Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method

doi:10.1016/j.jcp.2006.08.021

Journal of Computational Physics

Volume 223, Issue 1, 10 April 2007, Pages 89-107

https://doi.org/10.1016/j.jcp.2006.08.021 Get rights and content

Abstract

The lattice Boltzmann method (LBM) was used to solve the energy equation of a transient conduction–radiation heat transfer problem. The finite volume method (FVM) was used to compute the radiative information. To study the compatibility of the LBM for the energy equation and the FVM for the radiative transfer equation, transient conduction and radiation heat transfer problems in 1-D planar and 2-D rectangular geometries were considered. In order to establish the suitability of the LBM, the energy equations of the two problems were also solved using the FVM of the computational fluid dynamics. The FVM used in the radiative heat transfer was employed to compute the radiative information required for the solution of the energy equation using the LBM or the FVM (of the CFD). To study the compatibility and suitability of the LBM for the solution of energy equation and the FVM for the radiative information, results were analyzed for the effects of various parameters such as the scattering albedo, the conduction–radiation parameter and the boundary emissivity. The results of the LBM–FVM combination were found to be in excellent agreement with the FVM–FVM combination. The number of iterations and CPU times in both the combinations were found comparable.

Introduction

In the recent years, usage of the lattice Boltzmann method (LBM) as an alternative approach to the conventional computational fluid dynamics (CFD) solvers such as the finite element method (FEM), the finite difference method (FDM) and the finite volume method (FVM) has gained momentum [1], [2], [3], [4], [5], [6], [7], [8], [9]. As a different approach from the CFD solvers, the LBM uses simple microscopic kinetic models to simulate complex transport phenomena. In comparison to the CFD solvers, the advantages of the LBM include among other simple calculation procedure, simple and efficient implementation for parallel computation, easy and robust handling of complex geometries and high computational performance with regard to stability and accuracy [1], [2], [3], [4], [5], [6], [7], [8], [9].

The LBM has found extensive usage in the fluid mechanics [3], [5], [6], [7], [8] and its recent application to problems involving conductive, convective and/or radiative heat transfer has been very encouraging [10], [11], [12], [13], [14]. Shan [10] and Mezrhab et al. [11] used the LBM to analyze the convective flows. Ho et al. [12], [13] solved a non-Fourier heat conduction problem in a planar medium using the LBM. Solidification of a planar medium using the LBM was analyzed by Jiaung et al. [14]. Chatterjee and Chakraborty [15] used the LBM to analyze solid–liquid phase transitions in the presence of thermal diffusion. Mishra and Lankadasu [16] applied the LBM to solve the energy equation of transient conduction and radiation heat transfer in a planar medium with or without heat generation. They used the discrete transfer method (DTM) [17] to compute the radiative information. Mishra et al. [18] used the LBM to solve the energy equation of a transient conduction–radiation heat transfer in a 2-D square enclosure. In their study, they used the collapsed dimension method (CDM) [19] to compute the radiative information. Application of the LBM to analyze the solidification of a semitransparent planar layer was extended by Raj et al. [20]. They used the DTM [17] to compute the radiative information. Gupta et al. [21] used the concept of a variable relaxation time in the LBM and then solved the energy equation of a temperature dependent transient conduction and radiation heat transfer in a planar medium. They used the discrete ordinate method (DOM) [22] for the determination of the radiative information. In all the previous applications to the conduction–radiation heat transfer problems, the LBM was found to provide accurate results and compatibilities of the LBM for solution of energy equation and the DTM, the CDM and the DOM for the determination of radiative information were established. The DTM in [18], [20] and the DOM in [21] with the LBM were applied to 1-D planar geometry.

In radiative heat transfer, the finite volume method (FVM) [23], [24] is extensively used to compute the radiative information. This method is a variant of the DOM [22]. However, unlike the DOM, it does not suffer from the false-scattering [24]. In this method, the ray-effect is also less pronounced [24]. Since the FVM for the radiative heat transfer utilizes the same concept as that of the FVM of the CFD, in conjugate mode problems, its computational grids are compatible with the FVM grids that are utilized in the solution of the momentum and energy equations [25], [26]. Thus, although the FVM for the radiative heat transfer is only a 15-year old method, it enjoys more popularity than the DTM, and the DOM.

In solving the energy equations of the combined radiation, conduction and/or convection heat transfer problems using the conventional CFD based methods such as the FDM and the FVM, robustness of the FVM in providing radiative information is well established [23], [24]. Since the LBM is relatively a new method and for the solution of combined radiation, conduction and/or convection mode problems its application is very recent, the FVM has so far not been used in conjunction with the LBM to solve any such problems. The present article is thus aimed at extending the application of both the LBM and the FVM to a relatively new class of problems.

The objective of the present work is to establish the compatibility of the LBM for the solution of the energy equation and the FVM for the determination of radiative information. One other objective is also to see how the LBM–FVM combination performs against the FVM–FVM combination in which the energy equation is solved using the FVM of the CFD and the radiative information is computed using the FVM of radiative heat transfer. Towards this goal, transient conduction and radiation heat transfer problems in 1-D planar and 2-D rectangular enclosures are considered. For various parameters like the scattering albedo, the conduction–radiation parameter and the boundary emissivity, results of the LBM–FVM and the FVM–FVM are compared with those reported in the literature and compared against each other. Effects of the spatial and angular resolutions on the results are also made. The number of iterations and CPU times for the converged solutions are also reported.

The paper is organized as follows. In Section 2, the formulation of the FVM to calculate radiative information required for the energy equation is provided first. The methodology of the LBM is discussed next. Implementation of the boundary conditions is briefly discussed after that. The solution procedure has been discussed in Section 3. The results of the parametric study and the comparison are presented in Section 4. Conclusions and recommendations are made at the end.

Section snippets

Formulation

In the absence of convection and heat generation, for a homogeneous medium, the energy equation is given by $ρ c_{p} \frac{\partial T}{\partial t} = k \nabla^{2} T - \nabla \cdot {\vec{q}}_{R}$ where ρ is the density, c_p is the specific heat, k is the thermal conductivity and ${\vec{q}}_{R}$ is the radiative heat flux. With radiative information $\nabla \cdot {\vec{q}}_{R}$ computed using any of the methods such as the DTM [17], the CDM [19], the DOM [22] and the FVM [23], [24], Eq. (1) can be solved using any of the conventional CFD methods such as the FDM, FEM and the FVM or an alternative

Solution procedure

The medium is divided into a finite number of lattices/control volumes. The control volumes of the FVM for computing the radiative information $\nabla \cdot {\vec{q}}_{R}$ and lattices in the LBM are staggered as shown in Fig. 2, Fig. 3, Fig. 4. Sizes of the lattices in the LBM and the control volumes in the FVM are taken the same.

In solving Eq. (35), $\nabla \cdot {\vec{q}}_{R}$ information is required at the lattice centers (Fig. 2, Fig. 3, Fig. 4). In the FVM for the radiative heat transfer, in any control volume, intensity

Results and discussion

To validate the usage of the LBM and to show the compatibility of the FVM for the radiative information with the LBM solver for the energy equation, we consider transient conduction and radiation heat transfer in a 1-D planar and 2-D square geometries. To compare the performance of the LBM, the energy equations of the two problems were also solved using the FVM of the CFD. In both the LBM and the FVM solvers of the energy equations, radiative information $\nabla \cdot {\vec{q}}_{R}$ was computed using the FVM of the

Conclusions

The LBM was used to solve the energy equations of transient conduction–radiation heat transfer problems in 1-D planar and 2-D square geometries containing an absorbing, emitting and isotropically scattering medium. Medium boundaries were considered diffuse-gray. To compare the performance of the LBM, energy equations of the problems were also solved using the FVM of the CFD. In the solution of the energy equations using the LBM and the FVM, the radiative information was computed using the FVM

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Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method

Abstract

Introduction

Section snippets

Formulation

Solution procedure

Results and discussion

Conclusions

Phys. Rep.

J. Comput. Phys.

Int. J. Multipase Flow

Int. J. Heat Mass Transfer

Phys. Lett. A

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Lattice gas dynamics with enhanced collisions

Europhys. Lett.

Boltzmann approach to lattice gas simulations

Europhys. Lett.

Exponential tails in two-dimensional Rayleigh–Bénard convection

Europhys. Lett.

Lattice Boltzamann method for fluid flows

Ann. Rev. Fluid Mech.

Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction

The Lattice Boltzmann Method for Fluid Dynamics and Beyond

Simulation of Rayleigh–Benard convection using a lattice Boltzmann method

Phys. Rev. E

Hybrid lattice-Boltzmann finite-difference simulation of convective flows

Comput. Fluids