Thiele modulus for spherical particles for first$-$order irreversible reaction is given by:

${}_s=\frac{R}{3}\sqrt[\phantom{2}]{\frac{k_1{}_p}{D_e}}$

Where, $R$ is the particle radius, ${}_p$ is the particle density, $k_1$ is the rate constant, and $D_e$ is the

effective diffusivity $($the constant $,3,$ can be joined with the rate constant or left out of the square

root sign, but let's keep it out$).$

For reversible first$-$order reaction $(AharrB),$ the surface reaction rate law is given by:

$r_{\text{z}\text{u}\text{r}\text{f}\text{a}\text{c}\text{e}}=\frac{k_1(K+1)}{K}(C_A-C_{A.\text{e}\text{q}\text{u}\text{l}\text{t}\text{b}\text{r}\text{i}\text{m}\text{m}})$

Where, $k_1$ is the forward reaction rate constant, $K$ is the equilibrium constant.

Thiele modulus for spherical particles in reversible first$-$order reaction is defined as:

${}_{s,\text{r}\text{e}\text{v}\text{e}\text{r}\text{a}\text{i}\text{b}\text{l}\text{e}}=\frac{R}{3}\sqrt[\phantom{2}]{\frac{k_1(K+1){}_p}{K{D}_e}}$

The internal effectiveness factor for both cases can be defined as:

$=\frac{1}{{}_s}(\frac{1}{tanh3{}_s}-\frac{1}{3{}_k})$

I. For the same reaction conditions, how would you compare the effectiveness factors and

Thiele modulus for the two cases.

II$.$ For the reaction AharrB that is carried out at $3800mmHg$ and $323\mathrm{.}15K$ in a fixed bed

particles density is assumed to be $1000k\frac{g}{m^3}.$ The measured overall rates when pure A

is used to contain the solid particles of diameters of $3\mathrm{.}175mm,6\mathrm{.}35mm,$ and $9\mathrm{.}525mm$

are $0\mathrm{.}000485,0\mathrm{.}000401,$ and $0\mathrm{.}000354$kmol$\frac{e}{k}$gcatalyst.$s,$ respectively. Assume there

are no external mass transfer resistances, and follow Wiesz criterion for internal mass

transfer diffusion that if Thiele modulus is less than or equal one third, then the internal

mass transfer diffusion resistance can be neglected, determine:

a$.$ The largest particle size $($among the given above$)$ to be used with no internal

mass transfer resistance.

b$.$ for the different provided particle sizes.