In gas treatment plant, a tray tower is used to remove $99\%$ of ammonia $17k\frac{g}{k}$gmol $)$ in the air $(MW=28\mathrm{.}97k\frac{g}{k}g$ mol$)$ stream which is coming from reactor outlet. The air stream contains $6$mol$\%$ ammonia at $30C$ and $101\mathrm{.}325$Kpa, which enter the tower at a flowrate of $($:$128\frac{kg}{h}-m^2\}.$ The flowrate of pure water $k\frac{g}{k}$gmol $)$ entering the tray is $188k\frac{g}{h-m^2},$ The equilibrium of the dilute stream is represented by:

$y=0\mathrm{.}76x$

Where: $y$ is the mole fraction of $N{H}_3$ in the gas phase And, $x$ is the mole fraction of $N{H}_3$ in the liquid phase

By assuming that the ammonia concentration is sufficiently dilute through the tower that both operating and equilibrium curves are linear.

a$)$ Draw a sketch of the tray tower showing the amounts of inlet and outlet streams including the mole fractions and mole ratio of each stream.

b$)$ Determine the number of theoretical trays $($N$)$ graphically.

c$)$ Determine the number of theoretical trays $($N$)$ analytically using the Kremser Brown Souders $($KBS$)$ equation.

$N=\frac{ln[\frac{y_{N+1}-m{x}_0}{y_1-m{x}_0}(1-\frac{1}{A})+\frac{1}{A}]}{lnA}$

$A_1=\frac{L_o}{m{V}_1},A_N=\frac{L_N}{m{V}_{N+1}},A_1{A}_N,A=\sqrt[\phantom{2}]{A_1{A}_N}$

$OR,A_1=\frac{L_o}{m{V}_1},A_N=\frac{L_2}{m{V}_2},A=\sqrt[\phantom{2}]{A_1{A}_2}$