Stefan Tube

This example illustrates the use of the Maxwell-Stefan diffusion model available with the Transport of Concentrated Species interface. It models multicomponent gas-phase diffusion in a Stefan tube in 1D. In this case, it is a liquid mixture of acetone and methanol that evaporates into air.

The concentration profiles are modeled at steady-state and validated against experimental data by Taylor and Krishna (Ref. 6).

Model Definition

The Stefan tube, shown in Figure 1, is a simple device used for measuring diffusion coefficients in binary vapors.

Figure 1: Schematic diagram of a Stefan tube.

At the bottom of the tube is a pool of mixture. The vapor that evaporates from this pool diffuses to the top of the tube, where a stream of air, flowing across the top of the tube, keeps the mole fraction of diffusing vapor there to be zero. The mole fraction of vapor above the liquid interface is at equilibrium. Because there is no horizontal flux inside the tube, you can analyze the problem using a 1D model geometry representing the distance between the liquid mixture surface and the top of the tube. The system composition of acetone, methanol, and air has been extensively investigated; both diffusion coefficients and composition have been measured at various positions within Stefan tubes. This makes it an ideal system for this model.

As a comparison, one experiment measured the mole fraction at the liquid interface to be xAc = 0.319 and xMe = 0.528 where the pressure, p, was 99.4 kPa and the temperature, T, was 328.5 K. The length of the diffusion path was 0.238 m. The respective Maxwell-Stefan diffusion coefficients, Dij, of the three binary pairs were calculated and are used in the model according to Table 1.

Table 1: labels and Maxwell-Stefan diffusion coefficients
component	label	Dij	value
Acetone	1	D12	8.48·10-6 m2/s
Methanol	2	D13	13.72·10-6 m2/s
Air	3	D23	19.91·10-6 m2/s

To model this problem, use the Transport of Concentrated Species interface with the Maxwell-Stefan diffusion model. It solves for the fluxes in terms of mass fractions for two of the three components. The mass fraction, ω, of the third is given by the two other ones. The three equations are:

where D is the diffusion coefficient (SI unit: m2/s), p is the pressure (SI unit: Pa), T is the temperature (SI unit: K), u is the velocity (SI unit: m/s), x and ω are mole and mass fractions, respectively, and the mixture density, ρmix (SI unit: kg/m3), is a function of the average mixture mole fraction, Mmix (SI unit: kg/mol), according to Equation 2:

(1)

(2)

In this case, there is no imposed fluid velocity. However, there will appear a fluid velocity due to the diffusive fluxes. At the top of the tube the mass fractions are fixed, with the fraction of air being unity. At the bottom (at the liquid interface), the fractions are also fixed according to the previously mentioned experimental conditions. The fact that there is no air flux at the interface results in the following relation for the convective velocity, at steady state:

where ndiff,3 is the diffusive mass flux of air (SI unit: kg/(m2·s)).

Results

Both the modeled and experimental steady-state mole fractions as a function of position are shown in Figure 2.

Figure 2: Modeled and experimental (Ref. 6) steady-state mole fractions of: acetone, methanol, and air, in the Stefan tube.

We can see that the model reproduces the results from Ref. 6 well, which means the Maxwell-Stefan equations can describe the mass transport process in the system accurately.

The Maxwell-Stefan diffusion formulation includes the conservation of mass. In the absence of chemical reactions (source terms) and convective contributions, the Maxwell-Stefan formulation results in zero net mass flux. In this example, the convective term is included, which you can see in the velocity profile in Figure 3.