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 $\theta_0$ 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:

$\begin{array}{ccccc} A & \overset{k_1}\rightarrow & B \overset{k_2}\rightarrow & C\\ 2A & \overset{k_3}\rightarrow & D \end{array}$

where the reaction rates $k_i$ are a function of the reactor temperature $\theta$ via an Arrhenius law $k_i(\theta) = k_{i0} \cdot \exp ( \frac{E_i}{\theta / ^\circ C + 273.15} ), \quad i=1,2,3.$

"The temperature $\theta_K$ 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

$\begin{array}{llcl} \displaystyle \max_{\dot{V}, \dot{Q}_K} & c_B & & \text{ at the end of reaction} \\[1.5ex] \mbox{s.t.} & \dot{c_A} & = & \frac{\dot{V}}{V_R} (c_{A0} - c_A) - k_1 c_A - k_3 c_A^2, \\[0.5cm] & \dot{c_B} & = & -\frac{\dot{V}}{V_R} c_B + k_1 c_A - k_2 c_B, \\[0.5cm] & \dot{\theta} & = & \frac{\dot{V}}{V_R} ( \theta_0 - \theta) + \frac{k_w A_R}{\rho C_p V_R} (\theta_K - \theta) - \frac{1}{\rho C_p} (k_1 c_A H_1 + k_2 c_B H_2 + k_3 c_A^2 H_3), \\[0.5cm] & \dot{\theta_K} & = & \frac{1}{m_K C_{PK}} ( \dot{Q}_K + k_w A_R (\theta - \theta_K)),\\[0.7cm] & c_A(0) & = & c_{A0},\\ & c_B(0) & = & 0. \end{array}$

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	$k_{10}$	$1.287 \cdot 10^{12}$	$h^{-1}$
Arrhenius coefficient	$k_{20}$	$1.287 \cdot 10^{12}$	$h^{-1}$
Arrhenius coefficient	$k_{30}$	$9.043 \cdot 10^{19}$	$h^{-1}$
Arrhenius coefficient	$E_1$	$-9758.3$	[-]
Arrhenius coefficient	$E_2$	$-9758.3$	[-]
Arrhenius coefficient	$E_3$	$-8560$	[-]
Reaction enthalpy	$H_1$	$4.2$	$\frac{kJ}{mol}$
Reaction enthalpy	$H_2$	$-11.0$	$\frac{kJ}{mol}$
Reaction enthalpy	$H_3$	$-41.85$	$\frac{kJ}{mol}$
Solution density	$\rho$	$0.9342$	$\frac{kg}{l}$
Capacity of aqueous solution	$C_p$	$3.01$	$\frac{kJ}{kg \cdot K}$
Heat transfer coefficient for cooling jacket	$k_w$	$4032$	$\frac{kJ}{h \cdot m^2 \cdot K}$
Reactor surface area	$A_R$	$0.215$	$m^2$
Reactor volume	$V_R$	$10$	$l$
Coolant mass	$m_K$	$5$	$kg$
Capacity of coolant solution	$C_{PK}$	$2.0$	$\frac{kJ}{kg \cdot K}$
Starting concentration of subs. $A$	$c_{A0}$	$5.1$	$\frac{mol}{l}$
Inflow temperature	$\theta_0$	$104.9$	$^\circ C$

Reference solution

"The result of a steady state optimization of the yield $= \frac{c_B |_S}{c_{A0}}$ with respect to the design parameter $\theta_0$ (feed temperature) and the two controls yields the steady stae and controls" $c_A = 2.1402 \frac{mol}{l}, c_B = 1.0903\frac{mol}{l}, \theta = 114.19^\circ C, \theta_K = 112.91^\circ C$ and $\frac{\dot{V}}{V_R} = 14.19 h^{-1}, \dot{Q}_K = -1113.5 \frac{kJ}{h}$.

Source Code

Model descriptions are available in

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