Mixed Solvent Extractioii Notes on an Analytical Method J4JIES H. WIEGAND, Traverse City, Mich.
Hunter recently presented a graphical method of attack for mixed solvent extraction problems involving four components. The equilateral tetrahedron used by Hunter is modified in this article to a rectangular type of tetrahedron, three of whose edges coincide with t h e three axes of a rectangular coordinate system, and three of whose faces are isosceles right triangles. With rectangular coordinates, t h e methods of analytical geometry are readily applied. The problem given by Hunter is solved analytically and t h e solution presented in the form of generalized equations in terms of values obtainable from the characteristics of the system and from the statement of t h e problem. The numerical example given by Hunter is solved by this method. The results compare well with t h e graphical method, and there is a large saving in the time required for solution of t h e problem.
Equations 1 and 2 tagether define the line of intersection, RT. It is desired to find the intersection of line RT with the straight line joining corresponding points on the two equilibrium curves. Except for the case of RT parallel to the z axis, the intersection can be found by intersecting plane p Y + - 2 2 = 1
with the planes defined by Equations 1 and 2. For the final solution, corresponding values of z, y*, and z* can then be obtained from the equilibrium data of the systems involved. Equations 1, 2, and 3 are solved simultaneously by first eliminating z between Equations 1 and 2, then solving the result with Equation 3 to obtain z,giving z =
(:
(kz -
kl)
- ( k ? - k1/u)y*
- kl) - (k2
- kl/a)y*/z*
(; (:
- k l ) y* - ( k ? - kl)y*/z*
y = -
- k,)
- (k2
- kl/a)y*/z*
(5)
These values for y and z are then placed in Equation 2 and the equation is solved for x, giving -z - - ( l / c u - l ) y * - ( l / c - 1) - ( l / a - l)y*/z*
- klk?
Four-Component System on Rectangular Coordinates The system of representation used by Hunter consisted of a regular equilateral tetrahedron, but such a system does not lend itself readily to the use of analytical geometry methods. Kinney (3) pointed out that the equilateral method of representstion for three components can easily be converted to a right-triangle system, using the ordinary x-y coordinates. I n such a case, the third component is defined by the other two, since the total of the weight fractions must equal unity. This method is easily extended to four components, using the ordinary x-y-z coordinates of analytical geometry. I n this case, three edges of the tetrahedron coincide with the three ayes, and x, y, and z become the weight fractions of three of the components. The fourth component is thus defined, since the total of the neight fractions of the four components must equal unity. Figure 1 slions the problem given by Hunter for the system acetone-chloroform-water-acetic acid, where x. y, and z represent the Tveight fractions of water, acetic acid, and acetone, respectively. For convenience in comparison with the Hunter article, the apexes of the tetrahedron are and S?.The line RT is that given by Hunter labeled A , B , 81, and is the intersection of tlvo planes, each of which is determined by the tie line in one face and the opposite apex.
( k ~ / c- k l ) - (k2 - kl/a)y*/z*
jiA
The plane containing the tie line in the z-y plane and the z apex has the equation
5-2
(4)
This value of z is then placed in Equation 3 and the equation solved for y, giving
R E C E S T article by Hunter (1) presented a graphical method of computation for four-component solvent extraction problems. This article shows the application of analytical geometry methods to such a problem, with large savings in the time and the difficulty of solution.
Similarly, the plane containing the tie line in the and the y apex is repre-ented by Equation 2.
(3)
FIGURE 1. FOCR-COMPOSEST SYSTEM REPRESENTED ON
plane
RECT.4SGFLA R COORDINATES
380
(6)
ANALYTICAL EDITION
June 15, 1943
381
Also 2
(14) so that
A
ZM
zo =
+
RMXS 1 From the similarity of right triangles in Figure 2, -C = -
ka
20
ki
(15)
(16)
- XU
Combining Equations 13, 15, and 16 and simplifying, gives
M
z=c.
z
