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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%.

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Abbreviations

\(A\) :

Cross-sectional area (m2)

\(C\) :

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/m3)

\(\bar{g}\) :

Average value of the volumetric rate of internal energy generation (W/m3)

\(g_{0}\) :

Heat generation coefficient (W/m3)

\(k\) :

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 (m3)

\(x, y, z\) :

Cartesian coordinates (m)

\(\beta\), \(\gamma\), \(\eta\) :

Eigen values of the heat equation

References

  1. Noda N (1999) Thermal stresses in functionally graded materials. J Therm Stresses 22(4–5):477–512

    Article  Google Scholar 

  2. Gallegos-Muñoz A, Violante-Cruz C, Balderas B, Rangel-Hernandez V, Belman-Flores J (2010) Analysis of the conjugate heat transfer in a multi-layer wall including an air layer. Appl Therm Eng 30:599–604

    Article  Google Scholar 

  3. Norouzi M, Rahman H, Birjandi A, Jone A (2016) A general exact analytical solution for anisotropic non-axisymmetric heat conduction in composite cylindrical shells. Int J Heat Mass Transf 93:41–56

    Article  Google Scholar 

  4. Howell J (1998) The Monte Carlo method in radiative heat transfer. J Heat Transf Trans ASME 120:547–560

    Article  Google Scholar 

  5. Naeimi H, Kowsary F (2017) An optimized and accurate Monte Carlo method to simulate 3D complex radiative enclosures. Int Commun Heat Mass Trans 84:150–157

    Article  Google Scholar 

  6. Naeimi H, Kowsary F (2017) Macro-voxel algorithm for adaptive grid generation to accelerate grid traversal in the radiative heat transfer analysis via Monte Carlo method. Int Commun Heat Mass Trans 87:22–29

    Article  Google Scholar 

  7. Sadiku MNO (2009) Monte Carlo methods for electromagnetics. CRC Press, Boca Raton

    Book  Google Scholar 

  8. Haji-Sheikh A, Sparrow E (1967) The solution of heat conduction problems by probability methods. J Heat Transf Trans ASME 89:121–130

    Article  Google Scholar 

  9. Kowsary F, Arabi M (1999) Monte Carlo solution of anisotropic heat conduction. Int Commun Heat Mass Transf 26(8):1163–1173

    Article  Google Scholar 

  10. Kowsary F, Irano S (2006) Monte Carlo solution of transient heat conduction in anisotropic media. J Thermophys Heat Transf 20(2):342–345

    Article  Google Scholar 

  11. Grigoriu M (2000) A Monte Carlo solution of heat conduction and Poisson equations. J Heat Transf Trans ASME 122:40–45

    Article  Google Scholar 

  12. Wong B, Francoeur M, Pinar Mengüç M (2011) A Monte Carlo simulation for phonon transport within silicon structures at nanoscales with heat generation. Int J Heat Mass Transf 54:1825–1838

    Article  Google Scholar 

  13. Bahadori R, Gutierrez H, Manikonda Sh, Meinke R (2018) A mesh-free Monte-Carlo method for simulation of three-dimensional transient heat conduction in a composite layered material with temperature dependent thermal properties. Int J Heat Mass Transf 119:533–541

    Article  Google Scholar 

  14. Hua Y-C, Cao B-Y (2017) An efficient two-step Monte Carlo method for heat conduction in nanostructures. J Comput Phys 342:253–266

    Article  MathSciNet  Google Scholar 

  15. Talebi S, Gharehbash K, Jalali HR (2017) Study on random walk and its application to solution of heat conduction equation by Monte Carlo method. Prog Nucl Energy 96:18–35

    Article  Google Scholar 

  16. Hua Y-C, Zhao T, Gua Z-Y (2017) Transient thermal conduction optimization for solid sensible heat thermal energy storage modules by the Monte Carlo method. Energy 133(C):338–347

    Article  Google Scholar 

  17. Haji-Sheikh A, Buckingham F (1993) Multidimensional inverse heat conduction using the Monte Carlo method. J Heat Transf Trans ASME 115:26–33

    Article  Google Scholar 

  18. Woodbury K, Beck J (2013) Estimation metrics and optimal regularization in a Tikhonov digital filter for the inverse heat conduction problem. Int J Heat Mass Transf 62:31–39

    Article  Google Scholar 

  19. Zeng Y, Wang H, Zhang S, Cai Y, Li E (2019) A novel adaptive approximate Bayesian computation method for inverse heat conduction problem. Int J Heat Mass Transf 134:185–197

    Article  Google Scholar 

  20. Ohmichi M, Noda N, Sumi N (2017) Plane heat conduction problems in functionallygraded orthotropic materials. J Therm Stresses 40(6):747–764

    Article  Google Scholar 

  21. Hsueh-Hsien LuH, Young D, Sladek J, Sladek V (2017) Three-dimensional analysis for functionally graded piezoelectric semiconductors. J Intell Mater Syst Struct 28(11):1391–1406

    Article  Google Scholar 

  22. Hahn D, Özişik M (2012) Heat conduction. Wiley, Hoboken

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Bojnord, P.O. Box 9453155111, Bojnord, North Khorasan, Iran

    Hooman Naeimi

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

    Farshad Kowsary

Authors
  1. Hooman Naeimi
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  2. Farshad Kowsary
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Correspondence to Hooman Naeimi.

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Technical Editor: Francis HR Franca, Ph.D..

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Naeimi, H., Kowsary, F. Finite Volume Monte Carlo (FVMC) method for the analysis of conduction heat transfer. J Braz. Soc. Mech. Sci. Eng. 41, 260 (2019). https://doi.org/10.1007/s40430-019-1762-3

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