$(10$ pts$.)$ Consider a liquid$-$phase endothermic reaction, $A+BC+D,$ running in a plug flow reactor with a heat exchanger. The feed stream contains A and $B$ with a molar ratio of $A$:$B=1$:$5,$ with $v_0=360\frac{L}{h}.$ The density of reactant mixture is $0\mathrm{.}9k\frac{g}{L}$ and can be assumed to be constant throughout the reactor. Both A and B have the molecular weights of $50\frac{g}{m}ol.$ A small amount of acid catalyst is added. Under the given condition, the reaction rate is found to be: $-r_A=k{C}_A^2,k_0=1\mathrm{.}59{10}^{-2}\frac{L}{\text{m}\text{o}\text{l}\text{s}},E_A=10,k\frac{J}{m}ol.$ Physicochemical properties relevant to this reaction are: $H_{rxn}=10k\frac{J}{\text{m}\text{o}\text{l}\text{A}}$ and $C_{PA}=C_{PB}=C_{PC}=C_{PD}=100\frac{J}{\text{m}\text{o}\text{l}\text{K}}.$ The heat exchanger has $Ua=1k\frac{J}{m^3(h)(K)}$ and $T_a=30C($constant over the whole heat exchanger$).$ Answer the following questions:

$($a$)$ Describe the method to find the volume of reactor required to convert $50\%$ of $A$ to $C.$ Note that, in this question, you do not have to actually find a number.

$($b$)$ What do you expect if $Ua$ is very very large? Can you calculate the volume of reactor required to convert $50\%$ of $A$ to $C,$ if $Ua$ is very very large? If you can, what is the volume?