706
Ind. Eng. Chem. Res. 1997, 36, 706-716
Inventory Control in Processes with Recycle Paul W. Belanger and William L. Luyben* Chemical Process Modeling and Control Research Center and Department of Chemical Engineering, Lehigh University, Iacocca Hall, 111 Research Drive, Bethlehem, Pennsylvania 18015
This paper explores one of the important yet unanswered questions associated with the plantwide control of processes with recycle, namely, how inventory control system tuning affects plantwide regulatory performance. The first portion of this paper presents a simple analytic treatment of the problem using simple transfer function models. In the second portion of the paper, a more realistic reactor/column process is studied. Recommendations are given for tuning inventory controllers in a plantwide environment. We show that tight level control of chemical reactors is not desirable from the standpoint of plantwide regulatory control. Introduction The dynamics and control of individual process units are well understood. Information pertaining to the configuration and tuning of control systems for unit operations that operate independently of each other is readily available. Much of this information can be directly applied to plants consisting of unit operations in series. The effects of a disturbance simply cascade from one unit to the next. What is less clear, however, is how to apply existing techniques to plants containing recycle streams. At present, the dynamics of processes with recycle are poorly understood. Most of the work over the past few decades concerning recycle dynamics has focused on the development of heuristics from the results of simulation studies. This section focuses on presenting some of the pertinent literature in the area of recycle dynamics. It is by no means an exhaustive survey, but it gives the reader a flavor for the state of technology in the field of plantwide control. The first methodology for controlling an entire plant was proposed by Buckley (1964). He proposed that for units in series the inventory control structure be designed first and then the product quality control structure be designed. In this way, the fast and slow plant dynamics are considered at different stages in the control system design. This set of heuristics was widely used for plantwide control for many years to control units in series; however, material and thermal recycle streams presented new problems that needed to be addressed. The traditional approach was to add large surge tanks to the process to attenuate disturbances and, hence, reduce the effect of the recycles on plant dynamics. This practice results in increased capital costs as well as higher safety and environmental hazards. Gilliland (1964) used a highly simplified reactor/ column model to demonstrate that the overall time constant of a process is increased by material recycle. He concluded that, although including material recycle can improve plant economics, it will often make control system performance more sluggish. This was one of the first published studies of the dynamic characteristics of recycle systems. Luyben and Buckley (1977) discussed how liquid level control in a plantwide environment often involves two conflicting objectives. First, the liquid level should be kept close to the desired operating condition. Second, * To whom correspondence should be addressed. Phone: (610) 758-4256. Fax: (610) 758-5297. Email:
[email protected]. S0888-5885(96)00608-2 CCC: $14.00
the flowrate changes out of the tank should be made as smooth as possible so as to avoid upsetting downstream units. The different control objectives suggest different controllers. A proportional-only controller gives gradual outlet flowrate changes; however, it is subject to steadystate offset. A proportional plus integral controller leads to no steady-state offset, but it will cause the outlet flowrate to fluctuate more. Luyben and Buckley demonstrated how a combination of proportional-only feedback control and feedfoward control could be used to obtain gentle control with no offset, thus satisfying both objectives. Verykios and Luyben (1978) explored some of the steady-state and dynamic issues concerning a simple reactor/column system with recycle. They showed that the recycle flowrate varied considerably at steady state for changes in feed conditions. They concluded that the best strategy was to ratio the feeds of the reactants, thus minimizing the possibility of large swings in recycle flowrates. They also demonstrated that the dynamic characteristics of the plant changed considerably for different recycle flowrates. Higher recycle flowrates generally yielded a more underdamped system. Denn (1982) showed that a system with a recycle will have a large characteristic time (larger time constant) and higher steady-state gains than a process without recycle. This conclusion was supported by the work of Kapoor and Marlin (1986). Denn also showed that if a plant with recycle contains time delays, then the transfer function will contain resonant peaks that can be as large as the steady-state gains. It was mentioned that the presence of material recycle makes a plant more sensitive to low-frequency disturbances. This is a direct result of the increase in the plant time constant. Papadourakis (1985) demonstrated that the coupling of a series of units with a recycle stream places any time delays present in the individual units into the denominator of the plant transfer function. This makes the application of traditional Laplace domain analysis difficult because the open-loop plant model will have an infinite number of poles. He stated that frequency domain analysis was of limited applicability for this type of system (primarily due to software limitations at the time the thesis was written). He demonstrated that Laplace domain techniques could be applied if the plant model were simplified with a model order reduction technique. Luyben (1988) discussed a concept that he named “eigenstructure”. This term refers to a control structure that is best at rejecting load disturbances, which, he states, is the most important job of a control system. © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 707
He also stated that variable interaction in a control system is not necessarily undesirable. What matters is how well the control system performs in the face of load disturbances. Thus, designing a control system with the purpose of minimizing interaction will not necessarily lead to the best control system. He went on to state that each process has an intrinsically selfregulating control structure and that the first step of any control system design is to find this structure. Downs (1992) presented a methodology for designing a quality control system in a plantwide environment. He noted that the operation of the separations system is central to the control of component inventories in a process. He suggested that the development of the control system for the separations system be consistent with the overall plantwide operating objectives. Several examples were presented, illustrating the necessity of designing the quality control system in such a fashion as to allow each component inventory in a plant to be self-regulating. This is an important point to consider when designing a column control system for plants containing recycle streams. Some control systems that appear effective for single unit operations may fail in a plantwide environment. In a series of papers Luyben (1993a-c, 1994) demonstrated that the behavior of a system with recycle is more strongly dependent on the overall gain of the recycle than on the dynamics of the units within the recycle. He explored the tradeoffs between steady-state design and controllability using simulations of reactorseparator systems. He reported on the phenomenon he called “snowballing” (high sensitivity of recycle flowrates to changes in loading). He concluded that this snowballing effect could be prevented if one stream within the recycle loop is fixed. Price and Georgakis (1993) proposed an extension of Buckley’s methodology for the design of plantwide control structures. Price recommended that the design of a plantwide control system be broken down by function (i.e., material balance, quality control, overrides, online optimization, etc.). In this way the “plantwide” nature of the design problem is preserved. It was demonstrated through an exhaustive set of case studies of a CSTR/stripper process that the design of the inventory control system has a significant impact on the performance of the quality control system. A set of heuristic guidelines was proposed for the selection of an inventory control structure and a quality control structure in a plantwide environment. Yi and Luyben (1995) demonstrated the application of a method developed by Luyben (1975) to the problem of selecting possible control structures in the design of a plantwide control system. The method is called the steady-state disturbance sensitivity analysis. In short, the method involves the use of a rigorous steady-state simulator to determine how far the manipulated variables have to move at steady-state in order to compensate for various changes in load disturbances. If excessive manipulated variable moves are seen as a result of load changes, then the control structure is poor. Lyman (1995) presented a group of heuristics that can be used by design engineers as guidelines to improve the controllability of plants with material recycle. This set of heuristics was developed from a systematic study of the dynamic characteristics of several process designs involving recycle streams. He found that the controllability of plants with recycle improves when the size of the reactor(s) is increased. Larger reactors tend to
Figure 1. Diagram of a typical plant with recycle.
be better at filtering out disturbances. He found that light recycles should be kept small. This is because recycle streams introduce positive feedback into the plant. There is still some uncertainty concerning how the inventory control system and the quality control system of a process with recycle interact. This paper focuses on the effects of inventory control tuning on plantwide regulatory performance. Specifically, the tradeoffs between inventory control and quality control are explored. The first portion of this paper presents a simple analytic treatment of the problem using simple transfer function models. The second portion of the paper presents a study involving a more realistic reactor/ column process. Heuristics are developed for tuning inventory controllers in a plant with recycle. Effects of Inventory Control System Tuning The problem of plantwide control is very complex. In order to treat it analytically and still maintain generality, the problem must be viewed in its simplest form. Figure 1 presents a simple view of a typical chemical plant. The plant consists of two main sections: the reaction section and the separation section. The sections of the plant are connected by two streams, the reaction system effluent stream and the recycle stream. A fresh feed is introduced either to the reaction section or to the separation section. The location of this stream is determined during the design phase of the plant. A product stream is withdrawn from the separation section. The major goal of a plantwide control strategy is the regulation of product quality (denoted in Figure 1 by XB). Thus, the operation of the separation section plays a central role in the plantwide control problem. The effects of disturbances to the separation section are amplified by the presence of material recycle. When a disturbance is introduced to the separation section, the quality control and inventory control systems in the separation section respond by changing flowrates in the separation section. This results in a change in the conditions of the recycle stream (denoted by D and XD in Figure 1). The change in recycle conditions presents a disturbance to the reaction section of the plant. The control system in the reaction section responds by adjusting flowrates in the reaction section, and this causes a change in the conditions of the reaction section effluent stream (denoted by F and XF in Figure 1). This,
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in turn, has an impact on the operation of the separation section. This illustrates an important concept in the field of plantwide control: A control system can never eliminate a disturbance from a plant; it can only shift the effects of a disturbance from one location to another. With this in mind, it is natural to assume that if the plantwide control system is designed properly, the effects of disturbances can be shifted from critical variables such as product composition to not so critical variables such as vessel holdups. As will be shown, the tuning of the inventory control system is a useful tool for accomplishing this task. In the discussion that follows, it is assumed that the quality control system in the separation section is designed and operating in automatic mode. The tuning of this control system is assumed tight (the tuning of the quality control system must be tight in order to regulate product qualitysstability considerations are used to determine the tuning of this system). General Dynamic Relationship Let the effects of changes in F and XF on recycle flowrate D and composition XD be given by transfer function matrix GSr. This transfer function is a closedloop regulator transfer function. Its form is governed by the design of the separation section as well as the structure and tuning of the inventory and quality control systems within the separation section. For convenience, the transfer function matrix GSr is partitioned into four sections (as shown by eq 1) to emphasize the separate effects of changes in composition and changes in flowrates.
[ ] [] [
g11 D F Sr ) G ) Sr X XD F g21 Sr
][ ]
g12 Sr F XF g22 Sr
(1)
Let the effects of changes in recycle on reaction section effluent conditions be defined similarly as GRr. Again, the transfer function matrix is partitioned as is shown by eq 2.
[] [ ] [ F D XF ) GRr XD )
g11 Rr g21 Rr
g12 Rr g22 Rr
][ ] D XD
(2)
A disturbance can be introduced to the system at many locations. The effects of a disturbance on product quality depend on the point at which it is introduced to the plant; however, due to the nature of the coupling between the units, there is one characteristic determining the effectiveness of the quality control structure that is independent of the point at which the load is introduced to the plant. This characteristic describes the effects of the presence of material recycle. Consider the case where the fresh feed is introduced to the reaction section of the plant. The block diagram of this system is given in Figure 2. If the effects of fresh feed flowrate (F0) and composition (Z0) on reactor effluent conditions are given by transfer function matrix GRs and the effects of changes in the reactor effluent stream on product quality are given by transfer function matrix GSs, then the effects of changes in fresh feed conditions on product quality are given by eq 3.
[ ]
F X ) GSs[I - GRrGSr]-1GRs Z0 0
(3)
Figure 2. Block diagram of plant dynamics.
The matrix inverse that appears in eq 3 characterizes the effects of the presence of material recycle. It can be shown that this term appears in the overall transfer function matrix regardless of the location and type of load disturbance that is introduced to the plant. For different types of load disturbances, the matrices GRs and GSs remain as well; however, the meaning and form of these matrices are dependent upon the nature of the load disturbance. It is interesting to note that the product of the terms GRs and GSs of eq 3 describes the effects of load disturbances on the product quality in the absence of material recycle (this characterizes the effect of the unit operations operating in series, hence the subscript “s”). The entire effect of the recycle is contained in the matrix inverse term. This term determines to what extent the effects of load disturbances are amplified by the presence of material recycle. The matrix inverse term of eq 3 can be rewritten as
[I - GRrGSr]-1 )
Adj[I - GRrGSr] |I - GRrGSr|
(4)
where the numerator is the adjoint of the matrix and the denominator is the determinant. After partitioning the matrix transfer functions as shown in eqs 1 and 2 and applying Schur’s formula, the following expression is obtained for the determinant in eq 4: 11 12 21 21 12 |I - GRrGSr| ) |I - g11 RrgSr - gRrgSr ||(I - gRrgSr 22 21 11 22 21 11 11 g22 RrgSr) - (-gRrgSr - gRrgSr )(I - gRrgSr 21 -1 11 12 12 22 g12 RrgSr ) (-gRrgSr - gRrgSr )| (5)
Simplifying Assumptions Equations 3-5 together give a general linear model of plant dynamics. In their present form they are not very useful for the analysis of the effects of inventory control system tuning in a plantwide environment; however, simplifying assumptions can be used to reduce these equations to a form that is useful for analysis. This section demonstrates how the equations can be reduced for the case of liquid-phase reactions. This, of course, limits the applicability of the following treatment; however, other assumptions can be made that would allow eqs 3-5 to be used to study the effects of inventory control in a plant where gas-phase reactions take place. It is assumed that the reaction taking place in the reaction section of the plant is equimolar, allowing
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g12 Rr to be set to zero. This assumption is not necessary if flowrates are given on a volumetric basis (the density of the reactor effluent stream for the case of a liquidphase reaction is roughly constant); however, it simplifies the treatment by allowing a molar basis to be used. Another assumption is that it takes significantly longer for composition changes in the recycle stream to propagate through the system than flowrate changes in the recycle stream. Again, this is justified by the assumption of a liquid-phase reaction. This means that composition fluctuations in the recycle stream are filtered out to a greater extent by the reaction section than flowrate fluctuations. Assuming that the compositions of the recycle stream do not have an impact on the flowrate of the reaction section effluent stream (g12 Rr ) 0), eq 5 can be rewritten as
|
11 21 12 22 22 |I - GRrGSr| ) |I - g11 RrgSr | I - gRrgSr - gRrgSr 11 (-g21 RrgSr
|
21 11 12 - g22 RrgSr )(-gRrgSr )
1-
11 g11 RrgSr
X) (6)
11 Equation 6 consists of two parts. The |I - g11 RrgSr | part describes the effects of flowrate recycling. The other terms represent the effects of composition recycling. It is assumed that composition recycling in the plant occurs at a much slower rate than flowrate recycling. The effects of composition recycling therefore do not add significantly to the magnitude of the determinant in eq 4 (at least not in the frequency range of importance). For this reason, the determinant in eq 4 can be approximated by:
11 |I - GRrGSr| ≈ |I - g11 RrgSr |
1 G L 11 Ls 1 - g11 RrgSr
(8)
Equation 8 describes the effects of disturbances on product quality. The equation consists of two parts. The first part is a scalar that describes the amplification of the effects of every load disturbance on every product quality variable due to the presence of the material recycle loop in the plant (specifically due to the presence of flowrate recycling). This term is independent of the nature and location of the load disturbance. The second term is a matrix that contains all other factors that determine the effects of specific load disturbances on specific product quality variables. This term is dependent upon the nature and location of the disturbance as well as the design of each section of the plant and the structure and tuning of the plant control system. The first term, being more general, is the focus of further study. It is referred to as the “recycling function” for the remainder of the paper. Reactor Inventory Control Tuning As an example application of eq 8, consider the control of reactor inventory in a plantwide environment. A source of considerable debate in the field of plantwide control is the type of controller and the tuning that
[
]
1 GLsL 11 1 - g11 RrkSr
(9)
Equation 9 can be used to explore the effects of the type and tuning of the inventory control system in the reaction section on quality control. Consider the case where the reaction section consists of a single liquid phase CSTR. Let the holdup (VR) be controlled by a proportional-only controller with gain Kc manipulating the flowrate of the reactor effluent stream (F). For this case, g11 Rr is given by eq 10.
g11 Rr )
(7)
If GRs, GSs, and the adjoint of the matrix inverse term of eq 3 are grouped into a single term (GLs) and eq 7 is used to approximate the determinant of the matrix inverse, then eq 3 can be rewritten as
X)
should be used. Proponents of proportional-only (P) inventory control claim that proportional-integral (PI) control does not provide adequate “flow-smoothing” in a plantwide environment. Advocates of PI control claim that the steady-state superiority of the PI controller (no steady-state offset) justifies its use. Aside from the question of which type of controller to use, the question of tuning remains unanswered. Regarding the question of level control of liquid-phase reactors, the common practice is to try to achieve tight level control. The discussion below will challenge this conventional wisdom. Assume that the dynamics of the separation section are fast when compared to the dynamics of the reaction section (the separation inventory control system is tuned tightly). In this case, the transfer function g11 Sr can be approximated by its steady-state gain k11 (the recycle Sr gain). Equation 8 can then be rewritten as
1 1 s+1 Kc
(10)
Note that this is the closed-loop transfer function relating input D to output F with the level controller an automatic. When eq 10 is substituted into eq 9, the following expression is obtained for the recycling function:
[
]
1 s+1 K 1 c GLsL X) 1 1 - k11 Sr s + 1 Kc(1 - k11 Sr )
(11)
It is interesting to study the shape of this recycling function. The gain of the function is given by:
Kcycle )
1 1 - k11 Sr
(12)
This equation shows that, as the recycle gain k11 Sr increases, the amplification of a load disturbance caused by flowrate recycling increases. This is an important generalization. It shows that if a plant and its control system are designed to yield small values of the recycle gain (k11 Sr ≈ 0), then the effects of the recycle on product quality will be minimized. This supports the findings of Luyben (1994). The recycling function has two breakpoint frequencies. These are given by eq 13.
ωbreak1 ) Kc(1 - k11 Sr )
ωbreak2 ) Kc
(13)
710 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
Figure 3. Effects of reaction system inventory control tuning on the recycling functionsP-only control.
For very small values of the recycle gain (k11 Sr ≈ 0) the breakpoint frequencies are nearly the same. For higher values of the recycle gain, the breakpoint frequencies are separated. Figure 3 shows the shape of the recycling function for a recycle gain of half (k11 Sr ) 0.5) and different values of controller gain Kc. For each controller tuning, the steady-state gain of the recycling function is the same. Note that for all possible values of the recycle gain (between 0 and 1), the steady-state gain is greater than 1. Also, as the frequency increases, the magnitude of the recycling function approaches unity (regardless of the values Kc and k11 Sr ). This shows that the presence of the recycle loop always increases the effects of a disturbance on product quality. It is important to note that for frequencies below 0.1 Kc(1 k11 Sr ) the full effects of the flowrate recycling on product quality are seen, whereas for frequencies higher than 10Kc they are not. This shows that Kc is a parameter that if chosen properly can significantly reduce the effects of material recycle on product quality. Now consider the case where PI control is used to regulate the holdup of a single liquid phase CSTR. For this case g11 Rr is given by eq 14.
g11 Rr )
Kc(τIs + 1)
(14)
2
τIs + Kc(τIs + 1)
When eq 14 is substituted into eq 9, the following expression is obtained for the recycling function:
X)
[
τIs2 + KcτIs + Kc
]
11 τIs2 + KcτI(1 - K11 Sr )s + Kc(1 - kSr )
GLsL
(15)
Although the form of the recycling function in eq 15 is more complicated than the recycling function in eq 11, it can be seen that the two recycling functions share the same characteristics. The steady-state gain of the recycling function is still given by eq 12. At high frequency, the magnitude of the recycling function is still unity. Thus it can be seen that the low-frequency and high-frequency effects of the presence of material recycle are the same regardless of whether P or PI control is used. What does change is the behavior of the recycling function in the intermediate frequency ranges. Figure 4 shows the form of the recycling
Figure 4. Effects of reaction system inventory control tuning on the recycling functionsPI control.
function for three different values of Kc (Kc ) 1.0, top; Kc ) 10.0, middle; Kc ) 100.0, bottom) and reset times selected to yield three different values of the damping coefficient (ζ ) 2.0, solid curve; ζ ) 1.0, dashed curve; ζ ) 0.4, dash-dotted curve). The relationship between controller reset time, controller gain, and damping coefficient is given by eq 16. The value of the recycle gain used is 0.5.
τI ) 4ζ2/Kc
(16)
A comparison between Figures 3 and 4 shows that the effects of changes in Kc are the same for both P and PI control. At frequencies lower than 0.1Kc(1 - k11 Sr ), the amplification of a load disturbance caused by flowrate recycling is roughly equal to the steady-state gain of the recycling function. Frequencies higher than 10Kc are not affected by flowrate recycling. This is independent of the damping coefficient of the inventory control loop. Studies performed for a wide range of recycle gains verified this fact. Figure 4 shows that the behavior of the recycling function for a PI inventory controller at intermediate frequencies is strongly affected by the selection of the damping coefficient. For high values of the damping coefficient, the recycling function for a PI controller resembles that of a P-only controller. As the damping coefficient decreases, the recycling function develops a resonant peak which amplifies the effects of flowrate recycling on product quality at intermediate frequencies. The only way to compensate for the effects of the resonant peak on product quality is to reduce the controller gain. This moves the peak to a lower frequency where the integral action of the product quality control system can effectively deal with the disturbance. Detuning the inventory controller in this fashion increases the magnitude of reactor holdup fluctuations at low frequencies. This defeats the purpose of using the PI inventory controller. Thus, it can be seen that the presence of a resonant peak in the recycling function is undesirable. Figure 4 shows that even an inventory control system
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that is tuned to be critically damped (traditionally thought of as conservative tuning) results in a significant resonant peak. A damping coefficient of 2.0, however, tends to produce a resonant peak of negligible magnitude. Studies were performed that verified this fact for a wide variety of recycle gains. Thus, for PI inventory control, the value of τI that should be used is 16/Kc. The question that needs to be answered at this point is, how should the gain of the inventory controller be selected? The answer depends on the economics of the process. As stated before, a control system can never eliminate a disturbance from a plant; it can only shift the disturbance from one location to another. So, the improvement in quality control must be paid for by sacrificing the control of some other process variable. For the current case, this variable is reactor holdup. As the gain of the reactor inventory controller is decreased, the fluctuations of the reactor holdup increase in magnitude. This means that a larger reactor must be used. Therefore, when choosing the gain of the reactor inventory controller, the increased revenue from tighter quality control must be weighed against higher capital costs due to reactor oversizing. Consider the case where a P-only controller is used to regulate reactor inventory. Assume the disturbance that impacts the liquid level in the reactor the most is a fresh feed flowrate change. The control law governing the flowrate of F can be written as
∆VR )
τIs
F0 (18)
11 τIs + KcτI(1 - k11 Sr )s + Kc(1 - kSr ) 2
The peak closed-loop log modulus from fresh feed flowrate to reactor volume can be calculated numerically from eq 18. If the worst case fresh feed flowrate disturbance is assumed to be a pure sine wave of a given amplitude at the frequency corresponding to the peak log modulus, then the required reactor volume can be estimated. In cases where reactor costs dominate process economics, it is desirable to have tight reactor inventory control. The opposite extreme is the case where product quality considerations dominate process economics. In this case, it is desirable to remove the effects of flowrate recycling on all the quality variables that have a significant economic impact. For all cases between these two extremes, it is necessary to choose the inventory controller gain that gives the best tradeoff between the two economic effects. For the case where it is desirable to minimize fluctuations in product quality, it is possible to generate a simple rule for the selection of Kc. Suppose that the effect of a particular load disturbance on a particular product quality variable is to be minimized. Let the closed-loop regulator transfer function from load disturbance to quality variable be given by eq 19.
]
1 s+1 K 1 c gijLLj xi ) 11 1 1 - kSr s+1 Kc(1 - k11 Sr )
∆D ) ∆F - ∆F0 The recycle gain gives another relationship between D and F:
∆D ) k11 Sr ∆F These three equations can be combined to yield an expression for the change in reactor holdup at steady state that can be expected for a given change in fresh feed flowrate:
1 ∆F0 Kc(1 - k11 Sr )
VR )
[
1 ∆F Kc
A material balance around the reactor yields
∆VR )
the effects of changes in fresh feed flowrate on reactor holdup are given by eq 18.
(17)
Equation 17 shows that, for a given change in the fresh feed flowrate to the plant, the change in reactor holdup increases as the value of Kc decreases. This comes as no surprise since lower values of Kc indicate less aggressive control of reactor inventory. Equation 17 also shows that the expected increase in reactor holdup increases as the recycle gain increases. Note that only the steady-state value of the change in reactor holdup is considered here. This is because, for fresh feed flowrate fluctuations, the largest variations in reactor holdup are seen at zero frequency (steady state). This is a direct result of using proportional-only inventory control. For the case of PI inventory control, there is no simple analytic expression such as eq 17 to determine the required size of the reactor. For this case it is necessary to determine the peak value of the closed-loop regulator log modulus curve numerically. It can be shown that
(19)
Let the effects of the load variable on the quality variable be characterized by the peak log modulus of the closed-loop regulator transfer function (this corresponds to the frequency component of the load disturbance that excites the closed-loop system the most). Figures 3 and 4 show that, as the value of Kc is reduced, the effects of flowrate recycling are moved to lower frequencies. The point at which a reduction in Kc no longer brings a significant improvement in quality control corresponds to where the frequency of the peak in the closed-loop regulator log modulus is roughly 10 times larger than the higher breakpoint frequency of the recycling function. A reduction of Kc past this point does, however, continue to increase the required size of the reactor, increasing capital costs. Thus, the best reactor inventory controller gain for this case is given by:
Kc ≈ ωpeak/10.0
(20)
Figure 5 demonstrates the improvement in quality control that can be expected by the use of eq 20. A log modulus plot of a typical closed-loop regulator transfer function (with perfect reactor inventory control) is shown as a solid curve. The frequency corresponding to the peak is 7.6 rad/h. The log modulus curve of the same system using a Kc of 152 is shown as a dashed curve. Notice that there is very little improvement in the region of the peak regulator log modulus. This is because the value of Kc was selected to place the lower breakpoint frequency of the recycling function at a point that is 10 times higher than the frequency of the peak
712 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
capital cost of the plant. The benefits of improved product quality must therefore be weighed against increased capital cost. Thus, it can be seen that the aggressiveness of the inventory control system is an economic variable. The tuning of the inventory control structure should take process economics into account. Application to a Realistic Reactor/Stripper Process
Figure 5. Effects of reaction system inventory control tuning on product quality control.
log modulus. A third curve, shown as a dash-dotted line, represents the log modulus curve of the system when a Kc of 0.76 is used. This corresponds to the case where eq 20 was used to place the upper breakpoint frequency of the recycling function at a point 10 times lower than the frequency of the peak log modulus. The frequency corresponding to the peak regulator log modulus can be found analytically or through the application of the LATV method of Belanger and Luyben (1996). For the case where several load/output pairs are of interest, the frequency that should be used in eq 20 is the lowest peak frequency. The application of this method is not limited to the case of a single liquid phase CSTR or to the case where the dynamics of the reaction section are considered slow with respect to the separation section. The method can be adapted to other cases by determining the form of the recycling function with respect to inventory control system tuning parameters. Results should be similar to those given in this section. That is, flowrate recycling amplifies low-frequency components of a load disturbance but not high-frequency components. The frequencies that are amplified by flowrate recycling are determined by the aggressiveness of the inventory controllers within the recycle loop. Summary of Conclusions What can be concluded from this section is that both the recycle gain and the aggressiveness of the inventory control system have a significant impact on the overall performance of the quality control system. The recycle gain determines to what extent the material recycle amplifies the effects of low-frequency components of a disturbance on product quality. The gain of the inventory control system determines the range of frequencies of the load disturbance that are amplified by the presence of the material recycle. The effects of disturbances on product quality can only be reduced to a certain extent by reducing the gain of the inventory controller. If PI control is used, the value of the reset time should be selected to yield a damping coefficient of 2.0. This keeps the effects of the resonant peak in the recycling function to a minimum. The improvement in quality control that is achieved by the relaxation of reaction section inventory control is not without its price. Larger reactors are required to handle larger fluctuations in reactor inventory. This increases the
In order to show how the results of the previous section apply to more complex processes, a reactor/ stripper system is investigated. This system is illustrated in Figure 6. A first-order irreversible chemical reaction (A f B) occurs in a single continuous stirred-tank reactor. Some of the reactant A is converted into product B. The reactor effluent stream is fed to a stripper. Nearly pure product B is removed from the bottom of the stripper. The overhead product of the stripper (mostly unreacted A) is recycled back to the reactor. Fresh feed consisting of reactant A and some product B is fed to the reactor at a given flowrate F0 and composition zA0. The plant is designed around a fixed feed composition of zA0 ) 0.9. The balance is made up of product B. The physical properties of the components are the same except for the relative volatilities. The reaction rate law may be written as
R ) VRkxAF where R is the rate of reaction (lb‚mol/h), VR is the holdup of the reactor (lb‚mol), k is the reaction rate constant (h-1), and xAF is the mole fraction of A in the reactor. From the column calculations, constant relative volatilities of RA ) 2, RB ) 1, equimolar overflow, theoretical trays, and a saturated liquid feed are assumed. A partial reboiler and a total condenser are used. Trays are numbered from the top down. The production rate is fixed at B ) 239.5 lb‚mol/h at a desired purity xAB ) 0.0105. The reactor is run isothermally at a temperature that yields a specific reaction rate constant of k ) 0.34 h-1. The reactor effluent has a flowrate of F lb‚mol/h and a composition of xAF. The vapor boilup rate in the stripper is designated as V. Liquid from the accumulator is recycled to the reactor at a rate of D lb‚mol/h and composition xAD. It can be shown that there are two design degrees of freedom associated with this process (reactor holdup and the number of stripping trays). For each design an economic performance measure (in this case the discounted cash flowrate of return or DCFROR) can be calculated. The DCFROR is the rate of return generated by the investment made in the project. The higher the DCFROR, the more desirable the economics of the project are. A detailed description of the DCFROR is given by Douglas (1988). The terms involved in the application of this equation (onsite costs, revenue, utility costs, and raw material costs) are calculated from the following set of correlations and assumptions: 1. For the calculation of column diameter, an F factor of 1.0 is assumed. The diameter of the column can be calculated from F factor ) VfaceFv0.5. Assuming a molecular weight of 50 lb/lb‚mol, a pressure of 44.7 psia, and a temperature of 760 °F, the diameter of the column can be calculated: Dc ) 0.1838V0.5.
Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 713
Figure 6. Reactor/stripper system.
2. The column height can be calculated by assuming a 2-ft tray spacing and allowing 20% more for base level volume: Lc ) 2.4NT. 3. A diameter to height ratio of 0.5 is assumed for calculation of reactor dimensions. 4. Reboiler and condenser heat-transfer rates are determined from the value of V using a heat of vaporization of 250 Btu/lb. The reboiler area Ar is calculated assuming an overall heat-transfer coefficient U ) 100 Btu/(h °F ft2) and a log mean temperature difference of 50 °F. Condenser area Ac is calculated using an overall heat-transfer coefficient U ) 150 Btu/(h °F ft2) and a log mean temperature difference of 20 °F. 5. Energy costs are calculated at a rate of $5/106 Btu. 6. Capital costs of the reactor, column, and heat exchangers are estimated using correlations given by Douglas (1988):
reactor cost ) (3.0)(1917)Dr1.066Lr0.802 column tray cost ) 36.36Dc1.55NT column shell cost ) 1917Dc1.066Lc0.802 heat exchanger cost ) 1557[Ac0.65 + Ar0.65] 7. Materials of construction are stainless steel. Design pressures are 300 psig. A Marshall and Swift index of 800 is used. 8. Raw material cost is calculated on a basis of $20/ lb‚mol of fresh feed F0. 9. Revenue is calculated on a basis of $23/lb‚mol of the product stream B. 10. The plant is assumed to operate for 8400 h/yr.
Table 1. Optimal Steady-State Design no. of trays zA0 F0 (lb‚mol/h) xAF F (lb‚mol/h) xAD DC
15 0.9 239.5 0.386 805.6 0.545
D (lb‚mol/h) xAB B (lb‚mol/h) V (lb‚mol/h) VR (lb‚mol)
Reactor and Stripper Diameters (ft) 4.4 DR
reboiler
Heat-Exchanger Areas 1415 condenser
reactor cost col. shell cost
566.1 0.0105 239.5 566.1 1623
10.1
(ft2)
Capital Costs ($1000) 752.1 col. tray cost 163.6 exchanger cost
2359 5.4 416.3
Utility and Raw Material Costs ($1000/yr) energy cost 297.2 raw material cost 40236.0 revenue ($1000/yr)
46271.4
DCFROR
0.46415
11. A 10-yr plant life is assumed. Table 1 summarizes the optimal steady-state design for the assumptions given above. The flowrate of the recycle stream is used to control the inventory of the accumulator. The bottoms product flowrate is used to control the inventory of the column base. Pressure is controlled through the manipulation of the cooling water flowrate to the condenser. Perfect control is assumed for all three of these loops. A PI controller manipulates the flowrate of steam to the reboiler of the stripper in order to regulate the purity of the bottoms product. A 3-min deadtime is associated with the measurement of product composition. TL (Tyreus and Luyben, 1992, 1993) settings were determined for this loop from the results of a relay-feedback test (Kc ) 0.1875, τI ) 0.4930 h). A proportional-only controller manipulates the flowrate of the reactor ef-
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Figure 8. LATV test results.
Figure 7. Closed-loop response of plant to designed disturbance.
fluent stream in order to regulate reactor inventory. The effects of the tuning of this controller on plant economics are the focus of the study in this section. In order to quantify the load rejection properties of this system on an economic scale, the capacity-based economic approach of Elliott and Luyben (1995, 1996) is used. This approach uses the frequencies of the peak closed-loop regulator log moduli to construct a “worst case” load disturbance (a signal designed to excite the peak frequencies in the plant). Acceptable bounds are chosen for product quality, and the plant is simulated with the worst case load disturbance. The revenue of the plant is reduced in proportion to the fraction of product that is outside of the specification band (“onaim” control). The peak closed-loop regulator log moduli for this process were determined through the use of the LATV method of Belanger and Luyben (1996). Results are shown in Figure 7 for several values of reactor inventory control gains. Here LM1 corresponds to the peak log modulus from fresh feed flowrate disturbances to product composition. LM2 corresponds to the peak log modulus from fresh feed composition to product composition. The value of the recycle gain for the separation section was determined experimentally to be k11 Sr ) 0.703. Both curves in Figure 7 are very similar. The shape of each curve is determined by the recycling function shown in Figure 3. For very high values of Kc (aggressive inventory control), the low breakpoint frequency of the recycling function is above the frequencies corresponding to the peak closed-loop regulator log moduli. In this region the full effects of flowrate recycling are seen. For very small values of Kc (loose inventory control) the high breakpoint frequency of the recycling function is below the frequencies of the peak log moduli. In this region, none of the effects of flowrate recycling are seen. Each curve has two breakpoints that mark the regions where changes in Kc have a significant impact on quality control performance. According to eq 12, the contribution of flowrate recycling at low frequencies to each disturbance is 10.54 dB (the recycle gain is 0.703). Figure 7 shows that the
maximum improvement in the peak log modulus for the fresh feed composition disturbance is close to this value. The maximum improvement in the peak log modulus for fresh feed flowrate disturbances, however, is much larger (approximately 33 dB). Each of the load disturbances affects the system in a different way. The fresh feed composition disturbance has a direct effect on reactor composition, but it does not have a direct effect on reactor level. Thus, the fresh feed composition disturbance does not directly affect the flowrate of the reactor effluent stream. A fresh feed flowrate disturbance, however, directly affects both the reactor level and reactor composition. Since the proportional-only controller immediately compensates for a change in the reactor level, the fresh feed flowrate disturbance has a direct effect on the flowrate of the reactor effluent. Smaller values of Kc tend to make the effects of fresh feed flowrate disturbances on reactor effluent flowrates less severe. This makes the effects of Kc on fresh feed flowrate disturbances stronger. To illustrate the application of the Elliott method, consider the case where Kc ) 10.0. Assume that the peak fluctuations in the fresh feed flowrate are 30 lb‚mol/h and that the peak fluctuations in the feed composition are 10 mol %. The specification band for product purity is (0.25 mol %. Assume that all offspec material is worth 95% of what onspec material is worth (either because it will sell for less or because it must be reprocessed). Figure 8 shows the load disturbances that were designed from the peak frequencies found via LATV tests as well as the closed-loop response of the plant to these disturbances. The fluctuations in fresh feed flowrate and fresh feed composition are shown on the left. The time profile of the product composition is shown in the upper right. The specification bands on product purity are shown as dashed lines. The value of the vapor boilup rate vs time is shown in the lower right figure. The percentage of offspec material calculated from the results of this simulation was 26.3%. Thus, the revenue predicted by the steady-state model should be reduced by 1.3%. Equation 17 was used to estimate the value of the reactor holdup that would be required in order to increase the fresh feed flowrate by 30 lb‚mol/h. For a Kc of 10.0, the reactor size must be increased by 10.1 lb‚mol beyond the steady-state value calculated. Thus, the reactor must be sized to hold at least 1633.1 lb‚mol. Taking into account the amount of offspec product that is likely to be generated and the required oversizing of the reactor, a more realistic value
Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 715
Figure 9. Economic impact of reaction system inventory control tuning.
of the DCFROR can be calculated. For this case the DCFROR is 0.424 84. It can be seen that dynamic performance can have a significant impact on the profitability of a plant. For this system, the steady-state model predicts a return on investment of 46.4%. Taking dynamics into account, it can be seen that the actual return on investment is more likely to be 42.5%. Figure 9 summarizes the effects of Kc on the overall profitability of the process. The top portion of the figure illustrates how the required reactor size is affected by the choice of Kc. For large values of Kc, the required reactor size is very close to the optimal steady-state holdup. For lower values of Kc, it is necessary to oversize the reactor significantly. This increases the overall capital cost of the plant. The middle portion of the figure illustrates the impact that Kc has on quality control. For high values of Kc, a significant portion of product violates product purity specifications. This decreases the amount of revenue generated by the plant. As the value of Kc is decreased, the amount of offspec material decreases until a point is reached where all of the material produced is onspec. For all values of Kc below this point, 100% of the product is onspec. The bottom portion of Figure 9 shows how the rate of return on investment changes with respect to Kc. As a result of the tradeoffs between capital cost (reactor oversizing) and revenue (product quality), the DCFROR vs Kc curve exhibits a maximum. The best value of Kc is the one that maximizes the rate of return on investment. As can be seen in Figure 9, the best value of Kc for this plant is approximately 5.0. This is the highest gain that could be selected such that 100% of the product would be onspec. The procedure followed above for the selection of inventory control gain can be fairly time consuming. It would be very useful to have some form of tuning heuristic. It has been noted that the controller tuning that yields the best quality control performance is given by eq 20. Therefore, the first step for determining the gain of the inventory controller is to determine the frequencies of the peak closed-loop regulator log moduli
from load disturbances to all outputs that have an economic impact. This can be accomplished by performing LATV tests with a high inventory control gain (it is assumed that the peak frequencies are not strongly influenced by inventory control gain). The gain that yields the best quality control performance will be 10 times lower than the smallest peak frequency. The second step is to determine the required reactor volume for this controller gain using assumed maximum bounds on feed flowrate fluctuations and either eq 17 or eq 18. If the expected fluctuations in reactor volume for this value of the controller gain are too large, then the gain of the inventory controller must be increased until the magnitudes of holdup fluctuations are acceptable. As an example of the above heuristic, consider the design of a PI inventory controller for the CSTR/stripper process. With an inventory control gain of 100.0 (τI selected to yield a damping coefficient of 2.0), LATV tests are performed to yield peak frequencies of 7.04 and 1.82 rad/h for feed flowrate and feed composition disturbances, respectively. According to eq 20, the inventory control gain that will minimize fluctuations in product purity is 0.18. Oversizing the reactor by 100 lb‚mol results in a half percent reduction in return on investment. Assume that 100 lb‚mol is the largest tolerable fluctuation in reactor holdup. Also, assume that the bounds on fresh feed flowrate fluctuations and fresh feed composition fluctuations are 30 lb‚mol/h and 10 mol % as before. For an inventory control gain of 0.18, eq 18 predicts holdup fluctuations of 561 lb‚mol. This means that the inventory control gain must be increased. An inventory control gain of 1.01 is high enough to keep inventory fluctuations below 100 lb‚mol. Therefore, the inventory control gain is selected to be 1.01. A reset time of 15.8 is used to yield a damping coefficient of 2.0. New LATV tests are performed with the new inventory control tuning. Results give peak log moduli of LM1 ) -91.73 dB at ω1 ) 1.92 rad/h and LM2 ) -51.87 dB at ω2 ) 0.97 rad/h. With the same bounds ((0.25 mol %) on product purity, the Elliott method predicts that none of the product produced will be offspec for the given inventory tuning. The predicted value of the rate of return on investment with dynamics taken into account is 46.0%. This is very close to the optimal value (46.3%) of the discounted cash flow obtained from a direct search for the best proportionalonly controller tuning. Thus, it can be seen that the heuristic does a fairly good job. It should be noted that the heuristic only locates the best tuning of the inventory controller approximately. Further adjustments in the tuning parameters may result in higher process profitability. Conclusions This study of the effects of inventory control tuning on plantwide regulatory performance has illustrated several important results. The amplification of the effects of load disturbances on controlled variables caused by the presence of material recycle is mainly due to flowrate recycling. The two factors that dictate the effects of flowrate recycling are the recycle gain and the tuning of the inventory controllers within the recycle loop. A bode plot of the recycling function reveals that, below certain frequencies, the full effects of flowrate recycling are seen, while above certain frequencies, none of the effects are seen. Only the recycle gain affects the amplification at low frequencies caused by material recycle. The gains of the inventory controllers in the
716 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997
recycle loop affect only the breakpoint frequencies of the recycling function. These breakpoint frequencies determine the frequency ranges where either none or all of the effects of material recycle are seen. Thus, reduction of inventory control gains can improve quality control only to a certain extent. There is a range of inventory control gains where an adjustment causes an improvement in product quality control. Outside of this range, no change in product quality control will be seen. This range is determined by the value of the recycle gain as well as the frequencies of the peak closed-loop regulator log moduli from each load disturbance to each product quality variable. The economic effects of changes in inventory control tuning have also been illustrated. As an inventory controller is detuned, the regulation of product quality improves, increasing revenue; however, the regulation of vessel inventories deteriorates, requiring vessels to be oversized, increasing capital costs. The optimal inventory control tuning is determined by the economics of the plant. Searching for the inventory control gains that optimize plant economics can be a time-consuming task. For this reason, a general heuristic was presented that allows inventory controllers to be tuned with the aid of a few tests. Nomenclature B ) product flowrate (lb‚mol/h) D ) recycle flowrate (lb‚mol/h) F ) reactor effluent flowrate (lb‚mol/h) F0 ) fresh feed flowrate (lb‚mol/h) GLd ) transfer function matrix representing “series effect” GRr ) transfer function matrix from recycle to reactor effluent GRs ) transfer function matrix from fresh feed to reactor effluent GSr ) transfer function matrix from reactor effluent to recycle GSs ) transfer function matrix from reactor effluent to product k ) reaction rate constant (1/h) Kc ) reactor inventory control gain Krecycle ) steady-state gain of the recycling function k11 Sr ) recycle gain L ) general load disturbance LM ) log modulus (dB) VR ) reactor inventory (lb‚mol) X ) general product quality variable XB ) product composition XD ) recycle composition XF ) reactor effluent composition Z0 ) fresh feed composition Greek Symbols RA ) relative volatility of component A with respect to B τI ) reactor inventory controller integral reset time (h) ωbreak1 ) low-frequency breakpoint of the recycling function (rad/h) ωbreak2 ) high-frequency breakpoint of the recycling function (rad/h) ωpeak ) peak frequency (rad/h) ζ ) damping coefficient
Buckley, P. S. Techniques of Process Control; Wiley: New York, 1964. Denn, M. M.; Lavie, R. Dynamics of plants with recycle. Chem. Eng. J. 1982, 24, 55-59. Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988. Downs, J. J. Distillation control in a plantwide control environment. In Practical Distillation Control; Luyben, W. L., Ed.; Van Nostrand Reinhold: New York, 1992; pp 413-439. Elliott, T. R. Quantitative Assessment of Controllability During Design. Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 1996. Elliott, T. R.; Luyben, W. L. Capacity-based economic approach for the quantitative assessment of process controllability during the conceptual design stage. Ind. Eng. Chem. Res. 1995, 34 (11), 3907-3915. Gilliland, E. R.; Gould, L. A.; Boyle, T. J. Joint Automatic Control Conference, 1964. Kapoor, N.; Marlin, T. E.; McAvoy, T. J. Effect of recycle structure on distillation tower time constants. AIChE J. 1986, 32 (3), 411-418. Luyben, W. L. Steady-state energy conservation aspects of distillation column control system design. Ind. Eng. Chem. Fundam. 1975, 14 (4), 321-325. Luyben, W. L. The concept of eigenstructure in process control. Ind. Eng. Chem. Res. 1988, 27, 206-208. Luyben, W. L. Dynamics and control of recycle systems. 1. Simple open-loop and closed-loop systems. Ind. Eng. Chem. Res. 1993a, 32 (3), 466-475. Luyben, W. L. Dynamics and control of recycle systems. 2. Comparison of alternative process designs. Ind. Eng. Chem. Res. 1993b, 32 (3), 476-485. Luyben, W. L. Dynamics and control of recycle systems. 3. Alternative process designs in a ternary system. Ind. Eng. Chem. Res. 1993c, 32 (6), 1142-1153. Luyben, W. L. Snowball effects in reactor/separator processes with recycle. Ind. Eng. Chem. Res. 1994, 33 (2), 299-305. Luyben, W. L.; Buckley, P. S. A proportional-lag level controller. Instrum. Technol. 1977, 24, 65-68. Lyman, P. R. A Method for Assessing the Effect of Design Parameters on Controllability. Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 1995. Papadourakis, A. Stability and Dynamic Performance of Plants With Recycle. Ph.D. Dissertation, University of Massachusetts, Amherst, MA, 1985. Price, R. M.; Georgakis, C. Plantwide regulatory control design procedure using a tiered framework. Ind. Eng. Chem. Res. 1993, 32, 2693-2705. Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integrator/ dead time processes. Ind. Eng. Chem. Res. 1992, 31, 26252628. Tyreus, B. D.; Luyben, W. L. Dynamics and control of recycle systems. 4. Ternary systems with one or two recycle streams. Ind. Eng. Chem. Res. 1993, 32 (6), 1154-1162. Verykios, X. E.; Luyben, W. L. Steady state sensitivity and dynamics of a reactor/distillation column system with recycle. ISA Trans. 1978, 17 (2), 31-41. Yi, C. K.; Luyben, W. L. Evaluation of plant-wide control structures by steady state disturbance sensitivity analysis. Ind. Eng. Chem. Res. 1995, 34, 2393-2405.
Received for review October 1, 1996 Revised manuscript received December 23, 1996 Accepted December 24, 1996X IE960608+
Literature Cited Belanger, P. W.; Luyben, W. L. A new test for evaluation of the regulatory performance of controlled processes. Ind. Eng. Chem. Res. 1996, 35, 3447-3457.
X Abstract published in Advance ACS Abstracts, February 1, 1997.
