A protein is present in $8$ m$3$ of fermentation liquor at a concentration of $0\mathrm{.}6$ g$/$l$.$

Ultrafiltration under gel polarization conditions is used to concentrate the protein by a factor of five. The gelation concentration is estimated as $25$ g$/$l$.$ the filtration is carried ouusing a spiral$-$wound module with a mass transfer coefficient of $3\mathrm{.}4\backslash$times $10-6$ m$/$s$.$ The protein is rejected completely by the membrane and the effects of fouling are assumed to be negligible.

a$)$ What membrane area is required for batch ultrafiltration if the process must be completed within $1$ hour?

b$)$ What membrane area is required if the filtration is carried out as a continuous operation over $1$ hour with a feed flow rate of $8$ m$3/$h$?$