Radial integral boundary element method for simulating phase change problem with mushy zone

doi:10.1007/s10483-021-2760-8

Abstract

Abstract: A radial integral boundary element method (BEM) is used to simulate the phase change problem with a mushy zone in this paper. Three phases, including the solid phase, the liquid phase, and the mushy zone, are considered in the phase change problem. First, according to the continuity conditions of temperature and its gradient on the liquid-mushy interface, the mushy zone and the liquid phase in the simulation can be considered as a whole part, namely, the non-solid phase, and the change of latent heat is approximated by heat source which is dependent on temperature. Then, the precise integration BEM is used to obtain the differential equations in the solid phase zone and the non-solid phase zone, respectively. Moreover, an iterative predictor-corrector precise integration method (PIM) is needed to solve the differential equations and obtain the temperature field and the heat flux on the boundary. According to an energy balance equation and the velocity of the interface between the solid phase and the mushy zone, the front-tracking method is used to track the move of the interface. The interface between the liquid phase and the mushy zone is obtained by interpolation of the temperature field. Finally, four numerical examples are provided to assess the performance of the proposed numerical method.

Key words: phase change, mushy zone, boundary element method (BEM), precise integration method (PIM)

2010 MSC Number:

65M38
80A22

Hongxiao YAO, Weian YAO, Chong ZUO, Xiaofei HU. Radial integral boundary element method for simulating phase change problem with mushy zone. Applied Mathematics and Mechanics (English Edition), 2021, 42(8): 1155-1170.

TrendMD

References

[1] HONG, Y., YE, W., DU, J., and HUANG, S. Solid-liquid phase-change thermal storage and release behaviors in a rectangular cavity under the impacts of mushy region and low gravity. International Journal of Heat and Mass Transfer, 130, 1120-1132(2019)
[2] CRANK, J. Free and Moving Boundary Problems, Clarendon Press, Oxford (1984)
[3] HE, Y., DONG, X., and YANG, H. A new adaptive algorithm for phase change heat transfer problems based on quadtree SBFEM and smoothed effective heat capacity method. Numerical Heat Transfer, Part B:Fundamentals, 75(2), 111-126(2019)
[4] ZHAO, X. and CHEN, X. The temperature analysis of the billet with phase change during continuous casting. Applied Mathematics and Mechanics (English Edition), 15(2), 109-118(1994) https://doi.org/10.1007/BF02451045
[5] LI, F., LIU, J., and YUE, K. Exact analytical solution to three-dimensional phase change heat transfer problems in biological tissues subject to freezing. Applied Mathematics and Mechanics (English Edition), 30(1), 63-72(2009) https://doi.org/10.1007/s10483-009-0107-x
[6] YAO, W., WANG, Z., ZUO, C., and HU, X. Precise time-domain expanding boundary element method for solving phase change problems. Numerical Heat Transfer, Part B:Fundamentals, 76(4), 203-223(2019)
[7] YU, B., YAO, W. A., GAO, X. W., and ZHANG, S. Radial integration BEM for one-phase solidification problems. Engineering Analysis with Boundary Elements, 39, 36-43(2014)
[8] LI, Y. and ZHANG, Z. Solving the free boundary problem in continuous casting by using boundary element method. Applied Mathematics and Mechanics (English Edition), 16(12), 1201-1208(1995) https://doi.org/10.1007/BF02466990
[9] CHESSA, J., SMOLINSKI, P., and BELYTSCHKO, T. The extended finite element method (XFEM) for solidification problems. International Journal for Numerical Methods in Engineering, 53(8), 1959-1977(2002)
[10] ZHOU, J. and QI, L. Treatment of discontinuous interface in liquid-solid forming with extended finite element method. Transactions of Nonferrous Metals Society of China, 20, s911-s915(2010)
[11] HE, M., YANG, Q., LI, N., FENG, X., and LIU, N. An extended finite element method for heat transfer with phase change in frozen soil. Soil Mechanics and Foundation Engineering, 57(6), 497-505(2021)
[12] LIAO, J. and ZHUANG, Z. A consistent projection-based SUPG/PSPG XFEM for incompressible two-phase flows. Acta Mechanica Sinica, 28(5), 1309-1322(2012)
[13] YU, B., XU, C., YAO, W., and MENG, Z. Estimation of boundary condition on the furnace inner wall based on precise integration BEM without iteration. International Journal of Heat and Mass Transfer, 122, 823-845(2018)
[14] YANG, K., LI, H., PENG, H., and GAO, X. New interface integration BEM for solving multi-medium nonlinear heat transfer problems. Engineering Analysis with Boundary Elements, 117, 66-75(2020)
[15] ANG, W. T. and WANG, X. A numerical method based on boundary integral equations and radial basis functions for plane anisotropic thermoelastostatic equations with general variable coefficients. Applied Mathematics and Mechanics (English Edition), 41(4), 551-566(2020) https://doi.org/10.1007/s10483-020-2592-8
[16] YU, B., ZHOU, H. L., GAO, Q., and YAN, J. Geometry boundary identification of the furnace inner wall by BEM without iteration. Numerical Heat Transfer, Part A:Applications, 69(11), 1253-1262(2016)
[17] CHOLEWA, R., NOWAK, A. J., and WROBEL, L. C. Application of BEM and sensitivity analysis to the solution of the governing diffusion-convection equation for a continuous casting process. Engineering Analysis with Boundary Elements, 28(4), 389-403(2004)
[18] WANG, Z., YAO, W., ZUO, C., and HU, X. Solving phase change problems via a precise time-domain expanding boundary element method combined with the level set method. Engineering Analysis with Boundary Elements, 126, 1-12(2021)
[19] YU, B., TONG, Y., HU, P., and GAO, Q. A novel inversion approach for identifying the shape of cavity by combining Gappy POD with direct inversion scheme. International Journal of Heat and Mass Transfer, 150, 119365(2020)
[20] GAO, X. The radial integration method for evaluation of domain integrals with boundary-only discretization. Engineering Analysis with Boundary Elements, 26(10), 905-916(2002)
[21] YANG, K., LIU, Y., and GAO, X. Analytical expressions for evaluation of radial integrals in stress computation of functionally graded material problems using RIBEM. Engineering Analysis with Boundary Elements, 44, 98-103(2014)
[22] AL-BAYATI, S. A. and WROBEL, L. C. Radial integration boundary element method for two-dimensional non-homogeneous convection-diffusion-reaction problems with variable source term. Engineering Analysis with Boundary Elements, 101, 89-101(2019)
[23] ZHONG, W. X. On precise time-integration method for structural dynamics (in Chinese). Journal of Dalian University of Technology, 34(02), 131-136(1994)
[24] ZHANG, H., CHEN, B., and GU, Y. An adaptive algorithm of precise integration for transient analysis. Acta Mechanica Solida Sinica, 14(3), 215-224(2001)
[25] OZISIK, M. N. and UZZELL, J. C. Exact solution for freezing in cylindrical symmetry with extended freezing temperature range. Journal of Heat Transfer, 101(2), 331-334(1979)
[26] YU, B., YAO, W. A., and GAO, Q. A precise integration boundary-element method for solving transient heat conduction problems with variable thermal conductivity. Numerical Heat Transfer, Part B:Fundamentals, 65(5), 472-493(2014)
[27] YAO, W., YAO, H., ZUO, C., and HU, X. Precise integration boundary element method for solving dual phase change problems based on the effective heat capacity model. Engineering Analysis with Boundary Elements, 108, 411-421(2019)
[28] GU, Y. X., CHEN, B. S., ZHANG, H. W., and GUAN, Z. Q. Precise time integration method for solution of nonlinear transient heat conduction (in Chinese). Journal of Dalian University of Technology, 40(1), 24-28(2000)
[29] O'NEILL, K. Boundary integral equation solution of moving boundary phase change problems. International Journal for Numerical Methods in Engineering, 19(12), 1825-1850(1983)
[30] LIN, J. Y. and CHEN, H. T. Numerical analysis for phase change problems with the mushy zone. Numerical Heat Transfer, Part A:Applications, 27(2), 163-177(1995)