In this screen cast we're going to look at the arrangement of two isothermal reactors
in series, a plug flow reactor and a continuous stirred tank reactor, and the question we want
to answer is, does the order that we arrange them, whether it's the plug flow reactor or
the CSTR that's first, affect the final conversion? And the short answer is yes it does. For example,
in a second-order reaction we're going to get a higher conversion if the plug flow reactor
is first so we can carry out reaction at the high concentration of reactant where we get
a high rate of reaction. However, for a first-order reaction it does not matter. This is a result
of the equations being linear for the mass balances and what we want to do in this screencast
is work through and look at those equations. So here's the two situations that we want
to analyze. A CSTR followed by a plug flow reactor, a plug flow reactor followed by a
CSTR, the volumes of the plug flow reactor in the CSTR are not the same, and the reaction,
first-order and so what we're going to do is write the mass balances. First we're going
to write the mass balance for the CSTR. So it's steady state so there's no accumulation.
There's flow of A into the system, feed concentration C_A0, small v is the volumetric flow rate.
So this is the molar flow rate in. The molar flow rate out, we're looking at a liquid phase
system so the volumetric flow rate's the same, C_A1 and the difference is a result of the
fact that we have reaction which is at concentration C_A1 because it's a CSTR and then the volume
of the CSTR. Now we can arrange this equation to solve for C_A1 and so the concentration
at the exit of the CSTR is linearly proportional to the concentration entering the CSTR because
it's first-order. For the plug flow reactor we have a differential equation. The volumetric
flow rate, the change in the concentration as a function of cumulative volume as we
move down the plug flow reactor and again a first-order reaction. Now this we're going
to rearrange and integrate, and so note that I've integrated along the cumulative volume
V_P. The inlet concentration is C_A1, the outlet concentration is C_A2 and so this gives
me an equation relating concentration to the volume and the volumetric flow rate. So after
integrating then I've rearranged and again you'll notice that C_A2 is linearly related
to C_A1 and then C_A1 we obtained from the CSTR so we're going to write down the result
for C_A1 and now we have an equation that relates the exit concentration from the two
reactors in terms of volumetric flow rates, the volumes of the two reactors and the rate
constant, but the important thing to notice is this linear relationship. We're going to
do the same calculations for the reverse order and observe similar behavior. So I've written
down the equation now for the system here, the plug flow reactor. So the inlet concentration,
C_A0 and the outlet, C_A1. So I get a similar relation to what I've derived before and then
similarly C_A2 is going to be related to C_A1 by the same equation as here. But now the inlet
to the CSTR is C_A1, the outlet is C_A2 and then I can substitute for C_A1 into this equation
and you can see that we get the exact same equation for the outlet concentration of the
reactant when we have the plug flow reactor first as when we have the CSTR first. And
you can also observe from looking at the equation the linear relationships we have here that
C_A1 is linearly proportional to C_A0 and then we have here that C_A2 is linearly proportional
to C_A1. Thus switching the order does not make a difference and we're going to get the
same conversion for these two systems for a first-order reaction, only for a first-order
reaction. The only other type of reaction where we get the same conversion would be
a zero-order reaction because that doesn't depend on concentrations at all.