Transient Free Convection Flow Between Two Long Vertical Parallel Plates with Constant Temperature and Mass Diffusion

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1 Proceeings of he Worl Congress on Engineering 8 ol WCE 8, Jul - 4, 8, Lonon, U.K. Transien Free Convecion Flo Beeen To Long erical Parallel Plaes ih Consan Temperaure an Mass Diffusion Narahari Marneni Absrac Transien free convecion flo of a viscous an incompressible flui beeen o infinie verical parallel plaes in he presence of consan emperaure an mass iffusion has been invesigae analicall. The meho of Laplace ransform is use o solve he imensionless governing parial ifferenial equaions. The veloci, emperaure an concenraion profiles have been presene for ifferen parameers like Pranl number, Schmi number an for muliple buoanc effecs aiing an opposing. The values of he skin-fricion an volume flux are abulae. The ransien soluion approaches he sea sae hen he non-imensional ime becomes comparable ih he acual Schmi an Pranl numbers. Keors Transien free convecion, erical parallel plaes, Hea ransfer, Mass ransfer, Asmmeric heaing.. NTRODUCTON Free convecion flos in verical channels have been suie exensivel because of is imporance in man engineering applicaions. Osrach [] iniiae he su of full evelope free convecion beeen o verical alls ih consan emperaure. The firs exac soluion for free convecion in a verical parallel plae channel ih asmmeric heaing for a flui ih consan properies as presene b Aung []. Osrach [3], Booia an Oserle [4], Aung e al. [5], Miaake an Fujii [6-8], Miaake e al. [9], Lee an Yan [], Higuera an Razansev [], Camp e al. [], Panokraoras [3] have presene heir resuls for a sea free convecion flo beeen verical parallel plaes uner ifferen coniions on he all emperaure. The combine effec of hermal an mass buoanc forces on laminar free convecion flos beeen verical parallel plae channels has receive less aenion. This effec is foun o be imporan in man engineering siuaions, such as in he esign of hea exchangers, nuclear reacors, solar energ collecors, hermo proecion ssems an man chemical processes. Yan e al. [4] have suie he effec of laen hea ransfer associae ih he liqui films vaporizaion on he hea ransfer in he naural convecion flos riven b he combine buoanc forces of hermal an mass iffusion. Nelson an Woo [5-7] have presene numerical analsis Manuscrip receive Februar, 8. Narahari Marneni is ih he Elecrical an Elecronic Engineering Deparmen, Universii Teknologi PETRONAS, 375 Tronoh, Banar Seri skanar, Perak, Malasia ( marneni@peronas.com.m). of eveloping laminar flo beeen verical parallel plaes for combine hea an mass ransfer naural convecion ih uniform all emperaure/concenraion an uniform hea/mass flux bounar coniions. The also have presene an analical soluion for he full evelope combine hea an mass ransfer naural convecion beeen verical parallel plaes ih asmmeric bounar coniions. Lee [8] performe a combine numerical an heoreical invesigaion of laminar naural convecion hea an mass ransfer in open verical parallel plaes ih unheae enr an unheae exi for various hermal an concenraion bounar coniions. Desraau an Lauria [9] have examine he hea an mass ransfer analog for conensaion of humi air in a verical parallel plae channel. These papers iscuss he sea free convecion flos b consiering ifferen phsical siuaion of ranspor processes. Hoever, ver fe papers eal ih unsea flos in verical parallel plae channel. Transien consieraions ma be imporan if a cooling arrangemen is o be esigne using parallel plaes. Thus he knolege of he ransien an he sea-sae componens is significan o unersan he exac naure of hese siuaions. Singh e al. [] have suie he ransien free convecion flo of a viscous incompressible flui in a verical parallel plae channel hen he alls are heae asmmericall. Narahari e al. [] have suie he ransien free convecion flo beeen o verical parallel plaes ih consan hea flux a one bounar an he oher mainaine a a consan emperaure. Jha e al. [] have suie he ransien free convecion flo in a verical channel as a resul of smmeric heaing of he channel alls. Recenl, Singh an Paul [3] have presene an analsis for he ransien free convecive flo of a viscous an incompressible flui beeen o verical alls as a resul of asmmeric heaing or cooling of he alls. Bu he ransien free convecion flo beeen o infinie verical parallel plaes ih consan emperaure an mass iffusion a one bounar has no been suie in he lieraure, hence he moivaion. n Sec., he mahemaical analsis is presene an in Sec. 3, he conclusions are summarize.. MATHEMATCAL ANALYSS Here an unsea flo of a viscous incompressible flui beeen o verical parallel plaes ih consan emperaure an mass iffusion is consiere. The x -axis is aken along one of he plaes in he vericall upar SBN: WCE 8

2 Proceeings of he Worl Congress on Engineering 8 ol WCE 8, Jul - 4, 8, Lonon, U.K. irecion an he -axis is aken normal o he plaes. niiall, a ime, he o plaes an he flui are assume o be a he same emperaure T an concenraion. A ime >, he emperaure an concenraion of he plae a are raise o T an respecivel, causing he flo of free convecion currens. Then he flo can be shon o be governe b he folloing equaions uner usual Boussinesq s approximaions: u * u gβ ( T T ) gβ ( ) () T T ρ C p k () D (3) The iniial an bounar coniions are as follos: : u, T T, for, C > : u, T T, C a, u, T T, a. (4) C Here u is he veloci of he flui, g he acceleraion ue o gravi, β volumeric coefficien of hermal expansion, ime, he isance beeen o verical plaes, T he emperaure of he flui, T emperaure of * he plae a, β volumeric coefficien of concenraion expansion, species concenraion in he flui, species concenraion a he plae, he kinemaic viscosi, he coorinae axis normal o he plaes, ρ he ensi, C p he specific hea a consan pressure, k he hermal conucivi of he flui, D he mass iffusion coefficien, T emperaure of he plae a, species concenraion a he plae. We no inrouce he folloing non-imensional quaniies: u u,, u, gβ ( T T ) Gr gβ ( T T ) Gr C 3 T T, θ, T T * gβ ( T T ), Gm μc Pr k 3 p, Sc, D Gm N. (5) Gr, Where u he imensionless veloci, imensionless coorinae axis normal o he plaes, imensionless ime, θ he imensionless emperaure, C he imensionless concenraion, Gr hermal Grashof number, Gm mass Grashof number, μ he coefficien of viscosi, Pr he Pranl number, Sc he Schmi number, an N is he buoanc raio parameer. Then in vie of equaions (5), equaions () (4) reuce o he folloing non-imensional form of equaions: u u θ NC (6) θ θ Pr (7) C C Sc (8) The iniial an bounar coniions are : u, θ, C for, > : u, θ, C a, u, θ, C a. (9) The soluions o Eqs. (6) (8) saisfing he iniial an bounar coniions (9) are erive b he usual Laplace-ransform echnique as follos: Case : Sc (Sc ) N ) u (, ) )(Sc ) ( a a )erfc a b a b ) b Pr b ( b a Pr b Pr (b )erfc b Pr Pr )erfc a Pr ( a a Pr Pr )erfc N b Sc ( b Sc )erfc (Sc ) n b Sc b Sc ( a a Sc Sc )erfc SBN: WCE 8

3 Proceeings of he Worl Congress on Engineering 8 ol WCE 8, Jul - 4, 8, Lonon, U.K. a Sc a Sc Where a n, b n. () a Pr b Pr θ (, ) erfc erfc () a Sc b Sc C (, ) erfc erfc () Case : Sc u (, ) ) ( a a )erfc a b a ( b )erfc b a Pr b a Pr a erfc ( a a Pr Pr )erfc b Pr ( b Pr )erfc b Pr b Pr N a a Where c n. b b erfc b c c( n )erfc 4( n ) c b (3) compue for ifferen parameers like Pranl number, Schmi number, buoanc raio an ime. The buoanc raio parameer N represens he raio beeen mass an hermal buoanc forces. When N, here is no mass ransfer an he buoanc force is ue o he hermal iffusion onl. N > means ha mass buoanc force acs in he same irecion of hermal buoanc force, hile N < means ha mass buoanc force acs in he opposie irecion. Since he o mos commonl occurring fluis are amospheric air an aer, he resuls are limie o Pranl numbers of.7 (air) an 7. (aer). The effec of buoanc raio N for boh aiing an opposing flos are shon in Fig.. is observe ha he veloci increases in he presence of aiing flos hereas i ecreases in he presence of opposing flos. is also observe ha he veloci increases ih increasing he ime. eloci(u) eloci(u) Fig.. eloci profiles for ifferen N an Pr.7, Sc.6 N N. Sc Pr Sea sae a b C (, ) erfc erfc (4). The series in Eqs. () (4) can be shon o be absoluel convergen because of he presence of sanar mahemaical funcions. The numerical values of he veloci, emperaure, concenraion, skin-fricion an volume flo rae are Fig.. eloci profiles for ifferen Sc an SBN: WCE 8

4 Proceeings of he Worl Congress on Engineering 8 ol WCE 8, Jul - 4, 8, Lonon, U.K. To erive he soluions for sea sae, e pu ( ) / in Eqs. (6) (8) hich hen reuces o u θ NC (5) θ (6) C (7) These are solve using he bounar coniions (9) an hese sea-sae veloci, emperaure an concenraion profiles are compue an shon in Figs. o 4 as oe lines. When Temperaure(θ) Concenraion(C) Fig. 3. Temperaure profiles Pr Sea sae Fig. 4. Concenraion profiles Sc Sea sae compuing sea-sae soluions for veloci, emperaure an concenraion from Eqs. (6) (8), i is observe ha for., 5 he values of u for fixe N, θ an C for Sc.6, 5; Pr.7, 7. respecivel coincies ih hose erive from he soluion of Eqs. (5) (7). Hence he ransien soluion approaches he sea-sae hen he non-imensional ime becomes comparable ih he acual Schmi an Pranl numbers. n Fig., he veloci profiles are shon for ifferen values of Schmi number an ime. is observe ha an increase in Schmi number leas o a fall in he veloci. Also, he veloci increases ih increasing ime. The emperaure profiles are shon in Fig. 3 for ifferen values of Pranl number an ime. From his figure i is evien ha he emperaure increases ih increasing ime bu i falls oing o an increase in he Pranl number. The numerical values of he concenraion profiles are compue from Eqs. () an (4) an hese values are epice in Fig. 4 for ifferen values of Schmi number an ime. The effec of Schmi number is ver imporan in concenraion fiel. is observe ha he concenraion increases ih increasing ime bu ecreases ih increasing he value of he Schmi number. We no su he skin-fricion, hich is given in non-imensional form b Case : Sc gβ u ( T T ) ((Sc ) N )) )(Sc - ) ) N (Sc ) n n erfc n / n ( n ) erfc ( n ) / n Pr n Pr erfc Pr ( n ) ( n ) Pr erfc n n Sc erfc Pr Sc ( n ) Pr/ Pr Sc n Pr/ n Sc / SBN: WCE 8

5 Proceeings of he Worl Congress on Engineering 8 ol WCE 8, Jul - 4, 8, Lonon, U.K. an u ( n ) ( n ) Sc erfc ((Sc - ) N )) )(Sc - ) Sc( / π ) e Sc ( n ) Sc / (n ) erfc n (n ) (8) N ) ) ) n (n ) erfc (n ) Pr erfc Pr (n ) (n ) Pr (n ) Pr n n 3 N ( n ) erfc erfc () Case : Sc ) N (Sc ) N ) ) (n ) (n ) Pr erfc Pr (n ) Sc erfc Sc n erfc n (n ) Pr Pr (n ) Sc (n ) Sc n ( n ) erfc n n Pr erfc ) n / ( n ) / Pr Pr ( n ) Pr ( n ) Pr erfc Pr n Pr/ ( n ) Pr/ (9) n N ( n ) erfc () The numerical values of an are evaluae an hese are lise in Table. From his able, i is observe ha he skin-fricion increases ih increasing ime bu ecreases ih increasing he value of he Schmi an Pranl numbers. Phsicall, his is possible because fluis ih high Schmi an Pranl numbers move slol an hence here is less fricion a he plaes. Moreover, he skin-fricion increases in he presence of aiing flos an ecreases in he presence of opposing flos. is also compue he sea-sae value of he skin-fricion b calculaing an from Eqs. (8) an (9) for large values of ime for a fixe buoanc raio, for example N., an i is seen ha. 4 an. hich agree ell ih hose compue from heir sea-sae soluion obaine from Eq. (5). Table. Numerical values of, an Q Pr Sc N Sea sae Anoher ineresing phenomenon in his su is o unersan he effecs of, Sc, Pr an N on he volume flo rae hich is given b Q an SBN: WCE 8

6 Proceeings of he Worl Congress on Engineering 8 ol WCE 8, Jul - 4, 8, Lonon, U.K. Q 3 Q gβ ( T T ) u () Where Q is he non-imensional volume flux. We subsiue for u from () in Eq. (), an compue he inegral numericall using Simpson s rule. The numerical values of Q are lise in Table. is observe from his able ha he volume flux increases ih increasing ime an i ecreases ih increasing he value of he Schmi an Pranl numbers. is also observe ha he volume flux increases in he presence of aiing flos an ecreases in he presence of opposing flos.. CONCLUSONS An exac soluion of he ransien free convecion flo beeen o long verical parallel plaes ih consan emperaure an mass iffusion a one bounar is presene. The imensionless governing couple linear parial ifferenial equaions are solve b he usual Laplace-ransform echnique. The effec of ifferen parameers like buoanc raio, Schmi number, Pranl number an ime are suie. Conclusions of he su are as follos:. The veloci of he flui increases in he presence of aiing flos ( N > ) an ecreases ih opposing flos ( N < ).. The veloci increases ih increasing ime an i ecreases ih increasing he value of he Schmi number. 3. The emperaure increases ih increasing ime bu falls oing o an increase in he Pranl number. 4. The concenraion increases ih increasing ime bu ecreases ih increasing he value of he Schmi number. 5. The skin-fricion increases ih increasing ime bu ecreases ih increasing he value of he Schmi an Pranl numbers. Also, he skin-fricion increases in he presence of aiing flos an ecreases ih opposing flos. 6. The volume flux increases ih increasing ime an i ecreases ih increasing he value of he Schmi an Pranl numbers. Also, he volume flux increases in he presence of aiing flos an ecreases ih opposing flos. REFERENCES [] S. Osrach, Laminar naural-convecion flo an hea ransfer of fluis ih an ihou hea sources in channels ih consan all emperaures, NASA, Repor No. NACA-TN-863, 95. [] W. Aung, Full evelope laminar free convecion beeen verical plaes heae asmmericall, n. J. Hea Mass Transfer, vol. 5, 97, pp [3] S. Osrach, Combine naural an force convecion laminar flo an hea ransfer of fluis ih an ihou hea sources in channels ih linearl varing all emperaure, NASA, Repor No. NACA-TN-34, 954. [4] J. R. Booia an J. F. Oserle, The evelopmen of free convecion beeen heae verical plaes, ASME J. Hea Transfer, vol. 84, 96, pp [5] W. Aung, L. S. Flecher, an. Sernas, Develope laminar free convecion beeen verical flae plaes ih asmmeric heaing, n. J. Hea Mass Transfer, vol. 5, 97, pp [6] O. Miaake an T. Fujii, Free convecive hea ransfer beeen verical parallel plaes One plae isohermall heae an he oher hermall insulae, Hea Transfer-Jpn. Res., vol. (), 97, pp [7] O. Miaake an T. Fujii, Naural convecion hea ransfer beeen verical parallel plaes a unequal uniform emperaures, Hea Transfer-Jpn. Res., vol. (4), 973, pp [8] O. Miaake an T. Fujii, Naural convecion hea ransfer beeen verical parallel plaes ih unequal hea fluxes, Hea Transfer-Jpn. Res., vol. 3(3), 974, pp [9] O. Miaake, H. Tanaka, T. Fujii, an M. Fujii, Naural convecive hea ransfer beeen verical parallel plaes One plae ih a uniform hea flux an he oher hermall insulae, Hea Transfer-Jpn. Res., vol. (), 973, pp [] K. T. Lee an W. M. Yan, Laminar naural convecion beeen pariall heae verical parallel plaes, Wärme-un Soffüberragung (Hea an Mass Transfer), 9, pp. 45-5, 994. [] F. J. Higuera, an Yu. S. Razansev, Naural convecion flo ue o a hea source in a verical channel, n. J. Hea an Mass Transfer, vol. 45,, pp. 7-. [] A. Campo, O. Manca, B. Morrone, Numerical invesigaion of he naural convecion flos for lo-pranl fluis in verical parallel-plaes channels, ASME J. Hea Transfer, vol. 73, 6, pp [3] A. 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Lee, Naural convecion hea an mass ransfer in pariall heae verical parallel plaes, n. J. Hea an Mass Transfer, vol. 4, 999, pp [9] G. Desraau, an G. Lauria, Hea an mass ransfer analog for conensaion of humi air in verical channel, Hea an Mass Transfer, vol. 37,, pp [] A. K. Singh, H. R. Gholami an. M. Sounalgekar, Transien free convecion flo beeen o verical parallel plaes, Wärme-un Soffüberragung (Hea an Mass Transfer), vol. 3, 996, pp [] M. Narahari, S. Sreenah an. M. Sounalgekar, Transien free convecion flo beeen long verical parallel plaes ih consan hea flux a one bounar, J. Thermophsics an Aeromechanics, vol. 9(),, pp [] B. K. Jha, A. K. Singh, an H. S. Takhar, Transien free convecion flo in a verical channel ue o smmeric heaing, n. J. Applie Mechanics an Engineering, vol. 8(3), 3, pp [3] A. K. Singh, an T. Paul, Transien naural convecion beeen o verical alls heae/coole asmmericall, n. J. Applie Mechanics an Engineering, vol. (), 6, pp SBN: WCE 8