$x($g$)$ Define the following nondimensional variables and parameters

$y=\frac{C}{C_0}$;$,x=\frac{z}{L}$;$,Pe=\frac{u_{0}L}{D}$;$,N=\frac{Wk}{F}$

and show that the resulting modeling equations are

$\frac{1}{Pe}\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}-Ny=0$

$x=0$;$,1=y{|}_x|=0-\frac{1}{Pe}\frac{dy}{dx}{|}_x|=0$

$x=1$;$,\frac{dy}{dx}=0$

Compare this model $($hereafter called Model $2)$ with Model $1,$ and show that the axial diffusion may be ignored when $Pe1($this can be accomplished several ways: by decreasing porosity $$ or by reducing $D,$ or by increasing velocity or length$).$

$($h$)$ To study the effect of the mass transfer inside the catalyst particle, we need to remove the assumption of no diffusion resistance inside the particle. This means that the mass balance within the particle must be linked with the external composition. To investigate this effect, we shall ignore the axial diffusion $($which is usually small for packing made up of finely granulated solid$)$ and the external film resistance surrounding the particle.

Set up a thin spherical shell $($control volume$)$ inside the particle, and show that the mass balance equation is

$D_e\frac{1}{r^2}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r^2\frac{del{C}_p}{\text{d}\text{e}\text{l}\text{r}})-{}_{p}k{C}_p=0$

where $C_p$ is the toxic gas concentration within the particle, and $D_c$ is the effective diffusivity and is defined as Fickian$-$like:

$J_p=-D_e\frac{del{C}_p}{\text{d}\text{e}\text{l}\text{r}},\{\frac{\text{m}\text{o}\text{l}\text{e}\text{s}\text{t}\text{r}\text{a}\text{n}\text{s}\text{p}\text{o}\text{r}\text{t}\text{e}\text{d}by\text{d}\text{i}\text{f}\text{f}\text{u}\text{s}\text{i}\text{o}\text{n}}{\text{c}\text{r}\text{o}\text{s}\text{s}-sectional\text{a}\text{r}\text{e}\text{a}-\text{t}\text{i}\text{m}\text{e}}\}$

and suitable boundary conditions for negligible film resistance and particle symmetry are

$C_p(R)=C$;$,$del$\frac{C_p}{d}$elr$=0,at,r=0$

where $R$ denotes particle radius.

$($i$)$ Next, set up the mass balance around the thin element spanning the whole column cross section $($as in Model $1),$ but this time the control volume will exclude the catalyst volume. This means that material is lost to the various sinks made up by the particles. Show that the mass balance equation on this new control volume is

$-u_0\frac{dC}{dz}=(1-)\frac{3}{R}{D}_e\frac{del{C}_p}{\text{d}\text{e}\text{l}\text{r}}{|}_r|=R$