In this screencast we're going to look at a reaction taking place in a steady-state
CSTR, but we're going to have multiple solutions to the mass and energy balances.
This is a CSTR with heat exchange, so we're given various properties and flow rates.
Note that in addition to two reactants, A and B, we also have an inert, methanol, that
we need to take into account.
The reaction is first order in A and zero order in B, and we're given a rate constant,
an activation energy, and a heat of reaction, and so we should look at all this information
about what's known about, in this case the heat transfer coefficient times the area for
heat transfer is given, and the heat transfer temperature which we're going to assume constant
as a reasonable first approximation.
So the question is, what are these steady-state operating conditions?
What we'll calculate is the temperature and the conversion that are the multiple solutions
to this particular CSTR.
Now, the first thing we want to do is make a drawing that represents the process to make
sure we have all the information, and list the important parameters that we're going
to use.
So I've drawn a diagram to represent the process, listed the feed conditions, the unknowns,
namely the temperature and the conversion leaving the reactor.
I should also indicate the volume of the reactor, remember this is steady-state, and then I've
listed the various parameters that we're going to use in the solution.
So now the first thing I want to do is write down a mole balance, because my approach to
solve this when I know I have multiple solutions is to write down the mole balance and the
energy balance, and to plot the conversion versus temperature that I obtained from each
of those equations, and the intersections are the steady-state solutions and that's
an easy way then to get the solution using something like Excel to make the plot.
But first let me look at the mole balance, and I'll do the balance on A, the reactant,
steady-state, no accumulation, A flows in, A leaving is what we don't know, then we have
a rate of reaction per volume times the volume.
So we can write this, since we want to be able to calculate the conversion, a function
of essentially temperature, F_A0, that's a constant, I'm going to write this as the volumetric
flow rate times a concentration.
Since this is liquid phase, the volumetric flow rate is going to stay constant at 25
cubic meters per hour, and then the rate of reaction, this is first order, minus sign
because reactant, k, concentration times volume.
Now, I've written it this way so now I can solve for the concentration of A, so let me
just rearrange and solve for the concentration of A. So I've solved for concentration of
A in terms of k, flow rate in, the volume of the reactor and volumetric flow rate.
Notice that the concentration leaving is the same as the concentration in the reactor with
a well-stirred CSTR.
Now I want to calculate conversion.
That turns out to be an easy parameter to plot as a function of temperature to solve
this, so first I'm going to calculate the flow rate.
So the flow rate of A leaving is going to be the volumetric flow rate times the concentration
of A, substituting in here F_A0 over the volumetric flow rate, k times the volume.
And then I can rearrange this, so I've written the flow rate leaving the reactor in terms
of the entrance flow rate, the rate constant, and now the space time, the volume in the
reactor divided by the volumetric flow rate.
Now I can then calculate the conversion, inlet flow rate of A, exit flow rate of A, over
the inlet flow rate of A, and if I substitute this into the equation, and now for an expression
for X, when I do the algebra, and tau, my space time, is 20 meters cubed over 25 cubic
meters per hour, so then tau is 0.8 hours, and the rate constant times tau, I'm going
to calculate that, so I have a value of k times tau which is dimensionless, and we're
going to substitute this into this expression.
I'm going to make a plot of the conversion versus temperature, because temperature is
the only variable now in k times tau, and I can plot that as one half of the plots that
I need to determine the solution.
So now I need to solve the energy balance.
I'm going to write the steady-state energy balance.
Steady-state, so there's no accumulation term, there's a term due to the fact that we're
flowing A, B and M into the reactor, so flow rate of A, heat capacity of A, then likewise
for B, and then likewise for the inert M, heat capacity of M, so we can look at this
term as the energy that it would take to heat up, if the reactor temperature was higher
than the feed temperature it would require some energy to heat up the feed, and then
we have the heat of reaction times the rate of reaction times volume of the reactor, and
then we have a heat transfer term, cooling temperature minus the reactor temperature.
I'm going to try and simplify this and make the terms a little easier.
F_A0, F_B0, F_M0 are equal, so I can write them all as F_A0, then times alpha, where
alpha is just the sum of these three heat capacities, r_A times V is F_A minus F_A0.
R_A times V is a negative number because r_A is minus k C_A, so F_A minus F_A0 is negative,
and that's if we plug in a mass balance that's the equality we would get.
Now I'm going to divide this by F_A0 and simplify.
Now I'm going to do a few things.
I'm going to calculate the value of alpha, I'm going to calculate the value of UA over
F_A0.
Also notice that this term here is minus the conversion.
And so I'm going to calculate these values and then substitute in.
So I'm going to substitute alpha, and then this ratio and X into the equation, and then
I'll rearrange it.
I've made the substitution, checked the units, they're joules per mole for each of the terms
we're adding together and now let's just simplify this equation.
And so now, I have an equation for X terms of T for an energy balance, I also have X
in terms of temperature from the mole balance.
What I'm going to do is just plot those with Excel, and then I'll show you the plot that
I get.
So I've plotted conversion versus temperature with Excel.
This is the energy balance, straight line.
This is the mole balance.
These are the steady-state solutions.
This is an unstable one, but for now we're just interested in, you can read off the graph
at these three temperatures, here are the corresponding conversions.
This is the objective of the problem, to determine what the steady state solutions to the mass
and energy balances are.