1822
ANDRZEJWITKOWSKI
Vol. 64
THE THEORY OF THE DISTILLATION COLUMN BY ANDRZEJWITKOWSKI~ Departmat of Chemistry, Harvard Umiversity, Cambridge, Massachusetts Received Dscsmbsr 80, 1968
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+
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+
It will be shown that concentration in a simple distillation column obeys the differential equation p - (Y at az 3C -y = 0, where constant coefficients of the equation are explicitly expressed by diffusion constants and relative p volatility. In particular one can get the time of the equilibrium establishment in the column and can expect the existence of B phenomenon similar to Debye’s effect in the thermogravitational column.
2
Many papers2 have been written in connection with the theories of distillation but most of these have been concerned only with equilibrium separation. The well known theory of WesthaVera involving the equilibrium separation generally agrees favorably with experiment. Recently a complete analysis of the hydraulics of the problem has been given by M i ~ h a l i k . ~ The problem of approaching to equilibrium especially for the columns without packing has not been studied in as great detail as the equilibrium case. On the other hand it has been stressed6 that this problem is very important in the separation by the distillation of isotopes and other components with only slightly different boiling points. Thus we will make an attempt to formulate a theory of the distillation column with particular concern for the time dependence and which is valid for the problem of the separation of such components. We will apply the analysis to the simple type of column composed of an opened tube with a thin film of liquid flowing down the wall around the vapor streaming up. Separation is obtained by virtue of the existence of a concentration difference between the liquid and the vapor together with relative motion; the changes of concentration are governed by the continuity equation in which diffusion plays an important role. Usually the problem must be formulated using 3 variables: time, coordinate along and across the column, with specific boundary conditions for the contact between vapor and liquid. This problem as stated is too complicated for the exact treatment; however, we will make certain approximations which enable solution to a very good degree of approximation. Owing to the fact that changes of the concentration across the column are usually small compared to changes along the column, it will be shown that it is possible to include the “horizontal problem” in a vertical one. Thus, we may transform a linear 3-dimensional differential equation with non-linear boundary conditions at interface to a 2-dimensional (‘vertical” non-linear diff erential equation with known solution with only a small percentage error later to be determined. 2. Consider an open tube distillation column. (1) Department of Theoretical Chemistry, Jsgellonian University, KraMw, Poknd. ( 2 ) A. Rose, et ol.. “Distillation,” Interscience Publ. Inc., New York, N. Y.,1951. (3) J. W . Weathaver, l n d . Eng. Chem., 84, 126 (1942). (4) Michalik, Am. lnst. Chem. Env., 3, 276 (1957). (5) Ref. 2, p. 94.
~
Consider a two component mixture. Define the cylindrical coordinate system with Z-axis vertically upwards corresponding to the axis of the cylinder. We denote by t the coordinate of the surface of the liquid and by a the inner radius of the tube. Denoting by J , radial and by J, vertical components, at once for both phases, of the vector of the mass flux J we have
where D denotes the diffusion coefficient, 6 density, v velocity and where concentration c is measured cz = 1. For in fractional molar units, thus c1 simplicity we will denote c1 = c, thus c2 = 1 - c. Writing equation 1 we are concerned with the isothermal distillation problem. The problem of the distillation in a system where a horizontal temperature gradient exists, thus when distillation combines with thermal diffusion, was considered by the author elsewhere.” We have continuity equation for the concentra-
+
+ div J at
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0
(3)
tion with boundary conditions (Jr)r-o (Jr)r-s
