Applicability of Desirability Function for Control Structure Design in the

Article pubs.acs.org/IECR

Applicability of Desirability Function for Control Structure Design in the Frequency Domain Máté Haragovics,*,† Hajnalka Kencse,† and Péter Mizsey†,‡ †

Department of Chemical and Environmental Process Engineering, Faculty of Chemical Technology and Biotechnology, Budapest University of Technology and Economics, Budapest H-1521, Hungary ‡ Research Institute of Chemical and Process Engineering, University of Pannonia, Veszprém, H-8200, Hungary ABSTRACT: In this work, the application of desirability function on controllability indices in the frequency domain is studied and a methodology is presented for its proper application. To design control structures, controllability indices are often used. Different indices represent different controllability attributes of the investigated system or systems; thus, their results might not be in agreement. With the desirability function, it is possible, however, to aggregate the different indices and a final ranking can be obtained, which may substitute time domain studies. Two different fields are examined: control structure variable pairing and selecting a distillation scheme that shows the best controllability features. On one hand, for the variable pairing step of the control structure design, the methodology does not deliver acceptable results, because controllability indices prove to be inaccurate in such cases on which the calculation is based. On the other hand, the methodology delivers similar result for the selection of the right distillation scheme from the controllability point of view to that of the load rejection investigations. This result shows the right area for the applicability of the methodology based on the desirability function.

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INTRODUCTION Chemical process design is a complex process engineering problem that involves synthesis, analysis, optimization activities, and the evaluation of the design alternatives. Control structure synthesis has traditionally followed the process design, because problems in each area alone are complex, the interactions between design and control have been poorly understood, and computational possibilities of a simultaneous synthesis have been below the requirements. To overcome these difficulties, empirical overdesign factors, large storage tanks, and surge tanks have been installed in the technology. Control structure synthesis is a complex design step. It involves the definition of control loops, that is, the pairing of controlled variables (CVs) and manipulated variables (MVs). The proper selection and pairing of MVs and CVs is an important task of the design process. The pairing can be a determinative step, especially when comparing process design alternatives, such as different energy integrated distillation systems. In the industry, decentralized control systems are used for many reasons, e.g., they are simple, easy to use, and direct. In addition, if a control loop is easy to operate, the supervisory control can be smoother. Skogestad1 also prefers this type of controller over multivariable controllers in his work about control structure design, because their simplicity yields in robustness and lower cost. Morari2 has shown that the best performance that any controller may achieve on a plant is a function of the plant itself. Investigation of the design alternatives from the view of controllability can predict their controllability features in advance, not knowing anything about the controller structure. Hence, the first controllability considerations should begin at the conceptual stage of process design, where emphasis is placed on the screening of the flowsheet alternatives and such a © 2012 American Chemical Society

control structure should be designed that shows good control performance and fulfils the process requirements. Controllability Indices. According to Skogestad et al.,3 open-loop simulations themselves are not interesting, but they can predict how sensitive the closed-loop system can be. The reason for this is, if a system is not sensitive to loads without control, it is even more robust with a closed-loop controller. Open-loop investigations and indices have been applied over the last 20 years, but their usage is not universally accepted, they warn. Earlier Luyben presentations4,5 have stated that, regarding the large number of column configurations and their possible control structures, it is very probable that no universal tool exists that can simply differentiate between such configurations. The controllability investigations start at the early stage of process design and, in this step, the controllability indices are frequently used. There are several indices that can rely on steady-state or dynamic behavior. The typical unit operation that has brought the controllability indices into use is distillation. Birky et al.6 have compared the methods of two column control experts and found that those methods have only a few things in common. The relative gain array (RGA) by Bristol7 and Shinskey8,9 is one of the mentioned indices, but they also note that a steady-state RGA cannot predict the dynamic behavior. Skogestad et al.10 have discussed the defects of the steady-state RGA through an example with DB control structure, which can work despite its bad steady-state RGA value. Gross et al.11 have investigated a thermally coupled Received: Revised: Accepted: Published: 16007

April 13, 2012 October 19, 2012 November 16, 2012 November 16, 2012 dx.doi.org/10.1021/ie300973b | Ind. Eng. Chem. Res. 2012, 51, 16007−16015

Industrial & Engineering Chemistry Research

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deliver the same results as simulations in the time domain, by applying load rejection analysis. Such investigations may be much faster, simpler, and more automatic than modeling in the time domain.16,25 On the one hand, our goal is to compare results regarding frequency domain indices and time domain indices, and determine their usability. On the other hand, we follow the idea introduced by Kencse and Mizsey.25 They have proved that multicriteria evaluation of process alternatives is desirable and a method can be applied for the ranking of design alternatives based on an indicator number (i.e., the desirability function); however, they have not used it for controllability evaluation purposes.

distillation structure. They have compared three alternatives of control structure. Their work has covered steady-state and dynamic investigations. They have calculated singular values, condition numbers, and RGA-based indices in the frequency domain. From their results, it can be concluded that frequency domain indices do not show unambiguous results for controllability, i.e., the ranking of configurations is different according to different indices. To design the controller for a distillation system, experiments are necessary. In present practice, this means modeling rather than real-life experiments. For modeling purposes, rigorous, nonlinear model of the distillation column is necessary. Usually, different modeling concepts are applied. Dynamic and steadystate indicators are used in different articles. Some authors have used steady-state indices;11−14 others have preferred frequencydependent indices15−21 Controllability indices can be also calculated by singular value decomposition. These indices are the condition number (CN) and the Morari resiliency index (MRI). The CN of the transfer function matrix is the ratio between the maximum and minimum singular values (eq 1): γ (G ) =

σ ̅ (G ) σ̲ (G)

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SYSTEMS INVESTIGATED Our goal is to study the applicability of the desirability function26 on the evaluation of controllability features and the ranking of different control structures of different process design alternatives. Two types of investigations are considered: • determination and selection of the best control structure for an individual process, and • ranking of process design alternatives according to their controllability features. For the first research task, single distillation columns separating azeotropic binary mixtures are examined to find the best CV−MV pairing. For the second research task, alternative energy-integrated distillation systems separating azeotropic ternary mixtures are selected for comparison and ranking, according to their controllability features. Only composition control loops are considered during the control structure design. Distillation Columns Selected for Control Structure Design. Controlling a distillation column means several control loops, including liquid level control, temperature control, composition control, flow control, pressure control, etc. No matter how good the separation is in a given structure, if it is not controllable, it might do more harm than good. In our examinations, binary azeotropic mixtures are separated: ethanol−propanol, n-heptane−n-hexane, and benzene−toluene mixtures. CVs are product compositions, MVs can be seen in Figure 1. Structure names denote their corresponding manipulated variables: for example, “RB” means that distillate composition is controlled by reflux ratio and “bottoms product composition” is controlled by the bottoms flow rate. Other structures are analogues to this notation. The following pairs are investigated: RB, RQ, DQ,

(1)

where σ̅(G) is the maximum singular value and σ̲(G) the minimum singular value of the transfer function matrix. A value of CN < 10 indicates a well-controllable process.22 A matrix with CN > 103 is said to be ill-conditioned, which implies that the system is sensitive to unstructured input uncertainty. Theoretically, if the CN is small, the effects of the input uncertainty are always negligible, but large CN generally indicates model sensitivity; however, this rule is not applicable in every case. Therefore, the control structure with large CN cannot be excluded from the applicable pairings without investigating other features. The MRI is also called the minimal singular value of the open-loop transfer function, which represents a specific input and output direction. A control system that presents a large MRI value is preferred. Large MRI values indicate that the process can handle disturbances without saturation of the manipulated variables.23 Despite the fact that controllers are frequently used to keep the controlled variables at a constant value (i.e., the set point), the disturbances force the system into dynamic behavior. That is why we investigate the dynamic behavior of the possible process design alternatives. It is important to mention here that Bezzo24 has shown the benefits of using process simulators, especially for dynamic simulation. Their work supports our opinion: the best results can be achieved by dynamic simulations of the entire system with the control loops closed (i.e., all control devices are working). In this manner, the behavior of the system can be predicted with simple step disturbances in the compositions or flow rates, etc. In the course of such disturbances, the load rejection ability of a control structure is investigated. However, such investigations require plenty of time. The investigations follow the real-life situations: the control loops must be tuned properly, then the load rejection analysis is applied. The simulation is much faster and less expensive than experiments on the real process; however, simulation results must be carefully interpreted. For such dynamic behavior study, investigations in the frequency domain can be also selected, and that choice may

Figure 1. Controlled variables (CVs) and manipulated variables (MVs) of a single column. 16008

dx.doi.org/10.1021/ie300973b | Ind. Eng. Chem. Res. 2012, 51, 16007−16015


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LB, and LQ. The feed stream is an equimolar mixture of two components, and product purity is 95% in every case. Energy-Integrated Distillation Systems Selected for Controllability Investigations. The following energy integrated distillation systems are examined: direct distillation sequence with backward heat integration (DQB, Figure 2), fully

Figure 5. Sloppy distillation system with forward heat integration (SQF) configuration indicating its CVs and MVs.

simulations in the time domain, since controllability indices are not considered to be reliable.11 The selected group of MVs for the investigated distillation systems are presented in Table 1. These groups prove to be the best groups of the MVs, according to load rejection investigations.

Figure 2. Direct distillation sequence with backward heat integration (DQB) configuration indicating its CVs and MVs.

thermally coupled distillation column (FTCDC, Figure 3), conventional direct distillation scheme (Conv. Dir., Figure 4), and sloppy distillation system with forward heat integration (SQF, Figure 5).

Table 1. Manipulated Variables of Different Distillation Schemes distillation system

manipulated variable (MV) pairings

DQB FTCDC SQF Conv. Dir.

D1L2B2 LSQ LSB D1L2B

The operating conditions selected for controllability study are as follows: pentane−hexane−heptane ternary mixture, 95% product purity. Feed is an equimolar mixture of the components.

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APPLICATION OF DESIRABILITY FUNCTION Our methodology is summarized in a flowchart in Figure 6 . Basically, it consists of three levels: defining competing configurations, frequency domain investigations, and aggregation of the controllability indices (ranking). First Level: Defining Competing Configurations. Defining Systems To Satisfy Specifications. Before the analysis takes place, the possible design alternatives must be elaborated as usual. The models must meet the specifications to produce the desired products. This means steady-state short-cut calculations and rigorous steady-state modeling. The steadystate model is then transformed to a dynamic model, which is needed for this type of analysis. Product specifications are the same in every case. Selection of Controlled and Manipulated Variables. The first step of the analysis is the selection of the controlled and manipulated variables for the investigated systems. The common distillation column with two products is a 5 × 5 system. The CVs are levels on the top and in the bottom, pressure, top and bottom compositions. Since the levels and pressure must always be controlled to ensure stable operation, the level and pressure loops, as the so-called material balance loops, are designed first.3 Then, in the next step, the composition control loops can be designed for the distillation column. In our research, we focus only on the composition control loops.

Figure 3. Fully thermally coupled distillation column (FTCDC) configuration indicating its CVs and MVs.

Figure 4. Conventional direct distillation scheme (Conv. Dir.) configuration indicating its CVs and MVs.

Composition control of the three product streams is realized at the studied distillation systems; therefore, mole fraction is selected as the controlled variable. Selection of the MV pairings from the possible combinations is based on closed-loop 16009



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Based on these individual functions, the overall desirability function, Dfct, is constructed (eq 3), and this makes the ranking of the process alternatives possible. The Dfct is defined as the geometric average of the k individual desirability functions and it also includes the designer priorities and desires, using the m weight factor.29 k

m

m

Dfct = (∏ di j)1/ ∑ j = (d1m1 × d 2m2 × ... × dk j)1/ ∑ j i=1

(3)

where k is the number of indicators and m is the weight factor. However, in practice, mj weight factors are generally not considered; instead, d-functions are determined in a manner to reflect the importance of the given property. It is more straigthforward and just as easy to express the importance in the form of the conversion function to d, instead of using weight factors in eq 3. Moreover, the form of the d-functions can have a greater effect on the ranking than weight factors; therefore, it is more important to use proper functions. A high value of Dfct shows that all dk are toward the target value that indicates the most adequate process alternative in our methodology. The idea of the application of the desirability function comes from the fact that controllability indices may show opposite characteristics for a given case. Often, the results of the controllability study are difficult to interpret, because of the controversial results of the indices of the investigated system; for example, the value of the CN is close to the desirable band but the RGA value indicates strong interactions. In order to aggregate the different controllability indices, we propose and apply the desirability function in our methodology to get consistent results. Ranking is based on the desirability function where the input data are the controllability indices calculated on the previous level. Although indices provide important information about process performances and allow the comparison of process alternatives, with respect to specific aspects, they are not suitable for direct comparison of the different analysis types. Hence, the overall desirability function is used to summarize the different indicators, since it aggregates information obtained with help of the controllability indices. Using Desirability Functions with Indices in the Frequency Domain. Overall desirability function Dfct is defined as the geometric average of individual desirability functions. We follow the aforementioned descipline and do not use weight factors. Since controllability indices describe the transfer matrices of the process and their values can be classified immediately, independent of the process, the task is only to create functions that convert desirable index values to d-values close to 1, while discriminating worse index values by assigning values close to 0 to them. Indices, in some cases, also can have very high or low values; therefore, a linear function is not eligible to transform these values in every possible case. However, properly chosen exponential functions can transform indices to fit in the [0...1] interval. Exponential functions are even better to make the distinction between indices in the desired range. Other modifications of the exponential base functions are made empirically to fulfill the requirements of the given controllability index. Our individual desirability functions have the following specifications: (a) A value of CN < 10 indicates a well-conditioned system; in this range, dCN approaches 1, which is desirable. A transfer matrix with CN > 103 is said to be ill-

Figure 6. Methodology for advanced frequency domain evaluation.

Second Level: Frequency Domain Investigations. Calculation of the Controllability Indices in the Frequency Domain. Controllability analysis is carried out by calculating the transfer function matrices ( G(iω)) for open-loop control systems, based on the work of Gábor and Mizsey.27 These transfer function matrices are subjected to singular value decomposition (SVD).28 Based on SVD and transfer function matrices, controllability indices can be calculated. In our investigations, we used the MRI, the CN, and a derivative of the relative gain array matrix: RGA-number (RGAno). These controllability indices are used to predict the degree of directionality and the level of interactions in the system. Analysis is conducted in the frequency domain calculating frequency-dependent indices, which may yield additional information about the investigated system.10 It is important to note that the values of the controllability indices should be read in the range of the critical frequency that is calculated for each system in an open loop, as a function of the time constant, which results from the response of a step disturbance (eq 2): 2π ωcrit = Tol (2) Third Level: Aggregation of the Controllability Indices (Ranking). The last step of the methodology is making the decision based on the desirability function (Dfct). The desirability function approach is a useful statistical method to optimize multiple characteristic problems. This method, which was proposed by Harrington,26 converts the multiple quality characteristics to a single characteristic problem by maximizing the total desirability. An indicator is then transformed to a individual desirability value (d) , which is part of the desirability function model. If the response exceeds the acceptable value, the value of d becomes 0; if the response is at the target value, the desirability value d becomes 1. Individual desirability functions (d) are continuous functions and they are chosen from among a family of linear or exponential functions. 16010



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RESULTS OF THE SELECTION OF BEST CONTROL STRUCTURE FOR INDIVIDUAL PROCESS ACCORDING TO INTEGRAL ABSOLUTE ERROR Structure names denote the MVs and CVs of distillate composition and bottoms composition (Table 2). For example,

conditioned. Between 10 and 100, it can be both good or poorly conditioned, depending on the investigated structure.30 The value of dCN is close to 0 at large CN values (>103) but it is not zero; thus, the control structure is not excluded, because of its large CN (see eq 4). The chosen dCN function, which describes the previously mentioned criteria, has the following form: dCN = exp[− (a + b × CN )]

Table 2. MVs of CVs (xD and xB) for the Investigated Pairingsa

(4)

According to previous investigations, values of a = 1.11084 × 10−4 and b = 6.967 × 10−3 are good choices to transform CNs to values between 0 and 1. (b) The large MRI value predicts a resilient process that can deal effectively with disturbances. Maximization of this individual response is desirable. Thus, the function of dMRI has the form presented in eq 5. The dMRI function gives low d-values for MRI < 10 −2 and highly discriminates the MRI parameters between 10−2 and 1, since most of the investigated control structure has its MRI in this range. dMRI = 1 − exp( −10 × MRI)

xD xB a

∫0

(5)

LQ

DQ

LB

R Q

R B

L Q

D Q

L B

Variable names are as given in Figure 1.

mixture

Component 1

Component 2

feed x1

dist. x1

bott. x1

aromatic apolar polar

benzene n-pentane ethanol

toluene n-hexane propanol

0.5 0.5 0.5

0.95 0.95 0.95

0.05 0.05 0.05

applied in the feed flow rate and feed composition. First, the load rejections are calculated. Simulations in the frequency domain are also carried out. In the case of load rejection calculations, the integral absolute errors (IAEs) of the two composition control loops are summarized for easy comparison of the different structures. Time constants are determined with the control loop tuning tool of Aspen Dynamics in dynamic simulation mode. A step disturbance is applied and, from the settling characteristics, the time constant is calculated. After estimating the time constants of the open-loop dynamic load rejection for an individual process, critical frequencies are found to be in the range of 7.29 × 10−4 and 5.85 × 10−3rad/s in the frequency domain. All indices are constant in this range; thus, the corresponding indices are easy to read. Results are summarized in Table 4. High MRI, low CN, and low RGAno values yield better results. MRI values show the same situation for all mixtures: DQ structure has the highest value; all other structures have their MRI values close to each other significantly lower than in the case of DQ structure. CN values generally are all acceptable, showing good controllability. LQ is the only structure, the value of which exceeds 10, but only in one case being significantly higher than 10. For all mixtures, the DQ structure has the lowest values and the LQ structure has the highest. RGAno structures also present results showing similar properties. For all mixtures, the RB structure has the lowest value, while the LQ structure had the highest. This one is the only index predicting worse controllability of DQ than most of the other structures, because the DQ structure has relatively high RGAno values. Clearly, the three controllability indices deliver results that do not support each other unquestionably. MRI and CN ranks have the same tendencies, but that is no surprise, since they share a common origin (i.e., the singular value). As soon as we compare them to the RGAno-s, we cannot point similarities out. These facts encourage us to make an attempt to aggregate

(6)

∞

|e(t )| dt

RB

Table 3. Compositions of the Separated Mixtures of Variable Pairing Investigations

The process alternative with the highest Dfct value is the most favorable solution for the specific separation task, based on the criteria mentioned in the second level of the evaluation methodology. The selected process alternative can be subjected to the detailed engineering design as the next step of the process design project. Load Rejection Analysis To Verify Frequency Domain Results. Control loops in time domain are closed and tuned. The PI controllers are tuned with the Ziegler−Nichols method, using the automatic tuning tool of Aspen Dynamics in decentralized control structure to maintain the three product compositions at the set point. Performance of the control loops is evaluated by load rejection analysis, which is based on the integral of absolute error (IAE; see eq 7). IAE =

RQ

RQ means distillate composition control by reflux ratio and bottoms composition control by reboiler duty. The following control structures are investigated: RQ, RB, LQ, DQ, LB for the mixtures shown in Table 3. A load disturbance of 1% is

(c) Small values of RGAno indicate preferable variable pairings with less interaction. Therefore, the minimization of this individual response is advantageous. The chosen function dRGAno (eq 6) favors RGAno < 1 and its value abruptly decreases at values of >1. ⎛ RGAno ⎞ ⎟ dRGAno = exp⎜ − ⎝ 10 ⎠

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(7)

Applied Tools. The steady-state model is built into the Aspen Plus software. The steady-state model is then exported to Aspen Dynamics, where the open-loop matrices of the state space model of the investigated system are obtained using the built-in Control Design Interface. The closed-loop dynamic simulations are also performed in this software. The matrices of the state-space model are then processed with the MATLAB software suite. Open-loop simulations in time domain are also carried out. For our investigations, these functions are interesting at the eigenfrequency, which is calculated from the time constant of the loop responses. 16011



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LQ is the worst controller structure, according to the frequency domain results; however, in the time domain, it often has one of the lowest values. IAE values of the LQ structure are lower in all cases than those of the DQ. The conclusion is that the MRI and CN indices fail to predict the closed-loop behavior in the time domain. RGAno is the only index that does not show DQ as a good control structure; however, it also proves to be inappropriate, since it suggests that RB is an appropriate control structure. RB, in fact, is ranked third at best in the time domain simulations. According to RGAno, LQ should perform worse than the majority; meanwhile, time domain simulations show good controllability with this structure. Indices suggested that DQ is the best controller structure; thus, we would expect the short control time and small overshoot. Figure 7 shows the load rejection of the DQ structure, together with the LQ structure. The results show that DQ eliminates load much slower and with a much higher overshoot. This experience contradicts the frequency domain results thoroughly. The last possibility could be to take advantage of weighting factors in the desirability function to give more weight to the index that shows ranks closer to time domain ranks. Unfortunately, no strategy would emphasize that the DQ configuration is hardly controllable. Results show that investigation of controllability indices in the frequency domain, for the sake of ranking, is not able to deliver acceptable results. Conclusions of Control Structure Design Investigations. Results of frequency domain simulations are almost similar to each other, some small differences can be pointed out, but otherwise the ranks of the control structures are generally the same. Comparing the frequency domain and time domain results it can be seen, that frequency domain results do not deliver the same result as time domain investigations, even the aggregated form based on the application of the overall desirability function is ineligible to replace time domain

Table 4. Frequency Domain Results and Rankings of Different Control Structures for Variable Pairing DQ

LB

RB

RQ

LQ

Benzene−Toluene MRI CN RGAno Dfct rank

0.502 1.34 2.19 0.925 I

0.179 0.152 4.66 6.42 1.17 0.51 0.895 0.892 II III Ethanol−Propanol

0.149 7.69 0.85 0.877 IV

0.172 11.81 3.97 0.797 V

MRI CN RGAno Dfct rank

0.310 1.97 2.01 0.917 I

0.147 0.131 5.45 6.81 1.72 1.20 0.854 0.852 II III n-Pentane−n-Hexane

0.124 8.30 3.04 0.789 IV

0.133 20.54 12.78 0.562 V

MRI CN RGAno Dfct rank

0.560 1.18 2.00 0.932 I

0.158 6.44 0.34 0.902 IV

0.186 8.22 2.15 0.863 V

0.193 4.51 1.04 0.907 II

0.160 6.55 0.29 0.905 III

the different indices with the desirability function. The results speak for themselves: for all three mixtures, the rank is the same. The best controllable structurewith the highest Dfct valuesis DQ. It is followed by LB, RB, and RQ, in that order, and finally the worst controllable pairing is LQ. For comparison, time domain simulations are also carried out for both feed flow rate load rejection and feed composition load rejection. Their results can be seen in Table 5. In every case, RQ earns first place for all cases with the lowest IAE values; meanwhile, DQ is the worst, with an IAE value tha tis an order of magnitude larger than that of the better controllers.

Table 5. IAE Values and Rankings of Different Control Structures for Variable Pairing Benzene−Toluene control structure IAExD + IAExB

RQ 2.06 × 10−5

control structure IAExD + IAExB rank

RQ 3.86 × 10−5 I

control structure IAExD + IAExB

RQ 1.61 × 10−5


RQ 2.90 × 10−5 I

control structure IAExD + IAExB

RQ 2.39 × 10−5


RQ 4.21 × 10−5 I

Composition, 1% Disturbance LQ LB 3.73 × 10−5 4.92 × 10−5 Feed Flow Rate, 1% Disturbance LB RB 6.73 × 10−5 1.19 × 10−4 II III Ethanol−Propanol Composition, 1% Disturbance LQ RB 2.23 × 10−5 3.55 × 10−4 Feed Flow Rate, 1% Disturbance LQ RB 1.45 × 10−4 3.20 × 10−4 II III n-Pentane−n-Hexane Composition, 1% Disturbance LQ LB 3.13 × 10−5 4.47 × 10−5 Feed Flow Rate, 1% Disturbance LB RB 4.73 × 10−5 9.70 × 10−5 II III 16012

RB 1.41 × 10−4

DQ 1.96 × 10−3

LQ 1.68 × 10−4 IV

DQ 1.99 × 10−3 V

LB 3.01 × 10−3

DQ 6.20 × 10−3

LB 2.84 × 10−3 IV

DQ 6.21 × 10−3 V

RB 1.22 × 10−4

DQ 1.26 × 10−3

LQ 1.05 × 10−4 IV

DQ 1.37 × 10−3 V



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Figure 7. Product composition deviation from,,,,, set point in the case of DQ and LQ structures, after a step change in feed composition.

investigations. This conclusion of ours is in full agreement with Gross et al.11 In our case the DQ structure - which is the best controllable according to frequency-domain results - proves to be almost useless in the dynamic simulations. Meanwhile LQ is at the end of the rank list in the frequency domain, however, in many cases in time domain it is one of the best performing structures. RQ structure is undoubtedly the best control structure in time domain, but in the frequency domain this excellent behavior is not predicted. Finally, it can be declared that frequency domain examinations, even with the application of the overall desirability function, are not able to replace time domain simulations when used for variable pairing in the case of a single process alternative.

Table 6. Frequency Domain Indices and D-Values of Different Distillation Structures MRI CN RGAno Dfct

DQB

FTCDC

SQF

Conv. Dir.

0.404 2.99 0.921 0.957

0.464 2.77 2.810 0.902

0.405 4.09 5.093 0.831

0.023 48.28 0.082 0.526

Table 7. Ranking of the Distillation Systems Based on Individual Controllability Indices Controllability Indices

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RESULTS OF THE RANKING OF PROCESS DESIGN ALTERNATIVES, ACCORDING TO THEIR CONTROLLABILITY FEATURES The operating conditions selected for controllability study are: pentane - hexane - heptane ternary mixture and 95% product purity. Feed is an equimolar mixture of all components. The values of the critical frequency of each distillation system differ and they range between 10−3 and 10−2rad/sec. Values of controllability indices are read in the range of the critical frequencies that is calculated for each system with open control loops. Frequency dependent controllability indices (CN, MRI, RGAno) are calculated and presented in Table 6. Based on the CN numbers the ranking of the studied distillation systems in sense of controllability would be: the easiest to control systems are the DQB and SQF followed by the FTCDC, and conventional direct sequence shows the worst control properties (Table 7). If the MRI is taken into account, the abovementioned ranking is modified as follows: the more resilient process is the FTCDC that is followed by the DQB and SQF and last is the Conv. Dir. from the point of view of the RGAno

ranking

CN

MRI

RGAno

Dfct

I II III IV

DQB SQF FTCDC Conv. Dir.

FTCDC DQB SQF Conv. Dir.

Conv. Dir. DQB FTCDC SQF

DQB FTCDC SQF Conv. Dir.

- that indicates the interactions between the control loops of the distillation system - the ranking is totally different: the Conv. Dir. shows less interactions followed by the DQB, FTCDC, SQF. Table 7 also shows that ranking of the distillation systems from controllability point of view is hardly possible if the controllability indices are taken into account individually. In order to help the decision of the control designer the overall desirability function is applied. Last column of Table 7 indicates that the DQB has the best control properties taking the three indices simultaneously into account. Proposed controllability analysis makes it easy to select the easyest to operate distillation systems among the studied ones. The controllability analysis is verified by closed loop simulations. Verification contains a load rejection analysis that is carried out with the application of different disturbances on the distillation systems with closed-loop control structures. Closed loop simulations are made for two types of 16013



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disturbances: 1% step change in feed flow rate and 1% in feed composition. IAE measured as the responses after applying load on the feed flow rate is presented in Table 8 . As the table shows, DQB has the best dynamic behavior that is followed by FTCDC, SQF, and Conv. Dir.

that can predict closed loop behavior. Aggregating with desirability function can not help in this case. In the case of process design alternatives, controllability indices in the frequency domain aggregated with the overall desirability function can deliver the same results as time domain simulations. The best performing control structure of each system can be compared to the control structures of alternative distillation systems, and a rank can be set up based on controllability indices aggregated by desirability function. Therefore, the application of desirability function can be recommended for the ranking and selection of process design alternatives according to their controllability features.

Table 8. IAE Values in Case of Feed Flow Step Disturbance IAE Values of Product Streams DQB pentane hexane heptane summa rank

7.41 1.54 3.74 1.65 I

× × × ×

FTCDC −5

10 10−3 10−5 10−3

1.14 4.57 7.72 2.37 II

× × × ×

−3

10 10−4 10−4 10−3

SQF 1.92 1.42 3.51 3.53 III

× × × ×

Conv. Dir. −4

10 10−6 10−2 10−2

1.35 5.68 1.07 5.80 IV

× × × ×

10−4 10−2 10−3 10−2

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Corresponding Author

*E-mail: [email protected].

Responses for the feed composition loads are shown in Table 9. Based on the composition disturbance responses the rank of the distillation systems does not change.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by project OTKA No. 76139, as well as project Nos. TÁ M OP-4.2.2.B-10/1-2010-0009 and TÁ MOP-4.2.1-08/1/KMR-2008-0001.

Table 9. IAE Values in the Case of Feed Composition Step Disturbance IAE Values of Product Streams DQB pentane hexane heptane summa rank

1.31 2.77 1.71 4.25 I

× × × ×

10−4 10−4 10−5 10−4

FTCDC 9.10 3.40 5.19 1.77 II

× × × ×

10−4 10−4 10−4 10−3

SQF 6.33 4.34 1.26 2.33 III

× × × ×

10−4 10−4 10−3 10−3

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Conv. Dir. 3.62 1.50 4.87 2.35 IV

AUTHOR INFORMATION

× × × ×

10−4 10−3 10−4 10−3

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CONCLUSION OF RANKING OF PROCESS DESIGN ALTERNATIVES In the case of ranking the process design alternatives based on frequency domain indices, aggregated with the overall desirability function, deliver the same results as time domain investigations. The results of the comparative study of the different energy integrated distillation systems, from controllability point of view, show that the direct sequence with backward heat integration (DQB) is the best choice. The second is the FTCDC and it is followed by the SQF. However, the conventional direct distillation sequence, that is considered as a simple configuration, has the worst control properties. This result can be found odd but it is in correspondence with the results of other authors, e.g. Alatiqi and Luyben.31 They have compared the control properties of the conventional direct sequence with a thermally coupled distillation system and they found that the conventional sequence presented the worst control properties in the frequency domain controllability analysis and closed loop simulations as well.

NOTATION B = bottom product flow rate of the column (kmol/h) D = distillate flow rate of the column (kmol/h) e = error of control, difference of set point and actual value of the controlled variable i = imaginary unit L = reflux flow rate of the respective column (kmol/h) Q = heat duty of the column’s reboiler (kW) R = reflux rate of the column; R = L/D S = side product flow rate (kmol/h) Tol = time constant of open controller loop (s) V = vapor flow rate (kmol/h) x = composition, mole fraction ω = frequency (rad/s) ωcrit = critical frequency (rad/s)

Indices

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A = lightest component B = second-lightest component or bottoms C = heaviest component D = distillate

REFERENCES

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FINAL CONCLUSION Our research proposes a methodology to carry out control structure design. The methodology is tested on two cases: (I) evaluation of control structures for an individual process, (II) ranking of process design alternatives according to their controllability features. The determination of the best control structure according to controllability indices in the frequency domain proves to be ambiguous, none of them provides results 16014



Article

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Applicability of Desirability Function for Control Structure Design in the

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