Optimization of an Evaporation Process

The flowsheet below consists of a forced-circulation evaporator, used extensively over the years as a benchmark for plantwide control studies. The system is depicted in the figure below. The objective of this system is to increase the concentration of dilute liquor by evaporating the solvent from the feed, using a vertical heat exchanger.

Evaporation process schematic.

From the variables described, \(X_1=5\%\),\(T_1=40°C\),\(T_{200}=25°C\) are disturbances at these reported fixed values and are not considered as optimization variables. All flow rates are in \(kg/min\), temperatures in \(°C\), pressures in \(kPa\) heat duties in kW and compositions in \(%\). The main objective of this case-study is to find a maximum operating point from an economic perspective, defined as:

\[Cost = 600F_{100} +0.6F_{200}+1.009(F_2+F_3 )+0.2F_1- 4800F_2\]

The following equations describe the mass and energy balances for the evaporator process, and can be considered equality constraints in the optimization problem:

\[\begin{split} \begin{gathered} \frac{\mathrm{d} L_2}{\mathrm{~d} t}=\frac{F_1-F_4-F_2}{20} \\ \frac{\mathrm{~d} X_2}{\mathrm{~d} t}=\frac{F_1 X_1-F_2 X_2}{20} \\ \frac{\mathrm{~d} P_2}{\mathrm{~d} t}=\frac{F_4-F_5}{4} \\ F_4=\frac{Q_{100}-0.07 F_1\left(T_2-T_1\right)}{38.5} \end{gathered} \end{split}\]

\[\begin{split} \begin{gathered} T_2=0.5616 P_2+0.3126 X_2+48.43 \\ T_3=0.507 P_2+55.0 \\ T_{100}=0.1538 P_{100}+90.0 \\ Q_{100}=0.16\left(F_1+F_3\right)\left(T_{100}-T_2\right) \\ F_{100}=\frac{Q_{100}}{36.6} \\ Q_{200}=\frac{0.9576 F_{200}\left(T_3-T_{200}\right)}{0.14 F_{200}+6.84} \\ T_{201}=T_{200}+\frac{13.68\left(T_3-T_{200}\right)}{0.14 F_{200}+6.84} \\ F_5=\frac{Q_{200}}{38.5} \end{gathered} \end{split}\]

And the following bounds are true, which are considered inequality constraints:

\[\begin{split} \begin{gathered} X_2 \geq 35.5 \% \\ 40 \mathrm{kPa} \leq P_2 \leq 80 \mathrm{kPa} \\ P_{100} \leq 400 \mathrm{kPa} \\ 0 \mathrm{~kg} / \mathrm{min} \leq F_{200} \leq 400 \mathrm{~kg} / \mathrm{min} \\ 0 \mathrm{~kg} / \mathrm{min} \leq F_1 \leq 20 \mathrm{~kg} / \mathrm{min} \\ 0 \mathrm{~kg} / \mathrm{min} \leq F_3 \leq 100 \mathrm{~kg} / \mathrm{min} \end{gathered} \end{split}\]

Note that the equations are to be solved in steady state (i.e., derivative terms are zero)

Additionally, the description of each variable can be found in the table below:

Description of all process variables for the evaporation process.

Find the optimal operating point of this process flowsheet using constrained optimization techniques covered in Lecture 3. You should:

1. Formalize (write-down) the optimization problem, making the necessary assumptions/simplifications

2. Use scipy.optimize.minimize appropriately for the constrained optimization problem

3. Compare your results against the optimum found in the reported literature.

4. Discuss your code implementation and results during the presentation.