Alright in this problem we're going to show how to use a Levenspiel, or reciprocal rate
plots, to solve for the volume that's required in CSTR and PFR reactors, or the required batch
time if we're dealing with a constant volume batch reactor. Okay we're going to start by
considering a CSTR and we'll write a material balance for this CSTR where we just feed in
components A, B, might have other components, and then we'll expect it to react our reactants
A and B and form some products. Now as a reminder, what helps us in writing material
balances for CSTRs is that we have a well-mixed tank and the conditions everywhere in the
tank are uniform. So the rate of reaction is uniform everywhere in the tank, temperature,
concentration and so forth. We'll write a material balance in the form, in minus out
plus generation equals accumulation and we'll write this in terms of the component A. So
we have an inlet flow rate of A, F_A0, and an outlet flow rate, F_A and so our out in
minus out is F_A0 minus F_A. Our rate of generation is our rate per unit volume, r_A times the
volume of the reactor and then at steady state our accumulation term is equal to 0. We can
now use the definition of conversion. That, it's equal for a flow reaction to F_A0 minus F_A
over F_A0, so the amount reacted over the amount fed of component A, and this allows
us to simplify the above expression to F_A0 X is equal to negative r_A times V and we
can rearrange things a little more into a different form, where the volume divided by
F_A0 is equal to the conversion times the reciprocal rate of consumption of A, the negative
of the rate of generation which is the rate of consumption. Okay we see that we can determine
the volume, which is often what we want to solve for, in terms of the conversion and then the
reciprocal rate multiplied together. The reciprocal rate we're going to need to get from experimental
data and sometimes we even plot that experimental data in terms of the conversion, if we're at
a fixed temperature for example, and so we can plot just the rate verse conversion but
it's often convenient, because of the form of this equation, to plot the reciprocal rate
as a function of conversion. So that plot will often look something like this, that
the rate tends to go down as we convert more and more of our products because, there will
be more and more reactants to products because there are fewer reactants to react. Okay and
so therefore the reciprocal rate actually goes up with conversion. If we take one over
that rate that's decreasing with conversion then we get a function that's increasing with
conversion, and here we're assuming that we have constant temperature because the reaction
rate of course is also a very strong function of temperature. And so if we want to size
a CSTR we know that we need the conversion times the reciprocal reaction rate and that's
basically defined by the area of the rectangle that we just drew with the dashed line. Alright
and it's a reciprocal reaction rate which is the length of the side of the rectangle
and then the length of the base of the rectangle is the conversion that's achieved in the reactor.
Alright and this works because the reciprocal reaction rate is uniform everywhere within
the CSTR and we'll see that things are different when we look at a tubular reactor. And so we'll
just draw that the volume divided by the inlet flow rate of A is equal to the area of
that rectangle when we plotted up our experimental data in terms of the reciprocal rate verses
conversion of component A. Alright this was for a CSTR. Let's look at a PFR or a tubular
reactor. We'll have the same form for the material balance except that it's going to
be in terms, as we've shown before, of an integral function, where we get dX over negative
r_A, integrated from 0 to the final conversion in the reactor and so the reciprocal reaction
rate here changes down the length of the reactor because the reactor is not perfectly mixed
down the entire length of the reactor, and so we have to integrate the reciprocal reaction
rate over dX and it's not at a fixed value as it is for a CSTR and so we can look at
this graphically as the area under the curve for this reciprocal reaction rate plot evaluated
from 0 conversion up to the final conversion, and we see that what's typical, although not
always observed to be the case, is that that area is less than the area associated with
the CSTR, and therefore the volume required for a PFR to achieve a given conversion is
usually less than the volume required for a CSTR. Alright so the CTSR is again the area
of the rectangle. The PFR is the area under the curve and this difference is a consequence
of the fact that the reaction rate is uniform everywhere in a CSTR and varies down the length
of a PFR. Now we can also use this same form of experimental data to evaluate the required
time for a constant volume batch reactor system. We can again do a material balance on A which
we've shown previously in terms of conversion, and it can be written in this fashion for
a batch reactor, that we have actually an accumulation term shown on the left and no
inlet or outlet terms because it's a batch reaction system but we have a net generation term on
the right. We can rearrange our terms a little bit and put it into an integral form and we
get the expression shown here where again we get this integral of 1 over the reaction
rate, or in other words the reciprocal reaction rate, times dX. This looks exactly the same
as you'll notice as a PFR on the right hand side and so if we can evaluate the area under
the curve on a Levenspiel plot, then we can evaluate the required batch time. Alright
for a constant volume batch reactor system we'll usually write this N_A0 over V as a
C_A and then again we can use the batch time divided by the concentration, inlet concentration
or beginning concentration, C_A is equal to that area under the curve.