We will explain multiple steady states and how they arise for a simple system, a continuous
stirred tank reactor operating at steady state, and some of the notation, C_A is the exit
concentration, the volumetric flow rate, v_0, and v_0 leaving, so we can just call this
v then.
C_A0 is the inlet concentration of reactant, and C_A is the outlet concentration.
Reactor volume is V, and we'll look at a first order reaction, a simple case, and liquid
phase, exothermic, exothermic is very important.
If we're going to have multiple steady states for simple-type kinetics like this, then it
has to be exothermic, and we'll also look at the case of adiabatic.
Certainly there can be multiple steady states when there's heat transfer, but in terms of
simplifying the equations so we can see the behavior, that's the situation we'll look
at.
And so, we also want to compare this to a plug-flow reactor, and so in just a very free-hand
drawing, let's look at how we would imagine a plug-flow reactor.
Reactant concentration changes as we move down the reactor, so tau is residence time,
we increase the residence time as we move further down the reactor.
So what's going to happen, for an adiabatic reactor, the rate is going to be relatively
low, which means the concentration doesn't change very much, but as the reaction takes
place, the heat of reaction heats up the reactants which increases the rate constant exponentially,
so the rate starts speeding up, which means the concentration drops faster, and then we
have a compensating effect that we're running out of reactants.
So even though the temperature is increasing, the concentration of reactant now is not decreasing
as fast.
And as we'll see with the equation, there's a linear relation between how much of A reacts
and what the temperature is, if you'd like, the conversion.
So the temperature, forget about the starting value, but this would be the inlet temperature,
just like this here is C_A0 coming in, and this is 0, this is T_0 coming in, so the temperature
initially is not changing very fast, but as reaction proceeds and the temperature gets
hotter, the rate becomes faster, so the temperature rate increases, and exactly the shape, although
I won't be able to draw it that well, goes up and then levels out.
If I flip this upside down, it should look like this curve.
Alright, so that's plug flow reactor, and we want to compare the plug flow to the CSTR.
For plug flow, there's only one solution, given residence time, there's only one concentration.
CSTR is different.
If we were to make the same plot for a CSTR, it's going to, and again we're coming in at
the same concentration, it's going to look like this.
And this means the temperature again is going to mimic this, flipping it upside down, so
the temperature is going to show the same type of behavior.
So here's a CSTR, exothermic reaction, and adiabatic.
When I say these are exact, but just inverted, that's assuming constant heat capacities and
constant heats of reaction as we'll see shortly.
But the thing to notice is that if I look at some residence time, there's actually three
concentrations for one residence time.
We don't see that for a plug flow reactor, but we see that for a CSTR, and what that
means is there's three solutions to the reactor equation, three concentrations, and correspondingly
three temperatures, if I draw that better, three concentrations, three temperatures for
the same residence time.
So the important thing that we see is multiple solutions.
That's for the CSTR.
There's only one solution for the plug flow reactor, and the difference between these
two is the feedback.
Because we're operating at the final temperature, the final temperature can be high for an exothermic
reaction, that makes the rate fast, and that kind of feedback allows us, depending on where
we start, but which condition we end up with at steady state, because we're looking at
steady-state solutions, right, these are all steady-state, but the actual operating conditions,
by operating conditions I mean the concentration at the exit, the temperature at the exit,
the operating conditions depend on how we start the reactor, meaning that we could start
with a low temperature in the reactor, we could start with pure reactant, or we could
start with 90% product, or we could start with a high temperature in the reactor.
Depending on where we start is going to determine where we're going to end up.
Okay, now these equations, this feedback, the multiple solutions result because of the
non linear equations, namely the mass and energy balances.
So non linear equations and feedback due to mixing.
So, as we saw, we could have three solutions for this simple case, but not all solutions
however are the same.
In other words the solution, so there are actually three types of solutions.
It doesn't mean those three solutions are one of each type, it means we have the possibility
of each of those solutions being one of these types, and the one you might expect is stable
points, namely, that will run at steady state at that condition.
But a solution can also be an unstable point, meaning, unless we use some sort of controller,
it's not going to stay at that location.
It's a solution of the mass and energy balances, but any perturbation kicks it away from this
solution.
But, we can certainly operate at this point, but it takes some sort of controller.
And then, the third type of possible solution is something known as a limit cycle, which
means we have oscillations around the point and they're sustained, so they'll continue
for a long time.
So this is just the introduction to the idea that we can have multiple states for a CSTR.
If we look at the system, we'll see for a simple system, this would be a stable point,
this will be a stable point, and this would be an unstable point.
It's not obvious how we know that from just this plot, but we'll see when we look at more
details to explain this, we can solve the unsteady state equations to determine whether
something is a stable point or not.