Analysis of Non-Thermal Equilibrium in Porous Media

Transcription

1 Analysis o Non-Thermal Equilibrium in Porous Media A. Nouri-Borujerdi, M. Nazari 1 School o Mechanical Engineering, Shari University o Technology P.O Box , Tehran, Iran anouri@shari.edu Abstract This paper is concerned with the heat transer between two parallel plate illed with a porous medium under non-equilibrium condition. A two-equation model is utilized to represent the luid and solid energy transport.analytical solution is used or both luid and solid temperature ields incorporating the eect o thermal conductivity ratio and porosity. Keywords: Heat Transer, Porous Media, Non-Thermal equilibrium, Analytical solution Introduction Analysis o luid low and heat transer in porous medium has been a subject o continuous interest during the past decades because o the wide range o engineering applications such as heat exchangers, energy storage units, drying technology and geothermal system. Although the local thermal equilibrium is used in many o these applications, this assumption breaks down when a substantial temperature dierence exists between the solid and the luid phases. Thermally developing orced convection in a porous media with axial conduction and viscous dissipation eects in local thermal equilibrium state is obtained by Nield et al. [1], in which, the brinkman model is employed and the analysis leads to expression or the local Nusselt number as a unction o the dimensionless longitudinal coordinate and other parameters. They also treated eect o local thermal non-equilibrium on thermally developing orced convection in a porous medium. Analytical solution is used or obtaining local Nusselt number or various values o the luid-solid conductivity ratio and or constant values o porosity, Pecelt number and Darcy number []. The two-energy equation model or conduction and convection is proposed by Nakayama et al. [3], in which,the two energy equations or the individual phases are combined into a ourth order ordinary dierential equation. Then the exact solutions are obtained or two undamental cases, namely, one-dimensional steady conduction in a porous slab with internal heat generation within a solid, and also thermally developing unidirectional low through a semi-ininite medium. Amiri and vaai [4] have employed a general luid low model and two-phase energy equations to investigate the orced convection heat transer within a channel with constant wall temperature. This work involves the numerical simulation o orced convective incompressible low through porous media and the associated transport processes. Also they have used ull general model or the momentum equation. The problem o orced convection in a porous medium channel or duct is a classical one. In this present paper we concentrate on the case o a parallel plate channel with uniorm temperature or constant heat lux at the boundary walls. The two energy equation model is used to represent the eect o conductivity ratio and porosity change on the luid and solid phases. Criterion or local thermal equilibrium At the pore level a local temperature dierence between solid and luid is assumed to be Td. Similarly across the representative elementary volume and over the system dimension, the maximum temperature dierences are Tl and respectively, the assumption o local thermal equilibrium is imposed by requiring that [5]: T >> T >> T (1 L l d 1 - M.Sc Student

2 With this assumed negligible local temperature dierence between the phases, it is assume that the heat transer rate rom the solid phase to the luid phase in the overall system is equal to the heat transer rate which is carried by the luid lowing through the porous medium. The heat transer rate rom solid phase to luid phase in the overall system is: q h av ( Ts T = hav Tl & ( a, h and V are interacial surace area per unit volume, interstitial heat transer coeicient between the luid phase and the solid phase, and system volume, respectively. The amount o heat which is transerred to or rom the luid lowing through porous medium can be expressed as: p ( Tout Tin mc & p TL q& mc. = (3 By substituting Eqs.(,3 into Eq.(1, the criterion or local thermal equilibrium is expressed as ollows equation[6]:. mc p 1 h a V This criterion implies that the eect o local thermal equilibrium becomes dominant in a porous medium as either the interstitial heat transer coeicient or the interacial surace area or heat transer increases. On the other hand, the criterion presented by Carbonell and Whitaker [7] is expressed as: εk l A o L K K s And K s l 1 + A o L K << 1 1 K s << 1 (5 (6 (4 Modeling and Formulation Consider a channel low with two parallel lat plates illed with porous medium. The temperatures o the plates are held constant at T w.(fig.1. Using a steady-state and one dimensional low, the energy equations o the solid and luid phases are as ollows respectively [8]. T ρ C p u = εk T + ha( Ts T (7 x ( 1 K T ha( T T + ( 1 ε q = 0 ε s s s s (8 Where ε is the porosity o the medium. q s is heat generation per unit solid volume and takes place within the solid phase. Dierent models o the luid to solid heat transer coeicient and the luid to solid speciic area has been presented in Alazmi and Vaai [9]. Assuming high thermal conductivity at the boundary, the temperature o the solid and luid at the wall interace will be the same, and: T s(x, y H = T (x, y = H = Tw = (9 The symmetry condition at the center line will be: T Ts (y = 0 = (y = 0 = 0 (10 The convection term in Eq.(7 is neglected and using Eq(9: T Ts (y = H = T (y = H = 0 (11 Also by neglecting source term and applying irst derivation rom Eq.(8 and using symmetry condition rom Eq.(10: 3T 3 (y = 0 = 0 The dimensionless orm o Eq.(7,8 are: θ H + * ε k s θs H k s θs θ Where ( T Tw θ = q s, Y* = ( θ θ = 0 H k s ( + = 0 y H (1 (13 (14 (15 Solution Procedures Eliminating θs between the above two equations, the inal result is: 4 H θ σh θ k = (16 4 K * ε k s K ε The appropriate boundary conditions are:

3 θ 3 θ * * ( X,0 = 0, ( X,0 = 0 *3 θ * * * ( X,1 = 0, ( X,1 = 0 (17 θ (18 Where, K s + εk σ = 1 ε K (19 ( s Solving the above ourth order ordinary dierential equation yields the luid temperature as ollows: θ ( AY* ( A ε K Cosh = 1 + σ K s Cosh K H σk s K The solid temperature will be: ε θs = σ Where, K ( 1 Y K 1 Cosh 1 K s σ Cosh K H + σ K s K ( A Y ( A ( 1 Y (0 σ A = H ( ε (1 Result and Discussion The eects o variation in the conductivity ratio ( K s / K on the luid and solid phases are shown in ig.. In this igure ε =0.4. As the conductivity ratio increases, the order o magnitude o temperature dierence decreases. The temperature distribution or both solid and liquid phases ollows the same trend. the igure clearly shows that heat generation in the solid phase transer in to the luid phase such that the local thermal equilibrium assumption may not be valid or the case o small conductivity ratio( K s / K. The inluences o porosity on temperature o luid and solid phases are shown in Fig.3 (or K / =5. s K As is shown, increase the porosity leads to increase the temperature dierence between luid and solid phases and local thermal equilibrium assumption may not be correct. Conclusion In this work, convection heat transer in a channel illed with a porous medium was investigated analytically. Energy equation or the solid and the luid phases were used employing a non-thermal equilibrium model and source term in solid phase. An analytical representation o temperature ield or both phases was obtained incorporating the eect o conductivity ratio and porosity variation. The temperature dierence between luid and solid phases was ound to increase with decrease in conductivity ratio or increase in porosity. Reerences 1 Nield, A., Kuznetsov, A. V., Xiong, Ming, Thermally developing orced convection in a porous medium: parallel plate channel with walls at uniorm temperature, with axial conduction and viscous dissipation eects, Int. J. Heat Mass Transer, Vol.46, 003, pp Nield, D. A., Kuznetsov, A. V., Xiong, M., Eect o local thermal non-equilibrium on thermally developing orced convection in a porous medium, Int. J. Heat Mass Transer,Vol.45, 00, pp Nakayama, A., Kuwara, F., Sugiyama, M., Xu, G., A two-energy equation model or conduction and convection in porous media, Int. J. Heat Mass Transer, Vol.44, 001, pp Amiri, A., Vaai, K., Analysis o dispersion eects and non-thermal equilibrium, non- Darcian, variable porosity incompressible low through porous media, Int. J. Heat Mass Transer, Vol.37, 1994, pp kaviany, M., Principle o heat transer in porous media, second ed., Springer,Berlin, 1995 (chapter 3 6 Kim, S. J., Seok Pil Jang, Eect o the Darcy number, the Prandtl number and the Reynolds number on local thermal nonequilibrium, Int. J. Heat Mass Transer, Vol.45, 00, pp Carbonell, R. G., Whitaker, S., Heat and Mass Transer in porous media, Martinus Nijho, 1984 (in: J. Bear, M.Y. Corapcigolu (Eds., Foundamentals o transport phenomena in porous media 8 Nield, D. A., Bejan, A., Convection in porous media,second ed.,springer,berlin,1998 (chapter 4

4 9 Alazmi, B., Vaai, K., Analysis o variants within the porous media transport models, Int.J.Heat Transer, Vol.1, 000, pp Fig. 1 - channel illed with porous medium Fig. - Fluid and Solid temperature distribution with ε = 0.4, H = 5 K

5 k Fig. 3 - Fluid and Solid temperature distribution with s = 5, k H = 5 K