Transient natural convection between two vertical walls filled with a porous material having variable porosity (Q2775716)

The effects of variable porosity and inertial force on transient free convection flow between two vertical walls filled with a porous medium is studied theoretically using the Brinkman-Forchheimer-extended Darcy model. It is assumed that the walls are differentially heated and that the variable porosity of the porous medium is described by the following equation NEWLINE\[NEWLINE\varepsilon= \varepsilon_\infty \bigl[1+ \lambda_1 \exp(-\lambda_2y'/d') \bigr],NEWLINE\]NEWLINE where \(y'\) is the Cartesian coordinate normal to vertical planes, \(\varepsilon\) is porosity, \(d'\) is the sphere diameter, and \(\lambda_1\) and \(\lambda_2\) are constants depending on the sphere diameter. It is assumed that the flow is fully developed so that the velocity in the \(y'\) direction is zero, while the velocity along the channel and the temperature field depend only on time \(t'\) and coordinate \(y'\). Under these conditions the momentum and energy equations are written in non-dimensional form introducing three parameters, namely, Grashof number, Prandt number and the nondimensional sphere diameter \(d=d'/H\), where \(H\) is the distance between the walls. These equations are then solved numerically by using an implicit finite difference method. The numerical results are presented in the form of graphs and a table. It is found that the Darcy model predicts an enhancement in the air velocity near the heated wall compared to the general flow model. It is also concluded that the formation of boundary layer near the heated wall is a results of Brinkman term, and its effect is strongly dependent on the diameter of spherical beads. Finally, it is shown that the steady-state flow occurs when nondimensional time is close to Prandtl number of fluid. The problem is important and interesting, and the numerical solution is clear. However, the results are not compared with any data from the literature.