Free Convective Flow of Immiscible Permeable Fluids in a Vertical Channel with First Order Chemical Reaction

Transcription

1 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: Free onvective Flow of Immiscible Permeable Flids in a Vertical hannel with First Order hemical Reaction J. Prathap Kmar J.. Umavathi Shreedevi Kalan 3 Professor Department of Mathematics Glbarga Universit Kalabragi Karnataka India Professor Department of Mathematics Glbarga Universit Kalabragi Karnataka India 3 Research Scholar Department of Mathematics Glbarga Universit Kalabragi Karnataka India. orresponding athor ( p_rathap@ahoo.com Abstract In this article the effect of first order chemical reaction on free convective flow of immiscible permeable flids in a vertical channel is stdied. he flow in the poros medim is modeled sing Brinkman eqation. he channel walls are maintained at two different constant temperatres. Viscos and Darc dissipation terms are inclded in the energ eqation. he copled ordinar nonlinear differential eqations governing the heat and mass transfer are solved analticall b sing pertrbation method and nmericall b sing finite difference method. Separate soltions for the poros medim in both the regions are obtained and these soltions are matched at the interface sing sitable matching conditions. he soltions are evalated nmericall and the reslts are presented graphicall for varios vales of flow governing parameters sch as thermal Grashof nmber mass Grashof nmber poros parameter viscosit ratio width ratio and condctivit ratio. In addition closed form expressions for volmetric flow rate Nsselt nmber species concentration and total energ added to the flow are also derived. It is also fond that both analtical and nmerical soltions agree ver well for small vales of pertrbation parameter. Ke Words: hemical reaction poros medim pertrbation method finite difference method.. INRODUION onvection in poros media is applied in tilization of geothermal energ the control of polltant spread in grondwater the design of nclear reactors compact heat exchangers solar power collectors heat transfer associated with the deep storage of nclear waste and high performance inslators for bildings. onsiderable progress in this area was made b Nield and Bejan [] Kavian [] and Vafai and ien [3] also analzed the effects of a solid bondar and the inertial forces on flow and heat transfer in poros media. he copled flid flow and heat transfer problem in a fll developed composite region of two parallel plates filled with Brinkman-Darc poros medim was analticall investigated b Kavian []. Rdraiah and Nagraj [5] stdied the fll developed free-convection flow of a viscos flid throgh a poros medim bonded b two heated vertical plates. Beckerman [6] stdied natral convection in vertical enclosre containing simltaneosl flid and poros laers. Singh et al. [7] analzed heat and mass transfer phenomena de to natral convection in a composite cavit containing a flid laer overling a poros laer satrated with the same flid in which the flow in the poros region was modeled sing Brinkman-Forchheimer extended Darc model that incldes both the effect of macroscopic shear (Brinkman effect and flow inertia (Forchheimer effect. Forced convection in composite channel is a sbject of intensive investigation. his is de to the rapid development of technolog and nmeros modern thermal applications relevant to this area sch as cooling of microelectronic devices. Polikakos and Kazmierczak [8] presented analtical soltions for forced convection flow in dcts where the central part is occpied b clear flid and the peripheral part is occpied b a Brinkman-Darc flidsatrated poros medim. he reslts of Polikakos and and Kazmierczak [8] were extended b Kznetsov [9] to accont for the Forchheimer (qadratic drag effects. Prasad [] have made an excellent review for composite sstems. Alzami and Vafai [] reviewed different tpes of interface conditions between a poros medim and flid laer. 5 IRJE.NE- All Rights Reserved Page 86

2 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: Some novel designs of heat sinks for cooling microelectronic devices tilize highl poros materials sch as alminm foam (Paek et al. []. Nield and Kznetsov [3] considered a forced convection problem in a channel whose center is occpied b a laer of isotropic poros medim (poros laer and whose peripheral part is occpied b another laer of isotropic poros medim (poros laer each of the laers with its own permeabilit and thermal condctivit. he tilized the Darc law for the flow in poros laers. Malashett et al. [-6] stdied two-flid flow and heat transfer in an inclined channel containing a poros-flid laer and composite poros medim. Recentl Umavathi et al. [7-3] Umavathi and Manjla [] Umavathi [5] and Prathap Kmar et al. [6 7] stdied mixed convection in a vertical poros channel. ombined heat and mass transfer problems with a chemical reaction are of importance in man processes and have received a considerable amont of attention in recent ears. 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 occrs simltaneosl. Natral convection processes involving the combined mechanisms are also encontered in man natral processes sch as evaporation condensation and agricltre dring and in man indstrial applications sch as the cring of plastics cleaning and chemical processing of materials relevant to the manfactre of the printed circitr and the manfactre of plp-inslated cables. In man chemical engineering processes chemical reactions take place between a foreign mass and a working flid mass which moves de to the stretch of a srface. he order of the chemical reactions depends on several factors. One of the simplest chemical reactions is the first order reaction in which the rate of the reaction is directl proportional to the species concentration. hamkha [8] stdied the analtical soltions for heat and mass transfer b the laminar flow of a Newtonian viscos electricall condcting and heat generating /absorbing flid on a continosl moving vertical permeable srface in the presence of a magnetic field and the first order chemical reaction. Mthcmaraswam and Ganeshan [9] stdied the nmerical soltion for the transient natral convection flow of an incompressible viscos flid past an implsivel started semi-infinite isothermal vertical plate with the mass diffsion taking into accont a homogeneos chemical reaction of the first order. he analtical soltion of the free convection heat and mass transfer from a vertical plate embedded in a flid-satrated poros medim with the constant wall temperatre and concentration was obtained b Singh and Qeen [3]. he heat and mass transfer characteristics of the natral convection abot a vertical srface embedded in a satrated poros medim sbjected to a chemical reaction taking into accont the Soret and Dfor effects was analzed b Postelnic [3]. Prathap Kmar et al. [3-3] stdied the effect of homogenos and heterogeneos reaction on the dispersion of a solte for an immiscible flid. Keeping in view the wide area of practical applications on mlti flid flow and effects of chemical reaction as mentioned the objective of this std is to investigate the heat and mass transfer of two immiscible permeable flids between vertical parallel plates.. MAHEMAIAL FORMULAION OF HE PROBLEM he geometr nder consideration illstrated in figre consists of two infinite parallel plates maintained at eqal or constant temperatre taking X axis along the midsection of channel and Y axis perpendiclar to walls. he region-i ( Y h is filled with a homogeneos isotropic poros material having permeabilit densit viscosit thermal condctivit K thermal expansion coefficient concentration expansion coefficient and diffsion coefficient D. he region-ii ( h Y is filled with another homogeneos isotropic poros material having permeabilit. his region is satrated with different viscos flid of densit viscosit thermal condctivit K thermal expansion coefficient concentration expansion coefficient and diffsion coefficient D. he flids are assmed to have constant propert except the densit in the boanc term in the momentm eqation. A flid rises in the channel driven b boanc force. he temperatre properties of both the flids are assmed to be constant. We consider the flids to be incompressible; flow is stead laminar and fll developed. It is assmed that the flid viscosit and Brinkman viscosit (i. e effective viscosit are same. he flow in both the regions is assmed to be driven b a dp dx and common constant pressre gradient temperatre gradient. It is also assmed w w that at an given instant the temperatre of the flid and the temperatre of solid are same. he temperatre and concentration of bondar at Y h is w and w while at Y h is w and w respectivel. Under these assmptions the governing eqations of motion energ and concentration for incompressible flids ields Region I: 5 IRJE.NE- All Rights Reserved Page 86

3 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: dp d U U w dx dy g ( g ( ( d du U dy dy ( d D K dy (3 Region II: dp du g ( w g ( dx dy ( U d du U dy dy (5 d D K dy (6 Y h Poros X Poros Fig-: Phsical configration Y h he bondar conditions on velocit are no-slip conditions and the two bondaries are held at constant different temperatres. In addition continit of velocit shear stress temperatre heat flx concentration and mass flx at the interface are assmed. U( h U( h U( U( du du ( ( ( h w ( h w dy dy d d ( ( ( ( dy dy Y ( h ( h ( ( d d D ( D ( (7 dy dy he non-dimensional parameters are U i i U Y i i hi 3 w w g h Gr Re Uh h w w w U Br K ( w w 3 g h Gc D D h h p U dp dx w w w h K h (8 he governing eqations ( to (6 can be written in a dimensionless form b emploing the dimensionless qantities (8 Region I: d GR GR p d (9 d d Br d d ( d ( d Region II: d GR mr h bt GR mr h bc mh p d d k d Br d m d ( (3 d ( d where Gr Gc h GR GR h m Re Re h w D bt r bc d D k w w 5 IRJE.NE- All Rights Reserved Page 863

4 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: he bondar and interface conditions in non-dimensional form become ( ( d d ( ( ( ( d mh d ( ( ( ( ( ( ( ( 3. MEHOD OF SOLUIONS 3. Pertrbation Method d d ( ( d kh d d d d ( ( (5 d h d Eqations (9 to ( are copled and highl non-linear eqations becase of viscos and Darc dissipation terms hence exact soltions cannot be fond. he approximate analtical soltions can be fond sing reglar pertrbation method. he Brinkman nmber can be exploited as the pertrbation parameter. herefore the soltions are assmed in the form ( ( Br ( Br (... (6 i i i i i ( i( Br i ( Br i(... (7 Using eqations (6 and (7 in eqations (9 ( ( and (3 and eqating the coefficients of like powers of Br to zero and one we determine zeroth and first order eqations as follows Region I: Zeroth order eqations: d (8 d d GR GR p d (9 First order eqations: d d d d d GR d ( ( Region II: Zeroth order eqations: d ( d d d GR mr h b GR mr h b mh p t c First order eqations: (3 d k d d m d d GR m r h bt d ( (5 he corresponding bondar and interface conditions as given in eqation (5 can be written as Zeroth order bondar and interface conditions d d ( ( d mh d ( ( ( ( d d ( ( (6 d kh d First order bondar and interface conditions ( ( ( ( d d ( ( d mh d ( ( ( ( d d ( ( (7 d kh d he soltions for eqations ( and ( are obtained directl B osh( B Sinh( (8 B osh( B Sinh( (9 3 he soltions of zeroth and first order eqations (8 to (5 are obtained b sing bondar and interface conditions as defined in eqations (6 and (7 respectivel and are given b c c (3 c c (3 3 ( Aosh( A Sinh( r r r osh( r Sinh( 3 ( A osh( A Sinh( r r r osh( r Sinh( 7 8 (3 (33 5 IRJE.NE- All Rights Reserved Page 86

5 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: ( E E q q q q osh( 3 3 q Sinh( q osh( q Sinh( q osh( q Sinh( q osh( 8 9 q Sinh( q osh( q Sinh( 3 q osh( q Sinh( 5 q6osh( 7 q osh( q Sinh( q Sinh( 8 9 ( E E F F F F osh( F Sinh( F osh( F Sinh( F osh( F Sinh( F osh( 8 9 F Sinh( F osh( F Sinh( 3 F osh( F Sinh( F osh( 5 6 F osh( F Sinh( 7 8 F Sinh( 9 ( E osh( E Sinh( H H H H H H osh( H Sinh( H osh( H Sinh( H osh( 8 9 H Sinh( H osh( H Sinh( 3 H cosh( H Sinh( 5 H6osh( H 7 osh 8 9 ( H Sinh( H Sinh( H osh( H Sinh( ( E osh( E Sinh( H H H H H H osh( H Sinh( H osh( H Sinh( H osh( H35 osh( H36Sinh( H37osh (3 (35 (36 H Sinh( H osh( H Sinh( (37 ( H osh( H Sinh( H Sinh( H osh( H Sinh( N ( h c Br E q 3q q 3 qsinh( q osh( q Sinh( q osh( q Sinh( q osh( q Sinh( 8 9 ( ( ( q osh( q Sin h( osh( q o sh( Sinh( q S inh( 3 osh q osh Si nh 5 q ( Sinh( q ( Sinh( 6 7 q ( osh( 8 q ( osh( 9 N _ h c Br E F 3F F F Sinh( F osh( F Sinh( F osh( F Sinh( F osh( F Sinh( 8 9 ( ( ( F osh( F Sinh( osh( F osh( Sin h( F Sin h( 3 osh F osh S inh 5 F ( Sinh( F ( Sinh( 6 7 F ( osh( F ( osh( 8 9 (38 (39 he constants appeared in the soltions are not presented as the can be obtained while finding the soltions. he dimensionless total volme flow rate is given b Qv Qv Qv ( where Qv d Qv d he dimensionless total heat rate added to the flid is given b E E E ( where E d E d Heat ransfer he wall heat transfer expression in terms of the Nsselt nmber becomes d N h at d d N _ h d at he dimensionless total species rate added to the flid is given b s s s ( where s d s d 5 IRJE.NE- All Rights Reserved Page 865

6 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: Eqations (9 to ( are evalated for different vales of the governing parameters and the reslts are presented graphicall. 3. Finite Difference Method he approximate analtical soltions obtained in the sbsection are valid for vales of Brinkman nmber less then one. However in man practical problems especiall when viscos dissipation dominates the Brinkman nmber takes the vales greater than one. In sch sitations it is reqired to find the approximate soltions nmericall. he governing eqations ( to (6 with the bondar and interface conditions (5 are solved sing FDM. In nmerical iterations comptation domain is divided into a niform grid sstem. he second derivative and the sqared first derivatives terms are discritized with central difference of second order accrac. B replacing the derivatives with the corresponding finite difference approximation we obtain a set of n algebraic eqations where n is the nmber of divisions from - to.o validate the present nmerical method compted soltions are compared with analtical soltions. he nmerical and analtical soltions agree ver well in the absence of Brinkman nmber and as the Brinkman nmber increases error between FDM and PM also increases. he soltions obtained b FDM and PM are depicted in able and percentage error between FDM and PM is also evalated.. RESULS AND DISUSSION in the vale of Grashof nmber means an increase of the boanc force which spports the motion. Frther figres a and b also reveal that the magnitde of velocit and temperatre is large in the absence of chemical reaction when compared with vales in the presence of the chemical reaction = p =. GR = Fig-a: Velocit profiles for different vales of thermal Grashof nmber GR.. = p =. he problem concerned is with the heat and mass transfer in a vertical channel for composite poros medim in the presence of homogeneos first order chemical reaction. he flow is modeled with Darc-Lapwood-Brinkman eqation. he viscos and Darc dissipation terms are inclded in the energ eqation. he continit of velocit temperatre shear stress heat flx concentration and mass flx at the interface is assmed. he eqations governing the flow which are highl nonlinear and copled are solved analticall sing pertrbation method (PM and nmericall sing finite difference method (FDM.he pertrbation soltions are valid for small vales of Brinkman nmber and nmerical soltions are valid for all vales of Brinkman nmber. he effect of thermal Grashof nmber GR on the velocit and temperatre fields is shown in figres a and b respectivel in the presence ( and in the absence ( of first order chemical reaction. As GR increases the flow increases in both the regions. Phsicall an increase. 6 GR = GR = 6 Fig-b: emparetre profiles for different vales of thermal Grashof nmber GR. he effect of mass Grashof nmber GR on the velocit and temperatre fields shows the similar effect as that of thermal Grashof nmber as shown in figres 3a and 3b respectivel. hat is to sa that as GR increases flow increases in both the regions. he mass Grashof nmber is the ratio of species boanc force to the viscos force. As expected the flid velocit and temperatre increases de 5 IRJE.NE- All Rights Reserved Page 866

7 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: to the increase in the species boanc force. he effects of GR and GR on the flow were the similar reslts observed b Shivaiah and Anand Rao [35] for the flow past a vertical poros plate and Malashett et al. [-6] in the absence of chemical reaction = p =. GR = Fig-3a: Velocit profiles for different vales of mass Grashof nmber GR. he variation of velocit and temperatre for different vales of poros parameter ( is shown in figres a and b respectivel. prononced and as a reslt velocit is redced in both the regions. he effect of viscosit ratio m ( is to increase the velocit and temperatre fields in both the regions as shown in figres 5a and 5b respectivel. he viscosit ratio m is defined as the viscosit of the flid in region-i to the viscosit of the flid in region-ii. It is observed from figre 5b that the effect of viscosit ratio on the temperatre field is not ver significant Fig-a: Velocit profiles for different vales of poros parameter = p =.. = GR = GR = 6 Fig-3b: emperatre profiles for different vales of mass Grashof nmber GR. As the poros parameter increases the velocit temperatre decreases in both the regions. For large vales of the frictional drag resistance against the flow motion is Fig-b: emperatre profiles for different vales of poros parameter. he effect of width ratio h ( h h is to enhance velocit and temperatre field in both the regions as displaed in figres 6a and 6b respectivel. he width ratio h is defined as the ratio of width of the flid laer in region-ii to the width of the flid in region-i. It is well known that as h increases velocit increases which intern enhances the dissipation and hence reslts in enhancement of temperatre field also. 5 IRJE.NE- All Rights Reserved Page 867

8 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: m= Fig-5a: Velocit profiles for different vales of viscosit ratio m. he effect of condctivit ratio k ( K K on the flow is similar to the effects on viscosit ratio and width ratio as seen in figres 7a and 7b. he condctivit of the permeable flid laer in region-i is large compared to the condctivit of flid laer in region-ii larger the amont of heat transfer and hence velocit also increases h=.5. Fig-6b: emperatre profiles for different vales of width ratio h. m=.5.5 m= k= Fig-5b: emperatre profiles for different vales of viscosit ratio m Fig-7a: Velocit profiles for different vales of thermal condctivit ratio k h=.5 Fig-6a: Velocit profiles for different vales of width ratio h....5 k= Fig-7b: emperatre profiles for different vales of thermal condctivit ratio k. 5 IRJE.NE- All Rights Reserved Page 868

9 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: he effects of h and k in the presence of first order chemical reaction was the similar reslts observed b Malashett [] in the absence of first order chemical reaction..3. =.5.5 he effect of first order chemical reaction parameter on velocit temperatre and concentration fields is depicted in figres 8a 8b and 8c respectivel. It is evident from these figres that as increases the velocit temperatre and concentration are redced in both the regions. Phsicall an increase in the vales of increases in nmber of solte molecles that ndergoing chemical reaction reslting in decrease in the flid flow. his was the similar reslts observed b Damesh and Shannak [36] for viscoelastic flid and Krishnend Bhattachara [37] for viscos flid Fig-8a: Velocit profiles for different vales chemical reaction parameter. Qv m h k p = GR. = Fig-8b: emperatre profiles for different vales chemical reaction parameter. Fig-9: Effect of mass Grashof nmber viscosit ratio width ratio and condctivit ratio on the volme flow rate. Frther one can also come to the conclsion from figres 9 and that as m h and k increases the total volmetric flow rate species concentration and heat rate also increases. he vales of total volmetric flow rate species concentration and heat rate remains the same when m h k.... m h k p = -5. =.5.5 s Fig-8c: oncentration profiles for different vales chemical reaction parameter GR Fig-: Effect of mass Grashof nmber viscosit ratio width ratio and condctivit ratio on total species rate added to the flid. 5 IRJE.NE- All Rights Reserved Page 869

10 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: E 3 m h k p = GR.5 Fig-: Effect of mass Grashof nmber viscosit ratio width ratio and condctivit ratio on total heat rate added to the flid. his is the valid reslt becase considering all the ratios to be eqal to one implies the channel is filled with same poros flids in both the regions. However variation of m h and k for vales not eqal to one shows the different profiles for total volmetric flow rate species concentration and heat rate. In all the three graphs the magnitde of volmetric flow rate species concentration and heat rate is large for k when compared with m and h. he magnitde of volmetric flow rate species concentration and heat rate is optimal for m when compared with h. he Nsselt nmber at the cold ( N and hot walls ( N is shown in figre for variations of mass Grashof nmber GR. It is seen that as GR increases N and N increases in magnitde. N 6 p = N_ N GR Fig-: Effect of mass Grashof nmber on the Nsselt nmber. he effect of Brinkman nmber on the velocit and temperatre field is shown in table. It is seen that as the Brinkman nmber increases the velocit and temperatre increase in both the regions. An increase in Brinkman nmber reslts in increase of dissipation effects which reslt in an increase of temperatre and as a conseqence velocit increase for the increase in boanc force in the momentm eqation. his table also shows a comparison of nmerical and analtical soltions. It is seen that analtical and nmerical soltions are exact to the order of in the absence of Brinkman nmber and the difference increases as the Brinkman nmber increases. Frther the percentage of error is also calclated and shown in table. 5. ONLUSIONS he problem of heat and mass transfer in a vertical channel filled with poros immiscible flids was analzed analticall b sing reglar pertrbation method and nmericall b finite difference method. he following conclsions are drawn. he effect of thermal Grashof nmber and mass Grashof nmber was to enhance the velocit and temperatre fields.. he effect of poros parameter is to sppress the flow in both regions. 3. he larger the vales of viscosit ratio width ratio condctivit ratio the larger the flow field.. he flow field was fond to be less in the presence of first order chemical reaction parameter when compared in the absence of chemical reaction parameter. Frther as the chemical reaction rate parameter increases heat and mass transfer decreases. 5. he volmetric flow rate species concentration and heat rate added to the flow was to increase for increasing vales of mass Grashof nmber viscosit ratio width ratio and condctivit ratio. 6. Nsselt nmber at the hot and cold wall increases in magnitde for increasing vales of mass Grashof nmber. 7. he percentage of error between analtical and nmerical soltions increases as the Brinkman nmber increases. able-: Velocit and temperatre vales for different vales of Brinkman nmber with = = = GR p = - GR =. 5 IRJE.NE- All Rights Reserved Page 87

11 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: Velocit Br Br.5 Br.5 FD PM %Error FD PM %Error FD PM %Error emperatre AKNOWLEDGMEN One of the athor Dr. J. Prathap Kmar wold like to thank Universit Grants ommission New Delhi for the financial spport nder UG- Major Research project (Project No. - 77/ (SR. REFERENES [] D. A. Nield A. Bejan onvection in Poros Media. New York Springer-Verlag 6. [] M. Kavian Principles of heat transfer in poros media. Springer-Verlag New York 99. [3] K. Vafai. L. ien Bondar and inertia effects on flow and heat transfer in poros media Int. J. Heat Mass ransfer vol. pp [] M. Kavian Laminar flow throgh a poros channel bonded b isothermal parallel plates International Jornal of Heat Mass ransfer vol. 8 pp [5] N. Rdriah S.. Nagraj Natral convection throgh vertical poros stratm Int. J. Eng. Sci. vol. 5 pp [6]. Beckermann R. Viskanta S. Ramadhani Natral convection in vertical enclosres containing simltaneosl flid and poros laers he Jornal of Flid Mechanics vol. 86 pp [7] A. K. Singh. Pal G. R. horpe Natral convection de to heat and mass transfer in a omposite sstem Heat and Mass ransfer vol. 35 pp [8] D. Polikkos M. Kazmierczak Forced convection in a dct partiall filled with a poros material ASME J. Heat ransfer vol. 9 pp [9] A.V. Kznetsov Analtical std of flid flow and heat transfer dring forced convection in a composite channel partl filled with a Brinkman-Forchheimer poros medim Flow rblence ombst vol. 6 pp [] V. Prasad onvective flow interaction and heat transfer between flid and poros laers.in: Proceeding of NAO advanced Std Institte on onvective Heat and Mass ransfer in Poros Medim Izmir rke 99. [] B. Alzami K. Vafai Analsis of flid flow and Heat transfer interfacial conditions between a poros medim and flid laer Int. J. Heat Mass ransfer vol. pp [] J. W. Paek B. H. Kang S. Y. Kim J. M. Han Effective thermal condctivit and permeabilit of alminm foam materials Int. J. hermophs vol. pp IRJE.NE- All Rights Reserved Page 87

12 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: [3] D. A. Nield A. V. Kznetsov Effect of heterogeneit in forced convection in poros medim: Parallel plate channel or circlar dct Int. J. Heat Mass ransfer vol. 3 pp [] M. S. Malashett J.. Umavthi J. Prathap Kmar onvective flow and heat transfer in a composite poros medim J. Poros Media vol. pp. 5-. [5] M. S. Malashett J.. Umavthi J. Prathap Kmar wo flid flow heat transfer in an inclined channel containing poros and flid laer Heat Mass ransfer vol. pp [6] M. S. Malashett J.. Umavthi J. Prathap Kmar Flow and heat transfer in an inclined channel containing a flid laer sandwiched between two poros laers J. Poros Media vol. 8 pp [7] J.. Umavthi J. Prathap Kmar A. J. hamaka I. Pop Mixed convection in a vertical poros channel ransp. Poros Media vol. 6 pp [8] J.. Umavthi A. J. hamaka Abdl Mateen A. AlMdhaf Oscillator flow and heat transfer in a horizontal vertical channel with asmmetric wall heating conditions Jornal of poros media vol. 33 pp [9] J.. Umavthi A. J. hamaka Abdl Mateen A. AlMdhaf Unstead oscillator flow and heat transfer in a horizontal composite poros medim channel Nonlinear Analsis: Modeling and ontrol vol. pp [] J.. Umavthi J. Prathap Kmar K. S. R. Sridhar Flow and heat transfer of Poiseille- oette flow in a inclined channel for composite poros medim Int. J. Applied Mechanics and Enginring vol. 5 pp [] J.. Umavathi J. Ali. hamkha K. S. R. Sridhar Generalized plain coette flow and heat transfer in a composite channel ransp. Poros Media vol. 85 pp a. [] J.. Umavathi I.. Li J. Prathap Kmar D. Shaik- Meera Unstead flow and heat transfer of poros media sandwiched between viscos flids Appl. Math. Mec. - Engl. Ed. vol. 3 pp b. [3] J.. Umavathi I.. Liand H. H. Wang Poiselle-oette flow and heat transfer an inclined channel for composite poros medim Jornal of Mechanics vol. 8 pp. 3. [] J.. Umavathi M. H. Manjla onvective two flid flows throgh a vertical channel AMSE Modeling Measrement and ontrol vol. 75 pp [5] J.. Umavathi Free convection of composite poros medim in a vertical channel Heat ransfer-asian Research vol. pp [6] J. Prathap Kmar J.. Umavathi I. Pop M. Basavaraj Biradar Fll developed mixed convection flow in a vertical channel containing poros and flid laer with isothermal or isoflx bondaries ransp. Poros Med. vol. 8 pp [7] J. Prathap Kmar J.. Umavathi I. Pop M. Basavaraj Biradar Mixed convection of a composite poros medim in a composite poros medim channel Int. J. Heat and ech. vol. 5 pp [8] A. J. hamkha MHD flow of a niforml stretched vertical permeable srface in the presence of heat generation/absorption and chemical reaction Int. omm. Heat Mass ransfer vol. 3 pp [9] R. Mthcmaraswam P. Ganeshan Natral convection on a moving isothermal vertical plate with chemical reaction Eng. Phs. hermophs vol. 75 pp [3] P. Singh K. Qeen Free convection heat and mass transfer along a vertical srface in a poros medim Acta Mech. vol. 3 pp [3] A. Postelinc Inflence of chemical reaction on heat and mass transfer b natral convection from vertical srfaces in poros media Heat and Mass ransfer vol. 3 pp [3] J. Prathap Kmar J.. Umavathi Shivakmar Madhavarao Dispersion in composite poros medim with homogeneos and heterogeneos chemical reactions Heat ransfer-asian Research vol. pp [33] J. Prathap Kmar J.. Umavathi Shivakmar Madhavarao Effect of homogeneos and heterogeneos reactions on the solte dispersion in composite poros medim Int. J. Eng. Sci. and ech. vol. pp a. [3] J. Prathap Kmar J.. Umavathi M. Basavaraj Biradar Effects of homogeneos and heterogeneos reactions on the dispersion of a solte for immiscible viscos flids between two plates Jornal of Applied Flid Mechanics vol. 5 pp. 3- b. 5 IRJE.NE- All Rights Reserved Page 87

13 International Research Jornal of Engineering and echnolog (IRJE e-issn: Volme: Isse: Ma-5 p-issn: [35] S. Shivaiah J. Anad Rao hemical reaction effects on an n stead MHD free convection flow past a vertical poros plate in the presence of sction or injection App Mech. vol. 39 pp [36] R. A. Damesh B. A. Shannak Viscoelastic flid flow past an infinite vertical poros plate in the presence of first order chemical reaction Applied Mathematics and Mechanics vol. 3 pp [37] Krishnend Bhattachara Slip effects on bondar laer flow and mass transfer with chemical reaction over a permeable flat plate in a poros medim Frontiers in Heat and Mass ransfer (FHM 3 36 DOI:.598/hmt.v3..36:-6. 5 IRJE.NE- All Rights Reserved Page 873