Page 256 - Distillation theory

P1: JPJ/FFX P2: JMT/FFX QC: FCH/FFX T1: FCH
0521820928c07 CB644-Petlyuk-v1 June 11, 2004 20:18

230 Trajectories of the Finite Columns and Their Design Calculation

For the above-described conditions of validity of trial calculations, the con-
centrations of the non-key impurity components in the feed cross-section are
monotonously increasing functions of the concentrations of these components
◦
in the separation products. Besides that, in the vicinity of points x ◦ and x , the
f −1 f
concentration of each non-key impurity component is a linear function of the con-
centrations of all non-key impurity components in the corresponding separation
product.
Therefore, no calculation difficulties arise at the stage of preliminary search
under consideration.
We note, nevertheless, that points x ◦ and x do not meet the conditions
◦
f −1 f
of material balance in the feed cross-section. Therefore, the following step of
the algorithm of determination of the concentrations of the non-key impurity
components in the separation products is necessary, as it will ensure the material
balance.
Validity of this specifying step becomes easier by the fact that sought for points
x f−1 and x f are sufficiently close to already found points x ◦ and x .
◦
f −1 f
Therefore, at the specifying step, one may accept that the concentration of each
non-key impurity component in the feed cross-section is a linear function of the
little concentrations of all the non-key impurity components in the corresponding
separation product:

x f −1,i ≈ a i + a i,k+2 x D,k+2 + a i,k+3 x D,k+3 +· · · + a i,n x D,n (7.5)
x f,i ≈ b i + b i,1 x B,1 + b i,2 x B,2 +· · · + b i,k−1 x B,k−1 (7.6)

To determine coefficients a i , a i,k+2 , a i,k+3 ,... a i,n and b i , b i,1 , b i,2 ,... b i,k−1 , in-
crements are given to the concentrations x D,i or x B,i , while the concentrations
of the rest of the components in the separation products x ◦ or x ◦ are fixed, and
D, j B, j
the calculation of section trajectories is carried out.
Then, using Eqs. (7.5) and (7.6), the system of equations for discrepancies
of material balance in the feed cross-section is solved for all non-key impurity
components in both separation products (in the described algorithm, the validity
of material balance at non-key impurity components leads to balance validity at
all components):

L r (a i + a i,k+2 x D,k+2 + a i,k+3 x D,k+3 +· · · + a i,n x D,n) + L F x F,i
−L s (b i + b i,1 x B,1 + b i,2 x B,2 +· · · + b i,k−1 x B,k−1) = 0
(i = 1, 2,..., k − 1, k + 2, k + 3,..., n) (7.7)

By means of solution of Eq. (7.7), we determine more precise values of little
concentrations of non-key impurity components in the separation products
x 1 , x 1 ,..., x 1 , x 1 , x 1 ,..., x 1 , ensuring smaller values of discrep-
D,k+2 D,k+3 D,n B,1 B,2 B,k−1
ancies of material balance in the feed cross-section than the preliminarily found
concentrations x ◦ , x ◦ ,..., x ◦ , x ◦ , x ◦ ,..., x ◦ .
D,k+2 D,k+3 D,n B,1 B,2 B,k−1
To obtain solutions with a set precision, the above-described specifying step
should be taken a few times, because Eqs. (7.5) and (7.6) are rough linear and