Given system:

$\backslash$epsi $($dc$)/($dt$)=($D$\_($e$))/($r$^(2))($d$)/($dr$)($r$^(2)($dc$)/($dr$))-$S$($c$,$T$)$

$\backslash$rho C$\_($S$)($dT$)/($dt$)=($K$)/($r$^(2))($d$)/($dr$)($r$^(2)($dT$)/($dr$))+(-\backslash$Delta H$)$S$($c$,$T$)$

The initial conditions are given by:

c$(0,$r$)=$c$\_(0)$;T$(0,$r$)=$T$\_(0)$

The boundary conditions at the external boundary of the pill r$=$r$\_(0)$ are given by:

c$($t$,$r$\_(0))=$c$\_(0)$;T$($t$,$r$\_(0))=$T$\_(0)$

The boundary conditions at the center of the pill are.

$($dc$)/($dr$)|\_($r$)=0=0$;$($dT$)/($dr$)|\_($r$)=0=0$

$1.$ Introduce the dimensionless concentration $Y=\frac{c}{c_0},$ the dimensionless temperature $=\frac{T}{T_0},$ and

the dimensionless time$-$ and position variables $=D_e\frac{t}{r_0^2}$ and $z=\frac{r}{r_0}.$ Show that the equations

for temperature and concentration in the dimensionless form become:

$\frac{dY}{d}=\frac{1}{z^2}\frac{d}{dz}(z^2\frac{dY}{dz})-{}^2$Yexp$((1-\frac{1}{}))$

$Le\frac{d}{d}=\frac{1}{z^2}\frac{d}{dz}(z^2\frac{d}{dz})+{}^2$Yexp$((1-\frac{1}{}))$;

The initial and boundary conditions become:

$Y(0,z)=Y(,1)=1$;

$(0,z)=(,1)=1$;

$\frac{dY}{dz}(,0)=\frac{d}{dz}(,0)=0$

Provide expressions for the dimensionless parameters $Le($the Lewis number$){?}^1,($the Thiele modulus $-$explain it$!),$ and $.$