We are going to start a new chapter today
which deals with design of non-ideal reactors.
So, we going to look at reactor models for
non-ideal reactors let us take, a review of
what we have learnt so far as per as non-ideality
in reactors is concerned as you know, there
are ideal reactors like: CSTR, PFR in which
the flow pattern is well define. In these
reactor flow pattern is well defined whereas,non-ideal reactors the flow pattern can take any particular shape
in term like if you look at e curve.
Now, this flow pattern tells you, how the
flow is flowing inside a reactor or somehow
like it does not follow a very particular
or specific pattern. Now, in order to get
a converse final aim is to have a reactor
design. And this reactor design somehow you
have to incorporate this non-ideality. So,
I cannot assume any reactor to be plug flow
reactor or a CSTR. So, they can be non-ideal
flow patterns which will affect the rate reactor
design.
So, in the last chapter will look at 0 dimensional model 
for non-ideal reactors. So, in this you have
typical e curve right and this e curve of
course, is as you know e is the exit age distribution;
that means, I have a reactor this inlet going
outlet. And what you seeing in outlet gives
you, this particular pattern. Now, they can
be many different possibilities, which of
the flow patterns which would give raise to
this e curve. And what happens inside a reactor
is, not exactly reflected in the exit age
distribution is something like that we already
learned.
So, so there are different possibilities in
terms of what happens inside a reactor, the
mixing pattern inside a reactor. So, mixing
is not well incorporated in the e curve. So,
then we looked at 2 different extremes: so
1 extreme is a completes aggregation model
and another extreme is the maximum mixedness
model. Now what let me quickly revise, so
it founds a nice platform of a further discussion
as for is this chapter is concerned.
So, let me quickly revise what these 2 models
are, so maximum mixedness model means, that
mixing is maximum. So, mixing between what?
So, if you look at this e curve there are
different segments every segment right. So,
these segments whether will have will be flowing
through the every segment will have a residence
time. And that is why like will have a fluid
element, we will have a specific residence
time and it is have a distribution of residence
time.
So, will have different segments spending
different residence times in the reactor right
and the extent of in terms of a mass or volume
of this particular element spending this much
residence time the reactor that may vary right
that is nothing but e curve. So, you have
this different segments or different elements
spending different residence time. Now we
talking about, a mixing between these 2 elements
or several elements.
Now, there is an extreme where these elements
then mix there well mix right. And there is
another extreme where, these elements they
do not mix at all. So, they going parallel,
they do not talk to each other, they do not
interact with each other. So, that is segregation;
completes segregation and there is another
extreme where these are completely mixed.
Now, how it happens and all you will not going
to look at a in detail, but that is a meaning
of it. Why we look at these extremes, because
these 2 extremes would give us a bound on
conversion.
So, it gives a range given an e curve like
this I make an assumption that is completes
aggregation of these elements. And, then calculate
a conversion and, then another extreme where
there mix thoroughly mix before the come out
inside a reactor. And calculate a convergents
and these 2 convergents are likely to be different
and they are going to different for most of
the reactions, but they are going to same
for just 1 case remember what is it, when
the reaction is first order.
And I think it has been discussed to well
before why first order reaction, internal
mixing does not matter whereas, any other
order internal mixing matters a lot that.
So, if you go for the complete segregation
model and maximum mixedness model. These 2
models are going to give you different convergents,
different extend of reactions. Intrinsic current
is same, rest all volume the reactor everything
is same, e curve is same, but internal mixing
will matter.
So, what happen in this case I am not going
to get a exact conversion why? Because, the
0 dimensional model I am just going to look
at a extremes. So, I get bounds, I get a range,
I may say that, fine for given volume this
is the possible range of conversion that I
am likely to get if the e curve is like this
for a non-ideal reactor. So, it may vary from
0.45 to 0.55, so that is a idea I get right
I do not get a exact conversion that is a
limitation of this model.
So, this is 0 dimensional model it talks about,
the extremes as for as a mixing inside a reactor
is concerned e curve only gives you partial
information of the flow pattern it does not
tell you about a internal mixing. And for
the first order reaction of course, it does
not matter. So, it inside getting a bound
and I just getting a single value I do not
have to worry about, whether it is complete
segregation or whether it is maximum mixedness.
So, this is a quick revision of what you learned
before. And now let us, go ahead now this
is a problem with 0 dimensional model; where
it gives most of the times for in non first
order reaction the range of convergent. Because,
we consider 2 extremes we not looking at what
is happening inside.
Now, in this chapter we are going to look
at different models: 1 of them is 1 parameter
model. So, I have adsorption additional parameter
that is going to give me some idea about,
the convergent. It will help me get the exact
value of convergent for the given e curve.
Now, remember that this model that we are
going to discuss or other the 2 types of 1
parameter model we going to discuss they applicable
to particular geometry I will elaborate this
point later, but remember that.
In 1 parameter model, we have 2 models that
we going to discuss that are tank in series
and dispersion model. Now let us consider
a tube that is why I said it is applicable
to a particular geometry. Again I tell you
why am tell later tell just talk more on this
a tube this is flow that is taking place.
And you may have different extends of back
mixing that is occurring. Now, if you are
very flat profile what it means, is a you
have a plug flow reactor, what is a e curve
that I am going to get is a direct delta function.
So, the e curve for the plug flow reactor
is direct delta function, but this tube a
reactor that I am talking about, is not a
plug flow reactor not necessarily behaving
close to a flow reactors. You know not a single
real reactor is behaves like adsorption ideal
CSTR or PFS it is close to those reactors
in extreme situations. But they do not exactly
follow the patterns that we assume, but anyway
lie for if you are very close to a plug flow
reactor, if you getting a e curve is matching
with direct delta function.
So, some extend like a make adsorption assumption
that be behave like a plug flow reactor and
I can design a reactor accordingly. Considering,
some possibility of conversion being plus
minus, but that is ok, because by making assumption
of plug flow reactor it reduces the complexity
or legal in calculation it makes my calculations
simpler; it helps me to get some quick estimates
of convergent for a given volume or for given
convergent the estimate for the volume. So,
let me get back to this you have a tubular
a reactor in which there is possibility of
back mixing; that means, I am not going to
get adsorption e curve which is similar to
direct delta function or a plug flow reactor.
So, what is likely to happen see will have
e curves e t versus t. Now, for a plug flow
reactor I am going to get something like this
infinite right you know the meaning of it.
And this is nothing but tau which is volume
divided by volumetric flow rate it is an ideal
situation. Now, what I am going to see in
the reality is somewhat like this possible
I may see something like this; I may see something
like this why does this happen?
It happens, because there is mixing in axial
direction, if there is no mixing I am going
to see something like this right. But because
of mixing some fluid elements may spend more
time, because they go back and then forth.
So, it possible that day spend more time in
a reactor and they come later; some fluid
elements may spend a less time just componset
for those for gone a head or those lagging
behind.
So, it is possible that you get a distribution
right. So, it’s I am talking about a tubular
reactor where there is possibility of back
mixing, you may have packing it may provide
some because tutorcity and all it’s quite
possible that you are you do not have the
exactly plug flow type behavior. Now so these
are the different e curves how do I interprets
these e curves and this is happen only because
the back mixing. So, I need to a corporate
effect of back mixing in this particular behavior
that I have observed all right.
So, let’s consider any e curve right once
this e curve is obtained, I have a tubular
reactor I do a pulse injection experiment
and look at an exit age distribution I am
going to see this. Now, this is something
given to you from this you are going to come
up with a model I am going to determine a
parameter; now mixing is characterized, by
like you have a say let us have a hypothetical
mixture or agitator in the reactor.
So, in the tube itself I can say that tube
is consisting of various compartments. If
you have infinite such comportments in a given
volume total volumes small small compartments.
What is it mean? Infinite it means that, you
have flow similar to a plug flow reactor.
Now, you reduce a number of compartments what
is it mean? That means, this some back mixing
happening you go on reducing, go on reducing;
mixing extend of mixing would increase right.
Consider an extreme, where you have just 1
compartment there has to be compartment.
So, 1 compartment what is it mean? A compartment
with agitator good mixing complete back mixing
in that compartment what is it mean its CSTR,
this is an another extreme. So, what is it
mean? So, you have on 1 side you have a CSTR,
on the other side you have PFR; CSTR means
are 1 compartment, PFR means finite compartments
in between you have a reactor which has partial
back mixing right.
So now, you would have guess what is a parameter
that I am talking about, it is a number of
compartments, every compartment is CSTR perfect
back mixing right. So, its number of CSTR
or number of tanks which are in series that
are going to that number is going to be a
parameter for this model.
Fine, so why I am saying it is a tubular reactor.
So, let us a consider reactor which is a very
regular geometry say I have a reactor like
this, this is an inlet, outlet. Now, there
are flow patterns, then there are possibilities
of some isolated zones right. And you have
the output here and what you going to see,
is a behavior like this any general e curve.
Now, this particular e curve does not look
like the e curve of it tubular reactor why.
Because, for tubular reactor like let us,
look at all the possibilities now, you have
a plug flow reactor, CSTR e curve right CSTR.
Once CSTR is this what will happen to 2 CSTR
like a in the case of 2 CSTR you are going
to get something like this. Then 3 CSTRs,
4 CSTRs see what is happening now? As you
go on increasing number of CSTRs you are approaching
the plug flow reactor this is a of course,
going to infinity this is becoming narrow,
it starts its seeing a delay here.
So, what is happening as we go on increasing
the value of n that is number of compartments,
number of CSTRs in series. The behavior it
goes it follows this particular the train
goes from CSTR to PFR, but look at these curves
the nature of these curves. These curves and
to compare it compare these curves with this
particular curve I am getting a very irregular
shape here or other very an unusual like there
are many punctuations are up down and all
that. Why does that happen? It happens because
of the regular geometry here and that define
the geometry flow pattern is a quite complex.
So, this flow pattern and this flow pattern
they are not matching. So, can I apply a tank
in size model for this reactor? I cannot do
that Why because there is no curve that fits
well in this particular shape. So, again 1
parameter model has its own problems, it is
applicable to most of the times to tubular
reactor, where the e curve is like this or
1 of these rather. Its continuous thing, it
is continuous things, the no ups and downs,
no recirculations or recycle with have behavior.
So I will go back to my statements that, 1
parameter models are good for tubular reactors
where you get e curve with a nature like this
clear?
So, let us go ahead and do some mathematical
derivations how do a find out number of tanks
that I have in series for a given tubular
reactors say I have an e curve. How do a find
it? So, let us derive an expression for the
e curve for a tank in series suppose I have
n number of tanks in series let us try with
3 tanks in series first. And, then explain
this concept to n times, so let us have 3
tanks in series fine.
So, let me write it write a let expression
for a e curve I mean I am going to give pulse
here, I am going to see what happens here.
So, this is my e curve and this is what I
am going to get likely to get fine. Now, can
I get explanation for this? But I know CSTR
behaves. So, let us write e t into delta t
this is a fraction you know meaning of this
I am not going to repeat.
So, e delta t is a fraction that comes in
between t and t plus delta t that is equal
to the volumetric flow rate I am going to
assume an volumetric flow rate remains constant
throughout into c3 t. So that means, c3 is
a concentration that at the outlet of tank
number 3, so 1 2 and 3 right tank number 3
c3t into delta t divided by N0. What is N0?
N0 is a total amount of pulse that have injected.
How do a calculate N0? N0 is nothing but based
on c3 t, because now I am get read of N0 I
am just looking at concentration outlet see
and visualize a realize realistic experiment
that I am doing, I am looking at a concentration
at outlet. So, tank expires everything in
terms of concentration at outlet. Now, N0
is a total amount of pulse or other the tracer
rather that have injected. N0 is equal to
0 to infinity c3 is t dt total amount that
is come out into of course, the volumetric
flow rate see, concentration right concentration
into volumetric flow rate into time. So, I
just integrate right I just integrate from
0 to infinity. So, that much amount have injected
as tracer, so let becomes N0, so let us go
ahead fine.
Let me substitute for N0, so what I get is
e t is equal to v C3 t right it says this
is delta t. So, e t is equal to c3 t divided
by so I have got a patience for e t based on
the concentration at the outlet. Now, the
c3 I need to get that in terms of the residence
time, initial concentration or inlet concentration.
And I know these are all CSTR, let me write
an steady state balance for CSTR.
CSTR 1 V1 that is a volume of CSTR d c1 by
dt you learned this unsteady state balance
for CSTR is equal to minus v C1 right. Shall
I write here? This is what this is going out
this is going out, this is accumulation coming
in Shall I write this? No, because what I
am looking at see traces experiment, pulse
experiments, I am injecting a tracer. And,
then after that at 0 time an injecting tracer
and after then I am observing the response.
So, after that is there anything that is coming
in as per as a tracer is concerned no right.
So, I have injected and stopped it now it
is only the inlet that is flowing, the solvent
that is flowing. The tracer is not coming
in, after 0 plus and this is equation I have
written is for time 0 plus onwards let me
0 onwards. So, this term is not there, this
term is not there.
So, I have only this equation for the concentration
in the first time I am writing it for a first
time. Now, later on I will do it first, second,
third and then finally, I will get expressions
for c3 that is my objective. So, here from
this I get a concentration and outlet of tank
1, where I have the boundary condition at
time is equal to 0 time is equal to 0 C1 is
equal to C0, but solve this equation its very
simple right.
So that is nothing but C1 is equal to c0 e
raise to minus t by tau 1 what is tau 1? tau
1 is equal to v1 divided by v. This is a volumetric
flow rate, this is a volume of first tank
this is a expression for C1 that is the concentration
and outlet of first tank. So, let us continue
now, so this acts as a inlet for a second
tank. So, let us a write an expression for
the second tank.
Tank number 2 what is it? v2 d c2 by dt is
equal to v C1 coming in minus v c2 I cannot
neglect this. This is changing with respect
to time and 0 plus C1 will have some value.
So, there will be 2 terms as per as tank 2
is concerned, but tank 1 inlet was 0, because
tracer was 0 at tank 0 plus right. Before
we go ahead let me, make an assumption that
v1 is equal to v2 is equal to v3 all v’s
are equal same V just was simplicity right.
Let me say vi, because I am not use v for
something and hence later I am not use v for
the total volume let me call this is vi. So,
let becomes vi here fine, so let me simplify
or other solve this further for C1 I have
the expression. What is that? This 1. So,
this expression I will substitute for C1 what
I get is this dc2 by dt is plus c2 by now
I will say, tau i. Why tau i.? So, what is
tau i? tau i is equal to Vi divided by v which
is constant that is volumetric flow rate given
to you.
So, C2 divided by tau i is equal to C0 divided
by tau i into e raise to minus t by tau i
I am just substituting for C1 and expressing
everything in terms of tau i now. So, you
know what is tau i? So, I get an expression
for differential equation for C2 right now
again, for this I need a boundary condition
at t is equal to 0, C2 is equal to 0 right.
So, this equation I solve this using this
boundary condition, so this is famous or this
is a couple of method of integrations factor
e raise to t by tau i. So, this expression
can be solving using integration factor or
integrating factor e base to tau by.
So, t by tau i and I get the solution for
it; which is given by C2 is equal to C0 t
divided by t tau i into e raise to minus t
by tau i. See concentration and outlet of
the second tank is a express in terms of initial
concentration time anyway, I want to know
how it changes with time and of course, tau
i right. I will eliminated C1 what is my aim?
I am my aim is to get C3 in terms of C0 t
and tau i.
So, let us do it now I follow same methodology
for tank 3 substitute for C2, because now
in tank 3 in inlet is C2 right and I will
solve this equation fine. So, what I get is,
C3 is equal to, so you can try it out again
a same methodology I get what I get is this
e raise to minus t by tau i. So, you have
2 appearing here right see the difference,
not much difference but of course, you have
the you have the exponential term there. And,
but instead of t now it becomes t square i
have 2 appearing here tau instead of tau i
I have tau i square. So, let us go ahead now
I have the expression for the e curve or other
the e for the entire reactor that tanks in
series.
What is that? This 1 now I have the expression
for e, I have the expression for C3; substitute
this C3 in e curve or other the e expression
for e.
So, what I get is see final, so see objective
is to get e expression for e. So, I substitute
for C3 and let me write this first, so that
C3 right. Now, what is this? This is the total
amount of the tracer that I have injected,
can I get it in terms of initial concentration
C0? Possible what is it?
If you look at it what happens at time 0 you
have C0 is equal to N0; that is a total number
of moles or whatever unit can be grams or
moles whatever. You have that much amount
of tracer divided by V1. So, this is something
present at time is equal to 0 sharp in the
reactor V1 is a volume and it can be Vi also
in our case.
So, let me call this is Vi; which is nothing
but the total amount right is V0 which is
nothing but v rather into C3 t dt right divided by Vi. So, from this what is this v by Vi tau 1 by tau. So, this
till is tell me that this integration is nothing
but tau i into C0 right. And I am just expressing
it in terms of initial concentration, because
this is appearing the final equation right.
So, let me get back to this equation this
is for the e curve I am substituting for C3
that I have just derived C3 that have derived
here.
I will substitute it here and then I will
have a very simple equation e t is equal to
C0 C0 will get cancel. And I will have t square
divided by 2 tau i now it cubed, because it
is 1 tau I have just come from that denominator
integration term e raise to minus t by tau
i. Now, this is for 3 tanks if you applies
same logic do it 4 tanks, 5 tanks, 6 tanks,
n tanks then what I am going to get is this
is for n tanks is t raise to see 3 tanks 2
here n minus 1 divided by see n minus 1 factorial
into tau i raise to n see the logic, then
e raise to minus t by tau i this is for n
tanks in series.
So, I have got an expression for e curve for n tanks in series; a general expression for e curve for the n tanks in
series and that is what I want right. So,when I get e curve I have e curve, so in this is 1 parameter
that is n that I can calculate from this.
So, this will this will obtain from experiment
in the laboratory for a reactor; it tabular
reactor under a flow conditions that are desired
for the reaction. But of course, I do the
experiment nonreactive condition you know
that just look at a residence time distribution
right.
So, from this experiment I get e curve from
e curve I get n, because I had I have an expression
for tanks in series e curve and once again
get n then I can calculate a conversion. What
is n? n tells you the extent of back mixing
n infinity means, if PFR, n1 means its CSTR.
And in between 1 and infinity you have the
extent of back mixing characterized by the
number n fine. Let’s go ahead and simplified
further equation looks bit a complicated.
So, from this you have going to defining a
term called a variance of course, you know
what is variance when we had distribution
the variance that means, how much at particular
time in the values of away from the average.
So, the something called as variance before
that will define a term non-dimensional term
theta equal to t by tau, where tau is that
total residence time what is it mean? That means, tau is
n into tau i is individual residence time for every tank n into tau i is a total residence time.
So, if tau i is equal to Vi by v then tau
is equal to V which is nothing but V1 plus
V2 and so on divided by V that why I said
now I have been using V for something else
the total volume.
Why? Because see I do not have tanks, I am
just assuming a tubular reactor to be set
of tanks. So, all I know is the total volume
right, so I need to expression in terms of
total volume finally, right this total volume
is V1 plus V2 plus V3 plus V4 and so on. So,
the residence time based on total volume is
tau right for individual reactor is what tau
i now it is tau. And, then I have a dimension
less number theta express in terms of tau
that is t by tau.
So, I have the expression for e t I will write
it again to raise to n minus 1 divided by
n minus 1 factorial tau i n right e raise
to minus t by tau i. Now, this will get reduce
to e theta now t will be express in terms
of dimension less time theta while do that,
then all these will get converted to n into
n theta raise to n minus 1 see t will become
n theta. Because, see t right then e raise
to minus n theta divided by n minus 1 factorial.
So, this is the expression for e in terms
of dimensional less time instead of actual
time you going to use this further that why
you expressed like that.
Now, there is a variance that we define now
how do we define variance if you have a distribution
the expression for the variance is, in terms
of dimensional less time variance square is
equal to actual variance in time divided by
tau square. Now, so this is nothing but 0
to infinity theta minus 1 square e theta d
theta right. I hope it is clear if you write
just variance in time you will have t minus
tau you have to just divided by tau square
and then you get this.
So, it is a very as I said like how much you
go away from tau mean residence time. So,
if you expand this further what you get is
0 to infinity theta square e theta d theta
minus 2. I am just expanding this, so theta
minus 2 theta plus 1 so you get this. So,
what is the value of this? This is equal to
1 this particular term. What about this? This
particular term this is dimensionless average
residence time dimensionless.
So, it is going to be 1 because it is the
reference is tau for the residence time it
was t e t dt, then I was tau now it is theta
e theta d theta; that means, it is tau divided
by tau that means, 1. So, this is this is
going to be again is integration is going
to be 1, this is 1, this is 1. So, minus 2
plus 1 is minus 1 0 theta minus 1, so this
is an expression I am going to simplify this
further.
So, let us solve this integration what I get is or before that let me just write it again right substitute for e theta
see we have derived equation for e theta. So, this is e theta I just substitute for it n
n theta n minus 1 divided by n minus 1 factorial
e raise to minus n theta d theta minus 1.
Now, if you do all this what you get is, because
take n, so will have n raise to n divided
by n minus 1 factorial it come out right.
And you have 0 infinity theta raise to n plus
1 e raise to minus n theta right dt that is
solve this further what you get, you can do
it on your own.
So, it get simplified to very simple term
1 by n and this is what I want variance is
1 by n; variance is obtained from the e curve.
So, once you get variance right you get value
of and number of tanks in series. So, n is
equal to 1 by sigma theta square which is
nothing but tau square divided by sigma square
right. So, I get a value of n this is what
I want right.
So, given e curve now will see how to calculate
sigma from e curve, I have already told you
that expression for rate. But will solve or
I will tell you the procedure to get sigma
from that or this sigma they from that I get
n. So, e curve will give you sigma and sigma
will give you the value of n right.
How to calculate conversion for the given
n, now we talking about reactor tank in series
there is very simple know. I have n number
of tanks how do I calculate conversion for
a CSTR? I have n number of CSTR are in series
the conversion say for first order reaction
the conversion is 1 minus 1 by 1 plus tau
i k raise to n tau i is the again see do not
get confusion this 2 tau.
So, this 2 tau is the residence time for single
tank what is k? k is the rate constant. And
this is something like that you done before.
So, let me summarize I have a tubular reactor
I want to get a convergent problem can be
other around for given convergent, find out
a length a volume the tubular reactor, when
I have a tubular reactor let us talk about
given reactor in calculating convergent.
So, I have a tubular reactor what I will have
to? I will not assume it has a PFR. Now we
have talking about, non-ideality that can
be possibility of back mixing. So, what I
will do is I just pass of fluid I will do
residence time distribution experiment. I
will inject a tracer look at its falls it
outlet I will get a e curve form it right.
Once I have the e curve, then I can get a
variance from the e curve; from the variance
I will get a number of tanks in series for
that particular tubular reactors reactor other
if it is close to PFR the value of n will
be very large.
If there is so much back mixing happening
for some reasons value of n will be close
to 1 1 2 3 whatever. So, I get a value of
n once I know the value of n, I have the expression
for convergent and that tells me how much
is the convergent based on n. Because, rest
all you know what is this rate constant; tau
i is the residence time for the individual
reactor how do i calculate tau i? So, I know
the total volume for the tube.
So, know the total residence time from a total
residence time I calculate individual residence
time; residence time individual reactor. But
dividing it by n; n is known right. So, I
get an expression for a convergent that is
for the first order, but I am of course, for
second order, third order, but I have different
expression it is just solving a problem for
CSTR. So, I have converted a non-ideal reactor to
a an ideal reactor problem tanks in series;
ideal reactor in series. So, that n varies
n is the parameter thank you will continue
discussion will solves a small problem of
course, not numerically, but I will tell you
the procedure, so let things will be clear
to you fine.
Thank you
