ANALYTICAL AND NUMERICAL STUDIES OF NATURAL CONVECTION IN A CAVITY: A THERMAL NON-EQUILLIBRIUM APPROACH

Abstract

Natural convection in a fluid-saturated porous medium has received considerable attention during the past decades, because of its frequent presence in nature and its importance in many industrial and geophysical applications, such as: fluid flow in geothermal reservoirs, energy-related engineering problems, migration of moisture in grain storage system, crude oil production, contaminant transport [40], storage and preservation of grains and cereals, computer chips via use of porous metal foams [23, 116], drying/freezing of foods [117], geophysics and petroleum engineering [50], etc. It is important to mention here that the fluid flow and heat transfer in a porous medium is basically a two phase problem. The solid portion of the porous matrix is one phase and the saturated fluid in the void part of the medium is another phase. The majority of the available studies on convection in a porous medium are relevant to the situation where thermal equilibrium between solid porous matrix and saturated fluid is assumed. The corresponding fluid-saturated porous medium is said to be in local thermal equilibrium (LTE) state. However, at sufficiently large Rayleigh number or rapid heat transfer for high speed flow the solid porous matrix may have a different temperature from that of the saturating fluid [91] and the porous medium is said to be in local thermal non-equilibrium (LTNE) state. In-spite of the wide applications [23, 33, 97, 116, 117] of fluid flow as well as heat and mass transfer under LTNE state, the theoretical investigations in this direction is largely overlooked. Therefore, in the present thesis a step has been taken in this direction by investigating natural (thermal and thermosolutal) convection in an enclosure filled with porous i ii medium under LTNE state for different situations: (i) natural convection in a porous enclosure, subject to side heating by applying a constant heat flux, (ii) double-diffusive convection in a porous enclosure in which a uniform temperature and concentration flux is imposed along the side walls, (iii) double-diffusive natural convection in a porous enclosure, whose side walls are maintained at a constant temperature and concentration of solute and (iv) natural convection in an inclined porous enclosure. The complete thesis has been split into 7 chapters. Chapter 1 is introductory and it contains some basic definitions which are required to understand the flow dynamics in a porous medium. Chapter 2 highlights a survey of the existing literature related to the study of natural convection in a porous medium under the assumption of LTE as well as LTNE state. The entire literature survey has been split mainly into three sections, namely, (i) natural convection in a porous enclosure with LTE approach, (ii) double-diffusive natural convection in a porous enclosure with LTE approach, and (iii) natural convection in a porous medium with LTNE approach. Chapter 3 reports the influence of LTNE state on natural convection in a porous enclosure. The two dimensional steady flow is induced by a constant heat flux on side walls of the enclosure, when both horizontal walls are insulated. Porous medium is assumed to be both hydro-dynamically and thermally isotropic. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. Analytical solution, valid for slender enclosure, has been obtained using parallel flow assumption. To give a validation of the analytical solution, the complete problem has been solved numerically by ADI method. A comparative study has been made between convection in a square cavity and the same in a slender enclosure. Numerical experiments indicate that in comparison with a square cavity, where a sharp decrease of heat transfer rate for fluid (Nuf ) takes place upto a certain small value of inter-phase heat transfer coefficient (H), in a slender enclosure a smooth decrease of Nu f has been observed in the entire range of H iii when the conductivity ratio (g ) is very small. However, for g = 10, Nuf is almost independent of H in both geometries. For a given value of g , when the value of H is relatively very high, up to a certain value of Rayleigh number (Ra) the difference between both solid and fluid temperature rates is negligible. The corresponding temperature contours of two phases become almost identical in magnitude as well as pattern, which indicates that the LTE state is achieved. Finally, in contrast to LTE state, where the maximum heat transfer rate takes place in an enclosure for a threshold value (Ao) of aspect ratio (A) lying in between 1 and 1.5, and is almost independent of all LTE parameters, in LTNE state Nuf acts as an increasing function of g . Chapter 4 addresses the influence of LTNE state on double-diffusive natural convection in a fluid saturated porous enclosure subjected to constant flux of heat (for both the phases) and constant flux of concentration at both the vertical walls, while other walls of the enclosure are kept as adiabatic. For this the Darcy model has been adopted. The governing equations are solved numerically by ADI method and analytically by using parallel flow assumption. A comparative study has been made between buoyancy aiding flow (i.e., buoyancy ratio, N,> 0) and buoyancy opposed flow (i.e., buoyancy ratio, N,< 0). It was found that, same as LTE state particular oscillations and multiple solutions have also been observed on considering LTNE state for a certain range of H and g for buoyancy opposed flow. The range of the interval for buoyancy ratio, INM (in which multiple solution exists), increases on increasing the value of H as well as g , while the range of the interval for buoyancy ratio, INO (in which oscillation exists), vary in a subtle way. It was also reported that, same as LTE state, INO is a subset of INM under LTNE state for all the values of H as well as g considered in this study. Apart from this, increment of A enhances Nuf and Nus upto a threshold value A0 of A, beyond that threshold value the impact of same is negligible, while Sh acts in a reverse way. Chapter 5 presents the influence of LTNE state on double-diffusive natural convection in a square porous cavity. The two dimensional steady state flow is induced due to maintenance of constant temperature and concentration on the vertical walls and other walls are iv assumed to be adiabatic. Non-Darcy (Darcy-Brinkman-Forchheimer)model has been taken and the complete governing equations are solved by standard SIMPLER algorithm. A comparative study of the effect of the presence of Brinkman term in the momentum equation showed that the results under the Darcy model are very close to those for the non-Darcy Brinkman model for relatively low permeable medium (e.g., in this study Darcy number, Da = 10−4). From our numerical experiments it has been found that the impact of Lewis number (Le) on Nuf and Nus as well as on the thermal distribution of fluid and solid is not straightforward. It depends on the value of N and H. However, Le increases the average mass transfer rate (Sh). Also, for each Le there exist a point in the domain of N where Nuf is minimum. Similar points also exist for Nus and Sh. In general, these points are different and depend on the LTNE state parameters (H and g ), except at Le = 1. For any relatively large value of H, when almost LTE state is achieved, the point at which Nuf and Nus are minimum is same due to similar thermal distribution of fluid and solid. Also, it has been found for the buoyancy aided flow (N > 0), increase in H upto a threshold value (H0) of H decreases Sh as well as Nuf but increases Nus. This H0 is found to be a decreasing function of g of fluid and solid phases. Overall, the impact of LTNE state on the heat transfer rates and thermal distribution is significant but it is negligible on the Sh and solute distribution. In chapter 6, a comprehensive numerical as well as analytical investigation of the natural convection in an inclined porous enclosure is presented. A constant heat flux is applied for heating and cooling of the long side walls of the inclined slender enclosure, while the other two walls are insulated. Numerical solutions are obtained with the help of ADI method, while parallel flow approximation has been used to obtain analytical solution. Special attention is given to understand the effect of anisotropic parameter (permeability ratio, K∗) and angle of inclination (f ) on the flow dynamics as well as Nuf and Nus as a function of both H and g . A significant impact of K∗ and f has been observed on the heat transfer rates for both the phases for relatively large values (1, 100) of both H and g . Chapter 7 presents the summary and conclusions of this thesis and the possible directions for future work.