# Finite Volume Monte Carlo (FVMC) method for the analysis of conduction heat transfer

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## Abstract

The numerical solution of the heat equation is a particularly challenging subject in complex, practical applications such as functionally graded materials for which analytical solution is not available or hardly attainable. The Monte Carlo method is a powerful technique with some advantages compared to the conventional methods and is often used when all else fail. In this paper, we introduce the Finite Volume Monte Carlo (FVMC) method for solving 3D steady-state heat equation where, instead of using the usual finite difference scheme for discretization of the heat equation, the finite volume scheme is used. The FVMC method is tested for three problems to assess the robustness of the method, first one in a simple geometry for validation and evaluation of the predictive performance, the second one in a complex geometry with unstructured mesh and the last one in a problem with a variable heat source and different kinds of boundary conditions. Comparisons were made to the analytical solution in the first test case, whereas for the remaining test cases, the CFD methods were utilized in the absence of the analytical solutions. It was observed that the FVMC temperature distribution agrees perfectly with analytical and CFD solutions in all problems. Despite expecting computational accuracy to improve by increasing total number of particles in the FVMC method, a very good accuracy was obtained for all considered problems after a small number of walks, and the calculated relative root-mean-square errors were below 1%.

## Keywords

Monte Carlo Conduction Heat transfer Finite volume Numerical simulation## List of symbols

- \(A\)
Cross-sectional area (m

^{2})- \(C\)
Nondimensional temperature coefficient

- \(a\), \(b\) and \(c\)
Length, width and height of the box (m)

- \(\dot{E}\)
Energy (W)

- FVMC
Finite Volume Monte Carlo

- \(g\)
Volumetric rate of internal energy generation (W/m

^{3})- \(\bar{g}\)
Average value of the volumetric rate of internal energy generation (W/m

^{3})- \(g_{0}\)
Heat generation coefficient (W/m

^{3})- \(k\)
Thermal conductivity (W/m K)

- \(m_{i}\)
Total number of steps for each particle before reaching the boundary

- \(N\)
Total number of particles

- \(q\)
Heat flow rate (W)

- \(R\)
Random number

- \(S_{ }\)
Source term (W)

- \(T\)
Temperature (K)

- \(V\)
Volume (m

^{3})- \(x, y, z\)
Cartesian coordinates (m)

## Greek letters

- \(\beta\), \(\gamma\), \(\eta\)
Eigen values of the heat equation

## Notes

## References

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