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\textdegreeC 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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