$2\mathrm{.}2$ An adsorption column is packed with an adsorbent that has an adsorption area of $1500m^2.g^{-1}.$ The packed voidage is $40\%$ and the bulk density of the adsorbent is $400kg*m^{-3}.$ Assume the pores are all cylindrical with an average pore diameter of $1\mathrm{.}6nm.$

$2\mathrm{.}2\mathrm{.}1$ Determine the porosity of the adsorbent.

$[4]$

The same adsorbent is now used to adsorb a component with a concentration of $100mg.L^{-1}$ from a fermentation broth. The fermentation broth is pumped at $4m^3.h{r}^{-1}$ to the adsorption column. The isotherm that described the equilibrium adsorption is given by:

where

$q=g$ adsorbed per $g$ adsorbent

$C=$ concentration of solute $(\frac{g}{L})$

$2\mathrm{.}2\mathrm{.}2$ Estimate the amount of adsorbent as well as the volume of the adsorption column required for $2$ days of operation if the outlet concentration of the solute must be $0\mathrm{.}5mg*L^{-1}.$

$2\mathrm{.}3$ In a cross$-$flow ultrafiltration unit, a protein with a molecular mass of $3*{10}^{5}Da$ is separated from a fermentation mixture in a crossflow filtration cell with an ultrafiltration membrane. The membrane has a diameter $D=2cm$ and a length $L=50cm.$ The flow rate through the tube is $Q=2L*mi{n}^{-1}.$ The flow is turbulent, $f=0\mathrm{.}0005$ and $C4=2((atm)(\frac{s}{c}m{)}^2).$ The inlet pressure is $P_i=2$atm. The protein concentration in the solution and the gel film are $C_B=30mg*L^{-1}$ and $C_G=100g.L^{-1}$ respectively.

$2\mathrm{.}3\mathrm{.}1$ Determine the outlet pressure $(P_0)$

$[3]$

$2\mathrm{.}3\mathrm{.}2$ Determine the trans membrane pressure drop $(\frac{?}{{?}_{\frac{?}{?}}{P}_M})$

$[3]$

$2\mathrm{.}3\mathrm{.}3$ If the mass transfer coefficient $(k)$ for the protein is $5cm.s^{-1},$ determine the flux of the liquid through the UF membrane $(J).$

$[3]$

$2\mathrm{.}3\mathrm{.}4$ If the resistance of the filter is $R_M=0\mathrm{.}002$ atm. $c{m}^2.s.m{g}^{-1},$ determine the cake resistance, $R_G.$

$[3]$