Transient Behavior of an Adiabatic CSTR
Consider an irreversible first-order reaction, , taking place in an adiabatic CSTR (continuous stirred-tank reactor). The relevant equations derived from the mass and energy balances are the following:[more]
where and are the reactor's temperature and conversion fraction, is the reactor's residence time, is the rate constant, , , where is the heat of reaction, is the fluid density, and is the fluid heat capacity. It is assumed that and .
The reactor's feed has a concentration, , and a temperature, , chosen equal to 2 moles/liter and 300 Kelvin. The reactor's initial temperature is set by the user. The reactor is initially filled with either pure solvent (i.e., ) or with pure feed (i.e., ). Three steady states are found if one solves the nonlinear system of equations, which is obtained by setting the left-hand side of the governing equations equal to zero. These steady states are the following:
SS1: high conversion with and
SS2: intermediate conversion with and
SS3: low conversion with and
A stability analysis reveals that only the first and last steady states are stable (i.e., SS1 and SS3). Indeed, the Jacobian matrix is the following:
Eigenvalues for the three steady states can be computed and are equal to:
1/ for SS1: and (both are negative)
2/ for SS2: and (eigenvalues have opposite signs)
3/ for SS3: and (both are negative)
This Demonstration plots the transient behavior of the reactor's concentration and temperature. It is clear from the snapshots that only SS1 and SS3 can be reached depending on the initial condition. One snapshot shows that for certain initial conditions the trajectory approaches SS2, then goes to SS1. Indeed, SS2 is a saddle point. One can conclude that adiabatic CSTR reactors show very interesting behavior such as steady state multiplicity.[less]
L. D. Schmidt, The Engineering of Chemical Reactions, New York: Oxford University Press, 1998.