Forced convection in porous media

Forced convection is type of heat transport in which fluid motion is generated by an external source like a (pump, fan, suction device, etc.). Heat transfer through porus media is very effective and efficiently. Forced convection heat transfer in a confined porous medium has been a subject of intensive studies during the last decades because of its wide applications.

The basic problem in heat convection through porous media consists of predicting the heat transfer rate between a deferentially heated, solid impermeable surface and a fluid-saturated porous medium. Beginning with constant wall temperature.^[1]

In 2D steady state system

$\partial u/\partial x+\partial v/\partial y=0$

According to Darcy's law

$u=-(K/\mu )\partial P/\partial x$

$v=-(K/\mu )\partial P/\partial y$

$u\partial T/\partial x+v\partial T/\partial y={\boldsymbol {\alpha }}{\partial ^{2} \over \partial x^{2}}T$

$u=$ $U_{\infty }$ $v=0$

$P(x)=-(\mu /K)U\infty x+constant$

$\delta _{t}$ is the thickness of the slender layer of length x that affects the temperature transition from $T_{0}$ to $T_{\infty }$ .

Balancing the energy equation between enthalpy flow in the x direction and thermal diffusion in the y direction

$U_{\infty }\partial T/\partial x\sim \alpha \Delta T/\delta _{t}^{2}$

boundary is slender so $\delta _{t}<<x$

$\delta _{t}/x\sim Pe_{x}^{-}.5$

$Nu=hx/K\sim x/\delta _{t}\sim Pe_{x}^{0}.5$

The Peclet number is a dimensionless number used in calculations involving convective heat transfer. It is the ratio of the thermal energy convected to the fluid to the thermal energy conducted within the fluid.

$Pe_{x}$ $=$ Advective transport rate $/$ Diffusive transport rate