Solution of the Isotropic Heat Equation Using the Finite Volume Monte Carlo Method

Document Type : Research Article

Authors

1 Department of Mechanical Engineering, Faculty of Engineering, University of Bojnord, Bojnord, North Khorasan, Iran

2 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

Abstract

The solution of the heat diffusion equation in most practical applications involving complex geometry, thermophysical properties, and boundary conditions is not simply possible and there are some limitations for available numerical solutions. In this research, the finite volume Monte Carlo method was used for the solution of the isotropic heat equation due to two intrinsic capabilities of the finite volume method; first, each cell is energy conserved and second, the grid transformation is not necessary for complex geometries. The Monte Carlo method is a statistical approach based on the physical simulation of the problem capable to solve heat equation with any degree of complexity. First, a simple problem was investigated for validation of the method by comparing results with the analytical solution. Second, the prediction performance of the finite volume Monte Carlo method was evaluated in a problem with complex geometry, varying properties, and boundary conditions. Finally, the performance of the finite volume Monte Carlo method was investigated in estimating the temperature distribution of a three-layer body with different thermal conductivities and convection boundary condition. In all of the considered test cases, the predicted results were in good agreement with analytical and computational fluid dynamics solutions. It was also indicated that for a relatively small number of particles, it is possible to achieve acceptable accuracy with a low computational cost.

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[1] M. Norouzi, H. Rahmani, A.K. Birjandi, A.A. Joneidi, A general exact analytical solution for anisotropic non-axisymmetric heat conduction in composite cylindrical shells, International Journal of Heat and Mass Transfer, 93 (2016) 41–56.
[2] A. Gallegos-Muñoz, C. Violante-Cruz, B.J.A. Balderas, V.H. Rangel-Hernandez, J.M. Belman-Flores, Analysis of the conjugate heat transfer in a multi-layer wall including an air layer, Applied Thermal Engineering, 30 (2010) 599–604.
[3] N. Noda, Thermal stresses in functionally Graded materials, Journal of Thermal Stresses, 22(4-5) (1999) 477-512.
[4] S.V. Patankar, Numerical heat transfer and fluid flow, Hemisphere Publishing Corporation, 1980.
[5] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford Science Publications, 1986.
[6] F. Wu, W.X. Zhong, A modified stochastic perturbation method for stochastic hyperbolic heat conduction problems, Computer Methods in Applied Mechanics and Engineering, 305 (2016) 739-758.
[7] F.S. Loureiro, W.J. Mansur, V. C.A.B., A hybrid time/Laplace integration method based on numerical Green’s functions in conduction heat transfer, Computer Methods in Applied Mechanics and Engineering, 198 (2009) 2662–2672.
[8] A. Haji-Sheikh, E.M. Sparrow, The Solution of Heat Conduction Problems by Probability Methods, Journal of Heat Transfer, Transaction of ASME,  (1967) 121-130.
[9] F. Kowsary, M. Arabi, Monte Carlo solution of anisotropic heat conduction, International Communications in Heat and Mass Transfer, 26(8) (1999) 1163-1173.
[10] M. Grigoriu, A Monte Carlo solution of heat conduction and Poisson equations, Journal of Heat Transfer, Transaction of ASME, 122 (2000) 40-45.
[11] B.T. Wong, M. Francoeur, M. Pinar Mengüç, A Monte Carlo simulation for phonon transport within silicon structures at nanoscales with heat generation, International Journal of Heat and Mass Transfer, 54 (2011) 1825–1838.
[12] K. Chatterjee, A new Green’s function Monte Carlo algorithm for the estimation of the derivative of the solution of Helmholtz equation subject to Neumann and mixed boundary conditions, Journal of Computational Physics, 315 (2016) 264-272.
[13] Y.F. Zhang, O. Gicquel, J. Taine, Optimized Emission-based Reciprocity Monte Carlo Method to speed up computation in complex systems, International Journal of Heat and Mass Transfer, 55 (2012) 8172–8177.
[14] K.P. Keadya, M.A. Clevelanda, An Improved Random Walk Algorithm for the Implicit Monte Carlo Method, Journal of Computational Physics, 328(C) (2017) 160-176
[15] W.J. Yao, R.N. Yang, N. Wang, Monte Carlo simulation of thermophysical properties of binary Co–Gd liquid alloys, Journal of Alloys and Compounds, 627 (2015) 410–414.
[16] A. Karchani, R.S. Myong, Convergence analysis of the direct simulation Monte Carlo based on the physical laws of conservation, Computers and Fluids, 115 (2015) 98-114.
[17] A. Haji-Sheikh, F.P. Buckingham, Multidimensional inverse heat conduction using the Monte Carlo method, Journal of Heat Transfer, Transaction of ASME, 115 (1993) 26-33.
[18] K.A. Woodbury, J.V. Beck, Estimation metrics and optimal regularization in a Tikhonov digital filter for the inverse heat conduction problem, International Journal of Heat and Mass Transfer, 62 (2013) 31–39.
[19] M. Ohmichi, N. Noda, N. Sumi, Plane heat conduction problems in functionally graded orthotropic materials, Journal of Thermal Stresses, 40(6) (2017) 747-764.
[20] D.W. Hahn, M. Necati Ӧzişik, Heat conduction, 3rd ed ed., John Wiley and Sons, Hoboken, New Jersey, 2012.
[21] H. Naeimi, F. Kowsary, Finite Volume Monte Carlo (FVMC) method for the analysis of conduction heat transfer, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 41 (2019) 260.