Page 218 - Separation process principles 2

5.7 Degrees of Freedom and Specifications for Countercurrent Cascades 183

summary of possible variable specifications for each of these pressure) and all element and unit pressures and heat trans-
two cases for a number of separator types discussed in Chap- fer rates (except for condensers and reboilers) are specified.
ter 1 and shown in Table 1.1. For all separators in Table 5.4, Thus, only variable specifications for satisfying the remain-
it is assumed that all inlet streams are completely specified ing degrees of freedom are listed.
(i.e., C - 1 mole fractions, total flow rate, temperature, and

EXAMPLE 5.5
Consider a multistage distillation column with one feed, one side- element from Table 5.3 and then subtracting the redundant vari-
stream, a total condenser, a partial reboiler, and provisions for heat ables due to interconnecting flows. As before, redundant mole-
transfer to or from any stage. Determine the number of degrees of fraction constraints are subtracted from the summation of indepen-
freedom and a reasonable set of specifications. dent relationships for each element (NE), This problem was first
treated by Gilliland and Reed [9] and more recently by Kwauk [8].
SOLUTION Differences in ND obtained by various authors are due, in part, to
their method of numbering stages. Here, the partial reboiler is
This separator is assembled as shown in Figure 5.22, from the cir- the first equilibrium stage. From Table 5.3, element variables and
cled elements and units, which are all found in Table 5.3. The total relationships are as follows:
variables are determined by summing the variables (Nv), for each

Element or Unit
Total condenser
Reflux divider
(N - S) stages
Sidestream stage
(S - 1) - F stages
Feed stage
(F - 1) - 1 stages
Partial reboiler

Subtracting (C + 3) redundant variables for 13 interconnecting
streams, according to (5-68), with NA = 0 (no unspecified repeti-
tions), gives
(Nv)"nit = C(~v)e - 13(C + 3) = 7N + 2NC + 5C + 20
Subtracting the corresponding 13 redundant mole-fraction con-
straints, according to (5-69),

(NE)"nt = C(NE)e - 13 = 5N + 2NC + 4C + 9
Therefore, from (5-71),
ND=(7N+2NC+5C+20)-(5N+2NC+4C+9)
=2N+C+11
Note that the coefficient of C is only 1, because there is only one
feed, and, again, the coefficient of N is 2.
A set of feasible design variable specifications is
Variable Specification Number of Variables
1. Pressure at each stage N
(including partial reboiler)
2. Pressure at reflux divider outlet 1
3. Pressure at total condenser outlet 1
4. Heat transfer rate for each stage (N - 1)
(excluding partial reboiler)
5. Heat transfer rate for divider 1
6. Feed mole fractions and C
Figure 5.22 Complex distillation unit. total feed rate