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.

The present work investigates, numerically, the process of melting and solidification in hollow vertical cylinders, filled with air and phase change material (PCM). The PCM used is sodium nitrate, which expands upon melting. Therefore, a void must be present within the cylinder, which is filled with air. The influence of cylinder shape on melting time is determined. The numerical model takes both conductive and convective heat transfer into account during the melting process. The Volume-of-Fluid (VOF) model is used to track the interface between the PCM and air as the PCM melts. Three dimensionless numbers represent the characteristics of the problem, which are the Grashof, Stefan, and Prandtl numbers. The Stefan and Prandtl numbers are held constant, while the Grashof number varies. Inner Aspect Ratio (AR) is used to characterize the shape of the cylinder, which is defined as the ratio of the height to the diameter of the vertical cylinder. In this study, a range of AR values from 0.23 to 10 is investigated. Cylinders with small AR, corresponding to high Grashof numbers, lead to lower melting times compared with cylinders with high AR. The molten PCM velocity was also influenced greatly by this difference between solid PCM shape between high and low AR cases. Cylinders with small AR, corresponding to high Grashof numbers, lead to higher solidification times compared with cylinders with high AR. It was found that the velocity decreased during the solidification process, but the shape of the cylinder had an effect on the decrease. Natural convection velocity was found to decrease during the solidification process and, therefore, its effects diminish as solidification proceeds.

This study solves the problem of unsteady free convection from an inclined heated tube both numerically and analytically. The tube is taken to have an elliptic cross-section having a constant heat flux applied to its surface. The surrounding fluid is viscous and incompressible and infinite in extent. The Boussinesq approximation is used to describe the buoyancy force driving the flow. The underlying assumptions made in this work are that the flow remains laminar and two-dimensional for all time. This enables the Navier-Stokes and energy equations to be formulated in terms of the streamfunction, and vorticity. We assume that initially an impulsive heat flux is applied to the surface and that both the tube and surrounding fluid have the same initial temperature. The problem is solved subject to the no-slip and constant heat flux conditions on the surface together with quiescent far-field and initial conditions. An approximate analytical-numerical solution was derived for small times, t and large Grashof numbers, Gr. This was done by expanding the flow variables in a double series in terms of two small parameters and reduces to solving a set of differential equations. The first few terms were solved exactly while the higher-order terms were determined numerically. Flow characteristics presented include average surface temperature plots as well as surface vorticity and surface temperature distributions. The results demonstrate that the approximate analytical-numerical solution is in good agreement with the fully numerical solution for small t and large Gr.

This study solves the problem of unsteady free convection from an inclined heated tube both numerically and analytically. The tube is taken to have an elliptic cross-section having a constant heat flux applied to its surface. The surrounding fluid is viscous and incompressible and infinite in extent. The Boussinesq approximation is used to describe the buoyancy force driving the flow. The underlying assumptions made in this work are that the flow remains laminar and two-dimensional for all time. This enables the Navier-Stokes and energy equations to be formulated in terms of the streamfunction, and vorticity. We assume that initially an impulsive heat flux is applied to the surface and that both the tube and surrounding fluid have the same initial temperature. The problem is solved subject to the no-slip and constant heat flux conditions on the surface together with quiescent far-field and initial conditions. An approximate analytical-numerical solution was derived for small times, t and large Grashof numbers, Gr. This was done by expanding the flow variables in a double series in terms of two small parameters and reduces to solving a set of differential equations. The first few terms were solved exactly while the higher-order terms were determined numerically. Flow characteristics presented include average surface temperature plots as well as surface vorticity and surface temperature distributions. The results demonstrate that the approximate analytical-numerical solution is in good agreement with the fully numerical solution for small t and large Gr.