Consider the isothermal reaction in a plate$-$type catalyst with thickness $2$a with nonlinear

kinetics. External mass transfer is not to be neglected.

hat$(r)(c)=\frac{(hat(k))c}{(1+c{)}^2},u-=\frac{c}{c_b}$;$R(u)-=\frac{(hat(r))(c)}{(hat(r))(c_b)}$

where $c_b$ is the concentration in the bulk $($i$.$e$.,$ at the outer edge of the film around the

particle$).$

Obtain the transport equation below

;$,u-=\frac{c}{c_b},{}^2=\frac{k{a}^2}{D_e(1+c_b{)}^2},$ where $k-={}_b{S}_g$hat$(k)$

When mass transfer at the pellet surface is important, the boundary condition at the surface

becomes

$[\frac{du}{d}{]}_{}=1=$

$u(1-u_1),u_1-=\frac{c(a)}{c_b},$

$u,$ Biot number $-=\frac{k_{c}a}{D_e},k_c-=$ mass transfer coefficient

It is desired to develop a chart of effectiveness factor $$ vs$.$ for various values of $v.$

$($i$)$ Show how you will extend the development in class to the above case to get

$F(u)-={\displaystyle \underset{0}{\overset{u}{}}}R(x)dx=\frac{(1+c_b{)}^2}{c_b}[ln(1+c_{b}u)+\frac{1}{1+c_{b}u}-1]$

$v(1-u_1)=\sqrt[\phantom{2}]{2[F(u_1)-F(u_o)]}=v_1,v_1-=v(1)$

where $u_o-=u(0)$ is $($clearly$)$ unknown.