The elementary reversible gas$-$phase reaction $AB+C$ is carried out adiabatically

in a packed bed membrane reactor. The maximum catalyst weight that can be packed into

the reactor is $Wkg.$ Pure A enters the reactor at a volumetric flow rate of $v_{0}d\frac{m^3}{s},$ at a

pressure of $P_0$ atm and temperature $T_{{?}_{?}}(0)C.$ Assume $P=0.$

The rate constant $k$ is given by $0\mathrm{.}2e^{\frac{30,000}{R}}\frac{1}{T_0}-\frac{1}{T^{\text{'}}}\frac{d{m}^3}{kkk\text{c}\text{a}\text{t}\text{.}s}$

The specific heat capacities for each component are:

$C{p}_A=40\frac{J}{m}ol*K,C{p}_B=25\frac{J}{m}ol*K,Cp{C}_C=15\frac{J}{m}ol*K$

The heat of formations for each component are:

All heats of formation are referenced to $273K.$

a$)$ Derive the energy balance for the packed bed membrane reactor, with species $C$

diffusing out in order to obtain a differential equation for Temperature with respect to

catalyst Weight $(d\frac{T}{d}W).($Refer back to your textbook for more information$).$

Hint: Start your derivation with the energy balance provided below and state all your

assumptions for cancelling out any terms.

$\frac{d{}^{\text{l}\text{a}\text{r}\text{r}}}{dt}=Q^{}-W_s+{\displaystyle \underset{?}{\overset{?}{}}}F{|}_{ii\text{i}\text{i}\text{i}}-{\displaystyle \underset{?}{\overset{?}{}}}{E}_{iiii\text{o}\text{u}\text{t}}^{F|}$

b$)$ Using Polymath, plot the concentration profiles and temperature profile for different

values of $k_c=1,5,10s^{-1}$ where the equilibrium constant referenced at $300K$ is $K_{c0}=$

$0\mathrm{.}01mo\frac{l}{d}{m}^3$

Discuss the effect of varying $k_c,$ what changes do you observe in the results?

W $=71$ kg v$0=32$ dm$3/$s Po $=17$ atm To $=202$C