WALAILAK JOURNAL OF SCIENCE AND TECHNOLOGY

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1 HAPER-III RADIAIO EFFES O A USEADY MHD OVEIVE FLOW PAS A SEMI-IFIIE VERIAL PERMEABLE MOVIG PLAE EMBEDDED I A POROUS MEDIUM WIH VISOUS DISSIPAIO OES OF HE HAPER-III IS PUBLISHED I WALAILAK JOURAL OF SIEE AD EHOLOGY 34

2 3. IRODUIO Smltaneos heat and mass transfer from dfferent geometres embedded n poros meda has man engneerng and geophscal applcatons sch as geothermal reserors drng of poros solds thermal nslaton and enhanced ol recoer paced-bed cataltc reactors coolng of nclear reactors and ndergrond energ transport. Bean and Khar 7 (985 treated one of the most fndamental cases namel boanc- ndced heat and mass transfer from a ertcal plate embedded n a satrated poros medm. heng and Mnowcz 4 (977 presented smlart soltons for free conecton from a ertcal plate n a fld satrated poros medm. La and Klac 35 (99nestgated copled heat and mass transfer b mxed conecton from an sothermal ertcal plate n a poros medm. here has been a renewed nterest n stdng magnetohdrodnamc (MHD flow and heat transfer n poros and non-poros meda de to the effect of magnetc felds on the bondar laer flow control and on the performance of man sstems sng electrcall condctng flds. Rapts 53 et al (98 analzed hdromagnetc free conecton flow throgh a poros medm between two parallel plates. Grbben 7 (965presented the bondar laer flow oer a sem-nfnte plate wth an algned magnetc feld n the presence of pressre gradent. He obtaned soltons for large and small magnetc Prandtl nmber sng the method of matched asmptotc expanson. Helm 8 (998 presented an nstead two-dmensonal lamnar free conecton flow of an ncompressble electrcall condctng (ewtonan or polar fld throgh a poros medm bonded b nfnte ertcal plane srface of constant temperatre.gregantopolos 6 (98 stded two-dmensonal nstead free conecton 35

3 and mass transfer flow of an ncompressble scos dsspate and electrcall condctng fld past an nfnte ertcal poros plate. For some ndstral applcatons sch as glass prodcton and frnace desgn and n space technolog applcatons sch as cosmcal flght aerodnamcs rocet proplson sstems plasma phscs and spacecraft re-entr aerothermodnamcs whch operate at hgher temperatres radaton effects can be sgnfcant. hambre and Yong (958 hae presented a frst order chemcal reacton n the neghborhood of a horzontal plate. Deha 8 et al (994 nestgated the effect of the frst order homogeneos chemcal reacton on the process of an nstead flow past a ertcal plate wth a constant heat and mass transfer. Mthcmaraswam 4 ( presented heat and mass transfer effects on a contnosl mong sothermal ertcal srface wth nform scton b tang nto accont the homogeneos chemcal reacton of frst order. Mthcmaraswam and Meenashsndaram 4 (6 nestgated theoretcal std of chemcal reacton effects on ertcal oscllatng plate wth arable temperatre and mass dffson. In all these nestgatons the scos dsspaton s neglected. he scos dsspaton heat n the natral conecte flow s mportant when the flow feld s of extreme sze or at low temperatre or n hgh gratatonal feld. Gebhar 4( 96 shown the mportance of scos dsspate heat n free conecton flow n the case of sothermal and constant heat flx n the plate. Sondalgear 65 (97 analzed the effect of scos dsspate heat on the two dmensonal nstead free conecte flow past an ertcal poros plate when the temperatre oscllates n tme and there s constant scton at the plate. Israel ooe 3( 3 et al. Inestgated the nflence of scos 36

4 dsspaton and radaton on nstead MHD free conecton flow past an nfnte heated ertcal plate n poros medm wth tme dependent scton. he role of thermal radaton on the flow and heat transfer process s of maor mportance n the desgn of man adanced energ conerson sstems operatng at hgher temperatres. hermal radaton wthn these sstems s sall the reslt of emsson b hot walls and the worng fld. Baer and Gorla 5( 996 stded thermal radaton effects on mxed conecton from horzontal srfaces n poros medm. Baer 4 (reported the effect of radaton on the mxed conecton flow on an sothermal ertcal srface n asttrated poros medm and has obtaned self- smlar solton. Hossan and ahar 3 (996 analzed the effect of radaton on mxed conecton along a ertcal plate wth nform srface temperatre. Km and Fedoro 34 (3 analzed transent mxed radate conecte flow of a mcropolar fld past a mong sem-nfnte ertcal poros plate.radaton effects on an nstead MHD conecte heat and mass transfer flow past a sem-nfnte ertcal permeable mong plate embedded n a poros medm was stded b Ramachandra Prasad and Bhasar Redd 49 (8. he obecte of the present chapter s to analze the radaton and mass transfer effects on an nstead two-dmensonal lamnar mxed conecte bondar laer flow of a scos ncompressble electrcall condctng fld along a ertcal mong semnfnte permeable plate wth scton embedded n a nform poros medm n the presence of transerse magnetc feld b tang nto accont the effects of scos dsspaton. he eqatons of contnt lnear momentm energ and dffson whch goern the flow feld are soled b sng a fnte element method. he behaor of the 37

5 eloct temperatre concentraton sn-frcton sselt nmber and Sherwood nmber has been dscssed for aratons n the goernng parameters. 3. FORMULAIO OF HE PROBLEM Fg 3.. Phscal model and coordnate sstem of the problem An nstead two-dmensonal free conecton flow of a scos ncompressble electrcall condctng and radatng fld n an optcall thc fld past a sem- nfnte ertcal permeable mong plate embedded n a nform poros medm n the presence of thermal and concentraton boanc effects has been consdered. he x - axs s taen along the plate n the pward drecton and the - axs s taen normal to the plate. A nform magnetc feld s appled n the drecton perpendclar to the plate. he fld s assmed to be slghtl condctng and hence the magnetc Renolds nmber s mch less than nt and the ndced magnetc feld s neglgble n comparson wth the appled magnetc feld. he foregn mass present n the flow s 38

6 39 assmed to be at low leel and hence Soret and Dfor effects are neglgble. Frther de to the sem-nfnte plane srface assmpton the flow arables are fnctons of normal dstance and the tme t onl. ow nder the sal Bossneqe s approxmaton the goernng bondar laer eqatons of the problem are: (3. ( ( ρ σ ν β β ρ B K g g x p t (3. c q c t p p ν ρ ( q q α σ α (3.4 K D t r (3.5 Where are the eloct components n x drectons respectel. t - the tme ρ -the fld denst p - the pressre ν - the nematc scost σ - fld electrcal condctt s c - concentraton ssceptblt p c - the specfc heat at constant pressre g -the acceleraton de to grat β and β - the thermal and concentraton expanson coeffcent respectel B - the magnetc ndcton α - the fld thermal dffst K - the permeablt of the poros medm - temperatre of the fld n the bondar laer - speces concentraton n the bondar laer -the thermal condctt q - the radate heat flx σ - the Stefan-Boltzmann constant D - the

7 mass dffst r - the chemcal reacton parameter. he thrd and forth terms on the rght hand sde of the momentm eqaton (3. denote the thermal and concentraton boanc effects respectel. Also the second and thrd terms on rght hand sde of the energ eqaton (3.3 represent the radate heat flx and scos dsspaton respectel. It s assmed that the permeable plate moes wth a constant eloct n the drecton of the fld flow and the free stream eloct follows the exponentall ncreasng small pertrbaton law. In addton t s assmed that the temperatre and concentraton at the wall as well as the scton eloct are exponentall arng wth tme. Eqaton (3.4 s the dfferental approxmaton for radaton nder farl broad realstc assmptons. he bondar condtons for the eloct temperatre and concentraton felds are p U U ε ε n t ( e ε ( w at n t ( e as ε w e n t (3.6 Where p s the plate eloct w and w are the wall dmensonal temperatre and concentraton respectel and are the free stream dmensonal temperatre and concentraton respectel U the free stream eloct U and n -the constant. From the eqaton (3. t s clear that scton eloct normal to the plate s ether a constant or fncton of tme. Hence t s assmed n the form: ( n t Ae V ε (3.7 4

8 4 Where A s a real poste constant ε and A ε are small ales less than nt and V s scale of scton eloct whch s non zero poste constant. Otsde bondar laer eqaton (3. ges: ρ σ ν ρ B U K U dt du x p (3.8 Snce the medm s optcall thn wth relatel low denst and << α the radate heat flx gen b eqaton (3.3 wth reference to cogle 5 (968 et al. becomes: ( q 4α (3.9 Where B δλ α where B s Planc s fncton. In order to wrte the goernng eqatons and the bondar condtons n dmensonless form the followng non-dmensonal qanttes are ntrodced. V t t V n n U U U U U V V U w w p p θ ν ( ( Pr c U Ec U c R V B M D Sc V K K w p p w p ρ α ρ ν σ α ν ρ ν * ( ( V K K U V g Gm U V g Gr r r w w ν β β (3. In ew of eqatons (3.4 and ( eqatons (3. (3.3 and (3.5 redce to the followng dmensonless form: ( ( U Gm Gr dt du Ae t nt θ ε (3. Pr ( Ec R Ae t nt θ θ ε θ (3.

9 nt ( ε Ae Kr (3.3 t Sc where M and K Gr Gm M K Pr R Ec Sc and K r are the thermal Grashof nmber soltal Grashof nmber magnetc feld parameter permeablt parameter Prandtl nmber radaton parameter Ecert nmber Schmdt nmber and chemcal reacton parameter respectel. he correspondng bondar condtons are: U p U θ εe εe nt nt εe θ nt at as ( MEHOD OF SOLUIO B applng Galern fnte element method for eqaton (3. oer the element (e ( s: ( e ( e ( e ( e P t ( e Rd (3.5 Where nt P Aεe R nεe Grθ Gm U M K nt ( e ( ( ( ( ( e e e e ( e e P t eglectng the frst term n eqaton (3.6 then ( e R d (3.6 ( e ( e ( e P ( e ( e t ( e R d 4

10 43 Let ( ( ( e e e φ be the lnear pecewse approxmaton solton oer the element ( e ( where [ ] [ ] e e ( ( φ and are the bass fnctons. One obtans: d d P d ' ' ' ' ' ' ' ' ' ' ' ' d R d Smplfng we get ( 6 6 ( R l P l e e Where prme and dot denote dfferentaton w.r.t and tme t respectel. Assemblng the element eqatons for two consecte elements and followng s obtaned: ( ( R l P l e e (3.7 ow pt row correspondng to the node to zero from eqaton (3.7 the dfference schemes wth h l e ( s:

11 [[[ P 4 h h 6 6 Applng ran-cholson method to the aboe eqaton then we gets A [ ] [ ] 4 [ ] R A A 3 A 4 A 5 A 6 R (3.8 Applng smlar procedre to eqaton (3. and (3.3 then we gets B θ B θ B3 θ B4 θ B5 θ B6 θ R ( Where (3. A 6r 3Phr A r 4 8 A 6r 3Phr A 6r 3Phr A r 4 A 6r 3Phr 6 B Pr 6r 3PrPhr Pr R B 8Pr r 4PrR B 3 Pr 6r 3Pr Phr Pr R B4 Pr 6r 3PrPhr PrR B5 8Prr 4Pr R B6 Pr 6r 3Pr Phr Pr R Sc6r 3PScrh ScK 8Sc r 4ScK r r Sc 6r 3PScrh ScK Sc 6r 3PScrh ScK 3 r 4 r 8Sc r 4ScK Sc 6r 3PScrh ScK 5 r 6 r R * ( Gr ( Gm U θ nεe nt R r Pr Ec( [ ] [ ] 44

12 Here r and h h are the mesh szes along drecton and tme t drecton respectel. Index refers to the space and refers to the tme. In eqatons ( tang ( n and sng ntal and bondar condtons (3.4 the followng sstem of eqatons are obtaned: A X B (3. Where A s are matrces of order n and X B s colmn matrces hang n components. he soltons of aboe sstem of eqatons are obtaned b sng homas algorthm for eloct temperatre and concentraton. Also nmercal soltons for these eqatons are obtaned b -program. In order to proe the conergence and stablt of fnte element method the same -program was rn wth slghtl changed ales of h and and no sgnfcant change was obsered n the ales of θ and. Hence the fnte element method s stable and conergent. he sn-frcton sselt nmber and Sherwood nmber are mportant phscal parameters for ths tpe of bondar laer flow. he sn-frcton at the plate whch n the non-dmensonal form s gen b f τ w ρ U V (3. he rate of heat transfer coeffcent whch s n the non-dmensonal form n terms of the sselt nmber s gen b x w Re x θ (3.3 he rate of mass transfer coeffcent whch s n the non-dmensonal form n terms of the Sherwood nmber s gen b 45

13 46 Re x w Sh x Sh (3.4 Where ν x V x Re s the local Renolds nmber.

14 3.4 ABLES AD GRAPHS.3. Fg 3.4.: Effect of Gron eloct Gr Gm. M. K.5 Pr.7 R.5 Ec. Sc.6 n. t. A.5 K r.5 U p Fg 3.4.: Effect of Gm on eloct.3. Gm Gr. M. K.5 Pr.7 R.5 Ec. Sc.6 n. t. A.5 K r.5 U p

15 .3. Fg 3.4.3: Effect of M on eloct M GrGm. K.5 Pr.7 R.5 Ec. Sc.6 n. t. A.5 K r.5 U p Fg 3.4.4: Effect of K on eloct K GrGm. M. Pr.7 R.5 Ec. S c.6 n. t. A.5 K r.5 U p

16 .3. Fg 3.4.5: Effect of Pr on eloct Pr GrGm. M. K.5 R.5 Ec. Sc.6 n. t. A.5 K r.5 U p Fg 3.4.6: Effect of Pr on temperatre GrGm. M. K.5 R.5 Ec. Sc.6 n. t. A.5 K r.5 U p.5 Pr

17 .4.3. Fg 3.4.7: Effect of R on eloct R GrGm. M. K. Pr.7 Ec. Sc.65 K r.5 n. t. A.5 U p Fg 3.4.8: Effect of R on temperatre GrGm. M. K. Pr.7 Ec.Sc.65 K r.5 n. t. A.5 U p.5 R

18 .4.3. Fg 3.4.9: Effect of Ec on eloct Ec GrGm. M. K.5 Pr.7 R.5 Sc.6 n. t. A.5 K r.5 U p Fg 3.4.: Effect of Ec on temperatre GrGm. M. K.5 Pr.7 R.5 Sc.6 n. t. A.5 K r.5 U p.5 Ec

19 .3. Fg 3.4.: Effect of Sc on eloct Sc GrGm. M. K.5 Pr.7 R.5 Ec. n. t. A.5 K r.5 U p Fg 3.4.: Effect of Sc on concentraton GrGm. M. K.5 Pr.7 R.5 Ec. n. t. A.5 K r.5 U p.5 Sc

20 Fg Effect of K r on eloct K r GrGm. M. K.5 Pr.7 R.5 Ec. Sc.6 n. t. A.5 U p Fg Effect of K r on concentraton GrGm. M. K.5 Pr.7 R.5 Ec. Sc.6 n. t. A.5 U p.5 K r Fg Effect of U p on eloct 5 5 GrGm. M. K.5 Pr.7 R.5 Ec. Sc.6 K r.5 n. t. A.5 U p

21 able 3.4(a: Effects of Gr on Gr f f able 3.4(b: Effects Gm of on f Gm f able 3.4(c: Effects of R on and R f f Re x. Re x able 3.4(d: Effects of Sc on and f Sh Re x. Sc f Sh Re x

22 able 3.4(e: Effects of Ec on and Ec 3.5 RESULS AD DISUSSIO f f Re x. Re x he formlaton of the problem that acconts for effects of radaton on an nstead MHD conecte flow past a sem-nfnte ertcal permeable mong plate embedded n a poros medm wth scos dsspaton has been performed n the precedng sectons. he goernng eqatons of the flow feld were soled analtcall sng fnte element method and the expressons for the eloct temperatre concentraton sn-frcton sselt nmber and Sherwood nmber were obtaned. In order to get a phscal nsght of the problem the aboe phscal qanttes are compted nmercall for dfferent ales of the goernng parameters z. thermal Grashof nmber Gr soltal Grashof nmbergm magnetc parameter M permeablt parameter K Prandtl nmber Pr radaton parameter R Ecert nmber Ec Schmdt nmber Sc chemcal reacton parameter.effect of Grashof nmber on eloct K r and the plate eloct U p. Fg 3.4. presents tpcal eloct profles for aros ales of the Grashof nmber Gr. For the case of dfferent ales of thermal Grashof nmber Gr the eloct profles n the bondar laer are shown n Fg 3.4. As expected t s obsered that an ncrease n Gr leads to decrease n the ales of eloct de to enhancement n boanc force. Here the poste ales of Gr correspond to coolng of the srface. In addton 55

23 the cre show that the pea ales of the eloct decreases rapdl near the wall of the poros plate as Grashof nmber ncreases and then decas to the free stream eloct.. Effect of Soltal Grashof nmber on eloct Fg 3.4. presents tpcal eloct profles n the bondar laer for aros ales of the soltal Grashof nmber Gm whle all other parameters are ept at some fxed ales. he eloct dstrbton attans a dstncte maxmm ale n the cnt of the plate srface and then decrease properl to approach the free stream ale. As expected the fld eloct ncreases and the pea ale more dstncte de to ncrease n the concentraton boanc effects represented b Gm. hs s edent n the ncrease n the ale of as Gm ncreases n Fg Effect of magnetc feld parameter(m on eloct Fg presents tpcal eloct profles for aros ales of the magnetc feld parameter. he effect of magnetc feld on eloct profles n the bondar laer s depcted n Fg From ths fgre t s seen that the eloct starts from mnmm ale at the srface and ncrease tll t attans the pea ale and then starts decreasng ntl t reaches to the mnmm ale at the end of the bondar laer for all the ales of magnetc feld parameter. It s nterestng to note that the effect of magnetc feld s to decrease the ale of the eloct profles throghot the bondar laer. he effect of magnetc feld s more promnent at the pont of pea ale.e. the pea ale drastcall decreases wth ncreases n the ale of magnetc feld becase the presence of magnetc feld n an electrcall condctng fld ntrodce a force called the 56

24 Lorentz force whch acts aganst the flow f the magnetc feld s appled n the normal drecton as n the present problem. 4. Effect of permeablt parameter on eloct Fg3.4.4 Shows the eloct profles for dfferent ales of the permeablt parameter K clearl as K ncreases the pea ales of the eloct tends to ncrease. hs tpe of resstng force slows down the fld eloct as shown n ths fgre Fg Veloct and temperatre profles for dfferent ales of Prandtl nmber Fgs and llstrate the eloct and temperatre profles for dfferent ales of Prandtl nmber Pr. he nmercal reslts show that the effect of ncreasng ales of Prandtl nmber reslt n an ncreasng eloct. he nmercal reslts show that an ncrease n the Prandtl nmber reslts a decrease of the thermal bondar laer and n general lower aerage temperatre wthn the bondar laer. he reason s that smaller ales of Pr are eqalent to ncrease n the thermal condctt of the fld and therefore heat s able to dffse awa from the heated srface more rapdl for hgher ales of Pr. Hence n the case of smaller Prandtl nmber the thermal bondar laer s thcer and the rate of heat transfer s redced. 6.Veloct and temperatre profles for dfferent ales of Radaton parameter(r For dfferent ales of the radaton parameter R the eloct and temperatre profles are plotted n Fgs and s obos that an ncrease n the radaton parameter R reslts n a decrease n the eloct and temperatre wthn the bondar laer as well as decreased thcness of the eloct and temperatre bondar laer. 57

25 7. Veloct and temperatre profles for dfferent ales of Ecert nmber he effect of the scos dsspaton parameter.e. the Ecert nmber Ec on the eloct and temperatre are shown n Fgs and 3.4. Grater scos dsspate heat cases a rse temperatre as well as the eloct. 8. Veloct and concentraton profles for dfferent ales of Schmdt nmber Fgs 3.4. and 3.4. dspla the effects of Schmdt nmber Sc on eloct and concentraton profles respectel. As the Schmdt nmber ncreases the concentraton decreases. hs cases the concentraton boanc effects to decreases eldng a redcton n the fld eloct. he redctons and the eloct and concentraton profles are accompaned b smltaneos redctons n the eloct and concentraton bondar laers. 9. Effect of chemcal reacton on Veloct and oncentraton Fg shows that the eloct dstrbton aganst for dfferent ales of notced that the eloct decreases wth ncreases K r. We K r.fg dsplas the effects of the chemcal reacton K r on concentraton profles. We obsere that concentraton profles decreases wth ncreases K r.. Effect of plate mong eloct U p Fg llstrates the araton of eloct dstrbton across the bondar laer for seeral ales of plate mong eloct U p n the drecton of the fld flow. Althogh we hae dfferent ntal plate mong eloctes the eloct decrease to the constant ales for gen materal parameters. 58

26 . Profles of Sn frctonsselt nmber and Sherwood nmber able 3.4(a-3.4(e present the effects of the thermal Grashof nmber soltal Grashof nmber radaton parameter Schmdt nmber and Ecert nmber on the sn-frcton f sselt nmber and Sherwood nmber Sh. From able 3.4(a and 3.4(b t s obsered that as Gr or Gm ncreases the sn-frcton coeffcent ncreases. Howeer from able 3.4(c t can be seen that as the radaton parameter ncreases the snfrcton decreases and the sselt nmber ncreases. From able 3.4(d t s notced that an ncrease n the Schmdt nmber redces the sn-frcton and ncreases the Sherwood nmber. Fnall t s obsered from able 3.4(e that as Ecert nmber ncreases the sn-frcton ncreases and the sselt nmber decreases. 59