In this problem, we're going to look at determining residence time distribution for making tracer
measurements. The idea is that we inject something that we can detect its concentration at the outlet,
could be a dye, a radioactive tracer. Inject the slug to a feed stream for a continuous
reactor, a steady-state reactor, in this case a stirred tank, and we measure concentration
at the outlet as a function of time and then what we want to do from these measurements
is determine the residence time for the reactor and the residence time distribution. So here
are the data for measurement as a function of time, every 120 seconds, and the tracer
concentration, so low concentration measured at these times. And we get the averaged residence
time by using this equation. So the average time, it's essentially just a weighted average,
integral of time times the concentration which is a function of time over all times from
zero to infinity divided by the integral of the concentration. What we're doing here is
we're normalizing because our average residence time should not depend on which tracer we
use and which concentration of tracer. As long as it's low concentration, it has the
same properties and mixes with our feed, then it should not depend on which tracer we use.
We can also get then the spread. In this case, we're looking at integrating the time minus
the average time and then this is squared so it's always a positive value, again times the concentration
as a function of time and then normalized by the concentration as a function of time.
This will give us sigma squared, the square root of that is the standard deviation. We're
going to do these measurements with a spreadsheet and Simpson's rule and so I'll give you the
answers. We determine the time is 371 seconds and sigma squared is 3.23 X 10^4 s^2 and sigma
is 180 seconds. Keep in mind this is not accurate to three significant figures when we look
at what the data looks like. So we look at the plot of the data. We don't have measurements
often enough, enough time intervals, to really get the kind of accuracy we might like of
3 significant figures but this gives us a good idea of what the average residence time
is. And so to do the integration, so we mentioned we used a spreadsheet, and let's look at how
that works. So what we'll do, we'll divide it up into increments and we'll first calculate
this area and with Simpson's rule, the way we're going to do that calculation, we're
going to take the value at time=0 plus 4 times the concentration at 120 seconds. So this
notation means we measured at 120 seconds, and then the concentration at 240 seconds,
so 1 times this value, 4 times this, 1 times this. We'll multiply by 120 seconds, that's
our step size if you like, and then we'll divide by 3 because we've actually got 6 concentrations
here, divide by 3 it's 2, so if you like, it's 240 times some average value and to get
that average value we count the middle one the most. We'll then do the same thing for
this next interval. This concentration, 4 times this, plus this using the same type
formula. We'll then add those areas up, in this case 4 areas and we'll get the total
area. So what I've shown here is the spreadsheet that's used to apply Simpson's rule. We've
entered the data in two columns. We do this integration as I mentioned, and again, here's
that plot where we're getting the area of this first section. So 120 divided by 3 and
then D4 is this concentration plus 4 times this concentration plus this concentration.
D4 plus 4 times D5 plus D6, we get the area of the first section. The same idea now for
the area of the second section, and the third and the fourth and then this just is a sum
of all these values. Then to get the average residence time, we multiply C and D columns
times the concentration as a function of time and we get the area again for these data using
the same formula, Simpson's rule. 4 times for the 4 sections. Get the total. We take
2.23 X 10^6, divide it by 6000 and we end up with the average time, 371 seconds. To
get the standard deviation, this is time minus time average and this is squared in the actual
calculation it's squared times D5 where D5 is the concentration. Apply Simpson's rule
one more time, same idea, 4 sections, get the total. This total divided by this area
for the concentrations will give us sigma squared and then the square root, 180 seconds
for sigma. And so we can use numerical data and a spreadsheet to calculate an average
residence time and a distribution. And certainly if we had more data, we'd apply the exact
same idea and get a more accurate value of the residence time distribution.