# Unstable systems

## Nonideal continuous stirred-tank reactor

- Details
- Last Updated on Thursday, 24 October 2013 08:41
- Published on Wednesday, 23 May 2012 13:58
- Written by Jaroslav Kolařík
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**System description**

This process is represented by a continuous stirred-tank reactor (CSTR) with nonideal mixing described by Cholette’s model. The process can be illustrated according to the following figure.

**Parameters:**

Presented parameters are described in detail in the mathematical model given below.

n = m = 0.75, V = 1 l, k1 = 10 s-1, k2 = 10 l/mol, q = 0.033 l/s

A nominal operating point (for the linearization):

cf = 6.484 mol/l, ce = 1.8 mol/l, c = 0.2387 mol/l

**Mathematical model**

A simplified mathematical model of the process dynamics can be described by the following nonlinear differential formulas (Liou & Chien, 1991; Padma Sree & Chidambaram, 2006):

where *C*(*t*) is the concentration of the reactant in the well-mixed zone, *C _{e}*(

*t*) is the concentration in the exit stream (controlled variable), and

*C*(

_{f}*t*) is the feed concentration (manipulated variable). Further,

*n*is the fraction of the reactant feed that enters the zone of perfect mixing, and

*m*is the fraction of the reactor total volume

*V*where the reaction occurs. Constants

*k*and

_{1}*k*describe the reaction rate, and

_{2}*q*is the inlet flow rate.

Linearization of the nonlinear model around the above given nominal operating point gives the transfer function model as:

**Simulation files**

Downloading files is possible only for registered users.

**Information sources**

PADMA SREE, R., CHIDAMBARAM, M. *Control of Unstable Systems*. Oxford: Alpha Science, 2006, 297 p. ISBN 1-84265-287-7.