Diffusion Through Variable Cross Sectional Area | Mass Transfer - Chemical Engineering PDF Download

2.3 Diffusion through variable cross-sectional area
2.3.1 Diffusion through a conduit of non-uniform cross-sectional area
Consider a component A is diffusing at steady state through a circular conduit which is tapered uniformly as shown in Figure 2.3. At point 1 the radius is r₁ and at point 2 it is r₂. At position Z in the conduit, A is diffusing through stagnant, nondiffusing B.

Figure 2.3: Schematic of diffusion of A through a uniformly tapered tube
At position Z the flux can be written as
                              (2.23)
Using the geometry as shown, the variable radius r can be related to position z in the path as follows: 
                                                  (2.24)
The Equation (2.24) is then substituted in the flux Equation to eliminate r and then the Equation is integrated and obtained as:

Or

                          (2.25)

2.3.2 Evaporation of water from metal tube
Suppose water in the bottom of a narrow metal tube is held at constant temperature T. The total pressure of air (assumed dry) is P and the temperature is T. Water evaporates and diffuses through the air. At a given time t, the level is Z meter from the top as shown in Figure 2.4. As diffusion proceeds, the level drops slowly. At any time t, the steady state Equation holds, but the path length is Z.

Figure 2.4: Schematic of evaporation in metal tube
 

Thus the steady state Equation becomes as follows where NA and Z are variables:
                                   (2.26)
Where
           (2.27)
Assuming a cross sectional area of 1 m² , the level drops dZ meter in dt sec, and ρ_A(dZ.1)/M_A is the kmol of A that has been left and diffused. Then
                                                (2.28)
Substituting the Equation (2.28) in Equation (2.26) and integrating, one gets
                      (2.29)
                                (2.30)
The Equation (2.30) represents the time t_F for the level to drop from a starting point of Z_o meter at t = 0 to Z_F at t = t_F.

2.3.3 Diffusion from a sphere 
There are lots of examples where diffusion can take place through the spherical shape bodies. Some examples are:
1. Evaporation of a drop of liquid 
2. The evaporation of a ball of naphthalene
3. The diffusion of nutrients to a sphere-like microorganism in a liquid

Assume a constant number of moles Ñ_A of A from a sphere (area = 4πr² ) through stagnant B as shown in Figure 2.5.

Figure 2.5: Schematic of diffusion from a sphere
 

From the Fick’s law of diffusion, the rate of diffusion can be expressed as:
                                 (2.31)
where
                                                      (2.32)
                       (2.33)
Integrating with limits of P_A2 at r₂ and P_A1 at r₁ gives:
             (2.34)

As r₁ << r₂, then 1/r2 ≈ 0:
  the flux at the surface                  (2.35)
This Equation can be simplified if P_A1 is small compared to P (a dilute gas phase), P_BM ≈ P. Also setting 2r₁ = D₁, diameter, C_A1 = P_A1/RT
                                  (2.36)