In this problem, we're given information about the rate of reaction in terms of a Levenspiel
plot, and we see it has sort of a funny shape, and that's to make the analysis easier in terms of the
math. This was previously an exam problem. We have 3 different reactors: a 20 liter PFR,
15 liter CSTR, and 25 liter CSTR and we're asked first what single reactor, if we use
only one of these, would result in the largest conversion and what would be the final conversion
for that reactor. We can write down the material balances for CSTR's and PFR's because a CSTR
operates only, within its entire volume, only at the outlet conditions, the required volume
to achieve a given conversion is a function only of the exit conditions. And so therefore,
for example, if we're going to determine what conversion we can achieve in a 15
liter CSTR, then we can put 15 liters as the volume, and then we can find the value of
x that we can multiply x by the experimental value of FA0*-rA at the exit in order to get
15 liters. This involves a trial and error procedure, but fortunately this is a fairly
simple process for what's given in the problem statement. I happen to know that if I pick
a conversion of 30%, what I can read off the plot is that experimentally, FA0 over the
reaction rate is equal to 50, and so 0.3*50L is equal to 15 L. And so therefore, we know
that the 15 liter CSTR gives us a conversion of 30%. Next, we can consider the 25 liter
CSTR. it turns out that if we pick a conversion of 50%, again, we use only the exit conditions,
we go over and read off the same reaction rate. And so we see that we achieve a conversion of 50%
in a 25 liter CSTR which is greater, not surprisingly, than in the smaller CSTR of 15 liters. Now,
for a PFR, we have a different material balance expression where the volume of the PFR, the
required volume for a given conversion, does not depend just on the exit conditions, but
as the integral of all the conditions up to the final conversion. And that corresponds
to the area under the curve up to the final conversion. Alright and so it turns out that
again, if we pick a conversion of 30% and we look at the area under this curve, that
will be equal to, we can basically break up this evaluation of the area under the curve
into a rectangle that's shown here and then a triangle that's above the rectangle. Okay,
and so the rectangle will be 0.3 conversion times the 50 for the height and so that's
the first term we show here. The second term, again, we have a height of 50 and a length
of 0.2. And since this is a triangle, we take half the base times the height, and so that
gives us the 20 liters. So we see that the 20 liter PFR gives us a conversion of 30% which
is well below that which is achieved in the 25 liter CSTR. In part b of the problem, we
are asked if you were to use two of the three reactors in series, which two would you use
and in which sequence would you use them. We've already seen that if we use either of
the CSTR's, it's probably going to make sense to use the 25 liter CSTR because it allows
us, since it's larger, to achieve a higher conversion than its counterpart. It actually
allows us to go all the way to this conversion of 50% and we notice there is a discontinuity
here and that beyond this point, what's happening is that the reaction rate decreases with conversion.
And for situations where the reaction rate decreases with conversion, it's usually advantageous
to use a PFR rather than a CSTR. The reason for that is that, say we were hypothetically
going out to this conversion. Alright, then the volume that would be required for a CSTR
would be much greater than that required for a PFR, so if we were trying to achieve this
extra bit of conversion, because the volume required for a PFR is proportional to the
area under the curve, where as the volume required for a CSTR would be proportional
to the area of this entire rectangle including under the curve. Using a PFR in this region,
beyond 50% conversion, is advantageous, whereas a CSTR is clearly advantageous in this
region where the opposite is true, where the rate actually increases as the conversion
increases. In this range, it doesn't matter which we use, zero-order type behavior. By
using first, the 25 liter CSTR and coming to a conversion of 50%, you utilize a CSTR
where it is most advantageous to do so, and then where it doesn't matter and then after
that, we can use our 20 liter PFR. And so our solution to part b of this problem is
that we will use the larger CSTR first then the PFR.