In this screencast we're going to look at how to solve reactor design problems, just
the mole balance part of those problems, and this is just an overview of the approach for
ideal reactors and then we'll provide links to more details for each of the types of reactors
and links to example problems that go through the actual calculations. So a few things that
we want to keep in mind about most chemical reactors and that affects the approach we're
going to use to model these reactors is that most reactors have multiple reactions taking
place simultaneously, and the implication of this is that the calculations can become
complicated and you end up solving most of them numerically because we have a number
of mole balances. Also most, not all, but most large scale reactors are non-isothermal
which means we also have to write an energy balance. Because of these complications most
reactor models are going to be solved numerically. We can't get an analytical solution. So then
a couple important aspects, some things we need to keep in mind when we write down the
mole balances for these reactors are the following. One is that we write mole balances on individual components.
That is we don't write just one mole balance for each reaction, so if we have
three reactions we have three moles balances, instead we might have six or seven mole balances
depending how many components are in each of these reactions. So the mole balances are
written on individual components. So if I have species A, B and C, I'll write down a
mole balance for A, another mole balance for B and another mole balance for C. That is
we're writing one balance for each component. So let's look at the best variables to use
when we write down these mole balances and we'll do it for each of the ideal types of reactors.
And then we'll look at what's the starting equation we should use for these mole balances.
We'll start off with the batch reactor and also the semibatch because they
really have a lot of similarities. And so the best variables to use are the number of
moles of each species and that's for the batch reactor. Also then for the semibatch we need
to use the molar flow rate. So this is total number of moles of course in the reactor and
this is moles per time and the general form then of the starting equations, say for component
A, change in the number of moles of A respect to time, this is an accumulation term, is
equal to the rate of reaction and that's for a batch reactor. For a semibatch we may have
additional terms such as flow of A into the reactor or flow of A out. We may have one
of these terms we could have both of these terms or we could not have any flow in of
A but we might have flow of B into the reactor. For a continuous-stirred tank reactor we can
certainly use the molar flow rates of A but often in most cases these are liquid phase
so using the concentration is also a reasonable variable. The general mass balance for
a steady state CSTR is a molar flow rate of n of A the molar flow rate out and then the
rate of reaction times volume. And keep in mind this rate expression r_ A can be positive
or negative depending on if A is a reactant or a product. The third type of ideal reactor
is a plug-flow reactor. Here the best variable typically to use is the molar flow rate and
the typical form of the starting equation, again we're looking at a steady state reactor,
is the change in that molar flow rate as a function of cumulative volume is equal to
the rate of reaction. So one thing to note, if we look at the equation just for the batch
reactor which is what I've circled, compared to a plug-flow reactor, these equations are
essentially of identical form where the time spent in the batch reactor, the time we have
available for reaction, corresponds to the time it takes to move through the plug-flow
reactor. So these equations are really quite similar and are solved essentially the
same way. Now I've indicated the variables that we could use. Well we could use fractional
conversion. So fractional conversion is the flow rate of A in, for example in a plug-flow
reactor, flow rate leaving over the flow rate in. Well we could use this where this is the
flow rate at any point in the reactor and then that's the fractional conversion at any
point. We could use this as out variable instead and that works for the simple cases but it
does not work as soon as we introduce additional complications. If we have multiple reactions
for example, it doesn't work. If we have a semibatch reactor that's not a good variable.
If we look at the start up of a CSTR we've indicated here the steady state equation but
if we look at the start up and what's going to be different for the start up is we'd have
instead of 0 on the left side of this equation we have the change in the number of moles
of A with respect to time, an accumulation term. Or if we look at a plug-flow reactor
where there's a side stream either entering or leaving, such as a membrane reactor, then we
can't use conversion. So our general approach then to solving mole balances in reactors
is first we start with the mole balances, and we write as many mole balances as we have
species in the reactor. Next we substitute into these equations for the rate of reaction.
For example, if we have A goes to 2B and the rate of reaction is the rate constant times
the concentration of A squared, there's a minus sign for A cause it's a reactant. The
rate of B plus, and there's 2 for the stoichiometric coefficient, we would substitute these into
the appropriate equation keeping in mind the stoichiometric coefficients. The third thing then
that we would then do, if we have gas phase reactions, a flow reactor which is typically
how we carry out gas phase reactions, we have to account for the fact that often the volumetric
flow rate changes as we move through the reactor. For example, in a plug-flow reactor the volumetric
flow rate changes because the temperature changes. As temperature changes we think of
ideal gas law then the volume: as temperature increases the volume increases. If it's a
packed-bed reactor and the pressure is dropping, this means the gas expands and occupies more volume.
This affects the concentration therefore it affects the rate of reaction, and then if
we have mole change. So this reaction A gas phase goes to 2B in the gas phase this mole
change is going to affect concentrations. As we move through the reactor the concentration
of A is decreasing because of the reaction but it's also decreasing because it's getting
diluted by B and this is going to slow down the reaction more than it would be otherwise
and we have to account for that. So the way we account for this is if we write down
for example we can use an equation of state, I'll just write it for ideal gas equation of
state, the concentration is a function of pressure, temperature, ideal gas concentration and the
mole fraction of A which in a flow reactor we can write for the simple case of just A
and B present. The ratio flow rate of A over the flow rate of A plus B is the mole fraction
of A and then pressure over RT. So we'd use this for each of the components that we have
in the rate expressions and then the next thing we do is solve these equations numerically.
Of course this also makes it easier to change conditions and look at what happens. So keep
in mind when we use concentrations that these are local concentrations in the reactor. What
I mean by that, if we had a liquid phase reaction that formed a solid product. If we had A liquid
goes to B solid and the solid does not dissolve in the liquid so it separates out or it's
carried through in the slurry form then the concentration of A actually would not decrease as the reaction
took place and this would affect the rate of reaction. So in this screencast we
provided links for the various reactors that goes through details of where these mass balances
came from and then how to use them in calculations, but here we're trying to give you an overview
of writing mass balances for chemical reactors.