## Info

Intermediate Residue 1

Bottom Residue or Product

Figure 6.1. STN with One Main-Cut and One Off-cut (Binary Mixture). [Mujtaba and Macchietto, 1993]a

Main-cut 1

Initial Charge Int. Res. 1 Int. Res. 2 Bottom Residue or Product

Figure 6.2. STN with Two Main-cuts and One Off-cut (Ternary Mixture). [Mujtaba and Macchietto, 1993]a

6.2.1. Binary Operation

### 6.2.1.1. Degrees of Freedom Analysis

Refer to the STN shown in Figure 6.1. Given, B0 and xB0 as the initial amount and composition of a mixture, we wish to obtain the main-cut 1 with specified purity in terms of the mole fraction of component 1 ( xpX ). The intermediate residue (Bh xB1)

is further distilled off to obtain the off-cut 1 so as to satisfy the specification on the final bottom product composition for component 2 ( xf*2 ). Let D, denote the amount a Reprinted from Computers & Chemical Engineering, 17, Mujtaba, I.M. and Macchietto, S., Optimal operation of multicomponent batch distillation- multiperiod formulation and solution, 1191-1207, Copyright (1993), with permission from Elsevier Science .

of main-cut 1, (R^ xR1) denote the amount and composition of off-cut 1 and B2 denote the amount of final bottom product (or residue).

The system can be considered with one input state (initial charge) and three output states (main-cut 1, off-cut 1, and bottom residue) defined by (B0, xB0), (D,, XdiX (Ri. *ri) and (B2, xB2), respectively. For a mixture with nc components each state is characterised by total amount of material and component mole fractions (nc

+ 1 variables), with one summation equation (22*:' = /), hence by nc independent variables (two in this binary case). Since there is no accumulation of the intermediate residue an overall mass balance for this system gives nc equations:

B0xiB0=D1xiDl+R1xim+B2xiB2-, i = 12,...,ne, nc =2 (6.1)

For a given feed charge (wc specifications), a degree of freedom analysis shows that there are 4nc (state variables) - nc (feed specifications) - nc (mass balance) = 2nc degrees of freedom.

Additional variables and defining equations may be introduced, if necessary, without changing the degree of freedom of this system. For example, a recovery of component 1 (Re^i) in main-cut 1 over the distillation task 1 can be defined as: