×

A novel alternate approach for multiscale thermal transport using diffusion in the Boltzmann transport equation. (English) Zbl 1218.80008

Summary: The Boltzmann Transport Equation (BTE) with the single Relaxation Time Approximation (RTA) can be employed to simulate energy transport in the sub-continuum regime, as long as the particle assumption for the heat carriers is valid. Here, analytical and numerical solutions of the BTE in one space dimension are obtained for two different sets of boundary conditions. For the steady case, numerical solutions obtained are found to be identical (to within purely numerical error) to the analytical solutions. Transient solutions are obtained using the Lattice Boltzmann Method (LBM). Additionally, the impact of introducing a diffusive term in the BTE is investigated. When the diffusive model is solved, an increasing gradient (but no jump) in the temperature at the boundaries is observed as the Knudsen number \((Kn)\) increases. The results for the transient case are compared to those obtained from the Fourier solution for small values of \(Kn\). It is observed that the BTE with an added diffusion term is applicable for all \(Kn\).

MSC:

80A20 Heat and mass transfer, heat flow (MSC2010)
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C70 Transport processes in time-dependent statistical mechanics
76R50 Diffusion
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cahill, D.; Ford, W.; Goodson, K. E.; Mahan, G.; Majumdar, A.; Maris, H.; Merlin, R.; Phillpot, S.: Nanoscale thermal transport, J. appl. Phys. 93, No. 2, 793-818 (2003)
[2] Sverdrup, P. G.; Ju, Y. S.; Goodson, K. E.: Sub-continuum simulations of heat conduction in silicon-on-insulator transistors, J. heat transfer 123, 130-137 (2001)
[3] Narumanchi, S.; Murthy, J. Y.; Amon, C. H.: Simulation of unsteady small heat source effects in sub-micron heat conduction, J. heat transfer 125, No. 5, 896-903 (2003)
[4] Gladrow, W.; Dieter, A.: Lattice-gas cellular automata and lattice Boltzmann models: an introduction, (2000) · Zbl 0999.82054
[5] Succi, S.: Lattice Boltzmann equation for fluid dynamics and beyond, (2001) · Zbl 0990.76001
[6] W. Zhang, T.S. Fisher, Application of the lattice Boltzmann method to sub-continuum heat conduction, IMECE2002-32122, 2002.
[7] S.S. Ghai, R.A. Escobar, C.H. Amon, M.S. Jhon, Sub-continuum heat conduction in electronics using the lattice Boltzmann method, InterPack 2002-35258, 2003.
[8] Escobar, R. A.; Ghai, S. S.; Jhon, M. S.; Amon, C. H.: Multi-length and time scale thermal transport using the lattice Boltzmann method with application to electronics cooling, Int. J. Heat mass transfer 49, 97-107 (2006) · Zbl 1189.80049 · doi:10.1016/j.ijheatmasstransfer.2005.08.003
[9] Christensen, A.; Graham, S.: Multiscale lattice Boltzmann modeling of phonon transport in crystalline semiconductor materials, Numer. heat transfer B 5, 89-109 (2010)
[10] Lai, J.; Majumdar, A.: Concurrent thermal and electrical modeling of sub-micrometer silicon devices, J. appl. Phys. 79, 7353-7361 (1996)
[11] Kittel, C.: Introduction to solid state physics, (1986) · Zbl 0052.45506
[12] Tien, C. L.; Majumdar, A.; Gerner, F. M.: Microscale heat conduction, (1998)
[13] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Harcourt, Fort Worth, TX, 1976.
[14] Narumanchi, S. V.; Murthy, J. Y.; Amon, C. H.: Comparison of different phonon transport models in predicting heat conduction in sub-micron silicon-on-insulator transistors, J. heat transfer 127, No. 7, 713-723 (2005)
[15] Narumanchi, S. V.; Murthy, J. Y.; Amon, C. H.: Sub-micron heat transport model in silicon accounting for phonon dispersion and polarization, J. heat transfer 126, No. 6, 946-955 (2004)
[16] R.A. Escobar, S.S. Ghai, M.S. Jhon, C.H. Amon, Time-dependent simulations of sub-continuum heat generation effects in electronic devices using the lattice Boltzmann method, IMECE2003-41522, 2003.
[17] Ghai, S. S.; Kim, W. T.; Escobar, R. A.; Amon, C. H.; Jhon, M. S.: A novel heat transfer model and its application to information storage systems, J. appl. Phys. 97, 10P703 (2005)
[18] Alvarez, F. X.; Jou, D.: Size and frequency dependence of effective thermal conductivity of nanosystems, J. appl. Phys. 103 (2008)
[19] Alvarez, F. X.; Jou, D.: Boundary conditions and evolution of ballistic heat transport, ASME J. Heat transport 132 (2010)
[20] Kennard, E. H.: Kinetic theory of gases, (1938)
[21] Gomes, C. J.; Madrid, M.; Goicochea, J. V.; Amon, C. H.: In-plane and out-of-plane thermal conductivity of silicon thin films predicted by molecular dynamics, ASME J. Heat transfer 128, No. 11, 1114-1121 (2006)
[22] Liu, W.; Asheghi, M.: Phonon-boundary scattering in ultra thin single-crystal silicon layers, Appl. phys. Lett. 84, No. 19, 3819-3821 (2004)
[23] W. Liu, M. Asheghi, Thermal conductivity of ultra thin single crystal silicon layers, Part 1 — Experimental measurements at room and cryogenic temperatures, in: Proceedings of ASME International Mechanical Engineering Congress and Exposition, Anaheim, CA, IMECE2004-62105, 2004.
[24] Rohsenow, W. M.; Hartnett, J. P.; Cho, Y. I.: Handbook of heat transfer, (1998)
[25] Tien, C. L.; Majumdar, A.; Gerner, F. M.: Microscale energy transport, (1998)
[26] Nayfeh, A. H.: Pertubation methods, (1973) · Zbl 0265.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.