Question $3(35$ points$)$

The catalytic reaction $A(g)longrightarrowB(g)$ takes place in a porous catalytic bed shaped like a cylindrical

shell. The reactor is designed such that the reaction mixture flows radially in the shell at a very low

mass flow rate of $w_r$ so that pressure drop to induce the flow is very small. The reaction is

exothermic and the rate of heat released per unit volume of the catalytic shell, which can be

assumed constant throughout, to be $H_R.$ Physical properties of A and $B$ can be assumed very

similar to one another. Effective thermal conductivity and heat capacity of the catalyst and gaseous

mixture can be assumed to be constant values of $k$ and $c_p$ respectively. If the inner surface of the

catalyst bed is kept constant at $T_k$ and outer surface at $T_{\text{I}},$ find an expression for temperature

distribution $T(r)$ in the catalytic shell. Your answer should include combined parameters;

$N=\frac{w_r{c}_p}{2kL}$ and $M=\frac{H_R}{2}k$

Note the solution of linear O$.$D$.$E$.$

$d\frac{y}{d}x+P(x)y=Q(x)$ is

$y=e^{-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}Pdx}({\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}Q{e}^{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}Pdx})+c{e}^{-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}Pdx}$