Finite Difference Solution of Mixed Convective Heat Transfer Transient Flow along a Continuously Moving Cooled Plate

Transcription

1 Annals of Pue and Applied Mathematis Vol 7, No, 14, 18-6 ISSN: 79-87X (P), (online) Published on 7 August 14 eseahmathsiog Annals of Finite Diffeene Solution of Mixed Convetive Heat Tansfe Tansient Flo along a Continuously Moving Cooled Plate Sade, M M Haque * and T Fedaus Mathematis Disipline Khulna nivesity, Khulna-98, Bangladesh * Coesponding autho s mmhaque@mathkuabd Reeived 1 August 14; aepted 6 August 14 Abstat Numeial study of a mixed onvetive heat tansfe tansitoy flo of a visous inompessible fluid along a ontinuously moving semi-infinite vetial ooled plate is ompleted hee It is also assumed that the plate is embedded in a poous medium This investigation is pefomed fo ooling poblem ith the both ai and ate A mathematial model elated to the poblem is developed fom the basis of studying Fluid Dynamis(FD) To solve the poblem, an expliit poedue of finite diffeene method ith stability and onvegene iteion has been used in this ok Both the loal and aveage shea stess ith Nusselt numbe is also omputed hee The obtained numeial values of veloity, tempeatue, shea stess and Nusselt numbe ae plotted in gaphs fo diffeent values of assoiated paametes as ell as the physial aspets of the poblem ae disussed in details Finally, some impotant findings ae onluded hee Keyods: Tansient Flo, Heat Tansfe, Mixed Convetive, Finite Diffeene AMS Mathematis Subet Classifiation (1): 58D3, 35Q35 1 Intodution The heat tansfe flo of an eletially onduting visous fluid is onsideed to be of signifiant impotane due to its appliation in many engineeing poblems suh as nulea eatos and those dealing ith liquid metals The onvetive heat tansfe poblems play a deisive ole in geothemal enegy eovey, oil extation, themal enegy stoage and flo though filteing devies Finston [1] studied a natual onvetive heat tansfe flo of fluid A simila solution fo lamina fee onvetion fom a non-isothemal vetial plate as omputed by Spao and Gegg [] A geneal seies solution of fee onvetive heat tansfe flo fom a non-isothemal vetial flat plate has been obtained by Kuiken [3] Quit eently, a numeial study is pefomed fo a fee onvetive heat tansfe flo of a visous fluid by Fadzilah et al [4] The flo though a poous medium is of speial inteest in many industial appliations Poous media ae vey idely used to insulate a heated body to maintain its tempeatue To make the heat insulation of the sufae moe effetive, it is neessay to study the fee 18

2 Finite Diffeene Solution of Mixed Convetive Heat Tansfe Tansient Flo onvetion flo though a poous medium Raptis et al [5] have obseved the steady fee onvetive flo though a poous medium bounded by an infinite sufae by use of the model of Yamamoto and Iamua [6] fo the flo nea the sufae Thee dimensional fee onvetive heat tansfe flo though a poous medium has been studied by Ahmed and Sama [7] Chaudhuy and Chand [8] futhe investigated the same poblem Reently, the analyti solutions fo unsteady fee onvetion in poous media have been obtained by Magyai et al [9] All the above oks ae elated to the stationay vetial plate Hoeve, the flo past a ontinuously moving plate has many appliations in manufatuing poesses suh as hot olling, metal and plasti extusion, ontinuous asting, glass fibe and pape podution Sakiadis [1] as the fist autho to eognize this bakad bounday laye situation and used a similaity tansfomation to obtain a numeial solution fo the flo field of a ontinuously moving plate The steady heat tansfe flo past a ontinuous moving plate ith vaiable tempeatue as analyzed by Soundalgeka and Ramana Muty [11] A solution of mixed onvetive heat tansfe flo along a ontinuously moving heated vetial plate ith sution o inetion has been omputed by Sami and Al-Sanea [1] The mixed onvetive fluid flos play an impotant ole in a numbe of industial appliations suh as fibe and ganula insulation, geothemal systems et Hene, ou main aim is to investigate a mixed onvetive heat tansfe unsteady flo along a ontinuously moving plate suounded by a poous medium Mathematial model of flo A time dependent mixed onvetive heat tansfe flo of an eletially onduting visous inompessible fluid past an eletially non-onduting semi-infinite vetial ooled plate embedded in a poous medium is onsideed hee The flo is also assumed to be in the x -dietion hih is taken along the plate in the upad dietion and y - axis is nomal to it Initially, e onside that the plate as ell as the fluid patiles ae at est at the same tempeatue T = T at all points, hee T be the fluid tempeatue of unifom flo It is assumed that the plate be at est afte that the plate is to be moving ith a onstant veloity in its on plane Within the fameok of the above stated assumptions, the equations elevant to the pesent poblem ae govened by the folloing system of oupled non-linea patial diffeential equations, u v Continuity Equation + = x y u + t u + x u = y u υ β α + υ y K Momentum Equation u v g ( T T ) u Enegy Equation T T T + u + v t x y κ T υ u = + ρ p y p y 19

3 Sade, M M Haque and T Fedous The oesponding initial and bounday onditions ae given belo, t =, u = v = T = T eveyhee t >, u = v = T = T at x = u = v = T = T at y = u = v = T = T as y hee x and y be the Catesian oodinates, u & v ae veloity omponents, t denotes time, g is the loal aeleation due to gavity, β is themal expansion oeffiient, υ is kinemati visosity, ρ is density, K is the pemeability of the poous medium, κ is themal ondutivity, p is speifi heat at onstant pessue and T be the fluid tempeatue nea at plate 3 Mathematial fomulation To find the solution of the poblem, it is equied to tansfe the system of equations into a non-dimensional system, so e take the folloing dimensionless quantities, x y u v t T T X =, Y =, =, V =, τ = and T = υ υ υ T T hee τ epesents the dimensionless time, X & Y be the dimensionless atesian oodinates, and V be the dimensionless veloity omponents and T be the dimensionless tempeatue sing the above elations, e obtain the folloing non-dimensional oupled patial diffeential equations, V + = X Y V = GT + τ X Y Y K T T T 1 T V E τ X Y P Y Y + + = + hee G υgβ = ( T T ) 3 (Gashof Numbe), 3 K K = (Pemeability Numbe) and υ E = p υρ p P = (Pandtl Numbe) κ ( T T ) Also the assoiated initial and bounday onditions beome τ =, =, V =, T = eveyhee τ >, =, V =, T = at X = = 1, V =, T = 1, at Y = =, V =, T = as Y (Eket Numbe)

4 Finite Diffeene Solution of Mixed Convetive Heat Tansfe Tansient Flo 4 Shea stess and nusselt numbe Sine the quantities of hief physial inteest ae shea stess and Nusselt numbe, hene fom the veloity field, e study the effets of vaious paametes on the loal and aveage shea stess The folloing equations epesent the loal and aveage shea stess at the plate Loal shea stess, τ L = µ and aveage shea stess, τ A = µ dx hih ae popotional to Y = Y = 1 and Y = dx espetively And fom the tempeatue field, e investigate the effets of vaious paametes on the loal and aveage heat tansfe oeffiients The folloing equations epesent the loal and aveage heat tansfe ate that is ell knon Nusselt numbe T Loal Nusselt numbe, NuL = µ and Y = T Aveage Nusselt numbe, NuA = µ dx T hih ae popotional to 5 Numeial solution Y Y = and Y = 1 T Y = dx espetively The expliit finite diffeene method has been used to solve the govened seond ode nonlinea oupled dimensionless patial diffeential equations ith the oesponding initial and bounday onditions To obtain a system of finite diffeene equations, the flo egion is divided into a gid o meshes of lines paallel to X and Y axes hee X- axis is taken along the plate and Y-axis is nomal to the plate Hee it is onsideed that X max ( = 1 ) ie X vaies fom to 1 and egad Y max ( = 5 ) as oesponding to Y ie Y vaies fom to 5 It is also onsideed that m = 1 and n = 1 gid spaing in the X and Y dietions espetively We have the onstant mesh size along X X = 1 X 1 and the onstant mesh size along Y dietion, dietion, ( ) Y 5( Y 5) = ith the smalle time-step τ = 1 Y = Let, V and T denote the values of, V and T at the end of a time-step espetively sing the finite diffeene appoximations, e obtain the folloing appopiate set of finite diffeene equations, i 1, Vi, Vi, 1 + = X Y 1

5 i 1, V τ Sade, M M Haque and T Fedous X Y = G T + T T T T T T 1 T T + T + + V = ( ) ( Y ) i 1, τ X Y P Y + E + and the initial and bounday onditions ith the finite diffeene sheme ae, =, V =, T = =, V =, T = n n n,,, = 1, V =, T = 1 n n n =, V =, T = n n n L L L + 1 K Hee the subsipts i and designate the gid points ith x and y oodinates espetively and the supesipt n epesents a value of time, τ = n τ hee n =,1,, Fom the initial ondition, the values of and T ae knon at τ = Then at the end of any time-step τ, the ne tempeatue T, the ne veloity and V at all inteio nodal points may be obtained by suessive appliations of enegy and momentum equations espetively This poess is epeated in time and povided the time-step is suffiiently small, hene, V and T should eventually onvege to values hih appoximate the steady-state solution of the poblem The stability ondition of finite diffeene method is τ V τ + + τ 1 and the X Y P Y onvegene iteia of the poblem is P 3 6 Results and disussion ( ) To disuss the effets of the assoiated non-dimensional paametes on the flo vaiables, the numeial solutions of the poblem ae obtained by the expliit finite diffeene method ith the help of a ompute pogamming language Compaq Visual Fotan In ode to analyze the physial situation of the model, e have omputed the steady state numeial values of the non-dimensional veloity and tempeatue T ithin the bounday laye fo diffeent values of Gashof numbe ( G ) numbe ( K ), Pandtl numbe ( P ) and Eket numbe ( E ) Y, Pemeability It is obseved that the esults of the omputations, hoeve, sho little hanges afte τ = 5 Thus the solution fo τ = ae essentially steady state solutions Hene the veloity and tempeatue pofiles ae dan fo τ = 1, 5 &

6 Finite Diffeene Solution of Mixed Convetive Heat Tansfe Tansient Flo The values of the Gashof numbe ae taken to be positive ( > ) G fo the ooling poblem and G = 1, 11 and 1 ae onsideed hee Fo the most impotant fluids suh as atmosphei a salt ate and ate so the esults ae limited to P = 71(Pandtl numbe fo ai at C), P = 1 (Pandtl numbe fo salt ate at C) and P = 7 (Pandtl numbe fo ate at C ) Also the values of anothe paametes K and E ae hosen abitaily The pofiles of the tansient veloity and tempeatue vesus Y ae illustated in Figs The effet of the Gashof numbe on the veloity field is pesented in Fig 61 It is obseved that the veloity ineases ith the ise of G The same effet on the veloity uve is found in Fig 6 that is the veloity ineases in ase of stong Pemeability numbe It is obseved fom Fig 63, the veloity stongly deeases ith the inease of Pandtl numbe An ineasing effet of Eket numbe on the veloity pofiles ae found fom Fig 64 at the steady-state Fig 61: Veloity pofiles fo K = 3, Fig 6: Veloity pofiles fo G = 1, P = 71 and E = 1 P = 71 and E = 1 Fig 63: Veloity pofiles fo G = 1, Fig 64: Veloity pofiles fo G = 1, K = 3, and E = 1 K = 3, and P = 71 3

7 Sade, M M Haque and T Fedous Fig 65 shos that the steady state tempeatue of fluid deeases in ase of stong Gashof numbe In Fig66, e see that the fluid tempeatue deeases fo the ineasing values of pemeability numbe A stong deeasing effet of the Pandtl numbe on the tempeatue uves ae obseved fom Fig 67 The effet of the Eket numbe on the tempeatue pofiles ae displayed in Fig 68 It is shon that the tempeatue ineases ith the ise of E Fig 65: Tempeatue pofilesfo P = 71 Fig 66: Tempeatue pofiles fo G = 1, K = 3 and E = 1 P = 71 and E = 1 Fig 67: Tempeatue pofiles fo G = 1, Fig 68: Tempeatue pofiles fo G = 1, K = 3 and E = 1 K = 3 and P = 71 The pofiles of steady state loal and aveage shea stess fo diffeent values of G, K o P ae shon in Fig 69 & Fig 61 espetively It is obseved fom these figues, both the loal and aveage shea stess ineases ith the ise of Gashof numbe o Pemeability numbe hile deeases fo the inease of Pandtl numbe The effets of the paametes on the steady state loal and aveage Nusselt numbe ae displayed in Fig 611 & Fig 61 These figues sho that, both the loal and aveage Nusselt numbe deeases ith the inease of G o E but it ineases fo the ineasing values of Pandtl numbe 4

8 Finite Diffeene Solution of Mixed Convetive Heat Tansfe Tansient Flo Fig 69: Steady-state loal shea stess Fig 61: Aveage shea stess fo fo diffeent values of G, K o P diffeent values of G, K o P Fig611: Steady-state loal Nusselt numbe Fig 61: Aveage Nusselt numbe fo fo diffeent values of G, E o P diffeent values of G, E o P 7 Conlusions Some of the impotant findings obtained fom the gaphial epesentation of the esults ae listed belo; 1 The tansient veloity ineases ith the inease of G, K o E hile it deeases ith the inease of P The tansient tempeatue ineases ith the inease of E hile it deeases ith the inease of G, K o P 3 Both the loal and aveage shea stess inease ith the ise of G o 5 K hile it deeases ith the inease of P 4 Both the loal and aveage Nusselt numbe deease ith the ise of G o hile it ineases ith the inease of P E

9 Sade, M M Haque and T Fedous These findings may be useful in many engineeing appliations suh as nulea eatos, geothemal enegy eovey, oil extation, themal enegy stoage also in a numbe of industial appliations as fibe and ganula insulation, metal and plasti extusion, ontinuous asting, glass fibe and pape podution REFERENCES 1 MFinston, Fee onvetion past a vetial plate, J Appl Math Phy, 7 (1956) EMSpao and JLGegg, Simila solutions fo fee onvetion fom a nonisothemal vetial plate, ASME J Heat Tans, 8 (1958) HKKuiken, Geneal seies solution fo fee onvetion past a non-isothemal vetial flat plate, App Si Res, (1969) MAFadzilah, NRoslinda, and MANoihan, Numeial investigation of fee onvetive bounday laye in a visous fluid, Ameian J Si Res, 5 (9) ARaptis, GTzivanidis and NKafousias, Fee onvetion and mass tansfe flo though a poous medium bounded by an infinite vetial limiting sufae ith onstant sution, Lettes Heat and Mass Tansfe, 8(5) (1981) KYamamoto and NIamua, Flo ith onvetive aeleation though a poous medium, J Engi Math, 1(1) (1976) NAhmed and DSama, Thee dimensional fee onvetion flo and heat tansfe though a poous medium, Indian J Pue Appl Math, 6 (1997) RCChaudhuy and TChand, Thee dimensional flo and heat tansfe though a poous medium, Int J Appl Meh Engi, 7(4) () EMagya I Pop and B Kelle, Analyti solutions fo unsteady fee onvetion in poous media, J Eng Math, 48() (4) BCSakiadis, Bounday-laye behavio on ontinuous solid sufae: I Boundaylaye equations fo to-dimensional and axisymmeti flo, AIChE J, 7 (1961) 6 11 VMSoundalgeka and TVRamana Muty, Heat tansfe in flo past a ontinuous moving plate ith vaiable tempeatue, Wame, Stoffubetag, 14 (198) ASami and Al-Sanea, Mixed onvetion heat tansfe along a ontinuously moving heated vetial plate ith sution o inetion, Int J Heat Mass Tansfe, 47 (4)