Unsteady Mixed Convection Heat and Mass Transfer Past an Infinite Porous Plate with Thermophoresis Effect

Transcription

1 Unseady Mixed Convecion Hea and Mass Transfer Pas an Infinie Porous Plae wih Thermophoresis Effec TARIQ AL-AZAB Mechanical Engineering Deparmen, Al-Al-Sal Communiy College Al-Balqa Applied Universiy P.O.Box 7003 (19117) Tel: Fax: Sal-Jordan E mail: azab@bau.edu.jo Absrac: - An analysis has been developed in order o sudy he unseady mixed convecion flow of an incompressible fluid pas an infinie verical porous plae wih hermophorisis paricle deposiion effec. The governing equaions are solved numerically using an implici finie difference echnique. The seleced numerical mehod is validaed by comparing he resuls wih he analyical soluions. Numerical resuls for he deails of he velociy profiles which are shown on graphs have been presened. I is found ha he seady sae values of hermophoreic deposiion velociy reached faser as he hermophoresis consan decreased and he emperaure raio increased. Key-Words: - Mixed convecion, Hea and mass ransfer, Unseady, Thermophorisis, Plae 1 Inroducion The sudy of hea and mass ransfer in he boundary layer induced by a verical porous surface is imporan in several manufacuring processes in indusry, which include he boundary layer along maerial handling, he exrusion of plasic shees, he cooling of an infinie meallic plae. Many auhors (see, for example, Ellio [1], Sakiadis [], Kerboua and Lakis [3], Corcione [4]) have sudied he problem of combine hermal convecion from a semi-infinie verical plae. Ching [5] uses he inegral mehod o sudy he hea and mass ransfer by mixed convecion from verical plaes wih consan wall emperaure and concenraion in porous media sauraed wih an elecrically conducing fluid in he presence of a ransverse magneic field. Thermophoresis is a phenomenon, which causes small paricles o be driven away from a ho surface and oward a cold one. Small paricles, such as dus, when suspended in a gas emperaure gradien, experience a force in he direcion opposie o he emperaure gradien. This phenomenon has many pracical applicaions in removing small paricles from gas sreams, in deermining exhaus gas paricles rajecories from combusion devices, and in sudying he pariculae maerial deposiion on urbine blades. I has been also shown ha hermophoresis is he dominan mass ransfer mechanism in he modified chemical vapor deposiion process used in he fabricaion of opical fiber performance. Goren [6] sudied he role of hermophoresis of a viscous and incompressible fluid, he classical problem of flow over a fla plae is used o calculae deposiion raes and i is found ha he subsanial changes in surface deposiion can be obained by increasing he difference beween he surface and free sream emperaures. Gokoglu and Rosner [7], Park and Rosner [8] obained a se of similariy soluions for he wo dimensional laminar boundary layers and sagnaion poin flows respecively. Chio [9] obained he similariy soluions for he problem of a coninuously moving surface in a saionary incompressible fluid, including he combined ISSN: Issue, Volume 4, April 009

2 effecs of convecion, diffusion, wall velociy and hermophoresis. Grag and Jayaraj [10] discussed he hermophoresis of small paricles in forced convecion laminar flow over inclined plaes. Epsein e al. [11] have sudied he hermophoresis ranspor of small paricles hrough a free convecion boundary layer adjacen o a cold, verical deposiion surface in a viscous and incompressible fluid. Chiou [1] has considered he paricle deposiion from naural convecion boundary layer flow on isohermal verical cylinder. Consideraion in his work is given o he hermophoresis effecs on unseady mixed convecion hea and mass ransfer problems from infinie verical porous surfaces. Numerical resuls for he velociy, emperaure and concenraion profiles as well as he hermophoresis velociy, under he effec of differen dimensionless groups are presened. Analysis Consider an unseady convecion boundary layer flow of a fluid pas an infinie isohermal verical plae of consan emperaure T w and concenraion C w. The ambien emperaure is T and concenraion C. The plae emperaure T w and concenraion C w is higher han he ambien emperaure T and concenraion C. I is assumed ha he fluid properies are consan excep he influence of densiy variaion wih emperaure is considered only in he body force erm. The flow is assumed o be in he x- direcion, which is along he verical plae in he upward direcion, and y-axis is aken o be normal o he plae, Fig. 1. Allowing for boh Brownian moion of paricles and hermophoreic ranspor, he governing equaions are, Chiou, M. C. (1991), v = 0 u u + v u + υ = gβ ( T T T ) + gβ ( C C C ) (1) () T T T + v = α ( Cv ) C C + v + T = D (3) (4) The physical problem assumes he following boundary condiions: 0 f 0 f 0 u( y, = 0, T( y, = T C( y, = C u(0, = 0, v(0, = vw, T(0, = Tw, (5) C(0, = Cw u(, = u, T(, = T, C(, = C o Here x and y are he dimensional disance along and normal o he plae, respecively. (u, v) are he averaged velociy componens along he x and y, direcions respecively, vw is he sucion velociy, T is he emperaure, C is he concenraion, β T and β C are he coefficien of hermal expansion of emperaure and concenraion respecively. ν is he kinemaic viscosiy, α is he hermal diffusiviy and D is he Brownian diffusion coefficien. x y Velociy Temperaure Concenraion g T, C Fig. 1: Schemaic diagram for flow mode coordinae sysem. ISSN: Issue, Volume 4, April 009

3 The hermophoreic velociy v can be expressed in he from, v υ T = k T Y (6) Where k is he hermophoreic coefficien. In order o non-dimensionalize he governing equaions, we inroduce he following nondimensional parameers: u U=, uo uo τ =, l 1/ 1/ 1/ Y= Ra y/ l, λ= Ra vw / uo, V = Ra v / uo, (7) T T C C θ=, ϕ= T T C C w w Where u o = gk β T (T w -T )/ν is he characerisic velociy, Ra = gk β T (T w -T )/ν α is he Rayleigh number, l is he characerisic lengh of he plae and λ is he sucion parameer. The dimensionless form of he governing equaions and heir boundary condiions are reduced o β T (T w -T ) is he buoyancy parameer. The quaniy of physical ineres is he wall hemophoreic deposiion velociy ha can be express as: V w Pr θ = k N + 1 Y Y = 0 (13) 3 Soluion Mehodology The mass conservaion equaion is a nonlinear equaion. Furhermore, he mass and energy equaions are coupled wih he momenum equaion, for such reasons, he sysem of equaions wih he corresponding boundary condiions (8-1) are solved numerically using an implici finie-difference echnique similar o Crank-Nicolson mehod, which is discussed by Anderson [13]. All he firs-order derivaives wih respec o τ are replaced by using formula of he form: n+ 1 n ( C j C ) C = τ Δτ U U 1 U λ = + B( θ + Nϕ) ) All he second-order derivaives wih respec o τ Y Pr Y Y are replaced by using formula of he form: (8) (8) θ θ θ n+ 1 n λ = (9) (1/ ) ( C j+ 1 C j+ 1 ) + τ Y Y n+ 1 n (1/ ) ( (9) C j C j ) + ϕ ϕ ( ϕ + V ) 1 ϕ n+ 1 n λ + = (10) C (1/ ) ( C j 1 + C j 1 ) τ Y Y Le Y = (15) Y ( ΔY ) (10) Pr θ V = k (11) The concenraion equaion is ransformed o N + θ Y finie difference equaions by applying he cenral difference approximaions (11) o he firs τ 0 U( Y, = 0, θ( Y, = 0, ϕ( Y, = 0 and second derivaives. The finie difference τ f 0 U(0, = 0, θ(0, = 1, ϕ(0, = 1 (1) equaions form a ri-diagonal sysem can be τ f0 U(, = 1, θ(, = 0, ϕ(, = 0 solved by he ri-diagonal soluion scheme. The effec of he grid size ΔY and Δτ on he where B = Gr Ra/Pr is he dimensionless mixed numerical soluion had been sudied. The resuls convecion parameer and drawn here are independen on he grid size. Gr = ν g β T (T w -T )/u 3 o is he hermal Grashof The grid spacing used here are hose larges number. Pr = ν/α and Le = α/d are he Prandl values of ΔY and Δτ which does no aler he and Lewis numbers, N = (T w -T )/T is he soluion, his process have imporan effecs on hermophoresis parameer and N = β C (C w -C )/ compuaional ime. j (14) ISSN: Issue, Volume 4, April 009

4 In order o verify he accuracy of he seleced numerical mehod, he energy equaion is a linear equaion and can be solved analyically by Laplace ransform echnique; i is suiable o noe ha he nonlineariy in he concenraion equaion are due o he hermophoreic velociy. Furhermore, he code developed in his invesigaion was validaed by comparing he resuls obained by his numerical mehod wih analyical soluion of he energy equaion using he Laplace ransform echnique. The soluion of he energy equaion is given by: θ ( Y, τ ) = 1/ exp( λ Y ) Y 1/ exp( λ Y ) erfc + ( λτ ) 1/ τ Y 1/ + exp( λy ) erfc ( λτ ) 1/ τ (16) Fig illusraes a comparison beween he numerical and analyical soluions for he seady sae emperaure disribuion. I is seen ha he agreemen beween he resuls are excellen. This has esablished confidence in he numerical resuls o be repored in his paper. 4 Resuls And Discussion In he presen work, he numerical soluions were conduced o invesigae he influence of he hermophoresis coefficien k, he buoyancy raio N, he Lewis number Le and he emperaure raio N. For all numerical calculaions he Pandl number and he sucion parameer are assigned a value of 1.0 (Pr = 1.0, λ =1.0). The effec of sucion parameer λ on seady sae emperaure disribuion is shown in Fig.. I can be concluded ha he emperaure profiles decrease as he values of he sucion parameer λ increases. This leads he hermal boundary layer hickness o decrease. Figs. 3 show he effec of buoyancy raio N on seady sae velociy profiles for B = 1.0, N = 50.0, k = 0.5, Le = 10.0, λ = 1.0 and Pr = 1.0. Posiive values of N indicae aiding flow. The Figure shows ha as he buoyancy parameer increased he velociy increased due o favorable slip velociies near verical surfaces and concenraion conribuion in immigraion of fluid paricles from he verical surfaces. Fig. 4 display he effec of Lewis number Le on seady sae concenraion profiles for B = 1.0, N = 50.0, k = 0.5, N = 10.0, λ = 1.0 and Pr = 1.0. Increases in he Lewis number ends o increases he buoyancy-induced flow along he surface a he expense of reduced concenraion and is boundary layer hickness. Fig. 5 represens he ime dependen hermophereic deposiion velociy for B = 1.0, N = 50.0, k = 0.5, N = 10.0, λ = 1.0, Pr = 1.0 and a differen hermophoresis coefficien. The Figure shows ha as he hermophoresis coefficien is increased he wall hermophoresis velociy is also increased; his is due o favorable emperaure gradiens. Also he figure shows ha as he ime is increased he hermophoresis velociy is decreased up o seady sae condiions which is reached more fas for low values of hermophoresis coefficien. Fig. 6 shows he ime dependen hermophoresis deposiion velociy values for B = 1.0, k = 0.5, k = 0.5, N = 10.0, λ = 1.0, Pr = 1.0 and a differen values of emperaure raio N = 10, 0, 50, 100. I is clear ha he hermophoresis values are decreased when emperaure raios are increased; his is due o small emperaure differences beween verical surface and free sream condiions. Clearly, his is why he seady sae hermophoresis velociy need less ime for higher values of emperaure raion. 5 Conclusions The unseady hea and mass ransfer mixed convecion problem of a Newonian fluid over an infinie verical porous plae in he presence of hermophoresis paricle deposiion effec were sudied. Based on he obained resuls, he following conclusions can be repored. 1- I was found ha he hermophoreic deposiion velociy increased as he ISSN: Issue, Volume 4, April 009

5 hermophoresis consan k increased and as emperaure raion N decreased. - The increase of Le number concenraion boundary layer a seady sae condiions decreases due o an. 3- The seady sae values of hermophoreic deposiion velociy reached faser as he hermophoresis consan decreased and he emperaure raio increased. References: [1] L. Ellio, Unseady laminar flow of gas near an infinie fla plae, Zei, Angew, Mah Mech, Vol. 49, 1969, pp [] B. C. Sakiadis, Boundary layer behavior on coninuous solid surface. II, The boundary layer on a coninuous fla surface, Amer. Ins. Chem. Engrg. J. Vol. 7, 1961, pp [3] Y. Kerboua and A.A. Lakis, Dynamic behaviour of plaes subjeced o af fluid, WSEAS Transacions on Fluid Mechanics, Vol. 3, No., 008, pp [4] M. Corcione, Naural convecion hea ransfer above heaed horizonal surfaces, 5h WSEAS In. Conf. on Hea and Mass ransfer, Acapulco, Mexico, January, 008, pp [5] Cing-Yang Cheng, Magneic Field Effecs on Coupled Hea and Mass Transfer by Mixed Convecion along a Verical Surface Embedded in a Porous Medium by Inegral Mehods, Proceedings of he 4h WSEAS Inernaional Conference on Hea and Mass Transfer, Gold Coas, Queensland, Ausralia, January, 007, pp [6] S. L. Goren, Thermophoresis of aerosol paricles in he laminar boundary layer on a fla plae, J. Colloid Inerface Sci., Vol. 61, 1977, pp [7] S. A. Gokoglu and D. E. Rosner, Thermophoreically augmened mass ransfer raes o solid walls across boundary layers, AIAA J., Vol. 4, 1986, pp [8] H. M. Park and D. A. Rosner, Combined inerial and hermophoresis effecs on paricle deposiion raes in highly loaded dusy gas sysems, Chem. Eng. Sci., Vol. 44, no. 10, 1989, pp [9] M. C. Chiou, Effec of hermophoresis on submicron paricle deposiion from a forced laminar boundary layer flow ono an isohermal moving plaes, Aca Mech., Vol. 89, 1991, pp [10] V. k. Garg and S. Jayaraj, Thermophoresis of a crosol paricles in laminar flow over inclined plaes, In. J. Hea Mass Transfer, Vol. 31, 1998, pp [11] M. Epsein, G. M. Hauser and R. E. Henry, Thermophoresis deposiion of paricles in naural convecion flow from a verical plae, J. Hea Transfer, Vol. 107, 1985, pp [1] M. C. Chiou, Paricle deposiion from naural convecion boundary layer ono an isohermal verical cylinder, Aca Mech, Vol. 19, 1998, pp [13] J. Anderson, Compuaional Fluid Dynamics, chap. 9, McGraw-Hill, Nomenclaure C Fluid concenraion c p Specific hea capaciy D Brownian diffusion coefficien g Graviaional acceleraion Gr Grashof number k Thermophoresis coefficien Le Lewis number, α m D N Buoyancy raio, [ β C ( Cw C ) βt ( Tw T )] N Dimensionless emperaure raio, T [ Tw T ] Pr Prandl number, υ α Ra Local Rayleigh number, Kgβ ( Tw T ) x υα T Temperaure u,v Velociy componens in x-and y- direcions v Thermophoresis velociy v w Thermophoresis velociy a wall V Dimensionless hermophoresis velociy, v x / α m V w Dimensionless hermophoresis velociy a wall x,y Axial and normal coordinaes ISSN: Issue, Volume 4, April 009

6 Greek symbols: α Effecive hermal diffusiviy of he porous medium β T Coefficien of hermal expansion, ( 1/ ρ )( ρ T ) P Coefficien of concenraion expansion, β C ( 1/ ρ )( ρ C) P λ Sucion parameer θ Dimensionless emperaure Φ Dimensionless concenraion μ Dynamic viscosiy υ Kinemaic viscosiy ρ Fluid densiy Subscrips w Surface condiions Free sream condiion T Thermophoresis effecs ISSN: Issue, Volume 4, April 009

7 Analyical soluion Numerical soluion θ(y) λ Y 6 Fig. Temperaure disribuion for differen values of λ ISSN: Issue, Volume 4, April 009

8 N = 0,, 4, 10 U(Y) Y B = 1.0 N = 50.0 K = 0.5 Le = 10.0 λ = 1.0 Pr = 1.0 Fig. 3 Effec of buoyancy parameer on velociy disribuion ISSN: Issue, Volume 4, April 009

9 φ(y) Le = 1, 5, 10, 50 B = 1.0 N = 50.0 K = 0.5 N = 10.0 λ = 1.0 Pr = Fig. 4 Effec of Lewis number on concenraion disribuion Y ISSN: Issue, Volume 4, April 009

10 0.04 V w k = 0.1 k = 0. k = 0.4 k = 0.8 B = 1.0 N = 50.0 Le = 5.0 N = 10.0 λ = 1.0 Pr = τ Fig. 5 Effec of hermophoresis coefficien on ime dependen hermophereic deposiion velociy ISSN: Issue, Volume 4, April 009

11 V w N = 10 N = 0 N = 50 N = 100 B = 1.0 K = 0.5 Le = 5.0 N = 10.0 λ = 1.0 Pr = τ ISSN: Issue, Volume 4, April 009