7. Understanding Multiple Steady-state Behavior

In previous sections we found that there were three steady-state solutions for case 2 parameters. The objective of this section is to determine how multiple steady-states might arise. Also, we show how to generate steady-state input-output curves that show, for example, how the steady-state reactor temperature varies as a function of the steady-state jacket temperature.

7.1 Heat generation and heat removal curves

In section 3 we used numerical methods to solve for the steady-states, by solving 2 equations with 2 unknowns. In this section we show that it is easy to reduce the 2 equations in 2 unknowns to a single equation with one unknown. This will give us physical insight about the possible occurance of multiple steady-states.

Solving for Concentration of A as a function of Temperature

The steady-state concentration solution (dCA/dt) = 0) for concentration is


We can rearrange this equation to find the steady-state concentration for any given steady-state reactor temperature, Ts


Solving for Temperature

The steady-state temperature solution (dT/dt = 0) is


The terms in (17) are related to the energy removed and generated. If we multiply (17) by VrCp we find that



Energy Removed by flow and heat exchange Heat Generated by reaction

Note the form of Qrem


Notice that this is an equation for a line, where the independent variable is reactor tempeature (Ts). The slope of the line is and the intercept is . Changes in jacket or feed temperature shift the intercept, but not the slope. Changes in UA or F effect both the slope and intercept.

Now, consider the Qgen term


substituting (16) into (20), we find that


Equation (21) has a characteristic ìSî shape for Qgen as a function of reactor temperature.

From equation (18) we see that a steady-state solution exists when there is an intersection of the Qrem and Qgen curves.

7.2 Effect of Design Parameters

In Figure 6 we show different possible intersections of the heat removal and heat generation curves. If the slope of the heat removal curve is greater than the maximum slope of the heat generation curve, there is only one possible intersection (see Figure 6a). As the jacket or feed temperature is changed, the heat removal lines shifts to the left or right, so the intersection can be at a high or low temperature depending on the value of jacket or feed temperature.

Notice that as long as the slope of the heat removal curve is less than the maximum slope of the heat generation curve, there will always be the possibility of three intersections (see Figure 6b) with proper adjustment of the jacket or feed temperature (intercept). If the jacket or feed temperature is changed, the removal line shifts to the right or left, where only one intersection occurs (either low or high temperature). This case is analyzed in more detail in section 7.3.

a. b.

Figure 6. Possible intersections of heat generation and heat removal curves.

7.3 Multiple Steady-State Behavior

In Figure 7 we superimpose several possible linear heat removal curves with the S-shaped heat generation curve. Curve A intersects the heat generation curve at a low temperature; curve B intersects at a low temperature and is tangent at a high temperature; curve C intersects at low, intermediate and high temperatures; curve D is tangent to a low temperature and intersects at a high temperature; curve E has only a high temperature intersection. Curves A, B, C, D and E are all based on the same system parameters, except that the jacket temperature increases as we move from curve A to E (from equation (7) we see that changing the jacket temperature changes the intercept but not the slope of the heat removal curve). We can use Figure 7 to construct the steady-state input-ouput diagram shown in Figure 8, where jacket temperature is the input and reactor temperature is the output. Note that Figure 8 exhibits hysteresis behavior, which was first discussed in chapter 15.

Figure 7. CSTR Energy Generated and Energy Removed as a Function of Reactor Temperature.

Figure 8. Reactor Temperature as a Function of Jacket Temperature.

The term hysteresis is used to indicate that the behavior is different depending on the ìdirectionî that the inputs are moved. For example, if we start at a low jacket temperature the reactor operates at a low temperature (point 1). As the jacket temperature is increased, the reactor temperature increases (points 2 and 3) until the low temperature ìlimit pointî (point 4) is reached. If the jacket temperature is slightly increased further, the reactor temperature jumps (ignites) to a high temperature (point 8); further jacket temperature increases result in slight reactor temperature increases.

Contrast the input-output behavior discussed in the previous paragraph (starting at a low jacket temperature) with that of the case of starting at a high jacket temperture. If one starts at a high jacket temperature (point 9) there is a single high reactor temperature, which decreases as the jacket temperature is decreased (points 8 and 7). As we move slighly lower than the high temperature limit point (point 6), the reactor temperature drops (also known as extinction) to a low temperature (point 2). Further decreases in jacket temperature lead to small decreases in reactor temperature.

The hysteresis behavior discussed above is also known as ignition-extinction behavior, for obvious reasons. Notice that region between points 4 and 6 appears to be unstable, because the reactor does not appear to operate in this region (at least in a steady-state sense). Physical reasoning for stability is discussed in the following section.

7.4 Multiple Steady-states: Stability Considerations (steady-state analysis)

Consider the system described by the energy removal curve C in Figure 7. Notice that the steady-state energy balance is satisfied for the operating points 3, 5 and 7, that is, there are three possible steady-states.

What can we observe about the stability of each of the possible operating points, solely from physical reasoning?

Lower Steady-State (Operating Point 3)

Consider a perturbation from operating point that is colder than point 3, say T3 - dT. At this point we are generating more heat than can be removed by the reactor, so the temperature begins to rise - eventually moving to T3. If we start at T3 + dT more energy is being removed than we are generating, so the temperature begins to decrease - eventually moving to T3.

The lower temperature intersection, T3, may be a stable operating point.

The open-loop stability cannot be assured until an eigenvalue analysis is performed (see section 5).

Middle Steady-State (Operating Point 5)

If we start at T5 - dT, we are generating less energy than is being removed, causing T to decrease, eventually causing the temperature to go to T3. If we start at T5 + dT, more energy is being generated than is being removed, causing the temperature to rise, increasing until T7 is reached.

The middle temperature intersection, T5, is an unstable operating point.

Here the steady-state analysis is enough to determine that point 5 is an unstable operating point.

Upper Steady-State (Operating Point 7)

Using the same arguments as T3, we find

The high temperature intersection, T7, may be a stable operating point.

Again, the stability of the upper steady-state can only be determined by eigenvalue analysis (section 5).

We have used physical reasoning to determine the open-loop characteristics solely from steady-state analysis. Recall that when we studied the case 2 conditions in section 5, we used eigenvalue analysis to find that the lower and upper temperature steady-states were stable, while the intermediate steady-state was unstable.

7.5 Generating Steady-state Input-Output Curves

There are two different ways to find how the steady-state reactor temperature varies with the jacket temperature. One way is to solve the nonlinear algebraic (steady-state) equations for a large number of different jacket temperatures, and construct the curve relating jacket to reactor temperature. It turns out that there is another, simpler, way for this particular problem.

Here we can easily solve for concentration as a function of reactor temperature. We can also directly solve for the jacket temperature required for a particular reactor temperature. Then we simply plot reactor temperature vs. jacket temperature. We show the calculations in detail below.

As in section 7.1, we can find the steady-state concentration for any steady-state reactor temperature, Ts (from the material balance equation for reactant A)

From the steady-state temperature solution (dT/dt = 0) we can solve for the steady-state jacket temperature, Tcs

Similarly, if we assume that the jacket temperature remains constant, we can solve for the required feed temperature

The function file cstr_io.m shown in Appendix 4 is used to generate the following input-output curves for the case 2 parameters.

»Tempvec = 300:2.5:380;

»[Tjs,Tfs,conc] = cstr_io(Tempvec);



Figure 9. Steady-state reactor temperature as a function of jacket temperature, case 2 parameters.

Figure 10. Steady-state reactor temperature as a function of feed temperature, case 2 parameters.

7.6 Cusp ìCatastropheî

In chapter 15 we presented a simple example of a cusp catatrophe which occurs when the input-output behavior moves from monotonic through a hysteresis bifurcation to multiple steady-state behavior. This can be shown clearly in the following diagram where we find that, as we increase F/V, we move from monotonic to multiple steady-state behavior.

Figure 11. Cusp Catastrophe for the Diabatic CSTR, Case 2 conditions. The input/output (jacket temperature/reactor temperature) behavior changes as a function of the space velocity (F/V).