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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

1. HORVÁTH,M., MIZSEY, P., Decomposability of Control Structure

Design Problem of Recycle Systems, Ind. Eng. Chem. Res., 48 (13),

2009, pp. 6339

700

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Manuscript received: 23.02.2010

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