International Journal of Modern Engineering Research (IJMER) Vol. 3, Issue. 4, Jul - Aug pp ISSN:

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1 Vol. 3, sse. 4, Jl - Ag. 3 pp SSN: Effect of Chemical Reaction and Radiation Absorption on Unstead Convective Heat and Mass Transfer Flow in a Vertical Channel with Oscillator Wall Temperatre and Concentration J. Deepthi, Prof. D. R. V. Prasada Rao Lectrer, Department of Mathematics, Ragiv Gandhi Universit of KnowledgeTechnologies, APT, R.K.Valle,A.P,ndia. Professor, Department of Mathematics, S. K. Universit, Anantapr, A. P., ndia. Abstract: We investigate the combined inflence of chemical reaction and radiation absorption on mixed convective flow in a vertical channel with oscillator wall temperatre and concentration.the non-linear copled partial differential eqations governing the flow heat and mass transfer are solved b a pertrbation techniqe. The effect of varios forces acting on the flid sstem is analzed b graphical representation of the velocit, temperatre and concentration. Kewords: chemical reaction, heat and mass transfer, radiation absorption, variable temperatre and concentration, vertical channel.. NTRODUCTON Combined heat and mass transfer problems with chemical reaction are of importance in man processes and have, therefore, received a considerable amont attention in recent ears. n processes sch as dring, evaporation at the srface of a water bod, energ transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occr simltaneosl. We are particlarl interested in cases in which diffsion and chemical reaction occr at roghl the same speed. When diffsion is mch faster than chemical reaction, then onl chemical factors inflence the chemical reaction rate; when diffsion is not mch faster than reaction, the diffsion and kinetics interact to prodce ver different effects. The std of heat generation or absorption effects in moving flids is important in view of several phsical problems, sch as flids ndergoing exothermic or endothermic chemical reaction. De to the fast growth of electronic technolog, effective cooling of electronic eqipment has become warranted and cooling of electronic eqipment ranges from individal transistors to main frame compters and from energ sppliers to telephone switch boards and thermal diffsion effect has been tilized for isotopes separation in the mixtre between gases with ver light moleclar weight (hdrogen and helim ) and medim moleclar weight. Mthcmaraswam and Ganesan () stdied effect of the chemical reaction and injection on flow characteristics in an nstead pward motion of an nstead pward motion of an isothermal plate. Deka et at. (4) stdied the effect of the first order homogeneos chemical reaction on the process of an nstead flow past an infinite vertical plate with a constant beat and mass transfer. Chamkha (3) stdies the MHD flow of a nmerical of niforml stretched vertical permeable srface in the presence of heat generation/absorption and a chemical reaction. The effect of foreign mass on the free-convection flow past a semi-infinite vertical plate were stdied (5) Chamkha (3) assmed that the plate is embedded in a niform poros medim and moves with a constant velocit in the flow direction in the presence of a transverse magnetic field. Raptis and Perdikis (9) stdied the nstead free convection flow of water near 4 C in the laminar bondar laer over a vertical moving poros plate. n spite of all these stdies, the nstead MHD free convection heat and mass transfer for a heat generating flid with radiation absorption has received little attention. Hence, the main objective of the present investigation is to std the effects of radiation absorption, mass diffsion, chemical reaction and hear sorce parameter of hear generating flid.the process of free convection as a mode of heat transfer has wide applications in the fields of Chemical Engineering, Aeronatical and Nclear power generation. t was shown b Gill and Casal (6) that the boanc significantl affects the flow of low Prandtl nmber flids which is highl sensitive to gravitational force and the extent to which the boanc force inflences a forced flow is a topic of interest. Free convection flows between two long vertical plates have been stdied for man ears becase of their engineering applications in the fields of nclear reactors, heat exchangers, cooling appliances in electronic instrments. These flows were stdied b assming the plates at two different constant temperatres or temperatre of the plates varing linearl along the plates etc. The std of fll developed free convection flow between two parallel plates at constant temperatre was initiated b Ostrach (6). Combined natral and forced convection laminar flow with linear wall temperatre profile was also stdied b Ostrach (7). The first exact soltion for free convection in a vertical parallel plate channel with asmmetric heating for a flid of constant properties was presented b Ang (). Man of the earl works on free convection flows in open channels have been reviewed b Manca et al. (). Recentl, Campo et al. () considered natral convection for heated iso-flx bondaries of the channel containing a low-prandtl nmber flid. Pantokratoras (8) stdied the fll developed free convection flow between two asmmetricall heated vertical parallel plates for a flid of varing thermophsical properties. However, all the above stdies are restricted to fll developed stead state flows. Ver few papers deal with nstead flow sitations in vertical parallel plate channels. Transient free convection flow between two long vertical parallel plates maintained at constant bt neqal temperatres was stdied b Singh et al.(). Jha et al. (9) extended the problem to consider smmetric heating of the channel walls. Narahari et al. (5) analzed the transient free convection flow between two long vertical parallel plates with 89 Page

2 Vol. 3, sse. 4, Jl - Ag. 3 pp SSN: constant heat flx at one bondar, the other being maintained at a constant temperatre. Singh and Pal () presented an analsis of the transient free convective flow of a viscos incompressible flid between two parallel vertical walls occrring as a reslt of asmmetric heating / cooling of the walls. Narahari (4) presented an exact soltion to the problem of nstead free convective flow of a viscos incompressible flid between two long vertical parallel plates with the plate temperatre linearl varing with time at one bondar, the other bondar being held at constant. There are man reasons for the flow to become nstead. When the crrent is periodic de to on-off control mechanisms or de to partiall rectified a-c voltage, there exist periodic heat inpts. Hence, it is important to std the effects of periodic heat flx on the nstead natral convection, imposed on one of the plates of a channel formed b two long vertical parallel plates, the other being held at a constant initial flid temperatre. Recentl Narahari(5) has discssed the nstead free convection flow of dissipative viscos incompressible flid between two long vertical parallel plates in which the temperatre of one of the plates is oscillator whereas that of the other plate is niform. Haritha (7) has analsed nstead convective heat transfer of dissipative viscos flid throgh a poros medim confined in a vertical channel on whose walls an oscillator temperatre is prescribed. brahim et al. (8) have stdied the effect of chemical reaction and radiation absorption on the nstead MHD free convection flow past a semi infinite vertical permeable moving plate with heat sorce and sction Kesavaiah et al () have stdied the effect of the chemical reaction and radiation absorption on an nstead MHD convective Heat and Mass Transfer flow past a semi-infinite vertical permeable moving plate embedded in a poros medim with heat sorce and sction n this paper we analse the effect of chemical reaction and radiation absorption on nstead convective Heat and Mass Transfer flow of a viscos flid in a Vertical channel on whose walls an oscillator temperatre is prescribed. Approximate soltions to copled non-linear partial differential eqations governing the flow, heat and mass transfer are solved b a pertrbation techniqe. The velocit, temperatre, skin friction, concentration and rate of heat and mass transfer are discssed for different variations of Sc, N, K, Q, γ.. FORMULATON OF THE PROBLEM We consider the flow of a viscos incompressible chemicall reacting flid in a vertical channel bonded b flat walls in the presence of constant heat sorces. We choose a Cartesian coordinate sstem (x ) with walls at = b sing Bossinesq approximation we consider the densit variation onl on the boanc term. The eqation governing the flow to heat and mass transfer are Eqation of Linear Momentm g ( ) t k Eqation of Energ T T Cp( k f Q Q ( C C) t () Eqation of Diffsion C C D KC t ( 3) Eqation of State * T T ) ( C ) (4) ( C where is the velocit component in x-direction, T is the temperatre, p is the pressre, is the densit, is the electricall condctivit, e is the magnetic permeabilit, k is the coefficient of poros permeabilit, is dnamic viscocit, k f is coefficient of thermal condctivit is the coefficient of volme expansion, Q is the strength of heat sorce, * is the volmetric coefficient of expansion with mass fraction, D is the chemical moleclar diffsivit and K is the coefficient of chemical reaction. Q is radiation absorption parameter. The bondar conditions are =,T = T,C=Cat = -L () 9 Page

3 Vol. 3, sse. 4, Jl - Ag. 3 pp SSN: =, T = T + (T T ) cos( t), C = C + (C C ) cos( t)at = +L (5) On introdcing the non-dimensional variables. / L T T T T, = /L,, t = t, Eqations (.) (.3) redce to (dropping the dashes) ( c c) ( c c ) G[ N] D (6) t P S Q t C C KC t c (8) where 3 ( T T ) G gl (Grashoff nmber) P P C K (Prandtl nmber) f L ( T T ) (Heat sorce parameter) S c D K f (Schmidt nmber) L *( CL C) N ( T T ) (Wormsl nmber) L KL K D QCL Q (Boanc ratio) (Chemical reaction parameter) (radiation absorption parameter) K f The transformed bondar conditions are =, =, = at = - =, = + cos(t), c = + cos(t) at = + (9) (7). METHOD OF SOLUTON n view of the bondar conditions (5) we assme = + e it = + e it () = + e it sbstitting () in the eqations6-8 and comparing harmonic & Non harmonic terms we get D ( ) G N () ( D i ) G( N ) Q () ( 3) 9 Page

4 Vol. 3, sse. 4, Jl - Ag. 3 pp SSN: ip Q d K d d d ( K i Sc ). SOLUTONS OF THE PROBLEM The soltions of ()- (6) sbject to the bondar conditions are Ch Sh Ch Sh = b + b Ch + b 3 Sh + a 3 + a 4 = b b b 8 + a 5 + a 6 + (b 5 + b 9 ) Ch ) + (b 6 + b ) Sh. Ch3 Sh3 Ch Sh 3 3 = k 49 Ch 3 + k 5 Sh 3 + k 5 Ch + k 5 Sh = k 55 Ch 3 + k 56 Sh 3 + k 57 Ch + k 58 Sh + k 59 Ch 4 + k 6 Sh 4. Where K (4) i P i Sc 3 4 V. NUSSELT NUMBER and SHERWOOD NUMBER The rate of heat transfer (Nsselt nmber) at the walls = is given b d ( N) d and the corresponding expressions are (5) (6) ( N) a7 Ec[ a8 a9 a3sh a3ch] (.). E33. a ( N) a Ec[ a a3 a4sh a5ch] (.). E33. a The rate of mass transfer (Sherwood nmber) at the walls = is given b ( Sh ) dc d And the corresponding expressions are ( Sh) a35 (.) E33. a ( Sh) a33 (.) E33. a where a, a, a 36 are constants shown in appendix V. RESULTS AND DSCUSSON n this analsis we investigate the effect of chemical reaction on mixed convection heat and mass transfer flow of a viscos flid throgh a poros medim in a vertical channel on whose walls oscillator temperatre and concentration are prescribed. The velocit () is shown in figres -4 for different vales of N,Sc, K, Q. The variation of with boanc ratio N shows that when the moleclar boanc force dominates over the thermal boanc force, the velocit enhances in the left half and redces in the right half when the boanc forces act in the same direction and for the forces acting in opposite directions redces in the left half and enhances in the right half(fig.). Fig represents the variation of with Sc.Lesser the moleclar diffsivit smaller in the flow region, and for frther lowering of the diffsivit the velocit enhances in the left half and redces in the right half and for still lowering of the moleclar diffsivit the velocit enhances in the entire flow region. The effect of chemical reaction on is shown in fig 3. t is fond that an increase in k <.5 enhances in entire region and for higher k.5, enhances in the left half and redces in the right half. The variation of with radiation absorption parameter Q is shown in fig 4. Fixing the other parameters. t is fond that an increase in Q Page

5 Vol. 3, sse. 4, Jl - Ag. 3 pp SSN: leads to an enhancement in and for frther higher Q = 4, redces in the flow region and for still higher Q 6, redces in the left half and enhances in the right half V V Fig. : Variation of with N Fig. : Variation of with Sc V V N Sc V.5.5 V Fig. 3: Variation of with K Fig. 4 : Variation of with Q V V K Q 4 6 The Non-dimensional temperatre distribtion () is shown in fig 5-8 for different parametric vales N,Sc, K, Q. The variation of with the boanc ratio N shows that the actal temperatre experiences a depreciation with increasing in N irrespective of the directions of the boanc forces(fig.5). The variation of with Schmidt nmber Sc shows that lesser the moleclar diffsivit larger the actal temperatre in the left half and smaller in the right half, and for frther lowering of the moleclar diffsivit smaller the actal temperatre in the entire flow region and for still lowering of the moleclar diffsivit smaller the actal temperatre in the left half and larger in the right half (fig 6). From fig 7 we notice that for smaller and larger vales of the chemical reaction parameter K, the actal temperatre depreciates in the flow region and for an intermediate vale of K =.5, the actal temperatre experiences an enhancement in the flow region. The variation of with radiation absorption; parameter Q shows that the actal temperatre depreciates appreciabell with increase in Q in the entire flow region (fig.8). 93 Page

6 Vol. 3, sse. 4, Jl - Ag. 3 pp SSN: The non-dimensional concentration C is shown in fig 9- for different vales of Sc, K and. From fig 9 we notice that lesser the moleclar diffsivit larger the actal concentration in the flow region. An increase in the chemical reaction parameter K reslts in a depreciation in the concentration in the entire flow region (fig. ). An increase in the Wormsel nmber () leads to an enhancement in the actal concentration in the flow field(fig.)... C V C Fig. : Variation of C with K Fig : Variation of C with V K The rate of heat transfer at = are exhibited in tables -4 for different parametric vales. t is fond that the rate of heat transfer increases with increase in G (or) M (or) K, ths higher the Lorentz force larger the Nsselt nmber at both the walls. An increase in the radiation absorption parameter Q enhances N at =+ and redces at = - (tables and 3). From tables and 4 we find that the rate of heat transfer depreciates at = + and enhances at = -. The variation of N with heat sorce parameter shows that the rate of heat transfer redces at = + and enhances at = - with increase in >, while an increase in < redces N at =. The rate of mass transfer at = is shown tables 5-6 for different Sc, K and. The rate of mass transfer enhances with Schmidt nmber Sc.Ths higher the moleclar diffsivit larger Sh at =. The variation of Sh with chemical reaction parameter K shows that the rate of mass transfer enhances with K at = + while at = - it enhances with K.5 and redces with higher K.5. With reference to variation of Sh with Wormsel nmber () exhibits that Sh redces with increase in 4 and enhances with 6, while at = - it experiences an enhancement with (tables.5-7). 94 Page

7 Vol. 3, sse. 4, Jl - Ag. 3 pp SSN: Table Nsselt Nmber (N) at = + G V V V V V X X X M 3 4 K Sc Q 4 Table Nsselt Nmber (N) at = + G V V V N Table 3 Nsselt Nmber (N) at = G V V V V V M 3 4 K Sc Q 4 Table 4 Nsselt Nmber (N) at = G V V V Page

8 Vol. 3, sse. 4, Jl - Ag. 3 pp SSN: Table 6 S c V V V K V. CONCLUSON An attempt has been made to investigate the combined inflence of chemical reaction and radiation absorption on the nstead convective heat and mass transfer flow in a vertical channel techniqe sing a reglar pertrbation techniqe the non-linear copled eqations has been solved. The important conclsions of this analsis are. Lesser the moleclar diffsivit smaller in the flow region, and for frther lowering of the diffsivit the velocit enhances in the left half and redces in the right half and for still lowering of the moleclar diffsivit the velocit enhances in the entire flow region. An increase in k <.5 enhances in entire region and for higher k.5, enhances in the left half and redces in the right half.an increase in Q leads to an enhancement in and for frther higher Q = 4, redces in the flow region and for still higher Q 6, redces in the left half and enhances in the right half.. The actal temperatre experiences a depreciation with increasing in N irrespective of the directions of the boanc forces.lesser the moleclar diffsivit larger the actal temperatre in the left half and smaller in the right half, and for frther lowering of the moleclar diffsivit smaller the actal temperatre in the entire flow region and for still lowering of the moleclar diffsivit smaller the actal temperatre in the left half and larger in the right half. 3. For smaller and larger vales of the chemical reaction parameter K, the actal temperatre depreciates in the flow region and for an intermediate vale of K =.5, the actal temperatre experiences an enhancement in the flow region.the actal temperatre depreciates appreciabell with increase in Q in the entire flow region. 4. Lesser the moleclar diffsivit larger the actal concentration in the flow region. An increase in the chemical reaction parameter K reslts in a depreciation in the concentration in the entire flow region. 5. Higher the Lorentz force larger the Nsselt nmber at both the walls. An increase in the radiation absorption parameter Q enhances N at =+ and redces at = The rate of mass transfer enhances with Schmidt nmber Sc.The rate of mass transfer enhances with K at = + while at = - it enhances with K.5 and redces with higher K.5. REFERENCES [] Ang.W, Fll developed laminar free convection between vertical plates heated asmmetricall. nt. J. Heat and Mass Transfer, Vol 5, pp , (97). [] Campo. A, Manca O, and Marrone B, Nmerical investigation of the natral convection flows for low-prandtl flid in vertical parallel-plates channels. ASME Jornal of Applied Mechanics, Vol 73, pp. 6-7 (6). [3] Chamka A.J, MHD flow of a nmerical of niforml stretched vertical permeable srface in the presence of heat generation/absorption and a chemical reaction. nt Comm Heat and Mass transfer,vol 3, pp.43- (3). [4] Deka R. Das U.N, Sondalgekar V.M, Effects of mass transfer on flow past an implsivel started infinite vertical plate with constant heat flx and chemical reaction. Forschng in ngenierwesen,vol6,pp (994). [5] Gebhart.B, Pera. L,The natre of vertical natral convection flow reslting from the combined boanc effects of thermal and mass diffsion. J. Heat Mass Transfer Vol 4, pp.5-5(97). [6] Gill W.M and Casal,A.D, A theoretical investigation of natral convection effects in forced horizontal flows, Amer. nst. Chem. Engg. Jor., V.8,pp.53-5(96). [7] Haritha. A, Transient free convective Heat transfer of a viscos flid in channels/circlar dcts with heat sorces and radiation effects, Ph.D. Thesis, S.P. Mahila Universit, Tirpathi (). [8] brahim F.S., Elaiw A.M., Bakr A.A, Effect of chemical reaction and radiation absorption on the nstead MHD free convection flow past a semi infinite vertical permeable moving plate with heat sorce and sction. Commnications in Nonlinear Science and Nmerical Simlation Vol.3, pp (8). [9] Jha B.K, Singh A.K, and Takhar H.S, Transient free convection flow in a vertical channel de to smmetric heating. nternational Jornal of Applied Mechanics and Engineering, 8(3), pp (3). [] Kesavaiah DC.h, Satanaraana P.V and Venkataramana.S, Effects of the Chemical Reaction and Radiation absorption on an nstead MHD convective Heat and Mass transfer flow past a semi-infinite vertical permeable moving plate embedded in a poros medim with heat sorce and sction. nt.j. of Appl. Math and Mech.7(), pp.5-69 (). [] Manca.O, Marrone. B, Nardini.S, and Naso. V, Natral convection in open channels. n: Snden B, Comini G. Comptational Analsis of Convection Heat Transfer. WT press, Sothempton, pp (). [] Mthcmaraswam.R, Ganesan.P, Effect of the chemical reaction and injection on flow characteristics in an nstead pward motion of an isothermal plate, J.Appl.Mech Tech Phs, 4, pp (). [3] Narahari M, Free convection flow between two long vertical parallel plates with variable temperatre at one bondar. Proceedings of nternational Conference on Mechanical & Manfactring Engineering (CME 8), Johor Bahr, Malasia (8). [4] Narahari. M, Oscillator plate temperatre effects of free convection flow of dissipative flid between long vertical parallel plates. nt. J. of Appl. Math. And Mech. 5(3), pp (9). 96 Page

9 Vol. 3, sse. 4, Jl - Ag. 3 pp SSN: [5] Narahari.M, Sreenadh. S, and Sondalgekar V.M, Transient free convection flow between long vertical parallel plates with constant heat flx at one bondar. Thermophsics and Aeromechanics, 9(), pp (). [6] Ostrach.S, Laminar natral-convection flow and heat transfer of flids with and withot heat sorces in channels with constant wall temperatres. Technical Report 863, NASA, USA (95). [7] Ostrach.S, Combined natral and forced convection laminar flow and heat transfer of flids with and withot heat sorces in channels with linearl varing wall temperatre. Technical Report 34, NASA, USA (954). [8] Pantokratoras.A, Fll developed laminar free convection with variable thermo phsical properties between two open-ended vertical parallel plates heated asmmetricall with large temperatre differences. ASME Jornal of Heat Transfer, 8, pp (6). [9] Raptis.A, Perdikis.C,Free convection flow of water near 4 C past a moving plate. Forschng m ngenierwesen; Vol. 67, pp.6-8 (). [] Singh A.K and Pal. T, Transient natral convection between two vertical walls heated/cooled asmmetricall. nternational Jornal of Applied Mechanics and Engineering, (), pp (6). 97 Page