Polynomial cross-roots application for the exchange of radiant energy between two triangular geometries
Main Article Content
Abstract
The view factor between surfaces is a vital element in the radiative heat transfer. Currently, the technical literature does not have an analytical method that allows directly calculating the view factors for the exchange of radiant energy between triangular surfaces. Developing an analytical model for triangular geometries requires the addition of multiple integrals, due to the change in the integration contours of the surfaces, which makes it a complex task to obtain the solution of various configurations. In this work, the analytical development of an expression of the view factor for the exchange of radiative energy between 32 triangular geometric configurations with common edges and included angle theta is intended. To establish comparisons, 42 examples with different shape configurations were calculated for each geometry using the analytic solution (AS), the numerical solution of the quadruple integral using Simpson's multiple rule 1/3 with five intervals (SMR) and the view factor computed using Bretzhtsov's crossed roots (BCR). From eight basic geometries, the view factor for another 24 triangular geometries is obtained using the summation rule. In all the cases evaluated, the BCR showed the best fits. The practical nature of the contribution and the reasonable adjustment of the values obtained, establish the proposal as a suitable tool for its application in thermal engineering.
Keywords
triangular surfaces, Bretzhtsov cross-root, view factor superficies triangulares, raíz cruzada de Bretzhtsov, factor de visión
References
[1] J. R. Howell and M. P. Menguc, “Radiative transfer configuration factor catalog: A listing of relations for common geometries,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 112, no. 5, pp. 910–912, 2011. [Online]. Available: https://doi.org/10.1016/j.jqsrt.2010.10.002
[2] Y. F. Nassar, “Analytical-numerical computation of view factor for several arrangements of two rectangular surfaces with non-common edge,” International Journal of Heat and Mass Transfer, vol. 159, p. 120130, 2020. [Online]. Available: https://doi.org/10.1016/j.ijheatmasstransfer.2020.120130
[3] M. F. Modest and S. Mazumder, “Chapter 4 - view factors,” in Radiative Heat Transfer (Fourth Edition), fourth edition ed., M. F. Modest and S. Mazumder, Eds. Academic Press, 2022, pp. 127–159. [Online]. Available: https: //doi.org/10.1016/B978-0-12-818143-0.00012-2
[4] Y. Camaraza-Medina, A. Hernandez-Guerrero, and J. L. Luviano-Ortiz, “Analytical view factor solution for radiant heat transfer between two arbitrary rectangular surfaces,” Journal of Thermal Analysis and Calorimetry, vol. 147, pp. 14 999–15 016, 2022. [Online]. Available: https://doi.org/10.1007/s10973-022-11646-4
[5]Y. Camaraza Medina, Introducción a la termotransferencia. Editorial Universitaria (Cuba), 2020. [Online]. Available: https://bit.ly/44PcgsO
[6] J. R. Howell, A Catalog of Radiation Configuration Factors. McGraw-Hill, 1982. [Online]. Available: https://bit.ly/42qHzIJ
[7] J. R. Howell, M. Pinar Menguc, and R. Siegel, Thermal Radiation Heat Transfer. CRC Press, 2010. [Online]. Available: https://bit.ly/42u9jfy
[8] M. K. Gupta, K. J. Bumtariya, H. Shukla, P. Patel, and Z. Khan, “Methods for evaluation of radiation view factor: A review,” Materials Today: Proceedings, vol. 4, no. 2, Part A, pp. 1236–1243, 2017, 5th International Conference of Materials Processing and Characterization (ICMPC 2016). [Online]. Available:https://doi.org/10.1016/j.matpr.2017.01.143
[9] A. Narayanaswamy, “An analytic expression for Radiation view factor between two arbitrarily oriented planar polygons,” International Journal of Heat and Mass Transfer, vol. 91, pp. 841–847, 2015. [Online]. Available: https://doi.org/10.1016/ j.ijheatmasstransfer.2015.07.131
[10] A. Narayanaswamy and P. Meyappan, “An analytic expression for radiation view factors between two planar triangles with arbitrary orientations,” in Proceedings of CHT-15. 6th International Symposium on ADVANCES IN COMPUTATIONAL HEAT TRANSFER, 12 2015, pp. 1545–1550. [Online]. Available: https://doi.org/10.1615/ICHMT. 2015.IntSympAdvComputHeatTransf.1500
[11] R. Sudharshan Reddy, D. Arepally, and A. Datta, “View factor computation and radiation energy analysis in baking oven with obstructions: Analytical and numerical method,” Journal of Food Process Engineering, vol. 46, p. e14270, 01 2023. [Online]. Available: http://dx.doi.org/10.1111/jfpe.14270
[12] Y. Zhou, R. Duan, X. Zhu, J. Wu, J. Ma, X. Li, and Q. Wang, “An improved model to calculate radiative heat transfer in hot combustion gases,” Combustion Theory and Modelling, vol. 24, no. 5, pp. 829–851, 2020. [Online]. Available: https: //doi.org/10.1080/13647830.2020.1769866 https: //doi.org/10.1080/13647830.2020.1769866
[13] X.-J. Yi, L.-Y. Zhong, T.-B. Wang, W.-X. Liu, D.-J. Zhang, T.-B. Yu, Q.-H. Liao, and N.-H. Liu, “Near-field radiative heat transfer between hyperbolic metasurfaces based on black phosphorus,” The European Physical Journal B, vol. 92, no. 9, p. 217, Sep 2019. [Online]. Available: https://doi.org/10.1140/epjb/e2019-100274-y
[14] J. R. Ehlert and T. F. Smith, “View factors for perpendicular and parallel rectangular plates,” Journal of Thermophysics and Heat Transfer, vol. 7, no. 1, pp. 173–175, 1993. [Online]. Available: https://doi.org/10.2514/3.11587
[15] C. K. Krishnaprakas, “View factor between inclined rectangles,” Journal of Thermophysics and Heat Transfer, vol. 11, no. 3, pp. 480–481, 1997. [Online]. Available: https://doi.org/10.2514/2.6267
[16] H. J. Sauer Jr., “Configuration factors for radiant energy interchange with triangular areas,” ASHRAE Transactions, vol. 80, no. 2, pp. 268–279, 1974. [Online]. Available: https://bit.ly/3I0jXm0
[17] Y. Camaraza-Medina, A. A. Sánchez Escalona, O. Miguel Cruz-Fonticiella, and O. F. García- Morales, “Method for heat transfer calculation on fluid flow in single-phase inside rough pipes,” Thermal Science and Engineering Progress, vol. 14, p. 100436, 2019. [Online]. Available: https://doi.org/10.1016/j.tsep.2019.100436
[18] W. Boeke and L. Wall, “Radiative exchange factors in rectangular spaces for the determination of mean radiant temperatures,” Building services engineering, vol. 43, pp. 244–253, 1976.
[19] F. F. Sönmez, H. Ziar, O. Isabella, and M. Zeman, “Fast and accurate ray-casting-based view factor estimation method for complex geometries,” Solar Energy Materials and Solar Cells, vol. 200, p. 109934, 2019. [Online]. Available: https://doi.org/10.1016/j.solmat.2019.109934
[20] S. Francisco, A. Raimundo, A. Gaspar, A. V. Oliveira, and D. Quintela, “Calculation of view factors for complex geometries using stokes’ theorem,” Journal of Building Performance Simulation, vol. 7, pp. 203–216, 05 2014. [Online]. Available: http: //dx.doi.org/10.1080/19401493.2013.808266
[21] S.-A. Biehs, R. Messina, P. S. Venkataram, A. W. Rodriguez, J. C. Cuevas, and P. Ben- Abdallah, “Near-field radiative heat transfer in many-body systems,” Rev. Mod. Phys., vol. 93, p. 025009, Jun 2021. [Online]. Available: https: //doi.org/10.1103/RevModPhys.93.025009
[22] Y. Camaraza-Medina, A. Hernandez-Guerrero, and J. L. Luviano-Ortiz, “Experimental study on influence of the temperature and composition in the steels thermo physical properties for heat transfer applications,” Journal of Thermal Analysis and Calorimetry, vol. 147, no. 21, pp. 11 805–11 821, Nov 2022. [Online]. Available: https://doi.org/10.1007/s10973-022-11410-8
[23] M. Lakhi and A. Safavinejad, “Numerical investigation of combined force convective–radiative heat transfer in a horizontal channel with lattice boltzmann method,” Journal of Thermal Analysis and Calorimetry, vol. 146, no. 4, pp. 1911–1922, Nov 2021. [Online]. Available: https://doi.org/10.1007/s10973-020-10136-9
[24] M. Bonnici, P. Mollicone, M. Fenech, and M. A. Azzopardi, “Analytical and numerical models for thermal related design of a new pico-satellite,” Applied Thermal Engineering, vol. 159, p. 113908, 2019. [Online]. Available: https: //doi.org/10.1016/j.applthermaleng.2019.113908
[25] Y. Camaraza-Medina, “Methods for the determination of the heat transfer coefficient in air cooled condenser used at biomass power plants,” International Journal of Heat and Technology, vol. 39, no. 5, pp. 1443–1450, 2021. [Online]. Available: https://doi.org/10.18280/ijht.390505
[2] Y. F. Nassar, “Analytical-numerical computation of view factor for several arrangements of two rectangular surfaces with non-common edge,” International Journal of Heat and Mass Transfer, vol. 159, p. 120130, 2020. [Online]. Available: https://doi.org/10.1016/j.ijheatmasstransfer.2020.120130
[3] M. F. Modest and S. Mazumder, “Chapter 4 - view factors,” in Radiative Heat Transfer (Fourth Edition), fourth edition ed., M. F. Modest and S. Mazumder, Eds. Academic Press, 2022, pp. 127–159. [Online]. Available: https: //doi.org/10.1016/B978-0-12-818143-0.00012-2
[4] Y. Camaraza-Medina, A. Hernandez-Guerrero, and J. L. Luviano-Ortiz, “Analytical view factor solution for radiant heat transfer between two arbitrary rectangular surfaces,” Journal of Thermal Analysis and Calorimetry, vol. 147, pp. 14 999–15 016, 2022. [Online]. Available: https://doi.org/10.1007/s10973-022-11646-4
[5]Y. Camaraza Medina, Introducción a la termotransferencia. Editorial Universitaria (Cuba), 2020. [Online]. Available: https://bit.ly/44PcgsO
[6] J. R. Howell, A Catalog of Radiation Configuration Factors. McGraw-Hill, 1982. [Online]. Available: https://bit.ly/42qHzIJ
[7] J. R. Howell, M. Pinar Menguc, and R. Siegel, Thermal Radiation Heat Transfer. CRC Press, 2010. [Online]. Available: https://bit.ly/42u9jfy
[8] M. K. Gupta, K. J. Bumtariya, H. Shukla, P. Patel, and Z. Khan, “Methods for evaluation of radiation view factor: A review,” Materials Today: Proceedings, vol. 4, no. 2, Part A, pp. 1236–1243, 2017, 5th International Conference of Materials Processing and Characterization (ICMPC 2016). [Online]. Available:https://doi.org/10.1016/j.matpr.2017.01.143
[9] A. Narayanaswamy, “An analytic expression for Radiation view factor between two arbitrarily oriented planar polygons,” International Journal of Heat and Mass Transfer, vol. 91, pp. 841–847, 2015. [Online]. Available: https://doi.org/10.1016/ j.ijheatmasstransfer.2015.07.131
[10] A. Narayanaswamy and P. Meyappan, “An analytic expression for radiation view factors between two planar triangles with arbitrary orientations,” in Proceedings of CHT-15. 6th International Symposium on ADVANCES IN COMPUTATIONAL HEAT TRANSFER, 12 2015, pp. 1545–1550. [Online]. Available: https://doi.org/10.1615/ICHMT. 2015.IntSympAdvComputHeatTransf.1500
[11] R. Sudharshan Reddy, D. Arepally, and A. Datta, “View factor computation and radiation energy analysis in baking oven with obstructions: Analytical and numerical method,” Journal of Food Process Engineering, vol. 46, p. e14270, 01 2023. [Online]. Available: http://dx.doi.org/10.1111/jfpe.14270
[12] Y. Zhou, R. Duan, X. Zhu, J. Wu, J. Ma, X. Li, and Q. Wang, “An improved model to calculate radiative heat transfer in hot combustion gases,” Combustion Theory and Modelling, vol. 24, no. 5, pp. 829–851, 2020. [Online]. Available: https: //doi.org/10.1080/13647830.2020.1769866 https: //doi.org/10.1080/13647830.2020.1769866
[13] X.-J. Yi, L.-Y. Zhong, T.-B. Wang, W.-X. Liu, D.-J. Zhang, T.-B. Yu, Q.-H. Liao, and N.-H. Liu, “Near-field radiative heat transfer between hyperbolic metasurfaces based on black phosphorus,” The European Physical Journal B, vol. 92, no. 9, p. 217, Sep 2019. [Online]. Available: https://doi.org/10.1140/epjb/e2019-100274-y
[14] J. R. Ehlert and T. F. Smith, “View factors for perpendicular and parallel rectangular plates,” Journal of Thermophysics and Heat Transfer, vol. 7, no. 1, pp. 173–175, 1993. [Online]. Available: https://doi.org/10.2514/3.11587
[15] C. K. Krishnaprakas, “View factor between inclined rectangles,” Journal of Thermophysics and Heat Transfer, vol. 11, no. 3, pp. 480–481, 1997. [Online]. Available: https://doi.org/10.2514/2.6267
[16] H. J. Sauer Jr., “Configuration factors for radiant energy interchange with triangular areas,” ASHRAE Transactions, vol. 80, no. 2, pp. 268–279, 1974. [Online]. Available: https://bit.ly/3I0jXm0
[17] Y. Camaraza-Medina, A. A. Sánchez Escalona, O. Miguel Cruz-Fonticiella, and O. F. García- Morales, “Method for heat transfer calculation on fluid flow in single-phase inside rough pipes,” Thermal Science and Engineering Progress, vol. 14, p. 100436, 2019. [Online]. Available: https://doi.org/10.1016/j.tsep.2019.100436
[18] W. Boeke and L. Wall, “Radiative exchange factors in rectangular spaces for the determination of mean radiant temperatures,” Building services engineering, vol. 43, pp. 244–253, 1976.
[19] F. F. Sönmez, H. Ziar, O. Isabella, and M. Zeman, “Fast and accurate ray-casting-based view factor estimation method for complex geometries,” Solar Energy Materials and Solar Cells, vol. 200, p. 109934, 2019. [Online]. Available: https://doi.org/10.1016/j.solmat.2019.109934
[20] S. Francisco, A. Raimundo, A. Gaspar, A. V. Oliveira, and D. Quintela, “Calculation of view factors for complex geometries using stokes’ theorem,” Journal of Building Performance Simulation, vol. 7, pp. 203–216, 05 2014. [Online]. Available: http: //dx.doi.org/10.1080/19401493.2013.808266
[21] S.-A. Biehs, R. Messina, P. S. Venkataram, A. W. Rodriguez, J. C. Cuevas, and P. Ben- Abdallah, “Near-field radiative heat transfer in many-body systems,” Rev. Mod. Phys., vol. 93, p. 025009, Jun 2021. [Online]. Available: https: //doi.org/10.1103/RevModPhys.93.025009
[22] Y. Camaraza-Medina, A. Hernandez-Guerrero, and J. L. Luviano-Ortiz, “Experimental study on influence of the temperature and composition in the steels thermo physical properties for heat transfer applications,” Journal of Thermal Analysis and Calorimetry, vol. 147, no. 21, pp. 11 805–11 821, Nov 2022. [Online]. Available: https://doi.org/10.1007/s10973-022-11410-8
[23] M. Lakhi and A. Safavinejad, “Numerical investigation of combined force convective–radiative heat transfer in a horizontal channel with lattice boltzmann method,” Journal of Thermal Analysis and Calorimetry, vol. 146, no. 4, pp. 1911–1922, Nov 2021. [Online]. Available: https://doi.org/10.1007/s10973-020-10136-9
[24] M. Bonnici, P. Mollicone, M. Fenech, and M. A. Azzopardi, “Analytical and numerical models for thermal related design of a new pico-satellite,” Applied Thermal Engineering, vol. 159, p. 113908, 2019. [Online]. Available: https: //doi.org/10.1016/j.applthermaleng.2019.113908
[25] Y. Camaraza-Medina, “Methods for the determination of the heat transfer coefficient in air cooled condenser used at biomass power plants,” International Journal of Heat and Technology, vol. 39, no. 5, pp. 1443–1450, 2021. [Online]. Available: https://doi.org/10.18280/ijht.390505
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Published
Jul 6, 2023
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Section
Mechanical Engineering - Energies
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