Numerical simulation of microradioelectric element cooling by natural air convection

About the authors:

Pavel I. Tomchik, Postgraduate Student, Department of Fundamental Mathematics and Mecha­nics, School of Computer Sciences, University of Tyumen, Tyumen, Russia; p.i.tomchik@utmn.ru, https://orcid.org/0000-0001-6960-4097

Anatoliy A. Kislitsin, Dr. Sci. (Phys.-Math.), Professor, Department of Applied and Technical Physics, School of Natural Sciences, University of Tyumen, Tyumen, Russia; a.a.kislicyn@utmn.ru, https://orcid.org/0000-0003-3863-0510

Abstract:

The article presents a series of numerical experiments aimed at expanding the scope of application of a popular method of mathematical modeling of natural convection, the so-called “Boussinesq problem”. The studied area is a cube with impenetrable faces filled with air. On the lower face of the cube there is a microradioelement, which is a rectangular parallelepiped with a square base. The vertical faces of the cube are heat-insulated, the upper and lower faces are maintained at a constant temperature. Inside the microradioelement there is a constant heat source of a given power. Three-dimensional numerical modeling was performed taking into account the dissipative function with a change in the Grashof numbers from 36989 to 262051, Eckert numbers from 7,79 10–12 to 5,67 10–11, Rayleigh numbers from 51937 to 190974 and heat source powers from 0.7 mW/mm3 to 10 mW/mm3. With such parameters, the temperature of the microradioelement case changed from 20 °C to 32.66 °C. Under the condition of a small temperature difference, the thermophysical properties of air are taken equal to its properties at a temperature of 20 °C. Discrete analogs of the system of equations of the mathematical model are obtained by the control volume method, the solution is searched for by the software implementation of the modified SIMPLER algorithm. The obtained flow field in the central vertical section of the studied volume is compared with the results of experimental studies by other authors, and a qualitative agreement of the results is noted. At different powers of the internal heat source, the effect of natural convection on the decline (after the initial rise) and subsequent stabilization of the temperature of the microradioelement case is shown. The effect of reorientation of large-scale circulation is demonstrated using temperature fields as an example.

References:

Zubkov, P. T. (October 22–26, 2018). Proceedings of the seventh Russian National Conference on Heat Exchange: In 3 volumes. Volume 1. MEI Publishing House, pp. 323–326. [In Russian]

Korotaev, B. A., Gamolich, V. Ya., & Burov, A. A. (2011). Dissipative function of a closed flow of incompressible viscous liquid. Modern science: research, ideas, results, technologies, 2(7), 119–121. [In Russian]

Landau, L. D., & Lifshits, E. M. (1988). Theoretical Physics: in 10 volumes. Volume 6: Hydrodynamics. Nauka. [In Russian]

Patankar, S. V. (2003). Numerical solution of problems of thermal conductivity and convective heat transfer during flow in channels. MEI Publishing House. [In Russian]

Sukhanovsky, A. N. (2021). Convective flows of various scales in stationary and rotating enclosed volumes [Doctoral dissertation, Institute of Continuum Mechanics of the Ural Branch of the RAS]. [In Russian]

Tananaev, A. V. (1979). Current in the channels of MHD devices. Atomizdat. [In Russian]

Tomchik, P. I., Kislitsyn A. A. (2024). Numerical study of the stability of natural convection. Bulletin of the Tyumen State University. Physical and Mathematical Modeling. Oil, Gas, and Energy, 10(4)(40), 50–67. [In Russian]

Cherkasov, S. G., Moiseeva, L. A., & Ananiev A. V. (2018). Limitations of the Boussinesque model on the example of laminar natural gas convection between vertical isothermal walls. Thermophysics of High Temperatures, 56(6), 902–907. [In Russian]

Schlichting, G. (1974). Theory of the boundary layer. Nauka. [In Russian]

Bennacer, R., El Ganaoui, M., & Leonardi, E. (2006). Symmetry breaking of melt flow typically encountered in a Bridgman configuration heated from below. Applied Mathematical Modelling, 30(11), 1249–1261.

Benoit, M. R., Brown, R. B., Todd, P., Nelson, E. S., Klaus, D. M. (2008). Buoyant plumes from solute gradients generated by non-motile Escherichia coli. Physical Biology, 5(4),
art. 046007.

Bontoux, P., Roux, B., Schiroky, G. H., Markham, B. L., & Rosenberger, F. (1986). Convection in the vertical midplane of a horizontal cylinder. Comparison of two-dimensional approximations with three-dimensional results. International Journal of Heat and Mass Transfer, 29(2), 227–240.

Christon, M. A., Gresho, P. M., Sutton, S. B. (2002). Computational predictability of time-dependent natural convection flows in enclosures (including abenchmark solution). International Journal for Numerical Methods in Fluids, 40(8), 953–980.

Delgado-Buscalioni, R., & del Arco, E. C. (1999). Stability of thermally driven shear flows in long inclined cavities with end-to-end temperature difference. International Journal of Heat and Mass Transfer, 42(15), 2811–2822.

De Vahl Davis, G. (1983). Natural convection of air in a square cavity: A bench mark numerical solution. International Journal for Numerical Methods in Fluids, 3(3), 249–264.

El Ganaoui, M., Lamazouade, A., Bontoux, P., & Morvan, D. (2002). Computational solution for fluid flow under solid/liquid phase change conditions. Computers & Fluids, 31(4-7), 539–556.

Ferialdi, H., & Lappa, M. (2024). An experimental and numerical investigation into the sensitivity of Rayleigh–Bénard convection to heat loss through the sidewalls. Physica D: Nonlinear Phenomena, 464, art. 134190. https://doi.org/10.1016/j.physd.2024.134190

Gelfgat, A. Y. (1999). Different modes of Rayleigh–Bénard instability in two-and three-dimensional rectangular enclosures. Journal of Computational Physics, 156(2), 300–324.

Glatzmaier, G. A., & Roberts, P. H. (1997). Simulating the geodynamo. Contemporary Physics, 38(4), 269–288.

Heng, K., & Showman, A. P. (2015). Atmospheric dynamics of hot exoplanets. Annual Review of Earth and Planetary Sciences, 43(1), 509–540.

Hortmann, M., Peri´c, M., & Scheuerer, G. (1990). Finite volume multigrid prediction of laminar natural convection: Bench-mark solutions. International Journal for Numerical Methods in Fluids, 11, 189–207.

Lyubimova, T., Rushinskaya, K., & Zubova, N. (2020). Onset and nonlinear regimes of convection of a binary mixture in rectangular cavity heated from below. Microgravity Science & Technology, 32, 961–972. https://doi.org/10.1007/s12217-020-09823-x

Melnikov, D., Mialdun, A., & Shevtsova, V. (2007). Peculiarity of 3D flow organization in Soret driven instability. Journal of Non-Equilibrium Thermodynamics, 32(3), 259–270.

Santamaria, C., Šeta, B., Gavalda, J., Bou-Ali, M. M., & Ruiz, X. (2020). Determining diffusion, thermodiffusion and Soret coefficients by the thermogravitational technique in binary mixtures with optical digital interferometry analysis. International Journal of Heat and Mass Transfer, 147, art. 118935.

Seta, B., Errarte, A., Ryzhkov, I., Bou-Ali, M., & Shevtsova, V. (2023). Oscillatory instability caused by the interplay of Soret effect and cross-diffusion. Physics of Fluids, 35(2), art. 021702.

Lappa, M. (2005). On the nature and structure of possible three-dimensional steady flows in closed and open parallelepipedic and cubical containers under different heating conditions and driving forces. Fluid Dynamics and Materials Processing, 1(1), 1–19.

Lappa, M. (2007). Secondary and oscillatory gravitational instabilities in canonical three-dimensional models of crystal growth from the melt. Part 1: Rayleigh–Bénard systems. Comptes Rendus Mécanique, 335(5-6), 253–260.

Rotunno, R. (2013). The fluid dynamics of tornadoes. Annual Review of Fluid Mechanics, 45(1), 59–84.

Tian, Y. S., & Karayiannis, T. G. (2000a). Low turbulence natural convection in an air-filled square cavity: Part I: the thermal and fluid flow fields. International Journal of Heat and Mass Transfer, 43(6), 849–866.

Tian, Y. S., & Karayiannis, T. G. (2000). Low turbulence natural convection in an air-filled square cavity: Part II: the turbulence quantities. International Journal of Heat and Mass Transfer, 43(6), 867–884.

Yanagisawa, T., Yamagishi, Y., & Homano, Y. (2005). Rayleigh-Benard convection in spherical shell with infinite Prandtl number at high Rayleigh number. Journal of the Earth Simulator, 4, 11–17.