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

(15)

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

(16)

**Solving for Temperature
**

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

(17)

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

(18)

Qrem=Qgen

Energy Removed by flow and heat exchange Heat Generated by reaction

Note the form of Qrem

(19)

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

(20)

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

(21)

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.

**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.

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);

»plot(Tjs,Tempvec)

»plot(Tfs,Tempvec)

**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.