THERMOPHORESIS PARTICLE DEPOSITION ON FLAT SURFACES DUE TO FLUID FLOW IN DARCY-FORCHHEIMER POROUS MEDIUM

Transcription

1 Telfh Inernaional Waer Technology Conference, IWTC1 008, Alexandria, Egyp 1069 THERMOPHORESIS PARTICLE DEPOSITION ON FLAT SURFACES DUE TO FLUID FLOW IN DARCY-FORCHHEIMER POROUS MEDIUM Rebhi A. Damseh 1 and Kamel Alzboon 1 Mechanical Engineering Deparmen, Al-Husn Universiy College Al-Balqa Applied Universiy, P.O. Box 50, Irbid, Jordan rdamseh@yahoo.com Environmenal Engineering Deparmen, Al-Husn Universiy College Al-Balqa Applied Universiy, P.O. Box 50, Irbid, Jordan ABSTRACT An analysis is performed for non-darcy free convecion flo over a verical plae embedded in a hermally sraified, fluid sauraed porous medium and aking ino accoun he presence of hermophoresis paricle deposiion. A finie-difference scheme as used o solve he sysem of ransformed governing equaions. Numerical resuls for he deails of he velociy, emperaure and concenraion profiles hich are shon on graphs have been presened. I is shon ha he ineria forces have a significan influence on he flo characerisics in his problem. Comparison ih previous published ork is performed and he resuls are found o be in excellen agreemen. Keyords: Non-Darcy, Porous Medium, Thermophoresis, Naural Convecion, Fla Plae. INTRODUCTION Thermophoresis is a phenomenon, hich causes small paricles o be driven aay from a ho surface and oard a cold one. Small paricles, such as dus, hen suspended in a gas emperaure gradien, experience a force in he direcion opposie o he emperaure gradien. Thermophoresis is of pracical imporance in many engineering applicaions hen ho gases conaining small suspended paricles flo over cool surfaces. For example, hermophoresis can be effecive in removing or collecing small paricles from laminar gas sreams in air leaning and aerosol sampling devices [1]. There are several oher pracical siuaions here e come across his phenomenon as an origin for he deposiion of pariculae maer on surfaces of hea exchangers, causing scale formaion ih he aendan reducion of he hea-ransfer coefficien []. In cerain applicaions such as he microelecronics indusry, deposiion of conaminan paricles by hermophoresis on afers in clean rooms during manufacuring seps can be a major

2 1070 Telfh Inernaional Waer Technology Conference, IWTC1 008 Alexandria, Egyp cause of loss of produc yield [3]. Also, hermophoreic deposiion of radio acive paricles is considered o be one of he imporan facors causing accidens siuaion in nuclear reacors [4]. Goren [5] sudied he role of hermophoresis of a viscous and incompressible fluid. The classical problem of flo 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 beeen he surface and free sream emperaures. Gokoglu and Rosner [6] and Park and Rosner [7] obained a se of similariy soluions for he o dimensional laminar boundary layers and sagnaion poin flos respecively. Chiou [8] obained he similariy soluions for he problem of a coninuously moving surface in a saionary incompressible fluid, including he combined effecs of convecion, diffusion, all velociy and hermophoresis. Grag and Jayaraj [9] discussed he hermophoresis of small paricles in forced convecion laminar flo over inclined plaes; Epsein e al. [10] 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. Flo, hea and mass ransfer in porous media has been sudied exensively in recen years. This is due o he increasing need in undersanding he complicaed ranspor process for applicaion of diverse fields hich include geohermal engineering, building insulaion, energy conservaion, solid marix hea exchangers, filraion processes, underground disposal of nuclear ase maerials, and many more. These problems are ell documened in he books by Nield and Bejan [11] and Pop and Ingham [1]. Early sudies on flo hrough porous medium ere based on he Darcy la. The hermophoresis effec in Darcian porous medium is invesigaed by Chamkha and Pop [13]. I is generally recognized ha he Darcy model is valid under he condiion of lo velociies. Hoever, i is ell knon ha Darcy model is valid hen he Reynolds number based on he pore size is less han uniy. For high velociy flo siuaions and/or porous maerial of large pore radius, Forchheimer modificaion is inroduced by adding he quadraic inerial erm o he original Darcy model. More recen sudies can be found in Al-Hussien e al. [14] and Al-Oda e al. [15, 16]. This ork is exension o he discussed model by Chamkha and Pop [13], so consideraion is given o he influence of he ineria forces on fluid flo ih he presence of hermophoresis effecs on free convecion hea and mass ransfer problems from verical surfaces embedded in porous medium. PROBLEM FORMULATION Consider he o-dimensional, non-darcy, free convecion boundary layer flo of a viscous incompressible fluid pas an isohermal verical plae of consan emperaure T and concenraion C. The ambien emperaure is T and concenraion C. The plae emperaure T and concenraion C are higher han he ambien emperaure T and

3 Telfh Inernaional Waer Technology Conference, IWTC1 008, Alexandria, Egyp 1071 concenraion C. I is assumed ha he fluid properies are consan excep he influence of densiy variaion ih emperaure is considered only in he body force erm. The flo is assumed o be in he x-direcion, hich is along he verical plae in he upard direcion, and y-axis is aken o be normal o he plae, Fig. 1. Alloing for boh Bronian moion of paricles and hermophoreic ranspor, he governing equaions are, Chamkha and Pop [13], Chiou [8]: u v x (1) u F K υ u Kβ g υ T T Kβ C g υ C () u T T T v α x m (3) y C C v x ( Cv ) T D (4) x, u T C elociy Temperaure Concenraion T, C g Y, v u Fig. 1. Schemaic diagram for flo model and coordinae sysem

4 107 Telfh Inernaional Waer Technology Conference, IWTC1 008 Alexandria, Egyp The physical problem assumes he folloing boundary condiions: y u v T (0, ) T C(0, ) C (5) y u T T C 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, 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 m is he effecive hermal diffusiviy, K is he permeabiliy of he medium, F is he ineria coefficien and D is he Bronian diffusion coefficien. The hermophoreic velociy v can be expressed in he from, C v υ T k T (6) here k is he hermophoreic coefficien. In order o non-dimensionalize he governing equaions, e inroduce he folloing non-dimensional parameers: X x, U l T T T T θ, u u o, Ra C C ψ C C 1 v u o, Y Ra 1/ y / l, Ra 1 / v / u o, (7) here u o gk T (T -T )/ is he characerisic velociy, Ra gk T (T -T )/ is he Rayleigh number, l is he characerisic lengh of he plae. The dimensionless form of he governing equaions and heir boundary condiions are reduced o: U X (8) U U Γ θ N ψ (9) U θ X θ θ (10)

5 Telfh Inernaional Waer Technology Conference, IWTC1 008, Alexandria, Egyp 1073 U ψ X ψ ( ψ ) 1 ψ Le (11) k Pr θ N θ (1) Y Y U θ 1 θ ψ 1 ψ (13) here Γ FKgβ T K ( TW T ) υ is he dimensionless ineria coefficien expressing he relaive imporance of he ineria effec, Pr / m and Le m /D are he Prandl and Leis numbers respecively, N (T T )/T is he hermophoresis parameer and N C (C -C )/ T (T -T ) is he buoyancy parameer. Equaions (8-13) can be ransformed from he (X, Y) coordinaes o he dimensionless coordinae ( ξ, η) by inroducing he folloing non-dimensional variables: 1 1 ξ X, η Yξ, ψ f ( η) ξ, θ θ ( η), ψ ψ ( η) (14) In he equaions above, he sream funcion ψ saisfied he coninuiy equaion (8) ih U ψ and ψ X. Finally one can obain he folloing sysem of dimensionless equaions: f Γf f θ Nψ (15) 1 θ f θ (16) 1 1 k Pr ψ ψ fψ θ ψ ψθ θ Le N θ N θ (17) f (0), f 0, θ (0) 1, ψ (0) 1 θ 0, ψ 0 as η (18) The quaniies of physical ineres are he all hemophoreic deposiion velociy and he local Sherood number ha can be expressed as: Pr k θ (0) (19) N 1 Sh Ra ψ (ξ, 0) (0) x x

6 1074 Telfh Inernaional Waer Technology Conference, IWTC1 008 Alexandria, Egyp RESULTS AND DISCUSSION For his presen problem, numerical compuaions have been carried ou. I is clearly seen ha he resuls are given values of he parameers N (buoyancy raio), k (hermophoresis coefficien), N (emperaure raio) and (ineria parameer). The resuling ordinary differenial Equaions (15-17) under boundary condiions (18) have been solved by means of he fourh-order Runge-Kua mehod ih symmeric esimaion of f (0), θ (0), and ψ (0) by he shooing echnique. The basic sep size 3 used for he calculaion is η This value as arrived a afer performing many numerical experimens o access grid independence. An ieraion process is employed and coninued unil he desired resuls are obained ihin he folloing convergence crierion: j i 1 j i 1 j i 10 6 (1) here j sands for f, θ, orψ and i refers o space coordinae. Furhermore, he accuracy of he numerical resuls obained in his invesigaion as validaed by a direc comparison ih he soluions repored by Chamkha and Pop [13] and Bakier and Mansour [17]. Table 1 presens a comparison beeen he values of θ (0) and ψ (0) for obained by he presen numerical mehod ih hose corresponding o he above menioned references. I is clearly seen from Table 1 ha excellen agreemen beeen he respecive resuls is obained. Table 1 alues of θ (0) and ψ (0) for compared o hose obained by Chamkha and Pop (004) and Bakier and Mansour (007).. k, N, Le 1 k, N Le 1 Chamkha and Pop Bakier and Mansour Presen sudy Chamkha and Pop Bakier and Mansour Presen sudy θ (0) ψ (0) Figures, 3, and 4 sho he effec of ineria parameer on velociy, emperaure and concenraion profiles a differen values of buoyancy raio N and for values of N 10.0, k.4, Le 5.0, and Pr.7. The Figures sho ha as he buoyancy parameer is increased he velociy is increased due o favorable slip velociies near verical surfaces and concenraion conribuion in immigraion of fluid paricles from he verical surfaces. On he oher hand, i is clearly shon from he figures ha he velociy in he non-darcy model is dramaically decreased, hile he emperaure and concenraion in he non-darcy model is increased as he ineria parameer increased. This behavior

7 Telfh Inernaional Waer Technology Conference, IWTC1 008, Alexandria, Egyp 1075 aribued o he fac ha increasing he ineria parameer has a endency o resis he flo. Furhermore, hen emperaure and concenraion raios are increased; his is due o small emperaure differences beeen verical surface and free sream condiions, a decrease in hea and mass ransfer process is occurred as shon in Figure Pr.7 Le 5 N 10 K Pr.7 Le 5 N 10 K f () () N,, 4 0. N,, Fig.. Effec of ineria parameer on he velociy disribuion a differen values of buoyancy raio N Fig. 3. Effec of ineria parameer on he emperaure disribuion a differen values of buoyancy raio N Pr.7 Le 5 N 10 K Pr.7 Le 5 N 10 K.5 1 ψ () Sh N,, N Fig. 4. Effec of ineria parameer on he concenraion disribuion a differen values of buoyancy raio N Fig. 5. Effec of buoyancy raio on he Sherood number for boh Darcy and non-darcy models

8 1076 Telfh Inernaional Waer Technology Conference, IWTC1 008 Alexandria, Egyp Figure 6 represens he dependen hermophereic deposiion velociy on buoyancy raio N for boh Darcy and non-darcy models. The oher parameers are assigned values of Le 5.0, and Pr.7. I is clear from he figure, and as said before, he ineria force ends o slo don he flo. This in ern decreases he deposiion velociy a he all. On he oher hand, he decrease in emperaure difference beeen verical surface and free sream condiions plays, due o dependency of deposiion velociy on emperaure gradien a he all, a big role in decreasing he all deposiion velociy. The figure also shos ha as he hermophoresis coefficien k and buoyancy parameer increase he all hermophoresis velociy is also increased; his is due o favourable emperaure gradiens and concenraion velociies conribuion in immigraion of fluid paricles from he verical surfaces. Clearly, he hermophoresis values are decreased hen emperaure raios are increased; his is due o small emperaure differences beeen verical surface and free sream condiions k N Fig. 6. Effec of buoyancy raio on he all deposiion velociy for boh Darcy and non-darcy models N CONCLUSIONS The effec of ineria force on hea and mass ransfer free convecion problem of a Neonian fluid over a fla verical plae embedded in non-darcy medium in he presence of hermophoresis paricle deposiion effec ere sudied. In his problem, he non-darcy model hich conribues for he ineria force is employed o describe he flo in porous medium. I as found ha he inclusion of he ineria parameer in he calculaion can cause a sligh increase in he fluid emperaure and concenraion, in

9 Telfh Inernaional Waer Technology Conference, IWTC1 008, Alexandria, Egyp 1077 addiion o a significan decrease in he fluid velociy. In urns, he deposiion velociy a he all ill decrease. The hermophoreic deposiion velociy increased as he hermophoresis consan k and he buoyancy parameer N increased and as he emperaure raion N decreased. NOMENCLATURE C Fluid concenraion c p Specific hea capaciy D Bronian diffusion coefficien F Dimensional ineria coefficien g Graviaional acceleraion K Permeabiliy of he porous medium k Thermophoresis coefficien Le Leis number N Buoyancy raio, [ β C ( C C ) β T ( T T )] N Dimensionless emperaure raio, T [ T T ] Pr Prandl number Ra Local Rayleigh number, Kgβ ( T T x υα ) Sh Sherood number T Temperaure u,v elociy componens in x-and y-direcions v Thermophoresis velociy v Thermophoresis velociy a all Dimensionless hermophoresis velociy Dimensionless hermophoresis velociy a all x,y Axial and normal coordinaes Greek symbols: α Effecive hermal diffusiviy of he porous medium β T Coefficien of hermal expansion, ( 1/ ρ )( ρ T ) P β C Coefficien of concenraion expansion, ( 1/ ρ )( ρ C) P η Non-dimensional ransformed variable Dimensionless ineria parameer θ Dimensionless emperaure ψ Dimensionless concenraion υ Effecive kinemaic viscosiy ρ Fluid densiy

10 1078 Telfh Inernaional Waer Technology Conference, IWTC1 008 Alexandria, Egyp Subscrips Surface condiions Free sream condiion Thermophoresis effecs REFERENCES [1] Bachelor, G. K., and Shen, C., J. Colloid Inerface Sci., ol. 107 (1985), pp [] Monassier, N., Boulaud, D., and Renoux, A., J. Aerosol Sci. ol. (1991) pp [3] Ye, Y., Pui, D. Y. H., Liu, B. Y. H., Opiolka, S., Blumhors, and Fissan, S., J. Aerosol Sci. ol. (1991) pp [4] Williams, M. M. R., and Loyalka, S. K., Aerosol Science: Theory and Pracice, ih Special Applicaions o he Nuclear Indusry, Pergamon Press, Oxford, [5] Goren, S. L., J. Colloid Inerface Sci., ol. 61 (1977), pp [6] Gokoglu, S. A., and Rosner, D. E., AIAA J., ol. 4 (1986), pp [7] Park, H. M., and Rosner, D. A., Chem. Eng. Sci., ol. 44 (1989), No. 10, pp [8] Chiou, M. C., Aca Mech., ol. 89 (1991), pp [9] Garg,. K., and Jayaraj, S., In. J. Hea Mass Transfer, ol. 31 (1998), pp [10] Epsein, M., Hauser, G. M., and Henry, R. E., J. Hea Transfer, ol. 107 (1985), pp [11] Nield, D. A., and Bejan, A., Convecion in Porous Media, nd ed., Springer-erlag, Berlin, Germany, [1] Pop, I., and Ingham, D. B., Convecive Hea Transfer: Mahemaical and Compuaional Modelling of iscous Fluids and Porous Media, Pergamon, Oxford, 001. [13] Chamkha, A., and Pop, I., In. Comm. Hea Mass Transfer, ol. 31 (004), No. 3, pp [14] Fayez, M. Al-Hussien, Damseh, Rebhi A., and Al-Oda, M. Q., Inernaional Journal of Transpor Phenomena, ol. 7 (005), pp [15] Al-Oda, M. Q., Al-Hussien, F. M. S., and Damseh, R. A., Forsch Ingenieures, ol. 69 (005a), pp [16] Al-Oda, M. Q., and Damseh, Rebhi A., Dirase, Engineering Sciences, ol. 3 (005b), No., pp [17] Bakier, A. Y., and Mansour, M. A., Thermal Science, ol. 11 (007), No. 1, pp