Viscous flow in a packed bed $-$ The Brinkman Equation. During the early days of this

class, we talked about the Ergun equation. We showed that its viscous component is called

the Kozeny$-$Carman equation. The generalized form of that equation is Darcy's Law ${?}^5$ :

$gradientP-g$

Darcy's Law describes flow in porous media due to pressure gradients and gravitational

forces. However, viscous losses due to shear forces and therefore wall effects are neglected.

The Brinkman equation empirically includes them:

$-$gradP$+g+gra{d}^{2}v-\frac{}{}v=0$

For steady$-$state unidirectional flow in a horizontal and tubular packed bed of length $L,$

with a bed diameter small enough so that gravitational effects are negligible, the Brinkman

equation reduces to:

$-\frac{dP}{dz}+\frac{}{r}\frac{d}{dr}(r\frac{d{v}_z}{dr})-\frac{}{}{v}_z=0$

a$.$ Neglecting the end effects, solve this equation to obtain the velocity profile:

$v_z=\frac{}{}\frac{|P|}{L}[1-\frac{I_0(\frac{r}{\sqrt[\phantom{2}]{}})}{I_0(\frac{R}{\sqrt[\phantom{2}]{}})}]$

b$.$ Show that the volumetric flow rate can be expressed as:

$V^{}=\frac{|P|R^2}{L}[1-\frac{2\sqrt[\phantom{2}]{}}{R}\frac{I_1(\frac{R}{\sqrt[\phantom{2}]{}})}{I_0(\frac{R}{\sqrt[\phantom{2}]{}})}]$