## 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 amhttps://unacademy.com/lesson/development-of-concept-of-residence-time-distribution/N98PTATK

Go through this for better understanding

## TLB IZR

Sep 9, 2018, 3:35 pmHow 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 amHi, 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?