Towards integrated process and control system synthesis for heat-integrated plants.
One of the subjects that have received the most attention from researchers in this area is the steady state design of Heat Exchanger Networks (HENs). Several tools have been developed and are in use; however, the development of a tool for synthesis of HENs that takes into account network controllability is not available. Hence, the purpose of this paper is the development of a new methodology for design of heat-integrated chemical processes, particularly HENs where controllability and energy recovery are both balanced during the design synthesis stage.
La plupart des procedes chimiques sont des reseaux de differentes pieces d'equipement, comme des reacteurs, des colonnes a distiller, des compresseurs, des echangeurs de chaleur, etc. L'integration des procedes est un domaine du genie chimique qui s'occupe de la conception optimale de ces reseaux, en termes d'effi cacite energetique, de couts en capital, de reduction des emissions, de minimisation des eaux usees et de l'utilisation des matieres premieres. Jusqu'a recemment, les ingenieurs creaient des procedes conceptuels par experience et intuition, mais avec la venue des methodologies d'integration des procedes, ce travail peut se faire de facon systematique. L'un des sujets qui depuis peu attire le plus l'attention des chercheurs dans ce domaine, c'est la conception a l'etat stable de reseaux d'echangeurs de chaleur (HEN). Plusieurs outils ont ete mis au point et sont employes : cependant, on n'a pas encore mis au point un outil pour la synthese des HEN qui tienne compte de la controlabilite des reseaux. Ainsi, cet article vise a elaborer une nouvelle methodologie pour la conception des procedes chimiques integres thermiquement, en particulier les HEN pour lesquels la controlabilite et la recuperation d'energie sont toutes deux equilibrees durant la phase de synthese de la conception.
Keywords: multivariable control system analysis, disturbance analysis, heat-integration, Heat Exchanger Networks (HENs)
Design of HENs
Every process consists of hot and cold streams. Hot streams are those ones that have to be cooled down and, cold streams have to be heated up. Cold water (CW) is the most common cooling media used for cooling hot streams and steam is the most common fl uid used for heating cold streams. Figure 1a shows a process where all hot streams are cooled using cold water and all cold streams are heated using steam. In this case, no heat integration is employed and Figure 1b shows the grid diagram as proposed by Linnhoff and Flower (1978) for this process.
[FIGURE 1 OMITTED]
In order to save energy, hot streams should be used to heat cold streams (where possible); in other words, a HEN should be designed for this process. Several techniques have been proposed for synthesis of HENs. Some of them are based on thermodynamic insights, such as those methodologies based on Pinch Analysis (Linnhoff and Hindmarsh, 1983; Linnhoff and Ahmad, 1990), and others are based on mathematical programming (Cerda and Westerberg, 1983; Floudas et al., 1986; Briones and Kokossis, 1999). Figure 2a shows the fl ow sheet for a heat integrated process, where a HEN was designed for the maximum energy recovery and Figure 2b shows the corresponding grid diagram.
[FIGURE 2 OMITTED]
Use of a Controllability Index in Design of HENs
In order to ensure that target temperatures of all hot and cold streams are attained during the plant operation, a control system must be implemented. A control system is comprised of: a primary element (sensor/transmitter), controller, a final control element (usually control valves) and the process itself (Svrcek et al., 2000). In case of temperature control, the primary element is a device (often a thermocouple) that measures the target temperature of the stream and sends a signal (pneumatic, electrical or digital) to the controller, that compares this value with the specified set point, and according to a control law, the controller sends a signal to the final control element (a valve) that will regulate the flow rate of a hot utility stream (e.g. steam), for cold streams, or a cold utility stream (e.g. cold water), for hot streams. In this case, the process is the heat exchanger that will present a specific dynamic response to disturbances. Figure 3 shows the grid diagram of a process with no heat integration, where the temperature control loops of each stream are emphasized. The process shown in Figure 3 is very easy to control, because in order to keep the control of each target temperature, one utility (steam or cold water) flow rate can be manipulated, i.e., no integration or coupling/feedback of the energy streams takes place.
[FIGURE 3 OMITTED]
However, when a HEN is employed there is no longer a one to one correspondence for one target temperature and one utility flow rate. Therefore, for this situation process controllability and start-up of the processing unit is much more complex. Kotjabasakis and Linnhoff (1986) studied the flexibility of HENs, but the authors did not provide any information about the controllability of different configurations. The advantage of using this approach is that it provides a warning of the extreme variations happening in the target temperatures. Although this approach will often suggest to the design engineer the conservative design, over-designing the hot and cold utilities, this approach also suggests that modification of the heat transfer areas can make the HEN more flexible. This will generally require additional expenditure in order to have a flexible design.
As the number of hot and cold streams in a process increases, the number of possible HENs configurations increases dramatically. The use of stream splitting increases the degrees of freedom to almost infinity. In this context, the following question arises: how can different HEN configurations be compared in terms of controllability and start-up? The primary objective of this research is to propose a controllability methodology for HENs. This index should derive from current techniques for non-integrated plant found in the literature such as the Relative Gain Array--RGA (e.g. Seborg et al., 1989), Niederlinski Index (e.g. Svrcek et al., 2000) or Synergistic Parameters (Coselli, 2000).
Many authors have addressed the importance of control systems design during the early stages of the development of the process design (e.g. Luyben et al., 1997) and it has been demonstrated that good control design is an effective way to ensure that a plant will exhibit flexibility for changes and resiliency to disturbances to operating conditions. This situation also applies for the design of a HEN. Normally the classical HEN synthesis approaches assume and keep all design operating conditions at a constant value (e.g. Shenoy, 1995). Before further design for detailed engineering some analysis and modifications of the synthesized HEN have to be carried out, in order to guarantee good flexibility and resiliency of the HEN, since in reality operating conditions are expected to change. Flexibility is the ability of the HEN to readily adjust and meet the requirements of changing conditions. Resiliency is the ability of the HEN to tolerate and to recover from disturbances. The purpose of this paper is to present the appropriate controllability rules for heat-integrated plants, in order to assess process design developments and to propose control strategy alternatives appropriate and suitable for a HEN.
Nowadays many techniques are available for the synthesis of HENs. Many of them are based on the pinch point technology (Linnhoff and Hindmarsh, 1983), and are used to synthesize a preliminary arrangement and then on this base design evolutionary and optimization procedures are implemented. Colberg et al. (1989) present an approach with two resilient targets for HEN synthesis involving large uncertainties in supply temperatures and heat capacity fl ow rates. These two targets are the design uncertainties that will be studied in this paper.
In the search for the controllability of a HEN, Aguilera and Marchetti (1998) present a procedure for the optimization of a HEN on top of the base level control using linear programming and non-linear programming techniques. They proposed a seven-step-procedure to select appropriate bypass pairings. This approach will not necessarily comply with the overall performance of the HEN. Westphalen et al. (2003) pioneer an approach to determine a controllability index of a HEN based on the degrees of freedom approach of Glemmestad (1996) and the computation of the interactions in a multivariable dynamic system. All these approaches need to be integrated along with other important issues to come up with a general, effective and comprehensive methodology to design flexible and resilient HENs.
In the following section a comprehensive and systematic methodology to determine the controllability of a heat-integrated plant is presented. This approach is based solely on steady-state information, and its results are confirmed by dynamic simulations. This methodology can be utilized as a screening process to determine the more controllable plant amongst many alternative configurations of heat exchanger networks.
MULTIVARIABLE SYSTEM ANALYSIS OF A HEN
Decentralized Feedback Control System for a HEN
A HEN may be considered a multivariable system problem, involving at least one process variable or controlled variable (cv), in this case the target temperature (TT) attained through a process-to-process heat exchanger, and one or more potential manipulated variables (mv's). For one TT we have at least two candidate mv's, either the hot stream bypass or the cold stream bypass. The control of multivariable systems requires more complex analysis than single-variable systems; however, the concept of a decentralized feedback control system is the simplest approach to multivariable control design (Marlin, 2000). We find three major advantages using a decentralized feedback control system: flexible operation, simple design, and failure tolerance.
Pairings between Control Variables and Manipulated Variables
The general plant gain matrix at steady state, [G.sup.G], is an m x n matrix, with the number of rows determined by the number of target temperatures and the number of columns determined by the number of all potential cold and hot single and multiple bypasses. In general, [G.sup.G] can be computed through the following equation:
[g.sup.G.sub.ij] = [partial derivative] / [[partial derivative][Z.sub.j] T[T.sub.i][approximately equal to] [DELTA]/[DELTA][Z.sub.j]T[T.sub.i] (1)
[g.sup.G.sub.ij] = is the element ij of [G.sup.G], representing the variation of the i-th TT with respect to the variation of the j-th potential bypass in the HEN. The information needed for [G.sup.G] can be obtained easily from steady-state models, dynamic models or process identification models.
The first step to be taken towards the design of a decentralized control system is to determine the pairings between cv's and mv's. It is well known that for an n x n plant there are n! possible single-loop pairings. It is evident that the problem to determine the total number of bypass alternatives becomes a huge "brute force" effort that could mean a computational challenge. The number of possible pairings can be determined through the use of the non-square Relative Gain Array (ns-RGA). The ns-RGA is found to be an adequate tool to determine the needed pairings between single and multiple bypasses and target temperatures (e.g. Cao, 1995). The way this ns-RGA will be formulated is described in the following array for a number m of target temperatures attained using process-to-process heat exchangers; T[T.sub.1] to T[T.sub.m] and a number n of potential bypasses [Z.sub.1] to [Z.sub.n]. The ns-RGA can be evaluated using the following formula:
[[LAMBDA].sup.G.sub.HEN] = [G.sup.G] [cross product][[G.sup.G[dagger]].sup.T] (2)
where [cross product] is the Schur product between two matrices and [G.sup.G[dagger]] is the pseudo-inverse matrix of [G.sup.G]. Furthermore the rank of [[LAMBDA].sup.G.sub.HEN] is m x n. In order to determine the appropriate pairings between bypasses and target temperatures, elements nearest to 1 have to be chosen and this will ensure that the pairing values in the square RGA (Bristol, 1966) derived from the general ns-RGA will be positive and relatively close to one. Once the appropriate elements of ns-RGA are chosen, a square RGA is calculated using the selected columns of each of the rows of the ns-RGA.
If a multiple bypass is chosen, a single or lower level multiple bypass has to be avoided. In fact, a global bypass is preferred over a single bypass, because its action is faster and may minimize the effect of disturbances (Mathisen et al., 1992). Once the appropriate pairings are selected, a square RGA of the HEN,
[[lambda].sub.HEN], is calculated:
[[lambda].sub.HEN] = G [cross product] [[G.sup.-1].sup.T] (3)
Now the rank of [[lambda].sub.HEN] is m x m. This resulting RGA will confirm that the appropriate pairing will have positive large values, but it will not guarantee values close to 1. Using this procedure an exhaustive screening process is avoided and it is not necessary to screen a factorial number of potential alternatives (e.g. Mathisen et al., 1992).
Westphalen et al. (2003) proposed to determine the controllability of a HEN based on the analysis of the condition number ([gamma]). The value of [gamma] is calculated using the gain matrix G. If this condition number is small, the HEN is considered to be more controllable than a HEN with a large [gamma].
[gamma] = [[??].sub.G] / [[??].sub.G] (4)
The two singular values, [??]G and [??]G, can be determined from the singular value decomposition of G. A HEN with a large condition number is considered to be ill-conditioned (e.g. Seborg, 1989). Note that [gamma] is highly dependent on the scaling of the cv's and mv's (e.g. Skogestad and Postlethwaite, 1996). A large condition number implies that the magnitude of cv's is strongly stretched in certain direction of the mv's (e.g. Morari and Zafi riou, 1989).
Performance Relative Gain Array
In order to overcome the problem of one-way coupling that the RGA cannot assess, Hovd and Skogestad (1992) introduced the concept of the performance relative gain array (PRGA), which is defi ned as follows:
[[GAMMA].sub.HEN] = [G.sup.+][G.sup.-1] (5)
where [G.sup.+] is a diagonal matrix, having the same diagonal values of G. PRGA is useful to determine the existence of interactions in HENs for which G is triangular, or its off-diagonal elements may be very small. The RGA may indicate that interactions are not a problem for control, but PRGA will indicate that one-way coupling may exist. PRGA is independent of the mv's scaling but highly dependent on cv's scaling. PRGA is a good measure of the performance of the decentralized control system on the HEN giving an idea of the magnitude of the cv's. An svd analysis on a PRGA is useful to determine the interaction direction in a HEN (e.g. Grosdidier, 1990). Set point changes coming in the direction of the maximum singular value will affect adversely the performance of the control system because of the magnitude of interactions. Application of this concept is very useful to determine the flexibility of a HEN. The magnitude of interactions due to set point changes can be predicted and used to select the HEN having the lowest degree of interaction.
DISTURBANCE ANALYSIS OF A HEN
Disturbances Acting on a HEN
Flow rate upsets and supply temperature variations are considered the most important disturbances acting on a HEN. Kotjabasakis and Linnhoff (1986) use these disturbances in their proposed sensitivity table analysis. Glemmestad et al. (1996) also agreed that these are the most important disturbances that may be present and they even establish qualitative rules for the expected effect of such disturbances.
The Disturbance Gain Matrix
The disturbance gain matrix [G.sub.d] of a HEN can be determined similarly as [G.sup.G], based on steady-state information, dynamic models or system identification data of the HEN. Each element of [G.sub.d] can be determined using the following formula:
[g.sub.dik] = [partial derivative] / [[partial derivative]d.sub.k] T[T.sub.i] [approximately equal to] [DELTA]/[DELTA][d.sub.k]T[T.sub.i] (6)
Each element represents the variation of the i-th TT with respect to the variation of the k-th disturbance in the fl ow rate of each of the process streams and supply temperatures--in a similar way as for [G.sup.G].
Disturbance Condition Number
The disturbance condition number is interpreted as the ratio between the required mv's values for disturbance rejection and the mv's values needed if the same disturbance gain was aligned with the largest HEN gain direction. Disturbance condition number for a disturbance [d.sub.k], [[gamma].sub.dk], can be calculated from the following equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[g.sub.dk] is the column of [G.sub.d] corresponding to the disturbance k. [??] G is the largest singular value of G. In the formula the 2-norm of a vector is being used and indicated redundantly. [[gamma].sub.dk] may vary between 1 and [gamma]. The later situation is undesirable.
Relative Disturbance Gain
First introduced by Stanley et al. (1985), the use of the relative disturbance gain (RDG) is fundamental to determine the capacity of a HEN to accommodate disturbances. The numerator implies perfect control on all target temperatures, and the denominator implies perfect control on only T[T.sub.i]. This RDG will determine how the manipulated variable [Z.sub.j] is affected by a disturbance [d.sub.k]. Its magnitude is how much the decentralized control system will struggle to accommodate disturbances in the HEN. One limitation in the use of the RDG that has to be taken into consideration is that RDG can take very large values or, even worst, infinite values. In order to understand this implication, we must first see another way to calculate RDG, based on the following equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
A very large value of [[beta].sub.dki] means that the magnitude of the disturbance k is close to zero. In the worst case, [[beta].sub.dki] is zero, with no effect on T[T.sub.i] at all. Almost no effect on T[T.sub.i] may be a good property of the HEN, but it does not agree with the preferred criteria of having 0 < [beta][d.sub.k]I < 1.
Closed Loop Disturbance Gain
Skogestad and Hovd (1990) introduced the concept of the closed loop disturbance gain (CLDG), and this is an indicator for the performance of the decentralized control with respect to disturbance rejection (e.g. Groenendijk et al., 2000). The CLDG is calculated via the following formula:
[[DELTA].sub.HEN] = [[GAMMA].sub.HEN] [G.sub.d] = [G.sup.+][G.sup.-1][G.sub.d] (9)
where [[DELTA].sub.HEN] is CLDG of a HEN. CLDG is closely related to the RDG (e.g. Skogestad and Wolff, 1996). And if the values of its elements are below 1, it is said that no control is needed since the difference between the set point and the cv will never exceed its bounds (e.g. Groenendijk et al., 2000), this implies no controller is needed. The CLDG and the PRGA have the same mathematical origin; both were derived at the same time (e.g. Skogestad and Hovd, 1990). Through an svd analysis on the CLDG, similar to the one presented by Grosdidier (1990) for the PRGA, we can predict the magnitude and interaction direction of the mv's in order to reject disturbances in the HEN.
Partial Disturbance Gain
In the case of having some controlled variables uncontrolled, one or more control loops out of service, the partial disturbance gain (PDG) concept is a useful measure to determine the capability of the HEN to reject disturbances in all controlled variables simultaneously. For a particular disturbance, there might be a particular pairing of target temperature--bypass for which the PDG is less than 1 in magnitude. If we want to find the corresponding k-th disturbance sensitivity with all other inputs perfectly controlled, we use the following equation:
PD[G.sub.ijk] = [[G.sup.-1] [G.sub.d]].sub.ik] / [[G.sup.-1].sub.ji] (10)
For simultaneous disturbances the worst-case scenario has to be evaluated by taking for each pairing the sum of element magnitudes. This requires the calculation of a combined PDG matrix as follows:
[[G.sub.PDG].sub.ij] = [summation over (k)]|[PDG.sub.ijk]| (11)
It is advantageous to find uncontrolled pairings where the [G.sub.PDG] elements are less than one.
Magnitude of Manipulated Variables
Constraints on the mv's can bind the capability of the HEN to reject disturbances and track set point changes. It is also important to find the smallest possible change in the mv's needed to reject disturbances. One way to achieve this is to determine the specified control performance. For "perfect control," when the difference between the set point and the cv or error is zero, the values of the mv's for "perfect" disturbance rejection is calculated using the following formula:
u = [G.sup.-1] [G.sub.d] (12)
Considering each disturbance and the worst case scenario, when |[d.sub.k]| = 1, in order to verify if there will be no mv saturation, then the following should be attained:
[parallel][G.sup.-1][G.sub.d][[parallel].sub.[infinity]] [less than or equal to] 1 (13)
For "acceptable control," and if there is no set point changes, r = 0, it is possible to achieve [parallel]u[[parallel].sub.[infinity]] [less than or equal to] 1 for any [parallel]d[[parallel].sub.[infinity]] [less than or equal to] 1. Mathematically it is required to solve the following optimization:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
Subject to [parallel] G x u + [g.sub.d] [d.sub.k] [[parallel].sub.[infinity]] [less than or equal to] 1 (15)
|[d.sub.k]| = 1
|[d.sub.i]| = 1 represents the worst-case scenario. For the case of simultaneous disturbances, the optimization problem is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
Subject to [parallel]G x u + [G.sub.d] x [[parallel].sub.[infinity]] [less than or equal to] 1 (17)
[parallel]d[[parallel].sub.infinity]] [less than or equal to] 1
Because of the mv's constraints, mentioned above, for multivariable systems the infinite norm is the appropriate choice when considering mv's saturation, which also serves for scaling purposes. For the case of HENs, we will have as many bypasses needed as target temperatures. With this set-up we can achieve "perfect control."
Introduced by Morari et al. (1985), the resiliency of the HEN can be determined by the maximum disturbance range, also known as Resiliency Index (RI), such that for a maximum disturbance range, feasible operation is achieved. RI is mathematically denoted by the following:
[parallel]dk[parallel][less than or equal to] R1 (18)
Feasible operation is defined as an acceptable performance [parallel]e[[parallel].sub.infinity]][less than or equal to] 1 in spite of limitations on the manipulated variable [parallel]u[[parallel].sub.[infinity]] [less than or equal to] 1.
Scaling of Variables
All the above analyses can be carried out utilizing the scaled values of G and [G.sub.d]. This scaling is very important, and also will be for the plant gains due to disturbances represented by the matrix [G.sub.d], because it will allow us in further analysis to determine the saturation of the cv's and mv's, and also the disturbances d's (e.g. Skogestad and Wolff, 1996). The main objective of scaling variables is to keep them within the interval -1 and 1, making easier the interpretation of the results and ensuring the stability of calculations. This normalization process depends on the process nature and the design requirements. mv's will be normalized with respect to their allowed range of variation, which normally will correspond to a physical constraint of the control valves that will be implemented on the real plant or availability of utilities or other process variables. The cv's will be normalized with respect to the allowed error range or offset in a feedback control system. d's will be normalized with respect to their expected magnitude.
Case Study The following case study is the HEN 4S1 analyzed and studied by Shenoy (1995). This is the case of two hot process streams, H1 and H2, and two cold process streams, C3 and C4. The minimum temperature approach is 13[degrees]C. The stream data is shown in Table 1. Using the pinch-point design method (Shenoy, 1995) we have three potential HENs, shown in Figures 4, 5 and 6, denoted by HEN#1, HEN#2 and HEN#3 respectively.
[FIGURES 4-6 OMITTED]
The following two HENs are an evolutionary design of HEN#1, by loop breaking and path relaxation (Shenoy, 1995). Removing HX1 in HEN #1 generates HEN #4, and then some adjustment in the heat loads of each heat exchanger is made, in order to maintain the 13[degrees]C temperature approach. Removing HX4 in HEN #1, again some adjustments is made to maintain the specified temperature approach generates HEN #5. Figures 7 and 8 show HEN#4 and HEN#5 respectively. Finally, HEN#6 is generated using the fast match algorithm procedure (Shenoy, 1995). Figure 9 shows this HEN.
[FIGURES 7-9 OMITTED]
Table 2 summarizes the control objectives for each of the HENs:
Degrees of Freedom
The number of potential single and multiple bypasses available to control each of the target temperatures are indicated in Table 3.
Operation Rates and Conditions
The expected variations will be [+ or -]25% in the flow rate and [+ or -] 15[degrees]C in the supply temperature.
HEN's Gain Matrix
Based on a steady-state model for each HEN, Tables 4-9 show the process gain values for each HEN. Units are [degrees]C/bypass opening %. The tables also show the ns-RGA for each of the HENs. The bolded and underlined values are the ones considered most appropriate for pairing and will be selected for the square process gain matrix and for further analyses. Based on the above results, Figures 4 to 9 show the HENs with their corresponding set of bypasses for controllability.
Square Process Gain Matrix and RGA
Table 10 shows the results for the square RGA of each HEN, confirming that the bypasses selected are appropriate to control the HEN.
Scale Process Gain Matrix
According to the expected variations in the operating conditions and in the target temperatures, we scale the process gain matrix of each HEN multiplying by 0.05. The meaning of such factor is that we expect 5[degrees]C of variation in the target temperature and 0.25 flow rate fraction in the bypass. 0.25 flow rate fraction means that the maximum flow rate in the bypass is 50% of the flow rate through the heat exchanger.
Calculate the Condition Number
Table 11 summarizes the results from svd of G, showing the values of the smallest singular values of G, [??], and the [gamma]for each HEN.
For HENs #5 and #6 the [??] values are less than 1, therefore we expect problems in the controllability of such HENs.
Calculate PRGA and its svd Values
The PRGA for each HEN and its svd values are shown in Table 12, in order to determine the performance of the decentralized control system with respect to set point changes. HENs with smallest variations will be preferred.
Determine HEN's Disturbance Gain Matrix and Scale it.
Table 13 shows the scaled disturbance gain matrices, Gd, for each HEN.
The scaling factors for each of the disturbances are established according to their expected maximum variations. For disturbances in the fl ow rate, we expect a variation of [+ or -]25% and for variations in the supply temperatures we expect a variation of [+ or -]15[degrees]C.
Calculate the Disturbance Condition Number
Table 14 shows the disturbance condition number, for each of the HENs.
Disturbance condition numbers that are far from [gamma], are desired. Those close to [gamma] are considered to be in bad direction. By inspection, all disturbances have the same degree of effect on the HEN.
Table 15 shows the results from RDG calculations. For some HENs, some values are infinite, meaning that the disturbance gain is zero, and not necessarily that the HEN has no capacity to reject disturbances. However, values that are less than one will be preferred.
For all the HENs, the majority of the RDG values are above one that one which means the manipulated variables have to move far from the initial steady-state in order to reject the disturbances.
CLDG and its svd Values
Table 16 shows the values of the svd of CLDG. The direction of the disturbances that will affect the HEN the most is the same as the values found from svd of Gd, although the magnitudes of their effects are slightly different. Those having the least magnitude of effect will be preferred.
Calculate the Disturbance Sensitivity under Partial Control
Table 17 shows the matrices of the PDG, when all disturbances occur simultaneously. Those having the smallest values will be preferred.
Perfect and Acceptable Control
Table 18 shows the maximum value of the two manipulated variables in order to reject each and all the disturbances in the HEN to have perfect control. If numbers are above 1, then we have to consider that we only are able to have acceptable control. Table 18 shows the maximum value of the manipulated variables (bold and underlined) in order to achieve acceptable control, which means that the target temperatures will have a variation within a range of [+ or -]5 [degrees]C from the specified set point value. If some of the values of [U.sub.min] are greater than one that means that the bypass has no capacity to reject the disturbance and we might have a large offset. In the case of all disturbances occurring simultaneously, we take into account the direction of maximum effect.
Table 19 shows the values of the resiliency index for each of the HEN for each individual disturbance and when all disturbances occur simultaneously. Those with the largest values will be preferred.
Some Results From Dynamic Simulations
Analyzing all the values presented in the above tables, we can determine that overall HEN#1 is the most appropriate arrangement, because, overall, it presents the best values of all the controllability parameters. Figure 10 shows the dynamic behaviour of HEN#1 due to set point changes, this was confirmed by the maximum singular value of PRGA ([[SIGMA].sub.[GAMMA]ii]), which is the smallest among all HENs. Figure 11 shows its dynamic behaviour to disturbances in WC[p.sub.H2] and T[S.sub.H2], and Figure 12 shows its dynamic behaviour to disturbances in T[S.sub.H2] and in T[S.sub.C4] and the HEN on partial control. In Figure 11 we observe that WC[p.sub.H2] disturbances occurring in HEN#1 imply a saturation of the manipulated variables, something that is confirmed from the results shown in Table 18 for [d.sub.2]. All of these figures corroborate that the results from the controllability measurements based on steady-state information agrees with the dynamic simulation of the different operating scenarios when disturbances occur or when the plant is subject to partial control.
[FIGURES 11-12 OMITTED]
The above methodology using the controllability measures proved to be a useful screening tool to determine the controllability of a set of potential HEN for a particular heat integrated plant. The magnitude of such measures can be used to determine the more controllable HEN, and dynamic simulations support the results from this methodology. By keeping the methodology simple and practical, the hope is that it can be easily understood and exploited by process and control systems design engineers using commercial simulators, in the design of actual chemical processes.
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Manuscript received February 25, 2005; revised manuscript received October 27, 2005; accepted for publication November 6, 2005.
Brent R. Young (1*), Rodolfo Tellez (2) and William Y. Svrcek (3)
(1.) Department of Chemical and Materials Engineering, University of Auckland, Private Bag 92019, Auckland City, New Zealand
(2.) Aspen Technology, Inc., 900, 125-9 Avenue SE, Calgary, AB, Canada T2G 0P6
(3.) Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4
* Author to whom correspondence may be addressed. E-mail address: firstname.lastname@example.org
Table 1. Case study 4S1 stream data Stream WCp TS TT (kW/[degrees]C) ([degrees]C) ([degrees]C) H1 10 175 45 H2 40 125 65 C3 20 20 155 C4 15 40 112 Table 2. Control objectives for each HEN HEN TT #1, #2, #4, #5 and #6 H2 C4 #3 H1 C4 Table 3. Number of potential manipulated variables Single Double Triple Potential Hot Cold Hot Cold Hot Cold sets HEN#1 4 4 2 2 - - 54 HEN#2 4 4 2 2 - 1 60 HEN#3 4 4 2 1 - 1 52 HEN#4 3 3 1 1 - - 15 HEN#5 3 3 1 1 - - 15 HEN#6 3 3 1 1 - - 15 Table 4. Gain and ns-RGA matrices for HEN#1 [G.sup.G.sub.HEN#1] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.016 0.014 [Y.sub.2] 0.047 -0.041 [Y.sub.3] 0.025 -0.022 [Y.sub.4] -0.016 -0.016 [Y.sub.14] -0.025 -0.023 [Y.sub.23] 0.096 -0.084 Cold bypasses [X.sub.1] 0.005 0.004 [X.sub.2] 0.090 -0.019 [X.sub.3] 0.024 -0.056 [X.sub.4] -0.009 -0.008 [[X.bar].sub.12] 0.124 0.014 [[X.bar].sub.34] 0.053 -0.156 [[LAMBDA].sup.G.sub.HEN#1] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.012 0.010 [Y.sub.2] 0.040 0.027 [Y.sub.3] 0.012 0.008 [Y.sub.4] 0.013 0.012 [Y.sub.14] 0.029 0.028 [Y.sub.23] 0.167 0.112 Cold bypasses [X.sub.1] 0.001 0.001 [X.sub.2] 0.238 -0.015 [X.sub.3] -0.003 0.086 [X.sub.4] 0.004 0.003 [[X.bar].sub.12] 0.529 0.036 [[X.bar].sub.34] -0.041 0.694 Table 5. Gain and ns-RGA matrices for HEN#2 [G.sup.G.sub.HEN#2] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.015 0.000 [Y.sub.2] 0.042 -0.034 [Y.sub.3] 0.042 -0.034 [Y.sub.4] -0.016 0.000 [Y.sub.14] -0.025 0.000 [Y.sub.23] 0.108 -0.092 Cold bypasses [X.sub.1] 0.004 0.000 [X.sub.2] 0.067 -0.160 [X.sub.3] 0.062 0.000 [X.sub.4] -0.007 0.000 [X.sub.12] 0.082 0.000 [X.sub.24] 0.087 0.000 [[X.bar].sub.124] 0.124 0.000 [[LAMBDA].sup.G.sub.HEN#2] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.006 0.000 [Y.sub.2] 0.021 0.021 [Y.sub.3] 0.021 0.021 [Y.sub.4] 0.007 0.000 [Y.sub.14] 0.016 0.000 [Y.sub.23] 0.133 0.160 Cold bypasses [X.sub.1] 0.001 0.000 [X.sub.2] -0.062 0.798 [X.sub.3] 0.099 0.000 [X.sub.4] 0.001 0.000 [X.sub.12] 0.172 0.000 [X.sub.24] 0.194 0.000 [[X.bar].sub.124] 0.393 0.000 Table 6. Gain and ns-RGA matrices for HEN#3 [G.sup.G.sub.HEN#3] T[T.sub.H1] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.046 0.000 [Y.sub.2] 0.139 0.000 [Y.sub.3] -0.005 -0.031 [Y.sub.4] -0.005 -0.034 [[Y.bar].sub.12] 0.367 0.000 [Y.sub.34] -0.014 -0.087 Cold bypasses [X.sub.1] 0.015 0.000 [X.sub.2] 0.045 0.000 [X.sub.3] 0.058 0.000 [[X.bar].sub.4] 0.000 -0.161 [X.sub.23] 0.103 0.000 [X.sub.123] 0.298 0.000 [[LAMBDA].sup.G.sub.HEN#3] T[T.sub.H1] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.008 0.000 [Y.sub.2] 0.074 0.000 [Y.sub.3] 0.000 0.027 [Y.sub.4] 0.000 0.032 [[Y.bar].sub.12] 0.515 0.000 [Y.sub.34] 0.001 0.212 Cold bypasses [X.sub.1] 0.001 0.000 [X.sub.2] 0.008 0.000 [X.sub.3] 0.013 0.000 [[X.bar].sub.4] 0.000 0.729 [X.sub.23] 0.041 0.000 [X.sub.123] 0.340 0.000 Table 7. Gain and ns-RGA matrices for HEN#4 [G.sup.G.sub.HEN#4] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.2] 0.050 -0.027 [Y.sub.3] 0.027 -0.015 [Y.sub.4] -0.011 -0.011 [Y.sub.23] 0.100 -0.052 Cold bypasses [[X.bar].sub.2] 0.117 0.000 [X.sub.3] 0.021 -0.050 [X.sub.4] -0.007 -0.007 [[X.bar].sub.34] 0.057 -0.162 [[LAMBDA].sup.G.sub.HEN#4] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.2] 0.082 -0.001 [Y.sub.3] 0.024 0.000 [Y.sub.4] 0.009 0.009 [Y.sub.23] 0.333 -0.008 Cold bypasses [[X.bar].sub.2] 0.627 0.000 [X.sub.3] -0.005 0.082 [X.sub.4] 0.004 0.004 [[X.bar].sub.34] -0.074 0.914 Table 8. Gain and ns-RGA matrices for HEN#5 [G.sup.G.sub.HEN#5] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.000 0.000 [Y.sub.2] 0.015 -0.019 [Y.sub.3] 0.023 -0.032 [[Y.bar].sub.23] 0.048 -0.067 Cold bypasses [X.sub.1] 0.000 0.000 [X.sub.2] 0.016 0.000 [[X.bar].sub.3] 0.066 -0.163 [X.sub.12] 0.016 0.000 [[LAMBDA].sup.G.sub.HEN#5] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.000 0.000 [Y.sub.2] 0.084 -0.040 [Y.sub.3] 0.211 -0.101 [[Y.bar].sub.23] 0.827 -0.398 Cold bypasses [X.sub.1] 0.000 0.000 [X.sub.2] 0.255 0.000 [[X.bar].sub.3] -0.632 1.539 [X.sub.12] 0.255 0.000 Table 9. Gain and ns-RGA matrices for HEN#6 [G.sup.G.sub.HEN#6] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.000 0.000 [Y.sub.2] 0.007 -0.031 [Y.sub.3] 0.025 0.000 [Y.sub.23] 0.089 -0.031 Cold bypasses [X.sub.1] 0.000 0.000 [X.bar].sub.2] 0.039 -0.166 [X.sub.3] 0.093 0.000 [[X.bar].sub.13] 0.093 0.000 [[LAMBDA].sup.G.sub.HEN#6] T[T.sub.H2] T[T.sub.C4] Hot bypasses [Y.sub.1] 0.000 0.000 [Y.sub.2] -0.001 0.034 [Y.sub.3] 0.025 0.000 [Y.sub.23] 0.288 0.000 Cold bypasses [X.sub.1] 0.000 0.000 [[X.bar].sub.2] -0.023 0.966 [X.sub.3] 0.356 0.000 [[X.bar].sub.13] 0.356 0.000 Table 10. RGA for HEN #1-6 HEN #1 HEN #2 HEN #3 HEN #4 HEN #5 HEN #6 [[lambda].sub.11] 0.962 1.000 1.000 1.000 2.321 1.000 Table 11. [gamma] for each HEN HEN #1 HEN #2 HEN #3 HEN #4 HEN #5 HEN #6 [gamma] 1.42 1.71 2.28 1.69 11.11 1.92 [??] 1.19 1.07 1.61 1.06 0.17 0.90 Table 12. PRGA and its svd [GAMMA] [[SIGMA].sub.ii[GAMMA]] Vr HEN#1 0.962 0.329 1.096 0.623 0.782 -0.111 0.962 0.878 0.782 -0.623 HEN#2 1.000 0.421 1.233 0.630 -0.777 0.000 1.000 0.811 0.777 0.630 HEN#3 1.000 0.000 1.000 1.000 0.000 0.000 1.000 0.000 0.000 1.000 HEN#4 1.000 0.354 1.193 0.643 -0.766 0.000 1.000 0.838 0.766 0.643 HEN#5 2.321 0.943 4.688 0.850 -0.527 3.250 2.321 0.495 0.527 0.850 HEN#6 1.000 0.236 1.125 0.665 -0.747 0.000 1.000 0.889 0.747 0.665 Table 13. Scaled [G.sub.d] for all HENs [G.sub.dHEN#1] T[T.sub.H2] T[T.sub.C4] [d.sub.1] WC[p.sub.H1] 0.34 0.19 [d.sub.2] WC[p.sub.H2] 2.08 0.42 [d.sub.3] WC[p.sub.C3] -1.56 -0.11 [d.sub.4] WC[p.sub.C4] -0.67 -0.91 [d.sub.5] T[S.sub.H1] 0.11 0.06 [d.sub.6] T[S.sub.H2] 1.57 2.16 [d.sub.7] T[S.sub.C3] 1.03 0.01 [d.sub.8] T[S.sub.C4] 0.34 0.71 [G.sub.dHEN#2] T[T.sub.H2] T[T.sub.C4] [d.sub.1] WC[p.sub.H1] 0.33 0.00 [d.sub.2] WC[p.sub.H2] 2.10 0.43 [d.sub.3] WC[p.sub.C3] -1.54 0.00 [d.sub.4] WC[p.sub.C4] -0.83 -0.95 [d.sub.5] T[S.sub.H1] 0.10 0.00 [d.sub.6] T[S.sub.H2] 1.46 2.27 [d.sub.7] T[S.sub.C3] 0.67 0.00 [d.sub.8] T[S.sub.C4] 0.80 0.66 [G.sub.dHEN#3] T[T.sub.H2] T[T.sub.C4] [d.sub.1] WC[p.sub.H1] 1.79 0.00 [d.sub.2] WC[p.sub.H2] 0.06 0.41 [d.sub.3] WC[p.sub.C3] -1.34 0.00 [d.sub.4] WC[p.sub.C4] 0.00 -0.94 [d.sub.5] T[S.sub.H1] 0.85 0.00 [d.sub.6] T[S.sub.H2] 0.23 2.26 [d.sub.7] T[S.sub.C3] 2.00 0.00 [d.sub.8] T[S.sub.C4] 0.00 0.66 [G.sub.dHEN#4] T[T.sub.H2] T[T.sub.C4] [d.sub.1] WC[p.sub.H1] 0.12 0.07 [d.sub.2] WC[p.sub.H2] 2.07 0.30 [d.sub.3] WC[p.sub.C3] -1.45 0.00 [d.sub.4] WC[p.sub.C4] -0.71 -0.90 [d.sub.5] T[S.sub.H1] 0.29 0.18 [d.sub.6] T[S.sub.H2] 1.27 1.96 [d.sub.7] T[S.sub.C3] 1.03 0.00 [d.sub.8] T[S.sub.C4] 0.43 0.79 [G.sub.dHEN#5] T[T.sub.H2] T[T.sub.C4] [d.sub.1] WC[p.sub.H1] 0.00 0.00 [d.sub.2] WC[p.sub.H2] 1.58 0.49 [d.sub.3] WC[p.sub.C3] -0.14 0.00 [d.sub.4] WC[p.sub.C4] -0.71 -1.10 [d.sub.5] T[S.sub.H1] 0.00 0.00 [d.sub.6] T[S.sub.H2] 1.89 2.19 [d.sub.7] T[S.sub.C3] 0.47 0.00 [d.sub.8] T[S.sub.C4] 0.75 0.72 [G.sub.dHEN#6] T[T.sub.H2] T[T.sub.C4] [d.sub.1] WC[p.sub.H1] 0.00 0.00 [d.sub.2] WC[p.sub.H2] 2.19 0.17 [d.sub.3] WC[p.sub.C3] -1.22 0.00 [d.sub.4] WC[p.sub.C4] -0.54 -0.84 [d.sub.5] T[S.sub.H1] 0.00 0.00 [d.sub.6] T[S.sub.H2] 1.54 2.22 [d.sub.7] T[S.sub.C3] 1.04 0.00 [d.sub.8] T[S.sub.C4] 0.49 0.68 Table 14. Disturbance condition number HEN#1 HEN#2 HEN#3 HEN#4 HEN#5 HEN#6 [[gamma].sub.d1] 1.42 1.49 1.00 1.69 - - [[gamma].sub.d2] 1.37 1.60 2.27 1.60 11.02 1.88 [[gamma].sub.d3] 1.34 1.49 1.00 1.53 10.11 1.85 [[gamma].sub.d4] 1.37 1.69 2.28 1.62 9.37 1.62 [[gamma].sub.d5] 1.42 1.49 1.00 1.69 - - [[gamma].sub.d6] 1.37 1.65 2.27 1.59 10.10 1.65 [[gamma].sub.d7] 1.32 1.49 1.00 1.53 10.11 1.85 [[gamma].sub.d8] 1.31 1.71 2.28 1.55 10.48 1.67 Table 15. RDG for all HENs [B.sub.HEN#1] [B.sub.HEN#2] T[T.sub.H2] T[T.sub.C4] T[T.sub.H2] T[T.sub.C4] [d.sub.1] 1.15 0.76 1.00 [infinity] [d.sub.2] 1.03 0.41 1.09 1.00 [d.sub.3] 0.99 0.57 1.00 [infinity] [d.sub.4] 1.41 0.88 1.48 1.00 [d.sub.5] 1.15 0.76 1.00 [infinity] [d.sub.6] 1.42 0.88 1.66 1.00 [d.sub.7] 0.96 15.9 1.00 [infinity] [d.sub.8] 1.66 0.91 1.35 1.00 [B.sub.HEN#3] [B.sub.HEN#4] T[T.sub.H2] T[T.sub.C4] T[T.sub.H2] T[T.sub.C4] [d.sub.1] 1.00 [infinity] 1.22 1.00 [d.sub.2] 1.00 1.00 1.05 1.00 [d.sub.3] 1.00 [infinity] 1.00 [infinity] [d.sub.4] [infinity] 1.00 1.45 1.00 [d.sub.5] 1.00 [infinity] 1.22 1.00 [d.sub.6] 1.00 1.00 1.55 1.00 [d.sub.7] 1.00 [infinity] 1.00 [infinity] [d.sub.8] [infinity] 1.00 1.65 1.00 [B.sub.HEN#5] [B.sub.HEN#6] T[T.sub.H2] T[T.sub.C4] T[T.sub.H2] T[T.sub.C4] [d.sub.1] [infinity] [infinity] [infinity] [infinity] [d.sub.2] 2.61 12.80 1.02 1.00 [d.sub.3] 2.32 [infinity] 1.00 [infinity] [d.sub.4] 3.78 4.42 1.37 1.00 [d.sub.5] [infinity] [infinity] [infinity] [infinity] [d.sub.6] 3.41 5.12 1.34 1.00 [d.sub.7] 2.32 [infinity] 1.00 [infinity] [d.sub.8] 3.23 5.70 1.32 1.00 Table 16. svd of the CLDG of each HEN HEN#1 HEN#2 HEN#3 HEN#4 HEN#5 HEN#6 Maximum effect 9.60 10.56 7.58 9.80 33.22 8.85 Table 17. [G.sub.PDG] for all HENs [G.sub.PDGHEN#1] [G.sub.PDGHEN#2] [G.sub.PDGHEN #3] 9.3 35.0 [infinity] 9.6 6.3 [infinity] 27.1 4.0 4.0 22.9 [infinity] 4.3 [G.sub.PDGHEN#4] [G.sub.PDGHEN#5] [G.sub.PDGHEN# 6] 8.9 [infinity] 7.4 8.8 7.9 [infinity] 25.0 4.2 18.1 12.3 33.7 3.9 Table 18. Perfect and acceptable control for all HENs [[parallel]u[parallel].sub.[infinity]] and [U.sub.min] HEN#1 HEN#2 HEN#3 HEN#4 HEN#5 HEN#6 [d.sub.1] 0.315 0.266 0.488 0.124 0.000 0.000 [d.sub.2] 0.685 0.696 0.253 0.702 1.817 1.061 [d.sub.3] 0.588 0.625 0.366 0.616 0.661 0.601 [d.sub.4] 0.763 0.997 0.588 0.881 1.199 0.790 [d.sub.5] 0.100 0.084 0.232 0.297 0.000 0.000 [d.sub.6] 0.749 0.800 0.786 0.594 6.637 0.886 [d.sub.7] 0.801 0.544 0.545 0.885 0.436 0.031 [d.sub.8] 0.448 0.868 0.411 0.604 0.120 0.699 All d's 6.139 6.648 2.037 6.421 28.897 7.180 Table 19. Resiliency Index for all HENs HEN#1 HEN#2 HEN#3 HEN#4 HEN#5 HEN#6 [d.sub.1] 6.48 8.08 2.61 17.44 [infinity] [infinity] [d.sub.2] 1.18 1.16 6.42 1.16 0.91 0.97 [d.sub.3] 1.64 1.73 3.48 1.74 11.82 1.78 [d.sub.4] 2.67 2.16 2.76 2.45 1.39 2.94 [d.sub.5] 20.31 25.57 5.48 7.28 [infinity] [infinity] [d.sub.6] 1.14 1.10 1.15 1.28 0.58 1.05 [d.sub.7] 2.55 3.95 2.33 2.44 3.40 2.08 [d.sub.8] 3.72 2.48 3.95 3.34 1.54 3.32 all 0.28 0.28 0.61 0.28 0.22 0.27
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|Author:||Young, Brent R.; Tellez, Rodolfo; Svrcek, William Y.|
|Publication:||Canadian Journal of Chemical Engineering|
|Date:||Apr 1, 2006|
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