Difference between revisions of "Continuously Stirred Tank Reactor problem"

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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
Link to Google Scholar.

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^{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{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

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