Unsteady Convective Heat and Mass Transfer Flow of a Viscous Fluid in a Vertical Wavy Channel with Variable Wall Temperature and Concentration

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1 Aailable online a Aances in Applie Science Research :97-9 SS: 97-8 ODE USA: AASRF Unsea onecie Hea an Mass ranser Flow o a Viscos Fli in a Verical Wa hannel wih Variable Wall emperare an oncenraion. Linga Raj an P. Mralihar Deparmen o Engineering Mahemaics ollege o Engineering Anhra Uniersi Visakhapanam A.Pnia ABSRA We analse he eec o chemical reacion on nsea conecie hea an mass ranser low o a iscos li in a erical channel bone b wa walls. he walls are mainaine a oscillaor emperare an concenraion. he low akes place in he presence o emperare epenen hea sorces. he non-linear eqaions goerning he low hea an mass ranser are sole b emploing a reglar perrbaion echniqe. he eloci emperare an concenraion are iscsse graphicall or ieren ariaions o GMγ k an. he rae o hea an mass ranser on he bonar walls are ealae nmericall or ieren ariaions o he goerning parameers. Kewors: hemical reacion eec Wa channel Hea sorces. RODUO ranspor phenomena inoling he combine inlence o hermal an concenraion boanc are oen enconere in man engineering ssems an naral enironmens. here are man applicaions o sch ranspor processes in he insr noabl in chemical isilleries hea echangers solar energ collecors an hermal proecion ssems. n all sch classes o lows he riing orce is proie b a combinaion o hermal an chemical ision eecs. n amosphere lows hermal conecion o he earh b snligh is aece b ierences in waer apor concenraion. his boanc rien conecion e o cople hea an mass ranser in poros meia has also man imporan applicaions in energ relae engineering. hese incle moisre migraion ibros inslaion spreaing o chemical pollion in sarae soils eracion o geohermal energ an nergron isposal o naral wase. ombine hea an mass ranser b ree conecion ner bonar laer approimaions has been sie b Bejan an Khair[] Lai an Klacki[] an Mrh an Singh[8]. ople hea an mass ranser b mie conecion in Darcian li-sarae poros meia has been analse b Lai[]. he ree conecion hea an mass ranser in a poros enclosre has been sie recenl b Angirasa e al[]. he combine eecs o hermal an mass ision in channel lows has been sie in recen imes b a ew ahors noabl elson an Woo[] Lee a al[] an ohers[]. n recen ears energ an maerial saing consieraions hae prompe an epansion o he eors a procing eicien hea echanger eqipmen hrogh agmenaion o hea ranser. has been esablishe b Gagan[] ha channels wih ierging conerging geomeries agmen he ransporaion o hea ranser an momenm. As he li lows hrogh a oros pah iz. he ilae consrice geomer here will be more inimae conac beween hem. he low akes place boh aiall primar an ransersel seconar wih he seconar eloci being owars he ais in he li blk raher han conining wihin a hin laer as in sraigh channels. Hence i is aanageos o go or conerging ierging geomeries or improing he esign o hea ranser eqipmen. Vajrael an aeh[8] hae inesigae he inlence o he wall wainess on ricion an pressre rop o he generae coqee low. Vajrael an Sasr[] hae analse he ree conecion hea ranser in a iscos incompressible li conine beween long erical wa walls in he presence o consan hea sorce. Laer Vajrael an Debnah[9] hae eene his s o conecie low in a erical wa channel in or ieren 97

2 geomerical conigraions. his problem has been eene o he case o wa walls b McMichael an Desch[7] Deshikachar e al[] Rao e al[7] an Sree Ramachanra Mrh[]. Han Goo Kwon e al[8] hae analze ha he Flow an hea/mass ranser in a wa c wih arios corrgaion angles in wo imensional low regimes. Mah e al[] hae sie he mie conecion hea an mass ranser on a erical wa plae embee in a sarae poros meiaps/pse. omini e al[] hae analze he onecie hea an mass ranser in wa inne-be echangers. Jer-Han Jang e al [9] hae analze ha he mie conecion hea an mass ranser along a erical wa srace. Mhraj e.al[] hae consier mie hea an mass ranser hrogh a erical wa channel wih poros meim. Recenl Kmar[] has inesigae wo-imensional hea ranser o a ree conecie MHD low wih raiaion an emperare epenen hea sorces o a iscos incompressible li in a erical wa channel. Seeral ahors [89] hae iscsse he low in wa channels ner arie coniions. n man chemical engineering processes here oes occr he chemical reacion beween a oreign mass an he li in which he plae is moing. hese processes ake place in nmeros insrial applicaions iz. polmer procion manacring o ceramics or glassware an oo processing. Das e al[] hae sie he eecs o mass ranser on low pas an implsiel sare ininie erical plae wih consan hea l an chemical reacion. Mhkmaraswam [] has sie he eecs o reacion on a long srace wih scion. Recenl Gnaneswar[7] has sie raiaion an mass ranser on an nsea wo-imensional laminar conecie bonar laer low o a iscos incompressible chemicall reacing li along a semi-ininie erical plae wih scion b aking ino accon he eecs o iscos issipaion. he presen ren in he iel o chemical reacion analsis is o gie a mahemaical moel or he ssem o preic he reacor perormance. A large amon o research work has been repore in his iel. n pariclar he s o hea an mass ranser wih chemical reacion is o consierable imporance in chemical an hromeallrgical insries. hemical reacion can be coiie as eiher heerogeneos or homogeneos processes. his epens on wheher he occr a an inerace or as a single phase olme reacion. Freqenl he ransormaions procee in a moing li a siaion enconere in a nmber o echnological iels. A common area o ineres in he iel o aeronamics is he analsis o hermal bonar laer problems or wo imensional sea an incompressible laminar low passing a wege. Simlaneos hea an mass ranser rom ieren geomerics embee in a poros meia has man engineering an geophsical applicaion sch as geohermal reseroirs ring o poros solis hermal inslaion enhance oil recoer packebe caalic reacors cooling o nclear reacors an ner gron energ ranspor. A er signiican area o research in raiaie hea ranser a he presen ime is he nmerical simlaion o combine raiaion an conecion/concion ranspor processes. he eor has arisen largel e o he nee o opimize insrial ssem sch as rnaces oens an boilers an he ineres in or enironmen an in no conenional energ sorces sch as he se o sal-graien solar pons or energ collecion an sorage. n pariclar naral conecion ince b he simlaneos acion o boanc orces resling rom hermal an mass ision is o consierable ineres in nare an in man insrial applicaion sch as geophsics oceanograph ring process soliiicaion o binar allo an chemical engineering Mhkmarswam e al [9] Kanaswam e al[] hae iscsse he eecs o chemical reacion hea an mass ranser on bonar laer low oer a poros wege wih hea raiaion in he presence o scion or injecion. H osω osω - L /L L /L g onigraion o he Problem 98

3 99 n his paper we make an aemp o analse he nsea conecie hea an mass ranser low o iscos elecricall concing li conine in a erical channel on whose walls an oscillaor emperare an concenraion are prescribe. Approimae solions o cople non-linear parial ierenial eqaions goerning he low hea an mass ranser are sole b a perrbaion echniqe. he eloci emperare concenraion skin ricion an rae o hea an mass ranser are iscsse or ieren ariaions o GMEcP an. FORMULAO AD SOLUO OF HE PROBLEM We consier he nsea low o a iscos incompressible elecricall concing li in a erical wa channel bone b wa walls in he presence o consan hea sorces. he nseainess in he low is e o he oscillaor emperare an concenraion prescribe on he bonaries. We choose a aresian coorinae ssem wih walls a / L L ±. B sing Bossinesq approimaion we consier he ensi ariaion onl on he boanc erm also he kinemaic iscosi υ he hermal concii k are reae as consans. he eqaion goerning he low hea an mass ranser are g H p o o e σµ µ p µ Q k p K D where an are he eloci componens in an -irecions respeciel is a emperare is he oncenraion p is a pressre is a ensi k is he permeabili o he poros meim µ is namic iscosi k is coeicien o hermal concii is coeicien o olme epansion is he coeicien o epansion wih concenraion D is he moleclar isii K is he coeicien o chemical reacion qr is he raiaie hea l an Q is he srengh o hea sorce. he bonar coniions are a -L/L cosω cosω on L/L he low is rien b a consan l gien b L L L q 7 Eliminaing he pressre beween he eqaions& an inrocing he sream ncion eine b he resling eqaion is g g H o o e σµ ν 8 he eqaions o energ an ision are Q k p 9

4 9 K D On inrocing he non-imensional ariables / / / L L ν ω he goerning eqaions are in he non-imensional orm are G M γ γ P K Sc Sc γ where γ gl G Grasho nmber ν σµ L H M o e Harmann mber P K P µ Pranl nmber K L Q. Hea sorce parameer Boanc raio D Sc ν Schmi nmber ν ω γ L Wormsel mber ν K L K hemical reacion parameer M M he ransorme bonar coniions are a - cos ω cos ω a SOLUO OF HE PROBLEM On inrocing he ransormaions an 7 he eqaions - rece o γ G M F F F F 8 P γ 9

5 9 K Sc Sc γ n iew o he bonar coniions - we assme... ε e i... ε e i... ε e i Sbsiing he series epansion in eqaions 7- an separaing he sea an ransien erms we ge G M γ G M i P γ i P Sc K c Sc i k Sc γ 7 he eqaions -7 are non-linear in nare. Assming << we ake he solions as

6 9 Sbsiing 8 in eqaions -7 an separaing he like powers o we obain 9 Q Q G M G M Q 8 7 Q 8 7 G 9 7 G On soling he eqaions 9- sbjec o relean bonar coniions we obain. h h a h h a a

7 9 8 7 φ a a a h a φ h a a a a. h h. h h φ b h b 7 φ h b b h b b 8 8 φ e h e φ h h h h h h h h h h h 9 8 φ h φ h h h h h h h h h h h h h where M KSc P i Sc K i γ γ γ i M 7

8 an a a.a b b.b. 7 are consans. * FOR an or resls are in goo agreemen wih ARAHAR. USSEL UMBER an SHERWOOD UMBER he Rae o hea ranser ssel nmber a ± is gien b q w ± m w an he corresponing epressions are ε ± i i e e m m m he Rae o mass ranser erwoo mber a ± is gien b ± m w ± an he corresponing epressions are ε i i 7 e 9 8 e m m m where are consans. DSUSSO OF HE UMERAL RESULS We inesigae he eec o chemical reacion on he nsea conecie hea an mass ranser low o a iscos elecricall concing li in a erical wa channel which are mainaine a oscillaor emperare an concenraion. he walls are aken a ± ep. ε ε > correspons o ilaion o he channel walls an < represens consricion o he walls. We conine or aenion o he case o <. he aial eloci is shown in igs 9 or ieren ales o G MγK Sc an. he acal aial eloci is in he ericall ownwars. Fig. represens wih Grasho nmber. he magnie o enhances wih increase in G> an epreciaes wih G< wih maimm a. Higher he magneic orce lesser in he low region ig.. An increase in he srengh o he hea sorce enhances eerwhere in he region ig.. he ariaion o wih shows ha greaer he consricion o he channel walls smaller in he region ig.. An increase in Wormsel nmber γ leas o an enhancemen in. Also enhances wih increase in he chemical reacion parameer K ig.. When he moleclar boanc orce ominaes oer he hermal boanc orce he aial eloci enhances when he boanc orces ac in he same irecion an or he orces acing in he opposie irecions he eloci epreciaes in he low region ig.. he ariaion o wih Schmi nmber Sc shows ha lesser he moleclar isii larger ig.7. From ig.8 we in ha enhances wih higher hermal isii. Moing along he aial irecion he eloci enhances wih π an epreciaes wih higher π ig.9. εe i εe i 9

9 V V V Fig. : Variaion o wih G Fig. : Variaion o wih M V V V G M V Fig. : Variaion o wih Fig. : Variaion o wih V Fig. : Variaion o wih γ Fig. : Variaion o wih k γ... k... he seconar eloci is ehibie in igs -8 or ieren parameric ales. Fig. represens wih G. is on ha is owars he mi region in he righ hal an is owars he bonar in he le hal. enhances wih increase in G. Higher he Lorenz orce smaller in he enire low region ig.. An increase epreciaes an enhances wih ig.. Greaer he consricion o he channel walls larger in he low region ig.. From ig. we in ha he seconar eloci eperiences an enhancemen wih increase in γ. he ariaion o wih chemical reacion parameer k shows ha enhances wih increase in k in he le hal an in he righ hal epreciaes wih k. an enhances wih higher k.ig.. he ariaion o wih shows ha enhances wih > an reces wih < ig.. he ariaion o wih Sc shows ha enhances wih 9

10 increase in Sc. hs lesser he moleclar isii larger he magnie o eerwhere in he low regionig.7. Moing along he aial irecion he seconar eloci enhances wih ig V -.8 V Fig. 7 : Variaion o wih Fig. 8 : Variaion o wih Sc V V Sc V Fig. 9 : Variaion o wih V π/ π/ π π V V V Fig. : Variaion o wih G Fig. : Variaion o wih M V V V G M 9

11 V Fig. : Variaion o wih Fig. : Variaion o wih V Fig. : Variaion o wih γ Fig. : Variaion o wih k γ... k V V Fig. : Variaion o wih Fig. 7 : Variaion o wih Sc V V Sc

12 V Fig. 8 : Variaion o wih V π/ π/ π π he non-imensional emperare is shown in igs.9 8 or ieren parameric ales. We ollow he conenion ha he non imensional emperare is posiie or negaie accoring as he acal emperare is greaer / lesser han he. Fig.9 represens wih G. is on ha he acal emperare reces wih G > an enhances wih G in he enire region. he acal emperare enhances wih higher Lorenz orce ig V V.8. V Fig. 9 : Variaion o wih G Fig. : Variaion o wih M V V V G M V Fig. : Variaion o wih Fig. : Variaion o wih V

13 Fig. : Variaion o wih γ Fig. : Variaion o wih k γ... k V.8.. V Fig. : Variaion o wih Fig. : Variaion o wih Sc V V Sc.... Also an increase in leas o an enhancemen in he acal emperare in he low region ig.. Greaer he consricion o he channel walls larger he emperare ig.. From ig. we in ha epreciaes wih increase in γ. An increase in K leas o a epreciaion in he acal emperare ig.. he ariaion o wih shows ha he acal emperare reces wih > an enhances wih < ig.. Lesser he moleclar isii smaller he acal emperare in he low region ig.. Moing along he aial irecion he acal emperare eperiences an enhancemen eerwhere in he region ig.7. he ariaion o he emperare wih Pranl nmber P shows ha higher he hermal isii smaller he acal emperare ig V.8.. V Fig. 7 : Variaion o wih Fig. 8 : Variaion o wih P V V π/ π/ π π P

14 he non imensional concenraion is ehibie in igs.9-7 or ieren parameric ales. We ollow he conenion ha he non imensional concenraion is posiie or negaie accoring as he acal concenraion is greaer/lesser han. Fig.9 represens wih G. is on ha he acal concenraion reces wih increase in G > an enhances wih G. An increase M or reces he acal concenraion igs.&. Greaer he consricion o he channel walls higher he acal concenraion in he enire low region ig.. An increase in γ reces he acal concenraion ig.. From ig. i ollows ha an increase in he chemical reacion parameer k reces he acal concenraion an enhances wih higher k V V V Fig. 9 : Variaion o wih G Fig. : Variaion o wih M V V V G M V Fig. : Variaion o wih Fig. : Variaion o wih V V Fig. : Variaion o wih γ Fig. : Variaion o wih k γ... k... When he moleclar boanc orce ominaes oer he hermal boanc orce he acal concenraion enhances irrespecie o he irecions o he boanc orces ig.. he ariaion o wih Sc shows ha he acal 9

15 concenraion epreciaes wih lesser he moleclar isii ig.. Moing along he aial isance o he channel he acal concenraion reces wihin he enire low region ig V. V Fig. : Variaion o wih Fig. : Variaion o wih Sc V V Sc V Fig. 7 : Variaion o wih V π/ π/ π π he ssel nmber a ± is shown in ables - or ieren ales o G M γ K he rae o hea ranser epreciaes a an enhances a - wih increase in G > while or G < enhances a an reces a -. he ariaion o wih Harmann nmber M shows ha higher he Lorenz orce smaller a ±. An increase in he srengh o he hea sorce enhances a an reces a -. he inlence o srace geomer on is shown in ables &. is obsere ha greaer he consricion o he channel walls larger a an smaller a -. Also enhances wih increase in he Wormsel nmber γ a boh he wallsables &. he ariaion o wih chemical reacion parameer K shows ha higher he chemical reacion parameer K smaller a an smaller a -. When he moleclar boanc orce ominaes oer he hermal boanc orce he rae o hea ranser reces a boh he walls when he boanc orces ac in he same irecion an or he orces acing in opposie irecions i enhances a an reces a -ables &. able : ssel nmber a G V V V V V X X M γ

16 able : ssel nmber a G V V V K able : ssel nmber a G V V V V V X X M γ able : ssel nmber a G V V V K P he rae o mass ranser a ± is shown in ables 8 or ieren parameric ales. is on ha he rae o mass ranser enhances wih increase in G > an reces wih G a boh he walls. An increase in M leas o a epreciaion in a while a - i reces wih M an enhances wih higher M. he ariaion o wih shows ha greaer he consricion o he channel walls larger a an smaller a -. enhances a an reces a - wih increase in γ ables &7. he ariaion o wih K shows ha an increase in he chemical reacion parameer K enhances a an reces a -. able : erwoo mber a G V V V V V M γ able : erwoo mber a G V V V V V X K Sc

17 able 7 : erwoo mber a G V V V V V M γ able 8: erwoo mber a G V V V V V X K Sc he ariaion o wih boanc raio shows ha reces wih irrespecie o he irecions o he boanc orces. Also lesser he moleclar isii larger a an smaller a - ables &8. OLUSO n his analsis we inesigae he eec o chemical reacion an wall waness on he low characerisics. he imporan conclsions are:. he ariaion o wih shows ha greaer he consricion o he channel walls smaller in he region. enhances wih increase in he chemical reacion parameer K. When he moleclar boanc orce ominaes oer he hermal boanc orce he aial eloci enhances when he boanc orces ac in he same irecion an or he orces acing in he opposie irecions he eloci epreciaes in he low region.. Greaer he consricion o he channel walls larger in he low region. he ariaion o wih chemical reacion parameer k shows ha enhances wih increase in k in he le hal an in he righ hal epreciaes wih k. an enhances wih higher k.. he ariaion o wih shows ha enhances wih > an reces wih <.. Greaer he consricion o he channel walls larger he emperare. An increase in K leas o a epreciaion in he acal emperare. he ariaion o wih shows ha he acal emperare reces wih > an enhances wih <.. Greaer he consricion o he channel walls higher he acal concenraion in he enire low region. An increase in he chemical reacion parameer k reces he acal concenraion an enhances wih higher k... Greaer he consricion o he channel walls larger a an smaller a -. he ariaion o wih chemical reacion parameer K shows ha higher he chemical reacion parameer K smaller a an smaller a -.. Greaer he consricion o he channel walls larger a an smaller a -. enhances a an reces a - wih increase in γ ables 7&. he ariaion o wih K shows ha an increase in he chemical reacion parameer K enhances a an reces a -. REFEREES []. Angirasaa D Peerson G P an Pop ombine hea an mass ranser b naral conecion wih opposing boanc eecs in a li sarae poros meim n. J. Hea Mass ranser []. Bejan A an Khair K R Hea an Mass ranser b naral conecion in a poros meim n.j.hea Mass ransr []. omini.g onino an Saino S onecie hea an mass ranser in wa inne-be echangers nernaional jornals o merical Mehos or Hea an Fli low i []. Das U Deka R K an Sonalgekar V M Eecs o mass ranser on low pas an implsiel sare ininie erical plae wih consan hea l an chemical reacion Forschngim ngenierwesen []. Deshikachar K S an Ramachanra Rao A Eec o a magneic iel on he low an bloo ogenaion in channel o ariable cross-secion n. J. Engg Sci 98. 9

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19 [8]. Vajrael K an Ali eeh nlence o wall wainess on ricion an pressre rop in channels n. J. Mahs an Mah. Sc [9]. Vajrael K an Debnah L on-linear s o conecie hea ranser an li lows ince b raeling hermal waes Aca Mech []. Vajrael K an Sasr K S Hea ranser in a iscos incompressible li conine beween a long erical wa wall an a parallel la wall J.Fli Mech