Page 28 - Distillation theory

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2 Phase Equilibrium and Its Geometric Presentation

2
a) b)

2 1
x 1
x 3
x 1 x 2
x 2 x D
x F
x B
1 3
2 2
c) d)

x 1
x 4 x 3
x 1
x 2
1 1 x 2 4

3
Figure 1.1. Concentration simplexes (a) for binary mixtures,
(b, c) for three-component mixtures. and (d) for four-compo-
nent mixtures. x 1 , x 2 , x 3 , x 4 , concentrations of components.

Concentration space is the number of points representing all possible compo-
sitions of an n-component mixture. Concentration space of a binary mixture C 2 is
a segment of unit length; the ends correspond to pure components, and the inner
points correspond to mixtures of various compositions (Fig. 1.1a)
For a three-component mixture, it is convenient to present the composition
space C 3 as an equilateral triangle, the height of which equals one (Fig. 1.1b). The
triangle’s vertexes represent pure components, the points within its sides, repre-
sent the binary constituents of the three-component mixture, and the inner points
of triangle represent the three-component mixture compositions. The lengths of
the perpendiculars to the triangle’s sides correspond to the concentrations of the
components indicated by the opposite vertexes. The described system of coordi-
nates, which bears the name of the system of uniform coordinates, was introduced
by Mobius and was further developed by Gibbs.
Another way to present a three-component mixture’s composition space C 3
implies the use of an isosceles right-angle triangle (Fig. 1.1c), with a side equal
to one. In this method of representation the concentrations of components 1 and
2 are expressed by the length of perpendicular segments, as in the first case of
the composition’s representation, and the concentration of the third component
is defined in accordance with the formula: x 3 = 1 − (x 1 + x 2 ).
Four-component mixture composition can be represented by a point of an
equilateral tetrahedron C 4 (Fig. 1.1d). In this tetrahedron the vertexes represent
the pure components, the edges represent the binary constituents, and the faces
represent the three-component constituents.
In this book, we will often represent the mixture compositions corresponding
to the material balance (e.g., the compositions of feed flow and product flow of