Assumption $(6)$ is readily supported on the

basis of recent work by Bishoff$[13],$ whereas

assumption $(7)$ is justified a posteriori by the

fact that the length of the "affected" zone is

generally very much greater than the spacing

of the pores. By recalling the assumption made

for quasi$-$steady state, the system may be rep$-$

resented by two ordinary differential equations.

one for axial diffusion in the pore, and the other

for radial diffusion through the reacted zone.

These differential equations are coupled through

their boundary conditions.

Thus we have:

For radial diffusion in the solid:

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{dC}{(d)r})=0,R_{p}rR(y)$

$($for any given value of $y$ and $t)$

with

$C=C_p(y),atr=R_p$

and

$-D\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{r}}=kC,atr=R(y)$

where $R(y)$ is the radial position of the reaction

front

$C$ is reactant concentration in the

solid

$C_p$ is reactant concentration in the pore

$k$ is the reaction rate constant and

$D$ is the diffusion coefficient in the

solid product layer.

Equation $(2)$ is readily integrated and we obtain

the following expression:

$C=\frac{C_p(y)}{1+\frac{R(y)*k}{D}ln(\frac{R(y{)}^{+}}{R_p})},atr=R(y)$

and

$-\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{r}}=\frac{C_p(y)}{R_p(\frac{D}{R(y)k}+ln(\frac{R(y)}{R_p}))},atr=R_p.$

At this stage Eqs. $(5)$ and $(6)$ contain the un$-$

knowns $R(y)$ and $C_p(y),$ which have to be

obtained from the following relationships:

For the gas phase diffusion in the pore we

have:

$D_p\frac{de{l}^2{C}_k}{del{y}^2}+\frac{2}{R_p}D(\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{r}}{)}_r=R_s=0$

with the following boundary conditions:

$C_p=C_0,aty=0$

and

$C_{p}0$; $asy$

where $D_p$ is the $($gas phase$)$ diffusion coefficient

in the pore, and $$ is a correction factor through

which allowance is made for the restricted

availability of reactant surface in the region

where the reacted zones interact. Thus we have:

For

For $\frac{L}{\sqrt[\phantom{2}]{2}}>R(y)>\frac{L}{2},$ or $tan=\frac{2}{L}\sqrt[\phantom{2}]{?}(R^2(y)-(\frac{L}{2}{)}^2)D_n\frac{de{l}^2{C}_p}{del{y}^2}-\frac{2D{C}_p(y)}{R_p^2[\frac{D}{R(y)k}+ln(\frac{R(y)}{R_p})]}=0.=0Yyy$ where $Y$ signifies the

axial distance moved $by$ the reaction front due $to$ diffusion

through the product layer $in$ the $y$ direction. $As$ mentioned

earlier, the extent $of$ reaction resulting from this mechanism

$is$ small except for very short contact times.$=0,$ for $y$

$We$ may proceed $by$ substituting for the second

term $inEq.(7)$ from $Eq.(6)to$ obtain:

$D_n\frac{de{l}^2{C}_p}{del{y}^2}-\frac{2D{C}_p(y)}{R_p^2[\frac{D}{R(y)k}+ln(\frac{R(y)}{R_p})]}=0.$

$We$ also have that $=0,$ for $y$ where $Y$ signifies the

axial distance moved $by$ the reaction front due $to$ diffusion

through the product layer $in$ the $y$ direction. $As$ mentioned

earlier, the extent $of$ reaction resulting from this mechanism

$is$ small except for very short contact times.$y_1$

where

$tan=\frac{2}{L}\sqrt[\phantom{2}]{?}(R^2(y)-(\frac{L}{2}{)}^2)$

and finally

$=0,$ for $y$

$We$ may proceed $by$ substituting for the second

term $inEq.(7)$ from $Eq.(6)to$ obtain:

$D_n\frac{de{l}^2{C}_p}{del{y}^2}-\frac{2D{C}_p(y)}{R_p^2[\frac{D}{R(y)k}+ln(\frac{R(y)}{R_p})]}=0.$

$We$ also have that $=0,$ for $y$ where $Y$ signifies the

axial distance moved $by$ the reaction front due $to$ diffusion

through the product layer $in$ the $y$ direction. $As$ mentioned

earlier, the extent $of$ reaction resulting from this mechanism

$is$ small except for very short contact times.

Kindly help me reproduce equations $(2)$ through to $(14)$