Y. Dharmendar Reddy and V. Srinivasa Rao Department of Mathematics, Anurag Group of Institutions, Ghatkesar, R.R.Dist.,Telangana State, India.

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1 Jornal of Ald Mahmacs and Fld Mchancs. ISS Volm 8 mbr (6). 5-3 Inrnaonal sarch Pblcaon Hos h://.rhos.com Fn Elmn Solon of hrmal adaon and Mass ransfr Flo as Sm- nfn Movng Vrcal Pla h Vscos Dssaon n Prsnc of ha Sorc / Sn. Y. Dharmndar dd and V. Srnvasa ao Darmn of Mahmacs Anrag Gro of Insons Ghasar..Ds.langana Sa Inda. Absrac An analss s carrd o o nvsga h radaon ffcs on nsad ha and mass ransfr flo of a chmcall racng fld as a sm-nfn vrcal la h vscos dssaon n rsnc of ha sorc / sn. h mhod of solon can b ald for Fn lmn chnq. mrcal rsls for h vloc h mrar and h concnraon ar shon grahcall. h xrssons for h sn-frcaon ssl nmbr and Shrood nmbr ar oband. h rsls sho ha ncrasd coolng (Gr>) of h la and h Ecr nmbr lads o a rs n h vloc. Also an ncras n h Ecr nmbr lads o an ncras n h mrar Whras ncras n sorc/sn aramr lad o ncras n h vloc and mrar dsrbon hn h la s bng coold. Kords: Ha ransfr Vscos dssaon adaon hmcal racon ha sorc/sn Fn lmn chnq. IODUIO For som ndsral alcaons sch as glass rodcon and frnac dsgn n sac chnolog alcaons cosmal flgh arodnamcs roc rolson ssms lasma hscs hch ora a hghr mrars radaon ffcs can b sgnfcan. Sondalgar and ahar [] consdrd h radav fr convcon flo of an ocall hn gr-gas as a sm-nfn vrcal la. adaon ffcs on mxd convcon along an sohrmal vrcal la r sdd b Hssan and ahar[]. as and Prds[3] hav sdd h ffcs of hrmal radaon and fr convcon flo as a movng vrcal la.

2 6 Y. Dharmndar dd and V. Srnvasa ao hamha al. [4] analzd h ffcs of radaon on fr convcon flo as a smnfn vrcal la h mass ransfr. Km and Fdorov [5] sdd ransn mxd radav convcon flo of a mcro olar fld as a movng sm-nfn vrcal oros la. Praash and Ogl [6] hav nvsgad an nsad o-dmnsonal flo of a radang and chmcall racng fld h m dndn scon. In man chmcal ngnrng rocsss hr dos occr h chmcal racon bn a forgn mass and h fld n hch h la s movng. hs rocsss a lac n nmros ndsral alcaons vz. Polmr rodcon manfacrng of cramcs or glassar and food rocsson. Das al. [7] hav sdd h ffcs of mass ransfr on flo as an mlsvl sard nfn vrcal la h consan ha flx and chmcal racon. In all hs nvsgaons h vscos dssaon s nglcd. h vscos dssaon ha n h naral convcv flo s moran hn h flo fld s of xrm sz or a lo mrar or n hgh gravaonal fld. Gbhar [8] shon h moranc of vscos dssav ha n fr convcon flo n h cas of sohrmal and consan ha flx n h la. Sondalgar [9] analzd h ffc of vscos dssav ha on h o dmnsonal nsad fr convcv flo as an vrcal oros la hn h mrar oscllas n m and hr s consan scon a h la. Isral oo al [] nvsgad h nflnc of vscos dssaon and radaon on nsad MHD fr convcon flo as an nfn had vrcal la n oros mdm h m dndn scon. h ha sorc/sn ffcs n hrmal convcon ar sgnfcan hr hr ma xs a hgh mrar dffrncs bn h srfac (.g. sac craf bod) and h ambn fld. Ha gnraon s also moran n h conx of xohrmc or ndohrmc chmcal racon. In rcn ars MHD flo roblms hav bcom n v of s sgnfcan alcaons n ndsral manfacrng rocsss sch as lasma sds rolm ndsrs Magno hdrodnamcs or gnraor coolng of clar racors bondar lar conrol n arodnamcs. Man ahors hav sdd h ffcs of magnc fld on mxd naral and forc convcon ha and mass ransfr roblms. h ffc of fr convcon on h acclrad flo of a vscos ncomrssbl fld as an nfn vrcal la h scon has man moran chnologcal alcaons n h asrohscal gohscal and ngnrng roblms. h hang of rooms and bldngs b h s of radaors s a famlar xaml of ha ransfr b fr convcon. adaon ffcs on nsad MHD flo hrogh a oros mdm h varabl mrar n rsnc of ha sorc/sn s sdd b Vaa Kmar al. []. Vaa shar and Vsanadh rdd [] hav oband h analcal solon for h ffcs of ha sn and chmcal racon on MHD fr convcv oscllaor flo as a oros la h vscos dssaon. Grsh Kmar and Saanaraana [3] sdd h ha and mass ransfr ffcs on nsad MHD fr convcv alr s mmor flo h consan scon n rsnc of ha sn. hodhr and Paban Dhar [4] nvsgad h ffcs of MHD Vsco-lasc fld as a movng la h dobl dffsv convcon n rsnc of ha gnraon/absoron.

3 adaon Effcs on Mass ransfr Flo hrogh A Hghl Poros Mdm h 7 h obcv of h rsn ar s o analz h radaon and mass ransfr ffcs on an nsad o-dmnsonal lamnar convcv bondar lar flo of a vscos ncomrssbl chmcall racng fld along a sm-nfn vrcal la h scon b ang no accon h ffcs of vscos dssaon n rsnc of ha sorc/sn. h qaons of conn lnar momnm nrg and dffson hch govrn h flo fld ar solvd b sng fn lmn chnq. h bhavor of h vloc mrar concnraon has bn dscssd for varaons n h govrnng aramrs. MAHEMAIAL AALYSIS An nsad o-dmnsonal lamnar bondar lar flo of a vscos ncomrssbl radang fld along a sm-nfn vrcal la n h rsnc of hrmal and concnraon boanc ffcs s consdrd b ang h ffc of vscos dssaon no accon. h x -axs s an along h vrcal nfn la n h ard drcon and h -axs normal o h la.h lvl of concnraon of forgn mass s assmd o b lo So ha h Sor and Dfor ffcs ar nglgbl. o ndr Bossnsq s aroxmaon h flo fld s govrnd b h follong qaons: v () v v g g () v c v D r qr c S c Whr v ar h vloc comonns n x drcons rscvl. - h m -h fld dns - h nmac vscos c - h scfc ha a consan rssr g -h acclraon d o grav and - h hrmal and concnraon xanson coffcn rscvl - h dmnsonal mrar - h dmnsonal concnraon -h fld hrmal dffsv - coffcn of vscos D - h mass dffsv - h chmcal racon aramr. r h bondar condons for h vloc mrar and concnraon flds ar U n n (3) (4) a = as (5)

4 8 Y. Dharmndar dd and V. Srnvasa ao Whr U s h scal of fr sram vloc and ar h all dmnsonal mrar and concnraon rscvl and ar h fr sram dmnsonal mrar and concnraon rscvl n - h consan. B sng ossland aroxmaon h radav ha flx s gvn b K q s r (6) Whr s - h Sfan-Bolzmann consan and K - h man absoron coffcn. I shold b nod ha b sng ossland aroxmaon h rsn analss s lmd o ocall hc flds. If mrar dffrncs hn h flo ar sffcn small hn qaon(6) can b lnarsd b xandng 4 n h alor srs abo hch afr nglcng hghr ordr rms a h form (7) In v of qaons (6) and (7) qaon (3) rdcs o s c S c K c c v (8) From h conn qaon () s clar ha scon vloc normal o h la s hr a consan or fncon of m. Hnc s assmd n h form n A V v (9) Whr A s a ral osv consan ε and εa ar small vals lss han n and V s scal of scon vloc a h la srfac. In ordr o r h govrnng qaons and h bondar condon n dmnson lss form h follong non- dmnsonal qans ar nrodcd. V n n V V U Pr V U v g Gm U V v g Gr D Sc V S S K V c U Ec s r r () In v of h qaons (6) - () Eqaons () - (4) rdc o h follong dmnsonlss form. Gm Gr A n () S Ec A n Pr ()

5 adaon Effcs on Mass ransfr Flo hrogh A Hghl Poros Mdm h 9 r n Sc A (3) Whr Sc Ec Gm Gr Pr and r ar h hrmal Grashof nmbr solal Grashof nmbr Prandl nmbr radaon aramr Ecr nmbr Schmd nmbr and chmcal racon aramr rscvl. h corrsondng bondar condons ar n n.5 a = as (4) SOLUIO OF HE POBLEM h Galrn qaon for h dffrnal qaon () bcoms d P (5) Whr n P A Gr Gm L h lnar cs aroxmaon solon ) ( ) ( ) ( ) ( ) ( Whr d P (6) glcng h frs rm n Eqaon (6) gs d P 6 l l P l Whr h l ) ( and do dnos h dffrnaon h rsc o. W r h lmn qaons for h lmns and assmbl hr lmn qaons oban 4 6 l P l l

6 Y. Dharmndar dd and V. Srnvasa ao o ro corrsondng o h nod o zro from Eqaon (6) h dffrnc schms s P l l 4 ( ) ( ) 6 Alng ran-cholson mhod o h abov qaon () hn gs A A A3 A4 A5 A6 P (7) Whr A 6r 3Phr A 8 r A 6r 3Phr 3 A 6r 3Phr A 8 r A 6r 3Phr * P ( Gr) ( Gm) ; Alng smlar rocdr o qaon () and () hn gs ** B B B3 B4 B5 B6 P (8) (9) Phr 6Ar S B Ar 4S B3 6Ar 3Phr S 4 6Ar 3Phr S 5 Ar 4S B 6 6Ar 3Phr S B 8 B B 8 ** Whr P Ec A Pr Whr Sc 6r 3PSc rh Q 8Sc r 4Q Sc 6r 3PSc rh Q Sc 6r PSc rh Q 5 8Sc r 4Q 6 Sc 6r 3PSc rh Q 4 3 Hr Q ScK r r and h h ar h msh szs along drcon and m drcon rscvl. Indx rfrs o h sac and rfrs o h m. In Eqaons (7)-(9) ang =()n and sng nal and bondar condons 3 h follong ssm of qaons ar oband: A X B =()3 () Whr A s ar marcs of ordr n and X B s colmn marcs havng n comonns. h solons of abov ssm of qaons ar oband b sng homas algorhm for vloc mrar and concnraon. Also nmrcal solons for hs qaons ar oband b -rogram. In ordr o rov h convrgnc and sabl of Galrn fn lmn mhod h sam -rogram as rn h slghl changd vals of h and and no sgnfcan chang as obsrvd n h vals of and. Hnc h Galrn fn lmn mhod s sabl and convrgn. h sn-frcon ssl nmbr and Shrood nmbr ar moran hscal aramrs for hs of bondar lar flo.

7 adaon Effcs on Mass ransfr Flo hrogh A Hghl Poros Mdm h h sn-frcon a h la hch n h non-dmnsonal form s gvn b f U V ) h ra of ha ransfr coffcn hch n h non-dmnsonal form n rms of h ssl nmbr s gvn b x x () h ra of ha ransfr coffcn hch n h non-dmnsonal form n rms of h Shrood nmbr s gvn b Sh x Sh x (3) Whr Vx x s h local nolds nmbr. ESULS AD DISUSSIO In h rcdng scons h roblm of an nsad fr convcv flo of a vscos ncomrssbl radang and dssang fld as a sm- nfn la h chmcall racng as formlad and solvd b fn lmn chnq. h xrssons for h vloc mrar and concnraon r oband. o llsra h bhavor of hs hscal qans nmrc vals r comd h rsc o h varaons n h govrnng aramrs vz. h hrmal Grashof nmbr Gr solal Grashof nmbr Gm Ecr nmbr Ec radaon aramr Prandl nmbr Pr Schmd nmbr Sc and chmcal racon aramr r. h vloc rofls for dffrn vals of h hrmal Grashof nmbr Gr ar dscrbd n fg.. I s obsrvd ha an ncras n Gr lads o ars n h vals of vloc. Hnc h osv vals of Gr corrsonds o coolng of h la. In addon s obsrvd ha h vloc ncrass radl nar h all of h la as Grashof nmbr ncrass and hn dcas o h fr sram vloc.

8 Y. Dharmndar dd and V. Srnvasa ao Gm=. Pr=.7 Ec=. Sc=.6 = r =A=.5ε=.n=. =.S= Gr= Fg.. Vloc rofls for dffrn vals of Gr For h cas dffrn vals of h solal Grashof nmbr Gm h vloc rofls n h bondar lar ar shon n fg.. I s nocd ha an ncras n Gm lads o a rs n h vals of vloc(s=.). Fgs 3(a) and 3(b) shos h vloc and mrar rofls for dffrn vals of h adaon aramr b ang S=. s nocd ha an ncras n h radaon

9 adaon Effcs on Mass ransfr Flo hrogh A Hghl Poros Mdm h 3 aramr rsls dcras n h vloc and mrar h n bondar lar as llas dcrasd h hcnss of h vloc and mrar bondar lars. h ffcs of h vscos dssaon aramr.. Ecr nmbr on h h vloc and mrar (S=.) ar shon n fg.4(a) and fg.4(b). Grar vscos dssav ha cass a rs n h mrar as ll as h vloc.

10 4 Y. Dharmndar dd and V. Srnvasa ao h ffc of h Prandl nmbr on h vloc and mrar(s=.) ar shon n Fg 5(a) and 5(b). As h Prandl nmbr ncrass h vloc and mrar dcrass.

11 adaon Effcs on Mass ransfr Flo hrogh A Hghl Poros Mdm h 5 h ffc of h Schmd nmbr on h vloc and concnraon (S=.) ar shon n Fg 6(a) and 6(b). As h Schmd nmbr ncrass h vloc and concnraon dcrass. hs cass h concnraon boanc ffcs o dcras ldng a rdcon n h fld vloc. dcons n h vloc and concnraon dsrbons ar accomand b smlanos rdcons n h vloc and concnraon bondar lars.

12 6 Y. Dharmndar dd and V. Srnvasa ao Fgs. 7(a) and 7(b) llsras h bhavor vloc and concnraon for dffrn vals of chmcal racon aramr r b ang S=.. I s obsrvd ha an ncras n lads o a dcras n boh h vals of vloc and concnraon.

13 adaon Effcs on Mass ransfr Flo hrogh A Hghl Poros Mdm h 7 Fgr 8(a) shos h ffcs of ha absoron aramr (S) for condcng ar(pr =.7) n h cas of coolng la (Gr > ).. h fr convcon crrns conv ha aa from h la no h bondar lar. Wh an ncras n S from -. (ha absoron) hrogh. o. (ha gnraon) hr s a clar ncras n h vloc.. h flo s acclrad. Whn ha s absorbd h boanc forc dcrass hch rards h flo ra and hrb gvng rs o h dcras n h vloc rofls. h mrar rofls θ ar dcd n Fg. 8(b) for dffrn vals of ha absoron aramr S h fld mrar s nocabl nhancd h an ncras n S from -. hrogh. o.. hs ncras n h mrar rofls s accomand b h smlanos ncras n h hrmal bondar lar hcnss..5 S= Gr=. Gm=. Sc=.6 =. n=. = r =A=.5 ε=. Pr=.7Ec=..5 3 Fgr 8(a). Effc of Ha Absoron Paramr S onvloc rofls.

14 8 Y. Dharmndar dd and V. Srnvasa ao.5 S = -... θ.5 3 Fgr 8(b). Effc of ha absoron aramr S on mrar Profls θ abl -6 rsn h ffcs of h hrmal Grashof nmbr solal Grashof nmbr radaon aramr Schmd nmbr and Ecr nmbr on h sn-frcaon coffcn ssl nmbr and Shrood nmbr. From abls and s obsrvd ha as Gr or Gm ncrass h sn frcon coffcn ncrass. Hovr from abl 3 can b sn ha as h radaon aramr ncrass h sn-frcon coffcn ncrass and ssl nmbr dcrass. From abl 4 s nocd ha an ncras n h Schmd nmbr rdcs h sn-frcon coffcn and ncrass h Shrood nmbr. Fnall s obsrvd from abl 5 ha as Ecr nmbr ncrass h sn-frcon coffcn ncrass and h ssl nmbr dcrass. From abl 6 can b sn ha as S ncrass h sn frcon coffcn ncrass and ssl nmbr dcrass. abl : Effc of Gr on f frnc vals as n Fg.. Gr f abl : Effc of Gm on f frnc vals as n Fg.. Gm f

15 adaon Effcs on Mass ransfr Flo hrogh A Hghl Poros Mdm h 9 abl 3: Effc of on f and frnc vals as n Fg.3(a)..5.. f abl 4: Effc of Sc on f and Sh frnc vals as n Fg.3(a) Sc Sh f abl 5: Effc of Ec on f and frnc vals as n Fg.3(a) Ec f abl 6: Effc of S on f and frnc vals as n Fg.3(a) S f OLUSIOS W hav formlad and solvd aroxmal h roblm of o-dmnsonal fld flo n h rsnc of radav ha ransfr vscos dssaon and chmcal racon aramr. A fn lmn chnq s mlod o solv h rslng cold aral dffrnal qaons. h conclsons of h sd ar as follos:. h vloc ncrass h h ncras n hrmal Grashof nmbr and solal Grashof nmbr.. An ncras n h Ecr nmbr ncrass h vloc and mrar.

16 3 Y. Dharmndar dd and V. Srnvasa ao 3. An ncras n h Prandl nmbr dcrass h vloc and mrar. 4. An ncras n h radaon aramr lads o ncras n h vloc and mrar. 5. An ncras n h ha sorc/sn aramr lads o ncras n h vloc and mrar. 6. h vloc as ll as concnraon dcrass h an ncras n h Schmd nmbr. 7. h vloc as ll as concnraon dcrass h an ncras n h chmcal racon aramr. AKOWLEDGME hs rsarch s carrd o ndr h UG Mnor rsarch Proc (MP- 469/4(SEO/UG)). h Ahors ar hanfl o UG for fnancal sor. EFEEES [] Sondalgar V.M. and ahar H.S. (993). adaon Effcs on Fr onvcon Flo Pas a Sm-nfn Vrcal Pla Modllng Masrmn and onrol [] Hossan M.A. and ahar H.S. (996). adaon Effcs on Mxd onvcon along a Vrcal Pla h Unform Srfac mrar Ha and Mass ransfr [3] as A. and Prds. (999). adaon and Fr onvcon Flo Pas a Movng Pla Al.Mch.Eng [4] hamha A.J. ahar H.S. and Sondalgar V.M. (). adaon Effcs on Fr onvcon Flo Pas a Sm-nfn Vrcal Pla h Mass ransfr hm. Engg. J [5] KmY.J. and FdorovA.G. (3).ransn Mxd adav onvcon Flo of a Mcro olar Fld Pas a Movng Sm-nfn Vrcal Poros Pla Inrnaonal Jornal of Ha and Mass ransfr [6] Praash J. and OglA. (6) Unsad o-dmnsonal Flo of a adang and hmcall acng MHD Fld h m-dndn Scon Indan J. Pr and Ald Phscs [7] Das U.. Da.K. and Sondalgar V.M.(994). Effcs of Mass ransfr on Flo Pas an Imlsvl sard Infn Vrcal Pla h onsan Ha Flx and hmcal racon Forschng n Ing-Engg sach [8] Sondagar.V.M. Ga.S.K and Bradar.S.S.(979) Effcs of mass ranfr and fr convcon ffcs on MHD sos roblm for a vrcal la clar Engnrng Dsgn [9] Gbhar. B. (96) Effcs of vscos dssav n naral convcon J. Fld Mch

17 adaon Effcs on Mass ransfr Flo hrogh A Hghl Poros Mdm h 3 [] Sondalgar. V.M. (97) Vscos dssav ffcs on nsad fr convcv flo as an vrcal oros la h consan scon In. J. Ha Mass ransfr [] A.G Vaa Kmar. Y aashara God. S.V.K Varma and K aghnah hrmal Dffson and radaon ffcs on nsad MHD flo hrogh oros mdm h varabl mrar and mass dffson n h rsnc of ha sorc/sn Aca chnca orvnnss Blln of Engnrng Vol. 6 o [] D Vaa Shar and G Vsanadh dd Effcs of chmcal racon on MHD fr convcv oscllaor flo as a oros la h vscos dssaon and ha sn Advancs n Ald Scnc sarch Vol. 3 o [3] J Grsh Kmar and P.V Saanaraana Mass ransfr ffcs on MHD nsad fr convcv Walr s mmor flo h consan scon and ha sn In. J. of Al. Mah and Mch. Vol.7 o [4] hodhr and D Paban Kmar Effcs of MHD vsco-lasc flo as a movng la h dobl dffsv convcon n rsnc of ha gnraon WSEAS ransacons on Fld Mchancs Vol

18 3 Y. Dharmndar dd and V. Srnvasa ao

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ISSN (ONLINE): , ISSN (PRINT): lm-7 Iss-4 Jl-Ags 7 Inrnanal Jrnal f Engnrng and Managmn Rsarch Pag Nmbr: 7-43 havr f assn Fld Sl Fl Pas a rcall Inclnd Pla Flld n Prs Mdm Sbmd n Magnc Fld: Ha Absrn and hmcal Racn Effcs P. Srsh M.. Ramana