The problem of low Mach number (non-Boussin´esq) conjugate laminar natural convection combined with surface radiation in a vertical annulus with a centrally located vertical heat generating rod is studied numerically, taking into account the variable transport properties of the fluid. Such problems arise often in practical applications like spent nuclear fuel casks, cooling of electrical and electronic equipment, convection in ovens, cooling of enclosed vertical bus bars and underground transmission cables.
The physical model consists of a vertical heat generating rod, a concentric outer isothermal boundary and adiabatic top and bottom surfaces. The heat generation in the rod drives the natural convection in the annulus. Surface radiation is coupled to natural convection through the solid-fluid interface condition and the adiabatic condition of the top and bottom surfaces. A mathematical formulation is written using the governing equations expressing the conservation of mass, momentum and energy for the fluid as well as the energy balance for the solid heat generating rod. The governing equations are discretized on a staggered mesh and are solved using a pressure-correction algorithm. Steady-state solutions are obtained by time-marching of the time dependent equations. The discretized equations for the dependent variables are solved using the Modified Strongly Implicit Procedure. A global iteration is introduced on the variables at each time step for better coupling. The parameters of the problem are the heat generation and gap width based Grashof number, aspect ratio, radius ratio and the solid-to-fluid thermal conductivity ratio. The coupling of radiation introduces the wall emissivity and the radiation number as the additional parameters and also necessitates the calculation of radiation configuration factors between the elemental surfaces formed by the computational mesh. The radiant heat exchange is calculated using the radiosity matrix method.
A parametric study is performed by varying Grashof number from 106 to 1010 , aspect ratio from 1 to 15, radius ratio from 2 to 8, the solid-to-fluid thermal conductivity ratio from 1 to 100, with the Prandtl number 0.7 corresponding to air as the working medium. The characteristic dimension and the outer boundary temperature are fixed. For Radiative calculations, and the emissivity is varied between 0.25 and 0.75. Converged solutions with laminar model could be obtained for high Grashof numbers also as the heat generation based Grashof number is generally two orders of magnitude higher than the temperature difference based Grashof number. Results are presented for the flow and temperature distributions in the form of streamline and isotherm maps. Results are also presented for the variation of various quantities of interest such as the local Nusselt numbers on the inner and outer boundaries, the axial variation of the centerline and interface temperatures, maximum solid, average solid and average interface temperature variations with Grashof number and the average Nusselt number variation for the inner and outer boundaries with Grashof number. The results show that simplification of conjugate problems involving heat generation by the prescription of an isoflux boundary condition on the rod surface is inadequate because a truly isoflux condition cannot be realised on the one hand and because the solid temperature distribution remains unknown with such an approach. The average Nusselt numbers on the inner and outer boundaries show an increasing trend with the Grashof number. For pure natural convection, the Boussin´esq model predicts higher temperatures in the solid and lower average Nusselt numbers on the inner and outer boundaries, compared to the non-Boussin´esq model and the Boussin´esq approximation appears to be adequate roughly upto a Grashof number of 109, beyond which the non-Boussin´esq model is to be invoked. The average pressure in the annulus is found to increase with an increase in the Grashof number. Radiation is found to cause convective drop and homogenize the temperature distribution in the fluid.
| Identifer | oai:union.ndltd.org:IISc/oai:etd.ncsi.iisc.ernet.in:2005/780 |
| Date | 06 1900 |
| Creators | Reddy, P Venkata |
| Contributors | Rao, S V Raghurama |
| Source Sets | India Institute of Science |
| Language | en_US |
| Detected Language | English |
| Type | Thesis |
| Relation | G22910 |
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