Water at $24C$ is flowing by gravity through a filter bed of small particles having an equivalent diameter of $0\mathrm{.}0060m.$ The void fraction of the bed is measured as $0\mathrm{.}42.$ The packed bed has a depth of $1\mathrm{.}50m.$ The liquid level of water above the bed is held constant at $0\mathrm{.}40m.$ What is the water velocity $v^{\text{'}}$ based on the empty cross section of the bed?

a$.$ First find the friction losses across the bed using Bernoulli eqn. $P_1$ and $P_2$ are both atmospheric and the diameter is constant.

b$.$ Use the Ergun equation to find the interstitial velocity.

$F=1\mathrm{.}75\frac{V_S^2}{D_p}\frac{(1-)x}{{}^3}+150\frac{V_S(1-{)}^{2}x}{D_p^2{}^3}$