Page 175 - Distillation theory

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5.5 Trajectory Bundles for Four- and Multicomponent Mixtures 149

1
components in phase equilibrium coefficient in point S . Each of the other sad-
dle points (if they exist) is located at the boundary element, formed by product
components and by one of the additional components, except the heaviest (for top
section) or the lightest one (for bottom section) among the absent in the product
+
components. The stable node N is located at the boundary element, formed by
product components and by additional component, that is the heaviest (for top
section) or the lightest one (for bottom section). If the product contains num-
ber of components smaller by one than separated mixture, then the stable node
1
2
+
N is located inside simplex, there is tear-off point S , point S coincides with
point N , and the rest of stationary points are absent. The described regularities
+
are explained by the fact that reversible distillation trajectories at which all the
stationary points of the bundle are located can be found only at the boundary ele-
ments mentioned above. For nonideal mixtures (especially for azeotropic), saddle
1
2
points S or S can be located not only at the boundary elements, but also at α-lines,
α-surfaces, or α-hypersurfaces inside simplex. Only at those α-lines, α-surfaces, or
α-hypersurfaces, where phase equilibrium coefficients of the components, absent
intheproduct,areequal.Onlyinthiscasereversibledistillationtrajectory,atwhich
2
1
the point S or S can be located, goes through the mentioned lines, surfaces, or
hypersurfaces.
So far, discussing distillation trajectories and their bundles, we proceeded from
the fact, that separation stages are equilibrium (“theoretical” plates). In real sep-
aration process at plates of distillation columns equilibrium is not achieved and
the degree of nonequilibrium is different for different components. That leads to
decrease of difference between compositions at neighboring plates and to change
of curvature of distillation trajectories (Castillo & Towler, 1998), but does not in-
fluence the location of stationary points of distillation trajectory bundles because
in the vicinity of stationary points equilibrium and nonequilibrium trajectories
behave equally. Therefore, implemented above analysis of the structure and of
evolution of section trajectory bundles is also valid for nonequilibrium trajectory
bundles.
1 2 1
+
+
At sharp split separatrix sharp split region S → S →· · · → N ≡ S ⇒ N ≡
1
+
Reg sh,R (below simply S − N ), that is the boundary element of working sec-
sep
R
tion region Reg w,r , appears in concentration simplex. Its trajectories, including
1
the working one, go through the product point x D or x B and tear-off point S .
The dimensionality of this separatrix bundle is equal to the difference (n − k)
between dimensionality of concentration simplex (n − 1) and dimensionality of
1
1
the product boundary element (k − 1). In bundle S − N , point S is its unstable
+
node. As we see below, at discussion of joining of section trajectory bundles of
1
two-section columns (see next section), not only separatrix bundle S − N will
+
be of great importance for us, but also its boundary element, the most remote
2
2
from product point – separatrix bundle S →· · · →N (below simply S − N ),
+
+
having dimensionality (n − k − 1) smaller by one than dimensionality of bundle
1
2
+
S − N . Point S is the unstable node of this bundle (separatrix min-reflux region
Reg min,R ).
sep