and 0.0 dependng on the values of the parameters. The results have been obtaned for both acceleratng ; 0, 0 and deceleratng ; 0, 0 free stream veloctes of the flud. In order to verfy the accuracy of the presented approach, e have verfed steady state results of heat transfer rate 0 by drect comparson th the results prevously reported by Tsou et al. [3], Soundalgekar and Murty [5], Al [8], Moutsoglou and hen [9] and Patl et al. [4]. The results of ths comparson are presented n Table and are found to be n ecellent agreement. The effects of buoyancy parameter and Prandtl number (Pr) on velocty and temperature profles F,,, G,, for acceleratng flo, 0.5, hen Q = 0.5, = 0.5, Sc =.57, N = 0.5, 0.5 and 0.5 are dsplayed n Fgs. and 3. The acton of buoyancy assstng force 0 shos the overshoot n the velocty profles F,, near the all for loer Prandtl

5 Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 5 number (Pr) flud (ar, Pr = 0.7) hle for hgher Prandtl number (Pr) flud (ar, Pr = 7.0), the velocty overshoot s not observed as shon n Fg.. The magntudes of the velocty overshoot ncreases th buoyancy parameter 0 hle t decreases as Prandtl number (Pr) ncreases. The physcal reason s that the buoyancy force effect s larger n a smaller Prandtl number (Pr) flud (ar, Pr = 0.7) due to the loer vscosty of the flud, hch enhances the velocty profle thn the movng boundary layer as the assstng buoyancy force acts lke a favorable pressure gradent and the velocty overshoot occurs. For hgher Prandtl number (Pr) fluds (ater, Pr = 7.0), the overshoot s not present because hgher Prandtl number (Pr) fluds mples more vscous flud hch have less mpact on the buoyancy parameter. It s remarkable to note from the Fgs. and 3 that the buoyancy opposng force ( < 0) reduces the magntude of the velocty near the all sgnfcantly thn the movng boundary layer for loer Prandtl number fluds (Pr = 0.7, ar) as ell as for hgher Prandtl number fluds (Pr = 7.0, ater). The effect of tme s crucal for the velocty overshoot. In partcular, for 0.5, Q = 0.5, = 0.5, Sc = 0., N = 0.5, 0.5, 0.5 at Pr = 0.7 hen = 3.0, overshoot n the velocty profle reduced appromately by 8% as tme ncreases has relatvely from 0.0 to.0. The buoyancy parameter less mpact on the temperature profle G,, as shon n Fg. 3. Furthermore, Fg. 3 also shos that the hgher Prandtl number fluds (ater, Pr = 7.0) results nto a thnner thermal boundary layer snce the hgher Prandtl number (Pr) fluds (ater, Pr = 7.0) have a loer thermal conductvty. Fgure 4 llustrates the role of stream se coordnate and tme on the velocty and temperature profles F,,, G,, for acceleratng flo, 0.5, hen Q = 0.5, = 0.5, Sc =.57, N = 0.5, 0.5,.0 and Pr = 0.7. When the movng plate moves faster, the velocty overshoot s observed more near the plate thn the movng boundary layer because movng plate causes faster movements n the flud near the plate. Hence, velocty and temperature profles F,,, G,, are ncreasng th stream se coordnate hle they are decreasng as tme ncreases from 0.0 to.0. Ths clearly reveals that an ncrease n stream se coordnate acts as a favorable pressure gradent and hence flud flos faster. The overshoot near the plate reduces th tme. For eample, Q = 0.5, = 0.5, Sc =.57, N = 0.5, 0.5,.0 and Pr = 0.7 at.0, overshoot n the velocty profle reduced appromately by 5% as ncreases from 0.0 to.0.

6 Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 6 Fgure 5 depcts the relatve mportance of the rato of buoyancy force N and tme on the velocty F,, for acceleratng flo, profle 0.5, hen Q = 0.5, = 0.5, Sc = 0., 0.5,, Pr = 0.7 and 0.5. In the rato of buoyancy forces adng flo N N 0, the buoyancy forces shos the sgnfcant overshoot n the velocty profles near the all for steady flo case hereas the velocty overshoot s not present for unsteady flo case. Although, the magntude of the overshoot ncreases th the rato of buoyancy forces parameter N N 0, t decreases as the dmensonless tme ncreases from 0.0 to.0. The physcal reason s that the rato of buoyancy forces parameter N affects more n steady case 0. Hence, the velocty ncreases thn the movng boundary layer as the assstng buoyancy force acts lke a favorable pressure gradent and the velocty overshoot occurs. It s nterestng to note from the Fg. 5 that for the rato of buoyancy opposng flo,.e. N ( N < 0), the buoyancy opposng force reduces the magntude of the velocty sgnfcantly thn the boundary layer for steady as ell as unsteady case. The overshoot near the plate reduces th tme. For eample, Q = 0.5, = 0.5, Sc = 0., 0.5,.0 and Pr = 0.7 at 0.5 hen N =, overshoot n the velocty profle reduced appromately by 7% as ncreases from 0.0 to.0. force due to thermal gradents as ell as ncrease n acts lke a favorable pressure gradent hch accelerates the flud for unform moton hen = 0.0 (steady case) causng the velocty overshoot near the surface thn the movng boundary layer. The velocty overshoot reduces sgnfcantly at =.0 and also th the ncrease of. Furthermore, t s observed that the velocty profle ncreases th tme thn the movng boundary layer for deceleratng flo ; 0.5. For eample, =.0, N = 0.5, Sc = 0., 0.5, = 0.5, Q = 0.5 and Pr = 0.7 hen 0.5 at = 0.5, the velocty profle ncreases appromately about 74% hen tme ncreases from 0 to. Ths clearly ndcates that the effect of movng boundary on the deceleratng flo s very promnent. Fgure 6 dsplays the effects of (the rato of free stream velocty to the composte reference velocty) and tme F,, for, on the velocty profles 0.5 and 0.5 hen =.0, N = 0.5, Sc = 0., 0.5, = 0.5, Q = 0.5 and Pr = 0.7. The velocty profle s strongly dependng on because t occurs n the momentum equaton as ell as n the boundary condton F,,. It has been observed that the magntude of the for velocty thn the movng boundary layer ncreases th the ncrease of hle decreases as ncreases from = 0.0 to =.0. The physcal reason s that the assstng buoyancy Fgures 7 and 8 llustrate the nfluence of the heat generaton or absorpton parameter Q and Prandtl number (Pr) on velocty and temperature profles F,,, G,, for acceleratng flo, 0.5, hen = 0.5, =.0, N = 0.5, Sc = 0., 0.5 and.0. It s noted that ong to the presence of a heat generaton or a heat source effect (Q>0), the thermal state of the flud ncreases. Hence, the velocty and temperature of the flud ncrease thn the movng boundary layers. In the event that the strength of the heat source s relatvely large, the overshoot s observed n the velocty and temperature profles thn the momentum and thermal boundary layers as can be seen n Fgs. 7 and 8. Further, the effect of heat source s more pronounced on temperature profles for hgh Prandtl number fluds (Pr = 7.0, ater). Fgure 8 dsplays that for Pr = 7.0, temperature profle has appromately % more overshoot th a thn thermal boundary layer due to hgher thermal conductvty as compared to the loer Prandtl number fluds (Pr = 0.7, ar). onversely, the presence of heat absorpton or a heat snk effect (Q < 0) has the tendency to reduce the flud temperature. Ths causes the thermal buoyancy effects to decrease resultng n a net reducton n the flud velocty. These behavors are clearly observed n Fgs. 7 and 8 n hch both the magntude of velocty and

7 temperature felds decrease for (Q < 0). Moreover, t s also observed that both the thcknesses of the hydrodynamc (velocty) and thermal (temperature) boundary layers decrease as the heat absorpton or heat snk effect ncreases. As compared to the case of no heat generaton or absorpton (Q = 0), one can see that hen the heat s generated (Q>0) the temperature ncreases th Q. Hoever, the opposte trend s revealed for heat absorpton case (Q < 0). Also, the overshoot near the plate reduces th tme. For eample, for Q =.0, = 0.5, =.0, N = 0.5, Sc = 0., 0.5 and Pr = 0.7 at.0, overshoot n the velocty and temperature profles are reduced appromately by 7% and 3%, respectvely, as tme ncreases from 0.0 to.0. Fgures 9 and 0 shos the varatons of velocty,, H,, profles for varous F and concentraton values of Schmdt number Sc and chemcal reacton parameter for acceleratng flo, 0.5, hen = 0.5, =.0, N = 0.5, Pr = 0.7,.0 and Q =.0. It shos that the magntude of the velocty and concentraton dstrbutons ncrease, hen the chemcal reacton parameter 0 (speces consumpton or destructve chemcal reacton), s ncreased. An ncrease n the concentraton of the dffusng speces ncreases the mass dffuson and thus, n turn, the flud velocty ncreases. On the contrary, for >0 (speces generaton or constructve chemcal reacton), as ncreases the velocty dstrbuton decreases, so that the concentraton reduces and thus, n turn, the flud velocty reduces. The values of the Schmdt number (Sc) are chosen to be more realstc, 0., 0.94 and.57, representng dffusng chemcal speces of most common nterest lke ater, Propyl Benzene hydrogen, ater vapor and Propyl Benzene, etc., at 5 degree elsus at one atmospherc pressure. It s observed that the velocty and concentraton boundary layers are to decrease as the Schmdt number Sc s ncreased. The physcal reason s that the Schmdt number Sc leads to a thnnng of the concentraton boundary layer. As a result the concentraton of the flud decreases and ths leads to a decrease n the flud velocty. Further, the effect of destructve chemcal reacton s more pronounced on concentraton profles for lo Schmdt number fluds (Sc = 0.). Fgure 0 dsplays that for Sc = 0., concentraton profle has appromately 54 % more overshoot th a thck concentraton boundary layer due to lo mass dffusvty as compared to the hgher Schmdt number fluds (Sc =.57). As compared to the case of no destructve or constructve chemcal reacton ( = 0), one can see that hen the speces are generated ( >0) the concentraton decreases th. Hoever, the opposte trend s revealed for speces consumpton case ( < 0). Also, n case of the destructve chemcal reacton, the overshoot near the plate reduces th tme. For eample, for = -.0, = 0.5, Q =, =.0, N = 0.5, Pr = 0.7 and.0 at Sc = 0. overshoot n the velocty and concentraton profles are reduced appromately by 4% and %, respectvely, as tme ncreases from 0.0 to.0. Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 7

8 Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 8 Fgures and represent the nfluence of acceleratng, 0.75 and deceleratng free stream flos ( or 0.75 ) on the skn frcton coeffcent and heat / / transfer rate f Re, Nu Re hen Q =.0, = 0.5,.0, Sc = 0., N = 0.5 and Pr = 0.7. Results ndcate / / that f Re, Nu Re ncrease th the ncrease of buoyancy parameter. The physcal reason s that the buoyancy force 0 mples favorable pressure gradent, and the flud gets accelerated, hch results n thnner momentum and thermal boundary layers. When stream se co-ordnate ncreases from 0.0 to.0, the local Nusselt / number Nu Re decreases hle skn frcton coeffcent f Re / ncreases, as shon n Fgs. and. For eample, for 0.75, Q =.0, = 0.5,.0, Sc = 0., N = 0.5 and Pr = 0.7 hen = 0.5, f Re / ncreases appromately about 48% and 38% as ncreases from.0 to 3.0 at 0.0 and at.0 (see Fg.), respectvely. / Further, Fgure shos that Nu Re decreases appromately about 8% and %, respectvely, at = 0 and at = hen decreases from 3 to. Moreover, t s noted that n the case of acceleratng flos, the heat transfer / rate Nu Re decreases th ncreasng tme and / Nu Re decreases th ncreasng tme for the case of deceleratng flos. In contrast, the skn frcton coeffcent / Re decreases th the ncrease of tme for the f case of acceleratng flos hch can also be understood from the velocty profles dsplayed n Fgs. 4. In the case of / deceleratng flos, f Re ncreases th the ncrease of tme as can be seen n Fgs.. Fgures 3 and 4 dsplay the relatve nfluence of acceleratng and deceleratng free stream flos (, 0.75 or 0.75 ) on the skn frcton / / coeffcent and heat transfer rate f Re, Nu Re hen Q =.0, = 0.5,.0, Sc = 0., and =.0. / / Results reveal that f Re, Nu Re ncrease th the ncrease of rato of buoyancy forces parameter N. The physcal reason s that the rato of buoyancy forces N 0 mples favorable pressure gradent, and the flud gets accelerated, hch results n thnner momentum boundary layer. When Prandtl number Pr ncreases from 0.7 / to 7.0, the local Nusselt number Nu Re as ell as skn / frcton coeffcent f Re decreases, as shon n Fgs. 3 and 4. For eample, for 0.75, Q =.0, = 0.5,.0, Sc = 0., =.0 and.0 hen = 0.5,

9 f Re / Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 9 ncreases appromately about 54% and 77% as N ncreases from 0.5 to.5 at Pr = 0.7 and at Pr = 7.0 (see Fg.), respectvely. Further, Fgure 4 shos that / Re ncreases appromately about 53% and Nu 45%, respectvely, at Pr = 0.7 and at Pr = 7.0 hen N decreases from.5 to 0.5. In support of smlar effects of N, and Pr, some of the other numercal results pertanng to acceleratng and deceleratng flos for / / / Re, Nu Re, Sh Re are tabulated n Table f and 3, respectvely. Moreover, t s noted that n the case of / acceleratng flos, the heat transfer rate Nu Re ncreases th ncreasng tme and / Nu Re decreases th ncreasng tme for the case of deceleratng flos. In / contrast, the skn frcton coeffcent f Re decreases th the ncrease of tme for the case of acceleratng flos / hle n the case of deceleratng flos, f Re ncreases th the ncrease of tme as can be seen n Fgs.3 and 4. Fgures 5 and 6 dsplay the effects of the heat generaton or absorpton parameter Q and Prandtl number Pr / on skn frcton coeffcent f Re and heat transfer / rate Nu Re for acceleratng and deceleratng free stream flos (, 0.75 and 0.75 ) hen =.0, =.0, = 0.5 N = 0.5,.0 and Sc = 0.. / The skn frcton coeffcent f Re as ell as heat / transfer rate Nu Re decreases as Prandtl number Pr ncreases from 0.7 to 7.0 sgnfcantly. In partcular, t s found for acceleratng flo ( 0.75 ) that the percentage / decreases of f Re by about 3% and 5%, respectvely, for = 0.5 at Q = -.0 and.0, respectvely, as Prandtl number ncreases from Pr = 0.7 to Pr = 7.0 hle / Re decreases by about 8% and 95%, Nu respectvely at Q = -.0 and.0 hen Prandtl number decreases from Pr = 7.0 to Pr = 0.7. Further, t s also / observed that f Re ncreases for heat generaton Q>0) / and decreases for heat absorpton (Q < 0) hle Nu Re ncreases for heat absorpton (Q < 0) and decreases for heat generaton (Q>0). It may be noted that the effect of heat generaton or absorpton (Q>0 or Q < 0) s more promnent on heat transfer rate than on skn frcton coeffcent. In fact, for Pr = 7.0,.0, Sc = 0., N = 0.5, =.0, 0.75, =.0 and = 0.5 at = 0.5, / Nu Re ncreases appromately by 83% hen Q / changes from 0 to.0 hle f Re decreases only 6% as Q changes from 0 to.0.

10 Table. omparson of the steady state results Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) for 0, N = 0, Q = 0, Sc = 0, = 0, 0, 0 and selected values of Prandtl number Pr to prevously reported ork. Pr Tsou et al. [3] Soundalgekar and Murty [5] Al [8] Moutsoglou and hen [9] Patl et al. [4] Table. Skn-frcton coeffcent for acceleratng flo Present ork Table 3. Skn-frcton coeffcent / Sh Re / / f Re, heat transfer coeffcent Nu Re, 0.5, 0.5, Q = 0.5,.0, 0.5, 0.5 and Sc = 0.. N Pr / Re for deceleratng flo f Nu Re and mass transfer coeffcent / Sh Re / / / f Re, heat transfer coeffcent Nu Re, m = 0 and = 0 / Sh Re and mass transfer coeffcent, 0.5, 0.5, Q = 0.5,.0, 0.5, 0.5 and Sc = 0.. N Pr / Re f Nu Re / Sh Re /

11 Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 the mass transfer rate ncreasng tme and / Sh Re / Sh Re ncreases slghtly th decreases th ncreasng tme for the case of deceleratng flos. Hoever, the skn / frcton coeffcent f Re decreases th the ncrease of tme 0.4 monotonously for the case of acceleratng / flos. In the case of deceleratng flos, f Re ncreases th the ncrease of tme 0.4 as can be seen n Fgs.7 and 8. The effects of Schmdt number Sc and chemcal reacton parameter for acceleratng and deceleratng free stream flos (, 0.75 or 0.75 ) on the skn frcton coeffcent and mass transfer rate / / Re, Sh Re hen Q =.0, = 0.5, N = 0.5, Pr = f 0.7, and = are depcted n Fg. 7 and 8 respectvely. It s noted that hen Schmdt number Sc ncreases from 0. to.57, the local Sherood number / Re as ell as skn frcton coeffcent Sh f Re / decreases, as shon n Fgs. 7 and 8. For eample, for 0.75, Q =.0, = 0.5, N = 0.5, Pr = 0.7, =.0 and.0 / hen = 0.5, f Re decreases appromately about 5% and 3% as Sc ncreases from 0. to.57 at =.0 and at = -.0 (see Fg.7), respectvely. / Re decreases Further, Fgure 8 shos that Sh appromately about 70% and 03%, respectvely, at =.0 and at = -.0 hen Sc decreases from.57 to 0.. Moreover, t s noted that n the case of acceleratng flos, 5. ONLUSIONS A numercal nvestgaton s performed to study the unsteady med convecton flo of a combned heat and mass transfer over a movng vertcal plate n a parallel free stream n the presence of heat generaton or absorpton and chemcal reacton. The unsteadness s caused due to tme dependent free stream velocty as ell as movng plate velocty. Results

12 Patl and hamkha / Internatonal Journal of Energy & Technology (9) (00) 3 ndcate that the buoyancy force enhances the skn frcton coeffcent and local Nusselt number. From the fgures, t s found that n presence of buoyancy force 0, the velocty profle ehbts velocty overshoot 90% more for loer Prandtl number as compared to the magntude of the velocty overshoot for hgher Prandtl number. For a fed buoyancy force, the coeffcent of skn frcton decreases th Prandtl number hle local Nusselt number ncreases. Also, an ncrease n the Prandtl number results nto a strong reducton up to 45 % n the thckness of the thermal boundary layer. Further, an ncrease n the Schmdt number Sc leads to a strong reducton up to 60% n the concentraton boundary layer. The effect of heat generaton leads to a rse n the momentum and thermal boundary layers hle heat absorpton leads to a fall n the momentum and thermal boundary layers. The effect of destructve chemcal reacton leads to a rse n the momentum and concentraton boundary layers hle constructve chemcal reacton leads to a fall n the momentum and concentraton boundary layers. Profles n graphs dsplay that the tme dependent free stream velocty has 40 % more effect on the velocty profles as compared to the correspondng temperature and concentraton profles. AKNOWLEDGEMENT: We ould lke to thank all of the reveers for ther ecellent comments hch have mproved sgnfcantly the qualty of the paper. One of the Authors (Dr. P. M. Patl) epresses hs sncere thanks to the Unversty Grants ommsson, South-West Regonal Offce, Bangalore, Inda for the fnancal assstance under the Mnor Research Project No. MRP(S)-636/09-0/KAKA060/UG-SWRO. NOMENLATURE f p concentraton local skn frcton coeffcent specfc heat at constant pressure f dmensonless stream functon F dmensonless velocty component g acceleraton due to gravty G dmensonless temperature * Gr, Gr Grashof numbers due to temperature and concentraton, respectvely H dmensonless concentraton k thermal conductvty N rato of Grashof numbers Nu local Nusselt number Pr Prandtl number Q 0 heat generaton coeffcent Q heat generaton or absorpton parameter Re local Reynolds number Sc Schmdt number Sh local Sherood number t, dmensonal tme T temperature U composte reference velocty U movng plate velocty W U free stream velocty u v velocty component n - drecton velocty component n y- drecton Greek symbols unsteady parameter thermal dffusvty m T, volumetrc coeffcents of the thermal and concentraton epansons, respectvely chemcal reacton parameter dmensonless tme unsteady functon of, transformed varables, buoyancy parameters due to temperature and concentraton gradents, respectvely dynamc vscosty knematc vscosty densty streamfuncton Subscrpts e condton at the edge of the boundary layer ntal condton 0 value at the all for 0, condtons at the all and nfnty, respectvely t,, y denote the partal dervatves th respect to these varables, respectvely,, denote the partal dervatves th respect to these varables, respectvely. REFERENES: [] B.. Sakads, Boundary-layer behavor on contnuous sold surfaces, II. The boundary layer behavor on contnuous flat surface, AIhE J 7 967, 5. [] H. Blasus, Grenzchechten n flussgketen rnt klener rebung, Z. Math. Phys , [3] F.K. Tsou, E.M. Sparro, R.J. Goldsten, Flo and heat transfer n the boundary layer on a contnuous movng surface, Int. J. Heat Mass Transfer 0 967, [4] T. A. Abdelhafez, Skn frcton and heat transfer on a contnuous flat surface movng n a parallel free stream, Int. J. Heat Mass Transfer 8 985, [5] Sam A. Al- Sanena, Med convecton heat transfer along a contnuously movng heated vertcal plate th sucton or njecton, Int. J. Heat Mass Transfer , [6]. N. Afzal, A. Baderuddn, A. A. Elgarv, Momentum and heat transport on a contnuous flat surface movng n a parallel stream, Int. J. Heat Mass Transfer , [7]. H. T. Ln, S. F. Haung, Flo and heat transfer of plane surface movng n parallel and reversely to the free stream, Int. J. Heat Mass Transfer , [8] J. P. Abraham, E. M. Sparro, Frcton drag resultng from smultaneous mposed motons of a free stream and ts boundng surface, Int. Jl. Heat and Flud Flo, 6 005, [9] E. M. Sparro, J. P. Abraham, Unversal solutons for the stream se varaton of the temperature of a movng sheet n the presence of a movng flud, Int. J. Heat and Mass Transfer, ,

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