Flash Tank Problem

Description

For this example^[1], a flash tank wih feed flow $F$ and composition $z_i$ with $i = \mathcal{C}$ is considered. We want to analyze how the split between vapor $V$ and liquid $L$ products with composition $y_i$ and $x_i$ respectively vary with temperature for a fixed pressure $p$. We can use the Rachford-Rice equation

to calculate the fraction of the feed that goes to the vapor phase (represented by $a_t$) with $K_i$ given by Raoult's law

$p_i^\text{sat}(T)$ is the vapor pressure of the pure component $i$ at temperature $T$ calculated using Antoine's equation

where $A_i$, $B_i$ and $C_i$ are constants for each compound $i$.

A problem with this formulation is that, if T is lower than the mixture's bubble point or larger than its dew point, $a_t$ assumes negative values and values greater than one respectively and, since $a_t$ represents the ratio $V/F$, that would be physically impossible. We can deal with it by formulating this problem as an optimization problem with complementarity constraints that enforce $V/F$ to be $0$ if $T$ is lower than bubble point and $1$ if it is larger than dew point.

The optimization problem is then modeled as

The 5$^\text{th}$ and 6$^\text{th}$ constraints correspond to material balances, total and componentwise respectively. Constraints 7-10 describe the complementarity conditions, $s_L$ and $s_V$ are slacks variables that represent how much $a_t$ is lower than $0$ and larger than $1$ respectively. Variable $a$, then, represents the actual ratio $V/F$, which is enforced by the 7$^\text{th}$ and 10 $^\text{th}$ constraints. Note that $a = V/F$ is not enforced as a hard constraint and, instead, is used as the objective function.

Implementation

The implementation in flash.jl uses

$F = 1$,
$T \in [380,400]$ in increments of $1$ K,
and 3 compounds with the following Antoine's constants and inlet composition

This problem is solved using the solve_mpcc! function provided with the default parameters.