In class we derived the equation for the effectiveness factor for a porous spherical catalyst

pellet and a porous slab. The Levenspiel text derived the expression for diffusion and reaction

in a single cylindrical pore where the gas enters from one end of the pore. In this problem I

want you to derive the expression for the effectiveness factor for a porous cylindrical catalyst

pellet where diffusion occurs in the radial direction. Assume that the cylinder ends are

impermeable and that the length of the cylinder is much longer than the radius so end effects

can be ignored. You should assume a first order reaction $(AB)$ with kinetics $r=k_{\text{e}\text{f}\text{f}}{C}_A$

$($a$)$ Start by using a differential mass balance to derive the differential equation that

describes the system. You should also state the boundary conditions for the system

$($hint one of them is that the $C_A$ is finite at the center of the cylinder$).$ You may assume

that there are no external diffusion limitations$.$

$($b$)$ Solve the differential equation to obtain $C_A$ as a function $r.$

Some information that may be helpful:

The differential equation, $x^2(d^2\frac{y}{d}{x}^2)+x(d\frac{y}{d}x)-a{x}^{2}y=0$ has the solutions $y=I_0(ax)$

and $y=K_0(ax)$ which are the modified Bessel functions $($these functions look somewhat

like $sin(x)$ and $cos(x)$ multiplied by an exponential decay function$)$ and that $K_0(x)$ goes

to negative infinity at $x=0.$

$($c$)$ Using your result form part $($b$)$ derive an equation for the effectiveness factor for this

system.

Useful formula: