Water is used to remove $C{O}_2$ from the air by absorption in a column packed with Pall rings. At a

region of the column where the partial pressure of $C{O}_2$ at the interface is $10\mathrm{.}2$atm and the

concentration of $C{O}_2$ in the bulk liquid is negligible, the absorption rate is $1\mathrm{.}886{10}^{-4}mo\frac{l}{s}-c{m}^2.$

The $C{O}_2$ diffusivity in water is $2\mathrm{.}0{10}^{-5}c\frac{m^2}{s}.$ Henry's law for $C{O}_2$ is $p=Hx,$ where $H=612\mathrm{.}0$atm.

Calculate:

$(1)$ Liquid$-$phase mass$-$transfer coefficient $)$ and film thickness $(l)$

$(2)$ If the mass transfer coefficient on gas film $(k_y)$ is $0\mathrm{.}05mo\frac{l}{s}-c{m}^2$ and the total pressure of the

gas mixture is $51\mathrm{.}0$atm, calculate the overall liquid$-$side mass transfer coefficient, $K_x.$