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
∂ u / ∂ x + ∂ v / ∂ y = 0 {\displaystyle \partial u/\partial x+\partial v/\partial y=0}
According to Darcy's law
u = − ( K / μ ) ∂ P / ∂ x {\displaystyle u=-(K/\mu )\partial P/\partial x}
v = − ( K / μ ) ∂ P / ∂ y {\displaystyle v=-(K/\mu )\partial P/\partial y}
u ∂ T / ∂ x + v ∂ T / ∂ y = α ∂ 2 ∂ x 2 T {\displaystyle u\partial T/\partial x+v\partial T/\partial y={\boldsymbol {\alpha }}{\partial ^{2} \over \partial x^{2}}T}
u = {\displaystyle u=}
U ∞ {\displaystyle U_{\infty }} v = 0 {\displaystyle v=0}
P ( x ) = − ( μ / K ) U ∞ x + c o n s t a n t {\displaystyle P(x)=-(\mu /K)U\infty x+constant}
δ t {\displaystyle \delta _{t}} is the thickness of the slender layer of length x that affects the temperature transition from T 0 {\displaystyle T_{0}} to T ∞ {\displaystyle T_{\infty }} . Balancing the energy equation between enthalpy flow in the x direction and thermal diffusion in the y direction
U ∞ ∂ T / ∂ x ∼ α Δ T / δ t 2 {\displaystyle U_{\infty }\partial T/\partial x\sim \alpha \Delta T/\delta _{t}^{2}}
boundary is slender so δ t << x {\displaystyle \delta _{t}<<x}
δ t / x ∼ P e x − .5 {\displaystyle \delta _{t}/x\sim Pe_{x}^{-}.5}
N u = h x / K ∼ x / δ t ∼ P e x 0 .5 {\displaystyle 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.
P e x {\displaystyle Pe_{x}} = {\displaystyle =} Advective transport rate / {\displaystyle /} Diffusive transport rate
P e x = U ∞ x / α {\displaystyle Pe_{x}=U_{\infty }x/\alpha }
See also Darcy's law Nusselt Number Porous media Convective heat transfer Heat transfer coefficient Porous media
References
External links https://web.archive.org/web/20091211060057/http://www.me.ust.hk/~mezhao/pdf/20.PDF