Question 24.3: A new gas separation process is being developed to separate ......

For the \mathrm{CO_{2}- e t h y l e n e} binary pair, let species A represent CO_{2} with molecular weight of 44 g/mole, and species B represent ethylene with molecular weight of 28 g/mole. Let us first use the Hirschfelder equation. From Appendix K, Table K.2, the Lennard–Jones constants needed for the Hirschfelder equation are \sigma_{A}=3.996\ \mathring{A} , \sigma_{B}=4.232\ \mathring{A} , \epsilon_{A}/\kappa\equiv190\ K,\,\epsilon_{B}/\kappa\equiv205\ \mathrm{K}.

Consequently,

\sigma_{A B}={\frac{\sigma_{A}+\sigma_{B}}{2}}={\frac{3.996\,{\mathring{\mathrm{A}}}+4.232\,{\mathring{\mathrm{A}}}}{2}}=4.114\,{\mathring{\mathrm{A}}}

{\frac{\kappa T}{\epsilon_{A B}}}=\left({\frac{\kappa}{\epsilon_{A}}}{\frac{\kappa}{\epsilon_{A}}}\right)^{1/2}T=\left({\frac{1}{190\,\mathrm{K}}}{\frac{1}{205\,\mathrm{K}}}\right)^{1/2}(350\,\mathrm{K})=1.77

From Appendix K, Table K.1, \Omega_{D}=1.123. Therefore,

D_{A B}=\frac{~0.001858~T^{3/2}\left(\frac{1}{M_{A}}+\frac{1}{M_{B}}\right)^{1/2}}{~P\sigma_{A B}^{2}\Omega_{D}}={\frac{\mathrm{0.001858~}(350)^{3/2}\left({\frac{1}{44}}+{\frac{1}{28}}\right)^{1/2}}{(2.0)(4.114)^{2}(1.123)}}=0.077\,\,\mathrm{cm^{2}/s}

Now, let us compare the Hirschfelder correlation with the Fuller–Schettler–Giddings correlation. From Table 24.3, the atomic diffusion volume for CO_2 is 26.9; the atomic diffusion volume for ethylene (C_{2}H_{4}) is estimated the group contribution method using C and H building blocks also given in Table 24.3:

(\Sigma n_{i})_{B}=2\cdot n_{C}+4\cdot n_{H}=2(16.5)+4(1.98)=40.92

Therefore, the diffusion coefficient estimated by the Fuller–Schettler–Giddings correlation is

D_{A B}=\frac{\mathrm{0.001~}T^{1.75}\left(\frac{1}{M_{A}}+\frac{1}{M_{B}}\right)^{1/2}}{P\Big[(\Sigma n_{i})_{A}^{1/3}+(\Sigma n_{i})_{B}^{1/3}\Big]^{2}} = \frac{0.001\left(350\right)^{1.75}\left(\frac{1}{44}+\frac{1}{28}\right)^{1/2}}{\left(2.0\right)\left[\left(26.9\right)^{1/3}+\left(40.92\right)^{1/3}\right]^{2}}=0.082\,\mathrm{cm}^{2}/s

The two correlations agree within 7%.

For the CO-C_{2}H_{4} binary pair, with A CO , and B ethylene (C_{2}H_{4}) , from Appendix J, Table J.1, the measured diffusion coefficient is { D}_{A B}=0.151\,\mathrm{cm}^{2}/s at 1.0 atm and 273 K. By the Hirschfelder extrapolation, the diffusion coefficient at 2.0 atm and 350 K is

D_{A B}(T,P)=D_{A B}(T_{o},P_{o})\left(\frac{P_{o}}{P}\right)\left(\frac{T}{T_{o}}\right)^{3/2}\frac{\Omega_{D}(T_{o})}{\Omega_{D}(T)}

=\left(0.151{\frac{\mathrm{cm}^{2}}{\mathrm{s}}}\right)\left({\frac{1.0\,\mathrm{atm}}{2.0\,\mathrm{atm}}}\right)\left({\frac{350\,\mathrm{K}}{273\,\mathrm{K}}}\right)^{3/2}\left({\frac{1.112}{1.022}}\right)=0.119\,\mathrm{cm}^{2}/\mathrm{s}.