Continuously Stirred Tank Reactor problem

Continuously Stirred Tank Reactor problem
State dimension:	1
Differential states:	4
Continuous control functions:	2
Interior point equalities:	2

The Continuously Stirred Tank Reactor problem considers a chemical reaction that produces cyclopenthenol while using up cyclepentadiene "by an acid-catalyzed electrophilic hydration in aqueous solution", an exothermal reaction which needs to be cooled. This problem can e.g. be found in [Diehl2001]Author: M. Diehl
School: Universit\"at Heidelberg
Title: Real-Time Optimization for Large Scale Nonlinear Processes
Url: http://www.ub.uni-heidelberg.de/archiv/1659/
Year: 2001
.

The inflow into the tank contains only cyclopentadiene (substance $A$ ) with temperature $θ_{0}$ and the flow rate $\dot{V}$ can be controlled. The outflow rate is the same as the inflow rate to keep the liquid level in the tank constant. "The outflow contains a remainder of cyclopentadiene, the wanted product cyclepentenol (substance $B$ ) and two unwated by-products, cyclopentanediol (substance $C$ ) and dicyclopentadiene (substance $D$ ) with concentrations $c_{A}, c_{B}, c_{C}, c_{D}$ ." The latter two are not tracked in the problem as the substances are not of use. The reaction scheme is given as:

$\begin{array}{ccccc} A & \overset{k_{1}}{\to} & B \overset{k_{2}}{\to} & C \\ 2 A & \overset{k_{3}}{\to} & D \end{array}$

where the reaction rates $k_{i}$ are a function of the reactor temperature $θ$ via an Arrhenius law $k_{i} (θ) = k_{i 0} \cdot \exp (\frac{E_{i}}{θ /^{\circ} C + 273.15}), i = 1, 2, 3 .$

"The temperature $θ_{K}$ in the cooling jacket is held down by an external heat exchanger whose heat removal rate ${\dot{Q}}_{K}$ can be controlled."

Mathematical formulation

The problem is given by

$\begin{array}{llcl} \max_{\dot{V}, {\dot{Q}}_{K}} & c_{B} & at the end of reaction \\ s.t. & \dot{c_{A}} & = & \frac{\dot{V}}{V_{R}} (c_{A 0} - c_{A}) - k_{1} c_{A} - k_{3} c_{A}^{2}, \\ \dot{c_{B}} & = & - \frac{\dot{V}}{V_{R}} c_{B} + k_{1} c_{A} - k_{2} c_{B}, \\ \dot{θ} & = & \frac{\dot{V}}{V_{R}} (θ_{0} - θ) + \frac{k_{w} A_{R}}{ρ C_{p} V_{R}} (θ_{K} - θ) - \frac{1}{ρ C_{p}} (k_{1} c_{A} H_{1} + k_{2} c_{B} H_{2} + k_{3} c_{A}^{2} H_{3}), \\ \dot{θ_{K}} & = & \frac{1}{m_{K} C_{P K}} ({\dot{Q}}_{K} + k_{w} A_{R} (θ - θ_{K})), \\ c_{A} (0) & = & c_{A 0}, \\ c_{B} (0) & = & 0 . \end{array}$

where the various values are given in the Parameters section.

Parameters

These fixed values are used within the model.

Parameters
Name	Symbol	Value	Unit
Arrhenius coefficient	$k_{10}$	$1.287 \cdot 1 0^{12}$	$h^{- 1}$
Arrhenius coefficient	$k_{20}$	$1.287 \cdot 1 0^{12}$	$h^{- 1}$
Arrhenius coefficient	$k_{30}$	$9.043 \cdot 1 0^{9}$	$h^{- 1}$
Arrhenius coefficient	$E_{1}$	$- 9758.3$	[-]
Arrhenius coefficient	$E_{2}$	$- 9758.3$	[-]
Arrhenius coefficient	$E_{3}$	$- 8560$	[-]
Reaction enthalpy	$H_{1}$	$4.2$	$\frac{k J}{m o l}$
Reaction enthalpy	$H_{2}$	$- 11.0$	$\frac{k J}{m o l}$
Reaction enthalpy	$H_{3}$	$- 41.85$	$\frac{k J}{m o l}$
Solution density	$ρ$	$0.9342$	$\frac{k g}{l}$
Capacity of aqueous solution	$C_{p}$	$3.01$	$\frac{k J}{k g \cdot K}$
Heat transfer coefficient for cooling jacket	$k_{w}$	$4032$	$\frac{k J}{h \cdot m^{2} \cdot K}$
Reactor surface area	$A_{R}$	$0.215$	$m^{2}$
Reactor volume	$V_{R}$	$10$	$l$
Coolant mass	$m_{K}$	$5$	$k g$
Capacity of coolant solution	$C_{P K}$	$2.0$	$\frac{k J}{k g \cdot K}$
Starting concentration of subs. $A$	$c_{A 0}$	$5.1$	$\frac{m o l}{l}$
Inflow temperature	$θ_{0}$	$104.9$	$\circ C$

Reference solution

"The result of a steady state optimization of the yield $= \frac{c_{B} |_{S}}{c_{A 0}}$ with respect to the design parameter $θ_{0}$ (feed temperature) and the two controls yields the steady stae and controls" $c_{A} = 2.1402 \frac{m o l}{l}, c_{B} = 1.0903 \frac{m o l}{l}, θ = 114.1 9^{\circ} C, θ_{K} = 112.9 1^{\circ} C$ and $\frac{\dot{V}}{V_{R}} = 14.19 h^{- 1}, {\dot{Q}}_{K} = - 1113.5 \frac{k J}{h}$ .

Source Code

Model descriptions are available in

AMPL/TACO code at Continuously Stirred Tank Reactor (TACO)