An undesired compound $A$ is to be reduced from $0\mathrm{.}1\%($mol $\%)$ to $0\mathrm{.}02\%$ by absorption in water

which contains the reactant $B.$ The absorption occurs in a bubble column, whose length$/$width

relation is such that one can assume ideal mixing in both phases. The reaction

$A+BP$

takes place only in the liquid phase and is a second order process. The reaction rate can be

written as

$r=k{C}_A{C}_B,[mol{m}^{-3}{s}^{-1}].$

Calculate the column's length under the following conditions:

Total pressure $P=100$kPa;

Temperature $T=27C$;

Molar flow of liquid $L=700kmol{h}^{-1}{m}^{-2}$;

Molar flow of gas $G=100kmol{h}^{-1}{m}^{-2}$;

Concentration of $B$ in the liquid inflow $C_{B0}=10mol{m}^{-3}$;

Specific contact area $a=20m^2{m}^{-3}$;

Henry's constant $He=2\mathrm{.}5Pa{m}^{3}mo{l}^{-1}$;

Diffusivities of A and $B$ in liquid phase $D_A=D_B={10}^{-9}{m}^2{s}^{-1}$;

Mass transfer coefficients:

for $A$ in gas phase $(k_g{)}_A=0\mathrm{.}5m{s}^{-1}$;

for A and $B$ in liquid phase $(k_L{)}_A=(k_L{)}_B=0\mathrm{.}4mm{s}^{-1}$;

Rate constant $k=40000m^{3}mo{l}^{-1}{s}^{-1}.$