Diffusion Coefficient: Measurement And Prediction | Mass Transfer - Chemical Engineering PDF Download

2.4 DIFFUSION COEFFICIENT: MEASUREMENT AND PREDICTION
The proportionality factor of Fick’s law is called diffusivity or diffusion coefficient which can be defined as the ratio of the flux to its concentration gradient and its unit is m2/s. It is a function of the temperature, pressure, nature and concentration of other constituents. Diffusivity decreases with increase in pressure ( D_AB ∝1/ p for moderate ranges of pressures, upto 25 atm) because number of collisions between species is less at lower pressure. But the diffusivity is hardly dependent on pressure in case of liquid. The diffusivity increases with increase in temperature ( D_AB ∝T^1.5 ) because random thermal movement of molecules increases with increase in temperature. The diffusivity is generally higher for gases (in the range of 0.5×10^–5 to 1.0 × 10^-5 m²/s) than for liquids (in the range of 10^–10 to 10^-9 m²/s). The diffusivity value reported for solids is higher in the range of 10^–13 to 10^-5 m²/s. Diffusion is almost impossible in solids because the particles are too closely packed and strongly held together with no ‘empty space’ for particles to move through. Solids diffuse much slower than liquids because intermolecular forces in solid are stronger enough to hold the solid molecules together.

2.4.1 Measurement of gas-phase diffusion coefficient 

There are several methods of experimental determination of gas-phase diffusion coefficient. Two methods are (a) Twin-bulb method and (b) Stefan tube method. Predictive Equations are sometimes used to determine diffusivity. These may be empirical, theoretical or semi-empirical.

(a) Twin-bulb method 
Two large bulbs are connected by a narrow tube. The schematic representation is shown in Figure 2.6. In the beginning two bulbs are evacuated and all the three valves
[V1, V2 and V3] are kept closed. Then V2 is opened and bulb 1 is filled with pure A at a pressure P. After that V3 is opened and bulb 2 is filled with pure B at the same pressure P. At steady state
                            (2.37)
where, a is cross sectional area of the connecting tube of length l. If p_A1 and p_A2 are partial pressures of A in two bulbs at any time,
                                                 (2.38)
and                                              (2.39)

From Equations (2.38) and (2.39) we have
                     (2.40)
         (2.41)
Boundary conditions: 

applying the above boundary conditions, Equation (2.41) is integrated to obtain the expression of D_AB as follows:
                          (2.42)

Figure 2.6: A schematic of the twin bulb apparatus

(b) Stefan tube method 
Stefan tube consists of a T tube made of glass, placed in a constant temperature water bath. Air pump is used to supply the air, passed through the T tube as shown in Figure 2.7. Volatile component is filled in the T tube and air passed over it by the pump and change in the level is observed by the sliding microscope. Let, at any time t, partial pressure of A at the Z distance from the top of the vertical tube is p_A1 and that at the top it is p_A2. The diffusional flux of A is given as:
                           (2.43)
The rate of evaporation is given by
                      (2.44)

Boundary conditions:
t=0; Z=Z₀ 
t=t^/ ; Z=Z^/, 
Integration of Equation (2.44) using the above boundary conditions gives, 
                                 (2.45)
where, partial pressure of a at liquid surface, p_A1 is equal to vapor pressure at the same temperature. The partial pressure of A at the top of the vertical tube, p_A2 is zero due to high flow rate of B.

Figure 2.7: A representation of the Stafan tube

(c)Predictisve Equations:
(I) Empirical: Fuller, Schettler and Giddings
                   (2.46)
where, T is temperature in K
M_A, M_B are molecular weights of A and B
P is total pressure in bar
ν_A, ν_A are atomic diffusion volume in m³ .

(II) Theoretical: Chapman-Enskog Equation
                                      (2.47)
where, σ_AB is characteristic length parameter of binary mixture in Å, Ω_D is collision integral=f(kT/ε_AB)
                               (2.48)