# MATHEMATICAL MODELLING OF THE CONJUGATE NATURAL CONVECTION IN A CLOSED SYSTEM WITH THE RADIANT HEATING SOURCE UNDER CONDITIONS OF RADIANT ENERGY DISTRIBUTION BY LAMBERT’S COSINE LAW

## Main Article Content

## Abstract

Various types of emitters are often used as energy sources in real engineering systems and technological processes. Investigations of heat transfer basic laws in such systems are of interest. We conducted mathematical modelling of conjugate heat transfer in a closed rectangular cavity under conditions of radiant energy source operating. Two – dimensional problem of conjugate natural convection in vorticity-stream function-temperature dimensionless variables has been numerically solved by means of the finite difference method. Radiant energy distribution along the gas-wall interfaces was set by Lamberts’ cosine law. We obtained fields of temperature and stream functions in a wide range of governing parameters (Rayleigh number 10^{4} ≤ Ra ≤ 10^{6} , the length of radiant heating source 0.15 ≤ D ≤ 0.6 ). Then we analyzed how heat-retaining properties of finite thickness heat-conducting walls made of different materials affect the heat transfer intensity. Differential characteristics distribution showed significant nonuniformity and nonstationarity of the conjugate heat transfer process under study.

## Article Details

**Thermal Science**, [S.l.], mar. 2017. ISSN 2334-7163. Available at: <http://thermal-science.tech/journal/index.php/thsci/article/view/2266>. Date accessed: 24 feb. 2018.

Authors retain copyright of the published article and have the right to use the article in the ways permitted to third parties under the - Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) licence. Full bibliographic information (authors, article title, journal title, volume, issue, pages) about the original publication must be provided and a link must be made to the article's DOI.

The authors and third parties who wish use the article in a way not covered by the the -Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) licence must obtain a written consent of the publisher. This license allows others to download the paper and share it with others as long as they credit the journal, but they cannot change it in any way or use it commercially.

Authors grant to the publisher the right to publish the article, to be cited as its original publisher in case of reuse, and to distribute it in all forms and media.

Accepted 2017-03-13

Published 2017-03-13

## References

[2] Zhu, K., et. al., Experimental study on the optimization of heat and mass transfer of industrial drying of the TiO2 bulb by infrared radiation, Journal of Thermal Science, 5 (1996), pp 278-284

[3] El-Mesery, H.S., Mwithiga, G., Performance of a convective, infrared and combined infrared- convective heated conveyor-belt dryer, Journal of Food Science and Technology, 52 (2015), pp 2721-2730

[4] Sudarushkin, Yu. K., et. al., Convective-Infrared Drying of a Polyamide-based Composite in a Fluidized Bed, Russian Journal of Applied Chemistry, 78 (2005), pp 1977-1980

[5] Allanic, N., et. al., Convective and radiant drying of a polymer aqueous solution, Heat and Mass Transfer, 43 (2007), pp 1087-1095

[6] Franz, F. P., et. al., Investigations on the Heating and drying of wood with infrared radiation, Wood Science and Technology, 1 (1967), pp 149-160

[7] Seyam, S., et. al., Experimental and numerical investigation of the radiant panel heating system using scale room model, Energy and Buildings, 82 (2014), pp. 130-141

[8] Aich, W., et. al., Numerical analysis of natural convection in a prismatic enclosure, Thermal Science, 15 (2011), pp. 1 – 15

[9] Cho, C.- C., et. al., Numerical investigation into natural convection heat transfer enhancement of copper – water nanofluid in a wavy wall enclosure, Thermal Science, 16 (2012), pp. 1309 – 1316

[10] Jani, S., et. al., Numerical investigation of natural convection heat transfer in a symmetrically cooled square cavity with a thin fin on its bottom wall, Thermal Science, 18 (2014), pp. 1119 – 1132

[11] Ghachem, K., et. al., Numerical study of heat and mass transfer optimization in a 3D inclined solar distiller, Thermal Science, (2015), pp. 1 – 15

[12] Ahmanache, A., Zeraibi, N., Numerical study of natural melt convection in cylindrical cavity with hot walls and cold bottom sink, Thermal Science, 17 (2013), pp. 853 – 864

[13] Saravanan, S., Sivaraj, C., Combined natural convection and thermal radiation in a square cavity with a nonuniformly heated plate, Computers & Fluids, 117 (2015), pp. 125 – 138

[14] Montiel-Gonzales, M., et. al., Theoretical and experimental study of natural convection with surface thermal radiation in a side open cavity, Applied Thermal Engineering, 75 (2015), pp. 1176 – 1186

[15] Zhao F.-Y., et. al., Conjugate natural convection in enclosures with external and integral heat source, International Journal of Engineering Science, 44 (2006), pp. 148 – 165

[16] Saeid, N.H., Conjugate natural convection in a vertical porous layer sandwiched by finite thickness wall, International Communications in Heat and Mass Transfer, 34 (2007), pp. 210 – 216

[17] Kuznetsov, G. V., et. al., Heat transfer under heating of a local region of a large production area by gas infrared radiators, Journal of Engineering Physics and Thermophysics, 86 (2013), pp. 519- 524

[18] Kuznetsov G.V., Sheremet M.A., Conjugate natural convection in an enclosure with local heat sources, Computational Thermal Sciences, 1 (2009), pp. 341 – 360

[19] Hamimid, S., Guellal, M., Numerical analysis of combined natural convection – internal heat generation source – surface radiation, Thermal Science, (2014), pp. 1 – 14

[20] Kolsi, L., et. al., Combined radiation - natural convection in three-dimensional vertical cavities, Thermal Science, 15 (2011), pp. 327 – 339

[21] Montiel-Gonzales, M., et. al., Numerical study of heat transfer by natural convection and surface thermal radiation in an open cavity receiver, Solar Energy, 86 (2012), pp. 1118 – 1128

[22] Nounanegue, H.F., et. al., Heat transfer by natural convection, conduction and radiation in a inclined square enclosure bounded with a solid wall, International journal of Thermal Science, 48 (2009), pp. 871 – 880

[23] Saravanan, S., Sivaraj, C., Coupled thermal radiation and natural convection heat transfer in a cavity with a heated plate inside, International Journal of Heat and Fluid Flow, 40 (2013), pp. 54 – 64

[24] Kuznetsov G.V. et. al., Heat transfer under heating of a local region of a large production area by gas infrared radiators, Journal of Engineering Physics and Thermophysics, 86 (2013), pp. 519 – 524

[25] Nee A. E., Nagornova .T. A., Numerical investigation of conjugate heat transfer in a local working area in conditions of its radiant heating, IOP Conference Series: Materials Science and Engineering, 66 (2014), pp. 1-5

[26] Kuznetsov, G. V., et. al., Computational modeling of conjugate heat transfer in a closed rectangular domain under the conditions of radiant heat supply to the horizontal and vertical surfaces of enclosure structures, Journal of Engineering Physics and Thermophysics, 88 (2015), pp. 168-177

[27] Modest, M. F., Radiative Heat Transfer, Elsevier Science, New York, USA, 2003

[28] Roache, P. J., Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, NM, USA, 1976

[29] Samarskii, A.A., The Theory of Difference Schemes [in Russian], Nauka, Moscow, Russia, 1977

[30] De Vahl Davis, G., Natural convection of air in a square cavity: a benchmark numerical solution, International Journal of Numerical Methods in Fluids, 3 (1983), pp. 249 – 264

[31] Dixit H.N., Babu, V., Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method, International Journal of Heat and Mass Transfer, 49 (2006), pp. 727-739

[32] Markatos, N.C., Pericleous, K. A., Laminar and turbulent natural convection in an enclosed cavity, International Journal of Heat and Mass Transfer, 27 (1984), pp. 755-772