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.

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

$$Pe_x = U_\infty x/\alpha$$