Consider the elementary reaction $AB$ occurring within a given flow reactor with volume $V,$

and fixed inlet molar flowrate of $A,F_{A0}.$ The Damk$$hler number is a positive dimensionless

number given by:

$Da=\frac{-r_{A0}V}{F_{A0}}=\frac{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}\text{r}\text{a}\text{t}\text{e}of\text{r}\text{e}\text{a}\text{c}\text{t}\text{i}\text{o}\text{n}\text{f}\text{o}\text{r}A}{\text{f}\text{e}\text{e}\text{d}\text{f}\text{l}\text{o}\text{w}\text{r}\text{a}\text{t}\text{e}A}=\frac{\text{r}\text{e}\text{a}\text{c}\text{t}\text{i}\text{o}\text{n}\text{r}\text{a}\text{t}\text{e}}{\text{c}\text{o}\text{n}\text{v}\text{e}\text{c}\text{t}\text{i}\text{o}\text{n}\text{r}\text{a}\text{t}\text{e}},$

and is commonly used to estimate reactor conversion with the following rule of thumb:

$Da>10x_A>0\mathrm{.}9$

$($a$)$ Using the differential form of the PFR design equation, derive an expression for the

conversion $x_A$ in terms of the Damk$$hler number for a liquid$-$phase first order reaction,

$D{a}_1.$ Assume the reaction occurs as $AB.$

$($b$)$ Using the differential form of the design equation for a PFR again, derive a similar

expression for the conversion $x_A$ in terms of the Damk$$hler number for a liquid$-$phase

second order reaction, $D{a}_2.$ Assume the reaction occurs as $2AB.$

$($c$)$ Suppose we have a PFR with fixed reactor volume of $0\mathrm{.}75m^3,$ feed molar flowrate of

$F_{A0}=1000\frac{\text{m}\text{o}\text{l}}{hr},$ feed volumetric flowrate $Q_0=\frac{4000(L)}{hr},$ and first order rate constant $k_1=$

$0\mathrm{.}41\frac{1}{hr}.$ Calculate the Damk$$hler number for the reactor and the reactor's final conversion.

Is the conversion in line with the Damk$$hler number rule of thumb?

$($d$)$ Suppose the reaction proceeds with a second order mechanism instead, with a different

second$-$order rate constant $k_2=245\frac{L}{\text{m}\text{o}\text{l}**hr}.$ Calculate the Damk$$hler number for the

reactor and the reactor's final conversion. Is the final reactor conversion in line with the

Damk$$hler number rule of thumb?