Plantwide Control

CHAPTER

PLANTWIDE CONTROL

15

15.1 INTRODUCTION Chapter 7 presented the hierarchical synthesis of process systems. In this approach, which proved successful in a large number of applications, the designer addresses one fundamental problem at each level, by applying appropriate analysis and synthesis techniques. This chapter considers another important issue, namely the control of the chemical plant. This is a significant topic because the recycling of raw materials, energy integration and reduced size of buffer vessels are characteristics of modern plants. In these conditions, the interaction between units is so strong that controllability of parts (unit operations) does not guarantee the controllability of the whole (the entire plant). Therefore, we talk about the ‘plantwide control’ problem. The book by Luyben et al. (1999) emphasizes that ‘How a process is designed fundamentally determines its inherent controllability. In an ideal project, dynamic and control strategies would be considered during the process synthesis and design activities’. To provide an answer to this challenge, we take a systemic approach. We consider that a process plant consists of several sub-systems interconnected through material and energy streams (Figure 15.1). The sub-systems will be called basic flowsheet structures (BFSs), defined as parts of the plant for which (local) control objectives are assigned and can be achieved using only manipulated variables that are local to the BFSs. Unit operations are the simplest BFSs. Often, some units interact so strongly that they must be treated as one entity. The plantwide control coordinates the BFSs in such a way that the whole system operates in the required manner. This is achieved by assigning control objectives to BFFs. The controllability of each BFS is a necessary (but not sufficient) condition for the controllability of the entire plant. Consequently, integrating design and control consists of two steps: •

•

Design BFSs with good controllability properties. This is possible for unit operations (such as reactors, distillation columns, heat exchangers, etc.), where a lot of industrial experience exists. During the last decades, significant progress was achieved concerning the control of more complex sub-systems, such as heat-integrated reactors or reactive distillation. Section 15.2 is devoted to the basic control techniques that are typically employed, with emphasis on the structural aspects. Appendix 15.6 contains the mathematical tools necessary for control system analysis and controller tuning. Couple the BFS in such a way that a controllable system is obtained. In Section 15.3, we will show that understanding the interaction between the chemical reactor and the separation section, which occurs due to material recycle, is of paramount importance. The right plantwide control is able to

Computer Aided Chemical Engineering. Volume 35. ISSN 1570-7946. http://dx.doi.org/10.1016/B978-0-444-62700-1.00015-2 © 2014 Elsevier B.V. All rights reserved.

599

600

CHAPTER 15 PLANTWIDE CONTROL

Chemical plant Products

AB ABC

PF

HX1

B

C BC

C HX1

Raw materials

A

Heat-integrated reactors

By-products Heat-integrated distillation

Unit operation

Purge Unit operation

Connections Azeotropic distillation with solvent recycle

FIGURE 15.1 Systemic view of chemical plants.

manage these interactions such that the whole system is stable and flexible. The importance of controlling the mass balance of the plant will be illustrated by presenting the design and control of several plants of increasing complexity.

15.2 INTRODUCTION TO PROCESS CONTROL 15.2.1 FEEDBACK

To illustrate the principle of feedback control, let us consider the simple case of heating a process fluid (Figure 15.2). Knowing the flow rate, inlet and outlet temperature of the process stream and the steam condition, the design would provide the necessary steam flow rate and heat exchanger sizing (heat transfer area, number of tubes and their diameter, etc.). Mass, energy and momentum conservation will be applied along the way, making use of constitutive equations to calculate physical properties (e.g. specific heat, conductivity, viscosity) and parameters of the conservation equations (e.g. heat transfer coefficients, friction factor). However, it is unlikely that the required performance will be achieved during operation. The reasons are the inherent disturbances in the feed flow rate and temperature, the uncertainty of the design equations and their parameters or other phenomena that were not taken into account, for example, fouling. Rejecting these disturbances is one task of the control system.

15.2 INTRODUCTION TO PROCESS CONTROL

Steam

Control valve

601

Setpoint TC Controller

TT Transmitter

Process fluid Heat exchanger Thermocouple

FIGURE 15.2 Feedback control of heat exchanger outlet temperature.

Moreover, the control system must also deal with the change in the outlet temperature which could be required from time to time. Under such circumstances, feedback control is the simplest way to achieve the requisite performance. Thus, the process variable (outlet temperature) is measured, for example, by a thermocouple. The resulting electrical signal is amplified and transmitted to the controller, to which the required value of the outlet temperature, named setpoint, is provided by the operator. The controller compares the process variable and the setpoint. If they are equal, no change is needed. If the two do not agree, then the controller uses the rules of a control algorithm, to compute and send a signal, called controller output, which adjusts the opening of the control valve on the steam line. The term feedback originates from the fact that a variable related to the process output is processed and fed back to the process input. Feedback control has several advantages. Firstly, it does not require detailed information about the controlled process, as long as the adjustment is made in the correct direction: in the heat exchanger example, the control valve must open more when the temperature is lower than its setpoint. Moreover, information about the disturbances is not required. However, better knowledge of the process in form of a mathematical model allows designing a control algorithm with better performances. Note that an unstable process can be stabilized only by means of feedback. The reverse is also true, processes might become unstable under wrongly designed feedback control. Among the limitations, we note that feedback control takes action only after the disturbances have upset the process. Moreover, the performance is unsatisfactory when large and frequent disturbances occur. These limitations can be addressed by employing more complex control schemes, namely feedforward and cascade control. In the following, the main techniques used for choosing the control algorithm and calculating the control parameters (controller tuning) will be briefly presented. The techniques make use of Laplace transforms and transfer functions. For readers’ convenience, in Appendix 15.6.2, we give a quick overview of these mathematical tools. Figure 15.3 presents the block diagram of the heat exchanger feedback control. Each sub-system is represented by a transfer function, which relates the block output to its input. We observe that the controlled process consists of at least three dynamic elements, namely the control actuator (valve), the

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CHAPTER 15 PLANTWIDE CONTROL

Steam pressure, D1(s) Valve

Temperature controller

+ Setpoint, R(s)

Tube side

Shell side Gd1(s)

Control error, E(s)

-

Feed rate, D2(s)

C1(s)

Gv(s)

Gs(s)

Control signal, U(s)

Measured temperature, Ym(s)

Gd2(s)

+ +

Steam flow rate

Gt(s)

+ + Outlet temperature, Y(s)

Duty

Gm1(s) Temperature measurement

FIGURE 15.3 Block diagram of the heat exchanger feedback control.

process itself (shell and tube) and the measuring device. Therefore, any real-process transfer function is at least of order three.

15.2.1.1 Direct synthesis method In the direct synthesis method (Figure 15.4), the performance of the closed-loop system is specified as the closed-loop transfer function G0(s). Then, the controller transfer function C(s) is found as C ð sÞ ¼

1 G 0 ð sÞ GðsÞ 1  G0 ðsÞ

(15.1)

Equation (15.1) shows that anything that prevents inverting G(s) makes control difficult: positive zeros introduce unstable poles in the controller, dead-time introduces a prediction (see Section 15.6.2 for an overview of typical dynamic behaviour). Let us consider the control of a first-order system (Figure 15.35), described by the transfer function GðsÞ ¼

Kp sT p + 1

(15.2)

The closed-loop system is specified to behave also as a first-order system with the time constant T0. G0 ðsÞ ¼

1 sT 0 + 1

(15.3)

In general, one should use a closed-loop time constant larger than the process time constant, for example, Tp < T0 < 3Tp.

15.2 INTRODUCTION TO PROCESS CONTROL

603

G0(s) R(s)

E(s) +–

C(s) = ?

U(s)

G(s)

Y(s)

FIGURE 15.4 Feedback controllers – direct synthesis method.

Then, application of Equation (15.1) leads to: 1

 +1 sT p + 1 sT 0 Tp 1 1  Kc 1 + C ð sÞ ¼  1+ ¼ 1 K p T 0 Kp sT p sT I 1 sT 0 + 1

(15.4)

In time domain, Equation (15.4) translates to:


 ð 1 t uðtÞ ¼ K c eðtÞ + eðtÞdt TI 0

(15.5)

which is a Proportional–Integral (PI) controller. If the process G(s) or the specified closed-loop transfer function G0(s) have higher orders, then the controller is more complex, including, for example, derivative action. Most industrial controllers are PI-type. The integral action ensures zero stationary error but induces damped oscillations in the dynamic response. When precise control is not required (e.g. buffer vessels), the controller has only proportional action. Derivative action is used for slow, noise-free control loops (such as temperature control). It is important to note that the controller integration time TI is equal to the process time constant. This observation is often used as a quick estimation of this tuning parameter. Then, there is only one user-defined parameter, namely the desired closed-loop time constant T0, which determines the controller gain Kc. Kc ¼

1 Tp  ; TI ¼ Tp Kp T0

(15.6)

A form of internal model control, called l-tuning, gives the same settings. When applied to a first-order plus dead-time process (see Figure 15.36), l-tuning recommends: Kc ¼

1 Tp  ; TI ¼ Tp Kp T0 + t

(15.7)

l-Tuning aims at smooth, non-oscillatory response to setpoint changes. If faster response is desired or disturbance rejection is the main control objective, then other tuning techniques are more effective.

15.2.1.2 Process response curve The process response curve method approximates the behaviour of the system with that of a first-order plus dead-time element (Figure 15.36). Table 15.1 presents the recommended controller settings.

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CHAPTER 15 PLANTWIDE CONTROL

Table 15.1 Tuning Rules for PI Controllers (Ziegler–Nichols) Kc
 1 T K t
 0:9 T K t
 1:2 T K t

Controller Type P

PI

PID

TI

TD

–

–

3.3t

2.0t

0.5t

15.2.1.3 Stability limit A popular tuning technique calculates the controller parameters such that the closed-loop system is far enough from the stability limit. Consider G(s) the plant transfer function and K the gain of a proportional controller. Then the closed-loop transfer function is as follows: G 0 ð sÞ ¼

GðsÞK 1 + GðsÞK

Stability is given by the poles of the transfer function, which are the roots of the characteristic equation: PðsÞ ¼ 1 + GðsÞK ¼ 0

(15.8)

Root locus shows the dependence of the roots of the characteristic equation P(s) ¼ 0 when the controller gain K is varied from 0 to 1. Consider, for example, the transfer function (15.9). Note that it contains three first-order elements in series, which could be the control valve, the process and the measurement device. G ðs Þ ¼

1 ð0:5s + 1Þðs + 1Þð2s + 1Þ

(15.9)

The real and imaginary parts of the roots of the characteristic equation, for varying values of the controller gain K, are presented in Figure 15.5, called the root-locus plot. For K ¼ 0 (no feedback control, open loop system), the roots are real, being just the poles of the controlled system. As the controller gain is increased, one root moves to 1. However, the other two roots become imaginary and, when the gain exceeds the critical value Ku (ultimate gain, stability limit), move to the right-half plane. In practice, bringing the system close to its stability limit is dangerous and not acceptable. The ATV (auto-tune variation) technique provides an estimation of the stability limit by periodically reverting the direction of the manipulated variable. The control parameters can be set based on the ultimate gain Ku and the period of oscillation at the stability limit, Pu (Table 15.2).

15.2.2 FEEDFORWARD CONTROL Sometimes, the process is affected by large, frequent and measurable disturbances. This information allows taking action before the process is affected. Thus, the expected disturbances (feed rate and temperature in our heat exchanger example) are measured, and the required change of the control variable

15.2 INTRODUCTION TO PROCESS CONTROL

wu =

6

605

2p = 1.94 Pu

4

Im (p)

K=0

2

Ku = 12.5 0

–4

–3

–2

–1

0

1

2

–2 –4 –6 Re (p)

FIGURE 15.5 Root locus plot.

Table 15.2 Tuning Rules for PI Controllers Tuning Rule

Kc

TI

Ziegler–Nichols Tyreus–Luyben

Ku/2.2 Ku/3.2

Pu/1.2 2.2 Pu

(opening of the steam control valve) is computed based on a model of the process. This change is implemented before the controlled variable (outlet temperature) is affected (Figure 15.6), with the result of better control performance. A block diagram of the combined feedback–feedforward control is shown in Figure 15.7. The feedback controller can be designed based on the process and disturbance transfer functions, and the requirement that the disturbance is compensated by the control action. Gp ðsÞ ¼ Gt ðsÞGs ðsÞGv ðsÞ

(15.10)

  Y ðsÞ ¼ Gp ðsÞC2 ðsÞGm2 ðsÞ + Gd2 ðsÞ D2 ðsÞ ¼ 0

(15.11)

C2 ðsÞ ¼ Gp ðsÞ1 Gd2 ðsÞGm2 ðsÞ1

(15.12)

Note that feedforward control can deal only with measurable disturbances and it requires a good model of the process (so that the effect on the controlled variable is predictable). Because a perfect process model is seldom available and not all disturbances are measurable, feedforward is always combined with feedback as shown in Figures 15.6 and 15.7.

606

CHAPTER 15 PLANTWIDE CONTROL

Feedforward controller

Setpoint Steam

+ +

Control valve

TC

Controller (feedback)

TT

Transmitter

Heat exchanger Process fluid

Thermocouple

FIGURE 15.6 Combined feedback–feedforward control of the heat exchanger outlet temperature.

Flow measurement

Temperature controller (feedforward) C2(s)

Feed rate, D2(s)

Gm2(s) Steam pressure, D1(s)

Temperature controller (feedback)

+ Setpoint, R(s)

C1(s)

Valve Gd1(s)

+ +

Gv(s)

+ +

Shell side

Tube side

Gs(s)

Gt(s)

– Control signal, U(s)

Measured temperature, Ym(s)

Steam flow rate

Gd2(s)

+

+ Outlet temperature, Y(s)

Duty

Gm1(s)

Temperature measurement

FIGURE 15.7 Block diagram of the heat exchanger combined feedback–feedforward control.

15.2.3 CASCADE CONTROL Cascade control is used when there are more than one measurements, but only one control variable is available. For our example, we note that although the valve opening is used for control purposes, the steam flow rate is the variable which actually determines the process outlet temperature. As the steam flow rate is determined both by the valve opening and by the steam pressure, a flow controller is used

15.2 INTRODUCTION TO PROCESS CONTROL

Feedforward controller

Setpoint

Steam +

Control valve

Process fluid

607

FC

Flow controller

+

TC

Temperature controller

TT Transmitter

Heat exchanger Thermocouple

FIGURE 15.8 Cascade control of the heat exchanger.

to keep the steam flow rate constant despite steam pressure disturbances. The flow controller receives its setpoint from the temperature controller, in a cascade fashion (Figure 15.8). In general, cascade control is recommended for slow processes which are controlled by means of a relatively fast process. Cascade control is effective against disturbances having a measurable effect before the process output, as shown in Figure 15.9. Therefore, cascade control improves the control performance for disturbance in steam pressure, but there is no benefit with respect to feed rate and temperature. The disadvantages of cascade control are related to its increased complexity and the additional measurement device and controller. In general, the improved performance justifies the investment when the inner loop is at least three times faster than the outer loop. Figure 15.9 shows the bloc diagram of the heat exchanger cascade control, emphasizing the structural aspects of this control technique: the disturbance (steam pressure) enters the process at an intermediate point, its effect is fast and can be easily measured.

15.2.4 OVERRIDE CONTROL Override control is used to protect against violating various process-related constraints. Figure 15.10 shows heating of a process stream in a furnace. The outlet temperature is controlled by means of a temperature–fuel flow rate cascade (TC1, FC). Although there is no control objective associated to the temperature of the stack gases, it is not allowed to increase above a certain limit. The solution is to add a second controller (TC2), which attempts to keep the stack temperature close to the maximum allowed value by manipulating the fuel flow rate. As two controllers attempt to act on the same control input, a low-selector block is inserted, just before the control valve. During normal operation, the stack temperature is below the safety limit. Hence, the control signal coming from TC2 and asking for more fuel will be ignored. As soon as the safety limit is approached, TC2 will command reducing the fuel flow rate, signal that is passed to the valve.

608

CHAPTER 15 PLANTWIDE CONTROL

FIGURE 15.9 Block diagram of the cascade control of the heat exchanger.

Stack gases TC2

Maximum temperature

Process stream

Temperature SP

Furnace

TC1

LS Fuel

FIGURE 15.10 Override control.

FC

15.2 INTRODUCTION TO PROCESS CONTROL

609

PV RB/A

SP

SP x

FC

:

FA Reactant A

RC

FB Reactant B

LC

FA Reactant A

FB Reactant B

LC

FIGURE 15.11 Two alternatives for implementing ratio control. Left – correct design. Right – wrong design due to non-linearity FB/FA.

15.2.5 RATIO CONTROL Often, process operation requires that the ratio between two flows is kept constant. Figure 15.11 shows a chemical reactor, where reactants A and B should be added in a certain ratio. One of the flows (A), named the wild flow, is set to FA by an upstream unit or according to some other control objective. The controlled flow FB must be changed such that the ratio FA/FB is constant. The left drawing shows the correct way to implement this: measure the wild flow, multiply by the required ratio RB/A and send the result as the setpoint of a flow controller. The second set-up, in which both flows are measured and their ratio is used as process variable in a feedback control loop, does not work because of the non-linearity derived from FB/FA computation.

15.2.6 SELECTIVE CONTROL Selective control is employed when more than one measurement is available, for the same process variable. In critical applications, failure of a single measurement cannot be tolerated. In this case, the same variable is measure with more than one (e.g. three) devices, and the median value is used for control purposes. Another example is related to operation of tubular reactors (Figure 15.12), where the maximum value of the temperature (magnitude of the hot spot) should be controlled. However, the precise location along the bed is not known because of modelling inaccuracies. The hot spot position might also change in time because of the deactivating catalyst. Therefore, several

610

CHAPTER 15 PLANTWIDE CONTROL

Deactivating catalyst

T

z

Reactants

TT

TT

TT

HS

Coolant

TC

Tmax

FIGURE 15.12 Selective control.

measurements are placed along the bed, the highest value is selected and used in a temperature control loop. Note that the coolant flows co-currently with the reaction mixture.

15.2.7 EXAMPLE – CONTROL OF A DISTILLATION COLUMN We will use control of a distillation column as an example illustrating how different control structures can be combined. A distillation column has five degrees of freedom, corresponding to the five control valves that are available on distillate, bottoms, reflux, cooling water and steam flows. These can be used to fulfil five control objectives. Several pairings of controlled–manipulated variable are possible. One alternative is shown in Figure 15.13. First, the column pressure is controlled by means of condenser cooling duty. This can be achieved by employing the cooling water flow rate, as shown in Figure 15.13. Other alternatives are possible, for example, using a flooded condenser where cooling duty is affected by the heat transfer area available for condensing the vapour. Then, control of inventory is addressed. Thus, distillate and bottoms flows are used to control the liquid levels in the reflux drum and column sump, respectively. Because the absolute level values are not important (as long as the vessels do not overflow or run dry), proportional-only controllers are used.

15.2 INTRODUCTION TO PROCESS CONTROL

611

Level P-only PC Feedback

FC Feedback

LC

Ratio x L/D

Feedforward

TC +

LC-2

Combined Feedback–feedforward

LS

FC Cascade Override

Level P-only

Cascade

XC

LC

FIGURE 15.13 Control of a distillation column, illustrating feedback, feedforward, cascade, ratio and override control.

The next step concerns quality control. The set-up presented in Figure 15.13 assumes that the purity of the bottoms stream is the important objective. However, concentration measurements are expensive, require costly maintenance and are often characterized by large time delays. Therefore, the control structure uses the setpoint of the steam flow controller to keep constant one temperature in the stripping section, which is a good indication of composition (inferential control). When concentration measurements are available, the setpoint of the temperature controller can be updated, in cascade fashion, by a concentration controller. In order to cope with frequent, measurable disturbances in the column feed rate, feedforward control is employed. Note that the fast temperature controller might over-react when the fraction of the light component in feed suddenly increases. In these circumstances, due to increased vapourisation rate, the reboiler could run dry. This is a dangerous situation, as the bottoms pump might lose suction or the reboiler might be damaged. To guard against this, override control is implemented. The secondary level controller LC-2 tries to keep the sump level at a value close to the minimum allowable one, by means of the steam valve. A low-select block is inserted before the steam valve, and selects the lowest of the two signals coming from the flow controller and from the secondary level

612

CHAPTER 15 PLANTWIDE CONTROL

controller. During normal operation (enough liquid in the sump), LC-2 asks for more steam, while the flow controller does not require any change in the steam rate. The lowest of the two signals (‘no change’) is passed to the valve. When the level drops to a dangerous value, LC-2 commands ‘less steam’, which is preferred over the ‘no change’ signal sent by the flow controller. The control scheme from Figure 15.13 considers that fluctuation of the distillate composition is allowed. Therefore, no distillate quality control loop is implemented. This way, interactions with the temperature control of the stripping zone are avoided. The ratio reflux:distillate is kept constant, which often provides satisfactory control of distillate composition.

15.3 PLANTWIDE CONTROL Plantwide control can be defined as the ensemble of control loops needed to operate an entire process and achieve its design objectives (Luyben, et al., 1997, 1999). Another definition (Larsson and Skogestad, 2000) considers plantwide control as the control philosophy of the overall plant with emphasis on the structural decisions such as placement of measurements, selection of manipulated variables and decomposition of the overall problem into smaller sub-problems. We consider that the plantwide control must maintain the momentum, energy and mass balance of the plant. Flow measurements are cheap and give a good indication of momentum, while pumps, valves and compressors are available as manipulated variables. Similarly, temperature is an indirect but often a good measure of energy, which can be controlled by means of utility system (steam, cooling water). In contrast, controlling the mass balance is more difficult because it concerns every chemical species involved in the process. Evaluation of each species inventory requires composition measurements, which are expensive and often unreliable. On the other hand, because there are no manipulated variables which directly influence the mass balance, the control system must rely on complex phenomena such as chemical reaction or physical separation. Therefore, we consider that the main task of the plantwide control is related to the mass balance of the plant, by avoiding the accumulation or depletion of chemical species involved in the process. In this context, the plantwide control philosophy can be derived considering the reaction stoichiometry and the recycle structure (which depends on the feasible separations as determined by the physical properties of the reactants and products). Thus, we will address the problem by analysing the behaviour of Reactor/Separation/Recycle systems, aiming to derive generic rules useful for designing plantwide control systems. Usually, the separation section can be considered as a black box. This is reasonable because the existing experience allows design and control of separation units once that feasible separation targets are set, for example, as species recovery or product purities. The separation is then modelled based on simple input–output component balances. In the simplest case, only the stoichiometry of chemical reactions is known, but no kinetic information is available. This allows designing the control structure, namely choosing the controlled and manipulated variables and their pairings. When the kinetics (at least for the main reactions) and the type of the reactor are known, the performance of the control structure can be assessed. By considering a steady-state model which incorporates kinetic reactor and blackbox separation and performing sensitivity studies, it is possible to assess the feasibility of operating procedures, for example, changing the raw materials, production rate or product distribution in multi-product plants. Moreover, dangerous phenomena can be revealed. One example is the high sensitivity to disturbances or uncertain design parameters, known as the ‘snowball effect’ (Luyben, 1994).

15.3 PLANTWIDE CONTROL

613

A plant showing multiple steady states is another potential unsafe situation. The designer must ensure that the desired steady state is reached during start-up and maintained during operation. Using the steady-state model, it is possible to identify unstable operating points (however, stability is a property characterizing the dynamic behaviour and it cannot be guaranteed based on steady-state considerations). Note that in systems where the un-reacted raw materials are recycled, steady-state operation is not always possible because the reactor has a limited capacity of converting the reactants into products. For the designer, understanding the mass balance of the plant is a key requirement which can be fulfilled when the Reactor/Separation/Recycle structure is analysed. The main idea is that all chemical species that are introduced in the process (reactants, impurities) or are formed in the reactions (products and by-products) must find a way to leave the plant or to be transformed into other species which leave the plant (Downs, 1992). Usually, the separation units take care that the products are removed from the process. This is also valid for by-products and impurities, although in some cases inclusion of an additional chemical conversion step is necessary (Groenendijk et al., 2000; Dimian et al., 2001). The mass balance of the reactants is more difficult to maintain, because the reactants are not allowed to leave the plant but are recycled to the reaction section. If a certain amount of reactant is fed to the plant but the reactor does not have the capacity of transforming it into products, reactant accumulation occurs and no steady state can be reached. The reaction stoichiometry sets an additional constraint on the mass balance. For example, a reaction of the type A + B ! products requires that the reactants A and B are fed in exactly one-to-one ratio. Any imbalance will result in the accumulation of the reactant in excess, while the other reactant will be depleted. In practice, feeding the reactants in the correct stoichiometric ratio is not trivial, because there are always measurement and control implementation errors. It should be remarked that understanding the mass balance and the way it can be controlled is of great help during flowsheeting. To explain, we note the concept of design degrees of freedom, which is the number of variables that must be specified to completely define the process. It can be calculated by subtracting the number of equations from the number of variables. The control degrees of freedom is the number of variables that can be controlled. This is equal to the number of manipulated variables, namely the control valves in the process. It turns out (Luyben, 1996) that for many processes the number of design degrees of freedom is equal to the number of control degrees of freedom. While developing the simulation model of the flowsheet, different sets of degrees of freedom can be chosen as specifications. In general, convergence is easy when the specified variables correspond to the controlled variables of a feasible plantwide control structure, while lack of steady-state convergence usually indicates problems of the corresponding control structure. This problem was illustrated in Chapter 2 by the simulation of 1,1 diethoxy butane plant. In the rest of the chapter, we will present the control of several Reactor–Separation–Recycle systems where reactions with different stoichiometries take place. We will start by presenting two different concepts behind plantwide control structures, namely feedback control and self-regulation. Both of them are feasible for one-reactant systems, although relying on self-regulation is not recommended for the case of low-reactant conversion (small reactor). Sections 15.3.3 and 15.3.4 will introduce more complexity by considering two reactants. When the reactants are recycled by different streams, the solution of feedback control of reactant inventory is easy to apply. However, when the reactants are recycled together, making use of self-regulation is unavoidable. This leads to dangerous non-linear phenomena as state multiplicity and instability, which can be avoided only by a proper design of

614

CHAPTER 15 PLANTWIDE CONTROL

the chemical reactor. A key lesson from this chapter is to avoid relying on self-regulation of the mass balance and to use feedback for the purpose of controlling species inventory, as illustrated by the more complex control of the vinyl acetate plant.

15.3.1 FEEDBACK CONTROL AND SELF-REGULATING INVENTORY Consider the simple example presented in Figure 15.14, showing two different ways of controlling the inventory. The first strategy employs an inventory (level) controller. If the level increases/decreases above/below the setpoint, less/more liquid is added. We say that the inventory is controlled by feedback. The second strategy consists of setting the feed on flow control. When more fluid is fed to the vessel, the level increases. The outlet flow rate, which is proportional to the square root of the liquid level, will also increase. After some time, the feed and outlet flows are equal, and a state of equilibrium is reached. We say that the inventory is self-regulating.

15.3.2 ONE-REACTANT PROCESSES Let us consider a process where the first-order reaction A ! P takes place in a CSTR. We assume that the raw material is pure reactant A and the separation section achieves a perfect A/P split, such that only product P leaves the plant and only reactant A is recycled. The flowsheet of the plant is presented in Figure 15.15. Two different control alternatives can be imagined, similar to the example from Figure 15.14. The first strategy (Figure 15.15, left) is based on feedback control of the inventory. It consists of measuring the reactant inventory (level) and implementing a feedback control loop. Thus, the increase or decrease of the amount of reactant in the process is compensated by less or more reactant being added. In the second control structure (Figure 15.15, right), the flow rate of reactant at plant inlet is fixed. When the reactant accumulates, the consumption rate increases until (hopefully) it balances the feed rate. This strategy is based on the self-regulation property.

FIGURE 15.14 Alternatives for inventory control. (A) Feedback control; (B) Self-regulating inventory.

15.3 PLANTWIDE CONTROL

615

FIGURE 15.15 Plantwide control of one-reactant systems. Left: Fixed reactor-inlet flow rate, reactant inventory controlled by feedback. Right: Fixed plant-inlet flow rate, self-regulating reactant inventory.

Let us consider the chemical reactor. The reactant consumption rate is given by: r A ¼ kMzA, 2

(15.13)

1

where k is the reaction rate constant (s ); M is the reactor hold-up (kmol); zA,2 is the reactant mole fraction (equal to the outlet concentration due to perfect mixing assumption). The reactor mass balance can be written as follows: F1  F1 zA, 2 ¼ kMzA, 2

(15.14)

From Equations (15.13) and (15.14), the following expressions can be easily obtained: Production rate at fixed reactor  inlet flow rate : FP, 2 ¼

kM 1 + kM=F1

(15.15)

Sensitivity of FP, 2 with respect to F1 :

@FP, 2 1 ¼ @F1 ð1 + F1 =kMÞ2

(15.16)

Sensitivity of FP, 2 with respect to kM :

@FP, 2 1 ¼ @kM ð1 + kM=F1 Þ2

(15.17)

Maximum production rate : FP, 2 ¼ lim FP, 2 ¼ kM F1 !1

Reactant conversion : XA ¼

kM=F1 1 + kM=F1

Conversion at maximum production rate : XA ¼ lim XA ¼ 0 F1 !1

(15.18)

(15.19)

(15.20)

Equation (15.15) suggests two ways in which production rate changes can be achieved: change the reactor-inlet flow rate F1, or the reactor hold-up M. These options are implemented as the control structure fixing the reactor-inlet flow, shown in Figure 15.15 (left).

616

CHAPTER 15 PLANTWIDE CONTROL

2

3

F1 = 4

kM = 4

1.5 2

2

FP,2

FP,2

2

1

1

1

1

0.5 0

0 0

2

4

6

8

0

10

1

F1 1

1

0.8

0.8

0.6

2

XA

XA

4

4

0.4

2 1

0.2

3

F1 = 1

0.6

kM = 4

0.4

2 kM

0.2

0

0 0

2

4

6 F1

8

10

0

1

2

3

4

kM

FIGURE 15.16 First-order reaction in CSTR. Left – change in the inlet-flow; right – change in the reaction volume/reaction rate constant.

Figure 15.16 (left) presents the production rate (FP,2) and reactant conversion (XA) versus the reactor-inlet flow rate F1. It can be observed that increasing the reactor-inlet flow rate leads to a higher production rate, but to a lower conversion. Note that according to Equation (15.16), F1 is an effective throughput manipulator only when kM is large (fast reaction, large reactor). Figure 15.16 (right) presents the dependence of the production rate (FP,2) and reactant conversion (XA) versus the reactor hold-up, kM. As expected, larger reactors lead to higher production rate. The same result is obtained by changing the reaction rate constant k (e.g. by changing the reaction temperature). According to Equation (15.17), kM is an effective throughput manipulator only when kM is small (slow reaction, small reactor). In conclusion, feedback control of reactant inventory can be implemented through the following steps: – fix the reactor-inlet (recycle plus fresh feed) flow rate. – determine the reactant inventory somewhere in the recycle loop, by measuring the appropriate level, pressure or concentration. – adjust the fresh feed to keep the inventory at a constant value.

15.3 PLANTWIDE CONTROL

617

The important advantage of this strategy is that the reactor is decoupled from the rest of the plant. The production is manipulated indirectly, by changing either the reactor-inlet flow (for fast reactions or large reactors, high conversion) or the reaction conditions (for slow reactions or small reactors, low conversion), which could be seen as a disadvantage. However, it better handles non-linear phenomena, as for example, the snowball effect or state multiplicity. Additionally, this strategy guarantees the stability of the whole recycle system if the individual units are stable or stabilized by local control. When the whole plant is considered, the entire amount of reactant that is fed to the process, FA,0, is transformed into product (which leaves the plant through the separation section). Therefore: r A ¼ FA, 0 ¼ FP, 2

(15.21)

This suggests a direct way for setting the production rate, as implemented by the control structure fixing the plant-inlet flow and presented in Figure 15.15 (right). Note that the reactant inventory is selfregulating, as the production rate FP,2 does not depend on reactor size M or the reaction kinetics k. However, Equation (15.18) is still valid, meaning that the feasibility of this control structure is limited by the reactor capacity. From Equation (15.15), the following relationships can be derived: Reactor  inlet flow rate for a given production rate : F1 ¼

Sensitivity of F1 with respect to FP, 2 :

FP, 2 1  FP, 2 =kM

@F1 1 ¼ @FP, 2 ð1  FP, 2 =kMÞ2

(15.22)

(15.23)

It can be observed that as the maximum production rate is approached (FP,2 ! kM), F1 and @F1/@FP,2 become very large. This high sensitivity is known as the snowball effect (Luyben, 1994). In conclusion, the self-regulation strategy, which could be designated as ‘classical’, is recommended only if the reactor is large enough and the per-pass conversion is high. Non-linearity of the plant could lead to phenomena as state multiplicity and closed-loop instability (Bildea and Dimian, 2003; Bildea et al., 2004; Kiss et al., 2007), as well as high sensitivity to production rate changes, process disturbances or design-parameter uncertainty (snowball effect). The approach has the advantage of directly setting the production rate. A supplementary benefit is that product distribution is fixed in the case of some complex reactions.

15.3.2.1 Example: di n-pentyl-ether production Di n-pentyl ether (DNPE) is a linear symmetric ether that can be used as blending additive in reformulated diesel fuels. It can be produced from C4 feedstocks via n-pentanol, obtained in its turn by selective hydroformylation of linear butenes. The etherification of n-pentanol is catalysed by NaA and H-Beta zeolites, Amberlyst 70, Nafion NR50 (Bringue et al., 2006; Pera-Titus et al., 2009) 2C5 H9  OH ! C5 H9  O  C5 H9 + H2 O ð PÞ

ðDÞ

ðW Þ

618

CHAPTER 15 PLANTWIDE CONTROL

When Amberlyst 70 is used as catalyst, the reaction kinetics is described by the following expression, derived from an Eley–Rideal mechanism (Pera-Titus et al., 2009)
 1 aW aD 1 K eq aP  r¼  1=2 aP 1 + K W aW aP ka2P

where:



 1 1 k ¼ 4:6  106 exp 11, 595  kmol=ðkgcat sÞ T 438 K eq ¼ 8:9229exp


 778:69 T



 1 1 K W ¼ 4:306exp 6616  T 438

The reaction is slightly endothermic and can be performed in an adiabatic plug-flow reactor. Because the reaction is equilibrium limited, complete reactant conversion is not possible. In order to design the separation section, we consider the boiling points of pure components and azeotropes that can be found in the n-pentanol–water DNPE system (Table 15.3). DNPE is the high-boiling component and therefore can be easily separated. Water is involved in several heterogeneous azeotropes, which suggest the use of a liquid–liquid split in order to cross the distillation boundary. The full flowsheet is presented in Figure 15.17. Fresh and recycled n-pentanol are mixed, heated to reaction temperature (190  C) and fed to the reactor (diameter 1.5 m, length 2 m, catalyst particle density 1400 kg/m3, void fraction 0.5). The n-pentanol conversion is XP ¼ 0.65. The reaction mixture is fed to the first distillation column (10 trays, 1 m diameter, distillate-to-feed ratio ¼ 0.7). The distillate is condensed, cooled to 30  C and sent to liquid–liquid separation. The aqueous phase is withdrawn as product (0.998 purity). The organic phase is returned as reflux to the column. The bottom product, containing n-pentanol and DNPE, is sent to the second distillation column (20 trays, 1.25 m diameter, distillate-to-feed ratio ¼ 0.7, reflux ratio ¼ 1.075). The distillate and bottoms are n-pentanol (0.99 purity, recycled) and the DNPE product (0.99 purity). Table 15.4 contains the detailed stream results.

Table 15.3 Singular Points in the n-Pentanol–Water–DNPE System Component

Boiling Point

Ternary azeotrope: heterogeneous n-pentanol (0.115); Water (0.8732); DNPE (0.0116) Binary azeotrope: heterogeneous n-pentanol (0.126); Water (0.874) Binary azeotrope: heterogeneous water (0.9539); DNPE (0.00461) Water n-Pentanol DNPE

96.37  C (unstable node) 96.41  C (saddle) 98.7  C (saddle) 100  C (stable node) 137.8  C (saddle) 186.5  C (stable node)

15.3 PLANTWIDE CONTROL

619

Recycle

Alcohol PC LC

Mixer

TC

LC

LC

Heater COL-1

TC

Water

Reflux

FC

Reactor

Rin

Rout TC

PC

LC

FC

COL-2 LC

Alc+Eth TC

LC

Ether

FIGURE 15.17 Flowsheet and plantwide control of DNPE plant.

Table 15.4 DNPE Plant – Stream Table

T ( C) P (bar) Vapour Frac F (kmol/h) F (kg/h)

n-Pentanol

Rin

Rout

Dist

Reflux

Water

n-Pentanol DNPE

Recycle

DNPE

25 1 0

190 5 0

167.1 4.5 0.355

123 1 1

30 1 0

30 1 0

145.8 1.09 0

137.5 1 0

189 1.19 0

32.0 2822

50.1 4433

50.1 4433

79.8 4626

63.9 4336

15.9 290

34.2 4143

18.1 1611

16.1 2533

1

0.997 198 PPB 0.003

0.363 0.317

0.539 0.446

0.673 0.308

0.002 0.998

0.531 290 PPB

0.011 Trace

0.32

0.015

0.019

742 PPB

0.469

0.991 546 PPB 0.009

Mole fraction Alcohol Water Ether

0.989

620

CHAPTER 15 PLANTWIDE CONTROL

The control structure (Figure 15.17) fixes the reactor-inlet flow of n-pentanol, while the fresh alcohol flow rate is used to control the level in the mixer. The reactor-inlet temperature is controlled by the heater duty. Control of the first distillation column involves sump level, pressure and a temperature on the stripping section (stage 9), by means of bottoms flow, vapour distillate flow and reboiler duty. The temperature of the liquid–liquid separator is controlled by the condenser duty. Finally, the levels of the organic and aqueous phases are controlled by the reflux and distillate flow rates. Control of the second distillation column is standard: condenser duty controls the pressure, distillate and bottoms flow rates control the levels of the reflux drum and column sump. The temperature on the lower part of the column (stage 18) is controlled by the reboiler duty, while the reflux rate is constant. A dynamic simulation was built in Aspen Dynamics. For an unknown reason, the Aspen Dynamic simulation gave results that were very different from the steady-state results. After a careful investigation, it was found that in the PFR dynamic model the reaction proceeds at a much faster rate compared to the PFR steady-state model. For this reason, the catalyst density was changed to 430 kg/m3, for which the dynamic PFR gave the same results as the steady-state PFR. All vessels were sized based on 15 min residence time. The controllers were chosen as PI and were tuned by the direct synthesis method. Thus, for each control loop, the appropriate ranges of the controlled and manipulated variable were specified, and controller gain was set to 1%/%. The integral time was set equal to an estimated time constant of the process (temperature – 20 min; pressure – 12 min). No integral action was used for level control. Table 15.5 presents the details of the control loops and controller tuning. Results of dynamic simulation are given in Figure 15.18. The simulation starts from steady state. At time t ¼ 2 h, the reactor-inlet flow rate is increased by 20%, from 50 to 60 kmol/h. As more reactant is fed to the reactor, the production rate increases from 16 to 18 kmol/h in about 5 h. After an initial peak, the fresh alcohol flow rate settles close to the theoretical value of 36 kmol/h. The purities of DNPE and water products remain practically unchanged. Similar results were obtained when the reactor-inlet flow rate was decreased, with the result of lower production rate. It should be remarked that the DNPE reactor was designed for a rather high reactant conversion. Therefore, we expect the conventional control structure, fixing the plant-inlet flow rate, to work well. We leave this as an exercise for the reader.

15.3.3 TWO-REACTANT, TWO RECYCLES When the process involves two reactants (A and B) that are recycled by two different streams, the recommended control structure uses feedback to control the inventory of each reactant, as illustrated by Figure 15.19 (left). Both reactor-inlet streams are on flow control. One of them, for example, FA,1, can be used as a throughput manipulator. The measured value FA,1 is multiplied by the required FB,1/FA,1 and the result is sent as setpoint to the FB,1 flow controller. If the process was designed for low conversion (e.g. due to selectivity reasons), then the setpoint of the reactor hold-up controller can be used for production rate changes. Sometimes, one of the fresh feeds (e.g. A) is set by an upstream unit and cannot be used for inventory control. In this case, the plant should be designed for large conversion of reactant A, achieved by using a large reactor and a large excess of B. Then, the control structure shown in Figure 15.19 (right) works well. However, two steady states are possible. The steady state corresponding to high conversion

15.3 PLANTWIDE CONTROL

621

Table 15.5 DNPE Plantwide Control – Controller Tuning Controller

Kc (%/%)

Ti (min)

Alcohol feed ¼ 32 kmol/h 0    64 kmol/h

1

600

Temperature ¼ 190  C 180    200  C

Duty ¼ 0.426  106 kcal/h 0    1  106 kcal/h

1

20

Pressure ¼ 1 bar 0.9    1.1 bar Stage 9 temperature ¼ 142.6  C 130    150  C Condensate temperature ¼ 30  C 20    40  C Level, organic phase ¼ 1.25 m 0    2.5 Level, aqueous phase ¼ 0.14 m 0    2.5 m Level, sump ¼ 1.875 m 0    3.75 m

Vapour distillate ¼ 79.9 kmol/h 0    160 kmol/h Reboiler duty ¼ 0.85  106 kcal/h 0    1.7  106 kcal/h

2

12

1

20

Cooling duty ¼  1.11  106 kcal/h 2.22  106    0 kcal/h

1

20

Reflux ¼ 4343 kg/h 0    8672 kg/h Water product ¼ 289.6 kg/h 0    580 kg/h Bottoms product ¼ 4143 kg/h 0    8286 kg/h

1

600

1

600

1

600

2

12

1

600

1

600

1

20

PV, Value and Range

OP, Value and Range

Level ¼ 1.25 m 0    2.5 m

Mixer

Heater

COL-1

COL-2 Pressure ¼ 1 bar 0.9    1.1 bar Reflux drum level ¼ 1.875 m 0    3.75 m Sump level ¼ 2.5 m 0  5 m Stage 18 temperature ¼ 149.8  C 140    160  C

Condenser duty ¼  1.8  106 kcal/h 0    3.6  106 kcal/h Distillate product ¼ 1612 kg/h 0    3224 kg/h Bottoms product ¼ 2531 kg/h 0    5065 kg/h Reboiler duty ¼ 1.85  106 kcal/h 0    3.7  106 kcal/h

of reactant A is stable, while the one corresponding to low conversion of A is unstable. Because the state multiplicity can lead to start-up difficulties, this has to be carefully investigated by dynamic simulation. We emphasize that fixing both plant-inlet flow rates FA,0 and FB,0 or their ratio does not work. The reason is that the ratio between fresh reactants must strictly obey the reaction stoichiometry. This cannot be achieved in practice because of unavoidable measurement uncertainty and control action implementation errors.

622

CHAPTER 15 PLANTWIDE CONTROL

30

80

Water

20

Water

15

60

Rin

Ether

40

Alcohol

0.995

Purity

25

Flow rate (kmol/h)

1

Recycle

Recycle

0.99

Ether

10 20

0.985

0

0.98

5 0 0

2

4

6

8

10

12

0

2

Time (h)

4

6

8

10

12

Time (h)

FIGURE 15.18 DNPE example – dynamic simulation results.

FIGURE 15.19 Plantwide control of two-reactant, two recycles processes.

15.3.3.1 Example: solketal plant Solketal (2,2-dimethyl-1,3-dioxolane-4-methanol) is obtained by the condensation reaction between glycerol and acetone, in the presence of an acid catalyst. Solketal can be used as a fuel additive to reduce the particulate emission and to improve the cold flow properties of liquid transportation fuels. Glycerol + Acetone $ Solketal + Water ðGÞ

ðAÞ

ðSÞ

ðW Þ

The reaction follows an LHHW mechanism. When Amberlyst 35 is used as catalyst, the following relationships describe the reaction kinetics:
 1 cS cW cG cA 1  K c cG cA r¼k ð 1 + K W c W Þ2

where subscripts G, A, S and W refer to glycerol, acetone, solketal and water, respectively.

(15.24)

15.3 PLANTWIDE CONTROL

623

The reaction rate constants are given by (Nanda et al., 2014):



 55,600kJ=kmol 1 1 k ¼ 8  10 exp   m3 =kmol=s R T 308 4

ln K c ¼ 11:308 +

ln K w ¼ 25:1925 +

3615:4 T

7785:8 3 m =kmol T

The reaction can be carried on in a CSTR, at 2 bar and 35  C, in the presence of ethanol as solvent. As the reaction is equilibrium limited, products separation and reactants recycle are necessary. The singular points of the acetone–glycerol–water–solketal–ethanol system are given in Table 15.6. Accordingly, the separation system involves three distillation columns for the separation of acetone/ethanol/water mixture (to be recycled), water (product), solketal (product) and glycerol (recycled). The plant flowsheets are given in Figure 15.20. Fresh and recycled reactants are mixed, heated to the reaction temperature and fed to the CSTR (4 m3). The reactor effluent is sent to the first distillation column (25 stages, 0.9 m diameter, distillateto-feed ratio ¼ 0.78, reflux ratio ¼ 4). Acetone and ethanol are removed with the distillate, together with small amounts of water due to azeotrope formation. The bottom product is sent to the second distillation column (13 stages, 0.3 m diameter, distillate-to-feed ratio ¼ 0.434, reflux ratio ¼ 2), which removes water as distillate product at a purity exceeding 0.999. The bottoms is sent to the third column (pressure 0.05 bar, 8 stages, 0.45 m diameter, distillate-to-feed ratio ¼ 0.765, reflux ratio ¼ 0.169) which gives the solketal product (0.999 purity) and glycerol recycle (0.997 purity). Detailed stream results are given in Table 15.7. Note that the plant design anticipated the plantwide controllability. Thus, we expect the fresh glycerol to be delivered by an upstream unit, for example, a biodiesel plant; for this reason, this flow rate will not be available for controlling glycerol inventory, which must be self-regulating. Therefore, the

Table 15.6 Singular Points in the Acetone–Glycerol–Water–Solketal– Ethanol System Component

Boiling Point

Binary azeotrope, homogeneous Acetone (0.9866); Water (0.0134) Acetone Binary azeotrope, homogeneous Ethanol (0.9013); Water (0.0987) Ethanol Water Solketal Glycerol

56.13  C (unstable node) 56.14  C (saddle) 78.18  C (saddle) 78.31 (saddle) 100  C (saddle) 189  C (saddle) 287.7  C (stable node)

624

CHAPTER 15 PLANTWIDE CONTROL

Acetone recycle Acetone Water

LC PC

Mixer

LC

FC

Heater

FC

COL-1

TC

Reactor

Rin

X

LC PC

TC

FC

LC

PC

FC

LC

Rout TC

LC

Solketal

COL-2 COL-3

Glycerol FC

TC

LC

Mixer

TC

LC

LC

Glycerol recycle

FIGURE 15.20 Solketal plant – flowsheet and plantwide control.

plant was designed for high glycerol conversion (more than 0.75). This was achieved by a sufficiently large reactor and a large acetone excess (6:1 acetone:glycerol ratio at reactor-inlet). The plantwide control scheme is shown in Figure 15.20. The glycerol feed rate is given by an upstream unit. Therefore, the glycerol inventory cannot be controlled by feedback and we have to rely on self-regulation. The ratio between the reactor-inlet acetone and glycerol flow rates is set in a feedforward manner, as discussed in Section 15.2.5. Reaction temperature and reactor levels are controlled by the duty and outlet flow, respectively. Control of the distillation columns is standard: condenser duty controls the pressure; levels in the reflux drum and column sump are controlled by the distillate and bottoms rate, respectively; a temperature in the stripping zone is controlled by the reboiler duty; the reflux is constant. Details of the control loops are given in Table 15.8. Results of dynamic simulation are presented in Figure 15.21. Starting from steady state, the production rate is increased (top diagrams) or decreased (bottom diagrams) by changing the plant-inlet flow rate of glycerol (Gly-0), from the initial value of 3.2 to 4 kmol/h and 2.4 kmol/h, respectively. The new production rate is reached in about 8 h. As more glycerol is fed to the plant, the flow rate of fresh acetone (Ac-0) is adjusted to the stoichiometric value, although the duration of the transient regime is quite long. The purity of solketal and water products exceeds 0.998. It should be remarked that other control strategies work equally well. For example, one can keep constant the reactor-inlet flow rate of acetone (instead of flow ratio). However, the next example will show that the range of achievable production rates is much larger when the ratio controller is in place. When the flow rate of fresh glycerol is available as a manipulated variable, one can switch the flow and level control loops around the glycerol mixer, controlling the glycerol inventory by feedback instead of relying on self-regulation, as illustrated in Figure 15.19 (right). These alternatives are left as an exercise for the reader.

Table 15.7 Solketal Plant – Stream Table

T ( C) P (bar) Vapour Frac F (kmol/h) F (kg/h)

Glycerol

Acetone

Rin

Rout

Solketal

Water

Acetone recycle

Glycerol recycle

BTM-1

q

25 1 0

25 1 0

35 2 0

35 2 0

107.3 0.05 0

99.3 1 0

56.9 1 0

194.3 0.064 0

128.2 1.24 0

199.5 1.12 0

3.22 296.2

3.22 186.8

34.29 2048.7

34.29 2048.7

3.22 425.1

3.22 58.1

26.87 1474.5

0.99 91.0

7.42 574.2

4.20 516.1

1 0 0 0 0

0 1 0 0 0

0.122 0.729 0.026 0 0.122

0.029 0.635 0.12 0.094 0.122

0 0 0 1 0

0 0 0.999 0 0.001

0 0.811 0.033 0 0.156

0.997 0 0 0.003 0

0.133 0 0.433 0.434 0.001

0.234 0 0 0.766 0

Mole fraction Glycerol Acetone Water Solketal Ethanol

Table 15.8 Solketal Plantwide Control – Controller Tuning Controller

Kc (%/%)

Ti (min)

PV, Value and Range

OP, Value and Range

Level ¼ 0.625 m 0    1.25 m

Outlet flow rate ¼ 387.2 kg/h 0    774 kg/s

1

60

Level ¼ 1.125 m Acetone feed ¼ 3.21 kmol/h 0.5    1.5 m 0    6.4e kmol/h Outlet flow rate (kg/h) ¼ 4.24  glycerol mixer outlet flow rate

1

60

10

20

5

6

2

12

1

60

1

60

1

20

2

12

1

60

1

60

1

20

20

12

1

60

1

60

1

20

Glycerol mixer

Acetone mixer

Reactor Level ¼ 1.72 m 0    3.44 m Temperature ¼ 35  C 20   50  C

Outlet flow rate ¼ 2048.7 kg/h 0    4100 kg/h Duty ¼  0.0445  106 kcal/h 0.1  106    0 kcal/h

Pressure ¼ 1 bar 0.9    1.1 bar Reflux drum level ¼ 1.875 m 0    3.75 m Sump level ¼ 1.25 m 0    2.5 m Stage 23 temperature ¼ 99.6  C 90    110  C

Condenser duty ¼  1.0  106 kcal/h 0    2.0  106 kcal/h Distillate product ¼ 1474 kg/h 0    3000 kg/h Bottoms product ¼ 574.2 kg/h 0    1150 kg/h Reboiler duty ¼ 1.05  106 kcal/h 0    2.1  106 kcal/h

Pressure ¼ 1 bar 0.9    1.1 bar

Condenser duty ¼  0.950  106 kcal/h 0    0.18  106 kcal/h Distillate product ¼ 58.12 kg/h 0    116 kg/h Bottoms product ¼ 574.2 kg/h 0    1150 kg/h Reboiler duty ¼ 0.117  106 kcal/h 0    0.235  106 kcal/h

COL-1

COL-2

Reflux drum level ¼ 0.5 m 0  1 m Sump level ¼ 1.25 m 0    2.5 m Stage 11 temperature ¼ 179.6  C 160    200  C COL-2 Pressure ¼ 0.05 bar 0    0.1 bar Reflux drum level ¼ 0.75 m 0    1.5 m Sump level ¼ 0.5 m 0  1 m Stage 7 temperature ¼ 132.3  C 120    150  C

Condenser duty ¼  0.058  106 kcal/h 0    0.12  106 kcal/h Distillate product ¼ 425 kg/h 0    850 kg/h Bottoms product ¼ 90.98 kg/h 0    180 kg/h Reboiler duty ¼ 0.032  106 kcal/h 0    0.065  106 kcal/h

15.3 PLANTWIDE CONTROL

8

Solketal

627

50

1

Rin

0.996

Solketal

Gly-0 4

0.994

Water 2

Flow rate (kmol/h)

0.998

Ac-0

6

Purity

Flow rate (kmol/h)

Water 40

Ac-rec

30 20

0.992

10

0.99

0

Gly-rec 0 0

5

10

15

20

0

25

5

10

Solketal

8

15

20

25

15

20

25

Time (h)

Time (h) 1

50

0.998

40

Ac-0 0.996 4 Water

0.994

Gly-0

2

Solketal

0.992

Flow rate (kmol/h)

6

Purity

Flow rate (kmol/h)

Water

30

Rin

20

Ac-rec

10 Gly-rec

0

0.99 0

5

10

15

20

25

0 0

Time (h)

5

10

Time (h)

FIGURE 15.21 Solketal plant – dynamic simulation results. Top – production rate increase by 25%; bottom – production rate decrease by 25%.

15.3.4 TWO-REACTANT, ONE RECYCLE When the process involves two reactants (denoted here by A and B) that are recycled together, only one mixed (feed + recycle) stream can be fixed, thus only the inventory of one component can be controlled by feedback. Consequently, the inventory of one reactant (e.g. A) must be self-regulating. Two control structures are shown in Figure 15.22. In the first control structure, the reactor-inlet streams, namely fresh A (FA,0) and the stream containing fresh B, recycled A and recycled B (F1), are on flow control. The second control structure adds a ratio controller, therefore the ratio M ¼ F1/FA,0 is fixed. For such systems, it is essential that the plant is designed for high conversion of reactant A. Moreover, as will be shown by the next example, a large excess of B increases the range of achievable production rate.

15.3.4.1 Example: production of 1,1 diethoxy butane 1,1 Diethoxy butane is an oxygenated compound which have been proposed to be used as combustion enhancers for biodiesel. The conceptual design of a plant producing 10 kmol/h of 1,1 diethoxy butane, by the reaction of butanal and ethanol, was presented in Chapter 2. Here, we will extend the design by including controllability considerations. Moreover, we will consider that ethanol is available as water– ethanol mixture, with a composition close to the azeotropic one, which in Chapter 2 was left as an exercise. For readers’ convenience, we will repeat some design data.

628

CHAPTER 15 PLANTWIDE CONTROL

Fresh B

Fresh B

Recycle A & B

Recycle A & B

M = F1/F0A

SP

F1

x LC

LC

FC

FC

F1

F1

MSP

M SP

Fresh A

Fresh A

F0A

A +B ® P

LC

Separation

P A +B ® P

LC

Separation

P

FC

FC

F0A

FIGURE 15.22 Plantwide control of two-reactant, one recycle system. Left: Fresh feed of reactant A and second reactor-inlet stream are fixed; right: fresh feed of reactant A and the ratio between the reactor-inlet streams are fixed.

The chemical reaction is as follows: 2C2 H5 OH + C3 H7  CH ¼ O $ C3 H7  CHðOC2 H5 Þ2 + H2 O ðEÞ

ð BÞ

ð AÞ

ðW Þ

with reaction rate (Agirre et al., 2010) given by:

  r ¼ mcat  k1 c2E cB  k2 cA cW

(15.25)


  35,505  3 3  k1 ¼ 1:08 exp  m = kmol2 skgcat 8:31T

(15.26)


 59, 752  3 2 k2 ¼ 1:06  105 exp  m =ðkmols kgcat Þ 8:31T

(15.27)

Figure 15.23 presents the Reactor–Separation–Recycle structure of the plant. The separation of water from 1,1 diethoxy butane will be addressed later. Ethanol and butanal are the lightest components; therefore, they will be recycled together. Moreover, due to formation of several azeotropes (see Table 2.4), the recycle will contain some amounts of water. The model of the Reactor–Separation–Recycle systems consists of Equations (15.28)–(15.36), where a kinetic model was used for the reactor and separation targets were imposed. Mixer (4 equations) FK, 0  FK , 1 + FK , 3 ¼ 0, K ¼ E,B, A,W

(15.28)

FK, 1  FK , 2 + nK rðcE , cB , cA , cW Þ ¼ 0, K ¼ E,B, A,W

(15.29)

Reactor (4 equations) P where cK ¼ FK,2/ KVm,K  FK,2 and the molar volumes Vm,K (in 103 m3/kmol) are: 89.568 (butanal), 58.173 (ethanol), 150.342 (1,1 diethoxy butane) and 18.05 (water). Column 1 (four equations): FK, 2  FK, 3  FK , 4 ¼ 0, K ¼ E,B,A, W

(15.30)

15.3 PLANTWIDE CONTROL

629

Ethanol, butanal, water 3

Ethanol (E) 0

Mixer

1

Reactor

2

Column 1

0 Butanal (B) B+2E

A+W 4 Water 1,1 diethoxy butane

FIGURE 15.23 Reactor–Separation–Recycle structure of the 1,1 diethoxy butane plant.

Separation targets (4 equations) FE, 2  FE, 3 ¼ 0

(15.31)

FB, 2  FB, 3 ¼ 0

(15.32)

FA, 2  FA, 4 ¼ 0

(15.33)

0:34FB, 3 + 0:12FE, 3  FW , 3 ¼ 0

(15.34)

FW , 0 0:105 ¼ ðazeotrope compositionÞ FE, 0 0:895

(15.35)

FA, 0 ¼ 0

(15.36)

Feed rates (two equations)

To solve the model, the amount of catalyst mcat is required. Then, two degrees of freedom are left. The additional specification leads to different control structures: • •

CS1: fix the butanal feed rate FB,0 and the mixed ethanol–water–butanal flow rate F1, similar to Figure 15.22 (left). CS2: fix the butanal feed rate FB,0 and the ratio M ¼ F1/FB,0 similar to Figure 15.22 (right).

In the following, we aim to determine the amount of catalyst mcat and the setpoint of the second flow controller (F1 or M) such that reasonably large changes of the production rate can be achieved.

630

CHAPTER 15 PLANTWIDE CONTROL

The model equations can be written in the following condensed form: f ðx, p, qÞ ¼ 0

(15.37)

where x ¼ FK,I, K ¼ E, B, A, W and i ¼ 0–4 is the vector of unknowns; p ¼ FB,0 is the (distinguished) parameter; q is the vector of remaining parameters; q ¼ {mcat, F1} for control structure CS1; q ¼ {mcat, M} for control structure CS2. Figure 15.24 shows the dependence of the butanal conversion, defined as XB ¼ 1  FB,2/(FB,0 + FB,1), versus the distinguished parameter FB,0, for different values of the remaining parameters {mcat, F1} and {mcat, M}. These are called bifurcation diagrams. For both control structures, two steady states exist at small values of the butanal feed rate FB0 (low production rates). However, as FB0 increases over a certain critical value, no steady state exists. This can be explained by the fact that the capacity of the chemical reactor to convert reactants into products is limited. The range of feasible FB0 values increases when more catalyst is used and when the excess of ethanol reactant (F1 or M) is larger. Moreover, the flexibility of control structure CS2 (right diagrams) is better, as clearly it can handle much larger production rate changes. The ‘turning points’ of the diagrams are also called fold bifurcation points. These points are of special interest, as they are the boundary between feasible operation (two steady states exist) and unfeasibility (no steady exists). The fold points can be found by solving the following defining conditions: f ðx, p, qÞ ¼ 0

(15.38)

f x ðx, p, qÞu ¼ 0

(15.39)

jjujj  1 ¼ 0

(15.40)

where fx (x, p, q) is the Jacobian matrix, u is an auxiliary vector, and || || denotes a vector norm. The dependence of the distinguished parameter p at the fold bifurcation versus one of the model parameter from the set q (e.g. mcat) represents the bifurcation set. The bifurcation set, presented in Figure 15.25 for both control structures, divides the (FB0, mcat) space into a feasible and an unfeasible region. Clearly, control structure CS2 offers higher flexibility and it will be further investigated. From Figures 15.24 and 15.25, it appears that the production rate could be increased by about 75% (from 10 to 17.5 kmol/h) when mcat ¼ 25 kg of catalyst are used and the ratio between the reactor-inlet flows is set to M ¼ 12. However, other disturbances (such as purity of the ethanol feed xE,0 and the mole fraction of water in the recycle stream xW,3) are possible and their effect must be also investigated. The bifurcation set is presented in Figure 15.26, for the flow ratio M ¼ 12 and two different values of the catalyst amount mcat. The nominal operating point is denoted by a marker. The rectangles represent operating windows, corresponding to 10–14 kmol/h production rate and 0.8–0.9 feed ethanol purity. If the mole fraction of water in the recycle stream xW,3 increases from the design value 0.15 to 0.2, the design with 25 kg of catalyst cannot cope with the combined disturbances. The design employing mcat ¼ 50 kg appears more robust and will be further considered. Having decided about the values of the design variables, we proceed with the steady-state simulation in Aspen Plus. The flowsheet is presented in Figure 15.27. Fresh butanal and the fresh ethanol + recycle stream are mixed, heated and fed to the chemical reactor (CSTR, 1 m3, 50 kg of catalyst, 40  C). The reactor effluent is sent to distillation (25 stages, 1.2 m diameter, distillate-to-feed ratio ¼ 0.766,

15.3 PLANTWIDE CONTROL

1

1

F1 = 80 kmol/h

631

M=8

0.8

0.8 50

25

mcat (kg) =

200

100

0.6

XB 0.4

0.4

0.2

0.2

0

100

25 50

XB

0.6

mcat (kg) = 200

0

0

5

10

15

0

10

20

FB,0 (kmol/h) 1

30

40

50

60

FB,0 (kmol/h) 1

F1 = 120 kmol/h

0.8

M = 12

0.8 50

25

mcat (kg) =

100

0.6 25

XB

XB

0.6

200

0.4

0.2

0.2

0

5

10

15

0

20

0

FB,0 (kmol/h)

50

100

150

FB,0 (kmol/h) 1

F1 = 160 kmol/h

1

100

mcat (kg) = 200

0.4

0

50

M = 16

0.8

0.8

XB

mcat (kg) =

50

100

0.6

25

XB

0.6

200

100

0.4

0.4

0.2

0.2

0

25

50

0 0

5

10

15

FB,0 (kmol/h)

20

25

0

50

mcat (kg) = 200

100

150

200

FB,0 (kmol/h)

FIGURE 15.24 1,1 Diethoxy butane plant: conversion versus butanal feed rate bifurcation diagrams. Left: control structure CS1; Right: control structure CS2.

632

CHAPTER 15 PLANTWIDE CONTROL

25 20

120

F B,0 (kmol/h)

F B,0 (kmol/h)

200

FE,1 (kmol/h) = 160

Unfeasible

15 80 10 Feasible 5

150

Unfeasible

M = 16

100

12

50

8 Feasible

0

0 0

50

100

150

0

200

50

100

150

200

mcat (kg)

mcat (kg)

FIGURE 15.25 Bifurcation set, dividing the space of operating parameters into feasible and unfeasible regions. Left: control structure CS1; Right: control structure CS2.

40

20

F B,0 (kmol/h)

30

F B,0 (kmol/h)

Unfeasible

15

0.2 10

0.25

5

xW,3 = 0.15

Unfeasible

xW,3 = 0.15

0.2 0.25

20

10

Feasible

Feasible

mcat = 25 kg M = 12

0

mcat = 50 kg M = 12

0 0.5

0.6

0.7

0.8

xE,0

0.9

1

0.5

0.6

0.7

0.8

0.9

1

xE,0

FIGURE 15.26 Bifurcation set for control structure CS2. Ethanol purity xE,0 and water concentration in the recycle stream xW,3 are considered as uncertain parameter.

reflux ratio ¼ 2). The distillate, containing ethanol, butanal and water is recycled. The bottoms stream containing 1,1 diethoxy butane and water is sent to liquid–liquid separation which takes place at 30  C. The water phase is removed, while the organic phase containing significant amounts of water is sent to the second distillation column (20 trays, 0.3 m diameter, distillate-to-feed ratio ¼ 0.121, reflux ratio ¼ 2). The distillate product is the water–1,1, diethoxy butane azeotrope, which is sent back to liquid–liquid separation. The bottom product contains 1,1 diethoxy butane of 0.995 purity. The stream report is presented in Table 15.9. The control structure fixes the fresh butanal flow rate and the ratio between the reactor and inlet streams, as previously described. Reaction temperature and reactor level are controlled by the duty and outlet flow rate. Control of the distillation columns is standard: pressure, reflux drum and column sump levels are controlled by condenser duty, distillate and bottoms flows, respectively. One temperature in the stripping zone is controlled by the reboiler duty. The reflux rate is constant. Decanting

15.3 PLANTWIDE CONTROL

633

FIGURE 15.27 1,1 Diethoxy butane plant plant – flowsheet and plantwide control.

temperature and the level of the organic and aqueous phases are controlled by the duty and outlet flow rates. Details of the control loops and controller tuning are given in Table 15.10. Results of dynamic simulation are presented in Figure 15.28. The simulation starts from steady state. Various disturbances are introduced at time t ¼ 1 h. The top diagrams show the effect of increasing the butanal feed rate (FB,0) by 50%. The ethanol feed rate (FE,0) is quickly adjusted to the corresponding stoichiometric ratio. The new production level (FA, FW) is attained after about 8 h. The purities of both 1,1 diethoxy butane and water product streams remain above 0.995. The bottom diagrams show the effect of changing the fresh ethanol purity from 0.895 (azeotrope composition) to 0.8. The flow rate of fresh ethanol is quickly adjusted. After few hours, the flow rate of water product stream reaches the new steady-state value. There is no noticeable effect on product purity.

15.3.5 COMPLEX REACTIONS 15.3.5.1 Example – vinyl acetate plant The vinyl acetate process was proposed as a benchmark design and plantwide control problem (Luyben and Tyreus, 1998). Here, we will give a brief description of the plant and we will outline the development of a plantwide control structure. More details, including dynamic simulation result, can be found elsewhere (Dimian and Bildea, 2008).

Table 15.9 1,1 Diethoxy Butane Plant – Stream Table Temperature ( C) Pressure (bar) Total flow (kg/h) Total flow (kmol/h)

B0

E0

Rin

Rout

Recycle

AW1

AW2

AW3

A

W

19.9 1.00 721.1 10.00

19.9 1.00 938.3 20.91

40.0 1.50 4694.7 99.34

39.9 1.50 4694.7 89.34

78.5 1.10 3035.3 68.43

104.7 1.34 1659.4 20.91

30 1.30 1506.9 11.43

30 1.10 44.27 1.38

159.5 1.29 1462.1 10.04

90.0 1.30 197.4 10.87

0.000 0.000 1.000 0.000

0.000 0.958 0.000 0.042

0.000 0.703 0.176 0.121

0.112 0.558 0.084 0.246

0.000 0.728 0.109 0.162

0.478 0.001 0.000 0.521

0.886 0.006 0.000 0.107

0.098 0.05 0.000 0.851

0.995 0.000 0.000 0.005

0.001 0.002 0.000 0.997

Mole fraction Diethoxy Ethanol Butanal Water

15.3 PLANTWIDE CONTROL

635

Table 15.10 1,1 Diethoxy Butane Plantwide Control – Controller Tuning Controller

PV, Value and Range

OP, Value and Range

Kc (%/%)

Ti (min)

Level ¼ 1.625 m 0    3.25 m

Etanol feed ¼ 938 kg/h 0    2000 kg/h

10

60

Level ¼ 1.356 m 1    1.5 m Temperature ¼ 40  C 20    60  C

Outlet flow rate ¼ 6200 kg/h 0    12,000 kg/h Duty ¼ 0.1  106 kcal/h 0.5  106    0 kcal/h

10

60

5

6

Pressure ¼ 1.1 bar 1    1.2 bar

2

12

Stage 23 temperature ¼ 98.9  C 90    110  C Reflux drum level ¼ 1.875 m 0    3.75 Level, sump ¼ 1.875 m 0    3.75 m

Condenser duty ¼  2.57  106 kcal/h 0    4.7  106 kcal/h Reboiler duty ¼ 2.75  106 kcal/h 0    5  106 kcal/h

1

20

Distillate rate ¼ 4541 kg/h 0    20,000 kg/h Bottoms product ¼ 1659 kg/h 0    3400 kg/h

1

600

1

600

Level, organic phase ¼ 1.05 m 0.5    1.5 m Lever, aqueous phase ¼ 0.8 m 0.4    1.2 m Temperature ¼ 30  C 20    40  C

Organic product ¼ 1526 kg/h 0    3000 kg/h Water product ¼ 197 kg/h 0    400 kg/h Duty ¼ 0    0.066  106 kcal/h 0.15  106    0 kcal/h

10

60

10

60

Pressure ¼ 1.1 bar 1    1.2 bar

Condenser duty ¼  0.06  106 kcal/h 0    0.12  106 kcal/h Distillate product ¼ 64 kg/h 0    130 kg/h Bottoms product ¼ 1462 kg/h 0    3000 kg/h Reboiler duty ¼ 0.11  106 kcal/h 0    0.25  106 kcal/h

2

12

1

600

1

600

1

20

Mixer

Reactor

COL-1

Decanter

COL-2

Reflux drum level ¼ 0.5 m 0  1 m Sump level ¼ 1.25 m 0    2.5 m Stage 18 temperature ¼ 117.3  C 105    125  C

636

CHAPTER 15 PLANTWIDE CONTROL

F3

20 FB,0

15

100

FA

10

50

FE,0

5

1 zE,3

0.8

150

FW

Mole fraction

Flow rate (kmol/h)

1

200

25

0.998 zw

0.6

0.996

zA

0.4

0.994 zw,3

0.2

0.992

zB,3

0

0

0 0

2

4

6

8

0.99 0

10

2

8

10

FW

F3

20

0.8

FB,0

100

10 50

FE,0

5

Mole fraction

150 FA

1

1

200

25

Flow rate (kmol/h)

6

Time (h)

Time (h)

15

4

0.998

zE,3 zw

0.6

0.996

zA

0.994

0.4 zw,3

0.2

0.992

zB,3

0

0 0

1

2

3 Time (h)

4

5

0.99

0

0

1

2 3 Time (h)

4

5

FIGURE 15.28 Acetal plant – dynamic simulation results. Top – production rate is increased by 50%, from 10 to 15 kmol/h. Bottom – decrease in the fresh ethanol purity from 0.895 (azeotropic composition) to 0.8.

The manufacturing of vinyl acetate by the oxy-acetylation of ethylene is described by the following gas-phase catalytic reaction: C2 H4 + CH3 COOH + 0:5O2 ! C2 H3 OCOCH3 + H2 O Dr H ¼ 176:2kJ=mol

A highly undesired secondary reaction is combustion of ethylene to CO2: C2 H4 + 3O2 ! 2CO2 + 2H2 O Dr H ¼ 1322:8kJ=mol

The plant flowsheet is presented in Figure 15.29 (reaction section) and Figure 15.30 (separation section). The exothermic reactions take place in a multi-tubular, catalytic reactor. The reaction heat is removed by generating steam. Pressurized water is fed to the shell-side of the reactor and returned to a steam drum, to which make-up water is supplied. The reactor effluent is used to preheat the feed and further cooled with cooling water, before being submitted to vapour–liquid separation. The vapour stream contains significant amounts of vinyl acetate, which is recovered by absorption in acetic acid. The liquid bottom stream from the absorber is combined with the liquid stream from the vapour–liquid separator and sent to the distillation column. The azeotropic distillation column separates the vinyl acetate and water from the acetic acid. The overhead product is condensed, when two liquid phases separate. A fraction of the organic phase, in which the mole fraction of vinyl acetate exceeds 90%, is removed, while the rest is returned as reflux to the distillation column. Part of the overhead gas leaving the absorber is sent to a

15.3 PLANTWIDE CONTROL

637

XC

Oxygen feed

PC

Steam TC

TC

FFC

Reactor

+ +

LC XC

Ethylene feed

TT FC

Acetic acid recycle

TT

Evaporator

Steam generator

HS

TT

Gas recycle (from separation)

PC LC

Acetic acid to Separation

FEHE

LC

Acetic acid storage

Acetic acid (to absorption column)

FIGURE 15.29 Vinyl acetate plant: flowsheet and control of the reaction section.

XC

CO2

CO2 removal PC

Acetic acid (from storage)

PC

Vent

FC TC

x Absorption

TC

TC

LC

LC

Vinyl acetate

Water

PC

LC

From Reactor

TC TC

Vapour–liquid split

Distillation

LC LC

FIGURE 15.30 Vinyl acetate plant: flowsheet and control of the separation section.

Acetic acid (to storage)

638

CHAPTER 15 PLANTWIDE CONTROL

CO2 removal system. The recycled gas is preheated, then mixed with fresh ethylene and vapourized acetic acid. Finally, oxygen is added at a concentration which should not exceed 8%. Figure 15.29 presents the control loops around the chemical reactor, where the main safety issues arise. Firstly, the reactor-inlet mixture must not contain more than 8% oxygen in order to avoid explosion risks. Therefore, the oxygen is added under concentration control. Secondly, the cooling should avoid reaction runaway. In a runaway situation, the excessive temperature leads to danger of explosion, catalyst deactivation and drastic decrease of the selectivity. The coolant is circulated at a constant rate. Several temperature measurements are placed along the reactor bed and the highest value is selected as the process variable. The manipulated variable of the control loop is the steam drum pressure, which directly influences the coolant temperature. The water level in the steam drum is controlled by the water make-up. Note that using a simple feedback loop may not work. When the steam rate increases, the correct action is to add more water make-up. However, the pressure simultaneously decreases. The lower pressure means that, initially, the steam bubbles will occupy a larger volume, and the liquid level will increase (inverse response). A feedback level controller will wrongly decrease the water make-up rate. Therefore, the steam rate is measured and the required water make-up is calculated. This feedforward action is combined with the feedback provided by the level controller. The inventory of reactants in the plant is maintained by fixing the reactor-inlet flows. Acetic acid is evaporated at a constant rate. Evaporator pressure is maintained by the heating duty, while the fresh acetic acid is added to keep the constant level. The flow rate of fresh ethylene is used to control the ethylene mole fraction, after mixing with the recycle and the acetic acid. The fresh oxygen rate is manipulated by a concentration control loop, as previously explained. The control of the separation section is presented in Figure 15.30. Although the flowsheet seems complex, the control is rather simple. The separation must deliver recycle and product streams with the required purity: acetic acid, vinyl acetate and water. Therefore, the distillation column can be operated at constant reflux, while boilup rate is used to control a temperature in the lower section of the column. For the absorption column, the flow rate of the absorbent (acetic acid) is kept constant. The concentration of CO2 in the recycle stream is controlled by changing the amount of gas sent to the CO2 removal unit. The additional level, temperature and pressure control loops are standard. Note that production rate changes cannot be achieved by manipulating the reactor-inlet flows because the per-pass conversions are quite low (10% for ethylene, 25% for acetic acid), while the reactorinlet oxygen concentration is restricted by safety concerns. Therefore, the strategy manipulating the reaction temperature should be applied.

15.4 SUMMARY OF PLANTWIDE CONTROL METHODOLOGY The plantwide control principles presented and used in the previous sections can be summarized in the following steps: 1. Establish the steady-state and dynamic control objectives (usually the range of production rate and product purity), the expected disturbances (such as the variability of raw materials) and the process constraints (e.g. reaction temperature, reactants ratio, pressures). 2. Assign control objectives to the separation units. These must ensure that the products, by-products and inert species leave the plant. 3. Add control loops to manage the reactants inventory.

15.5 CONCLUDING REMARKS

4. 5.

6. 7.

639

3.1. Limiting reactant: Several cases can be distinguished: (a) Limiting reactant is recycled alone and the plant-inlet flow is available as manipulated variable: Set the reactor-inlet flow rate (fresh + recycled) on flow control. Use the plantinlet feed to control the inventory (level on a buffer vessel, pressure). (b) Limiting reactant is recycled mixed with other species: set the plant-inlet flow rate on flow control. The plant must be designed for high conversion (large reactor, excess of co-reactants), to ensure that the inventory is self-regulating with reduced sensitivity to disturbances. 3.2. Co-reactants: (a) Co-reactant is recycled alone or at high purity (e.g. mixed with the limiting reactant): set the reactor-inlet flow rate by means of a co-reactant/reactant ratio controller. Use plantinlet flow rate to control the inventory. (b) Co-reactants are recycled together: use concentration measurements and control loops to ensure appropriate reactants ratio at the reactor inlet. 3.3. Inert species that cannot be separated from reactants: use purge flow to keep their concentration under control. Use a Reactor–Separation–Recycle model to assess the feasibility and steady-state performance of the control structure. Decide how the production rate will change. Identify the dominant variable(s), which significantly affect the reactant transformation rate (kmol/h). If the plant was designed for high conversion of the limiting reactant, the reactor-inlet or the plant-inlet flow rate work well, the former one being preferred. For the low-conversion design, candidate variables are reactor hold-up (if possible), reaction temperature (assuming no safety and yield constraints) and amount of catalyst. Address the control of the separation units: inventory and quality (according to values specified at step 2). Use rigorous dynamic simulation to assess the dynamic performance of the control system.

15.5 CONCLUDING REMARKS The problem of designing the plantwide control structure can be solved by a systemic approach, in which modelling the plant as a Reactor–Separation–Recycle system proves very useful. Today, the industrial experience allows designing chemical reactors and separation units with good controllability properties. Various books on process control present the models and mathematical tools (Ogunnaike and Ray, 1994; Luyben and Luyben, 1997) and various control schemes that can be applied at the unit level (Smith and Corripio, 2005). Plantwide control must ensure that these sub-systems also work well when coupled together (Luyben et al., 1999). In this context, maintaining the mass balance of the plant (controlling the species inventory) is the main task of plantwide control. Because the separation section removes the products from the plant, the main issues arise from controlling the reactants mass balance, for which two strategies are possible. In the feedback control strategy, the inventory of each reactant is measured and controlled by means of plant-inlet flow rate. Sometimes this is not possible (reactants are recycled together, plant-inlet flow is set by an upstream unit). Then, the control structure must rely on self-regulation, which works well only for the limiting reactant. Additionally, high conversion is required, which can be achieved by a large reactor, fast kinetics (catalyst) or large excess of co-reactants. In this chapter,

640

CHAPTER 15 PLANTWIDE CONTROL

these principles were illustrated by three case studies of increasing complexity. More case studies can be found in Dimian and Bildea (2008).

15.6 APPENDIX

15.6.1 LAPLACE TRANSFORM AND TRANSFER FUNCTIONS Laplace transform and transfer functions are essential tools for analysing the behaviour of dynamic systems, designing control structures and tuning controllers. This section presents the most important properties of Laplace transforms and transfer functions. A function f(t) is called an original function if the following conditions are fulfilled: (a) f(t) ¼ 0 for t < 0 and f(t) is piecewise continuous for t > 0 (b) the constants M and a exist such that M eat > |f(t)|. The Laplace transform of an original function is defined as follows: L½ f ðtÞ ¼ FðsÞ ¼

ð1

f ðtÞest dt

(15.41)

0

As a result of this operation, the time-dependent function f(t) is transformed into a function F(s) depending on the complex variable s. For example, the Laplace transforms of the step, u(t), and decaying exponential, h(t), functions (Figure 15.31) are given by: U ð sÞ ¼

ð1

1e

st

0

1 dt ¼ and HðsÞ ¼ s

ð1

eat est dt ¼

0

1 s+a

(15.42)

If f(t) and h(t) have the Laplace transform F(s) and H(s), then the following useful relationships hold:

1.2

1.2

1

1

0.8

0.8

0.6

u ( t) =

再

h (t)

u (t)

L½af ðtÞ + bhðtÞ ¼ aFðsÞ + bH ðsÞ

0 for t < 0 1 for t ³ 0

0.6

0.4

0.4

0.2

0.2

0 –1

(15.43)

h( t) =

再

0 for t < 0 e− a.t for t ³ 0

0 0

1

2

t

FIGURE 15.31 Step and decaying exponential functions.

3

4

–1

0

1

2

t

3

4

15.6 APPENDIX

641

 df ðtÞ L ¼ FðsÞ  f ð0Þ dt

(15.44)

ð t FðsÞ f ðtÞ dt ¼ L s 0

(15.45)

L½f ðt  yÞ ¼ esy FðsÞ

(15.46)

lim f ðtÞ ¼ lim sFðsÞ, if all poles of sFðsÞ are in the left half  plane

(15.47)

t!1

s!0

Let us consider a linear, time invariant dynamical system (15.48): dxðtÞ ¼ AxðtÞ + BuðtÞ dt yðtÞ ¼ CxðtÞ xð0Þ ¼ 0

(15.48)

where x(t), u(t) and y(t) are the state, input and output variables, respectively, defined as deviations from the steady-state values. The Laplace transform allows replacing the state-space representation (15.48) by the transfer function: Y ðsÞ ¼ GðsÞUðsÞ

(15.49)

GðsÞ ¼ CðsI  AÞ1 B

(15.50)

where

Then, solution of the differential equations in the time domain is replaced by calculating the Laplace transform of the input u(t), multiplication by the transfer function G(s) and inversion of the output Laplace transform Y(s). The procedure is shown in Figure 15.32. Note that the transfer function is a strictly

u(t)

Laplace transform

U(s)

Linear differential equations Deviation variables initially at steady state

y(t) Inversion of Laplace transform

Transfer function

Y(s) = G(s) U(s)

FIGURE 15.32 Use of Laplace transform and transfer functions.

Y(s)

642

CHAPTER 15 PLANTWIDE CONTROL

proper rational function, that is, is the ratio of two polynomials Q(s) and P(s), the order of the denominator being larger than the order of the numerator. Therefore, the Laplace transform can be inverted by using partial fraction expansion: Y ðsÞ ¼

X Ak Q ðs Þ QðsÞ ¼ ¼ ð s  pk Þ PðsÞ ðs  p1 Þ. . . ðs  pN Þ k yðtÞ ¼

X

(15.51)

Ak epk t

(15.52)

k

where Ak ¼ lim ½ðs  pk ÞY ðsÞ

(15.53)

s!pk

Transfer functions are very convenient tools for analysing the stability of dynamical systems. Loosely speaking, a system is said to be (strictly) internally stable if, for an initial condition x(0) 6¼ 0 and u(t) ¼ 0, the system returns to the steady state, therefore lim xðtÞ ¼ 0. Internal stability requires that all the eit!1 genvalues of the system matrix A have negative real part. Similarly, a system is said to be (strictly) externally stable if input changes of bounded magnitude lead to output changes of bounded magnitude. External stability requires that all the poles of the transfer function (roots of the denominator) have negative real part. Internal stability implies external stability, while the reciprocal is not always true. The transfer functions of coupled systems, for example, in series, parallel and feedback, can be easily calculated from the transfer functions of the sub-systems (Figure 15.33).

15.6.2 TYPICAL DYNAMIC BEHAVIOUR The zero-order system (integrator) A tank fed with a rate Fin(t) and from which a constant flow Fout is removed is the simplest example of a zero-order (integrating) system (Figure 15.34). Considering the feed flow rate u(t) ¼ Fin(t)  Fout and tank hold-up m(t)  m0 as input and output variables, respectively, the transfer function of the zero-order system becomes: GðsÞ ¼

U1(s) G1(s)

G2(s)

Y ( s) = G2 (s )  G1( s )  U(s )

(15.54) Feedback connection

Parallel connection

Series connection

U(s)

1 s

Y(s) U2(s)

G1(s)

Y(s) +

U(s) + –

G1(s)

G2(s)

Y(s)

G2(s)

Y ( s ) = G1( s)  U1 (s ) + G2 (s )  U2 ( s )

Y ( s) =

G2 ( s)  G1 (s )  U (s ) 1 + G2 ( s )  G1 ( s)

FIGURE 15.33 Transfer function of systems in connected in series (left) and parallel (middle) and negative feedback (right).

15.6 APPENDIX

643

The zero-order system is not strictly externally stable, as any persistent deviation of the feed flow rate Fin(t) from the steady-state value F(t) will lead to an unbounded change of the tank hold-up m(t).

The first-order system A simple first-order system is shown in Figure 15.35. The first-order system is characterized by the transfer function G ðs Þ ¼

K sT + 1

(15.55)

Fin(t) – Fout

Fin(t)

m(t) – m0

m(t)

t

t

( Fin (t ) – Fout )dt

0

Fout = constant

t

FIGURE 15.34 Zero-order system (integrator).

Fin(t) – Fout

Fin(t)

Du t m(t) – m0

m(t)

0.632·Dy Fout ( t ) = k  m

FIGURE 15.35 First-order system.

T

Dy

t

644

CHAPTER 15 PLANTWIDE CONTROL

The process gain K is a measure of the ‘strength’ of the input variable. It can be determine experimentally as the ratio between the output and input changes, K ¼ Dy/Du. Alternatively, it can be calculated from the transfer function: K ¼ Gð0Þ

(15.56)

During one time constant, the process achieves 62.3% of the total movement. The time constant is an indication of the speed of response.

The non-interactive second-order system A non-interactive second-order system is obtained by coupling two first-order systems in series, as shown in Figure 15.36. Often, the behaviour of the non-interactive second-order system can be approximated by a dead-time t and a first-order element with time constant T.

The interactive second-order system When feedback is present, the interaction between the sub-systems occurs in both directions (Figure 15.37). The transfer function of the system is given by (15.57), where T is the time constant and z is the damping coefficient. GðsÞ ¼

K T 2 s2 + 2zTs + 1

(15.57)

Oscillatory behaviour occurs for z < 1. In this case, the overshoot s and settling time T0.02 are given by: pffiffiffiffiffiffiffi2ffi 4T s ¼ ezp= 1z ; T 0:02 ¼ z

(15.58)

For example, z ¼ 0.3 leads to s ¼ 0.37 and T0.02/T ¼ 10.8

Fin(t)

Fin(t) – Fout

Du

m1(t) F1 ( t ) = k1 m1

t

m2(t) – mo

m2(t)

Dy F2 ( t ) = k 2  m2

t

FIGURE 15.36 Second-order system (non-interactive).

T

t

15.6 APPENDIX

Fin (t ) = − K c  (mSP 2 − m2( t ) )

645

m SP 2

m1(t) t

F1( t ) = k 1 m1

m

SP 2

− m2 ( t ) Large Kc

LC m2(t) m SP 2

Small Kc

F2 ( t ) = k 2  m2

t

FIGURE 15.37 Second-order system (interactive).

Vapour Duty Liquid level Duty t Level

Duty

More bubbles 1 G1 ( s ) = 2s + 1

+

Level

+ Evaporation −10 40s + 1

G 2 (s ) =

t

FIGURE 15.38 Second-order system with inverse response.

The second-order system with inverse response The reboiler of a distillation column (Figure 15.38) is a typical system with inverse response. When the reboiler duty increases, more bubbles are generated. Due to the lower density of the vapour–liquid mixture, the level initially increases. After some time, when the bubbles leave the system, the level drops, below the initial value.

646

CHAPTER 15 PLANTWIDE CONTROL

Cin(t) Du

Cout(t)

Cin(t)

t G ( s) = e−t s

C(t)

Concentration measurement

Cout(t)

Cm(t)

Dy = Du t

t

FIGURE 15.39 Dead-time system.

Inverse response can be recognized by a positive zero in the transfer function, for example: G ðs Þ ¼

20s  9 ð2s + 1Þð40s + 1Þ

(15.59)

The inverse response makes the control difficult, because the controller has to wait until the process variable moves in the right direction.

The dead-time Dead-time occurs in pipe flow, plug-flow reactor and concentration measurements (Figure 15.39). The transfer function of a dead-time, G(s) ¼ ets can be expressed as a Pade´ approximation, for example: e

ts

1 1 2 ets=2 1  2 ts + 12 ðtsÞ ¼ ts=2 1 1 e 1 + ts + ðtsÞ2 2 12

(15.60)

pffiffiffi The Pade´ approximation (15.60) has two positive zeros, 3 3. Similarly to inverse response, the dead-time is detrimental for control.

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