$(50$ pts$.)$ A batch reactor can be operated non$-$isothermally. The temperature of the batch reactor can be

controlled with a heat exchanger. Answer the following questions:

$($a$)$ Starting from the general energy balance, derive the energy balance for a non$-$isothermal batch reactor:

$\frac{dT}{dt}=\frac{(-r_{A}V)(-H_{rxn})-UA(T-T_a)}{{\displaystyle \underset{?}{\overset{?}{}}}{N}_i{C}_{}},$ where $V$ is the volume of reactor $[m^3],H_{rxn}$ is the heat of reaction $\frac{J}{m}$olA,

$U$ is the overall heat$-$transfer coefficient $[\frac{J}{m^2}(s)(K)],A$ is the heat$-$exchange area $[m^2],T_a$ is the temperature of

fluid used for heat exchange $[K],N_i$ is the mole number of species $i[mol],$ and $C_{}$ is the mean heat capacity

of species i $\frac{J}{m}$olK.

$($b$)$ Suppose that you want to run a single reaction, which is reversible and exothermic, in a non$-$isothermal

batch reactor. Sketch equilibrium conversion $(x_e)$ as a function of temperature $(T).$

$($c$)$ What would be the optimal temperature variation $(T_{\text{o}\text{p}\text{t}})$ with time $(t)$ for the single reversible exothermic

reaction in a non$-$isothermal batch reactor? You can describe either what the optimal temperature profile

would be or how to obtain the optimal temperature profiles.

$($d$)$ What kinds of information would you need to find the quantitative solution of the optimal temperature

profile $(T_{\text{o}\text{p}\text{t}}$ Vs$.t)$ for the single reversible exothermic reaction in a non$-$isothermal batch reactor?

$k_1$

$($e$)$ Suppose that the reaction is: $AB,$ and both forward and backward reactions are elementary. The rate

$k_2$

constants $($that is$,k_1$ and $k_2)$ follow the Arrhenius relationship. At each conversion, we can find the

temperature for maximum rate of reaction $($that is$,\frac{d(-r_A)}{dT}=0).$ Find the temperature as a function of

conversion $(x).$

$($f$)$ Suppose that you want to run multiple non$-$isothermal reactions, instead of a single reaction, in a non$-$

isothermal batch reactor. Which one would be more important, conversion or selectivity? And, why?

$($g$)$ What would be the optimal temperature variation with time for multiple non$-$isothermal reactions running

in a non$-$isothermal batch reactor? How to obtain the temperature profile? What kinds of information would

you need to find the quantitative solution of the optimal temperature profile?

$($h$)$ Suppose that the multiple reactions are: $A{k}_{1}D{k}_{2}U,$ and $D$ is the desired product. The activation energy

for $k_2$ is larger than that of $k_1.$ Considering economic aspects, sketch the optimal temperature profiles.

$($i$)$ Suppose that the multiple reactions are: $A{k}_{1}D$ and $A{k}_{2}U,$ and $D$ is the desired product. The activation

energy for $k_2$ is larger than that of $k_1.$ Considering economic aspects, sketch the optimal temperature profiles.

$($j$)$ Suppose that, in the parallel reactions in $($g$),$ the reactant and the desired product are very valuable, and

that the reactor cost is not important. Sketch the optimal temperature profiles.

Solve the questions in the image.