Conjugate natural convection in an enclosure with a heat source of constant heat transfer rate Geniy V. Kuznetsov, Mikhail A. Sheremet ⇑ Faculty of Thermal Power Engineering, Tomsk Polytechnic University, 30, Lenin Avenue, 634050 Tomsk, Russia Faculty of Mechanics and Mathematics, Tomsk State University, 36, Lenin Avenue, 634050 Tomsk, Russia article info Article history: Received 12 November 2009 Accepted 21 September 2010 Available online 19 October 2010 Keywords: Conjugate heat transfer Transient natural convection Local heat source of constant heat transfer rate Rectangular enclosure Numerical simulation abstract A numerical study of two-dimensional transient natural convection in a rectangular enclosure having finite thickness heat-conducting walls with a heat source of constant heat transfer rate located on the inner side of the left wall in conditions of convection–radiation heat exchange with an environment on one of the external boundaries has been performed. Mathematical simulation has been carried out in terms of the dimensionless variables such as stream function – vorticity – temperature. Stream func- tion, vorticity and energy equations have been solved by finite difference numerical method. The relevant governing parameters were: the Grashof number from 10 6 to 10 8 , the Prandtl number, Pr = 0.7 and the conductivity ratio. Detailed results including streamlines and temperature profiles have been obtained. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Conjugate natural convection is a subject of intensive research due to its technological applications ranging from nuclear reactors, thermal storage systems, cooling of electronic components [1–10]. Sharma et al. [2] present the results of a numerical investigation of transient turbulent natural convection heat transfer from a volu- metric energy generating source placed inside a cylindrical enclo- sure filled with low Prandtl number fluid. Unsteady temperature fields of the typical block for liquid metal fast breeder reactor (LMFBR) have been received, allowing to estimate effect of the time factor at heat mode formation. Bilgen [3] presents a numerical and experimental study of conjugate heat transfer by conduction and natural convection on a heated vertical wall. It is found that Nus- selt number is a strong function of Rayleigh number and wall thickness, and a weak function of conductivity ratio. Moiseeva and Cherkasov [4,5] present the results of a numerical simulation of conjugate natural convection in a vertical cylindrical tank par- tially filled by a fluid with heat conducting walls. It is shown that the Grashof number determining essential convection effect de- pends on some similarity parameters such as the dimensionless tank height, the distance between heat sinks, and the heat conduc- tivity ratio. It is determined, that convection effect is augmented, first, with increase in the dimensionless distance between heat sinks and, secondly, with decrease in either the heat conductivity of the wall or the wall thickness. Muftuoglu and Bilgen [6] deter- mined the optimum position of discrete heat source at the vertical wall with finite thickness in an open cavity. It is found that the optimum position of a discrete heater in the cavity is usually its off center. Jaluria [7] has analyzed problems of designing and opti- mization of thermal systems by means of numerical methods of mathematical physics. Modern lines of development of the microelectronics, linked to reduction of the overall dimensions, lead to increase in density of specific heat fluxes at unit of a surface area. The latter is negatively reflected in the system performance [9,10]. Development and opti- mization of present engineering solutions concerning creation of models and methods of effective cooling of the electronic equip- ment completely depends on the correct multiparameter analysis of the thermal modes in typical units and blocks of electronics. The objective of the present study is mathematical simulation of conjugate natural convection in a rectangular enclosure with a heat source of constant heat transfer rate on the assumption of radiative–convective heat exchange with an environment. The present research is development of the solution method of conju- gate natural convection problems in enclosures having finite thick- ness heat conducting walls at presence of the local heat source of constant temperature [11,12]. Analysis of a heat source of constant heat transfer rate allows modeling typical units and blocks of elec- tronics. It should be noted that the dissipation power of the elec- tronic components is the initial information at analysis of thermal modes of the electronic equipment. At the same time the temperature of the electronic elements surface is the important 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.09.046 ⇑ Corresponding author. Tel.: +7 3822 412 462; fax: +7 3822 529 740. E-mail address: Michael-sher@yandex.ru (M.A. Sheremet). International Journal of Heat and Mass Transfer 54 (2011) 260–268 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt