Attainable region

Construction of attainable region using modeling tools

Fabiá n Ricardo Arévalo Vá squez

Chemical Engineer, MSc
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1. Reactor Network design using the attainable region

• The attainable region (AR) defines the

achievable compositions that may be obtained
from a network of chemical reactors.

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1. Reactor Network design using the attainable region

Figure illustrates the attainable region for

van de Vusse kinetics, based on the
reactions

3
1. Reactor Network design using the attainable region

The boundary of the attainable region is

composed by arcs each of which results
of the application of a distinct reactor
type.
 A CSTR with Bypass (curve C).
 A CSTR point O.
 A CSTR followed by a PFR, curve D.

Within the region bounded by the three arcs and the

horizontal base line CB=0, product compositions can be
achieved with some combination of these reactor
4
configurations.
1. Reactor Network design using the attainable region

Note that the maximum achievable concentration , is

obtained using a CSTR followed by a PFR

The appropriate reactor configuration along

the boundary of AR depends on the desired
effluent concentration of A.

When 1> Ca >0,38 kmol/m3, a CSTR with

bypass provides the maximum
concentration of B

when Ca=0.38 , this is achieven using a

CSTR (point O), followed by a PFR (curve D)
for Ca<0.38

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2. Construction of attainable region

Problem Statement

The following liquid phase, constant density, isothermal reaction is carried out with
the next conditions.

The end goal of this exercise is to determine the reactor configuration that maximizes the
production of B for a feed pure A.
2. Construction of attainable region

Problem Statement

The following liquid phase, constant density, isothermal reaction is carried out with
the next conditions.

The end goal of this exercise is to determine the reactor configuration that maximizes the
production of B for a feed pure A.
2. Construction of attainable region

Choose the fundamental processes

CSTR
2. Construction of attainable region

Choose the fundamental processes

PFR
2. Construction of attainable region

Choose the state variables

2. Construction of attainable region

Choose the state variables

CB is a state variable because it is the value that we wish to optimize

CA “The behavior of CB is entirely depend on the change of CA
t is not a state variable because it is the independent variable in the system
• CSTR

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2. Construction of attainable region

Solve the equations

CSTR
2. Construction of attainable region

Solve the equations

CSTR
2. Construction of attainable region

Choose the fundamental processes

CSTR
2. Construction of attainable region

Solve the equations We have to solve non linear equation system for each 𝜏
CSTR

𝜏   1

C
   𝐴 1
C
   𝐵1
 
𝜏

. . . . .
. . . . .
2. Construction of attainable region

Solve the equations We have to solve non linear equation system for each 𝜏
CSTR

𝜏   1

C
   𝐴 1
C
   𝐵1
 
𝜏

. . . . .
. . . . .
2. Construction of attainable region

Solve the equations We have to solve non linear equation system for each 𝜏
CSTR (option A) 𝜏   1

C
   𝐴 1
C
   𝐵1
 

https://www.wolframalpha.com/input/?i=systems+of+equations+calculator&assumption=%22FSe
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2. Construction of attainable region

CSTR (option A) We have to solve non linear equation system for each 𝜏
𝜏   1=5

  =5
  =?

https://www.wolframalpha.com/input/?i=systems+of+equations+calculator&assumption=%22FSe
lect%22+-%3E+%7B%7B%22SolveSystemOf4EquationsCalculator%22%7D%7D&assumption=%7B%
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2. Construction of attainable region

CSTR (option A) We have to solve non linear equation system for each 𝜏
𝜏   1=5

  =5
  =?
2. Construction of attainable region

CSTR (option A) We have to solve non linear equation system for each 𝜏
𝜏   1=5

  =5
  =?
2. Construction of attainable region

CSTR

CSTR
tau Ca Cb
0 1 0,000000000
0,01 0,618 0,000053700
𝜏   𝑖 0,05
0,125
0,3582
0,2456
0,000102300
0,000107000 1,2
CSTR
0,000120000
0,25 0,1809 0,000095000 1 0,000100000
0,375 0,1505 0,000085200

Cb((kmol/m3)
Ca (kmol/m3)
0,8 0,000080000
C
   𝐴 𝑖
0,5
0,625
0,1317
0,1187
0,000077500
0,000071500 0,6 0,000060000

C
   𝐵𝑖
0,75
0,875
0,109
0,1013
0,000067000
0,000062800
0,4 0,000040000

1 0,095 0,000059000 0,2 0,000020000

  1,125 0,0899 0,000057000 0 0,000000000
1,25 0,085 0,000054100 0 0,2 0,4 0,6 0,8 1 1,2 1,4
1,375 0,082 0,000051900 Space time (s)
1,5 0,078 0,000050000
1,625 0,075 0,000048300
1,875 0,07 0,000045300
2,25 0,064 0,000041700
4 0,049 0,000032000
5 0,044 0,000028700
2. Construction of attainable region

CSTR (Another available tools)

Compilador

https://pynative.com/online-python-code-editor-to-execute-py
thon-code/

Tutoriales

https://www.youtube.com/watch?v=S4Qg2CsiIj8
2. Construction of attainable region

Solve the equations

CSTR Solve the equation for different values of spatial time
CSTR
1.2 0

1 0

0.8 0

Cb((kmol/m3)
Ca (kmol/m3)

0.6 0

0.4 0

0.2 0

0 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4

Space time (s)

• PFR

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2. Construction of attainable region

Solve the equations

PFR
2. Construction of attainable region

Solve the equations

PFR
2. Construction of attainable region

Solve the equations

PFR
2. Construction of attainable region

Solve the equations

PFR Solve the equation for different values of spatial time
1.2 1.20E-04
PFR

1 1.00E-04

0.8 8.00E-05

Cb((kmol/m3)
Ca (kmol/m3)

0.6 6.00E-05

0.4 4.00E-05

0.2 2.00E-05

0 0.00E+00
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Space time (s)

2. Construction of attainable region

PFR (Another available tools)

Compilador

https://pynative.com/online-python-code-editor-to-execute-py
thon-code/

Tutoriales

https://youtu.be/BRe7qKIAa34
2. Construction of attainable region
CSTR (Another available tools)
2. Construction of attainable region
PFR (Another available tools)

Simulation of other multiple reaction using MATLAB

2. Construction of attainable region
PFR (Another available tools)

Simulation of other multiple reaction using MATLAB

code

syms CA(t) Cr(t) Cs(t)

tmax=100; % Maximun time of simulation
eqns = [diff(CA,t)==-0.1*CA, diff(Cr,t)==0.1*CA-0.2*Cr,
diff(Cs,t)==0.2*Cr]; % The equations
cond = [CA(0)==1, Cr(0)==0 Cs(0)==0]; %Initial conditions
[CASol(t) CrSol(t) CsSol(t)] = dsolve(eqns, cond); %Solve de
equations, and save then in solutions vectors
hold on
ezplot(CASol,[0,tmax])
ezplot(CrSol,[0,tmax])
ezplot(CsSol,[0,tmax]), legend('CA','Cr','Cs')
title('System Solution')
 
 
 
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2. Construction of attainable region

Solve the equations

CSTR Solve the equation for different values of spatial time
CSTR
1.2 0

1 0

0.8 0

Cb((kmol/m3)
Ca (kmol/m3)

0.6 0

0.4 0

0.2 0

0 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4

Space time (s)

2. Construction of attainable region

Constructing the region

State-Space diagram- CB= f(CA)
1.20E-04 CSTR and PFR

1.00E-04

8.00E-05
CB (kmol/m3)

6.00E-05
CSTR
PFR

4.00E-05

2.00E-05

0.00E+00
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

CA (kmol/m3)
2. Construction of attainable region

Constructing the region State-Space diagram- CB= f(CA)

CSTR and PFR

X represents an arbitrary CSTR

X effluent point.
CA=0.3582
CB=0.0001023
CB (kmol/m3)

CA (kmol/m3)
2. Construction of attainable region

Constructing the region State-Space diagram- CB= f(CA)

CSTR with Bypass

X
CB (kmol/m3)

CA (kmol/m3)
2. Construction of attainable region

Constructing the region State-Space diagram- CB= f(CA)

CSTR with Bypass

To achieve any concentration along line , you

can mix the oulet of a CSTR operating at
point X with the feed at point O.
X
CB (kmol/m3)

CA (kmol/m3)
2. Construction of attainable region

Constructing the region State-Space diagram- CB= f(CA)

CSTR followed by PFR

The complete succesive PFR profile is found by numerically solving the differential PFR balancing equations with
feed cncentrations of X point.
2. Construction of attainable region

Constructing the region State-Space diagram- CB= f(CA)

CSTR followed by PFR
1.40E-04

1.20E-04
CB (kmol/m3)

1.00E-04

CSTR
8.00E-05
PFR
CSTR-PFR
CSTR-BYPASS

6.00E-05

4.00E-05

2.00E-05

0.00E+00
0 0.1 0.2 0.3 0.4 3)
CA (kmol/m 0.5 0.6 0.7 0.8 0.9 1
2. Construction of attainable region

1.40E-04

1.20E-04

1.00E-04
CB (kmol/m3)

CSTR
8.00E-05
PFR
CSTR-PFR
CSTR-BYPASS

6.00E-05

4.00E-05

2.00E-05

0.00E+00
0 0.1 0.2 0.3 0.4
CA (kmol/m3)0.5 0.6 0.7 0.8 0.9 1
References

B. J. G. Matthew J. Metzger, “Teaching Reaction Engineering Using the Attainable

Region,” Chemical engineering education, 2007.

D. R. Seider, W.D. Seader, J.D. Lewin, “Product and Process Design Principles -
Synthesis, Analysis, and Evaluation.” pp. 505–556, 2003.

H. S. Fogler, Elementos de ingeniería de las reacciones químicas. 2004 .

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