Ref: Seider et al, Product and process design principles, 3rd ed., Wiley, 2010.
1
Attainable Region
 Attainable region (AR) defines the achievable compositions that
may be obtained from a network of chemical reactors.
 The attainable region in composition space was introduced by
Horn (1964), and extended by Glasser and co-workers (19871990).
 A systematic method for the construction of the attainable region
using CSTRs and PFRs, with or without mixing and bypass (as
presented by Hildebrant and Biegler, 1995), is demonstrated for
van de Vusse kinetics as follows:
2
Attainable Region
 The van de Vusse kinetics is:
k1
k3
k4
A  B  C , 2A  D
k2
where : r1  k1C A , r2  k 2CB , r3  k3CB , r4  k 4C A2
At a particular temperature:
k1  0.01s-1, k2  5 s-1, k3  10 s-1, k4  100 m3 /(kmol.s)
 Step 1: Begin by construction a trajectory for a PFR from the
feed point, continuing to the complete conversion of A or
chemical equilibrium.
dCA
  k1CA  k 2CB  k 4CA2
For this case we have:
d
dCB
 k1CA  k 2CB  k3CB
d
3
Attainable Region
4
Attainable Region
 Step 2: When the PFR trajectory bounds a convex region, this
constitutes a candidate AR. A convex region is one in which all
straight lines drawn from one point on the boundary to any
other point on the boundary lie wholly within the region or on
the boundary. If not, the region is nonconvex. When the rate
vectors, [dCA/dτ, dCB/dτ]T, at concentrations outside of
candidate AR do not point back into it, the current limits are the
boundary of AR and the procedure terminates.
 In this example,
, so
proceed to the next step.
5
Attainable Region
 Step 3: The attainable region is expanded by linear arcs,
representing mixing between the PFR effluent and the feed. Note
that a linear arc connecting two points on a composition
trajectory is expressed by the equation: c*   c1  (1   ) c2 ,where
c1 and c2 are vectors for two streams in the composition space,
c* is the composition of the mixed stream, and α is the fraction of
the stream with composition c1 in the mixed stream. The linear
arcs are then tested to ensure that no rate vectors positioned on
them point out of the AR. If there are such vectors, proceed to
the next step, or not return to step 2.
 In this example, a linear arc, ADB, is added, extending the AR to
ADBC. Since rate vectors computed along this arc are found to
point out of the extended AR, proceed to the next step.
6
Attainable Region
7
Attainable Region
 Step 4: Since there are vectors pointing out of the convex hull, it
is possible that a CSTR trajectory enlarges the attainable
region. After placing the CSTR trajectory that extends the AR the
most, additional linear arcs that represent the mixing of streams
are placed to ensure that the AR remains convex.
 The CSTR trajectory is computed by solving the CSTR form of
the kinetic equations as a function of the residence time, τ:

CA0  CA   k1CA  k2CB  k4CA2
C B   k1CA  k2CB  k3CB 

 For this example, the CSTR trajectory that extends the AR most
is that computed from the feed point (curve AEF), which passes
through point B.
8
Attainable Region
9
Attainable Region
 Since the union of the previous AR and the CSTR trajectory is
not convex, a linear arc, AGO, is augmented. This arc represents
a CSTR with a bypass stream.
10
Attainable Region
 Step 5: A PFR trajectory is drawn from the position where the
mixing line meets the CSTR trajectory. If this PFR trajectory is
convex, it extends the previous AR to form an expanded
candidate AR. Then return to step 2. Otherwise, repeat the
procedure from step 3.
 As shown in the next Figure, the PFR trajectory, OHI, leads to a
convex attainable region. The boundaries of the region are: (a)
the linear arc, AGO, which represents a CSTR with bypass
stream; (b) the point O, which represents a CSTR; and (C) the
arc OHI, which represents a CSTR followed by a PFR in series.
It is noted that the maximum composition of B is obtained at
point H, using a CSTR followed by a PFR.
11
Attainable Region
12
Example 7.4
 Maleic anhydride,C4H2O3, is manufactured by the oxidation of
benzene with excess air over vanadium pentoxide catalyst:
9
Reaction1 : C 6 H 6  O 2  C 4 H 2 O 3  2CO 2  2H 2 O
2
Reaction2 : C 4 H 2 O 3  3O 2  4CO 2  H 2 O
Reaction3 : C 6 H 6 
15
O 2  6CO 2  3H 2 O
2
 Since air is supplied in excess, the reaction kinetics are approxi-
mated using first-order rate laws:
k3
k1
k2
A 
P 
B, A 
C, r1  k1CA , r2  k 2CP , r3  k3CA
  12660
  15000
  10800
k1  4280exp
,
k

70100
exp
,
k

26
exp
 2
 T (K )  3
 T (K ) 
T
(
K
)






A is benzene, P is maleic anhydride, and B and C are byproducts
(CO2 and H2O). The ris have the units of m3/(kg catalyst.s).
13
Example 7.4
 Given that the available feed stream contains benzene at a
concentration of 10 mol/m3, with a volumetric flow rate , v0, of
0.0025 m3/s (the feed is largely air), propose a network of
isothermal reactors to maximize the yield of maleic anhydride.
 Solution: First, an appropriate reaction temperature is selected.
Following Heuristic 7 in chapter 6, the next Figure Shows the
effect of temperature on the three rate coefficients, and indicates
that in the range 366< T <850 K, the rate coefficient of the
desired product, k1, is lager. An operating temperature at the
upper end of this range is recommended, as the rate of reaction
increases with temperature.
14
Influence of temperature on rate constants for
MA manufacture
15
Example 7.4: Solution
 For this system, the attainable region is straight forward to
contract. This begins by tracing the composition space trajectory
for a packed-bed reactor (PBR), modeled as a PFR, which
depends on the solution of the molar balances:
dCA
 k1CA  k3CA , CA0  10 mol/m3
dW
dCP
v0
 k1CA  k 2CP , CP0  0, W  kg of cat alyst
dW
v0
 The next Figure represents solutions of these equations for
several operating temperatures. Since these trajectories are
convex, and rate vectors computed along their boundaries are
tangent to them, it is concluded that each trajectory bounds the
AR for its corresponding temperature.
16
Attainable regions for MA manufacture
17
Example 7.4: Solution
 Evidently, a single PFR provides the maximum production of
maleic anhydride, with the desired space velocity being that
which brings the value of CP to its maximum value.
 At 800 K, it is determined that the maximum concentration of
MA is 3.8 mol/m3, requiring 4.5 kg of catalyst. At 600 K it is 5.3
mol/m3, but at this low temperature, 1400 kg of catalyst is
needed.
 A good compromise is to operate the PBR at an intermediate
temperature, for example, 770 K, with a maximum concentration
of MA of 4.0 mol/m3, requiring 8 kg of catalyst.
18
Example 7.4: Solution
 The following Figure shows composition profiles for all species
as a function of bed length (proportional to the catalyst weight),
for isothermal operation at 770 k.
19
Example 7.4: Solution
 The following Figure indicates that the yield (the ratio of the
desired product rate and feed rate) under these conditions is
61%, while the selectivity (the ratio of the desired product rate
and total product rate) is only about 10%.
20
Principle of Reaction Invariants
 Because the attainable region depends on geometric construct-
ions, it is effectively limited to the analysis of systems involving
two independent species.
 However, as shown by Omtveit et al (1994), systems involving
higher dimensions can be analyzed using the two-dimensional
AR approach by applying the principle of reaction invariants of
Fjeld et al. (1974).
 The basic idea consists of imposing atom balances on the react-
ing species. These additional linear constraints impose a relationship between the reacting species, permitting the complete
system to be projected onto a reduced space of independent
species.
21
Principle of Reaction Invariants
 Let the reacting system consist of ni moles of each species i, each
containing aij atoms of element j. The molar changes in each of
the species due to reaction are combined in the vector Δn, and the
coefficients aij form the atom matrix A., nothing that since the
number of gram-atoms for each element remain constant, A n  0
Partitioning Δn and A into dependent, d, and independent, i,
components: A  Ad | Ai , nT  nTd | nTi  Ad nd  Ai ni
If Ad is square and nonsingular, Δnd can be obtained by :
nd  A1
d Ai ni
The dimension of i is equal to the number of species minus the
number of elements. When this dimension is two or one, the
principle of reaction invariants permits the application of AR.
22
Example 7.5: AR for Steam-Methane Reforming
 Construct the attainable region for the steam-methane reforming
(SMR) at 1050 K, and use it to identify the networks that
provide for the maximum composition and selectivity of CO.
 Solution: The main three equilibrium reactions of the SMR are:
Reaction(1): CH 4  H 2O  CO  3H2
Reaction(2): CO  H 2O  CO 2  H 2
Reaction(3): CH 4  2H 2O  CO 2  4H 2
 Xu and Froment (1989) provide the kinetic expressions for the
above reactions as follows:
PH32 PCO 
 kmol  k1 
 (DEN)2
  2.5 PCH 4 PH 2O 
r1 
K1 
 kg cat  h  PH 2 
23
Example 7.5: AR for Steam-Methane Reforming
PH 2 PCO 2
 kmol  k 2 
 
r2 
PCO PH 2O 

K2
 kg cat  h  PH 2 

 (DEN)2


4
P
 kmol  k3 
H 2 PCO 2
2
  3.5 PCH 4 PH 2O 
r3 
K3
 kg cat  h  PH 2 
DEN  1  K CO PCO  K H 2 PH 2  K CH 4 PCH 4

 (DEN)2


PH 2O
 K H 2O
PH 2
4

2
.
292

10
2
3 
K1 (bar )  exp  20.52 
 7.195ln T  2.94910 T 
T



5.319103
4 
K 2 ( )
 exp  12.11
 1.012ln T  1.14410 T 
T


4

1
.
779

10
2
3 
K 3 (bar )  exp  29.96 
 7.835ln T  2.75010 T 
T

 24
Example 7.5: AR for Steam-Methane Reforming
 Ei  1 1 
ki  ki ,Tr exp    , i  1, 2, 3
 R  T Tr 
 H j  1 1 
   , j  CO, H 2 , CH 4 , H 2O
K j  K j ,Tr exp
 R  T Tr 
k1,648
k2,648
k3,648
KCO,648
KH2,648
KCH4,823
KH2O,823
1.842*10
7.558
2.193*10
40.91
0.02960
0.1791
0.4152
ΔHCO
ΔHH2
-4
E1
-5
E2
E3
ΔHCH4
ΔHH2O
kJ/kmol
2.401*10 6.713*10 2.439*10
5
4
5
8.868*10
4
7.065*10 8.290*10 3.828*10
4
Where: T  K , Pi  bar ,4 Tr (reference
temp.)4K
25
Example 7.5: AR for Steam-Methane Reforming
 These kinetic equations, involving five spices and three
elements. By evoking the principle of reaction invariants, the
number of independent species is reduced to two so that the AR
can be shown in two dimensions.
CH 4  H 2O  CO  3H2 , CO  H 2O  CO 2  H 2
CH 4  2H 2O  CO 2  4H 2
nT  [nTd | nTi ]  [nH2 , nH2O , nCO2 | nCH4 , nCO ]T
0 0 1

A  [ A d | A i ]  2 2 0
0 1 2
1 1

4 0
0 1
 4  1
nCH 4 


1
 n d   A d A i n i   2
1 


n
  1  1  CO 
26
Example 7.5: AR for Steam-Methane Reforming
 Step 1: Begin by construction a trajectory for a PFR from the
feed point, continuing to the complete conversion of methane or
chemical equilibrium. Here, the PFR trajectory is computed by
solving the following equations:
dNCH 4
dNCO
kmol
 r1  r3 ,
 r1  r2 , N i 
dW
dW
hr
The partial pressure of each components can be calculated by:
Ni
Pi 
N total
N H 2  4N CH 4  N CO
N H O  2N CH  N CO
2
4
P, where 
N  N CH 4  N CO
 CO 2
N total  2N CH 4
This leads to trajectory (1), which tracks the composition from
the feed point, A, to chemical equilibrium at point B.
27
Example 7.5: AR for Steam-Methane Reforming
28
Example 7.5: AR for Steam-Methane Reforming
 Step 2: When the PFR trajectory bounds a convex region, this
constitutes a candidate AR. In this example, the PFR trajectory
is not convex, so proceed to the next step.
 Step 3: The attainable region is expanded by linear arcs,
representing mixing between the PFR effluent and the feed.
Here, two linear arcs are introduced to form a convex hull,
tangent to the PFR trajectory from below, connecting to the
chemical equilibrium point B (line 2), and from the feed point to
point tangent to the PFR trajectory from above (line 3). It is
found that rate trajectories point out of the convex hull, so
proceed to the next step.
29
Example 7.5: AR for Steam-Methane Reforming
2
3
30
Example 7.5: AR for Steam-Methane Reforming
 Step 4: Since there are vectors pointing out of the convex hull, it
is possible that a CSTR trajectory enlarges the attainable
region. After placing the CSTR trajectory that extends the AR the
most, additional linear arcs that represent the mixing of streams
are placed to ensure that the AR remains convex.
 Here, the CSTR trajectory is computed by solving the CSTR
form of the kinetic equations as a function of catalyst weight, W:
NCH4 ,0  NCH4  W (r1  r3 ), NCO  W (r1  r2 )
 This gives trajectory (4), augmented by two linear arcs,
connecting the feed point to a point tangent to the CSTR
trajectory (line 5), and an additional line (6) connecting the
CSTR to the PFR trajectories.
31
Example 7.5: AR for Steam-Methane Reforming
32
Example 7.5: AR for Steam-Methane Reforming
 Step 5: A PFR trajectory is drawn from the position where the
mixing line meets the CSTR trajectory. If this PFR trajectory is
convex, it extends the previous AR to form an expanded
candidate AR. Then return to step 2. Otherwise, repeat the
procedure from step 3.
 As shown in the next Figure, the PFR trajectory (line 7) leads to
a convex attainable region. The boundaries of the region are: (a)
the linear arc (line 5), which represents a CSTR with bypass
stream; (b) the point C, which represents a CSTR; and (C) the
line 7 from C to B, which represents a CSTR followed by a PFR
in series. It is noted that the maximum composition of CO is
obtained at point D, using a CSTR followed by a PFR.
33
Example 7.5: AR for Steam-Methane Reforming
34