Question 24.2: Evaluate the diffusion coefficient of carbon dioxide in air ...

From Appendix Table K.2, the values of s and e / k are obtained

\begin{array}{lll} & \boldsymbol{\sigma}, \textbf{ in } \overset{\circ }{\textbf{A}} & \varepsilon_{A} / \kappa, \textbf{ in } \mathbf{K} \\ \text { Carbon dioxide } & 3.996 & 190 \\ \text { Air } & 3.617 & 97\end{array}

The various parameters for equation (24-33)

D_{A B}=\frac{0.001858F^{3/2}\left[\frac{1}{M_{A}}+\frac{1}{M_{B}}\right]^{1/2}}{P\sigma_{A B}^{2}\Omega_{D}}       (24-33)

may be evaluated as follows:

\begin{aligned} \sigma_{A B} & =\frac{\sigma_{A}+\sigma_{B}}{2}=\frac{3.996+3.617}{2}=3.806  \overset{\circ }{\text{A}} \\ \varepsilon_{A B} / \kappa & =\sqrt{\left(\varepsilon_{A} / \kappa\right)\left(\varepsilon_{B} / \kappa\right)}=\sqrt{(190)(97)}=136 \\ T & =20+273=293 \mathrm{~K} \\ P & =1 \mathrm{~atm} \\ \frac{\varepsilon_{A B}}{\kappa T} & =\frac{136}{293}=0.463 \\ \frac{\kappa T}{\varepsilon_{A B}} & =2.16 \\ \Omega_{D}(\text { Table } \mathrm{K} .1) & =1.047 \\ M_{\mathrm{CO}_{2}} & =44 \end{aligned}

and

M_{\text {Air }}=29

Substituting these values into equation (24-33), we obtain

\begin{aligned} D_{A B} & =\frac{0.001858 T^{3 / 2}\left(1 / M_{A}+1 / M_{B}\right)^{1 / 2}}{P \sigma_{A B}^{2} \Omega_{D}} \\ & =\frac{(0.001858)(293)^{3 / 2}(1 / 44+1 / 29)^{1 / 2}}{(1)(3.806)^{2}(1.047)}=0.147 \mathrm{~cm}^{2} / \mathrm{s} \end{aligned}

From Appendix Table J.1 for \mathrm{CO}_{2} in air at 273 \mathrm{~K}, 1 atm, we have

D_{A B}=0.136 \mathrm{~cm}^{2} / \mathrm{s}

Equation (24-41)

D_{A B_{T_2, P_1}}=D_{A B_{T_1}, p_1}\left(\frac{P_1}{P_2}\right)\left(\frac{T_2}{T_1}\right)^{3 / 2} \frac{\left.\Omega_D\right|_{T_1}}{\left.\Omega_D\right|_{T_2}}       (24-41)

will be used to correct for the differences in temperature

\frac{D_{A B, T_{1}}}{D_{A B, T_{2}}}=\left(\frac{T_{1}}{T_{2}}\right)^{3 / 2}\left(\frac{\Omega_{D} \mid T_{2}}{\Omega_{D} \mid T_{1}}\right)

Values for \Omega_{D} may be evaluated as follows:

\begin{array}{ll} \text { at } T_{2}=273 & \varepsilon_{A B} / \kappa T=\frac{136}{273}=\left.0.498 \quad \Omega_{D}\right|_{T_{2}}=1.074 \\ \text { at } T_{1}=293 & \left.\Omega_{D}\right|_{T_{1}}=1.074 \quad \text { (previous calculations) } \end{array}

The corrected value for the diffusion coefficient at 20^{\circ} \mathrm{C} is

D_{A B, T_{1}}=\left(\frac{293}{273}\right)^{3 / 2}\left(\frac{1.074}{1.047}\right)(0.136)=0.155 \mathrm{~cm}^{2} / \mathrm{s} \quad\left(1.55 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\right)