This is an example of using a measured residence
time distribution and the idea of segregated
flow to calculate what the conversion would
be in a reactor system with that residence
time distribution.
So showing here this residence time distribution,
E(t), and it's represented by the green line
so it's a simplified residence time distribution
in order to make the calculations easier and
see the behavior and I've tried to indicate
a grid of dashed lines on the graphs because
one of the things we have to do is put a scale
on this in order to apply the idea of residence
time distribution to conversion.
So we want to know the conversion of A for
a second order reaction where we have the
feed concentration of A and we have the rate
constant of A. So the first thing we have
to do is understand what we mean by E(t) so
it's worth writing down the definition to
make this clear.
So here is what we mean by E(t).
So E(t)*dt looking at some sliver in this
diagram represents the fraction of fluid that's
exiting the reactor that has spent a time
between t and t+dt in the reactor.
And so E(t)dt if we integrate from 0 to infinity,
all these fractions should add up to 1.
And of course we're going to take advantage
of this scale that is given and we'll do this
in a simplified way just looking at the plot.
So I can look at this area and if this has
some scale that we're trying to determine,
call it A, then one of these cubes is 5a,
there's 3 of them, so 3 times 5a has to equal
1.
So a=1/15 and the unit's min^-1, so units
of time.
Units here are inverse time, the area then
would be dimensionless.
So I can replace a here by 1/15 and indicate
here that the units are minutes^-1.
I can use the residence time distribution
and the information from the reaction rate
and segregated flow.
So what's meant by segregated flow is that
we can imagine packets moving through the
reactor like little batch reactors flowing
through and they spend different amounts of
time, which is indicated of course by the
green plot here.
And we're going to collect at the end all
these packets to determine the average conversion.
So that means each packet we need to know
how much reaction takes place, if it spends
a certain amount of time in the reactor so
for segregated flow, we're envisioning we
have a large number of batch reactors that
are moving through the system and having certain
amounts of time to react.
And so we need to do a mass balance for a
constant volume batch reactor.
It's a second order reaction.
The rate constant was 0.2*CA and this is a
fairly standard calculation.
So we can separate variables and we can integrate
from 0 to some time t which will correspond
to a concentration at the outlet of CA and
the starting concentration of CA0.
So if we integrate and substitute in 1.5 mol/L
for CA0 and simplify.
So the integration is shown here with the
limits and we've rearranged to give the concentration
of A as a function of time in this batch reactor.
So now what we want to do is use this idea
of the residence time distribution of segregated
flow to calculate the average concentration
leaving the reactor and we use this notation.
We're going to integrate in general over all
times the concentration of A which is a function
of time * how long each of these little batch
reactors have to react, the residence time
distribution.
So if we go back and look at our plot of the
residence time distribution, the function
is zero everywhere except 5 to 10 and 10 to
15.
From 5 to 10, its value is 2/15, from 10 to
15, its value is 1/15.
And so we can just integrate over those two
ranges because 5 to 10 minutes, its value
is 2/15 and then we have a concentration as
a function of time, dt and then from 10 to
15, its value is 1/15, so we can do the integration
to get to log expression and this will end
up being first term, 2/15.
So I've substituted in the numbers here for
the integration and then using these limits,
get the average concentration leaving this
reactor assuming segregated flow.
So if we want the conversion, it's the inlet
concentration minus in this case the average
outlet concentration and this is (1.5-0.419)/1.5.
So the conversion is 72% of the feed is converted
in a system with this residence time distribution.
