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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 θ0 and the flow rate 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 cA,cB,cC,cD." The latter two are not tracked in the problem as the substances are not of use. The reaction scheme is given as:

Ak1Bk2C2Ak3D

where the reaction rates ki are a function of the reactor temperature θ via an Arrhenius law ki(θ)=ki0exp(Eiθ/C+273.15),i=1,2,3.

"The temperature θK in the cooling jacket is held down by an external heat exchanger whose heat removal rate Q˙K can be controlled."


Mathematical formulation

The problem is given by

maxV˙,Q˙KcB at the end of reactions.t.cA˙=V˙VR(cA0cA)k1cAk3cA2,cB˙=V˙VRcB+k1cAk2cB,θ˙=V˙VR(θ0θ)+kwARρCpVR(θKθ)1ρCp(k1cAH1+k2cBH2+k3cA2H3),θK˙=1mKCPK(Q˙K+kwAR(θθK)),cA(0)=cA0,cB(0)=0.

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 k10 1.2871012 h1
Arrhenius coefficient k20 1.2871012 h1
Arrhenius coefficient k30 9.043109 h1
Arrhenius coefficient E1 9758.3 [-]
Arrhenius coefficient E2 9758.3 [-]
Arrhenius coefficient E3 8560 [-]
Reaction enthalpy H1 4.2 kJmol
Reaction enthalpy H2 11.0 kJmol
Reaction enthalpy H3 41.85 kJmol
Solution density ρ 0.9342 kgl
Capacity of aqueous solution Cp 3.01 kJkgK
Heat transfer coefficient for cooling jacket kw 4032 kJhm2K
Reactor surface area AR 0.215 m2
Reactor volume VR 10 l
Coolant mass mK 5 kg
Capacity of coolant solution CPK 2.0 kJkgK
Starting concentration of subs. A cA0 5.1 moll
Inflow temperature θ0 104.9 C

Reference solution

"The result of a steady state optimization of the yield =cB|ScA0 with respect to the design parameter θ0 (feed temperature) and the two controls yields the steady stae and controls" cA=2.1402moll,cB=1.0903moll,θ=114.19C,θK=112.91C and V˙VR=14.19h1,Q˙K=1113.5kJh.


Source Code

Model descriptions are available in