Abstract
Surface melting of tungsten under exposure to a pulsed electron beam was simulated numerically, the evaporation process taken into account. The calculation is based on the experimental time dependence of the total beam power. The model of the tungsten heating process is based on solving the two-phase Stefan problem. The position of the phase boundary depends on discontinuous nonlinear coefficients. The aim of the study is to provide a detailed resolution of the heat flow deep into the material with a fine spatial grid step. As compared with the size of the tungsten plate, the heating depth is very small. The problem statement under consideration is multiscale. Further expansion of the model involves gas dynamics equations to simulate the dynamics of the liquid and gaseous phases of the metal. Two approaches to solving the equation for temperature are considered: the implicit run method and the explicitly solvable Konovalov-Popov model. The results of calculations correlate with the experimental data obtained at the experimental stand Beam of Electrons for materials Test Applications (BETA) at Budker Institute of Nuclear Physics (BINP) of the SB RAS.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Vyacheslavov, L., Arakcheev, A., Burdakov, A., Kandaurov, I., Kasatov, A., Kurkuchekov, V., Mekler, K., Popov, V., Shoshin, A., Skovorodin, D., Trunev, Y., Vasilyev, A.: Novel electron beam based test facility for observation of dynamics of tungsten erosion under intense ELM-like heat loads. AIP Conf. Proc. 1771, 060004 (2016)
Konovalov, A.N., Popov, YuP: Explicitly solvable optimal discrete models with controlled disbalance of the total mechanical energy for dynamical problems of linear elasticity. Siberian Math. J. 56(5), 872–878 (2015)
Apushkinskaya, D.: Free Boundary Problems: Regularity Properties Near the Fixed Boundary. LNM, vol. 2218. Springer (2018)
Caffarelli, L.A.: The smoothness of the free boundaries in higher dimensions. Acta Math. 139(3–4), 155–184 (1977)
Meirmanov, A.M.: The Stefan Problem. Nauka, Novosibirsk (1986)
Vynnycky, M., Mitchell, S.L.: On the numerical solution of a Stefan problem with finite extinction time. J. Comput. Appl. Math. 276, 98–109 (2015)
Mitchell, S.L., Vynnycky, M.: On the numerical solution of two-phase Stefan problems with heat-flux boundary conditions. J. Comput. Appl. Math. 264, 49–64 (2014)
Cialkowski, M.J., Grysa, K.: Trefftz method in solving the inverse problems. J. Inv. Ill-Posed Prob. 18, 595–616 (2010)
Lukyanenko, D.V., Shishlenin, M.A., Volkov, V.T.: Solving of the coefficient inverse problems for a nonlinear singularly perturbed reaction-diffusion-advection equation with the final time data. Commun. Nonlinear Sci. Numer. Simul. 54, 1339–1351 (2018)
Kabanikhin, S.I., Shishlenin, M.A.: Recovering a time-dependent diffusion coefficient from nonlocal data. Numer. Anal. Appl. 11(1,) 38–44 (2018)
Kanel, G.I., Zaretsky, E.B., Razorenov, S.V., Ashitkov, S.I., Fortov, V.E.: Unusual plasticity and strength of metals at ultra-short load durations. Phys. Usp. 60, 490–508 (2017)
Pikuz, S.A. (Jr.), Faenov, A.Y., Skobelev, I.Y., Fortov, V.E.: Production of exotic states of matter with the use of X-rays generated by focusing a petawatt laser pulse onto a solid target. Phys. Usp. 57, 702–707 (2014)
Anisimov, S.I., Prokhorov, A.M., Fortov, V.E.: Application of high-power lasers to study matter at ultrahigh pressures. Sov. Phys. Usp. 27, 181–205 (1984)
Trunev, YuA, Arakcheev, A.S., Burdakov, A.V., Kandaurov, I.V., Kasatov, A.A., Kurkuchekov, V.V., Mekler, K.I., Popov, V.A., Shoshin, A.A., Skovorodin, D.I., Vasilyev, A.A., Vyacheslavov, L.N.: Heating of tungsten target by intense pulse electron beam. AIP Conf. Proc. 1771, 060016 (2016)
Taluts, S.G.: Experimental study of the thermophysical properties of transition metals and alloys based on iron at high temperatures: abstract. dis. d-RA Fiz.-Mat. Sciences: 01.04.14. Publ. house UGGA, Yekaterinburg (2001). (in Russian)
Davis, J.W., Smith, P.D.: ITER material properties handbook. J. Nucl. Mater. 233, 1593–1596 (1996)
Ho, C.Y., Powell, R.W., Liley, P.E.: Thermal conductivity of elements. J. Phys. Chem. Ref. Data 1, 279 (1972)
Pottlacher, G.: Thermal conductivity of pulse-heated liquid metals at melting and in the liquid phase. J. Non-Cryst. Solids 250, 177–181 (1999)
Popov, V.A., Arakcheev, A.S., Burdakov, A.V., Kasatov, A.A., Vasilyev, A.A., Vyacheslavov, L.N.: Theoretical modeling of shielding for plasma flow and electron beam heating. AIP Conf. Proc. 1771, 060009 (2016)
Samarskii, A.A., Vabishchevich, P.N.: Computational Heat Transfer, p. 784. Editorial URSS, Moscow (2003). (in Russian)
Yanenko, N.N.: Method of Fractional Steps for Solving Multidimensional Problems of Mathematical Physics, p. 196. Science. Sib. Deposition, Novosibirsk (1967) (in Russian)
Maksimova, A.G., Lazareva, G.G., Arakcheev, A.S.: Computer Calculation of Heating of Various Geometry of Cracks Formed under Pulsed Heat Load. Bulletin of CC: Computer Science, vol. 41. NCC Publisher, Novosibirsk (2018)
Muratov, M.V., Petrov, I.B., Levant, V.B.: Development of mathematical models of cracks for numerical solution of seismic problems with the use of grid-characteristic method. Comput. Res. Model. 8(6), 911–925 (2016). (in Russian)
Galanin, M.P., Proshunin, N.N., Rodin, A.S., Sorokin, D.L.: The Solution of three-dimensional unsteady heat conduction equation by finite element method taking into account phase transitions, No. 66, p. 27 . Preprint IPM im. M. V. Keldysh, (2016). (in Russian)
Konovalov, A.N.: Numerical methods for the dynamical problems of elasticity. Siberian Math. J. 38(3), 471–487 (1997)
Soldatkina, E.I., Arakcheev, A.S., Bagryansky, P.A.: Experiments in support of the gas dynamic trap based facility for plasma–material interaction testing. Fusion Eng. Des. 88(11), 3084–3090 (2013)
Arakcheev, A.S., Apushkinskaya, D.E., Kandaurov, I.V., Kasatov, A.A., Kurkuchekov, V.V., Lazareva, G.G., Maksimova, A.G., Popov, V.A., Snytnikov, A.V., Trunev, YuA, Vasilyev, A.A., Vyacheslavov, L.N.: Two-dimensional numerical simulation of tungsten melting under pulsed electron beam. Fusion Eng. Des. 132, 13–17 (2018)
Acknowledgements
The problem statement the experimental data were supported by the Russian Science Foundation (project â„– 17-79-20203). Discrete models and computational algorithms were conducted within the framework of the budget project 0315-2016-0009 for ICMMG of the SB RAS.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Lazareva, G.G., Arakcheev, A.S., Vasilyev, A.A., Maksimova, A.G. (2019). Numerical Simulation of Tungsten Melting Under Fusion Reactor-Relevant High-Power Pulsed Heating. In: Petrov, I., Favorskaya, A., Favorskaya, M., Simakov, S., Jain, L. (eds) Smart Modeling for Engineering Systems. GCM50 2018. Smart Innovation, Systems and Technologies, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-030-06228-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-06228-6_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-06227-9
Online ISBN: 978-3-030-06228-6
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)