Continuous enzyme conversion in a fixed bed reactor

A system is being developed to remove urea from the blood of patients with renal failure. A prototype fixed bed reactor is set up using urease immobilised in $2-$mm$-$diameter gel beads. Buffered urea solution is recycled rapidly through the bed so that the system is

well mixed and external mass transfer effects are negligible. The urease reaction is:

$(N{H}_2{)}_{2}CO+3H_2$Olongrightarrow$2N{H}_4^{+}+HC{O}_3^{-}+O{H}^{-}$

$K_m$ for the immobilised urease is $0\mathrm{.}54g{l}^{-1}.$ The volume of beads in the reactor is $250c{m}^3,$ the total amount of urease is ${10}^{-4}g,$ and the turnover number is $11,000gN{H}_4^{+}(g$ enzyme$){?}^{-1}{s}^{-1}.$ The effective diffusivity of urea in the gel is $7{10}^{-6}c{m}^2{s}^{-1}.$ The reactor is operated continuously with a liquid volume of $1$ litre. The feed stream contains

$0\mathrm{.}42g{l}^{-1}$ of urea and the desired urea concentration after enzyme treatment is $0\mathrm{.}02g{l}^{-1}.$ Ignoring enzyme deactivation, what volume of urea solution can be treated in $30$min $?$