M. Sankar

Numerical Study of Natural Convection in a Vertical Porous Annulus with an Internal Heat Source: Effect of Discrete Heating

This article reports a numerical study of natural convection in a vertical annulus filled with a fluid-saturated porous medium, and with internal heat generation subject to a discrete heating from the inner wall. The relative importance of discrete heating on natural convection in the porous annulus is examined via the Brinkman-extended Darcy equation. The inner wall of the annulus has a discrete heat source and the outer wall is isothermally cooled at a lower temperature. The top and bottom walls and the unheated portions of the inner wall are kept adiabatic. The governing equations are numerically solved using an implicit finite difference method. A wide range of numerical simulations is conducted to understand the effects of various parameters like heat source length, heat source location, Darcy number, radius ratio, and Rayleigh numbers due to external and internal heating on the flow and heat transfer. The numerical results reveal that the placement of the heater near the middle portion of the inner wall yields the maximum heat transfer and minimum hot spots rather than placing the heater near the top and bottom portions of the inner wall. The heat transfer increases with an increase in the external Rayleigh number and Darcy number, while it decreases with an increase in the internal Rayleigh number, porosity of the porous medium, and the size of the heater. Further, we found that the size and location of the heater has a profound influence on the heat transfer rate and maximum temperature in the annular cavity.

Natural Convection in a Vertical Annuli with Discrete Heat Sources

In this article, we numerically study natural convection heat transfer in a cylindrical annular cavity with discrete heat sources on the inner wall, whereas the outer wall is isothermally cooled at a lower temperature, and the top wall, the bottom wall, and unheated portions of the inner wall are assumed to be thermally insulated. To investigate the effect of discrete heating on the natural convection heat transfer, at most two heating sources located near the top and bottom walls are considered, and the size and location of these discrete heaters are varied in the enclosure. The governing equations are solved numerically by an implicit finite difference method. The effect of heater placements, heater lengths, aspect ratio, radii ratio, and modified Rayleigh number on the flow and heat transfer in the annuli are analyzed. Our numerical results show that when the size of the heater is smaller, the heat transfer rates are higher. We also found that the heat transfer in the annular cavity increases with radii ratio and modified Rayleigh number, and can be enhanced by placing a heater with the smaller length near the bottom surface.

Cooling of Heat Sources by Natural Convection Heat Transfer in a Vertical Annulus

This article reports convection heat transfer in a short and tall annular enclosure with two discrete isoflux heat sources of different lengths. The discrete heat sources are mounted at the inner wall and the outer wall is maintained at a lower temperature, whereas the top and bottom walls and the unheated portions of the inner wall are kept at adiabatic. An implicit finite-difference method is employed to solve the vorticity–stream function formulations of the governing equations. The significant influence of the discrete heaters on the flow and heat transfer is analyzed for a wide range of modified Rayleigh numbers, aspect ratio, and length ratio (ϵ) of heat sources. Our numerical results reveal that the average Nusselt number decreases with aspect ratio, whereas the magnitude of maximum temperature increases with the aspect ratio. For most of the parametric cases considered in the present study, the heat transfer rate is found to be higher at the bottom heater than at the top heater except for ϵ = 0.5. The effect of heater length ratio on the heat transfer rate is noticeable for unit aspect ratio, whereas its effect is insignificant as the aspect ratio increases. Furthermore, it was found that the maximum temperature is found generally at the top heater except for the case ϵ = 0.5, where the maximum temperature is found at the bottom heater.

Constant-flux discrete heating in a unit aspect-ratio annulus

Natural convection in an annulus with a discrete heat source on the inner cylinder is studied numerically. The outer cylinder is isothermally cooled at a fixed low temperature, and the top wall, the bottom wall and unheated portions of the inner cylinder are thermally insulated. For low applied heat flux through the heater, as measured non-dimensionally by a Grashof number, Gr, the flow in the annular gap consists of a single-cell overturning meridional flow driven by the radial temperature gradient between the heater on the inner cylinder and the cold outer cylinder. In this regime, the flow is very weak and heat is transported primarily via conduction. The flow structure does not change until Gr ~ 104, although the flow strength steadily increases with Gr. As the nonlinear convection terms become more important, the meridional circulation sweeps the isotherms from being almost vertical near the outer cylinder to almost horizontal near the bottom wall. By the end of the transition from the conduction-dominated regime (Gr < 104) to the convection-dominated regime (Gr ~ 106), the flow becomes segregated into three distinct regions: (i) for vertical levels below that of the bottom of the heater, an essentially cold stagnant pool develops, with the heat flux through the outer cylinder dropping to zero. (ii) At vertical levels between the bottom and the top of the heater, most of the region in between the two cylinders is stably stratified with a relatively weak radial flow from the cold to the heated cylinder. The horizontal isotherms adjust to the temperatures on the cylinders in thin buoyancy boundary layers which drive fluid down the cold cylinder and up the heated cylinder segment. The boundary layer on the heater is about half as thick as that on the cold cylinder, but about twice as intense. (iii) The third region is above the heater top. The boundary layer flow from the heater continues upward where it meets the top endwall and bounces off of it. A wavy jet bounces on its way radially outward between the top insulated wall and the stably stratified region below. Flow separation on the top wall leads to the formation of a recirculation zone there. The vast majority of the heat flux through the outer cylinder occurs at this upper level, and is heavily concentrated near the very top. Geometric factors, such as the radius ratio of the cylinders and the heater placement, have quantitative effects, which are described, but the overall qualitative picture remains unchanged.