Every chemical engineer has been taught, or exposed to, the concept of multiple steady states for continuous stirred tank reactors (CSTR). However, it is very easy to ignore that concept when using a process simulation program.
You are designing a new process and the catalyst developer may have given you a recommended reactor temperature. The catalyst developer may have made that recommendation for a number of reasons, such as catalyst or product temperature sensitivity, or byproduct formation at higher temperatures. You have a kinetic model so you want to explore the process economics which means you need to simulate the process and see how energy and capital requirements change with the assumptions.
You build the flowsheet simulation and specify the CSTR reactor temperature, feed temperature, flow rate and reactor volume. The program computes the heat exchange required for those specifications. Eventually you recommend a design based on these results. By not checking the design for stability (i.e. multiple solutions), you may have just designed a reactor that will not perform as designed. Oops!
Why does the above error happen?
Besides forgetfulness, there are many reasons that can lead to this error.
First, the single example you were given in the undergraduate reaction engineering course probably used a very simple reaction network. The mathematics to execute a similar analysis with a complex reaction network was probably not covered. Thus, finding the unstable region for a complex, possibly non-ideal system seems overwhelming.
Second, the professor may not have shown how most (?) of the process simulators can be used to explore the multiple steady state potential without having to write your own program to develop the heat generation and removal curves. Another advantage of using the process simulator is that it readily includes all of the heat of solution, heat of vaporization etc. needed to compute the enthalpy of the reactor streams.
Third, the method commonly taught, using the intersection of heat generation and a heat removal curves is not the best way to present this material. Instead, a total enthalpy change curve and heat removal curve are better suited for a couple of reasons to be discussed below.
A better way
Before showing how you can use a process simulator to avoid unstable regions, let's explore the problem with Mathcad to show why the process simulator can produce impractical results. All that is needed to compute the outlet composition of a CSTR is the reactor temperature, the space velocity, and the feed composition. No heat balance is needed and thus the feed temperature is also not needed.
If we add the feed temperature as a specification, still no heat balance is needed to determine the outlet conversion. Instead, a process simulator will solve the mass balances without a heat balance, then it will determine the heat exchange needed to achieve the solution of the mass balances.
If your process simulator can form a case study with the feed and reactor temperatures as the independent variables, you can produce a contour plot of the heat exchange required as shown in the plot below.
The highlighted band shows the unstable region of outlet temperatures. This region is unstable because the slopes of the overall enthalpy change contours are negative. For a given feed temperature and a reactor (Tout) temperature, any deviation from the solution (say on a contour), will cause the Tout temperature to move up to the stable solution or down to a lower stable solution. A nice feature of this plot is that it readily illustrates that the unstable temperature range exists for any feed temperature or heat exchange rate.
The green arrow shows the minimum feed temperature needed to operate at the high conversion solution for adiabatic operation. The feed temperature is sufficiently high so that the green line only intersects the 0 enthalpy contour at one point. A similar method can determine the minimum temperature for other heat exchange conditions.
This plot is thus very useful for determining feasible operating conditions. The catch is that your process simulator may not allow the feed temperature as a variable in a case study so you can't use it to produce the plot. The simulator may assume that the temperature of the feed stream has already been set by other operations so that case studies must use the other reactor variables.
Using the simulator to examine stability
If your simulator can conduct case studies for the CSTR, it most likely will allow the reactor temperature to be an independent variable. If you have provided the feed conditions and the space time (i.e. feed rate and reactor volume), then the program can determine the enthalpy change for each outlet temperature. These results can be plotted in the following manner.
Using the same reaction system as in the previous plot, the red operating curve is developed. This curve is the enthalpy of outlet streams minus the enthalpy of inlet streams as a function of the outlet temperature on the y axis. As before, the unstable region (negative slopes) is between 330 and 360 K.
The vertical line is a chosen heat exchange rate. This exchange rate was not included in the simulator study. Instead, it was added to the plot after the study. This line can be moved in either direction to achieve a stable design. In the example, a cooling rate of -50,000 J/kg is shown. With this cooling rate and the 320 K feed temperature, three solutions are possible.
Suppose you had simulated the reactor with this feed temperature and specified a reactor temperature of 347 K (the middle solution). The program would have returned -50,000 J/kg as the needed heat exchange requirement. Without performing the case study, you may be in deep trouble.
Caution: know the reference values used by your simulator
Simulators often use a common enthalpy at a low temperature for all compounds instead of their heat of formation. Thus, the stream enthalpy can measure only sensible heat changes. You need to include the heat of reaction to develop the operating curve. The simulator can provide this for the case study and you should be able to generate the operating curve.
Another advantage of this method
The heat generated-heat removed plot often does not show the intersections (i.e. solutions) very well. The two curves in that method are slanted and the "S" curvature in the generation curve can be very narrow compared to the manner shown here. The results are intersections of nearly parallel lines which yield imprecise results by visual inspection. The vertical heat exchange line is also much easier to understand and use in my opinion.
FYI, comparing contours and operating curves
You probably have noticed that the heat exchange contours in the first plot look almost identical to the operating curve in the second plot. They are not the same curve because the first is a curve on a temperature-temperature plot and the second is a curve on an enthalpy-temperature plot. The operating curve is the enthalpy exchange values at a constant feed temperature (i.e. along a vertical line in the first plot). Both curves convey the same information regarding the unstable region.
There's no way around it...
You can't avoid doing a case study by specifying the heat exchange instead of the reactor temperature. Even though this method couples the heat balance with the mass balances, if the conditions you provide have a solution in the unstable region, the program may still find that solution. Depending upon the initial conditions, you may get lucky and the simulator converges to a stable solution, but that is not guaranteed. To be certain, examine the system using the case study method described above.
It is possible that some of the tutorial materials for your simulator may show this method for determining the stable and unstable regions. Alas, a lot of great tutorial materials get overlooked.