Diffusion through a non-isothermal spherical film

Input(s)

P: Pressure (atm)

$\boldsymbol{D}_{\boldsymbol{A} \boldsymbol{B}}$: Diffusivity $\left(\mathrm{cm}^{2} / \mathrm{s}\right)$

R: Gas Constant $(\mathrm{J} / \mathrm{mol} \mathrm{K})$

$\boldsymbol{T}_{1}$: Temperature (K)

$\mathrm{n}$: Exponent (dimensionless)

$r_{1}$: Radius of Gas Film $(\mathrm{cm})$

$\boldsymbol{r}_{\mathbf{2}}$: Radius of Film $(\mathrm{cm})$

$\boldsymbol{x}_{\boldsymbol{A} 1}$: Position of Gas (fraction)

$x_{A 2}$: Position of Film (fraction)

Output(s)

$\boldsymbol{W}_{\boldsymbol{A}}$: Mass Rate $(\mathrm{mol} / \mathrm{s})$

Formula(s)

$$ \mathrm{W}_{\mathrm{A}}=\frac{4 * \pi *\left(\frac{\mathrm{P} * \mathrm{D}_{\mathrm{AB}}}{\mathrm{R} * \mathrm{~T}_{1}}\right) *\left(1+\left(\frac{\mathrm{n}}{2}\right)\right)}{\left(\mathrm{r}_{1}^{\frac{\mathrm{n}}{2}}\right) *\left(\left(\frac{1}{\mathrm{r}_{1}}\right)^{1+\frac{\mathrm{n}}{2}}-\left(\frac{1}{\mathrm{r}_{2}}\right)^{1+\frac{\mathrm{n}}{2}}\right)} * \ln \left(\frac{1-\mathrm{x}_{\mathrm{A} 2}}{1-\mathrm{x}_{\mathrm{A} 1}}\right) $$

Reference(s)

Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley \& Sons, Chapter: 18, Page: 551 .