Journal of Applied Fluid Mechanics, Vol. 7, No. 3, pp , Available online at ISSN , EISSN

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1 Journal of Applied Fluid echanics Vol. 7 No. 3 pp Available online at ISSN EISSN Radiation Effect on HD Fully Developed ied Convection in a Vertical Channel with Asymmetric Heating R. Patra S. Das and R. Nath Jana 3 3 Department of Applied athematics Vidyasagar University idnapore 7 India Department of athematics University of Gour Banga alda 73 3 India Corresponding Author jana67@yahoo.co.in (Received January 7 3; accepted April 3 3) ABSRAC Effects of radiative heat transfer on an HD fully developed mied convective flow of a viscous incompressible electrically conducting fluid through a vertical channel with asymmetric heating of walls in the presence of a uniform transverse magnetic field has been studied. An eact solution of the governing equations has been obtained in closed form. It is observed that the velocity field is greatly influenced by the radiative heat transfer as well as bouyancy forces. he induced magnetic field decreases near the cool wall and it increases near the hot wall of the channel with an increase in radiation parameter. Further an increase in radiation parameter leads to a decrease in the fluid temperature in the channel. A limiting consideration of the solutions of the governing equations of the flow are analyzed for Ra. Keywords: HD mied convective flow Grashof number radiation parameter Prandtl number and asymmetric heating. NOENCLAURE B induced magnetic field component along - direction Pr Prandtl number b non-dimensional induced magnetic field q radiative heat flu r component B the magnetic field vector r temperature difference ratio e planck function Ra radiation parameter p g acceleration due to gravity fluid temperature Gr Grashof number temperature at entrance of channel Gr Gr Critical Grashof numbers temperature of hot wall K absorption coefficient temperature of cool wall k thermal conductivity u velocity component in -direction magnetic parameter u non-dimensional fluid velocity cartesian co-ordinates p fluid pressure y Greek symbols non-dimensional pressure gradient dimensionless temperature coefficient of thermal epansion fluid density magnetic permeability e fluid density at entrance of channel kinematic viscosity conductivity of fluid wave length shear stresses at cool and hot walls non-dimensional width of the channel

2 R. Patra et al. / JAF Vol. 7 No. 3 pp INRODUCION Heat transfer in free and force convection in vertical channels occurs in any industrial processes and natural phenomena. ost of the interest in this subject is due to its applications for instance in the design of cooling systems or electronic devices chemical processing equipment microelectronic cooling and in the field of solar energy collections. Since some fluids can also emit and absorb thermal radiation it is of interest to study the effects of magnetic field on the temperature distribution and heat transfer when the fluid is not only an electrical conductor but also when it is capable of emitting and absorbing radiation. Hence heat transfer by thermal radiation is becoming of greater importance in space applications and higher operating temperatures. Ozisik(989) has mentioned in an ecellent review article that heat transfer by simultaneous radiation and convection has applications in numerous technological problems including combustion furnace design the design of high-temperature gas-cooled nuclear reactors nuclear-reactor safety fluidized-bed heat echagers fire spreads advanced enegy conservation devices such as open-cycle coal and natural-gas-fired HD solar pondssolar collectors natural convection in cavities and many others. On the other hand it is worth mentionaing that heat transfer by simultaneous radiation and convection is very important in the contet of space technology and processes involving high temperature. An ecellent description of the fundamentals of thermal radiation has been presented in the book by odest (3). For a comprehensive treatment of the radiation transfer and the interractions with convection the interested reader can consult also the books by Sprrow and Cess 97; Ozisik 973 ; Siegel and Howell 99). (Aung 97; Aung et al.97; Aung and Worku 986; Barletta ; Boulama and Galanis 4) deal with the evaluation of the temperature and velocity profiles for the vertical parallel-flow fully develoved regime. As is well known heat echangers technology involves convective flows in vertical channels. In most cases these flows imply conditions of uniform heating of a channel which can be modelled either by uniform wall temperature or uniform heat flu thermal boundary conditions. any process in new engineering areas occur at high temperatures and knowledge of radiative heat transfer becomes very important for the design of the pertinent equipment. Nuclear power plants gas turbines and various propulsion devices for aircraft missiles satellites and space vehicles are eamples of such engineering areas. Cogley et al.(968) have studied the differential approimation for radiative heat transfer in a non-grey gas near equilibrium. he hydrodyamic fully developed laminar convective flow through a vertical channel in the optically thin limit has been studied by Greif et al. (97) as Gupta and Gupta (974) have studied the same problem in the presence of a transverse magnetic field. he effects of radiative heat transfer on HD flows in vertical channel have been studied a number of researches. Datta and Jana (976) have discussed the effect of wall conductances on hydromagnetic convection of a radiating gas in a vertical channel. Ogulu and otsa (5) have studied the radiative heat transfer to magnetohydrodynamic Couette flow with variable wall temperature. Effect of radiation on unsteady free convection flow bounded by an oscillating plate with variable wall temperature have been described by Pathak et al.(6). Sharma et al. (7) have discussed the radiation effect on temperature distribution in three-dimensional Couette flow with suction / injection. he radiation effect on HD free convection flow of a gas past a semi-infinite vertical plate have been studied by akhar et al. (996). he effects of wall conductance on HD fully developed flow with asymmetric heating of the walls has been studied by Guria et al. (7). Ghosh and Nandi () have discussed magnetohydrodynamic fully developed combined convection flow between vertical plates heated asymmetrically. HD fully developed mied convection flow with asymmetric heating of the walls have been described by Ghosh et al. (). Pantokratoras (6) has presented his results for a steady free convection flow between vertical parallel plates by considering different conditions on the wall temperature. he thermal radiation effect on fully developed mied convection flow in a vertical channel have been eamined by Grosan and Pop (7). Suneetha et al. () have presented the radiation and mass transfer effects on HD free convective dissipative fluid in the presence of heat source/sink. Effects of thermal radiation on hydromagnetic flow due to a porous rotating disk with Hall effect have been studied by Anjali Devi and Uma Devi (). Baoku et al. () have analyzed the influence of thermal radiation on a transient HD Couette flow through a porous medium. In the present paper we study the effects of radiative heat transfer on HD fully developed mied convective flow in a vertical channel with the asymmetric heating of walls in the presence of a transverse magnetic field. We assume that the radiative heat flu follows a relation in an optically thin limit for a non-gray gas near equilibrium. he closed form solutions for velocity temperature shear stresses rate of heat transfer and critical Grashof number are presented. Flow and heat transfer results for a range of values of the pertinent parameters have been reported. It is observed that the velocity field is greatly influenced by the radiative heat transfer as well as bouyancy forces. he induced magnetic field decreases near the cool wall and it increases near the hot wall of the vertical channel with an increase in radiation parameter. Further an increase in radiation parameter leads to decrease the fluid temperature in the channel.. FORULAION OF HE PROBLE AND IS SOLUION Consider a steady HD fully developed mied convective flow of a viscous incompressible electrically conducting fluid confined between 54

3 R. Patra et al. / JAF Vol. 7 No. 3 pp vertical walls. he channel walls are at a distance d apart. Choose a Cartesian co-ordinates system with -ais in the upward direction along the cool wall in the direction of flow and the ais of y is perpendicular to it. he wall at y has a uniform temperature while the wall at y d is subjected to a uniform temperature >. A uniform magnetic field of strength B is imposed perpendicular to the channel walls. he flow is due to buoyancy force difference in temperature and in the presence of pressure gradient. Since the channel walls are infinitely long along the -direction all physical quantities are functions of y only. he velocity components are ( uv ) relative to the Cartesian frame of reference. Fig.. Geometry of the problem he Boussinesq approimation is assumed to hold and for the evaluation of the gravitational body force the density is assumed to depend on the temperature according to the equation of state [ ( )] () is the fluid temperature the fluid density the coefficient of thermal epansion and and being the temperature and the density at the entrance of the channel. he solenoidal equation B = gives By constant = B every in the flow B ( B B ). he flow being fully developed the v relations v = = y p and = y apply here p is the fluid pressure. herefore the u continuity equation gives = and hence u = u( y ). Using the Boussinesq approimation () the momentum equation along - ais and the magnetic induction equation are d u db d p e B g ( ) = () dy dy d d B du e B = dy (3) dy and the energy equation is d q r = k (4) dy y is the kinematic viscosity e the magnetic permeability the conductivity of the fluid g the acceleration due to gravity k the thermal conductivity and q r the radiative heat flu. It has been shown by Cogley et al.(968) that in the optically thin limit for a non-gray gas near equilibrium the following relation holds q r e h = 4( ) K d y (5) K is the absorption coefficient is the wave length e p is the Planck's function and subscript ' indicates that all quantities have been evaluated at the temperature which is the temperature at entrance of the channel. On the use of (5) the Eq.(4) becomes d = k 4 I( ) dy (6) e I K d h = In asymmetric heating of the walls the velocity magnetic and temperature boundary conditions are respectively u = and B = at y u and B = at y = d (7) assuming the walls are electrically non-conducting. Introducing the non-dimensional variables y ud B u b (8) d B e Eqs. () (3) and (6) become d u db Gr (9) d d d b d d u d () = d Ra = d () number and = Bd ( ) = g Gr d 4 Ra = Id k 3 is the Hartmann the Grashof number the radiation parameter and 3 d p the non-dimensional pressure gradient. he boundary conditions given by (7) become u r and b = at u = = and b = at = () r = is the temperature difference ratio. Solution of Eqs. (9) to () subject to the boundary 55

4 R. Patra et al. / JAF Vol. 7 No. 3 pp conditions () are al.(7) in the case of without wall conductance. sinh Ra sinh Ra ( ) r Ra sinh Ra sinh Ra ( ) = sinh sinh ( ) = r Ra sinh sinh (3) sinh sinh ( ) c sinh sinh Gr sinh sinh ( ) [ r Ra sinh sinh sinh Ra sinh Ra ( ) r Ra sinh Ra sinh Ra u( ) = sinh sinh ( ) c sinh sinh Gr sinh [( r cosh ) sinh sinh cosh r cosh ( ) Ra = (4) cosh cosh ( ) c c c sinh sinh Gr cosh cosh ( ) r Ra sinh sinh cosh Ra cosh Ra ( ) r Ra Ra sinh Ra Ra sinh Ra b ( ) = cosh cosh ( ) c c c sinh sinh Gr cosh ( r cosh ) sinh sinh sinh r sinh ( ) cosh r cosh ( ) Ra = (5) Gr( r ) cosh cosh Ra Ra Ra sinh Ra sinh Ra c = Gr( r ) 3 4 sinh (6) ( cosh )( sinh ) Ra = sinh ( cosh ) Gr( r ) sinh cosh cosh Ra Ra c = Ra cosh sinh Ra sinh Ra sinh Gr( r ) ( cosh ) 4 sinh (sinh ) =. Ra (7) and is given by (9). It is seen from the epressions (3)-(5) that the velocity field and induced magnetic field depend on the Grashof number Gr as the temperature distribution is independent of Gr. In the absence of radiation ( Ra = ) the velocity distribution and the induced magnetic field coincide with Guria et 3. EXPRESSION FOR PRESSURE GRADIEN It is noticed from the Eqs. (4) and (5) that the parameter is still to be evaluated. Using the rate of mass flow ud = (8) we have on the substitution of the value of u which is obtained from Eq. (4) as (cosh ) sinh (cosh ) 3 Gr( r ) sinh Ra sinh ( cosh ) Ra cosh cosh Ra = sinh Ra sinh Ra (cosh ) sinh (cosh ) Ra sinh Gr( r ) 4 sinh 4. RESULS AND DISCUSSION =. (9) o study the effects of radiative heat transfer and the magnetic field on the HD fully developed flow with asymmetric heating of the walls the dimensionless velocity u the induced magnetic filed b and temperature distribution are depicted graphically against for several values of radiation parameter Ra magnetic parameter bounyancy parameter Gr and temperature difference ratio r in Figs. to 3. Figures. and 3 depicts the effects of the radiative heat transfer and magnetic field on the velocity field. It is seen from Fig. that the fluid velocity u increases in the range.38 while it decreases in the range.38 <.86 and again it increases in the range.86 < with an increase in radiation parameter Ra. It is manifested that there is a closeness of the curves near the hot wall. Fig.3 reveals that the fluid velocity u increases in the range.4 while it decreases in the range.4 < with an increase in magnetic parameter. It means that the Lorentz force imposed by a transverse magnetic field to an electrically conducting fluid which slows down the fluid motion near the hot wall and enhances the fluid motion near cool wall. It is also manifested that there is a closeness of the curves near the hot wall. It is noticed from Fig.4 that the fluid velocity u decreases in the region.53 and it increases in the region.53 < with increase in Grashof number Gr. Fig.5 shows that with an increase in r the fluid velocity u increases in the region.53 and it decreases in the region.53 <. It is 56

5 R. Patra et al. / JAF Vol. 7 No. 3 pp observed from Fig.6 that the induced magnetic field b decreases at any point near the cool wall and it increases near the hot wall with an increase in Ra. Fig.7 shows that the induced magnetic field b decreases with an increase in. It is seen from Fig.8 that the induced magnetic field increases with an increase in Gr. Fig.9 reveals that the induced magnetic field b decreases with an increase in r. he temperature profiles have been drawn for different values of Ra and r in Figs. and. It is seen from Figs. and that the fluid temperature decreases with an increase in radiation parameter Ra while it increases with an increase in temperature difference ratio parameter r. b Fig.. Variation of velocity for different Ra when =5 r =.4 and Gr = Fig. 4. Variation of velocity for different Gr when =5 Ra =4 and r =.4 Fig. 5. Variation of velocity for different r when =5 Ra =4 and Gr = Fig. 3. Variation of velocity for different when Ra =4 r =.4 and Gr = Fig. 6. Variation of induced magnetic field for different Ra when =5 r =.4 and Gr = 57

6 R. Patra et al. / JAF Vol. 7 No. 3 pp Fig. 7. Variation of induced magnetic field for different when Ra =4 r =.4 and Gr = Fig. 8. Variation of induced magnetic field for different Gr when =5 Ra =4 and Gr = Fig. 9. Variation of induced magnetic field for different r when =5 Ra =4 and Gr = Fig.. Variation of temperature for different Ra when r =.4 Fig.. Variation of temperature for different r when Ra =4 One of the important characteristics of this problem is the shear stress at the walls. he non-dimensional shear stress at the cool wall ( = ) and hot wall ( =) are respectively given by du du = and = d d = = du d and = () (cosh ) sinh (cosh ) Gr( r ) cosh cosh Ra Ra sinh Ra sinh Ra Gr Ra (r cosh ) Ra(r cosh Ra) Ra sinh sinh Ra = (cosh ) sinh (cosh ) (cosh )(sinh ) Gr( r ) 3 sinh Gr sinh ( r cosh ) ( r cosh ) Ra = sinh () 58

7 R. Patra et al. / JAF Vol. 7 No. 3 pp ( cosh ) sinh (cosh ) Gr( r ) cosh cosh Ra Ra sinh Ra sinh Ra Gr Ra (cosh r) Ra(cosh Ra r) Ra sinh du sinh Ra = d = ( cosh ) sinh (cosh ) (cosh )(sinh ) Gr( r ) 3 sinh Gr sinh ( r cosh ) (cosh r) Ra = sinh () he values of non-dimensional shear stresses at the walls and are shown graphically against r for different values of Ra and in Figs. and 3. It is observed from Fig. that for fied values of and Gr the shear stress at the cool wall increases while the shear stress at the hot wall decreases with an increase in radiation parameter Ra. On the other hand Fig.3 shows that the shear stress at the cool wall increases while the shear stress at the right wall decreases with increase in. It is interesting to note that the shear stress due to the flow does not vanish at the walls and when buoyancy forces Gr =. hus we arrive an interesting conclusion that there is no flow reversal in the absence of the buoyancy forces. he shear stresses at the cool and hot wall vanish if ( Ra ) (cosh ) ( A B)[ sinh ( cosh )] Gr = ( cosh ) ( A B)[ sinh ( cosh )] Ra Ra = (3) ( Ra ) (cosh ) ( A B)[ sinh ( cosh )] Gr = ( cosh ) ( A B)[ sinh ( cosh )] Ra Ra = Fig.. Shear stress for different Ra when =5 Ra =4 and Gr = 5 Fig. 3. Shear stress for different when =5 r =.4 and Gr = 5 (4) able he values of Critical Grashof number Ra Gr 3 at the cool wall. Gr r

8 R. Patra et al. / JAF Vol. 7 No. 3 pp able he values of Critical Grashof number Gr at the hot wall. Ra r he values of the critical Grashof numbers Gr and Gr due to the flow at the cool wall and hot wall are entered in the ables and for different values of magnetic parameter radiation parameter Ra and the temperature difference ratio parameter r. It is seen from the able that the critical Grashof number Gr at the cool wall ( = ) due to the flow increases with increase in either Ra or r or. able shows that the the critical Grashof number Gr at the hot wall ( =) due to the flow increases with an increase in either or r. On the other hand with increase in the radiation parameter Ra the critical Grashof number Gr decreases. In the absence of radiative heat transfer ( Ra = ) the critical Grashof numbers Gr and Gr are given by 4 (cosh ) Gr ( r )[ ( cosh ) 4cosh 4( cosh )] (5) Gr Gr (6) which are identical with the Eq. (39) of Guria et al. (7). 4.. Limiting Case Now we discuss the case when the radiation parameter is Ra. In this case Eqs. (3) to (5) become Ra 3 Ra ( ) = ( ) r ) ( )( ) 6 6 (7) sinh sinh ( ) u( ) = c sinh sinh Gr sinh sinh ( ) r Ra sinh sinh Ra 3 Ra ( ) r ) ( )( ) 6 6 (8) Gr( r ) cosh b ( ) c ( Ra ) sinh cosh cosh ( ) c sinh sinh Gr cosh cosh ( ) r Ra sinh sinh 3Gr ( r ) r ( ) (9) ( Ra )( Ra 6) Gr( r ) sinh ( cosh ) c = ( Ra ) (cosh ) (cosh ) Gr( r ) sinh ( cosh ) ( ) Ra (3) In limit Ra Eqs. (7) to (8) for the temperature the velocity and the induced magnetic field yield respectively ( ) = ( ) (3) r sinh sinh ( ) u( ) = Grr c sinh sinh sinh Gr( r ) sinh (3) cosh cosh ( ) b( ) ( Grr ) c sinh sinh cosh Gr r c c (33) sinh [ sinh (cosh )] c = rgr Gr( r) (cosh ) sinh ( cosh ) c ( r ) ( ) Gr Gr r sinh (cosh ) = Gr( r ). sinh (cosh ) (34) he velocity and the induced magnetic field given by (3) and (33) coincide with Eqs. (4) and (5) of Guria et al. (7) in the case of without wall conductance. 5. CONCLUSION An HD fully developed mied convection in a vertical channel in the presence of radiation have been studied. It is seen that the fluid velocity is strongly affected by the radiative heat transfer as well as bouyancy force. he induced magnetic field decreases near the cool wall and it increases near the hot wall of the channel with an increase in radiation parameter. Radiation decreases the fluid temperature. It is interesting to note that the shear stress due to the flow does not vanish at the channel walls when Gr and hence there is no flow reversal in the absence of the buoyancy force. his model finds applications in the design of cooling systems chemical processing equipments and the field of solar energy collections. REFERENCES Anjali Devi S.P. and R. Uma Devi (). Effects of thermal radiation on hydromagnetic flow 5

9 R. Patra et al. / JAF Vol. 7 No. 3 pp due to a porous rotating disk with Hall effect. J. Applied Fluid echanics 5() -7. Aung W. and G. Worku (986). Developing flow and flow reversal in a vertical channel with asymmetric wall temperature. J.Heat ransfer Aung W. and G.Worku (986). heory of fully develoed combined convection including flow reversal. J.Heat ransfer Aung W. L.S. Fletcher and V. Sernas (97). Developing laminer free convection between vertical flat plates asymmetric heating. Int.J.Heat ass ransfer Aung W. (97). Fully developed laminer free convection between vertical plates heated asymmetrically. Int.J.Heat ass ransfer Baoku I. G. C. Israel-Cookey and B. I. Olajuwon (). Influence of thermal radiation on a transient HD Couette flow through a porous medium. J. Applied Fluid echanics 5() Barletta A. (). Fully developed mied convection and flow reversal in a vertical rectangular duct with unifrom wall heat flu. Int.J. Heat ass ransfer Boulama K. and N. Galanis (). Fully developed mied convection and flow reversal in a vertical rectangular duct with uniform wall heat flu. Int.J. Heat ass ransfer Cogley A.C. W.C.Vincenti and S.E. Gilles (968). Differential approimation for radiative transfer in a non-grey gas near equilibrium. AIAAJ Datta N. and R.N. Jana (976). Effects of wall conductances on hydromagnetic convection of a radiating gas in a vertical channel. Int. J. Heat ass ransfer Ghosh S.K I. Pop and D.K. Nandi (). HD fully developed mied convection flow with asymmetric heating of the walls. Int. J. Appl. ech. and Engineering 7. Ghosh S.K. and D.K. Nandi (). agnetohydrodynamic fully developed combined convection flow between vertical plates heated asymmetrically. Journal echnical Physics Greif R. I.S. Habib and J.C. Lin (97). Laminar convection of a radiating gas in a vertical channel. J. Fluid ech. 46(3) echanik 7() Gupta P.S. and A. S. Gupta (974). Radiation effect on hydromagnetic convection in a vertical channel. Int.J. Heat ass ransfer Guria. Das B.K. Jana R.N. and Ghosh S.K. (7). Effects of wall conductance on HD fully developed flow with asymmetric heating of the walls. Int.J.Fluid ech.research 34(6) odest. F. (3). Radiative Heat ransfer ( nd edition). Academic Press New York. Ogulu A. and S. otsa (5). Radiative heat transfer to magnetohydrodynamic Couette flow with variable wall temperature Physica Scripta Ozisik.N. (973). Radiative ransfer and Interactions with Conduction and Convection. Wiley New York. Ozisik.N. (987). Interaction of radiation with convection. In: Handbook of Single-Phase Convection Heat ransfer (Kakac S. Shah R.K. and Aung W. eds.). Wiley New York. Pantokratoras A. (6). Fully developed laminar free covection with variable thermophysical properties between two open-ended vertical parallel plates heated asymmetrically with large temperature differences. ASE J. Heat ranfer (8) Pathak G. C. H. aheshwari and S. P. Gupta (6). Effect of radiation on unsteady free convection flow bounded by an oscillating plate with variable wall temperature Int. J. Appl. ech.and Engineering Sharma B.K.. Agarwal and R.C. Chaudhary (7). Radiation effect on temperature distribution in three-dimensional Couette flow with suction or injection. Appl. ath. ech. (English Edition) 8(3) Siegel R. and J.R. Howell (99). hermal Radiation Heat ransfer (3 rd edition). Hemisphere New York. Sparrow E.. and R. D. Cess (97). Radiation Heat ransfer. Brooks/Cole Belmont California. Suneetha S. N. Bhaskar Reddy and V. Ramachandra Prasad (). Radiation and mass transfer effects on HD free convective dissipative fluid in the presence of heat source/sink. J. Applied Fluid echanics 4() 7-3. Grosan. and I. Pop (7). hermal radiation effect on fully develoved mied convection flow in a vertical channel echnische akhar H.S. S. R. Gorla and V..Soundalgekar (996). Radiation effect on HD free convection flow of a gas past a semi-infinite 5

10 R. Patra et al. / JAF Vol. 7 No. 3 pp vertical plate. Int. J. Num. ethods for Heat and Fluid Flow (6) 77. APPENDIX ( r )(cosh ) cosh cosh Ra Ra sinh (cosh ) sinh Ra sinh Ra A = ( r )( cosh ) ( sinh ) sinh [ sinh (cosh )] Ra = (r cosh ) Ra(r cosh Ra) sinh sinh Ra B = r r sinh Ra (cosh ) sinh ( cosh ) Ra = ( r )( cosh ) cosh cosh Ra Ra sinh (cosh ) sinh Ra sinh Ra A = ( r )( cosh ) (sinh ) sinh [ sinh (cosh )] (cosh r) Ra(cosh Ra r) sinh sinh Ra B = r r sinh Ra = Ra ( cosh ) sinh ( cosh ) Ra = 5