DISTRIBUTIONS OF
RESIDENCE TIMES (RTD)
FOR CHEMICAL REACTORS
-PART 1By: Mdm. Noor Amirah Abdul Halim
THEORY OF RTD
Definition of Residence Time Distribution (RTD)
- Probability distribution function that describes the amount of
time that a fluid element could spend inside the reactor

Purpose of RTD Analysis
- To characterize the mixing and flow within reactors
- To compare the behavior of real reactors to their ideal models
Why?..
- For reactors troubleshooting
- Estimation of yield for given reaction
- Future reactor design

The theory of RTD generally begins with 3 assumptions:
1. The reactor is at steady-state,
2. Transports at the inlet and the outlet takes place only by
advection
3. The fluid is incompressible (v = constant).

Residence Time Distribution (RTD) Function E (t)

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The distribution of residence times is represented by
an external residence time distribution or an exit age
distribution, E(t). The function E(t) has the units of
time^-1 and is defined such that:
The fraction of the fluid that spends a given duration,
(t) inside the reactor is given by the value of
The Cumulative Distribution Function F(t)

The fraction of the fluid that leaves the reactor
with an age less than (t1) is;
Where F (t) is called ‘cumulative distribution’

Thus, The fraction of the fluid that leaves the
reactor with an age greater than (t1) is
Mean Residence Time tm

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The average residence time is given by the first
moment of the age distribution:
For no dispersion/diffusion in the reactor: space
time (τ) equal to mean residence time (tm)
Variance σ2

The behavior of the function E(t) also could
indicate the degree of dispersion around the mean
through the variance (σ2),
MEASUREMENT OF RTD
RTD can be determined experimentally by;
1. Injecting an inert chemical/molecule/atom called
tracer ,into the reactor at t=0
2. Measure the tracer concentration ,C in the
effluent stream as a function of time.

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Properties of tracer :
Non reactive species
Easily detectable
Physical properties similar to reacting mixture
Completely soluble in the mixture
Not adsorb on the wall or other reactor surface
Common type tracer- Colored an radioactive
materials along with inert gases
RTD can be determined by two experimental
methods (based on injection method: pulse or
step)
1. Pulse experiment
2. Step experiment
PULSE EXPERIMENT

This method required the introduction of a very small
volume of concentrated tracer at the inlet of the
reactor, .The outlet tracer concentration C (t) is then
measured as a function of time.
Example:
 From our experiment data of the exit tracer
concentration from pulse tracer test

We can obtain;
STEP EXPERIMENT

In a step experiment, the concentration of tracer at
the reactor inlet changes abruptly from 0 to C0. The
concentration of tracer at the outlet is measured
and normalized to the concentration C0 to obtain
the non-dimensional cumulative distribution curve
F(t) which goes from 0 to 1:
DISCUSSIONS
The RTD function E(t) can be determined
directly from a pulse input,
 The cumulative distribution F(t) can be
determined directly from a step input.


Relation of the step- and pulse-responses of
a reactor are given by ;

The value of the mean residence time (tm) and the
variance (σ2) can also be deduced from the
cumulative distribution function F(t):
EXERCISE 1
The following is an E curve calculated for reactor Y
1. What is the maximum value of E shown on this
curve?
2. What fraction of the molecules spend between 2
and 2.5 minutes in the reactor
3. What fraction spend between 3.5 and 4
4.Which curve below corresponds to F(t)?
EXERCISE 2
The F curves is shown below for a real reactor
What is the mean residence time?
EXAMPLE
1. C(t) curve

Find E (t)
1. Determine the area of C(t) curve
2. Construct the E (t) curve

Fraction of material spend for 15-20 s in reactor

F(t) and, the fraction of material that spends 25 s or
less in the reactor

Mean residence time ( tm)

Variance
aArea= 15 s^2
RTD FOR IDEAL REACTORS
The RTD of a reactor can be used to compare its
behavior to that of two ideal reactor models: the
PFR and the CSTR (or mixed-flow reactor).
 This characteristic is important in order to
calculate the performance of a reaction with
known kinetics.

BATCH & PLUG FLOW REACTORS (PFR)

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In an ideal PFR, there is no mixing and the fluid elements
leave in the same order they arrived.
Therefore, fluid entering the reactor at time t will exit the
reactor at time t + τ, where τ is the residence time of the
reactor.
The residence time distribution function,E(t) is therefore a
dirac delta function (δ) at τ.
E (t) for PFR is given by:
Where the dirac delta function (δ) at is given as
Assume (x=t)
PROPERTIES OF DIRAC DELTA
Thus:
Mean residence time: :
Variance:
Cumulative distributions:
CSTR (MIXED FLOW REACTOR)

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An ideal CSTR is based on the assumption that the flow at the
inlet is completely and instantly mixed into the bulk of the
reactor.
The reactor and the outlet fluid have identical, homogeneous
compositions at all times. An ideal CSTR has an exponential
residence time distribution:
Equation for CSTR
PREDICTING CONVERSION AND EXIT
CONCENTRATION
The RTD tells us how long the various fluid
elements have been in the reactor, but it does not
tell us anything about the exchange of matter
between the fluid elements.
 The length of time each molecule spends in the
reactor is all that is needed to predict conversion.

TYPE OF MIXING

-
Macromixing
Produces a distribution of residence times
Micromixing
- Describes how molecules of different ages encounter
one another in the reactor.
2 types:
1. Complete Segregation :
All molecules of the same age group remain together
as they travel through the reactor and are not mixed
with any other age until they exit the reactor
2. Complete Micromixing
Molecules of different age groups are completely mixed
at the molecular level as soon as they enter the reactor

CONVERSION (X)

For batch reactor :

For PFR :
tm = τ
CA = CA0 (1-X)

For CSTR
Reactor Modeling
Using RTD
Segregation
Model
Maximum
Mixedness Model
SEGREGATION MODEL
In the segregation model globules
behave as batch reactors operated for
different times
Mean conversion
for the segregation model
The segregation model has mixing at
the latest possible point.
The farther the molecules travel along the reactor
before being removed, the longer their residence
time. Each globule exiting the real reactor at
different times will have a different conversion.
(X1,X2,X3...)
MAXIMUM MIXEDNESS MODEL
Maximum mixedness:
Mixing occurs at the earliest possible point
As soon as the
fluid enters the reactor,
it is completely mixed
radially (but not
longitudinally)
with the other fluid
already in the reactor.
Let 𝜆 be the time it takes for
the fluid to move from a
particular point to the end of
the reactor. In other words, 𝜆
is the life expectancy of the
fluid in the reactor at that
point
Continue from previous example:
CHAPTER ENDED. ALL THE BEST =)