4. Dynamic Behavior

We noted in the previous section that were three different steady-state solutions to the case 2 parameter set. Here we wish to study the dynamic behavior under this same parameter set. Recall that numerical integration techniques were presented in chapter 4.

The m-file to integrate the modeling equations is cstr_dyn.m, shown in Appendix 2. The command to integrate the equations is

[t,x] = ode45('cstr_dyn',t0,tf,x0);

where t0 is the inital time (usually 0), tf is the final time, x0 is the initial condition vector. t is the time vector and x is the state variable solution vector. Before performing the integration it is necessary to define the global parameter vector CSTR_PAR. To plot only concentration or temperature as a function of time, use plot(t,x(:,1)) and plot(t,x(:,2)), respectively.

 You may also run a JAVA based dynamic behaviour simulator to check out the different steady state solutions !

Initial condition 1

Here we use initial conditions that are close to the low temperature steady-state. The initial condition vector is [conc, temp] = [9,300]. The curves plotted in Figure 2 show that the state variables converge to the low temperature steady-state.

Figure 2. State variable responses with initial condition x0 = [9;300].

Initial condition 2

Here we use initial conditions that are close to the intermediate temperature steady-state. The initial condition vector for the solid curve in Figure 3 is [conc, temp] = [5,350], which converges to the high temperature steady-state. The inital condition vector for the dotted curve in Figure 3 is [conc, temp] = [5,325], which converges to the low temperature steady-state.

If we perform many simulations with initial conditions close to the intermediate temperature steady-state, we find that the temperature always converges to either the low temperature or high temperature steady-states, but not the intermediate temperature steady-state. This indicates to us that the intermediate temperature steady-state is unstable. This will be shown clearly by the stability analysis in section 5.

Figure 3. State variable responses with initial condition x0 = [5;350] (solid) and x0 = [5;325] (dashed).

Initial condition 3

Here we use initial conditions that are close to the high temperature steady-state. The initial condition vector is [conc, temp] = [1,400]. The curves plotted in Figure 4 show that the state variables converge to the high temperature steady-state.

Figure 4. State variable responses with initial condition x0 = [1;400].

In this section we have performed several simulations and presented several plots. In section 6 we will show how these solutions can be compared on the same ìphase planeî plot.