 | by Kenneth Chase | 3 comments

## Average Residence Time

In this example, I will show how we calculate
the average residence time for a non-ideal reactor if we do a tracer experiment where
we inject a pulse and then measure the concentration as a function of time at the outlet of the
reactor. Now this is a simplified diagram to make the calculations easy and it does
not represent what we would expect in a real system. But we can measure concentration as
a function of time and then we can use this to calculate the residence time distribution
which is essentially taking this concentration time plot and normalizing it. What we are
interested in is p(t) plotted versus t where this is our residence time distribution. Once
we have that, we can calculate average residence time by integrating over all times. This will
be then what we’re interested in so the tracer plot to normalize it, the integral of the
residence time distribution should be 1 because this is a probability of how long molecules
spend in the reactor and eventually all the molecules leave. And so to normalize it, what
we’ll do is integrate this tracer concentration as a function of time and this of course,
is just the area under the curve which would be the height times the width times one-half
because it’s a triangle, so it’s half of the rectangular area and the units that correspond
to the x- and y- axis get an area. If we divide this plot by this area, we would end up with
a residence time distribution. So we divided the concentration plot by 2.5 so the maximum
is at 0.2. And then we can write an equation for this residence time distribution by first
calculating the slope of the line. Slope is -0.02. And so the equation for p(t) is that
it’s zero for t less than 5, t greater than 15 and then p(t) is equal to -0.02t plus the
intercept. Now the intercept we can calculate. Easiest way is to say when t=15, p(t)=0 which
means that b=0.30. So we have an equation for the residence time distribution. Now we
can substitute in to calculate our average residence time distribution. So here is the equation
that we wrote down earlier for our average residence time distribution. We only need
to integrate from 5 to 15 minutes because p(t) is zero everywhere else. Put in t, I
put in p(t). Integration. Then I substitute the numerical values in, I get an average
residence time of 8.2 minutes. #### Vidhi Shah

Aug 8, 2018, 4:35 am Reply

Go through this for better understanding #### TLB IZR

Sep 9, 2018, 3:35 pm Reply

How did you draw the p(t) vs t curve and how you get 0.2 max on y-axis? (It would be so kind of you if someone answers) #### Erik BerMont

May 5, 2019, 9:42 am Reply

Hi, I hope somebody answer this: If the average residence time is tavg=int(t*p(t)dt), this would mean that the average residence time is just the area under the curve t*p(t), right?