A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media

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Abstract

A numerical experiment has been conducted to determine the interfacial convective heat transfer coefficient in the two-energy equation model for convection in porous media, which is needed when the local thermal equilibrium between the fluid and solid phases breaks down. The similarity of periodically fully developed temperature profiles allows one to perform a numerical experiment using only a single structural unit for determining the fully developed heat transfer coefficient without any empiricism. A universal correlation for the Nusselt number, which agrees well with available experimental data, has been established using the results obtained for a wide range of porosity, Prandtl and Reynolds numbers.

Introduction

There are certain situations where net heat transfer from one phase to another phase takes place in saturated porous media such that the assumption of local thermal equilibrium breaks down. The need for two-energy equation models (allowing two temperatures in two different phases) has long been recognized [1].

When there is a significant heat generation occurring in any one of the two phases (solid or fluid), the temperatures in the two phases are no longer identical [2]. The assumption of local thermal equilibrium must be discarded when we analyze the entrance region of packed column where a hot gas flows at a high speed. Many of unsteady problems associated with saturated porous media need treatments, which allow heat transfer from one phase to another. When the temperature at the bounding surface changes significantly with respect to time, and when solid and fluid phases have significantly different heat capacities and thermal conductivities, the local rate of change of temperature for one phase differs significantly from that for the other phase [3].

Moreover, Quintard [4] argues that assessing the validity of the assumption of local thermal equilibrium is not a simple task, since the temperature difference between the two phases cannot easily be estimated, and suggests that the use of a two-energy equation model is a possible solution to the problem. Numerous other physical situations where local thermal equilibrium fails are cited by Quintard and Whitaker [5].

Two energy equation models have been introduced heuristically in the literature [6]. These heuristic model equations fit in the following form:ερfCpf〈T〉ft+〈uf·∇Tf=∇·k̄̄feff·∇Tf+hsfasfTsTf,1−ερsCsTst=∇·k̄̄seff·∇Ts−hsfasfTsTf,where the subscripts (and superscripts) f and s denote fluid and solid phases, respectively. u is the volume-averaged velocity (i.e., Darcian velocity), whereas 〈Tf and 〈Ts denote the intrinsically averaged temperature of fluid phase and that of solid phase, respectively, such that 〈Tf=〈Ts under local thermal equilibrium. Moreover, ε, k̄̄eff, asf and hsf are the porosity, effective thermal conductivity tensor, specific surface area and interfacial convective heat transfer coefficient, respectively. In the two-equation model, the interfacial heat transfer coefficient hsf and two thermal conductivity tensors k̄̄feff and k̄̄seff need to be determined. Quintard and Whitaker [5] obtained somewhat different form of the equations from the point of view of the volume-averaging theory. The major difference between their form and the foregoing heuristic form is the appearance of additional coupling terms. They claim that these coupling terms are not necessarily negligible. Although their model appears to be more general than the classical one, it requires four thermal conductivity tensors in addition to the interfacial convective heat transfer coefficient. Some theoretical attempts of determining these transport coefficients have been made by Quintard and Whitaker [5]. However, the unknown transport coefficients as well as part of the formulation itself make it difficult and awkward for simulating practical applications [7]. Viskanta [8] foresees that the classical form described by , will continue to be used with empirical transport coefficients.

In this paper, we shall propose a numerical procedure to determine the macroscopic transport coefficients such as hsf purely from a theoretical basis without any empiricism. Upon extending the numerical procedure for thermal equilibrium, developed by Nakayama et al. [9] and Kuwahara et al. [10], to the case of non-thermal equilibrium, we will conduct numerical experiments for a wide range of porosity, Prandtl and Reynolds numbers. Noting the similarity of fully developed temperature profiles, we shall use only a single structural unit to simulate a porous medium, and determine the interfacial heat transfer coefficient for the asymptotic case in which the conductivity of solid phase is infinite.

Section snippets

Volume-averaged energy equations and expressions for transport coefficients

Let us compare the foregoing , of the classical model with the intrinsic volume-averaged equations obtained by integrating the individual energy equations over a representative elementary volume V. (Note that V1/3 must be much smaller than a macroscopic characteristic length, but at the same time, much larger than a pore size.) Following Cheng [11] and Nakayama [12], we obtain the following macroscopic energy equations for the two individual phases:ερfCpfTft+〈uf·∇Tf=∇·εkfTf+1VAintkfTdA−ρ

Numerical model and periodic boundary conditions

Fluid particles experience complex three-dimensional motions as passing through a microscopic porous structure. The macroscopic hydrodynamic and thermodynamic behavior of practical interest can be obtained from the direct application of the first principles to viscous flow and heat transfer at a pore scale. In reality, however, it is impossible to resolve the details of the flow and heat transfer fields within a real porous medium, even with a most powerful super-computer available today.

Method of computation and preliminary numerical consideration

The governing equations are discretized by integrating them over a grid volume. SIMPLE algorithm for the pressure–velocity coupling, as proposed by Patankar and Spalding [16] is employed. Convergence is measured in terms of the maximum change in each variable during an iteration. The maximum change allowed for the convergence check is set to 10−5, as the variables are normalized by appropriate references. A fully implicit scheme is adopted with the hybrid differencing scheme for the advection

Results and discussion

The velocity and temperature fields obtained for three different Reynolds numbers are shown in Fig. 4, Fig. 5, respectively. When the Reynolds number is low (say Re = 1), the velocity field around a rod (except front and rear stagnation regions) appears very much similar to what we observe in a channel, namely the parabolic profile. As increasing Re, recirculation bubbles expand further behind the rod. When the Reynolds number is sufficiently high, the thermal boundary layers cover around the

Concluding remarks

The correlation for the interfacial convective heat transfer coefficient has been established from a series of numerical experiments based on a two-dimensional structural model of porous media. A macroscopically uniform flow through a periodic model of isothermal square rods was assumed, considering periodically fully developed velocity and temperature fields. Upon noting the similarity of the temperature profile, only a single structural unit has been taken for a calculation domain. Effects of

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