# Chapter 24: Fundamentals of Mass Transfer. Example 2

Welty, James R.; Wicks, Charles E.; Wilson, Robert E. Fundamentals of Momentum, Heat, and Mass Transfer, third edition. New York: John Wiley and Sons.

Example 2

Evaluate the diffusion coefficient of carbon dioxide in air at 20$\textdegree$C and atmospheric pressure. Compare this value with the experimental value reported in Appendix Table J.1.

Will be using the following diffusivity equation:

$D_{AB} = \frac{0.001858 T^{3/2} (\frac{1}{M_A} + \frac{1}{M_B})^{1/2}}{P \delta_{AB}^2 \Omega_D}$

We have temperature and pressure. We can calculate the molecular weights via a periodic chart. $\delta$ and $\Omega$ can be obtained from Tables K.1 and K.2.

From K.2 of the appendix values $\delta$ and $\frac{\epsilon}{\kappa}$ are obtained:

Carbon dioxide: $\delta$ in $\AA$, 3.996 and $\frac{\epsilon_{CO_2}}{\kappa}$ in K, 190

Air: $\delta$ in $\AA$, 3.617 and $\frac{\epsilon_{N_2}}{\kappa}$ in K, 97

$\delta_{AB} = \frac{(\delta_A + \delta_B)}{2} = \frac{(3.996 \AA + 3.617 \AA)}{2} = 3.806 \AA$

$\frac{\epsilon_{AB}}{\kappa}= \sqrt{(\frac{\epsilon_{A}}{\kappa})(\frac{\epsilon_B}{\kappa}}) = \sqrt{(190)(97)} = 136$

T = 20 + 273 = 293 K

P = 1 atm

$\frac{\epsilon_{AB}}{\kappa T} = \frac{136}{293} = 0.463$

$\frac{\kappa T}{\epsilon_{AB}} = \frac{1}{0.463} = 2.16$

$\Omega_D (Table K.1) = 1.047$

This value was obtained by interpolation

$\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}$

$y - y_0 = \frac{y_1 - y_0}{x_1 - x_0}(x - x_0)$

$y = \frac{y_1 - y_0}{x_1 - x_0}(x - x_0) + y_0$

From Table K.1

$y_i = \Omega$ and $x_i = \frac{\kappa T}{\epsilon_{AB}}$

and

$x = 2.16$

Interpolate

$y = \frac{(1.041-1.057)}{(2.20-2.10)}(2.16 - 2.10) + 1.057 = 1.047 = \Omega_D$

We have all variable except molecular weights. Considering the most prevalent gasses:

$M_{CO_2} = M_C + 2M_O = 12 + 2(16) = 12 + 32 = 44$

$M_{Air} = \%N_2 (M_{N_2}) + \%O_2(M_{O_2}) = 0.79(28) + 0.21(32) = 29$

Now, we have all the information needed to calculate the diffusivity of $CO_2$ in air when using:

$D_{AB} = \frac{0.001858 T^{3/2} (\frac{1}{M_A} + \frac{1}{M_B})^{1/2}}{P \delta_{AB}^2 \Omega_D}$

$D_{AB} = \frac{0.001858(293^{3/2})(\frac{1}{44} + \frac{1}{29})^{1/2}}{1 atm(3.806 \AA)^2 (1.047)} = 0.147 \frac{cm^2}{s}$

Now, we want to compare to the experimental value that is reported in Table J.1

$T, K = 273, D_{AB}P \frac{cm^2 atm}{s} = \frac{0.136 \frac{cm^2 atm}{s}}{1 atm} = 0.136 \frac{cm^2}{s}$

Since the value is reported at 273 K, must use a conversion equation to compare at 293 K

$\frac{D_{AB,T_1}}{D_{AB,T_2}} = (\frac{T_1}{T_2})^{3/2}(\frac{\Omega_{D,T_2}}{\Omega_{D,T_1}})$

at $T_1 = 293 K$ and $\Omega_{D, T_1} = 1.047$

at $T_2 = 273 K$ and $\Omega_{D, T_2} = ?$ from Table Table K.1

$\frac{\epsilon_AB}{\kappa}\frac{1}{T_2} = 136\frac{1}{273} = 0.498$

$\frac{\kappa T}{\epsilon_{AB}} = \frac{1}{0.498} = 2.01$

Once again, interpolation of Table K.1 is needed

$\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}$

$y = \frac{y_1 - y_0}{x_1 - x_0}(x-x_0) + y_0$

$x = 2.01$

$y = \frac{1.057 -1.075}{2.10-2.00}(2.01-2.00) + 1.075 = 1.074 = \Omega_{D, T_2}$

Since we have all the values for the conversion equation

$D_{AB, 293} = (\frac{293}{273})^{3/2}(\frac{1.074}{1.047})(0.136) = 0.155 \frac{cm^2}{s}$

The diffusivity of carbon dioxide in air

Calculated: $0.147 \frac{cm^2}{s}$ and Corrected Experimental: $0.155 \frac{cm^2}{s}$

Percent Difference

$\frac{Calculated - Corrected Experimental}{Corrected Experimental} x 100= \frac{0.147 - 0.155}{0.155} x 100=5.16\%$

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