Case1: Loschmidt tube

Validation of our solver for diffusion dominated flows involving mass transfer is undertaken in this section. Such flow scenarios are quite often validated by using diffusion tubes [27, 25, 23]. We use the Loschmidt tube for validation.

Duncan and Toor conducted a few experiments to examine diffusion in an ideal ternary gas mixture containing Hydrogen, Nitrogen and Carbon-di-oxide [24]. It basically consists of two bulbs connected by a capillary as shown in figure 5.1. Among the three gases, one gas is made stationary and two gases are set in motion. This is achieved by adding almost equal quantities of one gas in both the bulbs (q2) resulting in a very small or in fact a negligible gradient of the gas rendering it almost stationary. The remaining two gases are added one in each bulb resulting in large gradients thus driving the two gases in motion when the stop cock is removed in order to achieve homogeniety.In our simulation, Loschmidt tube with Methane (CH4), Argon (Ar) and Hydrogen (H2) is simulated.

q₁ q₂ q₃+

q₁+ q₂ q₃

-Bulb 1 Bulb 2

Figure 5.1: Experimetal set up of loschmidt tube

In such a scenario, the gas with a small gradient (q2) behaves in a certain way. Toor predicted the behaviour and named them; Osmotic diffusion, Reverse diffusion and Diffusion barrier [10]. It is important to elaborate these terms which occur in a ternary mixture.

1. Osmotic diffusion: At time t = a , though the gas has negligible driving force, it moves.

2. Reverse Diffusion: During a time interval a < t < b, the gas moves in a direction opposite to its existing trifle gradient thus defying Fick’s law which states that, gases diffuse in a direction normal to their concentration gradients.

3. Diffusion barrier: Between b < t < c, the gas does not diffuse even though a large gradient exists. The diffusion flux is zero despite a large driving force [24]

The experimental results have been compared with the Maxwell-Stefan equations by Duncan and Toor [12] and is in good agreement. Hence the analytical solution of the Maxwell-Stefan equation is obtained by assuming a 1−D flow and verified with the results generated by the solver.

5.1.1 Case Setup

In order to simulate the Loschmidt tube, the whole process has to be abstracted to a simple model. The first abstraction done is the consideration of only the capillary and eliminating the two bulbs. Since the whole system is closed and entites of interest are the phase fractions, it is justified. The geometry used for simulation is shown in figure 5.2

Wall

Left tube

-  -

Figure 5.2: Simulation model of loschmidt tube

The length of the capillary is reduced to 100µm. High gradients prevail initially at the division of left tube and right tube. In order to resolve them, edge lengths of 10⁻⁶ are required [27]. To save computation time, the length of the whole capillary is reduced. The experiment is conducted at atmospheric pressure of 101.3kP a and under isothermal conditions. Composition of the three gases in both left and right tube is given in table 5.1

It is observed that Ar behaves as gas q2explained earlier and must undergo the stated osmotic diffusion, reverse diffusion and diffusion barrier effects. All the boundary patches are considered as walls ensuring a closed surface. The internal field is set to 101.3kP a. The pressure at the walls are set to zero gradient condition. Zero gradient means that there is no change of a quantity in normal direction. Velocity

CH4 0.295 0.405

H2 0.4 0.3

Ar 0.305 0.295

Table 5.1: Initial composition

at the wall is set to slip condition for all components. The slip condition ensures that the tangential component of velocity is zero gradient as explained earlier and the normal velocity component near the wall is zero [5].

An analytical solution to the problem assuming a 1−D transient problem is devel-oped in [25] (pages 110 − 115). A sci-lab program written based on the solution method which solves the phase fraction of a three component mixture from its ini-tial composition for any time t in the left tube can be found in appendix A.1. A graph is generated for each component using the program to plot the variation of its phase fraction with time.

0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 calculated alphaAr

analytical alphaAr b c

a

Figure 5.3: Phase fraction ofArgon in left tube: Analytical vs. Calculated a − b : Reverse diffusion. b − c :Diffusion barrier. > c :Normal diffusion

Figure 5.3 shows the behaviour of Argon. From time a until time b, reverse diffusion takes place. Though the left tube has a higher concentration, there is an incoming flux to the left tube(0.305) from right tube(0.295) and hence an increase and decrease of phase fraction on left and right respectively. Between time b and c, a diffusion barrier exists resulting in stagnation of flux even though a considerable gradient exists. After c, Argon diffuses normally. This assures the physical reality of solution.

The other two component gases begin to start homogenizing following the Fick’s law of diffusion and diffuse normally, down their concentration gradients. The animation of the intial stage of the other all the gases can be seen in appendix A.2

0.29 0.3 0.31 0.32 0.33 0.34 0.35

0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 calculated alphaCH4

analytical alphaCH4

Figure 5.4: Phase fraction of CH4 in left tube : Analytical vs. Calculated Methane behaves normally and obeys Fick’s law of diffusion. The initial composition of Methane in the left tube was comparatively lower. Hence phase fraction gradually increases until homogenization as shown in figure 5.4.

0.35 0.355 0.36 0.365 0.37 0.375 0.38 0.385 0.39 0.395 0.4

0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 calculated alphaH2

analytical alphaH2

Figure 5.5: Phase fraction of H2 in left tube: Analytical vs. Calculated There is not much of a surprise in case of Hydrogen too. It diffuses normally down its gradient. The initial composition of Hydrogen being high in left tube in compar-ison to right tube reduces in order to effect the homogenization process as seen in figure 5.5. The end time shown in the graph is not the time required to completely homogenize. There is no further change in the behaviour till the homogenization of both tubes.

phase fraction of all three components calculated from the solver is in good agree-ment. Thus the solver is validated for diffusion dominated flows involving mass transfer.