Computational Aspects of Radiative Transfer Equation in Non-Orthogonal Coordinates

Saad Bin Mansoor

doi:10.18186/thermal.654191

Article Open Access1 January 2019

Computational Aspects of Radiative Transfer Equation in Non-Orthogonal Coordinates

Order Reprints Cite Share

Saad MANSOOR ¹

¹King Fahd University of Petroleum and Minerals

Journal of Thermal Engineering 2019, Vol. 5, Issue 6, pp. 162-170; doi.org/10.18186/thermal.654191

Download PDF View DOI record

Abstract

Non-equilibrium energy transfer takes place for thin films when thermal disturbance is introduced. In this case, phonon transport inside the film governs the heat transport and temperature distribution in the film. In the present study an attempt is made to formulate and illustrate the phonon transfer in micro-scale silicon film of various shapes incorporating the non-orthogonal coordinate system. Successful application of the discreteordinates method to the solution of the equation for phonon radiative transport in non-orthogonal coordinates requires the application of various numerical techniques connected to the finite-difference method. The numerical solution of the equation for phonon transfer in non-orthogonal coordinate is introduced via adapting the discrete ordinate method. Phonon intensity distribution in the thin film is presented in terms of equivalent equilibrium temperature. It is found that film shape has significant effect on equivalent equilibrium temperature distribution inside the film. The validation study demonstrates that the code developed solving the equation for phonon transport is also applicable to the phonon transport in non-orthogonal coordinate system.

Keywords: Phonon Transport; Thin Film; Non-Orthogonal Coordinate; Equivalent Equilibrium Temperatur

References

Yilbas, B.S. (1988). The validity of Fourier theory of radiation heating of metals. Res. Mechanica, 24, 377- 82.
Majumdar, A. (1993). Microscale Heat Conduction in Dielectric Thin Films. J. Heat Transfer, 115, 7-16.
Yilbas, B.S., Dweik, A.Y., Mansoor, S.B. (2014). Non-equilibrium energy transport in a thin metallic film: analytical solution for radiative transport equation. Physica B, 454, 15-22.
Mansoor, S.B., Yilbas, B.S. (2015). Non-equilibrium cross-plane energy transport in aluminum-silicon- aluminum wafer. Modern Physics B, 29(17), 1550112-1 - 1550112-21. 169 Journal of Thermal Engineering, Research Article, Vol. 5, No. 6, Special Issue 10, pp. 162-170, December, 2019
Ali, H., Yilbas, B.S. (2016). Phonon cross-plane transport and thermal boundary resistance: effect of heat source size on phonon characteristics. Continuum Mechanics and Thermodynamics, 28(5), 1373-1393.
Mansoor, S.B. ,Yilbas, B.S. (2015). Thermal transport across a thin film composite due to laser short-pulse heating. J. of Non-Equilibrium Thermodynamics, 40(2), 103-120.
Mansoor, S.B., Yilbas, B.S. (2015). Laser short-pulse heating of an aluminum thin film: energy transfer in electron and lattice sub-systems. Physica B, 470-471, 82-91.
Ali, H., Mansoor, S.B., Yilbas, B.S. (2015). Thermal characteristics of an aluminum thin film due to temperature disturbance at film edges. Int. J. of Thermophysics, 36, 157-182.
Yilbas, B.S., Bin Mansoor, S. (2014). Phonon transport in aluminum and silicon film pear: laser short-pulse irradiation at aluminum film surface. Canadian Journal of Physics, 92(12), 1614-1622.
Yilbas, B.S., Bin Mansoor, S. (2014). Size effect on phonon transport in two-dimensional silicon film. Optical and Quantum Electronics, 46(11), 1467-1479.
Bin Mansoor, S., Yilbas, B.S. (2013). Phonon transport in silicon thin film: effect of temperature oscillation on effective thermal conductivity. Transport Theory and Statistical Physics, 42(4-5), 179-201.
Vaillon, R., Lallemand, M., Lemonnier, D. (1996). Radiative heat transfer in orthogonal curvilinear coordinates using the discrete ordinates method. Journal of Quantitative Spectroscopy and Radiative Transfer, 55(1), 7-17.
Freimanis, J. (2011). On vector radiative transfer equation in curvilinear coordinate systems. Journal of Quantitative Spectroscopy & Radiative Transfer, 112, 2134–2148.
Mansoor, S.B., Yilbas, B.S. (2016). Phonon transport across nano-scale curved thin films. Physica B, 503, 130-140.
Mansoor, S.B., Yilbas, B.S. (2017). Phonon Transport in Curved Thin Film: Effect of Film Curvature and Radius on Transport Characteristics. Journal of Computational and Theoretical Transport, 46(4), 283-306.
Yilbas, S.B., Mansoor, S.B., Ali, H. Heat Transport in Micro- and Nanoscale Thin Films, Elsevier, 2018.
Heinbockel, J.H. Introduction to Tensor Analysis and Continuum Mechanics, Trafford Publishing: Canada, 2001. 170

Share and Cite

Mansoor, S.B. Computational Aspects of Radiative Transfer Equation in Non-Orthogonal Coordinates. Journal of Thermal Engineering 2019, Vol. 5, pp. 162-170. https://doi.org/10.18186/thermal.654191

Export:

Computational Aspects of Radiative Transfer Equation in Non-Orthogonal Coordinates

¹King Fahd University of Petroleum and Minerals

Abstract

References

Yilbas, B.S. (1988). The validity of Fourier theory of radiation heating of metals. Res. Mechanica, 24, 377- 82.
Majumdar, A. (1993). Microscale Heat Conduction in Dielectric Thin Films. J. Heat Transfer, 115, 7-16.
Yilbas, B.S., Dweik, A.Y., Mansoor, S.B. (2014). Non-equilibrium energy transport in a thin metallic film: analytical solution for radiative transport equation. Physica B, 454, 15-22.
Mansoor, S.B., Yilbas, B.S. (2015). Non-equilibrium cross-plane energy transport in aluminum-silicon- aluminum wafer. Modern Physics B, 29(17), 1550112-1 - 1550112-21. 169 Journal of Thermal Engineering, Research Article, Vol. 5, No. 6, Special Issue 10, pp. 162-170, December, 2019
Ali, H., Yilbas, B.S. (2016). Phonon cross-plane transport and thermal boundary resistance: effect of heat source size on phonon characteristics. Continuum Mechanics and Thermodynamics, 28(5), 1373-1393.
Mansoor, S.B. ,Yilbas, B.S. (2015). Thermal transport across a thin film composite due to laser short-pulse heating. J. of Non-Equilibrium Thermodynamics, 40(2), 103-120.
Mansoor, S.B., Yilbas, B.S. (2015). Laser short-pulse heating of an aluminum thin film: energy transfer in electron and lattice sub-systems. Physica B, 470-471, 82-91.
Ali, H., Mansoor, S.B., Yilbas, B.S. (2015). Thermal characteristics of an aluminum thin film due to temperature disturbance at film edges. Int. J. of Thermophysics, 36, 157-182.
Yilbas, B.S., Bin Mansoor, S. (2014). Phonon transport in aluminum and silicon film pear: laser short-pulse irradiation at aluminum film surface. Canadian Journal of Physics, 92(12), 1614-1622.
Yilbas, B.S., Bin Mansoor, S. (2014). Size effect on phonon transport in two-dimensional silicon film. Optical and Quantum Electronics, 46(11), 1467-1479.
Bin Mansoor, S., Yilbas, B.S. (2013). Phonon transport in silicon thin film: effect of temperature oscillation on effective thermal conductivity. Transport Theory and Statistical Physics, 42(4-5), 179-201.
Vaillon, R., Lallemand, M., Lemonnier, D. (1996). Radiative heat transfer in orthogonal curvilinear coordinates using the discrete ordinates method. Journal of Quantitative Spectroscopy and Radiative Transfer, 55(1), 7-17.
Freimanis, J. (2011). On vector radiative transfer equation in curvilinear coordinate systems. Journal of Quantitative Spectroscopy & Radiative Transfer, 112, 2134–2148.
Mansoor, S.B., Yilbas, B.S. (2016). Phonon transport across nano-scale curved thin films. Physica B, 503, 130-140.
Mansoor, S.B., Yilbas, B.S. (2017). Phonon Transport in Curved Thin Film: Effect of Film Curvature and Radius on Transport Characteristics. Journal of Computational and Theoretical Transport, 46(4), 283-306.
Yilbas, S.B., Mansoor, S.B., Ali, H. Heat Transport in Micro- and Nanoscale Thin Films, Elsevier, 2018.
Heinbockel, J.H. Introduction to Tensor Analysis and Continuum Mechanics, Trafford Publishing: Canada, 2001. 170

Share and Cite

Export:

[1] Yilbas, B.S. (1988). The validity of Fourier theory of radiation heating of metals. Res. Mechanica, 24, 377- 82.

[2] Majumdar, A. (1993). Microscale Heat Conduction in Dielectric Thin Films. J. Heat Transfer, 115, 7-16.

[3] Yilbas, B.S., Dweik, A.Y., Mansoor, S.B. (2014). Non-equilibrium energy transport in a thin metallic film: analytical solution for radiative transport equation. Physica B, 454, 15-22.

[4] Mansoor, S.B., Yilbas, B.S. (2015). Non-equilibrium cross-plane energy transport in aluminum-silicon- aluminum wafer. Modern Physics B, 29(17), 1550112-1 - 1550112-21. 169 Journal of Thermal Engineering, Research Article, Vol. 5, No. 6, Special Issue 10, pp. 162-170, December, 2019

[5] Ali, H., Yilbas, B.S. (2016). Phonon cross-plane transport and thermal boundary resistance: effect of heat source size on phonon characteristics. Continuum Mechanics and Thermodynamics, 28(5), 1373-1393.

[6] Mansoor, S.B. ,Yilbas, B.S. (2015). Thermal transport across a thin film composite due to laser short-pulse heating. J. of Non-Equilibrium Thermodynamics, 40(2), 103-120.

[7] Mansoor, S.B., Yilbas, B.S. (2015). Laser short-pulse heating of an aluminum thin film: energy transfer in electron and lattice sub-systems. Physica B, 470-471, 82-91.

[8] Ali, H., Mansoor, S.B., Yilbas, B.S. (2015). Thermal characteristics of an aluminum thin film due to temperature disturbance at film edges. Int. J. of Thermophysics, 36, 157-182.

[9] Yilbas, B.S., Bin Mansoor, S. (2014). Phonon transport in aluminum and silicon film pear: laser short-pulse irradiation at aluminum film surface. Canadian Journal of Physics, 92(12), 1614-1622.

[10] Yilbas, B.S., Bin Mansoor, S. (2014). Size effect on phonon transport in two-dimensional silicon film. Optical and Quantum Electronics, 46(11), 1467-1479.

[11] Bin Mansoor, S., Yilbas, B.S. (2013). Phonon transport in silicon thin film: effect of temperature oscillation on effective thermal conductivity. Transport Theory and Statistical Physics, 42(4-5), 179-201.

[12] Vaillon, R., Lallemand, M., Lemonnier, D. (1996). Radiative heat transfer in orthogonal curvilinear coordinates using the discrete ordinates method. Journal of Quantitative Spectroscopy and Radiative Transfer, 55(1), 7-17.

[13] Freimanis, J. (2011). On vector radiative transfer equation in curvilinear coordinate systems. Journal of Quantitative Spectroscopy & Radiative Transfer, 112, 2134–2148.

[14] Mansoor, S.B., Yilbas, B.S. (2016). Phonon transport across nano-scale curved thin films. Physica B, 503, 130-140.

[15] Mansoor, S.B., Yilbas, B.S. (2017). Phonon Transport in Curved Thin Film: Effect of Film Curvature and Radius on Transport Characteristics. Journal of Computational and Theoretical Transport, 46(4), 283-306.

[16] Yilbas, S.B., Mansoor, S.B., Ali, H. Heat Transport in Micro- and Nanoscale Thin Films, Elsevier, 2018.

[17] Heinbockel, J.H. Introduction to Tensor Analysis and Continuum Mechanics, Trafford Publishing: Canada, 2001. 170

Published	1 January 2019
Version	v1
Access	Open Access

Published	1 January 2019
Version	v1
Access	Open Access

Computational Aspects of Radiative Transfer Equation in Non-Orthogonal Coordinates

Abstract

References

Share and Cite

Related Articles

Computational Aspects of Radiative Transfer Equation in Non-Orthogonal Coordinates

Abstract

References

Share and Cite

Related Articles