Numerical study with OpenFOAM on heat conduction problems in heterogeneous media
Introduction
Heat conduction in heterogeneous media such as composite materials [1], [2], phase change medium [3], [4], metal foam [5], and biological tissues [6] has extensive industrial applications. In heterogeneous media, there is usually a quite small spatial region, in which the physical properties change dramatically and the derivatives of physical quantity do not exist anymore. Therefore, it can be taken as an interface. Under certain thermal conditions, the non-uniformity of the temperature and heat flux at the interface can damage high-temperature aerospace structures [7] and electronic packaging [8], [9]. Accurate and efficient prediction of the temperature distribution is key to the analysis and design of heterogeneous materials.
Analytical solutions for heterogeneous conduction can only be obtained in some simplified cases. Therefore, numerical methods have often been developed to obtain approximate solutions. Classical numerical methods such as the finite volume method (FVM) [10] and finite element method (FEM) [11] are widely used for many computational fluid dynamics (CFD) platforms. These have been used to derive numerical methods such as the boundary element method (BEM) [12] to solve the heat conduction problem in non-homogeneous and functionally gradient materials with high accuracy. In order to overcome the difficulties of traditional methods with meshing and re-meshing when dealing with moving boundary problems, a large family of meshless methods has been proposed in the last decade to deal with homogeneous [13] and strongly heterogeneous conduction problems [14], [15]. The new and popular lattice Boltzmann method (LBM), whose kinetic nature makes it suitable for interactions at the microscope scale, has recently been extended to heat conduction in fibrous materials [16] and heterogeneous media [4]. In general, the programming progress of “in-house” codes based on these methods is difficult and time-consuming. In addition, extending such codes to arbitrary complex geometries is not easy. Commercial CFD packages such as ANSYS Fluent are quite convenient and efficient but only provide intact solutions for common situations considered by the designers. The limited programming interface makes it difficult to use these packages to implement new models for specific needs. In contrast, OpenFOAM (Open Source Field Operation and Manipulation) is a free-source CFD package [17] that provides both tutorials and the C++ source code. Existing calculation modules can be used, similar to commercial CFD software, or the code can be modified to different purposes. So far, many researchers have implemented their own codes in OpenFOAM for heat transfer problems [18], [19], [20], [21], [22]. It is worth mentioning that Starikovičius and co-workers [22] had studied the conduction in electrical power cables and compared the efficiency of conjugate gradient solver.
In this work, we further consider the heat conduction in heterogeneous media with OpenFOAM platform. Section 2 briefly introduces the governing equations and discretization procedures in OpenFOAM. Section 3 presents various cases with a high degree of heterogeneity that were used to evaluate the accuracy of the C++ open-source libraries. The results were compared with analytical solutions or numerical solutions from other methods.
Section snippets
Governing equation and discretization schemes
The governing equation for transient heat conduction with varying thermal diffusivity can be written as:where is the thermal diffusivity, T is the temperature, and t is the time. In this work, no heat source is considered.
This equation includes the Laplacian term and the time derivative term. OpenFOAM CFD toolbox adopts finite-volume method (FVM) and both of the two terms can be integrated over a control volume, such as:
With Gauss formula applied,
Results
OpenFOAM was used to numerically study various cases of 1D and 2D steady heat conduction problems in media with different degrees of heterogeneity. Actually, the transient heat conduction equation was solved in this paper. In order to compare the results with references, we simulate heat conduction process till the equilibrium is established and the results in steady state are obtained. To facilitate the comparison, the average relative error was defined as following:
Conclusion
OpenFOAM was employed to simulate heat conduction problems in heterogeneous media. To check the performance of OpenFOAM, 1D and 2D heat conduction problems with different kinds of heterogeneity were considered. The influences of the heterogeneity and mesh distribution were fully investigated. The results were compared with references and other commercial software and were proven to be accurate and reliable. The great advantage of this work is that the discontinuity interface can be captured
Conflict of interest
The authors declare that they have no conflict of interests.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 51406042), Research project of education reform of Harbin Institute of Technology (JGYJ-201650, BKQN201609).
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