On the Question of Separation of Gas Mixtures

V. G. Kriger, L. V. Serikov

(Presented by the scientific seminar of the Department of Radiation Chemistry)

In initially homogeneous mixtures of gases of different molecular weights, under the action of gravity and temperature gradients, the homogeneity of the composition may be disturbed. The reason for this will be mutual diffusion of the components of the mixture, and in the established state the components of the mixture will be partially separated.

In the present article the question of separation of a binary gas mixture of the hydrogen–nitrogen type under the action of external forces and thermodiffusion is discussed.

It is shown that separation caused by the force of gravity and inertial forces is insignificant, and the main contribution to the separation is made by thermodiffusion. The magnitudes of thermodiffusive separation are calculated for various hydrogen concentrations in the temperature interval \(300^\circ\text{--}500^\circ K\). The data obtained agree well with experiment.

I. Separation under the Action of External Forces

Let us consider a mixture of gases \(H_2 - N_2\), located in a cylinder of height \(h\) and radius \(R\). The direction of the longitudinal axis of the cylinder coincides with the direction of the force of gravity; the rotation of the cylinder is carried out about the longitudinal axis.

In an external field with potential \(\Phi(x,y,z)\), the conditions of thermodynamic equilibrium of the mixture will have the form [1, 2]:

\[ \mu_i + m_i \Phi_i(x,y,z) = const \tag{1} \]

where \(\mu_i\) is the chemical potential of the \(i\)-th component of the mixture,

\(m_i\) is the mass of the \(i\)-th molecule,

\(i = 1,2\) (the index 2 here and below refers to hydrogen).

Let us use the expression for \(\mu_i\) for ideal gases [1, 2]:

\[ \mu_i = kT \ln p + kT \ln n_i + f_i(T). \tag{2} \]

\(p\) is the pressure, \(T\) is the absolute temperature.

\[ n_i = \frac{n_i}{\sum_i n_i} \]

is the molar concentration of the \(i\)-th component.

\(f_i(T)\) is a function only of temperature.

Substituting (2) into (1), we obtain for \(n_i\) the expression:

\[ n_i(x,y,z) = \frac{ n_i(0)\exp\left[-\dfrac{m_i\bigl(\Phi(x,y,z)-\Phi(0)\bigr)}{kT}\right] }{ \sum_i n_i(0)\exp\left[-\dfrac{m_i\bigl(\Phi(x,y,z)-\Phi(0)\bigr)}{kT}\right] } \tag{3} \]

where \(n_i(0)\) is the concentration at \(\Phi(x,y,z)=\Phi(0)\).

I. Separation in a gravitational field

If the gravitational field is directed along the \(z\)-axis, then:

\[ \Phi(x,y,z)-\Phi(0)=ah \]

where \(a\) is the acceleration in the gravitational field.

\[ h=z-z_0; \]

\(z_0\) is the height taken as the initial one. Then (3) is rewritten in the form:

\[ n_i(h) = \frac{ n_i(0)\exp\left(-\dfrac{m_i ah}{kT}\right) }{ \sum_i n_i(0)\exp\left(-\dfrac{m_i ah}{kT}\right) } \tag{4} \]

The calculation shows that, for a small height difference
(\(h \sim\) tens of meters), the separation will be very insignificant, even at large accelerations. For example:
for \(h = 10\) m, \(a = 10 \cdot g\), \(n_2(0) = 10\%\),
we obtain:

\[ n_2 = 10.02\%. \]

2. Separation in a rotating cylinder

The rotation of a cylinder with gas is equivalent to the introduction of a field with potential

\[ \varphi = \frac{(\omega \tau)^2}{2}, \]

where \(\tau\) is the distance from the axis of the cylinder,

\[ \omega \]

is the angular velocity. For this case, (3) is written in the form:

\[ n_i(\tau)= \frac{ n_i(0)\exp\left(\frac{m_i(\omega \tau)^2}{kT}\right) }{ \sum_i n_i(0)\exp\left(\frac{m_i(\omega \tau)^2}{kT}\right) } \tag{5} \]

The separation will also be very insignificant. For example, for

\[ \omega = 20\,\frac{1}{\text{sec}}, \]

\(\tau \sim 2\) m, the separation will be of the same order as in the first case.

Thus, the separation of a mixture, both in the case of the action of gravitational forces and in a rotating cylinder, will be very insignificant.

II. Separation of a mixture as a result of thermal diffusion

A somewhat greater separation is achieved in thermal diffusion, i.e., diffusion caused by the presence of a temperature gradient in the mixture. Consider a mixture of gases enclosed in a cylinder (standing vertically), between whose end faces a constant temperature difference is maintained. To exclude the influence of convection, the upper end has the higher temperature:

\[ T' > T. \]

The theory of this process was developed in the works of Chapman and Enskog [3]. From this theory it follows that homogeneity

of the composition is observed only when the temperature is uniform; the presence of a temperature gradient, however, gives rise to a concentration gradient, namely:

\[ \frac{dn_2}{dz}=k_T\,\frac{1}{T}\,\frac{dT}{dz}, \]

whence, taking \(k_T\) to be constant, we obtain:

\[ n_2' - n_2 = k_T \ln \frac{T'}{T}, \tag{6} \]

where \(n_2=\dfrac{n_2}{\sum_i n_i}\) is the volume concentration of the “light” component in the region with temperature \(T\),

\(n_2'\) — in the region with temperature \(T'\).

The quantity \(\Delta n_2=n_2'-n_2\) is called the separation, and from (6) it is seen that, for \(k_T>0\), \(\Delta n_2>0\), i.e., the concentration of the “light” component will be higher in the region [[unclear: missing temperature designation]].

The coefficient \(k_T\) is called the “thermal-diffusion ratio” and will be discussed below. Here we note that \(k_T\), and consequently \(\Delta n_2\), depends on:

1). the ratio of the masses and diameters of the two kinds of molecules (\(k_T\) increases as this ratio increases),

2). the nature of the intermolecular forces: \(k_T\) is maximal when the molecules interact as elastic spheres,

3). the relative content of the two components:

\[ k_T=\alpha_T n_1 n_2, \tag{7} \]

where the thermal-diffusion constant \(\alpha_T\) depends only weakly on \(n_1\) and \(n_2\). From (7) it is seen that \(k_T\) is maximal when \(n_1=n_2\) and decreases sharply as the concentration of either component is reduced.

Analysis of the experimental data shows that, over a fairly broad temperature range, for a given composition of the mixture, \(\Delta n_2\) increases linearly with increasing \(\ln \dfrac{T'}{T}\), i.e., in this temperature range \(k_T\) depends only on the composition of the mixture.

For example, according to the data of Grew [4], \(k_T\) does not depend on temperature in the range \(160\text{–}600^\circ\mathrm{K}\). In our consideration we restrict ourselves to the following temperature region: \(T=300^\circ\mathrm{K}\), \(T'=300\text{–}500^\circ\mathrm{K}\). Thus, to calculate the separation it is necessary to specify the composition of the mixture and to compute the thermodiffusion ratio \(k_T\).

The quantity \(k_T\) can be calculated only in the first approximation and only for the simplest types of interaction of molecules, which include, in particular, the following:

1). the molecules interact as hard spheres.

2). the molecules repel one another with a force \(F=N r^{-\nu}\), where
\(N\) is the constant interaction force,
\(r\) is the distance between molecules.

In the first approximation the expression for \(k_T\) has the form

\[ k_T = 5(c-1)\, \frac{S_1 n_1 - S_2 n_2} {Q_1 n_1^2 + Q_2 n_2^2 + Q_{12} n_1 n_2} \, n_1 n_2 = \alpha_T \, n_1 n_2 . \tag{8} \]

\(Q_1; Q_2; Q_{12}; S_1; S_2\) are rather complicated functions of the collision integrals. The calculation of these integrals presents great mathematical difficulties and can be carried out only for simple types of molecular interaction, for example, for the hard-sphere model.

The factor \(5(c-1)\), when this model is used, reaches its maximum value, equal to 1.

The quantities \(Q_1; Q_2; Q_3; S_1; S_2\) in (8) depend on the mass ratio and on the law of interaction of both identical and dissimilar molecules. To estimate the degree of approximation of the law of molecular interaction to the hard-sphere model, the ratio of thermodiffusion constants is introduced:

\[ R_T = \frac{k_T}{k_T^{(\infty)}} = \frac{\alpha_T}{\alpha_T^{(\infty)}} , \tag{9} \]

where \(k_T^{(\infty)}\) and \(\alpha_T^{(\infty)}\) are calculated according to the hard-sphere model. The quantity \(R_T\) depends on the relative content of the component even less than \(\alpha_T\), and therefore can serve as a characteristic of the mixture. The theoretical calculation of \(R_T\), unfortunately, is possible only for a mixture of isotopes and for the case with

with a very large ratio of the molecular masses,

\[ \frac{m_1}{m_2}\to\infty \quad [4]. \]

Nevertheless, \(R_T\) has repeatedly been estimated, both experimentally and theoretically, and for the mixture \(H_2-N_2\)

\[ R_T = 0.57 . \tag{9} \]

Fig. 1. Dependence of \(\Delta n_2\%\) on \(\ln \frac{T'}{T}\)

\(1\) — \(4\%\)  \(2\) — \(14\%\)

\(\times\) — calculated data

\(\Delta\) — experimental data

Thus, knowing the mass ratio, the volume concentrations of the components of the mixture, and calculating \(S_1\), \(S_2\), \(Q_1\), \(Q_2\), and \(Q_{12}\), one can calculate \(\alpha_T(\infty)\) and \(k_T(\infty)\), and, using (9 and \(9'\)), determine \(\alpha_T\) and \(k_T\) for the given composition of the mixture.

By the method presented above, \(\alpha_T\) and \(k_T\) were calculated for the mixture \(H_2-N_2\) with hydrogen concentrations of \(4\%\) and \(14\%\). The values of \(Q_1\), \(Q_2\), \(Q_{12}\), \(S_1\), and \(S_2\) were taken from the table in [4, Appendix 7]. The data obtained are given in Table I.

Table I

It is seen from the table that $\alpha_T$ depends only weakly on the composition of the mixture. These values of $\alpha_T$ agree with the values reported by various authors ([4], Appendix 5). From the values of $k_T$ given in Table I, the separations of the $H_2$–$N_2$ mixture were calculated for 4% and 14% $H_2$. The temperature used in the calculations was: $T = 300^\circ\mathrm{K}$, $T' = 300$–$500^\circ\mathrm{K}$. On the basis of the calculation data, graphs of the dependence of $\Delta n_2$ on $\ln \frac{T'}{T}$ were constructed (shown in Fig. 1). For comparison with experiment, the data of Ibbs [4] for hydrogen concentrations of 4.7% and 10.7% were used. The experimental data for 10.7% were recalculated for 14% $H_2$. It is seen from the graphs that the agreement of the calculated data with experiment is sufficiently good.

REFERENCES

1. L. D. Landau, E. M. Lifshitz, Statistical Physics. Moscow, 1964.
2. D. ter Haar, G. Wergeland, Elementary Thermodynamics. Moscow, 1968.
3. S. Chapman, T. Cowling, The Mathematical Theory of Non-Uniform Gases. IL, 1960.
4. K. E. Grew, T. L. Ibbs, Thermal Diffusion in Gases. Moscow, 1956.