The Loschmidt Tube

An analysis of multicomponent diffusion in a Loschmidt tube was presented by Arnold and Toor (1967). The salient results of their work are summarized below. 

The equation governing unsteady-state, one-dimensional, multicomponent diffusion in the Loschmidt tube is 

The initial condition states that the concentration in each tube is uniform 

After the tubes are brought together diffusion proceeds, but since the tubes are sealed and are of finite length there can be no mass transfer across the ends of the tube the boundary conditions therefore are 

Before presenting the solution to the multicomponent diffusion problem we note that for binary systems the differential Eq. 5.5.1 simplifies to 

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Another device used to study diffusion and to measure diffusion coefficients is the Loschmidt tube illustrated in Figure 5.6. Two tubes containing fluids with different concentrations are brought together at time t = 0 and the fluids allowed to interdiffuse. After some time the tubes are separated and the compositions measured. 

Example 5.5.1 Multicomponent Diffusion in the Loschmidt Tube Another Test... 

Equation 6.2.3 has exactly the same form as Eq. 5.1.3 for binary systems. This means that we may immediately write down the solution to a multicomponent diffusion problem if we know the solution to the corresponding binary diffusion problem simply by replacing the binary diffusivity by the effective diffusivity. We illustrate the use of the effective diffusivity by reexamining the three applications of the linearized theory from Chapter 5 diffusion in the two bulb diffusion cell, in the Loschmidt tube, and in the batch extraction cell. 

SOLUTION As in the preceding example, none of the simple effective diffusivity formulas are applicable to the situation in the Loschmidt tube. We will proceed with the effective diffusivity formula of Wilke that gives... 

The Loschmidt tube is simply a tube with an impermeable partition separating the two sections of the tube (Figure 8.3-1). Initially, the partition is in the position that gases in the two sections do not mix with each other. Here we shall assume that the total pressure is the same in both sections of the tube, and the initial compositions are different in the two parts of the tube. At time t = 0, the partition is removed and the diffusion process is started. 

Multiplying the above equation by Ac, where A is the cross-sectional area of the Loschmidt tube and c (the total molar concentration), we obtain what is known as the overall mass balance equation ... 

The constitutive Maxwell-Stefan flux equations are the same as those presented in the last section (Section 8.3) because in this case to maintain the constant pressure of the closed system the sum of all fluxes must be zero, the same requirement as that in the Loschmidt tube. The flux equations are given by eqs. (8.3-7) with the matrix B given by eq. (8.3-6). 

For the purpose of programming this problem in MATLAB language, we define the following concentration matrices for the two sections of the Loschmidt tube. 

Modified Loschmidt apparatus, to be used according to procedure B. Stopcock A should have a very large bore which is as close as possible to the diameter of the diffusion tube. Stopcocks Bto should be high-vacuum type. As a safety precaution, each gas lecture bottle should be connected to the system via a reducing valve or a pressure-relief valve. 

Arnold and Toor (1967) investigated diffusional interaction effects in a Loschmidt tube of the kind described above. The system they used was methane (l)-argon (2)-hydrogen (3). The diffusion tube had a length of = 60 m . At the temperature (34°C) and... 

Our task here is to derive an expression that describes how the composition of a multicomponent mixture changes with time in a Loschmidt diffusion apparatus of the kind described in Section 5.5. The composition profile for a binary system is given by Eqs. 5.5.5 and 5.5.6) the solution to the binarylike multicomponent problem is given by the same expressions on replacing the binary diffusivity in those equations by the effective diffusivity. The average composition in the bottom tube after time Z, for example, is given by... 

Figure 6.4. Comparison between Loschmidt tube experiments of Arnold and Toor (1967) and the composition trajectories predicted by the linearized theory and effective diffusivity methods.

In an isobaric closed system like the diffusion cell or Loschmidt tube (see Chapter 5), any... 

Section 8.2.6 shows the usage of the Stefan-Maxwell equations for the steady state analysis of the Stefan tube, in which we have shown the elegance of the vector-matrix presentation in dealing with steady state diffusion problem. Now we will show the application of the constitutive Stefan-Maxwell equation to an unsteady state problem. Here we shall take a transient diffusion problem of a Loschmidt s tube, which is commonly used in the study of diffusion coefficient. 

Figure 8.3-1 Schematic diagram of the Loschmidt s tube 8.3.1 The Mass Balance Equations...

Setting up the mass balance equation in either the sections of the Loschmidt s tube, we obtain the following equation for the conservation of mass... 

To illustrate the Stefan-Maxwell approach in the analysis of the Loschmidt diffusion tube, we apply it to the experimental results obtained by Arnold and Toor (1967). The system is a ternary mixture containing methane, argon and hydrogen. The tube length is 0.40885 m and the two sections of the tube are equal in length. The operating conditions are ... 

So far we have analysed the two systems using the Maxwell-Stefan approach the steady state analysis of the Stefan tube and the transient analysis of the Loschmidt s tube, and they are conveniently used to study the diffusion characteristic of the system. Here, we consider another example which is also useful in the determination of diffusion characteristics. This system is the two bulb method, in which a small capillary tube or a bundle of capillaries is bounded by two well-mixed reservoirs as shown in Figure 7.2-3. 

In this experiment of Graham and Loschmidt, the two bulbs are joined together by a tube containing either a porous medium or a capillary (Figure 7.2-3). The left bulb contains gas A, while the right bulb contains gas B, having the same pressure as that of gas A. 

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