Page 149 - Distillation theory

P1: JPJ/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
0521820928c05 CB644-Petlyuk-v1 June 11, 2004 20:15

5.3 Trajectory Tear-Off Theory and Necessary Conditions 123

As far as x Di > 0 and x Bi > 0, L/V < 1 for the top section and L/V > 1 for the
bottom section, it follows from Eqs. (5.11) and (5.12) that:
L/V < K i st (for top section) (5.13)
L/V > K i st (for bottom section) (5.14)

Comparison of these inequalities with equality (Eqs. [5.6]) for the stationary
points and with inequalities (Eqs. [5.9]) and ([5.10]) for pseudostationary and
other points of the trajectory leads to the important result: in all points of the
t
trajectory and, in particular, in its tear-off points x from the boundary element,
the following inequalities should be valid:
t
K > K t (for top section) (5.15)
i j
t
K < K t (for bottom section) (5.16)
i j
Equations (5.9), (5.10), (5.15), and (5.16) are necessary and sufficient condi-
tions of trajectory tear-off from the boundary element of concentration simplex.
Equations (5.9) and (5.10) can be called operating ones because they depend on
separation mode, and Eqs. (5.15) and (5.16) can be called structural ones because
they depend only on the structure of the field of phase equilibrium coefficients.

5.3.2. Trajectory Tear-Off Regions and Sharp Distillation Regions
t
In trajectory tear-off points of the top section x phase equilibrium coefficients of
r
t
the components present in the product K should be greater than those of the com-
i
t
ponents absent in the product K , and vice versa in the bottom section. Therefore,
j
tear-off of trajectories from the boundary elements of concentration simplex is
feasible only if in the vicinity of this boundary elements there are component-
ijk
order regions Reg that meet these conditions of trajectory tear-off (Fig. 5.8).
ord
We call the region where trajectory tear-off is feasible “trajectory tear-off region”
j j
Reg t(k) or Reg t(k) . Those α-manifolds, in points of which phase equilibrium coeffi-
r s
i i
cients of one of the present in the product component and of one of the absent in
the product component are equal, are boundaries of trajectory tear-off regions.
+ +
N r t t N s t t
a) min K > max K rj b) min K > max K si
sj
ri
i j j i
L V / > max K t L V / < min K t
r r j rj s s j sj
x > 0 x > 0
j j
r S x D s S x B
x j = 0 x j = 0

Figure 5.8. Tear-off conditions from boundary elements of the con-
centration simplex for the section trajectories: (a) rectifying section,
and (b) stripping section.