1 Ref Seider et al Product and process design principles 3 rd ed Wiley 2010 Attainable Region Attainable region AR defines the achievable compositions that may be obtained from a network of chemical reactors ID: 168533
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Slide1
Reactor Network Design Using Attainable Region
1
Ref:
Seider
et al, Product and process design principles, 3
rd
ed., Wiley, 2010.
Slide2
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 (1987-1990). 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:
2Slide3
Attainable Region
The van de
Vusse
kinetics is:
At a particular temperature:Step 1: Begin by construction a trajectory for a PFR from the feed point, continuing to the complete conversion of A or chemical equilibrium. For this case we have:
3Slide4
Attainable Region
4Slide5
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,
the PFR trajectory is not convex from A to B
, so proceed to the next step.
5Slide6
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: ,where
c
1 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
c
1
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.
6Slide7
Attainable Region
7Slide8
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: 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. 8Slide9
Attainable Region
9Slide10
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.
10Slide11
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.
11Slide12
Attainable Region
12Slide13
Example 7.4
Maleic
anhydride,C
4
H2O3, is manufactured by the oxidation of benzene with excess air over vanadium pentoxide catalyst:Since air is supplied in excess, the reaction kinetics are approxi
-mated using first-order rate laws:
A is benzene, P is
maleic
anhydride, and B and C are byproducts (CO
2
and H
2
O). The ris have the units of m
3/(kg catalyst.s
).
13Slide14
Example 7.4
Given that the available feed stream contains benzene at a concentration of 10 mol/m
3
, 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,
k
1
, is lager. An operating temperature at the upper end of this range is recommended, as the rate of reaction increases with temperature.
14Slide15
Influence of temperature on rate constants for MA manufacture
15Slide16
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:
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.
16Slide17
Attainable regions for MA manufacture
17Slide18
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.
18Slide19
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.
19Slide20
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%.
20Slide21
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 relat-ionship between the reacting species, permitting the complete system to be projected onto a reduced space of independent species.
21Slide22
Principle of Reaction Invariants
Let the reacting system consist of
n
i
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 a
ij
form the atom matrix
A
., nothing that since the number of gram-atoms for each element remain constant,
Partitioning
Δn
and A into dependent,
d, and independent, i, components:
If A
d
is square and nonsingular,
Δ
n
d
can be obtained by :
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.
22Slide23
Example 7.5: AR for Steam-Methane Reforming
Construct the attainable region for the steam-methane reforming (SMR) at 1050 K
, 40 bar, H
2
O/CH4 = 3, and use it to identify the networks that provide for the maximum composition and selectivity of CO. Solution: The main two equilibrium reactions of the SMR are:
Xu
and
Froment
(1989) provide the kinetic expressions for the above reactions as follows:
23Slide24
Example 7.5: AR for Steam-Methane Reforming
24Slide25
Example 7.5: AR for Steam-Methane Reforming
25
k
1,648
k
2,648
K
CO,648
K
H2,648
K
CH4,823
K
H2O,823
1.842×10
-4
2.193×10
-5
40.91
0.02960
0.1791
0.4152
E
1
E
2
Δ
H
CO
Δ
H
H2
Δ
H
CH4
Δ
H
H2O
kJ/
kmol
2.401×10
5
2.439×10
5
-
7.065×10
4
-
8.290×10
4
-
3.828×10
4
8.868×10
4Slide26
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.
26Slide27
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:
The partial pressure of each components can be calculated by: This leads to trajectory (1), which tracks the composition from the feed point, A, to chemical equilibrium at point B.
27Slide28
Example 7.5: AR for Steam-Methane Reforming
28Slide29
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.29Slide30
Example 7.5: AR for Steam-Methane Reforming
30Slide31
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: 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.31Slide32
Example 7.5: AR for Steam-Methane Reforming
32Slide33
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.
33Slide34
Example 7.5: AR for Steam-Methane Reforming
34