Attainable Region IIT B notes

8/9/2019 Attainable Region IIT B notes

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Lecture Notes

on

Attainable Region

by

Prof. Arun S Moharir

Production Support by

Satyanjay Sahoo

Anshu Shukla

Chemical Engineering Department

Indian Institute of Technology Bombay


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Introduction

Attainable Region or AR is an important concept in design of reactor systems. In

any process, it is the reactor or a reactor network where value is created by

converting relatively low priced reactants into high priced products. The value

thus created is partly consumed in preparing the feed for the reactor, operating

the reactor itself, and processing the product from the reactor. All these

consumptions must be minimized while the value creation in the reactor

maximized to make the overall process optimally profitable. This, in a nutshell, is

the task of chemical process design. Chemical engineering curricula world over,

especially the undergraduate ones, aim at designing or creating the process

designers for future who will churn out newer process designs. Process design

thus must take a philosophical and holistic view rather than getting constrained

by individual design tasks such as designing a heat exchanger, a distillation

column, a reactor etc. Individual designs of unit operations and reactors are

covered in dedicated courses such as heat transfer, mass transfer, reactor

design, fluid mechanics etc. It is the process design course which must integrate

all these concepts to illuminate the path to that one final all important activity of

process flowsheet development and design. The course on process design is

thus the culmination of all the courses to finally build something whole and

complete. No wonder it is one of the final year, final semester courses in most

chemical engineering curricula. IITB, for whose final year undergraduate students

this note is prepared, is no exception.

The note deals with Attainable Region, AR, an extraordinary concept developed

by Prof. Glasser [1, 2] for reactor system design. It is also one of the more recent

philosophical ideas which has immensely enriched chemical process design task.

While the concept is well understood and further researched upon by teachers,

its teaching has been a challenge because it has been difficult to separate

complex mathematics from its basically simple philosophical content. It would not

be wrong to say that AR entered the research arena without entering the

textbooks of basic chemical engineering curricula. This note is an attempt to


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present AR to undergraduate students of chemical engineering in palatable and

less mathematical form. A small percentage of these budding engineers, who

end up being researchers, can always build on this and appreciate and contribute

to the concept later and comprehend the vast literature available now on AR. The

idea is to keep aside the brute mathematical force used on this concept and

convey the simple structural beauty of the profound concept of AR to the

beginners.

At tainable Region

Let us call it AR from now on.

Stated simply, AR is a set of all product compositions that are attainable or

achievable from a given feed composition by employing any conceivable

combination of reactors in any series-parallel combination. A single reactor is a

minimal embodiment of a reactor network. If composition of a stream is stated in

terms of concentrations of all components in the stream, AR is a set of all

concentration vectors that can be achieved from the given feed concentration

vector. To begin with, we use a simplifying assumption that all the reactors in the

network operate isothermally and at the same temperature. This is the

temperature at which the feed is assumed to be available. So, generation or

absorption of heat in the reaction which could be exothermic or endothermic is

assumed to be handled by suitable heat exchange to maintain the reactors at

designated temperature. This is also the temperature at which reaction kinetics

should be available. As can be appreciated, AR will depend on reactor kinetics

apart from the feed composition.

Let us start with some simple examples to develop appreciation for the concept

of AR.

Example 1: For reaction A B, construct AR if the feed is pure A with

concentration 1 kmol/m3. The rate constant is 1 hr-1.


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Let us agree to some conventions for the rest of the note. An arrow like the one

used in representing the reaction as above indicates irreversible reaction.

Kinetics of the reaction can be either explicitly given or is easily inferable from

units of rate constant. For example, above reaction is first order as indicated by

the units of rate constant. Rate constant value will be temperature dependent.

The given value will be valid only at that temperature. Unless otherwise stated, it

is assumed that the isothermal reactor operates at this temperature.

With only two chemical species involved in the above case, the concentration

space is 2-Dimensional. Using a dominant component of the feed on X axis is a

preferred practice, although not necessary. In the present case, the other

component is on the Y axis.

For the given stoichiometry, the maximum concentration of A or B is obviously 1

kmol/m3. The feed point can thus be represented by the vertex (1, 0) of the 2-D

concentration space as shown in Figure 1.

Feed composition is always a part of the AR. This statement needs no

qualification. This is the concentration already available and hence achievable if

one chooses to have no reactor at all. Although trivial, let us emphasize this by

marking this fact as Rule 1 in constructing any AR in future.

Rule 1: A point in concentration space representing feed composition is apart of the AR.

Irrespective of the type of reactor we use, if it provides infinite space time, entire

A in feed will be converted to B and the product will have B at concentration 1

kmol/m3

in the present case. This is so because of the stoichiometry and the

irreversible nature of the reaction. This infinite time composition of the product is

also obviously attainable. Rule 1 says that zero-time composition, i.e. feed

composition, is a part of AR. Rule 2 could thus be:


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Rule 2: A point in concentration space representing infinite space timecomposition of product is a part of AR.

For the above reaction, the stoichiometry indicates that the sum of

concentrations of A and B in the reaction mixture during reaction must be the

same as the corresponding sum in the feed. This is so because one mole of A

produces 1 mole of B by reaction. In the present case, it means:

CA+ CB= 1 orCB= 1 CA

Any composition achievable through reaction carried out for any time should thus

lie on a straight line with intercept 1 and slope -1 in our 2-Dimensional

concentration space.

The two ends of the line were developed earlier through Rule 1 and Rule 2. Note

that the feed point (1, 0) and infinite time product point (0, 1) both satisfy above

equation of the line. Any concentration attainable through reaction is thus on a

straight line joining these two points.

Figure 2:Attainable region for A B; CA 0=1, CB 0= 0, k=1hr-1

Figure 1 shows the feed and infinite time product points as well as the above line.

You can verify that if the feed is passed through a reactor of some size and some

type (say CSTR, PFR) which gives some conversion, the resultant product

composition has to lie on the line in this figure. No other composition is possible

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

Figure 1:Attainable region for

AB; CA 0=1, CB 0= 0, k=1hr-1


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to achieve starting with this feed. The straight line is thus itself the desired AR in

this case.

Being a straight line, the AR in this case can be said to be 1-Dimensional. Please

note that the concentration space was 2-Dimensional, while the AR is 1-

Dimensional.

Also worth noting is that we did not use any particular reactor type in developing

the AR. Also, we made no use of the reaction kinetics or rate constant explicitly.

All we used in arriving at the AR is stoichiometry, nature of reaction

(reversible/irreversible) to arrive at infinite time composition and the feed

composition.

With this first acquaintance with the concept of AR, let us look at some variations

of the above example to consolidate and expand our appreciation of AR.

Example 1A:Same as Example 1 as far as the kinetics and stoichiometry go,

but with the only difference that the feed is an equimolar mixture of A and B with

each at concentration 0.5 kmol/m3.

Figure 2 shows the AR. The feed point is now (0.5, 0.5), infinite time product

point is (0,1) and the straight line joining these two points is the AR. Slope and

intercept of the line are the same as in Example 1.

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

Figure 3:Attainable region for

AB; CA 0= CB0 =0.5, k=1hr-1


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Example 1B: Same as Example 1 as far as the stoichiometry and feed

composition go. The only difference is that the reaction is second order, with say

some given rate constant value. Figure 3 shows the representation of AR for the

mentioned conditions.

All arguments put forth for Example 1 as regards feed composition, infinite time

product composition and the linear relation between concentrations of the two

components for any conversion are valid. The AR is thus the same as in Figure

1.

The kinetics does not seem to play any role and the AR is same for Example 1

and 1B although the kinetics was different. Neither the reaction order nor the rateconstant value had any bearing on AR which is a straight line, i.e. AR is 1-

Dimensional.

Example 1C: Same as Example 1 as far as the stoichiometry and feed

composition go. The difference is in the nature of reaction which is now a

reversible reaction with equilibrium constant given as 1. Let is agree to use = sign

to represent reversible reactions and say that the reaction is A = B.

The feed composition is a point (1, 0) in the 2-D concentration space. The Infinite

time product concentration (i.e. product concentration achievable by employing a

reactor providing infinite space time) will now be (0.5, 0.5) because A and B will

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

Figure 4:Attainable region forAB; CA 0=1, CB0 = 0, k=1m3/kmol/hr


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split as per equilibrium constant value (CB/CA = 1 as given). The linear

relationship between concentrations of A and B will be valid as per the

stoichiometry as earlier. The AR will thus still be a straight line as shown in

Figure 4.

If you compare the ARs developed by us so far, they are all part or whole of the

same straight line. We are not budging from this line in spite of changes in

reaction nature, reaction order, and feed composition. We still have a 1-D AR

and a 2-D concentration space. It is tempting to make a generalization and

proclaim Rule 3 stating that reaction kinetics and feed composition have no

bearing on the geometric nature of the AR, at least qualitatively speaking. Or

may be Rule 4 stating that dimensionality of AR is one less than the

dimensionality of the concentration space. Let is desist from doing that and

explore further. AR cannot be, after all, such a trivial concept!

So, let us cook up some more variations of the same example.

Example 1D:Same as Example 1C except that the feed is an equimolar mixture

of A and B with both concentrations as 0.5 kmol/m3.

The feed point is now (0.5, 0.5). The infinite time product point is also (0.5, 0.5)

given the equilibrium constant as 1.0. The AR is thus just a point in a 2-

Dimensional space. The AR is Zero-Dimensional so to say. Stated simply, the

composition remains same and cannot be changed. Whatever be the reactor

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3

)

Figure 5:Attainable region forA = B; CA0 =1, CB0 = 0, k=1hr

-1


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type and size, the feed emerges unchanged as a product. That is because the

feed is at equilibrium itself. The AR for the above system is shown in Figure 5.

This example at least gave us some deviation from something we were on the

verge of formulating as a rule. The dimensionality of AR is 1 in Examples 1, 1A,

1B and 1C and is 0 (Zero) for example 1D. The concentration space was 2-D in

all cases.

The 0-Dimensional AR, say the point AR, is still on the same lines which

emerged as ARs for the earlier cases.

All reactions so far were having same stoichiometry. Let us make some deviation

from the stoichiometry in the next example.

Example 1E:It is similar to Example 1 except that the reaction is 2A B + C.

There are now three species and the concentration space is 3-Dimensional.

What about the AR in this 3-D space?

The feed concentration in terms of (A, B, C) is represented by point (1, 0, 0) in

this 3-D rectilinear space. The infinite time product composition, i.e. product

composition for complete conversion of A is (0, 0.5, 0.5). Both these points must

be on AR. What is the shape of the region which includes all other possible

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

Figure 6:Attainable region forA = B; CA0 = CB0 = 0.5, k=1hr

-1


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concentrations of A, B and C by employing any reactor of any size? A little

thought will indicate that it is still a straight line joining the feed and infinite time

product points. The AR is 1-Dimensional in that sense in this case also. It is not a

surface but a line. It can be represented in the 3-D space as in Figure 6.

In engineering, one invariably presents 3-D objects in their various 2-D views,

such as plan, elevation or profile views. The 3-D perspective is created by the

viewer in his mind from such views. We can try doing so in this case also and

show the AR by its projection on the A-B, A-C and B-C planes. These

representations are also shown in Figure 7. The AR is a straight line in all such

representations.

Figure 7:Attainable region for 2AB+C; CA0 =1, CB0 = CC0=0, k=1hr-1

The two dimensional views give the attainable concentrations of any two of the 3

components. The concentration of the remaining component can be deducted

from the overall mass balance which is CA+ CB+ CC= CA0+ CB

0+ CC

0. The feed

is pure A with concentration 1 kmol/m3in this case. Therefore, CA+ CB+ CC= 1

irrespective of the conversion of A. The 2-D representations are thus adequate

and provide an idea of all attainable compositions.

0 0.25 0.5 0.75 1 00.25 0.5

0.75 1

0.5

0.75

1

CB ( Kmol/m3)

0

CC(Kmol/m

3)

0.25

CA ( Kmol/m3)


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You can now try constructing 2-D or 3-D representations of AR as shown in

Figure 8 for the following examples which are similar variations of basic reaction

as we tried earlier.

Figure 8: 2D Projection of attainable region for 2A B+C; CA0 =1, CB0 = CC0=0,

k=1hr-1

Example 1F:Same as Example 1E but the feed is (0.5, 0.25, 0.25)

Example 1G:Same as Example 1E but the reaction is second order in A and

reversible with equilibrium constant 0.25

Figure 9 shows the graph between CB versus CA with feed as CA0=1 and

CB0=CC0=0.

Example 1H:Sameas Example 1G but with the feed composition as (0.5, 0.25,

0.25): The representation in Figure 10 shows AR for Example 1H.

You get 2 ARs which are 1-D in whatever the representation you choose. In one

case, the AR is a point or is 1-D.

We thus got 0 and 1-D ARs in this 3-D concentration space.

You can increase the dimensionality of the concentration space by making the

reaction have 4 or more components. For example, try A B + C + D or A + B

C + D etc. Irreversible nature could then be converted to reversible nature of

reaction. If you play around a bit with feed concentration and equilibrium

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

0

0.5

1

0 0.5 1

CC

(Kmol/m3)

CA (Kmol/m3)

0

0.5

1

0 0.5 1

CC

(Kmol/m3)

CB (Kmol/m3)


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constant, you will realize that the AR is still 0-D or 1-D. Representation of AR in

4-D concentration space is not conceivable and the best choice is to use a 2-D

space with selected pair of components. AR will be a straight line or a point in

such a representation.

What is stopping us from getting higher dimensional ARs, say a 2-D or 3-D AR,

such as a surface or a volume? We have played with almost everything except

that we have been considering only one reaction. Let us try multiple reactions for

a change and see what happens.

Example 2:Reaction A B C is carried out with pure A as feed. Feed is pure

A with concentration of A (CA0

) is 1 kmol/m

3

. The rate constants are 1 hr

-1

forboth the reactions. Construct an AR.

We have a series of two reactions, both irreversible, first order and with given

rate constant values. This is a reaction handled as the first encounter with

multiple reactions in any worthwhile book on CRE or Chemical Reaction

Engineering. CRE deals with two ideal reactor types, a CSTR and a PFR. Let us

see what product compositions one can achieve by employing a CSTR to begin

with. It can be of any size offering space time of any magnitude (0 to ).

Mass balance on species A and some jugglery gives the following relation

between concentration of A in CSTR product stream as a function of reactor

space time.

k1

CC

1

0

AA

+

=

Similarly, mass balance on B gives:

k1

CkcC

2

A1

0

BB

+

+=


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If concentrations of A and B are calculated at different values of from = 0 to

= , and plotted in the CA-CBspace, one gets a curve as shown in Figure 8. The

line indicates product compositions obtainable or attainable from a CSTR if all

feed is pushed through a single reactor of this type (CSTR). So, is this the AR weare looking for?

Not really! AR is the set of all compositions attainable through any combination of

any type of reactors, and not just a single reactor of a particular type. So, let us

say, we are ready to have two CSTRs in parallel, one processing mass fraction x

of the feed and the other the remaining fraction of feed (1-x). Depending on the

size of individual reactors, the product from each reactor will be some point on

the CSTR performance curve we have generated. Consider points 1 and 2 on the

curve as one such pair. When a product of composition as represented by point 1

and with mass fraction x is mixed with the product of composition as represented

by point 2 and mass fraction (1-x), we will get a combined product of composition

as shown on the line joining points 1 and 2. The exact location of the point will

split the line in two parts, one of fractional length x towards point 2 and one of

fractional length (1-x) towards point 1. This can be appreciated by simple mass

balance. That means, by changing x from 0 to 1, any point on this line is

attainable. This is true also of any other pair of points on the reactor performance

line, such as points 3 and 4, or 5 and 6 etc. In fact, this is true of a line joining the

two ends of the performance line (1, 0) (feed point) and (0, 0) (the infinite time

product composition). The same logic holds if one point is the feed point and

0

0.125

0.25

0 0.5 1

CB

(Kmol/m3)

CA (Kmol/m3)

12

3

5

6

4

Figure 9: Performance curve for A

BC ,CA0 =1, CB0 = CC0=0, k1=k2=1

hr-1,in CSTR


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another any point on the performance curve. The line joining these two points is

also attainable. In practice, one would achieve any composition on any such line

by passing a part of the feed through the reactor and mixing the remaining with

the product of the reactor by bypassing the reactor.

If all such possible lines are drawn and accepted as representing attainable

compositions, the overall AR that is attainable using none, one or two CSTRs

with or without bypass is as shown in Figure 9. The AR now is surface and not a

straight line. We got an AR which is two dimensional for the first time. It is a

surface in a 2-D plane. The composition of the third species (i.e. C) can be

obtained by overall balance as seen earlier.

But is this then the AR for the present case?

Actually, we do not know. Whereas we have allowed more than 1 CSTRs while

arriving at attainable product compositions, we have not yet considered using

PFR instead of, or in addition to, the CSTR. Let us do it now.

Similar to the above treatment, one can develop a PFR performance curve in the

2-D concentration space. Mass balance on A and B for a PFR gives the following

relationships.

k0

AA1eCC =

0

0.125

0.25

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3

)

Figure 10:AR for ABC ,CA0

=1, CB0 = CC0=0 k1=k2=1 hr-1

,in

CSTR


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[ ]kk12

0

A1k0

BB212 ee

kk

CkeCC

+=

If concentrations of A and B are calculated for varying values (0 to ) and

plotted, we get this curve. It is shown in Figure 10 along with the attainable

region with CSTR constructed earlier. We thus see more product compositions

attainable outside the region we developed using CSTR.

Figure 12: Performance curve for ABC ,CA0 =1, CB0 = CC0=0, k1=k2=1 hr-1

,

in both CSTR and PFR

Extending the logic used earlier (if any two compositions are attainable, then the

compositions lying on the line joining these two points are also attainable), we

can see the AR as the entire surface enclosed by the PFR line. This region

includes the region attainable using only CSTRs.

For the given feed and the reactions (stoichiometry, nature and kinetics), the AR

is thus possible to construct. For the series reaction considered above, the AR is

as shown in Figure 11. Composition outside this AR cannot be attained by any

series-parallel combination of PFR/CSTR with or without bypass from the givenfeed.

At this stage, we can formulate a more comprehensive definition of AR as

follows.

0

0.125

0.25

0.375

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

CSTR

PFR

Figure 11: Performance curve for A

BC ,CA0 =1, CB0 = CC0=0,

k1=k2=1 hr-1,in both CSTR and PFR


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AR or Attainable Region is a set of all compos it ions poss ible to achieve

from a given feed compos ition by employing one or more reactors o f basic

ideal types such as CSTR and PFR in a network compr ising of any series-

parallel combination of them with or w ithout a bypass stream and for g iven

reaction scheme in terms of its stoichiometry, kinetics and

thermodynamics.

Let us restrict at this time to all reactors operating isothermally and at same

temperature. Also, we do not consider restrictions on reactor sizes or constraints

on feed availability in terms of its quantity.

All the ARs we developed so far fit into this definition.

We have argued during above development that if any two points representing

two compositions are a part of AR, then the line joining these points must also be

a part of AR. Mathematically, this means that the geometrical shape of the AR

must be convex

In the above example, we constructed on a 2-D concentration space an AR for a

reaction scheme involving 3 species. The same could have been depicted in the

actual 3-D concentration space. Verify that it would still have been a region

marked on a plane surface. What we saw on the 2-D space was only a projection

of this actual AR in 3-D space on the chosen 2-D view. The AR could have been

similarly depicted on a CA-CCor CB-CCspace. The looks could be quite different

and this is worth trying as an exercise. The expected AR looks in these

0

0.125

0.25

0.375

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

CSTR

PFR

Figure 13:AR for ABC ,CA0

=1, CB0 = CC0=0, k1=k2=1 hr-1,in both

CSTR and PFR


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alternative choices of depiction are shown in Figure 12. Also shown are the

corresponding reactor lines. Whatever may be the preferred representation style,

the information content of an AR is the same. This content is very profound and

can be used to design the best suited reactor network for a given task as will be

seen later.

(a) (b)

Figure 14:AR for A BC ,CA0 =1, CB0 = CC0=0, k1=k2=1 hr-1,in both CSTR

and PFR in (a) CB = CCand (b) CA = CC plane

As of now, you could construct AR for the following variations of the above

example.

Example 2A:Same as Example 2 except that equimolar mixture of A and B is

used as feed. Feed concentration of A (CA0) is 0.5 kmol/m3.

Example 2B: Same as Example 2 except that pure B is used as feed. Feed

concentration of B (CB0) is 1 kmol/m

3.

Example 2C: Reaction A = B = C is carried out with pure A as feed. Feed

concentration of A (CA0) is 1 kmol/m

3. The forward rate constants are 1 hr

-1 for

both the reactions. Equilibrium constants for both the reactions are 1.0.

0

0.5

1

0 0.25 0.5

CC

(Kmol/m3)

CB (Kmol/m3)

CSTR

PFR

0

0.5

1

0 0.5 1

CC

(Kmol/m3)

CA (Kmol/m3)

CSTR

PFR


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Example 2D:Same as Example 2C except that feed is equimolar mixture of A, B

and C with feed concentration of A (CA0) as 1.0 kmol/m3. The forward rate

constants are 1 hr-1 for both the reactions. Equilibrium constants for both the

reactions are 1.0.

Example 2E: Reaction A B C is carried out with pure A as feed. Feed

concentration of A (CA0) is 1 kmol/m3. The rate constants are 1 m3/kmol/hr for

both the reactions.

And so on. You can cook up your own examples and practice more.

Some of these examples required you to go through a lengthy derivation, some

called for numerical techniques to solve equations to generate data to be plotted

to construct AR. If you have been efficient and correct, the examples took you

through these steps to handle reaction kinetics. The end result was you got good

looking ARs of various dimensions. You can try more examples with

permutations and combinations of reaction types (irreversible, reversible),

kinetics, feed compositions etc. You could redo all examples by changing the

reaction to parallel instead of series (A B, A C) for variety. This will give you

a lot of feel for concentration space, dimensionality of AR, convexity of AR etc.

The mathematics involved even with simple looking and few reactions is often

formidable. We will now try something which will keep this reaction-related

mathematics at the lowest ebb and allow AR related things to be emphasized.

We will consider zero order reactions, without bothering as to how realistic such

kinetics is. This will help us appreciate AR and its geometrical content even

more.

Example 3: Reaction A B C is carried out with pure A as feed. Feed

concentration of A (CA0) is 1 kmol/m3. The rate constants are 1 and 0.5

kmol/m3/hr for the first and second reaction respectively. Construct an AR.


19/44

Let us create a CSTR performance curve. Mass balance on A will give the

change in concentration of A as follows.

CA= CA0 k1

This is valid only for reactors providing space time from 0 to * where * is

CA0/k1. For larger reactors providing > *, CAwill be zero.

Similarly, mass balance on B will give expression of CB in terms of . Till such

time A is available, concentration of B will build up in the reactor because

consumption rate of B is less than formation rate of B as can be seen from the

two rate constants. Once A is completely consumed, concentration of B will also

start dropping and eventually reach zero for reactor space time CA0/k2. C is

formed continuously at the rate of k2. The reactor performance curve is

comprised of two straight line sections in CA-CBspace. The feed point is (1, 0). At

* = 1 hr, CA= 0, CB= 0.5 and CC= 0.5. The straight line joining (1, 0) to (0, 0.5)

is valid till *. Then onwards, B drops from 0.5 to 0.0. By time = 2 hr, all B is

consumed. This is point (0, 0) on the CA-CB space. The straight line joining (0,

0.5) and (0,0) is the second part of reactor line. In the actual 3-D concentration

space for (A, B, C), the corresponding points are (1, 0, 0), (0, 0.5, 0.5) and (0, 0,

1.0).

Using the earlier logic to construct a convex AR from reactor lines, the AR in A-B,

A-C, B-C and A-B-C space are as shown in Figure 13. The triangle is a convex

region. The third line of the triangle in each case was arrived at by joining the

feed point and the point corresponding to 2 hr space time when both A and B

deplete to zero.

This was done assuming CSTR as a reactor. PFR should also have beenincluded. But is PFR different from CSTR for zero order reactions? Think and get

convinced that the ARs in Figure 13 are the ARs in totality and in compliance

with the definition given earlier.


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The zero order reactions simplified reaction mathematics and one could visualize

ARs with much less computation.

Consider two small variations of this example and construct ARs in various views

suitably

(a) (b) (C)

(d)

Example 3A: Reaction A B C is carried out with pure A as feed. Feed

concentration of A (CA0) is 1 kmol/m

3. The rate constants are 1 kmol/m

3/hr for

both the reactions. Construct an AR.

Example 3B:Reaction A B C is carried out with equimolar mixture of A and

B as feed. Feed concentration of A (CA0

) is 0.5 kmol/m3

. The rate constants are

0.5 kmol/m3/hr for both the reactions. Construct an AR.

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

0

0.5

1

0 0.5 1

CC

(Kmol/m3)

CB (Kmol/m3)

0

0.5

1

0 0.5 1

CC

(Kmol/m3)

CA (Kmol/m3)

Figure 15: AR for

reaction A B C;

CA0=1 CB0 =CC0=0, in

(a) CA-CB , (b) CA-Cc ,

(c) CB-Cc plane and (d)

CA-CB-CC space0

0.51

0

0.51

0

0.5

1

CB(Kmol/m3)

CA (Kmol/m3)

CC

(Kmol/m

3)


21/44

8 views, without label, are given in Figure 14. These include one 3-D view and

three 2-D views for each of the above two examples. Do your own development

and label these eight views of AR suitably with selections from following list.

(1) Example 3A, 3-D (2) Example 3A, CA-CB

(3) Example 3A, CA-CC (4) Example 3A, CB-CC

(5) Example 3B, 3-D (6) Example 3B, CA-CB

(7) Example 3B, CA-CC (8) Example 3B, CB-CC

Appreciate how the look and feel of the AR changes with the choice of 2-D

concentration space. Information content is of course the same. All ARs have

been represented with coordinates of vertices of the space clearly given. The

borders or edges of the AR were anyway straight lines and needed no equations

to be specifically mentioned. However, if the edges of AR are curves, it is good to

write the equations of all distinct curves. ARs with coordinates of vertices and

equations of edges mentioned become that much more readable and also usable

for reactor design. We will deal with reactor design using AR later.

You can try similar cases with parallel reactions: A B and A C and learn

more. Do not expect the ARs to be even remotely similar to those for series

reaction. The similarity between the problems is deceptive. Did you observe the

same thing whey you did similar thing earlier with reactions being first order?

ARs with zero order reactions are interesting. ARs with some reactions as zero

order and some as non-zero order are even more interesting. Let us try some.


22/44

(a) (b) (c)

(d) (e) (f)

(g)

(h)

Figure 16:8 views of example 3B

0

0.5

1

0 0.5 1

CC

(Kmol/m3)

CB (Kmol/m3)

0

0.5

1

0 0.5 1

CB

(Kmol/m3)

CA (Kmol/m3)

CA(Kmol/m )

CC

(K

mol/m

3)

0

0.5

10.5 1

0.5

0

CB(Kmol/m )

1

0

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

0

0.5

1

0 0.5 1

C

C

(Kmol/m3)

CB(Kmol/m3)

0

0.5

1

0 0.5 1

CC

(Kmol/m3)

CA (Kmol/m3)

0 0.5 1 00.50

0.5

1

CB (Kmol/m3)

CA (Kmol/m3)

CC

(Km

ol/m

3)

0

0.5

1

0 0.5 1

CC(Kmol/m3)

CA (Kmol/m3)


23/44

Example 4:Construct AR for the series reaction A B C. The first reaction is

zero order while the second reaction is a first order reaction. The rate constants

are 1 kmol/m3/hr and 5 hr-1 respectively. The feed is an equimolar mixture of A

and B with CA0= CB

0= 1 kmol/m3.

The procedure should be the same as earlier. We should construct CSTR and

PFR performance curves to begin with. The starting point of both the curves will

be the feed point i.e. point (1, 1) on the CA-CB space. For either reactor, the

infinite space time product concentration is (0, 0) in this plane. By overall mass

balance, the concentration of C for any combination of CAand CBcan be found

as CC= 2 - CA CB. Thus CC is 0 at feed point and 2 at the infinite space time

point.

For a PFR or CSTR, the mass balances can be solved to get CA and CB as

functions of and pairs of CAand CBfor same values can be plotted to get the

reactor (PFR, CSTR) curves.

For a CSTR, the mass balance on A gives:

CA= CA0 k1 (k1is the rate constant for the zero order reaction)

This is true till A reacts out completely. This happens for CA0/ k1

After this CA0 stays 0 and only second reaction is active. For the given values,

this time is 1 hr. Concentration of A thus drops linearly with for one hour and

then stays 0.

Mass balance on B can be written separately for 1 and > 1.

For 1, concentration of B varies with space time as follows.


24/44

k1

kCC

2

1

0

BB

+

+=

At = 1 hr, A has completely depleted (CA= 0), B has a concentration of 2/6 or

1/3 as per above equation (CB= 1/3) and concentration of C is found from overall

balance (CC= 5/3). Beyond this time, B depletes as per first order kinetics of the

second reaction giving the following.

1)-(k1

3/1C

2

B+

=

If we choose to represent AR on a 2-D CA-CB space, it is better to develop an

expression for CBin terms of CA. This can be easily done if above expression for

CAis used to express in terms of CAand this is substituted in the corresponding

expression for CB. For example, from the expression for CAderived above, one

can write

= (CA0 CA)/k1

This when substituted in the expression for CBvalid up to 1 hr, one gets

1A

0

A2

A

0

A

0

BB

k/)C-(Ck1

C-CCC

+

+=

This is the equation of the line joining feed point (1, 1) to the point (0, 1/3).

Beyond this point, CBalso drops to zero at infinite space time. That line is along

the ordinate in our 2-D representation. It is just a vertical line joining points (0,

1/3) to (0, 0). In any other representation (say CB-CCspace) one would have to

get the equation of that line too by similar procedure. The CSTR performanceline is shown in Figure 15.

The procedure can be repeated for the PFR also.


25/44

Balance on A yields the same expression for concentration of A as a function of

space time. This is so, because the type of reactor has no bearing on reactor

equation for a zero order reaction. A will thus deplete from feed concentration to

zero concentration for a space time of 1 hr as earlier.

The balance on B till such time is a first order ODE and its solution gives the

following expression for concentration of B.

[ ]k2

1k0

BB22 e1

k

keCC +=

At = 1 hr, A vanishes completely and concentration of B is obtained by

substituting = 1, k1= 1 k2= 5 and CB0

=1 in the above expression. This value is

CB(= 1) = 0.8 e-5+ 0.2

After this space time, B depletes solely by the second reaction as per first order

kinetics. The expression is given as follows.

CB= [0.8 e-5

+ 0.2] e-5(-1)

The equation for CBin terms of CAcan be obtained by substituting for in terms

of CA as was done for a CSTR. This will allow plotting the PFR line till A

vanishes. Beyond this, B drops to zero as per above expression. The PFR and

the CSTR curves are as shown in Figure 15. The PFR line is below the CSTR

line in the A-B space. What is then the AR?

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA(Kmol/m3)

CSTRPFR

Figure 17:Performance curve for

A B C, CA0 = CB0 = 1

kmol/m3, k1= 1 kmol/m3/hr, k2= 5

hr-1, both in CSTR and PFR


26/44

AR has to be convex. Therefore, feed point (1, 1) must be joined with CSTR point

(1/3, 5/3) by a straight line. The concavity of the CSTR curve is bridged by this

line which is attainable by a CSTR with bypass stream.

There is also a concave region below the PFR curve. All points below the PFR

curve which can be attained by 2 PFRs in parallel or a PFR in parallel with CSTR

should be enclosed into the AR. This region is retrieved by drawing a tangent to

the PFR line from point (0, 0). Since the point at which this line meets the PFR

line is attainable and (0, 0) is also attainable. All points on this line are attainable.

Now, one has a convex AR which is shown in Figure 16. The equation of the

tangent and coordinates of the point where it touches the PFR line must be

evaluated. That is left to you as an exercise.

This is the first time we encountered reactor lines which have non-convexity on

both sides. It was bridged on one side by joining two prominent vertices while on

the other side, one had to draw a tangent. This is something new we saw by

conjuring up a series reaction with combination of zero and first order reactions.

It is suggested that you construct AR for this case in the other two choices of

views (in A-C and B-C space). The looks will be different but the information

content the same.

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA(Kmol/m3)

CSTR

PFR

Figure 18:AR forA B C, CA

= CB0= 1 kmol/m

3, k1= 1 kmol/m

3/hr,

k2= 5 hr-1

, both in CSTR and PFR


27/44

For zero order reactions, one has to realize that one or more reactants can be

completely consumed with concentration(s) dropping to zero. Once that happens,

that particular reaction is as good as not there. This should be kept in mind while

developing the reactor curves on way to development of ARs.

Since we are with reaction schemes with a sprinkle of zero order and first order

reactions, the following 2 examples would be interesting to attempt. If you do

these yourself, you would have revised all concepts we developed so far, without

much mathematical complications. .

Example 5: 10 different cases are listed below. These are different in the

reaction scheme and/or the feed specification. Identify for each case whether the

AR is 1-D, 2-D, 3-D or of higher dimension. Construct the AR only for cases

where AR is 1-D or 2-D. Select the x-y axes for your AR judiciously. Indicate on

the AR itself the equation of the edges and coordinates of the corners of the AR.

0 1 0 1 0Reaction Scheme 1: ABCDEF

1 0 1 0 1

Reaction Scheme 2: ABCDEF

The integers above the reaction arrow are the orders of those reactions. For

example, the 5 reactions in scheme 1 are of order 0, 1, 0, 1, 0 respectively. For

reaction scheme 2, the orders of the 5 reactions involved are 1, 0, 1, 0, 1

respectively.

Case 1: Reaction Scheme:1, Feed: pure A with concentration 10 kmol/m3

Case 2: Reaction Scheme:1, Feed: pure B with concentration 10 kmol/m3

Case 3: Reaction Scheme:1, Feed: pure C with concentration 10 kmol/m3

Case 4: Reaction Scheme:1, Feed: pure D with concentration 10 kmol/m3

Case 5: Reaction Scheme:1, Feed: pure E with concentration 10 kmol/m3

Case 6: Reaction Scheme:2, Feed: pure A with concentration 10 kmol/m3


28/44

Case 7: Reaction Scheme:2, Feed: pure B with concentration 10 kmol/m3

Case 8: Reaction Scheme:2, Feed: pure C with concentration 10 kmol/m3

Case 9: Reaction Scheme:2, Feed: pure D with concentration 10 kmol/m3

Case 10: Reaction Scheme:2, Feed: pure E with concentration 10 kmol/m3

The rate constant values for the zero order reactions are 10 kmol/m3/hr.

The rate constant values for the first order reactions are 1 hr-1

.

At first sight, the cases look formidable. With 6 species involved, the

concentration space is of a much larger dimension than we discussed so far.

ARs could thus also be of higher dimensions than we are comfortable with.

However, you are asked to only state the dimension of AR in each case andconstruct only ARs of dimension 2 or less. As you think more, these cases are

not beyond your reach.

We will discuss the cases a bit more and then leave the rest to you once the

solution seems attainable.

The series reactions are alternately of zero and first order. Let us consider Case

1.

The first reaction is of zero order and feed is pure A. The rate of consumption of

A to form B is very high due to high value of the rate constant of the zero order

reaction. B will thus be formed rapidly and till A is present. B will simultaneously

react to C as per first order kinetics. B will not be consumed entirely at any time,

except in a reactor with infinite space time. Concentration of any species cannot

exceed the feed concentration of A for the given stoichiometry. Concentration of

C that is formed is thus always less than 10 kmol/m3. Rate of consumption of C

to form D by zero order reaction is more than rate of formation of C from B for the

given rate constant values. Therefore, C will not be there in the reaction mixture

at any time. The reaction C D is thus instantaneous for all practical purposes


29/44

and C will be converted to D as soon as it is formed. The reaction C D could

thus be dropped from the reaction scheme without any effect on the end result. D

converts to E as per first order reaction and will be there in the reaction mixture

all the time. E converts to F by zero order reaction. By arguments as put forth

above, E will immediately get converted to F for given feed concentration and

rate constants. The effective reaction in this case is thus A B D F. The

first reaction is zero order and the following two first order. The concentration

space is 4-D and the AR is 3-D (check this statement. It could be misleading).

So, we do not have to construct AR in this case.

Case 2 is similar but feed is pure B. There is no A in feed and A cannot be

formed because the reactions are irreversible. We are thus basically starting with

a curtailed reaction scheme BCDEF. If you argue on the same lines

as above, you would realize that C and E will be consumed as soon as they are

formed and will not be present in the reaction mixture any time. The effective

reaction is thus B D F with both reactions as first order. We have handled

this case earlier. The AR will be 2-D and can be constructed as we did earlier.

Case 3 has pure C as feed. The first two reactions and species A and B are thus

irrelevant. The curtailed reaction is C D E F which further reduces to C

D F. The first reaction is zero order and the second first order. We have

handled AR for this series reaction previously, although for different values of

rate constant and different feed composition. You have to only do it for given rate

constant values and feed composition.

Case 4 has pure D as feed and the first three reactions and species A, B, C are

thus redundant. The effective reaction scheme is a single zero order reaction D

F. AR is 1-D and a straight line joining points (10,0) and (0,10) in D-F



30/44

Case 5 is anyways a single zero order reaction E F with similar 1-D AR in E-F


You thus essentially develop ARs for cases 2 and 3. Both are series reactions

and have been handled by us earlier. See if you can explore case 1 more.

Cases 6-10 can now be tackled successfully by you. You will surmise that the

ARs are respectively 3-D, 3-D, 2-D, 2-D and 1-D (Be careful). The 2-D and 3-D

ARs are similar to corresponding ARs for the earlier 5 cases.

The definition of AR given earlier is the most commonly used. One can however

extend it further by removing some restrictions one by one. For example, can we

develop AR for a case where more than one different feed compositions are

available? Similarly, can we develop AR for the case where a reactor of given

size is available and must be the only reactor to be used? We are basically trying

to find out what happens to the AR if there are constraints on resources such as

feed and reactor vessels. This is important because we ultimately want to use AR

for reactor design. There is no practical situation where there are no constraints

on resources.

Let us get back to our first example and extend it a further.

Example 6: For reaction A B, construct AR. Two feed compositions are

available. One feed has only A and inert with concentration of A as 1 kmol/m3.

The second feed has A, B and inert with concentration of A and B as 1 and 0.2

kmol/m3respectively. The reaction rate constant is 1 hr-1.

On the A-B concentration space, the compositions attainable from the first feed is

a straight line joining (1,0) and (0,1) points. From the second feed, any

composition on line joining (1,0.2) and (0, 1.2) is attainable. For the individual

feeds, the ARs are 1-D as expected. But if both the feeds are available, the AR


31/44

will be a parallelogram bounded by these two straight lines (Figure 17). The AR

is thus 2-D. Any composition within the parallelogram is attainable by suitably

mixing the two feeds and processing the combined feed to a certain conversion

in a suitable reactor. It is also possible to process two feeds in two separate

reactors and then mix the products to achieve the desired composition.

If a restriction is put that the feeds should not be mixed at all prior to reaction, but

a feed could mix with product formed in reactor using another feed, what will be

AR? If an additional restriction is put that you can employ only one reactor, what

will be the AR? Think hard on this.

What is important to note is that the availability of alternative feeds has increased

the dimensionality of AR. 1-D AR for a single feed and for given reaction became

a 2-D AR if two feeds are available. So, what happens if we have 3, 4 or more

feeds available? What will happen if in the above case, we had another feed also

available with concentrations of A and B as 0.6 and 0.2 kmol/m3respectively? Do

we get a 3-D AR?

What is AR in this 3-feeds case if restrictions such as no mixing of feeds prior to

reaction and employing only one reactor are applied? Again, think hard and

develop this AR. What if we relax the restriction a bit and say that you could use

up to 2 reactors, but not 3? Think carefully and develop this AR.

0

0.4

0.8

1.2

0 0.5 1

CF(Km

ol/m3)

CE (Kmol/m3)

Feed 1

Feed 2

Figure 19:AR for A

B with twofeed composition


32/44

That was of course a stupid question. The AR is 2-D in this case as you can

verify. It is not a parallelogram, but is a polygon with three vertices as three feed

compositions and other points obtained by infinite time composition with

individual feeds. The ARs with other restrictions on mixing and on number of

reactors could be different.

Example 7:Feed stream of pure A with concentration 8 kmol/m3 is available. A

CSTR of volume 25 m3 is also available and must be the only reactor used for

carrying out the following reaction. A B C. Both the reactions are zero order

with rate constants 16 and 8 kmol/m3/hr for the first and the second reaction

respectively. Construct an AR.

We have handled series zero order reactions earlier. The only new thing here is

that the reactor has to be a CSTR of given size. In any case, reactor type (PFR

or CSTR) makes no difference for zero order reactions. So, the only effective

restriction is the reactor size. However, by adjusting the feed rate through the

reactor, any space time can be achieved. Therefore, as far as AR is concerned, it

is no restriction. Therefore, AR can be constructed as earlier.

Rate of formation of A is more than rate of consumption. B will thus build up in

the reactor till A is available. After that, B will start depleting and eventually reach

a zero concentration level. A will take 30 minutes or 0.5 hr to react out

completely. In this time, A will react to form C at the rate of 8 kmol/m3/hr. At 0.5

hr, the concentration of A, B and C in the reactor product will be 0, 4 and 4

kmol/m3. After this, B will deplete to zero over the next 30 minutes. At 60 minute

space time, the concentrations of A, B, C in product stream will be 0, 0, 8

kmol/m3. The AR is thus as shown in Figure 18.


33/44

Now let us consider a little variation in this example as follows and construct an

AR.

Example 7A:Same as Example 7 except that availability of feed is restricted to a

maximum of 50 m3/hr.

We are seeing a restriction on feed availability for the first time. This is not as

trivial a change as it appears. There is a devil in it. It would be nice if you mull

over this example for a day and then read ahead.

The AR without any restriction on feed availability was given in Figure 18. With

the given restriction on feed availability, the AR will be a part of this AR.

The convexity of AR was logically arrived at by arguing that if two parallel

reactors operating at different space times give two different product

compositions, then any point on the line joining these two attainable points on the

reactor line can be achieved by adjusting flow rates processed by the two

reactors and mixing the product streams. Another justification was that if any

composition on the reactor line is possible, the reactor product can always be

mixed with feed stream which bypasses the reactor and any composition on the

straight line joining any point on the reactor line to the feed point is also

attainable. AR became convex due to these logical arguments.

0

4

8

0 4 8

CB(Kmol/m3)

CA (Kmol/m3)

Figure 20:AR for reaction A B

C, k1=16 kmol/m3/hr, k2= 8

kmol/m

3

/hr, with a fixed reactorvolume.


34/44

In the present case, there is only one reactor vessel. Therefore there is no

possibility of reactors in parallel. What is possible is only a bypass reactor. But,

there is a restriction on how much feed can be mixed with the reactor product

due to limitation on the total feed availability. Both the basic arguments used in

constructing AR are thus challenged in this example.

Let us see what part of the reactor line is attainable in this case out of the reactor

line comprising of a straight line joining (8,0) with (0,4) and another straight line

joining (0, 4) with (0,0) in Figure 36.

The minimum reactor space time that is achievable is when the entire feed

passes through the available reactor. With maximum feed availability as 50 m3/hr

and reactor size fixed at 25 m3/hr, the minimum space time that can be achieved

is 30 minutes. Reactor operation at space times from 0 to 30 minutes are thus

not feasible. This range was represented by the straight line portion joining (8,0)

to (0,4) in the previous case. The only portion of the reactor line that is attainable

by processing part or full feed through the available reactor is thus the line (0,4)

to (0,0) in Figure 19.

Now consider the reactor operating different flow rates to achieve different space

times from 30 minutes (all feed through reactor) to higher space times.

With 30 minutes, the only point attainable is (0,4). Let us operate the reactor at

40 minutes space time. A feed rate of 37.5 m3/hr will thus pass through the

reactor. The product will be a point (0, 8/3) in the A-B concentration space of

Figure 19.

This reactor product can be mixed with a bypass flow rate not exceeding the

balance available feed (12.5 m3/hr). If the entire balance (12.5 m3/hr with A, B, C

concentrations of 8, 0, 0 kmol/m3 is mixed with a reactor product of 37.5 m3/hr

with concentrations of A, B, C as 0, 8/3, 16/3 kmol/m3. the resultant stream will


35/44

be a mixed cup concentration. It can be checked that the resultant concentrations

of A, B and C will be 2, 2, 4 kmol/m3. The bypass reactor can attain any point on

the line joining point (0, 8/3) on the reactor line with this point (2,2) by choosing to

bypass suitable feed quantities between 0-12.5 m3/hr.

If we now consider product from a reactor operated at space time of 60 minutes,

it will have the concentrations 0, 0, 8 kmol/m3. It is point (0,0) in Figure 19. The

volume processed through the reactor to achieve this space time is 25 m3/hr.

That leaves up to 25 m3/hr feed available for bypass and mixing with reactor

product. A product composition of 4, 0, 4 kmol/m3 or a point (4,0) in the

concentration space of Figure 19 is thus attainable by sending 25 m3/hr through

the reactor and mixing the rest with the reactor product. If bypass stream flow is

reduced appropriately, any point on the line joining (0,0) to (4,0) is attainable.

If these arguments are used for other reactor space time possibilities, the

attainable region for a reactor with bypass seems to be the triangle joining

vertices (0,4), (0,0) and (4,0) on the A-B concentration space. Is this then the

attainable region? It is convex alright. But can AR exclude the feed point which is

always attainable?

Let us look at vertex (0,0) of the concentration space. It is attainable with a

reactor of space time 60 minutes. However, it should be noted that for the zero

order reactions of concern here, this composition is attainable also by reactors

0

4

8

0 4 8

CB(Kmol/m3)

CA(Kmol/m3)

Figure 21:AR for reaction A B

C, k1=16 kmol/m3/hr, k2= 8

kmol/m3/hr, with a restricted feed

rate.


36/44

operating at space times larger than 60 minutes (i.e. flow rates smaller than 25

m3/hr). If smaller flow rates are processed through the reactor, larger quantities

will be available for bypass. This will mean that we can attain points on the CA

axis to the right of (4,0) point also. For example, if we process only 12.5 m 3/hr in

the reactor, we get a space time of 2 hr. All A and B would have reacted to give

C. The product will have concentrations of A, B and C as 0, 0, 8 in that case. This

will then be mixable with 37.5 m3/hr of feed to get a combined product of

concentrations (6, 0, 2). If lesser and lesser feed is processed through reactor

leaving larger and larger feed flow rates for bypass and mixing, we can attain

concentrations all the way up to the feed point on the X axis in Figure 19.

The Attainable compositions are thus captured by the AR as shown in Figure 19.

The AR is thus a triangle with a line attached to it. This region is not convex. And

still the triangle and the remaining line contain the only attainable product

compositions by employing a given CSTR with feed bypass.

For the first time, we are seeing an AR which is not convex (or did you get any

non-convex AR earlier?). This can happen when there are simultaneous

constraints on reactor sizes and feed availability.

The example has placed a question mark on our confident statement earlier that

AR must be convex. True, but only if the premises which led to the need for

convexity are satisfied. And these premises were the possibility of using more

than one reactors and also using a reactor with feed bypass with no limits on

feed availability.

Non-convex ARs are rarely talked about in the literature on AR. But keep these

restrictions in mind as the restrictions have a practicality about them.

A more challenging case will be the following variation of the above example.


37/44

Example 7B:Same as 7A except that both the reactions are first order with rate

constant 1 hr-1for both the reactions.

The AR will not be as easy to develop as for zero order reactions. Resorting to

computer calculations will be called for. However, one would expect similar

observations. Develop this AR.

We have dealt with a case of two feeds earlier. We also dealt with a case of AR

with limitation on feed availability. Let us combine the two in the following

example, which is amenable to simpler calculations but will require excellent

analytical skills.

Example 8: For the following reversible reactions, develop AR for given feed

compositions, feed availability and desired production rates. The reactions are all

reversible and reach equilibrium as soon as appropriate conditions (temperature,

presence of catalyst etc.) are created.

A = B (equilibrium constant K1)

B = C (equilibrium constant K2)

C = A (equilibrium constant K3)

It is given that 2 K1= K2= 2.

Four feed streams are available with following compositions (concentrations in

kmol/m3).

Feed I: CA0= 0, CB

0= 0, CC0= 4

Feed II: CA0= 1, CB

0= 0, CC0= 3

Feed III: CA0= 1, CB

0= 1, CC0= 2

Feed IV: CA0= 0, CB

0= 1, CC0= 3


38/44

In a concentration space of A-B, construct AR to produce 48 m3/hr of product. All

feeds are available at 12 m3/hr each.

It is actually a trivial problem. It may be noted that all feeds have the same total

concentration (4 kmol/m3). The given reactions do not change the number of

moles. Therefore, any product obtained by mixing the available feeds before

and/or after reaction will also have the total concentration of 4 kmol/m3. Given the

concentration of any two components in any feed or product stream, the third

concentration is thus easily calculated. Representation on a 2-D concentration

space (such as given A-B space) is thus adequate.

For this choice, the four feeds are at 4 points (0,0), (1,0), (1,1) and (0,1) as

shown in Figure 20. All are available at a maximum rate of 12 m3/hr.

Equilibrium constants are given for the first two reactions. The same for the third

reaction can be deducted easily as 0.5.

Any feed or any mixture in any proportion of any number of feeds, if pushed

through a reactor reaches equilibrium instantaneously. The equilibrium

composition can also be deducted from the equilibrium constants as CA0= 1, CB

0

= 1, CC0= 2. It is the same as Feed III corner in Figure 40, i.e. point (1,1) in the

A-B space. Any feed stream can thus be converted to a composition same as

that of Feed III by reaction.

Since, we must have a production rate of 48 m3/hr, we need to us all the four

feeds at their maximum availability. AR will be the region, any concentration in

which is attainable by reaction and or mixing.

The kinetics is not very important here as the equilibrium is reached

instantaneously as soon as we decide to carry out reaction and create suitable

conditions. Also, type of reactor (CSTR, PFR) is irrelevant because of the same


39/44

reason. To generate AR, we thus do not need much calculations, but some

geometrical interpretation of the feeds occupying four corners of our

concentration space. Think along the following lines to get the AR.

If we decide not to employ any reactor and simply mix all the feeds available, we

will get a stream at 48 m3/hr with concentration represented by point (0.5,0.5).

That is, concentrations of A, B and C as 0.5, 0.5 and 3 kmol/m3. This is thus an

attainable point. Similarly, if we react Feed I and II and mix the product with Feed

III, we have a stream represented by (1,1) at 36 m3/hr. If this is mixed with Feed

IV at 12 m3/hr, we get 48 m3/hr of a stream represented by point (0.75, 1.0) on

the A-B concentration space. Similarly, composition (1, 0.75) is attainable. If we

react Feed II to equilibrium, we have 24 m3/hr of stream with composition

represented by (1,1). If we mix Feed I and IV, we get 24 m 3/hr of stream with

composition represented by point (0,0.5). If we mix these two streams, we get 48

m3/hr of stream with composition (0.5, 0.75). Similarly point (0.75, 0.5) is

attainable. Thinking on these lines, you can arrive at the AR as shown in Figure

20.

You can check that no composition outside this AR is attainable by any scheme.

Using the above as a way to construct AR in this case, let us now look at some

variations of the problem. We only change the production rate that we want to

achieve.

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)

Figure 22: AR for system of

reaction A = B, B=C, C=A, with 4

different feed to 48 m3

/hr of

product.


40/44

Example 8A: Same as Example 8 except that the desired production rate is 36

m3/hr.

We thus have now surplus feed available. This will allow us more flexibility in

reaction and mixing and we expect the AR to be a larger area as compared to

AR for Example 8. Develop the AR and verify that it is as shown in Figure 21.

Example 8B: Same as Example 8 except that the desired production rate is 24

m3/hr.

Example 8C: Same as Example 8 except that the desired production rate is 12

m3/hr.

Example 8D: Same as Example 8 except that the desired production rate is 6

m3/hr.

The AR keeps on expanding as we reduce the product requirement. When the

production rate reaches 12 m3/hr, the AR fills the entire concentration space. For

any further reduction in production, the AR remains the same and occupies the

entire concentration space. The AR for Example 8B is shown in Figure 22. AR for

Examples 8C and D are squares joining the four feed points and are not shown

explicitly.

0

0.5

1

0 0.5 1

CB(K

mol/m3)

CA(Kmol/m3)



different feed to 36 m

3

/hr ofproduct.


41/44

Now let us make one small change to the above cases. So far, we have

developed AR for some reaction scheme or the other. What if the reactive

transformations are not possible. For example, what are the attainable

compositions if only mixing of streams is allowed and reactions are not possible.

In short what is the AR in the above 5 examples if reactive change is not possible

or disallowed?

The ARs for the case where production rates are 48, 36, 24, 12 and 6 m3/hr are

jumbled up in Figure 23. Identify which one belongs to which case.

If we carve out respective ARs which indicated compositions which can be

attained by simple mixing as developed above from the ARs which were

developed earlier for the case where reaction and mixing both were allowed, we

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

CB(Kmol/m3)

CA(Kmol/m3)

0

0.5

1

0 0.5 1

CB(Kmol/m3)

CA (Kmol/m3)



different feed to 24 m3/hr of

product.

Figure 25:AR for only mixing of

feed streams for a product of 48,

36, 24, 12 m3/hr .


42/44

are left with AR for the case where reaction must be resorted to or a reactor must

be employed. Note that these ARs are not convex. We thus got another example

of non-convex ARs.

We have so far played with almost all possibilities: single/multiple reactions (in

fact reaction and no reaction), irreversible/reversible reactions, different orders of

reaction, single/multiple feeds, constrained/unconstrained availability of feed,

variable/fixed reactor sizes etc. This must give confidence that AR can be

developed although at times calculations may be involved. The examples also

took us through zero, first, second and higher dimensional ARs. With this feel for

ARs, one can now appreciate reaction schemes and ARs that have caught

attention of researchers. Most of these reactions are series-parallel, a

combination we have not considered explicitly. To create challenging problems

for research, authors have played with the structure of reaction schemes, orders,

types of reactions, feed compositions and rate constant values. This has given

rise to extraordinary examples, some of which inherit the names of the

developers. These reaction schemes are called model reactions, the famous

ones being van de Vusse, Denbigh, Tramboze reactions etc.

It is not the purpose of this primer to dwell on these examples. The objective of

this compilation was to prepare you to appreciate these higher complexity

examples, should you choose to work in reactor network domain. In any case,

can anyone do better than Prof. Glassers creation of van de Vusse reaction and

its AR? That must be read in original and appreciated as such.

We would now revisit the dimensionality of AR to understand it better. Then we

will see how AR can be useful to design a reactor system for a given task.

We have seen geometrically ARs of dimensionality 0, 1, 2 and 3. Dimensionality

of AR for a given feed composition is primarily decided by the reaction network.

There is a simple way of deciding the dimensionality of AR before even one


43/44

launches into calculations. Let us understand this with some of the reaction

schemes we used in the examples discussed so far.

Example 1 had a simple irreversible reaction A B. To appreciate the

dimensionality of AR, one should simply write the net rate of formation of each

individual species from all the reactions. In the present case, we have two

species A and B. The rates of formation of A and B per unit volume are as

follows.

AB

AA

Ckdt

dC

Ckdt

dC

=

=

where k is the reaction rate constant. The dimensionality of the AR for a single

feed is the number of species whose concentrations are involved in some or all

the rate expressions for all the species together. In the above case, only

concentration of A is involved in the expressions and hence the AR is one

dimensional, i.e. a straight line. This is what we observed earlier also.

This holds true for Example 1A and 1B also. In Example 1C where the reaction

was second order, only the concentration of A will be raised to power 2 in both

the expressions. However, number of concentrations involved is still 1 and AR is

1-Dimensional.

Example 1C had a reversible reaction A = B and the rates of formation of A and

B should take the forms as follows.

B2A1

B

B2A1

A

CkCkdt

dC

CkCkdt

dC

=

+=


44/44

Two concentrations are involved on the right hand side of the rate expressions.

The AR should thus be 2-D. However, it should be noted that the stoichiometry

demands that the total concentration of A and B at any time must be the same as

total concentration of A and B in the feed. Therefore, concentration of B can be

expressed in terms of the feed concentration and the concentration of A.

Therefore, the rate expressions have only concentration of A on the right hand

side and the AR is 1-D as we found out earlier.

Same is true for other variations of Examples 1. In a couple of variations,

although the AR should be 1-D as per above analysis based on rate expressions,

it was actually zero order. Both these examples had reversible reactions and a

feed already at equilibrium. Further reaction was thus not possible and the AR

shrunk to the feed point and became 0-D.

We also considered AR for the cases when there were more than one feeds. In

such cases, the dimensionality of AR could increase by one as was seen in some

examples. Except for these special cases where reversible reactions are involved

and feed is at equilibrium itself or where there are multiple feeds, the

dimensionality of AR can be deducted from the rate expressions for all the

species.

You can check this logic with so many other examples we considered.

Now let us see how the AR can be used for designing the reactor networks for a

given task. This is the main practical reason why one would develop AR.

[1] Glasser, D.;Hildebarndt, D.; Ind. Eng. Chem. Res.1987,26,1803.

[2] Hildebarndt, D.; Glasser, D.;Crowe, M. C.;Ind. Eng. Chem. Res.1990,29,49.

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