*i = Vu »2i ••• vm]T / = [0-0 vn lFi 0 ••• 0]r
Note that by commencing with the top row and adding each row to the one below it and then replacing the latter by the result so obtained, an upper triangular matrix which is bidiagonal is obtained (see Prob. 10-8).
Similarly, the total-material balances are the same as those presented for conventional distillation columns except for the condenser-accumulator section, and for this stage,
The complete set of total-material balances may be represented in the usual way by
where the matrix R is readily formed from the matrix A, of Eq. (10-38) by replacing A}[ by R}, where
Rj = OjiLj/VX The vectors V and jF contain the elements v = [L, v2 v3 ••• vNy
The matrix equation RV = — BF is readily solved by converting R to an upper triangular matrix which is bidiagonal by the same procedure described above for A, .
Observe that in the solution of Eq. (10-40), Lx is regarded as a dependent variable, that is, on the basis of a given set of 0/s (j = 2, 3, ..., N), Lx and the complete set of V/s (j = 2, 3,..., N) may be computed. When the solution set of 0/s has been found, the value of Lx given by Eq. (10-40) must be equal, of course, to the specified value of Lx. This condition is assured by taking one of the functions to be si =7TT--1 00-42)
This is only one of the IN + 1 independent functions which are needed to solve this problem in terms of the IN + 1 independent variables enumerated by Eq. (10-36). The remaining functions are taken to be the equilibrium functions Fj (j= 1, 2, ..., N) and the energy balance functions G} (j = 1, 2, ..., N). Thus, the complete set f is t=[SlFlF2...FNG1G2... (10-43)
Except for the function Ft, the equilibrium functions are of the same form as those stated in Chap. 4. The bubble-point form of the function Fx is used, namely, fi=fl(K..-l)'.. (10-44)
Enthalpy balance functions which are somewhat simpler in form than those presented in Chap. 4 may be obtained by taking advantage of the fact that dt = 0. In this formulation of the enthalpy balance functions, the enclosures for the energy balances include the condenser-accumulator and each stage below it. [It should be observed that the Newton-Raphson method could have been formulated in terms of IN — 1 independent variables by omitting Qc, QR, and TN.] The condenser duty could have been omitted by enclosing the top stage instead of the condenser section. The temperature TN could have been omitted because the composition of the bottoms is known to be that of the feed, and the reboiler duty Qr could have been omitted because it appears in only one function, GN.
In Example 10-5, the specifications are taken to be D = 0, B = F and Li = 125 lb mol/h for the example stated in Table 10-1. Composition profiles for a column at this operation are presented in Fig. 10-4.
When the distillate rate is fixed and the reflux rate is varied, the compositions obtained for the example stated in Table 10-1 are shown in Fig. 10-5. As V2!Lx approaches 1, a continuous distillation column at total reflux in both sections is obtained. The range of operation displayed in Fig. 10-5 extends from total reflux, V2/Ll = 1, to the limiting condition of no liquid reflux (Li = 0) for
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