Salina Aktar 1, Md.Abdul. Alim 2, Ali J. Chamkha 3

Transcription

1 International Journal o Energy & Technology ISSN 3-9X International Journal o Energy & Technology 6 () () CONJUGATE EFFECTS OF HEAT GENERATION AND PRESSURE WORK ON THE COUPING OF CONDUCTION WITH MAGNETOHYDRODYNAMIC FREE CONVECTION FOW AONG A VERTICA FAT PATE Salina Aktar Md.Adul. Alim Ali J. Chamkha 3 Department o Mathematic Jagannath Univerity Dhaka-Bangladeh. addre: alina8@gmail.com Department o Mathematic Bangladeh Univerity o Engineering and Technology Dhaka- Bangladeh. addre: maalim@math.uet.ac.d 3 Manuacturing Engineering Department The Pulic Authority or Applied Education and Training Shuweikh 76 Kuwait. achamkha@yahoo.com ABSTRACT Heat generation and preure work eect are conidered on MHD conjugate ree convection low o an electrically conducting luid along a vertically placed lat plate. The developed governing oundary layer equation and the coupled oundary condition are tranerred into dimenionle orm and then olved numerically uing the implicit inite dierence method. Numerical reult or the eect o the Prandtl numer magnetic parameter preure work parameter and heat generation parameter on the velocity proile temperature ditriution coeicient o kin riction and the heat traner rate are reported. It i oerved that there i a igniicant inluence o the preure work and the heat generation on the low ield. Keyword: Magnetohydrodynamic; conjugate heat traner; preure tre work; heat generation; kin riction; Nuelt numer; Keller-o method.. INTRODUCTION Electrically conducting luid low in the preence o the eect o preure work with vicou joule heating and heat conduction prolem are important rom the technical point o view and uch type o prolem have received much attention y many reearcher. In electronic in particular and in phyic roadly ued thee term. It i etalihed that heat generation eect are generally rather more important oth or gae and liquid. A medium through which heat i conducted may involve the converation o electrical nuclear or chemical energy into heat (or thermal) energy. In heat conduction analyi uch converion procee are characterized a heat generation. For eample the temperature o a reitance wire rie rapidly when electric current pae through it a a reult o the electrical energy are converted to heat at a rate o I R where I i the current and R i the electrical reitance o the wire. The ae and eective removal o thi heat away rom the ite o heat generation (the electronic circuit) i the uject o electronic cooling which i one o the modern application area o heat traner. ikewie a large amount o heat i generated in the uel element o nuclear reactor a a reult o nuclear iion that erve a the heat ource or the nuclear power plant. The heat generated in the Sun a a reult o the uion o hydrogen into helium make the Sun a large nuclear reactor that upplie heat to the earth. Another ource o heat generation in a medium i eothermic chemical reaction that may occur throughout the medium. The chemical reaction in thi cae erve a a heat ource or the medium. In the cae o endothermic reaction however heat i aored intead o eing releaed during reaction and thu the chemical reaction erve a a heat ink. The heat generation term ecome a negative quantity in thi cae. Heat generation i a volumetric phenomenon. That i it occur throughout the ody o a medium. Thereore the rate o heat generation in a medium i uually peciied per unit volume. Heat generation i the aility to emit greater-than-normal heat rom the ody.convection i the mode o energy traner etween a olid urace and the adjacent liquid or ga that i in motion and it involve the comined eect o conduction and luid motion. The ater the luid motion the greater the heat traner due to convection. Model tudie o the ree and mied convection low have earned reputation ecaue o their application in geophyical geothermal and nuclear engineering prolem. Alo the prolem o variou type o hape over or on a ree convection oundary layer low have een tudied y many reearcher. Amongt them Ackroyd [] tudied the tre work eect in laminar lat plate natural convection low. The conduction in olid wall and the convection in the luid hould e determined imultaneouly. Thi type o convective heat traner i reerred to a conjugate heat traner(cht) proce and it arie due to the inite thickne o the wall. The early theoretical and eperimental work o CHT prolem or a vicou luid ha een reviewed y Gdalevich and Fertman []. Miyamoto et al. [3] tudied the eect o aial heat conduction in a vertical lat plate on ree convection heat traner. Pozzi and upo [] invetigated the coupling o conduction with laminar convection along a lat plate. The eect o preure tre work and vicou diipation in ome natural convection low ha een analyzed y Johi and Gehart [6]. Gehart [6] invetigated the eect o

2 Aktar et al. / International Journal o Energy & Technology 6 () () diipation natural convection. Hoain [7] analyzed the vicou and Joule heating eect on MHD ree convection low with variale plate temperature. Gehart and Johi [9] have hown the eect o preure tre work and vicou diipation in ome natural convection low. Zakerullah [] invetigated the vicou diipation and preure work eect in aiymmetric natural convection low. Mamun []tudied the eect o conduction and convection on magnetohydrodynamic low with and without vicou diipation rom a vertical lat plate. Alam et al. [3] howed the eect o preure tre work and vicou diipation in natural convection low along a vertical lat plate with heat conduction. Alim et al. [] invetigated joule heating eect on the coupling o conduction with MHD ree convection low rom a vertical lat plate. Alam et al. [] analyzed tre work eect on ree convection low rom a vertical lat plate with joule heating and heat conduction. Mamun et al. [6] analyzed comined eect o vicou diipation with conduction on MHD ree convection low rom a vertical lat plate. Alim et al. [7] analyzed the comined eect o vicou diipation & joule heating on the coupling o conduction with and without MHD ree convection along a vertical lat plate repectively. Rahman et al. [8] invetigated the eect o temperature dependent thermal conductivity on MHD ree convection low along a vertical lat plate with heat conduction. Mamun et al. [9] invetigated in dierent way to how MHDconjugate heat traner or a vertical lat with vicou diipation and heat generation. Narin and Alim [] tudied the comined eect o vicou diipation and temperature dependent thermal conductivity on MHD ree convection low with conduction and joule heating along a vertical lat plate. Azim et al. [] analyzed the eect o vicou diipation and joule heating MHD- conjugate heat traner or a vertical lat plate in preence o heat generation. Chamkha and Ramadan [] decried olving analytical olution or the two-phae ree convection low in a particulate upenion pat an ininite vertical plate. Umavathi et al. [6] tudied mied Convection o a micropolar luid in a Vertical Channel in cae o ully developed low. Depite a lot o work ha een done on olving the energy equation o conjugate heat traner along a vertical lat plate there i no olution availale conidering imultaneouly heat generation and preure work eect.vicou diipation on magnetic ield. In thi work we olve the energy equation along along a vertical lat plate y taking heat generation and preure work into account. The developed equation are made dimenionle y uing uitale tranormation. The non-dimenional equation are then tranormed into non-linear equation y introducing a non-imilarity tranormation. The reulting non-linear equation together with their correponding oundary condition aed on convection are olved numerically y uing the inite dierence method along with Keller Bo Scheme [3]. Graph and tale are preented to how the important eature o the olution.. FORMUATION OF THE PROBEM At irt we conider a teady two-dimenional natural convection low o an electrically conducting vicou and incompreile luid with variale thermal conductivity along a vertical lat plate o length l and thickne (Fig. ). It i aumed that heat i tranerred rom the outide urace o the plate which i maintained at a contant temperature T where T > T the amient temperature o the luid. A uniorm magnetic ield o trength H i impoed along the y -ai i.e. normal direction to the urace and - ai i taken along the lat plate. It i aumed that heat i tranerred rom the outide urace o the plate which i maintained at a contant temperature T where T > T the amient temperature o the luid. A uniorm magnetic ield o trength H i impoed along the y -ai i.e. normal direction to the urace and - ai i taken along the lat plate. The governing equation o uch laminar low with heat generation and preure work along a vertical lat plate under the Bouineq approimation or the preent prolem can e written a u v y () u u u u v y y H u g ( T T ) T T u v k y C p y T u P Q T T C c P Upper urace T=T ower urace p o k l k interace T(o) Fig.. Phyical coniguration and coordinate ytem. T y u v T g () (3) y

3 Aktar et al. / International Journal o Energy & Technology 6 () () 3 The appropriate oundary condition to e atiied y the aove equation are u v T y u T T k k T T T T on y () a y The temperature and the heat lu are eeked continuou at the interace or the coupled condition and at the interace we mut have k k T o y k T ( ) y () y where k and k are the thermal conductivity o the olid and the luid repectively. The temperature T o in the olid a given y A. Pozzi and M. upo [] i y T T ( ) T T ( (6) o ) where T() i the unknown temperature at the interace to e determined rom the olution o the equation. We oerve that the equation () - (3) together with the oundary condition () - () are non linear partial dierential equation. In the ollowing ection the olution method o thee equation are dicued detail. 3. TRANSFORMATION OF THE GOVERNING EQUATIONS It i oerved that Equation () and (3) together with the oundary condition () are non-linear partial dierential equation. Then Equation () to (3) will e non-dimenionalized y uing the ollowing dimenionle variale: T T T T y y Gr u u Gr g ( T Gr 3 T ) v v Gr For eterior condition we know hydrotatic preure P e g and e and the preure work parameter Ge g which i le than one a uggeted y C P Gehart [6]. Uing the aove relation (7) the non-dimenional orm o the governing equation are: u v y (7) (8) u u u u v Mu (9) y y u v y Pr Ge y T T T T T Q () where Pr i the Prandtl numer M i the magnetic parameter and Q i the heat generation parameter deined repectively a C Pr k P Gr CP Gr The correponding oundary condition () then take the ollowing orm: u = v= - = p on y= > y u v a y > () where k p Gr i the conjugate conduction k parameter.here the coupling parameter 'p' govern the decried prolem. The order o magnitude o 'p' depend actually on and Gr / eing the order o unity. The l term l attain value much greater than one ecaue o l eing mall. In cae o air ecome very mall when the vertical plate i highly conductive i.e. > > and or material o( ) =. uch a gla. Thereore in dierent cae 'p' i dierent ut not alway a mall numer.in the preent invetigation we have conidered p = which i accepted or o o( ). To olve the equation (9) and () uject to the oundary condition () the ollowing tranormation are propoed y Merkin & Pop [8] / ( ) / ( ) / / y ( ) / / ( ) h ( ) H M Here i the dimenionle imilarity variale and i the tream unction which atiie the equation o continuity and u v y and h ( ) i the dimenionle temperature. and Q Q

4 Aktar et al. / International Journal o Energy & Technology 6 () () Sutituting () into equation (9) and () we get ater ome algera the ollowing tranormed non dimenional equation. 6 6 ( ) ( ) / / M ( ) h ( ) 6 h h Pr ( ) ( ) Ge h (3) T () h T T h Q h ( h ) Here prime denote dierentiation with repect to η. The oundary condition a mentioned in equation () then take the ollowing orm ( ) ( ) ( ) h( ) / / ( ) ( ) h( ) / h( ) () The et o equation (3) and () together with the oundary condition () are olved numerically y applying implicit inite dierence method with Keller o [] cheme. Since a good decription o thi method and it application to the oundary layer low prolem are given in the ook y Ceeci and Bradhaw [].From the proce o numerical computation in practical point o view it i important to calculate the value o the rate o heat traner and urace hear tre in term o Nuelt numer and the kin riction coeicient repectively. Thee can e written in the non-dimenional orm a C 3 Gr w and Nu k Gr T q T w (6) are the hearing tre and the heat lu repectively. Uing the new variale decried in (7) Equation (6) can e written athe local kin riction co-eicient and the urace temperature are otained rom the ollowing relation C 3 ( ) (7) Nu ( ) h (8) Alo the numerical value o the urace temperature ditriution are otained rom the relation ( ) h (9) and or dierent value o the preure work parameter Ge the kin riction coeicient and the the urace temperature ditriution are diplayed in taular orm. Moreover we have dicued the velocity proile and the temperature ditriution or dierent value o prandtl numer Pr magnetic parameter M preure work parameter Ge and heat generation parameter Q. 3. NUMERICA APPROACH The numerical method ued i inite dierence method together with Keller o Scheme y Keller []. To egin with the partial dierential equation. (3) and () are irt converted into a ytem o irt order dierential equation. Then thee equation are epreed in inite dierence orm y approimating the unction and their derivative in term o the central dierence ormula. Denoting the meh point in the (η)-plane y i and η j where i =... M and j =... N center dierence approimation are made uch that thoe equation involving eplicitly are centered at ( i-/ η j-/ ) and the remainder at ( i η j-/ ) where η j-/ =/(η j + η j- )etc. The aove central dierence approimation reduce the ytem o irt order dierential equation to a et o non-linear dierence equation or the unknown at i in term o their value at i-. The dicretization o momentum and energy equation are carried out with repect to non-dimenional coordinate and η to convey the equation in inite dierence orm y approimating the unction and their derivative in term o the central dierence in oth the coordinate direction. The whole procedure namely reduction to irt order ollowed y central dierence approimation Newton Quai linearization method and the lock Thoma algorithm i well known a Keller-o method. Thi i well predictale y Ceeci and Bradhaw [] and roadly ued y Hoain et al. [].. RESUTS AND DISCUSSION The ojective o the preent work i to analyze the eect o preure work on electrically conducting luid on ree convection low along a vertical lat plate due to wall thickne in preence o heat generation. The reulting olution or the velocity proile and temperature ditriution are hown in Fig. -. Detailed numerical olution have een otained or dierent value o all parameter conidered or the analyi. The eect or dierent value o heat generation on the velocity proile and temperature proile in cae o Prandtl numer Pr =.73magnetic parameter M =. and preure work parameter Ge =. are hown in Figure and (). It i een that the velocity o the luid increae y the increaing value o Q ecaue o the eect o Q generate heat in the low region a a reult the temperature o the luid increae within the thermal oundary layer hence the temperature increae gradually or increaing Q and eceed the level o the urace temperature. Thee phenomenon are preented in the Fig. and () repectively. The maimum velocitie are and 6 at η =.3 or the irt maimum value at η =.3693 or the nd 3 rd and th maimum value and η =.38 or the th maimum value or Q =...36 and. repectively. It can e een that the velocity increae y.9% a Q increae rom to.. It i een rom Fig. 3 that an increae in the preure work parameter i aociated with a coniderale increae in velocity. Near the urace o

5 Veloity proile Velocity Proile Temperature proile Velocity proile Temperature proile Aktar et al. / International Journal o Energy & Technology 6 () () the plate (mall η) the velocity increae ecome maimum then decreae and inally (large η) approache to zero. The maimum value o the nondimenional velocitie ( η) are and 9 or Ge =.... and. repectively and each o which occur at η =.3693 or the irt maimum and nd value maimum value and η =.3 or the 3rd and th and alo η =.379 or the th maimum value. Here it i oerved that the ( η) increae y 8.3% a Ge increae rom. to.. Fig. 3() how the ditriution o the temperature proile h (η) againt η or the ame value o the preure work parameter Ge and each o which reache the maimum at the urace. Thu h ( η) increae y.% a Ge increae rom. to () Q= Q=. Q=. Q=.36 Q=. Q= Q=. Q=. Q=.36 Q= () Ge=. Ge=. Ge=. Ge=. Ge=. Fig. 3. Variation o velocity proile and () variation o temperature proile againt η or varying o Ge with Pr=.73M=. and Q =.. In Figure it i hown that when the Prandtl numer Pr increae the velocity and the oundary layer thickne decreae. It i oerved rom Figure () that or higher value o Prandtl numer Pr the maimum value o temperature i higher and the thermal oundary layer thickne decreae due to the higher temperature gradient. It i een that the velocity decreae y 3.3% and the temperature decreae y 93.% when Pr increae rom to.. Figure and () illutrate the ehavior o the velocity and the temperature proile or dierent value o M with Pr =.73Ge =. and Q=.. The peak velocity increae with the increaing M due to thi retarding eect a hown in Fig.. A a reult the eparation o the oundary layer occur earlier ince the momentum oundary layer ecome thicker. Conequently the temperature within the oundary layer raie or growing value o magnetic parameter M rom to.. The magnetic ield decreae the temperature gradient at the wall and rie the temperature in the low region. Thu the magnetic parameter increae the thickne o the thermal oundary layer Fig.. Variation o velocity proile and () variation o temperature proile againt η or varying o Q with Pr=.73M=. and Ge=.. Pr= Pr=.73 Pr=.7 Pr=. Pr=.. Ge=. Ge=. Ge=. Ge=. Ge=

6 Skin riction co-eicient Temperature proile Velocity proile Temperature proile Aktar et al. / International Journal o Energy & Technology 6 () () Fig.. Variation o velocity proile and () variation o temperature proile againt η or varying o Pr with M=. Ge=. and Q = () Pr= Pr=.73 Pr=.7 Pr=. Pr= M =. M = M =. M =3. M =. M =. M = M =. M =3. M =. The variation o the kin riction co-eicient C rate o heat traner N u and the urace temperature ditriution θ( ) are hown graphically or dierent value o heat generation parameter Q (= ) with ied parameter Ge=.Pr=.73 and M=.. in Fig. 66() and 6(c) repectively. It can e een that an increae o the heat generation parameter Q increae the luid velocity within the oundary layer that hown in Fig..So the correponding kin riction coeicient and urace temperature increae or the increaing value o Q. The oppoite reult i oerved or heat traner ditriution along with the increaing o the heat generation parameter Q. In Figure 7 7()and 7(c) how that the preure work ha great inluence on kin riction coeicient C the rate o heat traner N u and urace temperature θ( ) a well. Frictional orce at the wall ecome much higher toward the uptream or higher value o preure work parameter Ge and the rate o heat traner N u and urace temperature θ( )a hown in Figure 7() and 7(c) gradually increaed or higher value o preure work parameter Ge. It i een in Figure 8 the kin riction coeicient C decreae irt up to the certain poition o and then the kin riction coeicient C increae a the Prandtl numer Pr increae. In Figure 8()and 8(c) the rate o heat traner N u and the urace temperature θ() increae rom a certain poition o ξ and rom that the rate o heat traner increae monotonically or increaing value o the Prandtl numer Pr. Thi epected ince there i no vicoity with the increaing o the Prandtl numer Pr the kin riction coeicient the rate o heat traner and the urace temperature increae. From Fig. 9 it can e een that the magnetic parameter M hrink the wall hear tre. Thi mean that the magnetic orce retard the low and work with the advere preure to reit the low. From Fig. 9() and 9(c) it i oerved that the local heat traner rate and urace temperature decreae with increaing M. Thi i ecaue the magnetic orce work againt the inertia orce and a a reult the value o the rate o heat traner and urace temperature reduce. Here it oerved that at = 3 the kin riction co-eicient the rate o heat traner and the urace temperature decreae y 3.6%3.9% and 67.% a the magnetic parameter M change rom to..... Q= Q=. Q=. Q=.36 Q= () Fig.. Variation o velocity proile and () variation o temperature proile againt η or varying o M with Pr=.73 Ge=. and Q =...3 3

7 Rate o heat traner Surace temperature Skin riction co-eicient Rate o heat traner Skin riction co-eicient Surace temperature Rate o heat traner Surace temperature Aktar et al. / International Journal o Energy & Technology 6 () () 7. Q = Q =. Q =. Q =.36 Q =. 8 6 Ge=. Ge=. Ge=. Ge=. Ge= () 3 (c)..9 Q= Q=. Q=. Q=.36 Q=. Fig. 7. Variation o kin riction co-eicient () variation o rate o heat traner and (c) variation o urace temperature againt or varying o Ge with Pr=.73M=. and Q= (c) Fig. 6. Variation o kin riction co-eicient () variation o rate o heat traner and (c) variation o urace temperature againt or varying o Q with Pr =.73M=. and Ge= Pr= Pr=.73 Pr=.7 Pr=. Pr=. 9 6 Ge=. Ge=. Ge=. Ge=. Ge=.. Pr= Pr=.73 Pr=.7 Pr=. Pr= (). 8 6 Ge=. Ge=. Ge=. Ge=. Ge=..9.3 Pr= Pr=.73 Pr=.7 Pr=. Pr=. 3 () 3 (c ) Fig. 8. Variation o kin riction co-eicient () variation o rate o heat traner and (c) variation o urace temperature againt or varying o Pr with M=.Ge =. and Q=..

8 Surace Temperature Rate o heat traner Skin riction co-eicient Aktar et al. / International Journal o Energy & Technology 6 () () M = M =. M =. M =3. M =. Tale : Skin riction coeicient and urace temperature proile or Q with cheming parameter M =. Pr =.73 and Ge =. Q= Q=. Q=. () θ() () θ() () θ() () 3 (c) M = M =. M =. M =3. M =. M = M =. M =. M =3. M =. Fig. 9. Variation o kin riction co-eicient () variation o rate o heat traner and (c) variation o urace temperature againt or varying o M with Pr=.73Ge =. and Q=.. In Tale the numerical value o the local kin riction co-eicient C and the urace temperature θ( ) againt or dierent value o Q while M =. Pr =.73 and Ge=. are characterized. It i oerved that the value o C grow up rapidly at dierent poition o. The rate o increae in the value o the kin riction coeicient and the urace temperature proile ecome much higher than that o the uptream value.. COMPARISON WITH PREVIOUS WORK Tale depict the comparion o preent numerical reult o the kin riction coeicient C and the urace temperature θ ( ) with thoe otained y Pozzi and upo [] and Merkin and Pop [8]. Here the parameter M Ge and Q are ignored and Pr =.733 with / = ξ i choen. It i clearly hown that there i an ecellent agreement among the preent reult with the olution o Pozzi and upo [] and Merkin and Pop [8] or oth the proile inide the oundary layer. Thee validation oot the conidence in the numerical code to carry on with the aove tated ojective o the current invetigation. Tale : Comparion o C and θ() among the preent reult and that o Pozzi and upo [] and Merkin and Pop [8] Skin riction co-eicient C Merkin and Pop [8] Pozzi and upo [] Preent work Surace temperature θ() Pozzi and upo [] Merkin and Pop [8] Preent work CONCUSION The eect o preure work and heat generation on MHD conjugate heat traner along a vertical lat plate ha een invetigated or dierent value o relevant phyical parameter. From the preent invetigation the ollowing concluion may e drawn:

9 Aktar et al. / International Journal o Energy & Technology 6 () () 9 The velocity proile increae or increaing value o preure work parameter Ge and heat generation parameter Q and decreae or increaing value Prandtl numer Pr and magnetic parameter M. The temperature proile decreae or increaing value Prandtl numer Pr. But the temperature proile increae and decreae that i cro the ide or the value o the magnetic parameter M preure work parameter Ge and heat generation parameter Q. For increaing value o Prandtl numer Pr preure work parameter Ge and heat generation parameter Q the kin riction coeicient increae and or increaing value o magnetic parameter M the kin riction coeicient decreae. The rate o heat traner increae or increaing value o Prandtl numer Pr and preure work parameter Ge. In another cae o the rate o heat traner increae and then lead to decreae or magnetic parameter M and heat generation parameter Q. REFERENCES [] J. A. D. Ackroyd Stre work eect in laminar lat plate natural convection J. Fluid Mech. Vol. 6 pp []. B. Gdalevich V.E. Fertman Conjugate prolem o natural convection Inzh-Fiz. Zh. Vol. 33(3) pp [3] M. Miyamoto J. Sumikawa T. Akiyohi T. Nakamura Eect o aial heat conduction in a vertical lat plate on ree convection heat traner Int. J. Heat Ma Traner Vol. 3 pp [] A. Pozzi M. upo The coupling o conduction with laminar natural convection along a lat plate Int. J. Heat Ma Traner Vol. 3 pp [] H. B. Keller Numerical method in the oundary layer theory Annual Rev. Fluid Mechanic Vol. pp [6] B. Gehart Eect o vicou diipation in natural convection. J. Fluid Mech Vol. pp [7] M. A. Hoain Vicou and joule heating eect on mhd-ree convection low with variale plate temperature Int. J. Heat Ma Traner Vol. 3 () pp [8] J. H. Merkin I. Pop Conjugate ree convection on a vertical urace Int. J. Heat Ma Traner Vol. 39 pp [9] Y. Johi B. Gehart Eect o preure tre work and vicou diipation in ome natural convection low Int. J. Heat Ma Traner Vol. () pp [] Md. Zakerullah Vicou diipation and preure work eect in aiymmetric natural convection low. Ganit (J. Bangladeh Math. Soc.) Vol. () pp [] T. Ceeci P. Bradhaw Phyical and Computational Apect o Convective Heat Traner Springer New York 98. [] A. A. Mamun Eect o conduction and convection on magnetohydrodynamic low with and without diipation rom a vertical lat plate. M. Phil. thei Department o Mathematic BUET. [3] Md. M. Alam M. A. Alim and Md. M.K. Chowdhury Eect o preure tre work and vicou diipation in natural convection low along a vertical lat plate with heat conduction J. Naval Arch. and Marine Engineering Vol. 3() pp [] M.A. Alim Md. M. Alam A.A. Mamun Joule heating eect on the coupling o conduction with magnetohydrodynamic ree convection low rom a vertical lat plate Nonlinear Analyi: Modelling and Control Vol. (3) pp [] Md. M. Alam M. A. Alim and Md. M.K. Chowdhury Stre work eect on natural convection low along a vertical lat plate with Joule heating and heat conduction Journal o Mechanical Engineering Vol. 38 pp [6] A.A. Mamun A. Azim Md. N. Hoain Md. A. Maleque Comined eect o conduction and vicou diipation on MHD ree convection low along a vertical lat plate Journal o Naval Architecture and Marine Engineering Vol. () pp [7] M. A. Alim Md. M. Alam A. A. Mamun Md. B. Hoain Comined eect o vicou diipation & joule heating on the coupling o conduction & ree convection along a vertical lat plate Int. Communication in Heat & Ma Traner Vol. 3(3) pp [8] M. M. Rahman A. A. Mamun M. A. Azim M. A. Alim Eect o temperature dependent thermal conductivity on mhd ree convection low along a vertical lat plate with heat conduction Nonlinear Analyi: Modelling and Control Vol. 3() pp [9] A.A. Mamun Z.R. Chowdhury M.A. Azim M.M. Molla MHD-conjugate heat traner analyi or a vertical lat plate in preence o vicou diipation and heat generation Int. Communication in Heat & Ma Traner Vol. 3() pp [] R. Narin M. A. Alim Comined eect o vicou diipation and temperature dependent thermal conductivity on magnetohydrodynamic (MHD) ree convection low with conduction and joule heating along a vertical lat plate Journal o Naval Architecture and Marine Engineering Vol. 6() pp [] M. A. Azim A. A. Mamun M. M. Rahman Vicou Joule heating MHD conjugate heat traner or a vertical lat plate in the preence o heat generation International Communication in Heat and Ma Traner Vol. 37 pp [] A.J. Chamkha H. Ramadan Analytical olution or the two-phae ree convection low o a particulate upenion pat an ininite vertical plate. International Journal o Engineering Science Vol. 36 pp

10 Aktar et al. / International Journal o Energy & Technology 6 () () [3] H. B. Keller T. Ceeci Accurate numerical method or oundary layer low Part-I. Two dimenional laminar low Proceeding o the Second International Conerence on numerical Method in Fluid Dynamic p. 9 Springer New York 97. [] M. A. Hoain M. A. Alim Natural convectionradiation interaction on oundary layer low along a thin cylinder Int. J. Heat and Ma Traner Vol. 3 pp [] M. A. Hoain M. A. Alim D.A.S. Ree Eect o thermal radiation on natural convection over cylinder o elliptic cro ection Acta Mechanica Vol. 9 pp [6] J. C. Umavathi A. J. Chamkha S. Veerhetty Fully developed mied convection o a micropolar luid in a vertical channel with oundary condition o third kind. International Journal o Energy & Technology Vol. (3) pp. -9. NOMENCATURE C C p g Ge Gr h H l M Nu P Pr plate thickne local kin-riction coeicient peciic heat at contant preure dimenionle tream unction acceleration due to gravity Preure work parameter Graho numer dimenionle temperature applied magnetic ield trength length o the plate reerence length magnetic parameter local Nuelt numer conjugate conduction parameter Prandtl numer Q T T T heat generation parameter temperature o the low luid temperature at outide o the plate temperature o the amient luid Greek ymol volumetric coeicient tream unction dimenionle imilarity variale denity o the luid inide the oundary layer kinematic vicoity vicoity o luid dimenionle temperature w hearing tre q w urace heat lu Sucript w wall condition amient condition quantitie outide the oundary layer dierentiation with repect to Supercript ' dierentiation with repect to η THE MATERIA WITHIN THIS PAPER AT THE AUTHORS RESPONSIBIITY HAS NOT BEEN PUBISHED ESEWHERE IN THIS SUBSTANTIA FORM NOR SUBMITTED ESEWHERE FOR PUBICATION. NO COPYRIGHTED MATERIA NOR ANY MATERIA DAMAGING THIRD PARTIES INTERESTS HAS BEEN USED IN THIS PAPER AT THE AUTHORS RESPONSIBIITY WITHOUT HAVING OBTAINED A WRITTEN PERMISSION.