Difference between revisions of "Continuously Stirred Tank Reactor problem"

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The inflow into the tank contains only cyclopentadiene (substance <math> A </math>) with temperature <math> \theta_0 </math> and the flow rate <math> \dot{V} </math> can be controlled. The outflow rate is the same as the inflow rate to keep the liquid level in the tank constant.
 
The inflow into the tank contains only cyclopentadiene (substance <math> A </math>) with temperature <math> \theta_0 </math> and the flow rate <math> \dot{V} </math> can be controlled. The outflow rate is the same as the inflow rate to keep the liquid level in the tank constant.
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"The outflow contains a remainder of cyclopentadiene, the wanted product cyclepentenol (substance <math> B </math>) and two unwated by-products, cyclopentanediol (substance <math> C </math>) and dicyclopentadiene (substance <math> D </math>) with concentrations <math> c_A, c_B, c_C, C_D </math>." The latter two are not tracked in the problem as the substances are not of use.
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"The outflow contains a remainder of cyclopentadiene, the wanted product cyclepentenol (substance <math> B </math>) and two unwated by-products, cyclopentanediol (substance <math> C </math>) and dicyclopentadiene (substance <math> D </math>) with concentrations <math> c_A, c_B, c_C, c_D </math>." The latter two are not tracked in the problem as the substances are not of use.
 
The reaction scheme is given as:
 
The reaction scheme is given as:
  

Revision as of 10:24, 24 July 2016

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^{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

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