NUMBER OF MASS-TRANSFER UNITS AT THE POINT OF TANGENCY OF THE OPERATING LINE AND THE EQUILIBRIUM LINE

G. M. IZMAILOV

(Presented by the scientific seminar of the departments of PMAChP and OChT)

In the course “Processes and Apparatuses of Chemical Technology,” when presenting such questions of mass-exchange processes as the minimum reflux ratio or the minimum absorbent flow rate, the conclusion obtained in each separate case turns out, in essence, to be one and the same: the phase-contact surface required for carrying out the process at the minimum value of the liquid-phase flow rate must be infinitely large [1, 2, 3, 4].

This conclusion may also be reached by investigating the general expression for the driving force of a mass-exchange process. To do this, it is necessary to require that the operating line and the equilibrium line have a common point—the point of tangency. In this case the number of transfer units will be infinitely large, as a result of which the required phase-contact surface will also prove to be infinitely large. Further, since the minimum liquid-phase flow rate always corresponds to the point of tangency, we obtain the conclusion mentioned above, but already in the form of one of the results of mass-transfer theory.

Analysis of the mean driving force inevitably leads to investigation of the number of transfer units. For this reason preference is given to the study of the number of transfer units.

Let us choose such a method of expressing the composition of the phases as will ensure the linear form of the operating line. Denote the composition of one phase by \(X\), and the composition of the other by \(Y\). Let the operating line \(AB\), represented in the graph (Fig. 1) by the equation

\[ Y = aX + b, \tag{1} \]

be located on this graph above the equilibrium line \(OB\) and have with this line the common point \(B\).

The required number of transfer units \(m_Y\) cannot be determined directly because of the indeterminacy of the form of the integrand entering the expression for the number of transfer units:

\[ m_Y = \int_{Y_1}^{Y_2} \frac{dY}{Y - Y_p}, \tag{2} \]

which is due to the general form of the expression for the dependence between the equilibrium compositions of the phases

\[ Y_p = f(X). \]

Let us construct from point \(B\) a segment of the straight line \(BC\), the points of which are located no higher than the line \(OB\), while the abscissa of point \(C\) is equal to the abscissa of point \(A\). Suppose that we have succeeded in changing the conditions of carrying out the process in such a way that the equilibrium line has become the segment \(BC\). Therefore the dependence between the equilibrium compositions of phases \(X\) and \(Y\) will be expressed by the equation of a straight line

\[ \overline{Y}_{p}=cX+d . \tag{3} \]

Since the method of expressing the composition has remained the same, the operating line will not change its position.

Fig. 1. Graph of operating and equilibrium phase compositions

Owing to the fact that both the operating dependence (1) and the equilibrium dependence (3) are linear functions, the expression for the number of transfer units for the new conditions

\[ \overline{m}_{Y}=\int_{Y_{1}}^{Y_{2}}\frac{dY}{Y-\overline{Y}_{p}} \tag{4} \]

will contain under the integral sign a definite rational function, the integral of which over the given limits is equal to infinity

\[ \overline{m}_{Y}=\infty . \tag{5} \]

The selected position of the segment \(BC\) with respect to the line \(OB\) ensures the following inequality:

\[ \frac{1}{Y-\overline{Y}_{p}} \geq \frac{1}{Y-Y_{p}} . \tag{6} \]

Invoking the property of integrals determined by the inequality of the integrand functions [5], from (2), (4), and (6) we obtain

\[ m_{Y}\geq \overline{m}_{Y}. \tag{7} \]

Now, on the basis of (5) and (7), we conclude that the number of transfer units at the point of tangency of the operating line and the equilibrium line is equal to infinity,

\[ m_{Y}=\infty . \]

The same result is obtained when studying the case of tangency of the equilibrium line with the left end of the operating line; for example, when the equilibrium line \(OB\) is tangent at point \(D\) to the operating line \(DE\).

REFERENCES

1. A. G. Kasatkin. Basic Processes and Apparatuses of Chemical Technology. Goskhimizdat, Moscow, 1961.

2. A. N. Planovskii, P. I. Nikolaev. Processes and Apparatuses of Chemical and Petrochemical Technology. Moscow, 1960.

3. Ya. Sh. Shiforovskii. Fundamentals of Processes of Chemical Technology. Khimiya Publishing House, Leningrad, 1967.

4. Chemist’s Handbook. Vol. V, Khimiya Publishing House, Moscow–Leningrad, 1966.

5. V. A. Il’in, E. G. Poznyak. Fundamentals of Mathematical Analysis. Nauka Publishing House, Moscow, 1967.