Transient convective heat and mass transfer flow in an axially varying pipe with traveling thermal wave

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and Pasada Rao D. R. V. Depamen of Mahemaics KL

1 Available online a Pelaia Reseach Libay Advances in Applied Science Reseach 6 7:55-65 ISS: OD USA: AASR ansien convecive hea and mass ansfe flow in an aially vayin pipe wih avelin hemal wave Kamalaka P. V. S. Rahavende Rao R. and Pasada Rao D. R. V. Depamen of Mahemaics KL Univesiy Geenfields Gunu Andha Padesh India Depamen of Mahemaics S.K. Univesiy Ananhapuam Andha Padesh India ABSRA he effec of hemal adiaion and hemo-diffusion on he unseady combined hea and mass ansfe of a viscous incompessible fluid in a couaed pipe in he pesence of a consan hea souce. he unseadiness is due o a avelin hemal wave imposed on he bounday. akin he slope of he bounday wall of he pipe as a peubaion paamee he equaions ovenin he flow hea and mass ansfe have been solved. he velociy he empeaue and he concenaion have been evaluaed fo diffeen vaiaions. he effec of he waviness of he bounday on he flow phenomenon has been ehibied houh vaious pofiles of he velociy he empeaue and he concenaion. he shea sess he ae of hea and mass ansfe ae analyzed compuaionally fo diffeen paamees. Keywods: Hea and Mass ansfe hemal Radiaion hemo-diffusion hemal wave IRODUIO anspo phenomena involvin he combined influence of hemal and concenaion buoyancy ae ofen encouneed in many nineein sysems and aual envionmens. hee ae many applicaions of such anspo pocesses in he indusy noably in chemical disilleies hea echanes sola eney collecos and hemal poecion sysems. In all such classes of flows he divin foce is povided by a combinaion of hemal and chemical diffusion effecs. In amospheic flows hemal convecion of he eah by sunlih is affeced by diffeences in wae vapo concenaion. his buoyancy diven convecion due o coupled hea and mass ansfe in poous medium has also many impoan applicaions in eney elaed enineein. hese include moisue miaion fibous insulaion speadin of chemical polluion in sauaed soils eacion of eohemal eney and unde ound disposal of nuclea wase. his poblem of combined buoyancy diven hemal and mass diffusion has been sudied in paallel plae eomeies by a few auhos noably Gebha [8] Lai [5] hen Yuh and Mousolov [] Poulikakos [] Pop e al [] Anias e al [] evisan and Bejan [7]. Recenly Aniasa e al [] have pesened he analysis fo combined hea and mass ansfe by naual convecion fo aidin and opposin buoyancies in fluid sauaed poous enclosues. In mos of he sudies peainin o convecion flows houh he pipes he aial dependence of he flow vaiables [ and 9] is neleced and eihe he empeaue o is adien is mainained non-unifom on he bounday. Also he hea ansfe analysis is invesiaed in he absence of any inenal hea souces in he flow field. he hea ansfe in a flow houh a pipe in he pesence of addiional inenal hea souce has diec applicaion o he modified chemical vapo deposiion pocess. his MVD pocess is bein used o make hih qualiy opical Pelaia Reseach Libay 55

2 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7:55-65 lass fibes [589].In commecializaion of his poduc i is desiable o incease he efficiency of he hemophoeic deposiion ae since he cos of he fibes depends on he pedicion of poblems. Keepin hese facs in view Kishna e al [ ] have discussed he combined fee and foced convecion flow of a viscous incompessible fluid houh an aially vayin veical pipe. he poblem is analyzed as a eula peubaion poblem assumin he slope of he pipe wall o be small. he behavio of he velociy empeaue and hea ansfe coefficien is discussed based on numeical compuaions in deail. Recenly eeaja [8] has invesiaed he unseady mied convecion flow in a pipe of vayin ap in which he flow is mainained by a pescibed oscillaoy flu and he pipe is mainained a a consan empeaue. Seshasailaja e al [a] have sudied he effec of nonunifom empeaue on convecive hea ansfe flow in an aially vayin pipe In all hese sudies he hemal diffusion is no consideed. his assumpion is ue only when he flow akes place a low concenaion level. hee ae howeve some ecepions. he hemal diffusion effeccommonly known as Soe effec fo insances has been uilized fo isoope sepaaion and in miue beween ases wih vey lih molecula weihhhe and he medium molecula weihai he diffusion-hemo effec was found o be of a maniude such ha i canno be neleced [6]. In view of he impoance of his diffusion-hemo effec Jha and Sinh [] Kafoussias [] Ajay Kuma Sinh [] Rajpu e al [] Abdul e al [] have analyzed he convecion hea and mass ansfe wih Soe effec unde diffeen condiions. Reddy [5a] has discussed he unseady double diffusive convecive hea ansfe flow of a viscous fluid in a veical wavy pipe. ORMULAIO O H PROBLM onside he unseady aisymmeic flow of an incompessible viscous fluid in a veical pipe of vaiable cosssecion on which a avelin hemal wave is imposed. he Boussinesq appoimaion is used so ha he densiy vaiaion will be eained only in he buoyancy foce. he viscous dissipaion is neleced in compaison o he hea flow by convecion. he cylindical pola sysem O is chose wih -ais alon he ais of he pipe. he bounday of he pipe is assumed o be af / a whee a is chaaceisic adial lenh f is a wice diffeeniable funcion and is a small paamee popoional o he bounday slope. he flow is mainained by a consan volume flow ae fo which a chaaceisic velociy U is defined as U af / a a u d. he equaions ovenin he flow and hea ansfe ae q ρe ςq p µ q ρ.. q. q R e ρ p q. λ. q. D k.5 ρ ρe e e.6 Whee ρ e is he densiy of he fluid in he equilibium sae q is he velociy ς is he voiciy p is he pessue ae he empeaue and concenaion in he flow eion ρ is he densiy of he fluid p is he specific hea a consan pessue Q is he senh of he hea souce λ is he coefficien of hemal conduciviy is he coefficien of volume epansion * is he coefficien of epansion wih mass facion D is molecula diffusiviy k is he coss diffusiviy and q R is he adiaive hea flu. Invokin Roseland appoimaion Bewse a he adiaive hea flu is iven by Pelaia Reseach Libay 56

3 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7: Pelaia Reseach Libay R R q σ.7 and epandin by aylo s epansion afe nelecin hihe ode ems we e e e.8 In he equilibium sae e p e ρ.9 Whee p p e p d ; p d bein he hydodynamic pessue. Usin equaion.6 &.9 he equaion of momenum. educes o e e q p q q µ ς. akin cul on boh sides of equaion. and inoducin he seam funcion as v u. whee u is he aial velociy and v is he adial velociy componen he equaion in ems of is. whee On inoducin he non-dimensional vaiables a U m m a / he equaions..5&.afe doppin he dashes educe o Re R e G. P e P. ScSo R e Sc.5 Whee m a Aspec aio; a G Gashof numbe Ua e R Reynolds numbe; λ µ p P Pandl numbe

4 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7: Pelaia Reseach Libay e RP P Peclec numbe; D Sc Schmid numbe k S Soe paamee; Buoyancy aio e R σ λ Radiaion paamee Inoducin he ansfomaion f he ovenin equaions.-.5 educe o R e Gf fr e f.6 f fp P e.7 ScSo f fr e Sc Sc.8 Whee and we use he epansions as... o... o... o.9 Subsiuin.7 in equaions.-.6 and sepaain he like powes of he equaions coespondin o he zeoh ode ae f α. ScSo. o R e Gf. he coespondin condiions on and o ae.5 γ on Sin on. he equaions o he fis ode ae fp. ScSo fr e Sc.5 fr e R e Gf.6

5 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7:55-65 he coespondin bounday condiions ae on and ae n on on P P e α α whee SOLUIO O H PROBLM Solvin he coupled equaions.-. subjec o he bounday condiions. we e he epessions fo fis ode is αf ScSoαf A 6 A A o Sin γ Solvin he 9 6 coupled equaions.-.6 subjec o he coespondin bounday condiions.7 we e he epessions fo as A5 A 6 5 A7 7 A A 9 A 5 A 6 A 7 A 8 A 9 A A 9 6 A 7 A 8 A 9 A A A A 6 A 7 A 8 A Whee A A A 9 ae he consans USSL UMBR AD SHRWOOD UMBR he shea sess fo he moion on he pipe u v u v σ ij p ij ρe ij whee e e e.5 he shea sess on he pipe f in he non-dimensional fom is iven by τ σ f σ σ f / f In ems of he non-dimensional vaiables we obain he non-dimensional shea sess as P ' τ P εp ff n n f n P Whee P P f P f and he coespondin epession is τ B 8 B B f f f he local ae of hea ansfe ussel numbe on he bounday of he pipe is calculaed by usin he fomula u whee m.5 d f m w B B and he coespondin epession is u f B Sin D B he local ae of mass ansfe Shewood numbe on he bounday of he pipe is calculaed by usin he fomula Pelaia Reseach Libay 59

6 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7:55-65 Sh whee m.5 d and he coespondin epession is f m w B B. B ae consans B 5 B 6 Sh whee f B 7 B 8 RSULS AD DISUSSIO In his analysis we discuss he effec of adiaion on he unseady mied convecive hea and mass ansfe flow of a viscous fluid in a veical wavy cylinde on whose wall a avelin hemal wave is imposed. he ovenin equaions ae solved by usin a eula peubaion echnique. he velociy empeaue and concenaions ae discussed fo diffeen values of G R α and γ. able Shea sess τ a G I II III IV V VI VII VIII IX X XI XII XIII X X α Xγ π/ π/ π/ π/ π/ π/ π/ π/ π/ π/ π/ π π able ussel numbe u a G I II III IV V VI VII VIII IX X XI XII XIII X X α Xγ π/ π/ π/ π/ π/ π/ π/ π/ π/ π/ π/ π π able Shewood numbe Sh a G I II III IV V VI VII VIII IX X XI XII XIII X X α Xγ π/ π/ π/ π/ π/ π/ π/ π/ π/ π/ π/ π π he aial velociy u is shown in fis. - fo diffeen values of α and γ. he influence of he suface eomey on u in shown in fi. I is found ha hihe he dilaion of he pipe lae he velociy in he flow eion.6 and in he emainin eion he aial velociy educes wih.7 and enhances wih hihe.9. om fi we find ha an incease in he adiaion paamee leads o an enhancemen in u. he influence of hea souces on u is shown in fi. I is obseved ha he aial velociy enhances wih incease in α> and depeciaes wih α. An incease in he phase γ π of he avelin hemal wave enhances u and educes wih hihe γ π fi. he seconday velociy v which is due o he waviness of he pipe is shown in fi. 5-8 fo diffeen paameic values. Hihe he dilaion of he pipe lae v in he flow eion fi 5. he effec of adiaive hea ansfe on v is shown in fi 6. v epeiences depeciaion wih incease in he adiaion paamee. An incease in he senh of hea souce/sink esuls in a mainal depeciaion in he seconday velociy fi 7. om fi 8 we find ha v enhances wih incease in γ π and depeciaes wih hihe γ π. Pelaia Reseach Libay 6

7 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7:55-65 Pelaia Reseach Libay 6

8 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7:55-65 Pelaia Reseach Libay 6

9 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7:55-65 hen non-dimensional empeaue disibuion is shown in fis 9- fo diffeen values of he ovenin paamees. he influence of wall waviness on is shown in fi 9. Hihe he dilaion of he wavy cylinde lae he acual empeaue. om fi we find ha hihe he adiaive hea flu lae he acual empeaue. he acual empeaue enhances wih incease in α< and educes wih α fi. he vaiaion of wih phase γ shows ha he acual empeaue enhances wih γ π. hus i flucuaes wih γ. his is in view of he avelin hemal wave impose on fi. he non-dimensional concenaion is shown in fis. -6 fo diffeen paameic values. he vaiaion of wih dilaaion paamee > shows ha he hihe he dilaion of he pipe lesse he acual concenaion in he enie flow eion fi. om fi we noice ha lesse he adiaive hea flu lae he acual concenaion in he enie flow eion. he acual concenaion educes wih incease in α> and enhances wih α in he enie eion fi 5. he vaiaion of wih phase γ shows ha i educes wih γ π and enhances wih hihe γ πfi 6 he Shea sess τ he bounday of he wavy cylinde is ehibied in able fo diffeen values of α and γ. he vaiaion of τ wih hea souce paamee α shows ha an incease in he senh of he hea souce enhances τ a. An incease in he senh of hea sink α educes τ and fo α 6 τ educes in he heain case and enhances in he coolin case. I is found ha hihe he dilaion of he wavy pipe.7 lesse he sess a and fo fuhe hihe dilaion.9 lae he sess a he pipe. An incease in educes τ fo Pelaia Reseach Libay 6

10 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7:55-65 G> and enhances i fo G< a. An incease in he phase γ π hemal wave τ enhances in he heain case and educes in he coolin case while fo hihe γ π a evesed effec is noiced in he behavio of τ a. he aveae ussel numbe u a is shown in able fo diffeen paameic values. I is found ha u enhances wih α> and depeciaes wih incease α. We find ha hihe he dilaion of he wavy pipe smalle u a. Also u epeiences an enhancemen wih incease in he adiaion paamee. An incease in he phase γ π enhances u and educes u wih hihe γ π. he Shewood numbe Sh a is shown in able fo diffeen paameic values. he vaiaion of Sh wih hea souce paamee α is shows Sh enhances wih α> and educes wih incease in α. Hihe he dilaion of he wavy pipe smalle Sh a able. 8. We find ha he ae of mass ansfe educes wih incease in he adiaion paamee. he vaiaion of Sh wih phase γ shows ha he ae of mass ansfe enhances wih incease in γ π and educes wih hihe values of γ π. OLUSIO An incease in he adiaion paamee leads o an enhancemen in aial velociy acual empeaue and depeciaion in he seconday velociy. Lesse he adiaive hea flu lae he acual concenaion in he enie flow eion. An incease in educes τ fo G> and enhances i fo G< a. u Sh epeiences an enhancemen wih incease in he adiaion paamee. he aial velociy he acual empeaue enhances and he acual concenaion educes wih incease in α> and u depeciaes and enhances wih α in he enie eion. An incease in he senh of hea souce/sink esuls in a mainal depeciaion in he seconday velociy. An incease in he senh of he hea souce enhances τ a and wih an incease in he senh of hea sink α educes τ and fo α 6. τ educes in he heain case and enhances in he coolin case. u Sh enhances wih α> and depeciaes wih incease α. Hihe he dilaion of he pipe lae he velociy in he flow eion.6 and in he emainin eion he aial velociy educes wih.7 and enhances wih hihe.9. Hihe he dilaion of he pipe lae v he acual empeaue and lesse he acual concenaion in he flow eion. Hihe dilaion of he pipe lae he sess smalle u Sh a he bounday of he pipe. An incease in he phase γ π of he avelin hemal wave enhances u and educes wih hihe γ π. v enhances wih incease in γ π and depeciaes wih hihe γ π he acual empeaue enhances wih γ π. he vaiaion of wih phase γ shows ha i educes wih γ π and enhances wih hihe γ π. An incease in he phase γ π hemal wave τ enhances in he heain case and educes in he coolin case while fo hihe γ π An incease in he phase γ π enhances u and educes u wih hihe γ π ae of mass ansfe enhances wih incease in γ π and educes wih hihe values of γ π. RRS []Abdul Saa. Md and Mahmud Alam Md Inenaional Jounal of Pue and Applied Mahs []Ajay Kuma Sinh Inenaional Jounal of Pue and Applied Mahs -5. []Aniasa D and Peeson G.P and Pop I Inenaional Jounal of Hea and Mass ansfe [a]bewese [] hen.s. Yuh. and Mousolov H Inenaional Jounal of Hea and Mass ansfe [5]Dephew.A and Auus S. he Ameican Sociey of Mechanical ninees Jounal of Hea ansfe [6]cke L.R.G and Dake R.M Analysis of hea and mass ansfe McGaw-Hill Book o ew Yok 97 [7]ai M and Wely JR Inenaional Jounal of Hea and Mass ansfe [8]Gebha B Jaluia Y Mahajan R.L and Sammakia B: Buoyancy induced flows and anspo hape 5 Hemi Sphee Washinon Oc [9]Gill W.M and asal A.D Ameican hemical nineein Jounal []Heinbochal J.H and Ash R.L: ZAMP Pelaia Reseach Libay 6

11 Kamalaka P. V. S. e al Adv. Appl. Sci. Res. 6 7:55-65 []Jha B.K and Sinh A.K: Aso Physical Space Science []Kafoussias Aso Physical Space Science []Kishna D.V and Pasad Rao D.R.V: Poceedins of he Ameican Sociey of Mechanical ninees U.S.A []KishnaD.V Pasada Rao D.R.V and Saojamma G Inenaional ommunicaions in Hea Mass ansfe [5]Lai.J Inenaional ommunicaions in Hea Mass ansfe [6]Moon B.R J. Mech. Appl. Mahs [7]Moon B.R J. luid Mech [8]eeaja G: PhD hesis S.P. Mahila Univesiy iupahi India 99 [9]eild D.A and Bejan B: onvecion in poous media hape 9 Spine-Vela ew-yok 99 []Pop I and Hewin Inenaional ommunicaions in Hea Mass ansfe []Poulikakos Inenaional Jounal of Hea and Mass ansfe []Rajpu U.S Vashney.K and Debanana Aca iencia Indica 9M 5. []Rapis A.A Aso Physics and Space Science []Reynolds W. Ameican Sociey of Mechanical ninees Jounal of Hea ansfe [a]seshasailaja M and Saojamma G Advances in Applied Science Reseach [5]Simpkins P.G Geenbe-Kosinski and Maechsney J.B Jounal of Applied Physics [5a]Seenivasa eddy B PhD hesis S.K. Univesiy Ananapu A.P India 6. [6]evisan O.V and Bejan A Inenaional Jounal of Hea and Mass ansfe [7]evisan O.V and Bejan A Inenaional Jounal of Hea and Mass ansfe [8]Walke K.L Geoe M. Homsy and anz Geylin Jounal of colloid and ineface science [9]Walke. K.L Geylin. and ael S Jounal of Ameican eamic Sociey Pelaia Reseach Libay 65

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion