Question 24.5: A dilute mixture of carbon dioxide (CO2) in ethylene (C2H4) ......

Let species A represent CO_{2} with molecular weight of 44 g/mole, and species B represent C_{2}H_{4} with a molecular weight of 28 g/mole. From Example 3, the gas-phase binary molecular diffusion coefficient, D_{A B},\,\mathrm{is}\ 0.077\,\mathrm{cm}^{2}/\mathrm{s} at 2.0 atm total system pressure and 350 K absolute temperature. Given that the diffusion process is within a mesoporous material, which has pore diameters less than 1 0\ μm\ (1.0\ μm = 1.0\times 10^{-6}\ m), Knudsen diffusion may contribute to the molecular mass-transfer process. The Knudsen diffusion coefficient for CO_{2} is estimated by equation (24-58b), with d_{\mathrm{pore}}=2.0\times10^{-5}\,{\mathrm{cm}} and M_{A} =44\ g/gmole:

D_{K A}=4850\,d_{\mathrm{pore}}{\sqrt{\frac{T}{M_{A}}}}        (24-58b)

=4850\,(2.0\times10^{-5}){\sqrt{\frac{350}{44}}}=0.274\,\mathrm{cm}^{2}/s

Consequently, from equation (24-60), for CO_{2} diluted in ethylene, effective diffusion coefficient of species A in a dilute gas mixture within the pore is

{\frac{1}{D_{A e}}}={\frac{1}{D_{A B}}}+{\frac{1}{D_{K A}}}          (24-60)

D_{A e}=\frac{D_{A B}D_{K A}}{D_{A B}+D_{K A}}=\frac{(0.077\ \mathrm{cm}^{2}/s)(0.274\ \mathrm{cm}^{2}/{\mathrm{s}})}{(0.077\ \mathrm{cm}^{2}/{\mathrm{s}})+(0.274\ \mathrm{cm}^{2}/{\mathrm{s}})}=0.060\ \mathrm{cm}^{2}/{\mathrm{s}}

We note that D_{A e}\lt D_{A B} showing the effect of Knudsen diffusion. To this point, the mean-free path and Knudsen number for CO_{2}  inside the pore are

\lambda=\frac{\kappa T}{\sqrt{2}\pi\sigma_{A}^{2}P} = \frac{\left(1.38\times10^{-16}{\frac{\mathrm{erg}}{K}}{\frac{1\,N\,\mathrm{m}}{10^{7}\,\mathrm{erg}}}\right)\left(350\,K\right)}{\sqrt{2}\pi\left(0.3996\,\mathrm{nm}\frac{1\,\mathrm{m}}{10^{9}\mathrm{nm}}\right)^{2}\left(2.0\,\mathrm{atm}\frac{101,300\,\mathrm{N/m^{2}}}{\mathrm{atm}}\right)}

=3.36\times10^{-8}\,{\mathrm{m}}=0.0336\,{\mathrm{\mu m}}

and

K n={\frac{\lambda}{d_{\mathrm{pore}}}}={\frac{0.0336\ \mathrm{\mathrm{\mu m}}}{0.200\ \mathrm{\mathrm{\small{\mu m}}}}}=0.17

From this analysis, Knudsen diffusion plays only a moderate role at the conditions of the process even though the pore diameter is only 0.2 μm. Finally, the effective diffusion coefficient for CO_{2} , as corrected for the void fraction within the random porous material, is

D_{A e}^{\prime}=\epsilon^{2}D_{A e}=(0.45)^{2}(0.060\,\mathrm{cm}^{2}/s)=0.012\,\mathrm{cm}^{2}/s

The effect of total system pressure on the ratio D_{A e}/D_{A B} is shown in Figure 24.6. At high total system pressures, the effective diffusion coefficient (DAe) approaches the molecular diffusion coefficient, D_{A B}. At low total system pressures, Knudsen diffusion becomes important.