Residence Time Distribution Data

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Process Engineering Guide:
GBHE-PEG-RXT-802

Residence Time Distribution Data

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Process Engineering Guide:

Residence Time Distribution
Data

CONTENTS

SECTION

0

INTRODUCTION/PURPOSE

2

1

SCOPE

2

2

FIELD OF APPLICATION

2

3

DEFINITIONS

2

4

BASICS OF RESIDENCE TIME DISTRIBUTION DATA

2

5

USE OF RESIDENCE TIME DISTRIBUTION DATA

4

5.1
5.2
5.3
5.4
5.5

Micromixing and Macromixing
Example 1 - Reaction is First Order
Example 2 - Reaction is Second Order
Complex Reactions and Residence Time Distribution
Examples

4
5
7
8
9

6

RESIDENCE TIME MEASUREMENTS WITH
RADIOISOTOPES

11

6.1
6.2
6.3
6.4
6.5
6.6

11
12
13
16
16
16

General
Types of Reactors
Models Based on Method of Moments
Non-impulse Input
Diagnosis of Problems
Commercial Radioisotope Service


7

17

BIBLIOGRAPHY

APPENDICES
A

RESULTS FOR EXAMPLE 1

19

B


20

FIGURES

1

A TYPICAL RESIDENCE TIME MEASUREMENT SET UP

12

2

IDEAL RESIDENCE TIME DISTRIBUTIONS

13


0

INTRODUCTION/PURPOSE

This Guide has been produced under the auspices of GBH Enterprises.
1

SCOPE

This Guide covers the principles and use, as well as the measurement by
radioactive techniques, of reactor residence time distribution data.

2

FIELD OF APPLICATION

This Guide applies to process engineers in GBH Enterprises worldwide.

3

DEFINITIONS

For the purposes of this Guide, no specific definitions apply:
With the exception of terms used as proper nouns or titles, those terms with initial
capital letters which appear in this document and are not defined above are
defined in the GBHE Glossary.

4

BASICS OF RESIDENCE TIME DISTRIBUTION DATA

Implicit in the equations of a mathematical model of a continuous reactor, is an
assumption about the distribution of residence times within the reactor. If this
assumption is wrong, even though we have accurate chemical kinetics, the
answers produced by the model will be wrong to some degree.
Residence time distribution (RTD) is a measure of the degree of mixing of the
fluid in the space through which the fluid is flowing. If the reactor to be modeled
does not exist, then the RTD has either to be assumed or calculated in some
way. Computational Fluid Dynamics could be helpful here. If the reactor exists,
we have the option of measuring the RTD.
This experiment consists of injecting a tracer substance into the feed stream to
the reactor, and measuring the concentration of the tracer as a function of time in
the exit stream. The injection may be continuous, representing a step change, or
"instantaneous", representing an impulse.

(The chapters headed "Non ideal flow" and "Mixing of fluids" in Levenspiel's book
(Ref 1) is recommended, and we will use his nomenclature here). The measured
response to the step change is known as the F curve and that of the pulse as the
C curve. The relationship between them is:

where:
C

=

tracer concentration in exit stream at time t
tracer concentration if tracer impulse has been instantaneously
mixed with other fluid in the reactor at time t = 0

F

=

tracer concentration in exit stream at time t
tracer concentration in entry stream at time t 0


These dynamic responses for the ideals of plug flow and ideally back mixed
(CSTR) are shown below:

In practical cases, the measured response will deviate from these ideals to a
greater or lesser extent. Later clauses will go into further details of the
experimental technique but results should be obtained in both of two forms:
(a)

The equivalent combination of plug flow and CSTRs in series whose
theoretical response best matches the observed response.

(b)

The C curve, either as measured or as calculated from the measured F
curve, equation (1).


5

USE OF RESIDENCE TIME DISTRIBUTION DATA

RTD data, when reported in the form of 4(a), cannot indicate in which order the
plug flow and series of CSTRs should be considered. In other words, the
technique cannot distinguish between the responses of:

This is because the measurement technique is a first order process. For first
order reactions the distinction is unimportant but for reactions whose rate
expression is non-linear in concentration, this problem needs to be resolved.
This resolution is not possible systematically. The modeler should use technical
judgment and a knowledge of the system to be modeled to formulate the
assumption to be made.


5.1

Micromixing and Macromixing

Two concepts of mixing can be visualized:
(a)

Mixing of individual molecules, i.e. individual molecules free to move
about, collide and intermix. We will define this as micromixing. The
individual molecules will have a residence time distribution given by the F
or C curve. In the ideal back mixed reactor, all the molecules will have the
same environment.

(b)

Small but significant aggregates or "packets" of molecules retain their
integrity without mixing with other aggregates. We will define this as
macromixing. In this case the aggregates, each of which will behave like a
tiny batch reactor, will have a residence time distribution given by the F or
C curve. Even in the ideal back mixed reactor, all the molecules will not
have the same environment.

Because in the macromixed CSTR model the reactant concentration does not
immediately fall to the average exit concentration, the predicted conversion is
higher than that for a micromixed CSTR for reaction orders greater than unity
and vice versa for reaction orders less than unity (except for zero order, where
residence time distribution is irrelevant). The predictions of the macro and
micromixing models will give bounds between which the actual answer will lie.
Residence time distribution data in the form of the equivalent combination of plug
flow and CSTRs in series will allow prediction of the performance of a reactor
based on micromixing.
Plug flow part

Integrate from Aop at t = 0
to
t = ṫ plug
Aop = concentration of feed to the plug flow part.


CSTR part

A os = concentration of feed to the CSTR part.
Residence time distribution data in the form of the C curve will allow prediction of
the performance of a reactor based on macromixing. At steady state, the reactor
outlet concentration will be given by:

A(t) = concentration in a batch reactor after time t.

5.2

Example 1 - Reaction is First Order

Consider a simple chemical reaction in an isothermal reactor, with rate constant
k:

then


For a plug flow reactor of residence time t, the exit concentration of A is given by:

and for a CSTR of residence time t’:

Now for a plug flow-single CSTR combination, the concentration of A leaving the
plug flow part will be:

Where Ao is the concentration of A in the feed to the combination and this will be
the feed concentration for the CSTR. So concentration of A leaving the CSTR will
be:

For a single CSTR-plug flow combination, the concentration of A leaving the
CSTR part will be:

and this will be the feed concentration for the plug flow part. So concentration of
A leaving the plug flow part will be:


Equations (8) and (10) are identical, illustrating that for linear processes the order
of CSTRs and plug flow parts is irrelevant.
Now for a CSTR, the C curve is:

So for the macromixing model, the exit concentration from a CSTR of average
residence time tl is given by equation (4) as:

Equation (13) is identical with equation (6) illustrating that for first order
processes, the micro and macromixing models give identical results.
rate constant k


For a plug flow reactor of residence time t, the exit concentration of A is given by:

and for a CSTR of residence time t:

Repeating the previous algebraic operations will show that the expression for A
from a plug flow-CSTR combination is not identical to that for a CSTR-plug flow
combination.
For the macromixing model, the exit concentrations from A CSTR is given by
equation (4) as:

This can be manipulated to:

Where:


The integral is known as the exponential integral for a. It is not soluble
analytically but values are tabulated as a function of a in mathematical tables.
The point here is that equations!(15) and (17) are not identical, illustrating that for
non linear processes, different results are given by the two mixing models. The
two models will give bounds between which the actual answer will lie.
Given a set of kinetic rate equations and a functional form of the C curve, then
numerical integration of equation!(4) in the form:

for an isothermal reactor is straight forward.
What is a more likely situation is that while the kinetics are available in a
functional form, the C curve is not. It may be reported as a tabulation of C(θ) vs θ
or as a graph from which a tabulation could be prepared.
In this case the following procedure is recommended (see Appendix A for
Flowchart):
Prepare a computer program which executes the following stages.
(a)

Analysis of tabulated data C(θ) vs t by cubic spline fitting NAg routine.
A sufficient number of coordinates should be supplied to represent the C
curve adequately.
The application may be more accurate by analyzing loge C(θ) vs t (see
example).

(b) Integrate equations of two types:
(1) to calculate batch concentrations vs t i.e. the A(t) in equation!18 i.e.
equations of the type:


(2) to calculate the

in equation (18) with the local value of

provided by the NAg interpolation routine, which uses parameters
determined at stage 1.

i.e. equations of the type

Since the integration of equation!(18) is to infinity, decide a value of t above
which the function

will make no further significant contribution to the integral.
In practice, this would be the value of t at which the measured
curve had fallen to zero.

The NAg routine can be used for the numerical integration.


5.5

Examples

5.5.1 Example 1
Determine the exit concentration for a reaction

with simple first order kinetics:

Where:
A = 1 at t = 0
k = 0.75
A can be in any concentration units.
t & k should be based on the same time units, such that t × k is dimensionless.
in an isothermal ideal macromixed CSTR with average residence time

The tabulated

t=4

for this example, was calculated from:

See the program in Appendix A.
The C curve is stored by the DATA statement. Note that if the loge form is to be
used, enter


non-zero data for

Results are given in Appendix B.
Analytical solutions are:
A(t) = exp(-kt) = exp(-0.75t)

The computed results show the integral for A asymptotic to this value.

5.5.2 Example 2
Determine the exit concentration for a reaction

with simple second order kinetics

Where:
A = 1 at t = 0
k = 0.75
A is in any concentration unit.
t in any time unit.
k in (concentration unit)-1 (time unit)-1.
in an ideal CSTR with average residence time
ṫ= 4


The previous program (Appendix A) was modified as follows:
the instruction in Example 1
A(t) = -0.75*t
was replaced by
A(t) = -0.75*t*t
The results are listed in Appendix C.
Analytical solution are
.

The integral is the exponential integral for a, and is a tabulated function e.g. (see
Ref 3).

The value of the integral is 0.82889

Again the computed results show that the integral for A is asymptotic to this
value.


The analytical result for a micromixed CSTR is

This illustrates that for a second order reaction, the conversion predicted for a
macromixed CSTR is higher than for a micromixed CSTR.

6

RESIDENCE TIME MEASUREMENTS WITH RADIOISOTOPES

6.1

General

Most chemical processes involve a reaction stage. At some point, two or more
components are brought together, mixed to some degree and encouraged to
react. Several different conditions will contribute to the reaction. Temperature,
pressure, catalyst, mixing and residence time can all play a part in controlling the
reaction and specifying the products. Of these variable conditions, mixing and
residence time are usually the more difficult to measure.
The residence time and mixing characteristics of the vessel can determine the
efficiency of the reaction and the product to byproduct ratio. This being the case,
measuring the residence time and degree of mixing can be as important as
measuring the temperature, pressure, etc.
The injection of tracers is a very useful technique for measuring the residence
time and mixing characteristics of a vessel. Chemical tracers have proved
successful in some systems, but they can have problems. The chemical may not
adequately mimic the behavior of the component of interest, the sampling
procedure may not provide a representative sample, a representative sample, or
the components of the vessel may be hazardous or two phase, making sampling
an undesirable component of the measurement.
Radioisotopes are particularly suitable tracers for residence time measurement
and mixing studies. Very often a tracer can be made from the component of
interest, very low concentrations of activity are generally suitable, and
radioisotopes are unambiguous since there are not usually any radioisotopes
naturally in the process. Normally, the tracer is detected through the walls of the
exit pipe, eliminating the need to sample.

A typical residence time measurement set up is illustrated in Figure 1.

FIGURE 1

A TYPICAL RESIDENCE TIME MEASUREMENT SET UP


A suitable injection point is needed into an inlet line. A gamma radiation detector
is mounted on the inlet line far enough downstream of the injection point to
ensure adequate lateral mixing in the pipe. The best plug flow situation possible
in the inlet line is desired to keep the inlet pulse sharp. The second detector is
placed on the exit line. Both detectors are connected to recording devices. A
chart recorder provides a instantaneous record of the detector response. The
signal can also be split to a computer for data logging and analysis.
6.2

Types of Reactors

In ideal situations, there are two types of reactors. A plug flow reactor is like a
piece of pipe, where all of the elements of the liquid flow at the same rate. Even
though there may be lateral mixing, there is no forward or back mixing. In this
situation, a pulse of tracer injected into the inlet line would exit the reactor with
the same shape, as in Figure 2(a).
In a stirred tank reactor, the components are immediately distributed equally
through the total volume of the reactor. A pulse of tracer injected into the inlet line
would exit the reactor with the maximum concentration occurring at the moment
the tracer entered the vessel. With time, the tracer concentration would then
decline along an exponential line, as in Figure 2(b), producing a residence time
distribution (RTD).

FIGURE 2

IDEAL RESIDENCE TIME DISTRIBUTIONS


In practice of course, ideal reactors do not exist. All reactors are then operating in
some non-ideal state. In a non-ideal reactor, parts of the tracer will move through
the reactor along various paths, exiting the reactor at different times. The result
will be an RTD curve that deviates from the ideal curve.
Some reactors are operating so poorly that an RTD curve will deviate
significantly from how it was designed to look. A residence time measurement
can often indicate the cause of the poor performance. Comparisons of RTD
curves from tests before and after alterations to the vessel or mixers can indicate
whether or not the alterations have improved the system. If time is available and
alterations are relatively easy to make, then the production efficiency or the
product to byproduct ratio can be used to measure the effect of alterations. But, if
a scientific, quantitative approach is desired and the alterations need to be done
right the first time, then a thorough understanding of the present situation is
needed. For this, models can provide the correct approach.

6.3

Models Based on Method of Moments

6.3.1 General
In general, three simple models are used, although other, more complex models
can be used. The first model assumes that the vessel exhibits only stirred tank
characteristics. The second model assumes only plug flow characteristics. The
third model deals with plug flow and then stirred tank characteristics. These
models are all based on the Method of Moments (MoM) calculations. The MoM
calculation assumes an impulse input of tracer that results in an outlet pulse
given by C1(t). The successive moments about the origin, the origin being the
time where the centre of the inlet peak occurs, are then computed using the
equations:


A1
A2
A3

= the first moment about the origin.
= the second moment about the origin.
= the third moment about the origin.

For equally spaced data points, the integral reduces to a summation function,

The moments are still calculated using equation (20).

6.3.2 Stirred Tank Equivalent Model
The first model commonly used is known as the Stirred Tank Equivalent Model.
This model assumes that the vessel under investigation can be represented by a
series of equal sized ideal stirred tanks, the sum of which are equal to the
volume of the vessel under investigation. In the Stirred Tank Equivalent Model,
where the beginning of the exit peak occurs at or before the centre of the inlet
peak, the only parameters of interest are the mean residence time (MRT) and
the stirred tank equivalent number (STE).
The mean residence time is the elapsed time from the centre of the inlet peak to
the centre of the exit peak. The MRT and STE are defined by the following
equations.


The best STE value possible is 1.0. Since STE is defined as the number of
equally sized stirred tanks, dealing with fractional STE's seems inconsistent.
However, in the case where a reactor may well have an STE value of 2, the
measurement of improvement from an operational or design change is only
feasible when using fractional values of STE.
Theoretically, a perfect plug flow reactor would have an STE value of infinity. In
practice, when the STE gets over a value of 10, the system is probably more plug
flow than stirred tank. And, since the term "approaching infinity" is not very
quantitative for plug flow, the Dispersed Plug. Flow Model (See 6.3.3) is better for
systems that are designed to or are operating in a plug flow manner.
6.3.3 Dispersed Plug Flow Model
The Dispersed Plug Flow Model utilizes the Inverse Peclet Number (IPN), a
dimensionless number, to give a quantitative measure of the plug flow.
Calculating the values of A1, A2, and A3 in the same manner as in 6.3.1, the
parameters of MRT and IPN are calculated with the following equations:

As the IPN approaches 0.0, the model approaches ideal plug flow. Theoretically,
a perfect stirred tank reactor would have an IPN of infinity. In practice, once an
IPN value goes over 0.5 the system is acting more like a stirred tank system and
the STE value will probably be more useful.


6.3.4 Mixed Models
In some cases, the exit peak looks like a stirred tank system, but there is a delay
between the centroid of the inlet peak and the start of the exit peak. This is one
of the more common mixed models where there is an element of plug flow and
then a stirred tank response. This mixed model uses three parameters to define
the model, MRT, delay time (t1), and STE. They are calculated using A1, A2, A3,
and the following equations:

At this point, caution should be advised about the use of mixed models. There
are several drawbacks to using mixed models, some of which are:
*

Just because the data fits a particular model does not mean that the
model is correct – just that the model cannot be rejected. It is perfectly
feasible for several models to fit a set of data.

*

Each parameter in a model needs to be measurable and have
significance. Even with simple models, such as the one above, negative
values for the delay time often result - which are physically meaningless.


*

The data from a mixed model should be used with extreme caution.
Residence time data can be combined with kinetics data in order to predict
reactor yield. However, this should be done very carefully, since in most
cases, yield is dependent on the order in which the different elements in
the model occur. As a general rule, the order in which these elements
occur cannot be predicted by residence time distribution analysis.

6.4

Non-impulse Input

In some instances, the inlet peak cannot be described as an impulse input. This
most often occurs when two reactors are in series and a suitable injection point
between the reactors is unavailable. As in the models above, the interest is in the
change between the inlet pulse and the exit pulse. With an impulse inlet, there is
no need to subtract the inlet peak. However, with a non-impulse inlet, the inlet
peak should be subtracted from the exit peak.
This is done by calculating the moments for each curve and then subtracting
them. For example, the moments A1, A2, A3, B1, and B2 are calculated as before,
but for the inlet peak. Then, D1, D2, D3, E1, and E2 are calculated for the exit
peak. The differences in the moments of the inlet and exit peaks are then
calculated with the following equations.

Once these values have been calculated, any of the three models can be used
and the parameters computed using F1 instead of A1, G1 instead of B1, and G2
instead of B2.


6.5

Diagnosis of Problems

Residence time distribution measurements are used regularly for quantitatively
measuring the effect of alterations to a vessel or its mixing elements. They are
also used to identify the cause of poor performance. If the MRT is shorter than
expected, then the effective volume of the reactor is smaller than design. This
could be due to solids accumulation or "dead" zones in the reactor allowing
channelling and short circuiting. Multiple peaks, decreasing in size indicates a
recirculation problem. The appearance of two distinct peaks indicates two paths
through the reactor such as a channel in a packed bed. Any of these problems
can be found in a plug flow or well mixed reactor.

7

BIBLIOGRAPHY


APPENDIX A



Using values:

Ao

=

1

k
ṫ

=
=

0.75
4


APPENDIX B



Residence Time Distribution Data

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