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REV. CHIM. (Bucharest) ♦ 61♦ Nr.7 ♦ 2010http://www.revistadechimie.ro696
Decomposability Investigations for Control Structure
Design of Recycle Systems in the Frequency-domain
M. HORVÁTH*, P. MIZSEY
Budapest University of Technology and Economics, Department of Chemical and Environmental Process Engineering, Muegyetem
rkp. 3, H-1111 Budapest, Hungary
The control structure design for process synthesis tasks becomes more difficult if recycle is present in the
process to be controlled. Our previous work [1] cleared up that the control structure design for recycle
systems can be decomposed into subproblems including only one unit of the investigated system. The
objective of this work is to apply the decomposability in the frequency-domain. Our investigations prove that
the task of the control structure design for the investigated industrial system is decomposable in the frequency
domain too and this can facilitate the control structure design.
Keywords: control structure design, frequency-domain, decomposability, recycle processes
* email: mhorvath@mail.bme.hu; Tel.: +36-1-4632035
In the chemical engineering the recycling is a widely
used solution to utilize material and energy more efficiently,
especially in separation systems. From controllability
aspects, recycle can be represented as a positive feedback
in the system and it means not only process design
problems, but it needs also special controllability
considerations. In several cases the recycle can lead to
instability and other special problems. These kind of
controllability problems were exhaustively investigated [8-
10]. He pointed out that the recycle loop gain basically
determines the behaviour of the recycle systems and
reflected to the problem called “snowball-effect” [11] in
some reactor-separator systems. These results were
certified by the researchers in [14], their work also reflected
the importance of the control of the recycle path and the
application of different controls allowed to handle the
snowball-effect. Different theoretical approaches were
also published. It was investigated a reactor-separator
system from the point of the poles of the plant and declared
the recycle as a positive feedback, too [12]. A new
classification of the effects of the recycle was
recommended. The correlation between the dynamic
effect of the recycle and the recirculated material flows
was exhaustively investigated. It was found that when the
flow rate of the recycle is significantly higher than the feed
flow rate, the recycle network exhibits a time-scale
separation: in the fast scale the system shows fast
dynamics, and in the long scale there are weak
interactions. A nonlinear supervisory controller was
recommended. From the point of the process design the
work of Dimian et al. [15] is very considerable. They
investigated the integration possibilities of process design
and controllability analysis for large plants, with significant
recycles and the structures with open- and closed-loops,
and recommended different alternative structures and
shorter recycle paths. In case of extended systems with
more units and with more recycles, the control structure
design becomes more difficult, and this urges exhaustive
investigation for the possibility of structure decomposition.
A new plantwide decomposition method which can be
supported with the AHP (analytical hierarchical approach)
was recomanded [16]. This method allows prioritizing the
design objectives, the operability constraints, and the
alternative decompositions. In this paper the process design
method is based on the frequency-dependent controllability
indices, and the decomposability properties of the control
system are exhaustively investigated.
In our previous work [1] we successfully determined
the control structure for an ethylbenzene-producing
industrial system and proved the decomposability of the
task of the control structure design for some 2×2 recycle
systems and for the industrial system too (the system
operates with two controlled and two manipulated
variables). The method for the 2×2 system with
hypothetical transfer function matrices are based on
different load rejection simulations, and then the
investigations were extended to the industrial system and
the decomposability for the control structure design of the
ethylbenzene-producing system was also proved. In this
paper exhaustive investigations are carried out of the
previously investigated ethylbenzene producing system,
other different tools of control system design are applied
and the investigations are extended to the frequency- and
the time-domains.
Experimental part
The scheme of the investigated industrial system [6]
can be seen on figure 1. The feed of the continuously stirred
tank reactor are benzene (C6H6) and ethylene (C2H4), the
chemical reactions can be seen in Table 1, while the flow
rates are presented in Table 2. In the chemical reactor we
apply a temperature of 1800 and a pressure of 10 atm using
AlCl3 as catalyst. The rate constants follow the Arrhenius-
law dependence, that is:
(1)
The feed of the first column contains all of the alkylated
compounds (ethylbenzene, diethyl-benzene and triethyl-
benzene) and residual benzene. The first column separates
the benzene which is recirculated back to the reactor. The
second column separates the ethylbenzene with a high
(99.9%) purity, and the third column separates the heavier
components from each other. The diethyl-benzene is
REV. CHIM. (Bucharest) ♦ 61♦ Nr. 7 ♦ 2010 http://www.revistadechimie.ro 697
Table 2
THE MATERIAL FLOW RATES OF THE SYSTEM
Table 1
REACTIONS IN THE CSTR
Fig. 2/a, 2/b: Step responses of the composition of the distillate of the first column-the solid line represents the system with recycle,
the datshed line represents the system without recycle
Fig. 1: The investigated system
recirculated too, but the flow rate of this stream is
significantly smaller than the flow rate of the benzene-
recycle.
The conversion of the benzene in the chemical reactor
is 28%, while the ethylene is totally converted, so the feed
of the first column does not contain ethylene. Two recycles
are applied: the distillate of the first and the distillate of the
third column. The flow rates of the different streams of the
system are presented in Table 2.
Effect of the recycle in the time-domain
The most significant effect of the recycle on the
investigated system is the change of the time constants.
The step-responses of recycle systems are slower, and on
the second hand, have higher gains than without recycle.
The open loop investigation is an adequate tool for
preliminary investigations of the dynamics of the recycle
systems and, from the results, further information can be
obtained for tuning the composition control loops. During
open loop simulations, composition control loops are
switched off; hence the disturbances cause definitive shifts
in the product compositions. Figures 2, 3 and 4 show the
responses of the composition of the distillates of the three
columns to feed flow rate and feed composition
disturbances. Disturbances are applied at the column feeds
and considering the result of some linearity-investigation,
the extent of the disturbances is 1%. Bold curves show the
columns’ responses without recycle, while the thin curves
show the responses with recycles (both of them).
REV. CHIM. (Bucharest) ♦ 61♦ Nr.7 ♦ 2010http://www.revistadechimie.ro698
Fig. 3/a, 3/b: Step responses of the composition of the distillate of the second column - the solid line represents the system
with recycle, the dashed line represents the system without recycle
Fig. 4/a, 4/b: Step responses of the composition of the distillate of the third column the solid line represents the system with
recycle, the dashed line represents the system without recycle
Table 3
THE CONTROL STRUCTURES OF THE COLUMNS (L: REFLUX
RATE, R: REFLUX RATIO, B: BOTTOMS RATE, Q; REBOILER HEAT
DUTY)
Compositions of the key-components are shown only. The
key components are: benzene in the first, ethylbenzene in
the second and diethyl-benzene in the third column.
The open-loop responses clearly show the effect of the
applied recycles: in case of recycle, the time constants of
the columns are 3-5 times higher than without recycle.
The dynamic properties of the chemical reactor is not
investigated here, only the distillation columns.
Optimal control structures
In order to keep the product-compositions at their
prescribed values, composition control loops are designed
for the system. The details of the design process is
described in our previous work [1]: it is a load rejection
based control structure design, which operates with the
system properties in the time-domain. Table 3 shows the
optimal control structures for each column in case of all of
the possible recycles.
Decomposability analysis in the frequency-domain
The load rejection-based decomposability investigations
in the time-domain [1] proved that the task of the control
structure design is decomposable. Now the investigations
are extended to the frequency-domain and the
decomposability of the task of the control structure design
is investigated based on frequency-dependent
controllability indices.
In order to quantify the different composition control
loops, the state space representation of the system and
the frequency-dependent controllability indices are used.
With the help of the Control Design Interface of Aspen
Dynamics the state space matrices (A, B, C and D) are
obtained and different frequency-dependent controllability
indices (CN, MRI and RGA numbers) are calculated using
Matlab. In this way frequency-function are obtained for the
distillation columns. On Figure 5 an illustrative example
can be seen: the condition number of the first column in
case of both of recycles is plotted in the function of the
frequency. Similar frequency-functions are determined for
the investigated systems with different case of recycles.
The CN and the MRI represent the controllability and the
expectable stability of the system faithfully, while the RGA
numbers are closer to the steady-state representation. From
the CN and from the MRI we create two modified
frequency-dependent controllability indices: the average
CN and the average MRI. The average CN is averaged values
of all of the CN values in the whole frequency-range, while
the average MRI is similarly generated by the MRI values of
the investigated frequency-range. The average CN provides
information the applied pairing of the investigated
manipulated and controlled variables and the average MRI
values show the distance of the structure from the
singularity in the whole frequency-range. The unique values
of these indices represent the system only at some specific
frequency, but the average values give a good
approximation in the whole studied frequency-range. With
the help of this two modified indices representative
qualitative analysis can be carried out for the investigated
REV. CHIM. (Bucharest) ♦ 61♦ Nr. 7 ♦ 2010 http://www.revistadechimie.ro
Fig. 5. a frequency-dependent controllability index (CN) in the
function of the frequency – first distillation column, both of
recycles (benzene and diethyl-benzene)
Fig. 6. The average MRI values for the global system in the case of
recycles – in the function of the applied control structures of the
first and the second columns
Fig. 7. The average CN values for the global system in the case of
recycles – in the function of the applied control structures of the
first and the second columns
ethylbenzene producing recycle system in the frequency-
domain. The results of these investigations are presented
as a two-variable function and this methodology requires
a 3D-representation. With the help of this multivariable
function, the decomposability of the task of the control
structure design for the global system can be certified.
Results and discussions
In order to determine the global optimal control structure,
it is necessary to investigate if the individually determined
optimal control structures for each column can form the
overall optimum or not. If they can form the overall
optimum, it is possible to determine it for the recycle
system with the application of the optimal control structure
of the individual units. In such a case it results that the
control structure design can be decomposed. Based on
load rejection investigations it is proved that the control
structure design is decomposable and now, frequency-
dependent controllability indices are used to perform
analysis in the frequency-domain. Simultaneous
investigations are carried out, considering the first and the
second columns.
The first selected modified controllability index is the
average MRI, which is calculated from the MRIs in the
whole investigated frequency-range, it is an arithmetic
average. Figure 6 shows the average MRI values for the
global system. The horizontal axes contain the control
structures of the first and the second columns, while the
vertical axis contains the current average MRI.
On Figure 6 a monotonous surface can be seen. The
optimal control structure for the whole system (which has
the highest MRI) can only be constructed from the optimal
control structures of the first and the second columns, every
other pairing of the columns result a more unfavorable
global system. The effect of the third column is not
significant, because it has the smallest recirculated
material flow rate (nine-times less than the recycle from
the first column). Therefore the third column is represented
only as a parameter here. Replacing the optimal control
structure of the third column to the second best one,
increases the average CN values less than 1%.
The second selected controllability index is the average
CN number. This index represents the average of the
measured CN values in the whole investigated frequency-
range. Figure 7 shows the average CN values in case of
different pairing of the control structures of the first and the
second columns.
The MRI values represent how far the control system is
from the singularity, and the average MRI values approaches
it in the whole frequency-range. More the MRI values are
more the system is farer from the singularity. Each point of
the MRI-surface shows the average MRI value of the global
system in case of application of the first column with the
control structure on the x-axis, and the second column
with the control structure on the y-axis. The x and y axes
show the optimality of the applied control structures. The
CN numbers represent the interactions between the control
loops and indicate the conditionality of the control loops.
In the whole frequency range it is represented by the
averaged CN here. Both of the obtained surfaces are
monotonous surfaces, and represent that changing a
control structure to an unfavorable one, it causes a more
unfavorable behaviour for the global system concurrently.
In other words, the pairing of two more unfavorable control
structures never forms a more favorable global system.
Similarly to the results of the MRI values, only one point
can represent the optimal control structure for the global
system, and it can only be formed by pairing the best
control structure of the first, and best of the second
columns. This fact certificates the decomposability of the
task of the control structure design for this recycle system
in the frequency domain, too.
The decomposability investigations in the frequency-
domain reproduced the results of the load-rejection-based
decomposability investigations, but in a more effective and
easier way. The optimal control structures are the same,
and the decomposability feature is successfully certified
in the frequency- and in the time-domain too. Concerning
these results we investigate the effect of the recycle on
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REV. CHIM. (Bucharest) ♦ 61♦ Nr.7 ♦ 2010http://www.revistadechimie.ro
Table 4
THE AVERAGE VALUES OF THE CONTROLLABILITY INDICES IN
CASE OF DIFFERENT RECYCLES (MRI, CN AND RGA VALUS)
Fig. 8: The MRI values of the whole system
Fig. 9: The RGA numbers of the whole system
Fig. 10: The CN values of the whole system
the systems equipped with the previously determined
optimal control structures.
Effect of recycle on the controllability features in the
frequency-domain
The frequency-based decomposability investigations
appoint the control structure for the global system. The
system equipped with this control structure is investigated
in the frequency-domain: MRI values, RGAno’s and CN
values are measured in case of different recycles. The
system has two possible recycles, the benzene-recycle
from the first column, and the diethyl-benzene recycle from
the third column. These can also be applied independently,
and the indices can be measured with both of the recycles,
too. Figure 8, 9 and 10 show the different frequency-
dependent controllability indices in the function of the
frequency, in case of different recycles.
The correlation between the recycle flow rates and the
controllability properties can be seen from the frequency-
dependent indices: by increasing the flow rate of the
recycled streams, the controllability indices show more
unfavorable system behaviour. Table 4 shows the average
values of the different controllability indices and the flow
rates of the different recycled streams.
Conclusions
In this paper a frequency-domain based application is
presented to design optimal control structure for an
industrial, ethylbenzene-producing recycle system. The
system is investigated in the frequency-domain based on
different controllability indices which proves that the task
of the control structure design is decomposable and proves
that this is not necessary to investigate the controllability
features in the time-domain, it can be completed in the
frequency-domain on this easier way. The frequency
dependent controllability indices clearly show the effects
of the applied recycles: more the flow rates of the applied
recycles are, more the frequency-dependent controllability
indices are unfavorable; the systems with high flow rate of
recycle streams have smaller resiliency-indices, it
represents a closer state to the singularity. In the steady-
state-, and the time-domain the preliminary investigations
certified the correlation between the recirculated material
flow rates and the values of the different controllability
indices, and now these results are extended to the
frequency-domain. The composition control loops are
selected based on the frequency-dependent indices and
the system equipped with these control loops provides the
expected product compositions and is stable in each
recycle cases.
Nomenclature
2×2- two manipulated and two controlled variables
B - Benzene
B - Bottom flow rate
BR - Boilup ratio
CN - Condition number
D - Distillate flow rate
DEB - Diethyl-benzene
EB - Ethylbenzene
F - Feed
G - Transfer function (or transfer function martix)
IAE - Integral Absolute Error
L - Reflux flow rate
MIMO - Multiple input, multiple output
MRI - Morari Resiliency Index
Q - Reboiler heat duty
R - Reflux ratio
Rec. - Recycle
RGA - Relative Gain Array
TEB - Triethyl-benzene
x - input signal
xF - composition of the Feed
y - output signal
Refernces
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700
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Manuscript received: 23.02.2010
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