multivariate production systems optimization - Stanford University

Zi = Xi L + Yi V (5.3)
where V and L are the vapor and liquid mole fractions and L = 1 - V. Using the relation Yi
= Ki / Xi and solving for Xi yields
Xi =
Zi
L + 1 - L Ki
and letting Xi = Yi / Ki and solving for Yi yields
Noting the constraint of
Yi =
we must find a solution to the equation
F L =
∑ i
∑ i
Ki Zi
L + 1 - L Ki
Xi - ∑ Yi = 0
i
Zi 1 - Ki
= 0
Ki + 1 - Ki L
This equation can be efficiently solved with Newton-Raphson iteration where
and convergence is achieved when both
1)
2)
L k+1 = L k - F L k
∂F
∂L L k
abs L k+1 - L k < ε
F L k+1 < ε
54
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
where ε is a small tolerance. In this study, ε was set equal to 10-10 . Once L is determined,
the compositions of the liquid and vapor phases are obtained from Equations 5.4 and 5.5.
The number of phases present is determined by considering Equation 5.7. At the
dew point, where L = 0, Equation 5.7 yields