Comparison of configurations of a four-column simulated moving bed

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Comparison of configurations of a four-column
simulated moving bed process by multi-objective
optimization
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Yoshiaki Kawajiri* and Lorenz T. Biegler
Dept. of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave.,
Pittsburgh, PA, 15213
*Corresponding author, Phone:(412) 268-2238, Fax: (412) 268-7139, E-mail:
kawajiri@cmu.edu
June 21, 2007
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Abstract
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Configurations of a four-column simulated moving bed chromatographic
process are investigated by multi-objective optimization. Various existing column
configurations are compared through a multi-objective optimization problem.
Furthermore, an approach based on an SMB superstructure is applied to
find novel configurations which have been found to outperform the standard
SMB configuration. An efficient numerical optimization technique is applied to
the mathematical model of the SMB process. It has been confirmed that although
the optimal configuration highly depends on the purity requirement, the
superstructure approach is able to find the most efficient configuration without
exploring various existing configurations.
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1 Introduction
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Chromatographic separation processes have been applied to many products for large-
scale production. The most widely used operation mode is batch elution using a single

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column, where the feed mixture is loaded periodically and eluted with desorbent. In
this mode, the purified products are collected at the end of the column. On the other
hand, in the continuous mode, the feed can be continuously supplied and products
can be continuously withdrawn. The representative system is the simulated moving
bed (SMB) process, where multiple columns are connected in a cycle, as shown in
Figure 1. SMB mimics the countercurrent operation by intermittently switching the
four streams, desorbent, extract, feed, and raffinate, in the direction of the liquid
flow. Because of this continuous and pseudo-countercurrent operation mode, SMB
has the potential to increase the productivity and reduce the desorbent consumption.
However, numerical treatment of SMB processes comes with great difficulty because
of the complex operation.
In order to find efficient designs and operating parameters, various mathematical
programming techniques have been applied to SMB. Dünnebier et al. (2000) and
Toumi et al. (2002) have reported applications of sequential quadratic programming
(SQP), a Newton-based nonlinear programming technique. In our previous study, we
reported successful results of the full-discretization approach using an interior-point
method for SMB processes (Kawajiri and Biegler, 2006d).
In recent years, many modifications to the standard SMB have been attempted
to further improve the performance. Fig.2(b) shows the Three-zone configuration
(Ruthven and Ching, 1989; Ching et al., 1992), where the circulation from the right
end to the left is cut and all liquid is withdrawn as the raffinate stream. Fig.2(c)
shows the Three-zone configuration with purging, where the column at the left end
is isolated and the components in the column are purged into the extract stream.
As a result, the number of design alternatives for multi-column processes increases
rapidly. However, only a few studies consider the column configuration of SMB.
Zhang et al. (2002) used a genetic algorithm in their multi-objective optimization

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formulation. They investigated configurations of SMB with five columns. Ziomek
et al. (2006) explored many different configurations of five identical columns. For
this problem, we previously reported a superstructure based approach to find the
optimal zone configuration and operation (Kawajiri and Biegler, 2006b). Although
many novel configurations and operations have been found in previous studies, a
systematic comparison of configurations has never been considered.
In this work, we investigate extensively configurations of a four-column SMB.
Various existing column configurations are compared through a multi-objective optimization
study to analyze the trade-off of throughput and desorbent consumption.
The mathematical model is fully discretized and incorporated within a large-scale nonlinear
programming (NLP) problem, which is solved using an interior-point solver,
IPOPT (Wächter and Biegler, 2005). The superstructure approach, which embeds
many possible configurations such as Three-zone and standard SMB, is employed. A
discussion of the standard and nonstandard column configurations is given in Section
2. In Section 3, the mathematical model and superstructure approach are presented.
Finally, standard and nonstandard configurations are compared with the superstructure
approach in Section 4.
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2 SMB column configurations
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In the standard SMB configuration, the column are connected in a cycle as shown
in Figure 1. It is important to note that in a four-column process, there is only one
standard configuration, where one column comes between every inlet/outlet streams.
On the other hand, there are numerous non-standard configurations. One representative
example is Three-zone (Figure 2(a)), where the circulation is cut off and all
liquid is withdrawn in the raffinate stream. This configuration consumes more des-

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orbent than the standard SMB configuration, since there is no recycle stream from
the rightmost column to the leftmost column. Moreover, in Three-zone with purging
(Figure 2(b)), the leftmost column is isolated and all components in the column are
purged into the extract stream. This configuration is efficient if there is a component
which has a high affinity to the adsorbent (slow moving component).
Furthermore, in our previous study (Kawajiri and Biegler, 2006a), the F-shaped
configuration was extracted from the SMB superstructure as the optimal column
configuration (Figure 2 (d)). This configuration has been found to have the highest
throughput over other SMB configurations in our case studies of linear and nonlinear
isotherms, but at the expense of higher desorbent consumption. This novel structure
was an interesting finding; the desorbent stream is located downstream of the raffinate
stream, and the recycle stream is cut off. An analysis of this configuration is given in
Section 4.
It is also important to note that operations which involve multiple configurations
are possible. For instance, the operation of the Varicol system has multiple standard
SMB configurations in one step. This feature can be generalized to operations
which have multiple non-standard configurations. For this problem, we proposed a
superstructure-based approach where the dynamic optimal configuration is obtained
as a solution of an optimal control problem Kawajiri and Biegler (2006c). However,
such dynamic configurations can be hard to implement in large-scale processes, since
frequent changes of flow rates may lead to pressure instability. Therefore, investigations
of operations of “static column configuration,” where the flow rates are kept
constant in a step, also have significant importance.
In the following section, we formulate a multi-objective optimization problem to
compare the above-mentioned configurations. In this study, we only consider configurations
of four-column SMB. Discussion of SMBs with higher numbers of columns

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can be found in our previous studies Kawajiri and Biegler (2006b).
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3 Mathematical Modeling and Problem Formula-
tion
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3.1 Modeling of Chromatographic Column
For the modeling of each chromatographic column, the linear driving force (LDF)
model is employed, as in our previous study (Kawajiri and Biegler, 2006d). Here
both axial dispersion and diffusion into adsorbent particles, which cause band broadening,
are lumped into a mass transfer coefficient. The validity of such an assumption
can be found elsewhere (for example, Dünnebier and Klatt (2000); van Deemter et al.
(1956); Golshan-Shirazi and Guiochon (2003)). The mass balance equations in the
liquid and solid phases are given by:
∂C j i (x, t)
ǫ b
∂t
(1 − ǫ b ) ∂qj i (x, t)
∂t
+ (1 − ǫ b ) ∂qj i (x, t)
∂t
+ u j ∂Cj i (x, t)
= 0 (1)
∂x
= K appl i (C j i (x, t) − Cj,eq i (x, t)) (2)
The equilibrium between the liquid and solid phase is given by the linear isotherm:
q j i (x, t) = f(Cj,eq i (x, t)) (3)
Here ǫ b is the void fraction, C j i (x, t) is the concentration in the liquid phase of com-
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ponent i in column j, q j i is the concentration in the solid phase, u j 105
is the superficial
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liquid velocity in the jth column, C j,eq
i (x, t) is the equilibrium concentration in the
liquid phase, q j,eq
i
is the equilibrium concentration in the solid phase, andK appl i is the

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liquid-phase based mass transfer coefficient, respectively. The subscripts i correspond
to chemical components, superscript j is the column index, N c is the total number of
components, and N Column is the number of columns.
At the CSS, the concentration profiles at the end of a cycle must be identical to
those at the beginning of the cycle. The dynamics of the concentration profiles in
the whole cycle need to be considered. However, the problem size can be reduced by
taking advantage of the symmetry of the SMB operation. By enforcing the following
constraints, the profiles at the end of the next step are required to be identical to those
at the beginning of the current step, thus shifting the entire profiles downstream:
C j i (x, 0) = Cj+1 i (x, t step ), j = 1, ..., N Column − 1 (4)
q j i (x, 0) = qj+1 i (x, t step ), j = 1, ..., N Column − 1 (5)
C N Column
i (x, 0) = C 1 i (x, t step) (6)
q N Column
i (x, 0) = q 1 i (x, t step) (7)
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3.2 Column configurations and SMB superstructure
In this section, the configurations shown in Figure 2 are discussed. Note that these
configurations can be extracted from the SMB superstructure shown in Figure 3.
Referring to this figure, the following stream and column velocities are defined:
u j , u j R , uj E , uj D , uj F ≥ 0, j = 1..N Column (8)
where u j R , uj E
are velocities of raffinate and extract withdrawn from jth column, and
u j D and uj F
are velocities of desorbent and feed supplied to jth column. Note that all
velocities remain constant within each step. Furthermore, the volume and component

alance equations between the j − 1 th and j th columns are given by:
u j−1 − u j−1
E
u N Column
− u N Column
E
− uj−1 R + uj D + uj F = uj , j = 2, . . .N Column (9)
− u N Column
R
+ u 1 D + u1 F = u1 (10)
C j−1
i (L, t)(u j−1 − u j−1
E
− uj−1 R ) + C F,iu j F = Cj i (0, t)uj , j = 2, . . .N Column
(11)
C N Column
i (L, t)(u N Column
− u N Column
E
− u N Column
R
) + C F,i u 1 F = C1 i (0, t)u1 (12)
where C F,i is the concentration of component i in the feed. Further, in order to prevent
draining the desorbent and feed into the product streams without going through a
column, the following constraint is implemented:
u j − u j D − uj F ≥ 0 j = 1, . . .,N Column (13)
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If (13) is inactive, then the liquid from a column is transported to the next column.
Otherwise, the circulation loop is cut open and all liquid is withdrawn in either or
both of the product streams. We previously reported that this constraint is necessary
not only to prevent undesirable operating parameters, but to ensure the robustness of
optimization (Kawajiri and Biegler, 2006a,c). Note that this superstructure embeds a
number of different non-standard configurations such as Three-zone, Three-zone with
purging, and the standard SMB. Those configurations can be extracted by enforcing
the flow constraints in Table 1.

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3.3 Optimization problem formulation
In this study, we investigate the trade-off of throughput maximization and desorbent
minimization with fixed purity and recovery requirements by solving the following
multi-objective optimization problem:
(MOO) max ū F :=
min ū D :=
N Column
∑
j
N Column
∑
j
subject to : (1) − (13)
(Extract Product Purity) =
u j F
(14)
u j D
(15)
(Extract Product Recovery) =
N Column ∑
t step
∫
j=1 0
N Column ∑ ∑Nc
j=1
N Column ∑
i=1
u j E (t)Cj E,Prod (t)dt
t step
∫
0
t step
∫
j=1 0
N Column t
∑ step
∫
j=1
0
u j E (t)Cj E,i (t)dt ≥ Pur min
u j E (t)Cj E,Prod (t)dt
(16)
u j F (t)Cj F,Prod (t)dt ≥ Rec min
(17)
u l ≤ u j ≤ u u (18)
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where t step is the valve switching interval, or step time, Pur min and Rec min are the
purity and recovery requirements of the desired product which should be recovered
in the extract stream, respectively. The desired product is denoted by the index
Prod. u u and u l are the upper and lower bounds on the zone velocities, respectively.
In addition, to examine the performance of the individual configurations shown in
Figure 2, the constraints shown in Table 1 are enforced.

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3.4 Solution strategy
The problem formulated in the previous section is a nonlinear multi-objective optimization
problem. There have been many solution methods proposed for this type of
problem. For SMB optimization, Hakanen et al. (in press,s) used an interactive multiobjective
optimization approach for SMB optimization with four objective functions,
throughput, desorbent consumption, purity, and recovery. Their approach is useful
when the number of objective function is large. Zhang et al. (2002) used a genetic
algorithm for the zone configuration problem for a five-column SMB. In this study,
however, the ǫ-constrained method based on a Newton-based numerical optimization
approach is employed since there are only two objective functions. In this method,
parameter ǫ p and the two objective functions of MOO (14)-(15) are reformulated into
the following:
maxū F (19)
subject to : ū D ≤ ǫ p (20)
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By changing the value of ǫ p and solving the above single-objective optimization problem
repeatedly, the whole Pareto curve can be approximated. Although more computationally
expensive than the interactive approach in Hakanen et al. (in press),
this method allows us to compare the performance of different SMB configurations
by examining the entire Pareto curves.
The resulting set of single-objective optimization problems is handled by the approach
where the partial differential algebraic equations (PDAEs) are fully discretized
both in the spatial and temporal domains using centered finite difference and Radau
collocation on finite elements, respectively. Our previous study has found that this
full-discretization approach significantly reduces the computational effort (Kawajiri

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and Biegler, 2006d). It is important to note that although the resulting NLP problem
is large, the linear system to be solved in each iteration, i.e., the linearized Karush-
Kuhn-Tucker (KKT) conditions, has a sparse structure and can be solved very quickly.
To satisfy this requirement, we choose IPOPT, an interior-point solver with a filter
line search method which exploits the sparse structure of linearized KKT conditions.
Details of IPOPT can be found in Wächter and Biegler (2005).
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4 Case study
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4.1 Comparison of different configurations
For the numerical case study we consider the separation of fructose and glucose. The
parameters in Hashimoto et al. (1983), where the equilibrium is linear, i.e.,
q j i (x, t) = K iC j,eq
i (x, t) (21)
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are used in this study and are given in Table 2. Throughout this study, the number
of finite elements in the temporal domain N FET and collocation points N COL are set
to 5 and 3 respectively, and the number of finite elements N Column is set to 40 per
column. Details of the accuracy of the discretization method and choice of numbers of
finite elements were determined in Kawajiri and Biegler (2006d). This optimization
problem is implemented within the AMPL modeling environment (Fourer et al., 1992).
It should be also noted that because the optimization problem is non-convex, our
approach does not guarantee finding the global optimum. Therefore, we use several
different starting points in order to avoid local optimal solutions. The starting points
are obtained in the following manner: an initialization problem is solved where all operating
parameters (u j ,u j R ,uj E ,uj F ,uj D ,t step) are fixed in a standard SMB configuration.

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Note that this initialization yields a set of linear equations which has no degrees of
freedom. It should also be noted that such a starting point satisfies all constraints of
the optimization problem except the purity and recovery requirements (16)-(17). Furthermore,
multiple starting points can be easily obtained by changing the operating
parameters arbitrarily and resolving the initialization problem.
First, the throughput maximization problem (ǫ p = ∞ in (20)) is solved for each
configuration. The result is summarized in Table 3. In all cases, the computational
time takes only a few CPU minutes, which demonstrates the efficiency of our numerical
optimization technique.
The propagation of concentration profiles of the throughput maximization problem
are compared in Figure 4. In the standard SMB configuration, concentrations of both
of the faster and slower components are nearly zero at the right and left ends of the
column train (Figure 4a). In other words, the faster component should not catch
up with the slower component in order to achieve the purity and recovery demands.
The optimizer finds such an operation because of the following reasons: Before the
faster component reaches x = 8.0 in the rightmost column, switching must occur
(the inlet and outlet streams are switched in the direction of the liquid flow) to
avoid contaminating Zone I (leftmost column) by the faster component. At the same
moment of switching, the concentration of the slower component at x = 0 must
be nearly zero to avoid contaminating Zone IV (rightmost column) by the slower
component. This “zone contamination” in Zones I/IV would lead to contamination
of extract/raffinate by the faster/slower component.
On the other hand, in the optimal concentration profiles of the Three-zone configuration
shown in Figure 4b, the concentration of the faster component at the right
end (x = 8.0) is significantly higher. However, this would not lead to contamination
of the extract, since the recycle stream from x = 8.0 to x = 0 is cut off. Therefore the

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internal concentration profiles can be higher, which leads to the higher throughput.
It should be also noted that the desorbent consumption is higher because the recycle
stream is cut off.
The throughput of the Three-zone with purging has the lowest throughput; this
is because the leftmost column, where purging takes place, does not contribute to the
separation effectively, as the concentrations remain nearly zero all the time (Figure
4c). We note that this conclusion may not hold in cases where the Henry constant
of the slower component K 2 is large and removing the slower component from Zone
I requires a higher amount of desorbent.
Finally, note that the throughput of the F-shaped configuration is the highest.
The propagation of the concentration profiles in the F-shaped configuration is shown
in Figure 4d. This configuration allows such a situation where the slower component
is “left behind” by the switching, and remains in the rightmost column. The resulting
slower component left in the rightmost column is collected in the extract stream. This
leads to further increase of throughput, since the concentration in the leftmost column
can be significantly higher.
We note that the above comparison of configurations cannot be generalized to
other systems, and it is not easy to determine the configuration that attains the
highest throughput a priori (We will see this in the following case study where we
consider different values of Pur min ). Here, an alternative approach is to use the SMB
superstructure, which embeds all possible configurations and extracts the optimal
configuration. For Pur min = 95%, the optimal configuration extracted from the superstructure
turns out to be the F-shaped configuration (Table 3). This configuration
and optimal operating parameters can be found simultaneously in 2.87 CPU minutes.
These configurations can be compared by changing ǫ p . Figure 5(a) shows the
Pareto sets of different configurations with Pur min = 0.95. The solid line shows the

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Pareto set of the optimal solution of the superstructure formulation. As can be seen
in the figure, it overlaps that of the standard SMB when the desorbent consumption is
low. On the other hand, the part of the higher desorbent consumption overlaps that of
the F-shaped configuration. Also note that the Pareto curves of other configurations,
Three-zone and Three-zone with purging, are below that of the superstructure. In
the region of 1.5 ≤ ū D ≤ 3.0, however, none of the fixed configurations achieves the
throughput of the superstructure solution; this optimal configuration is similar to the
F-shaped configuration (Figure 6(a)) except that it has a recycle stream from x = 8
to x = 0 in order to reduce desorbent consumption.
Finally, the multi-objective optimization problem is solved for different values of
Pur min . In all cases, Pareto curve of the superstructure optimization becomes higher
than those of other configurations. When Pur min = 0.90 (Figure 5b), the Pareto
curves show a similar trend to those of Pur min = 0.95. The optimal configuration at
ū D = 3.0 is shown in Figure 6 (b). However, when Pur min is decreased to 0.85 (Figure
5c), the highest throughput is achieved by the Three-zone configuration at the expense
of higher desorbent consumption. The optimal configuration and concentration profiles
at ū D = 7.0 are shown in Figure 6(c). Finally, in the case of Pur min = 0.80, the
F-shaped configuration never reaches the Pareto curve of the superstructure (Figure
5d). Instead, the Three-zone with purging has the highest throughput at the highest
desorbent consumption; this is closely followed by the Three-zone configuration with
lower desorbent consumption. The optimal configuration at ū D = 5.0, shown in Figure
6(d), is again similar to the F-shaped configuration, but the feed is supplied at
x = 2.0 and the liquid is recycled from x = 8 to x = 0.
From this multi-objective optimization study, we conclude that the optimal configuration
is highly dependent on the purity specification, although the standard SMB
configuration generally has good trade-offs when desorbent consumption should be

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kept low. Nevertheless, our optimization approach using the SMB superstructure is
able to find the most efficient configuration without exploring various configurations.
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5 Conclusions and Future Work
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Standard and nonstandard configurations of four-column SMB, under optimal operating
conditions, have been compared through a case study. The optimal configuration
is different depending on the purity requirement, throughput, and desorbent consumption.
Here, we determined that although the standard SMB configuration has
good trade-offs when desorbent consumption is low, optimal configuration is highly
dependent on the purity requirement. Nevertheless, our multi-objective optimization
scheme using an SMB superstructure is able to find the optimal configuration
efficiently without exploring various existing configurations. Also, our numerical optimization
technique has been found to be efficient and reliable.
Our future work will further extend our optimization approach to find optimal
operating schemes for multi-component separations, where more than two components
are fractionated into multiple streams. As our superstructure formulations embed a
number of both standard and nonstandard operating schemes, we aim to develop
further efficient designs and operating schemes.
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Acknowledgment
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Funding from the National Science Foundation under Grant CTS-0314647 is gratefully
acknowledged.

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Figure 1: Schematic diagram of 4 column SMB
Figure 2: Examples of configurations: (a) standard SMB, (b) Three-zone (c) Threezone
with purging (d) F-shaped.

Figure 3: SMB superstructure

Figure 4: Comparison of concentration profiles. Vertical dotted lines show the ports

Figure 5: Pareto curves

Figure 6: Configurations in Figure 5

Table 1: Configuration constraints
Standard SMB 3-zone 3-zone purging F-Shaped
u j D = 0, j ≠ 1 uj D = 0, j ≠ 1 uj D = 0, j ≠ 1, 2 uj D = 0, j ≠ 1
u j F = 0, j ≠ 3 uj F = 0, j ≠ 3 uj F = 0, j ≠ 3 uj F = 0, j ≠ 3
u j E = 0, j ≠ 1 uj E = 0, j ≠ 1 uj E = 0, j ≠ 1 uj E = 0, j ≠ 4
u j R = 0, j ≠ 3 uj R = 0, j ≠ 4 uj R = 0, j ≠ 4 uj R = 0, j ≠ 3
u 4 = u 4 R u 4 = u 4 R , u1 = u 1 E u 4 = u 4 E
Table 2: Parameters of Fructose / glucose separation
parameter value parameter value
ǫ b 0.389 L [m] 2.0
K 1 0.518 K appl 1 [1/s] 6.84 × 10 −3
K 2 0.743 K appl 2 [1/s] 6.84 × 10 −3
C F,1 [%] 50.0 C F,2 [%] 50.0
u l [m/h] 0.0 u u [m/h] 8.0
N Column 4 Rec min [%] 80.0
N FET 5 N COL 3
N FEX 40
Table 3: Statistics of throughput maximization problem for Pur min = 0.95
Objective function Φ Standard SMB 3-zone 3-zone purging F-Shaped Superstructure
ū F [m/h] 0.410 0.521 0.344 0.737 0.737
ū D [m/h] 1.199 4.836 2.083 4.669 4.669
Extract purity [%] 95.0 95.0 95.0 95.0 95.0
Extract recovery [%] 80.0 80.0 80.0 80.0 80.0
Number of iterations 39 44 48 57 54
CPU time [min] 1.76 1.65 1.87 3.12 2.87