8.21

Distillation: Optimization and Advanced Controls

QT V L

H. L. HOFFMAN, D. E. LUPFER B. A. JENSEN

(1995)

L. A. KANE

(1970)

B. G. LIPTÁK

D

(1985) Li

F

(2005)

V

LF

V QB LF

B

Flow sheet symbol

INTRODUCTION Section 8.19 described the basic, single-input single-output (SISO) distillation control systems. These simple control schemes do keep the operation stable, but they cannot optimize it and they do necessitate that the operator, as plant conditions change, periodically readjust the set points of these SISO loops. In Section 8.20, it was noted that a two-product distillation tower has five controlled and five manipulated variables. Because pressure is usually controlled to close the heat balance and the two levels are controlled to close the material balance around the column, eight configurations are possible to control product compositions (Table 8.20b). Interaction always exists between the material and energy balances in a distillation column. Section 8.20 describes how the interaction between the two composition control loops can be minimized by calculating the eight corresponding relative gain (RG) values and selecting the pairing, which gives an RG closest to 1.0. Control of distillation towers involves the manipulation of the material and energy balances in the distillation equipment to affect the composition of the products. This section builds upon the previous two, while focusing on optimization and on the use of multivariable advanced process controls 1 (APC). In today’s competitive market, it is necessary to push equipment to operating limits to maximize production rate or minimize the energy cost of production. Advanced process controls are usually distinguished from regulatory SISO controls by being multivariable in nature (multiple input/multiple output) and by utilizing some model of the process. The APC products on today’s market can be distinguished on the basis of their approach to modeling the process. They can be grouped into three categories: The white box models apply to well understood processes, such as distillation, where theoretical dynamic models of the pro1866 © 2006 by Béla Lipták

cess can be derived based on mass, energy, and momentum balances of the process. The fuzzy logic and black box models are used for processes that are poorly understood or when it is acceptable to use a complete mechanistic empirical model constructed solely from a priori knowledge. Because of the well-understood nature of distillation, this section will give emphasis to the white box approach to modeling. The goal of this section is to provide instrument engineers with the tools necessary to design unique advanced control strategies that will match the requirements of the specific distillation columns they encounter. The section will first discuss the various APC control strategies, and after that it will describe a variety of optimization schemes. After a listing of APC-related definitions, the discussion of APC in distillation will first discuss the black box and the fuzzy logic techniques, which are less applicable to this well-understood process. After this brief treatment, a more detailed discussion of the development of the white box models will be presented. Definitions (ANN): ANNs can learn complex functional relations by generalizing from a limited amount of training data; hence, they can thus serve as black box models of nonlinear, multivariable static and dynamic systems and can be trained by the input/output data of these systems. ANNs attempt to mimic the structures and processes of biological neural systems. They provide powerful analysis properties such as complex processing of large input/output information arrays, representing complicated nonlinear associations among data, and the ability to generalize or form concepts-theory.

ARTIFICIAL NEURAL NETWORKS

8.21 Distillation: Optimization and Advanced Controls

BLACK BOX MODEL:

See EMPIRICAL MODEL. EMPIRICAL MODEL: This type of model can be used for processes for which no physical insight is available or used. This model structure belongs to families that are known to have good flexibility and have been “successful in the past.” The parameters of the models are identified based on measurement data. A complete mechanistic model is constructed from a priori knowledge. FUZZY LOGIC MODELING: This type of model is used for processes that are not fully understood. It is a linguistically interpretable rule-based model that is based on the available expert knowledge and measured data. MODEL-BASED CONTROL (MBC): In model-based control, a process model is used to make control decisions. The controller uses this model of the process to calculate a value for the manipulated variable, which should make the controlled variable behave in the desired way. The “inverse” nomenclature arises from how the model is used. In a normal modeling approach, one specifies the process input, and the model predicts the process output response. By contrast, MBC determines the process input (manipulated variable) that will cause a desired process output response (controlled variable value) to occur. This is the model inverse. MODEL PREDICTIVE CONTROL (MPC): is a model-based control technique that uses process output prediction and calculates consecutive controller moves in order to satisfy control objectives. OPEN-LOOP GAIN: The steady-state gain of a control loop when the other control loop(s) is (are) in manual. (Their control valve opening is constant.) RELATIVE GAIN: RG is the ratio of the steady-state gain of the loop with other loops in manual, divided by the steady-state gain of the loop when the other loops are in automatic. RELATIVE GAIN ARRAY: A matrix of dimensionless gain ratios giving one RG value for each pairing of manipulated and controlled variables. WHITE BOX MODELING: This type of modeling is feasible if a good understanding of the process exists. In such cases, the dynamic models are derived based on mass, energy, and momentum balances of the process.

ADVANCED PROCESS CONTROL Fuzzy logic- and black box-type model-free expert systems can be compared to the behavior of tennis players. The players do not necessarily understand Newton’s laws of motion or the aerodynamic principles that determine the behavior of a tennis ball, but they have simply memorized the results of a large number of past responses. This is also the basis of human learning. All the neural network software packages on the market mimic this method of learning.

© 2006 by Béla Lipták

1867

Neural networks, fuzzy logic, and statistical process control are all such methods, which can be used without the need for knowing the mathematical model of the process. The major difference between fuzzy logic and neural networks is that the latter can only be trained by data, but not with reasoning. Fuzzy logic is superior from this perspective, because it can be modified both in terms of the gain (importance) and also in terms of the functions of its inputs. The main limitations of all model-free expert systems is their long learning period (which can be compared to the growing up of a child) and the fact that their knowledge is based solely on past events. Consequently, they are not prepared to handle new situations, and therefore if the process changes, they require retraining, because they are not well suited to anticipation. Model-based control, model predictive control, and internal model control (IMC) are all based on white box modeling and are all suited for the optimization of such unit processes that are well understood, such as heat transfer or distillation. Their performance is superior to that of the model-free systems (fuzzy logic and black box), because they are capable of anticipation and, thereby, can respond to new situations. In this sense their performance is similar to that of feedforward control systems, while the model-free systems behave in a feedback manner only. In this section, the APC control strategies that are based on fuzzy logic and black box models will be discussed first. This discussion will be followed by a more in-depth explanation of the white box model-based controls. The Goals of APC Advanced control strategies attempt to compensate for process deviations in the shortest time possible by accounting for process dynamics, dead times, time delays, and loop interactions. The benefits of better control 2 are: • • • • • • •

Increased throughput Increased product recovery Energy conservation Reduced disturbances to other processing units Minimum rework or recycle of off-spec products Reduced operating personnel Increased plant flexibility

For example, good product composition control of distillation towers can save 5–15% of the energy required to 3 achieve the required separation. The goal of basic distillation controls is to keep the unit running. The objective of advanced control is to keep it running at maximum profitability. The techniques available to implement advanced control include feedforward control; optimization, including constraint control; and model-based and multivariable control (MVC). The challenge is to utilize the technique, the tools, and the available resources to design unique advanced control

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strategies that will match the specific objectives for the distillation columns. The choice between any of these control techniques depends upon factors such as preference and familiarity, complexity of scheme, degree of optimization, hardware for application, and number of variables monitored and controlled by single strategy. Often, additional instrumentation is not needed when implementing advanced controls by building upon basic control designs. However, in many cases, new measurements are needed for calculation or compensation in order to implement an advanced control strategy. These must be retrofitted to the process. Unlike basic distillation control, in which much of the control can be implemented by analog control systems, advanced control strategies usually require the use of higher-level computing systems. Optimization programs and model-based controls require large amounts of computing power. It is for this reason that APC control systems can be distributed over a variety of control equipment types in some kind of hierarchical or distributed fashion.

Influence of disturbances

Set point

Controller

Process

Output

Influence of disturbances

Set point

IMC controller

Process

Output

Model

FIG. 8.21a The configuration of a PID control loop (top) and an internal model controller loop (bottom).

Model-Based Control The strategies presented in Sections 8.19 and 8.20 implement distillation control using PID controllers. Efforts have been made to improve PID performance by considering the dynamic nature of the fractionator, the nonlinearity of the system, and the decoupling of interactions. Model-based controls have been gaining increasing popularity and have been discussed in detail in Sections 2.13 to 2.18 in Chapter 2. These use alternatives to the PID algorithms 12 such as the internal model controller, model algorithmic con13 14 15 trol, dynamic matrix control, and neural controllers. Process model-based control uses an approximate process model directly for control in order to overcome the coupling effects in the distillation tower. Most of these methods are nonlinear, all are predictive, and many are multiple–input multiple-output (MIMO). All depend upon the availability of some process model. Once a process model has been established, it is possible to build the inverse of that model, which can be used as a controller. In that sense, the PID controller is a linear inverse model of a single loop. All control design is basically a model-based activity. This is true even with the PID controller, which uses firstand second-order lag approximations of the process to determine tuning parameters. An alternative to the PID controller is a linear model built into the controller. A simple model-based controller is the internal model controller. The difference between the PID and IMC controller is shown in Figure 8.21a. Note that the IMC looks like it has the same structure as a Smith predictor in Figure 8.19x in Section 8.19. The difference is that the process model is explicitly an internal part of the controller model in the IMC. For a first-order system

© 2006 by Béla Lipták

with dead time, a Smith predictor (Figure 8.19x) with a PI controller is equivalent to an IMC. Nonlinear approximate models include algebraic representation of the McCabe-Theile diagram for both rectifying and stripping sections, short-cut fractionator calculations, and 16 others. These methods require the power of a computer to solve the equations. A number of control strategies also exist once the process model is known. For dual composition con17 trol, one method is the generic model control (GMC), whose control law is described by the following equations:

∫

8.21(1)

∫

8.21(2)

yss = yo + K1,1 ( ysp − yo ) + K 2,1 ( ysp − yo ) dt xss = x o + K1,2 ( xsp − x o ) + K 2,2 ( xsp − x o ) dt

where xsp and ysp = the target set points for bottoms and overhead products xss and yss = the specifications for bottoms and overhead products xo and yo = the current compositions K = tunable parameters for disturbance rejection For example, if K1,2 = 2.5, K2,2 = 0, xsp = 2%, and xo = 1%, then xss = 3.5%. xss is then used in the process model as the basis to compute V/B or any other output (manipulated) variable. Because the same action can be performed for yss with yss being substituted into a process model equation, such as D/L, the model-based control can be multivariable, handling nonlinearity, disturbances, and coupling, by tuning the K values.

8.21 Distillation: Optimization and Advanced Controls

1869

PIC

PIC

LIC

SP

TIC

SP

F

Li

F

(%HKD)

Multivariable controller SP FIC

Stripper

FT

Tower 1

Calculated internal flow

FIC

(%LKB) FIC

LIC Q

LIC

Q AT

AT B

B

FIG. 8.21b Fractionator control using a multivariable controller.

Multivariable Control Multivariable control is a technique that services multipleinput, multiple-output algorithms simultaneously as opposed to the single-input, single-output ones. MVC is particularly well suited for highly interactive multivariable fractionators, where several control loops need to be decoupled. In general, the more difficult the process, the greater are the benefits of multivariable control. Multivariable control techniques can take safety constraints, process lags, and economic optimization factors all into consideration. Like the model-based controller, the MVC-type controller is a predictive controller that uses information from the past plus dynamic models of the process to predict future behavior. Based upon predicted responses, the controller plans future moves to manipulated variables that will minimize the errors in each dependent controlled variable (Figure 8.21b). The control diagram shown in Figure 8.21b illustrates an application of a multivariable controller. In the example, two products and an impurity stream are separated using two towers. The objective is to control the composition of both products. The two composition control loops are coupled so that when any single control action is taken to control one composition, that action also affects the other composition. In this example, the controlled variables are the two product compositions as measured by process analyzers. Feed flow rate is a disturbance variable. The steam to the first

© 2006 by Béla Lipták

column and the temperature at the top of that column are the manipulated variables. A constraint variable is an internal flow as calculated from other tower temperatures and flows. The multivariable controller will take the appropriate steps to control both compositions, subject to the calculated constraint, by adjusting the two manipulated variables while accounting for the dead time caused by the stripper. 13 The identification and command (IDCOM) method is a type of multivariable model algorithmic control. It is based on a process impulse response, which utilizes a predictive heuristic scenario technique to calculate the manipulated variable. The technique is to use a dynamic model to determine future values of the controlled variables. These calculated future values are compared to a desired reference set point trajectory. The manipulated variables are then adjusted to force future controlled variable values to follow the desired reference trajectories. The technique of multivariable control requires the development of dynamic models based upon fractionator testing and data collection. Multivariable control applies the dynamic models and historical information to predict future fractionator characteristics. Predicted fractionator responses result in planned controller actions on the manipulated variables to minimize error for the dependent controlled variable, while considering constraints in the present and the future. This controller is similar to a PID controller, except that the multivariable controller accepts several controlled variable

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Control and Optimization of Unit Operations

set points and load variable measurements and, subject to constraints, outputs several manipulated variables. All multivariable techniques require some sort of process model. Differences between various multivariable techniques lie in their calculation of internal models (whether nonlinear or linear), their method of predicting the future, their method of constraint handling, and their method for minimizing the controller’s error. Multivariable control may be considered to be an “overkill” and at worst a poor controller, if simpler techniques are adequate. However, for towers that are subject to constraints, towers that have severe interactions, and towers with complex configurations, multivariable control can be a valuable tool.

adaptive mechanism for learning from examples and to adjust its parameters based on the knowledge that is gained through this process of adaptation. During the “training” of these networks, the weights are adjusted until the output of the ANN matches that of the real process. Naturally, these networks do need “maintenance,” because process conditions change, and when they do, the network requires retraining. The hidden layers help the network to generalize and even to memorize. The ANN is capable to learn input/output relationships and inverse relationships, and hence it is useful in building internal model control based on the ANN-constructed plant models and their inverses. In a neural controller (Figure 8.21d), the ANN is used in calculating the control signal.

Dynamic Matrix Control

Neural Control The PID controller is the basic feedback mechanism for correcting errors between the current condition (measurement) and what is desired (set point). The PID assumes a linear process. Adaptive control and other techniques are used when nonlinearities are encountered (see Section 2.19 in Chapter 2). However, because of the structure of neural networks with their distributed representation, the neural controller promises the ability of adaptation, 18 learning, and generalization to nonlinear problems. In the single-input, single-output configuration, instead of utilizing the basic PID equation, the network builds an internal nonlinear model, relating the controlled and corresponding manipulated variable. It builds this model by learning or “training” from a data set of known measurements and process responses. Often, a primary disturbance variable is included in this model. The dynamic response is recorded for the training data set. This makes the neural controller more useful and more robust than the standard PID. These controllers most often use the back-propagation method of training to relate controlled, manipulated, and load variables (see Section 2.18 in Chapter 2). Figure 8.21c illustrates the simple back-propagation neural network used to create the nonlinear model. Because the neural network paradigm can accommodate multiple inputs and multiple outputs, an entire fractionator model can be built into a single controller. The neural controller can be thought of in the same terms as model-based control algorithms, whereby the neural network is used to obtain the inverse of the process model. A back-propagation network can be trained to obtain the inverse model by considering load and controlled variables in its input vector and manipulated variables in its output vector. An example of a neural network controller on a distillation tower control application is shown in Figure 8.21e.

A multivariable predictive controller is based on dynamic 14 matrix control. DMC is a predictive control technique that uses a set of linear differential equations to describe the process. The DMC method is based upon a process step response and calculates manipulated variable moves via an inverse model. Coefficients for the linear equations describing the process dynamics are determined by process testing. A series of tests are conducted whereby a manipulated or load variable is perturbed and the dynamic response of all controlled variables is observed. This identification procedure is time-consuming and requires local expertise because of the experimentation involved. Once the models are obtained, the controller design can be designed. The least-squares approach is taken to minimize the error of the controlled variables from their set points. Weighting constants scale controlled variable errors and influence which controlled variables are allowed to deviate from their set points if a constraint is encountered. The controller considers constraints in its plan for both present and future moves in each manipulated variable. Other factors affecting the response of the DMC controller are parameters that govern the relative amount of movement in the manipulated variables and the rate at which errors are reduced. This is analogous to the tuning parameters in a PID controller. Artificial Neural Networks As was discussed in detail in Section 2.18 in Chapter 2, one of the tools used in building internal models is the Artificial Neural Network, which can usually be applied under human supervision or integrated with expert or fuzzy logic systems. Figure 8.21c shows a three-layer, back-propagation ANN that serves to predict the manipulated variables of a column. Such predictive ANN models can be valuable, because they overcome the limitations of analyzers, which include both availability and dead time. The process model’s knowledge is stored in the ANN by the way the processing elements (nodes) are connected and the importance that is assigned to each node (weight). The ANN is “trained” by example, and therefore it contains the

© 2006 by Béla Lipták

Building the Neural Model To build such a model, all inputs and outputs must first be normalized based upon expected minimum and maximum values and are presented to the network as the training set. All weights and processing element offsets are initially set to small random values. A recursive algorithm starting at the output processing elements

8.21 Distillation: Optimization and Advanced Controls

1871

Manipulated variables Steam flow (Q)

Reflux flow (L)

Output node #1

Output node #2 Output layer Wj,4

Wj,0

Wj,3

Wj,2 Wj,1

Hidden node #1

Hidden node #2

Hidden node #3

Hidden Hidden layer node #4

Bias Input node #1

Input node #2

Input node #3

Input node #4

Top temp

Feed flow

Input node #5

Input node #6

Bottoms temp

Bottoms composition

Reflux temp

Feed temp

Disturbance variables

Input node #7

Distillate composition

Input node #8

Input layer

Tower pressure

Controlled variables

FIG. 8.21c Back-propagation neural network.

is used and repeated until the input processing elements are reached. The weights are adjusted by w +

− ef

ANN controller

u

y

Process

+ ANN model

−

em

filter

FIG. 8.21d The configuration of artificial neural network (ANN) being used as an internal model controller (IMC).

© 2006 by Béla Lipták

Wij (t + 1) = Wij (t ) + ηδ j xi

8.21(3)

where Wij (t) = the weight from hidden node i or from an input to node j at time t xi = either the output of node i or is an input η = a gain term δj = an error term for node j The error term is

δ j = dE/dx j

8.21(4)

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Control and Optimization of Unit Operations

PT

PIC AT LIC

TT

Set point (L) FIC

FT

Tower

TT

D

F T0 P

T1 Y

F (%HKD)

Neural controller Set point (Q) FIC

X

LIC

(%LKB) Tb

Q TT AT

B

FIG. 8.21e Fractionator control using a back-propagation neutral network.

where E is the difference between the desired output and the actual output. A transfer function (also known as a squashing function) is applied to the weighted sum of the normalized inputs at each processing element to calculate each processing element’s output. An often-used transfer function known as the sigmoid is f ( x ) = 1/(1 + e − x ) −1

8.21(5)

dE/dx j = x j (1 − x j )

8.21(6)

giving

If node j is an output node, then

δ j = yi (1 − yi )(d j − vi )

8.21(7)

where dj is the desired output of node j and yi is the actual output. If node j is a hidden node, then:

δ j = x j (1 − x j ) Σδ k w jk

8.21(8)

where k is over all nodes in the layers before node j. Convergence is sometimes faster if a momentum term is added and weight changes are smoothed by a filter: Wij (t + 1) = Wij (t ) + ηδ j xi + α [Wij (t ) − Wij (t − 1)] where 0 < α < 1.

© 2006 by Béla Lipták

8.21(9)

As with any gradient descent method, back-propagation could find a local minimum instead of the global minimum. The momentum term is designed to help the training algorithm overcome the small valleys of local minima. The learning procedures require that the change in weights be proportional to rate of change of error with respect to changes in weights. The constant of proportionality is called the learning rate, η (or learning coefficient). The larger the value of η, the faster the learning rate. Convergence is reached when the root mean square (RMS) error reaches a defined threshold value. By using the same historical data required for the multivariable controller, the network can be trained and a nonlinear internal model can be created. In fact, the single neural controller is just a subset of the overall network used to build the entire fractionator model. During the recall mode of operation, the network responds to the current values of the load and of the controlled variables by adjusting all manipulated variables accordingly. Each node sums the values of its weighted inputs and applies a transfer function. Thus, each output is attained by I j = Σ W ji xi

8.21(10)

y j = 1/(1 + e − Ij ) −1

8.21(11)

The network’s ability to do the prediction of the dynamics of the fractionator improves as more data become available for training. This approach assumes no explicit feedforward or feedback control actions because the control is totally integrated as part of the internally generated model (Figure 8.21e). Thus, the neural controller can be considered a specific type of nonlinear, multivariable, model-based control algorithm. Instead of creating the nonlinear process model with explicit equations that are dependent upon various sets of assumptions (such as equimolal overflow, constant relative volatilities at differing conditions, and constant efficiencies), the neural controller builds its own process model from actual tower operation. Because the neural controller is an empirical model as opposed to a theoretical model, it is susceptible to errors if operated outside the conditions of the training set. Data for the training set need to be continually gathered and the network retrained whenever novel conditions occur in order to increase the robustness of the neural controller throughout its life of operation.

SISO CONTROL ADVANCES Before proceeding to the subject of distillation optimization, some of the advances in single-input, single-output control strategies will be reviewed. These generally PID-based strategies involve the development of a process model and the use

8.21 Distillation: Optimization and Advanced Controls

of feedforward and supervisory control techniques to achieve better control quality and localized optimization goals.

Control Equations Listed below are some of the key material and energy balance equations that define the distillation model, as they have been developed in Section 8.19:

Process Model

F=D+B

The process model equations that were developed in Section 8.19 will also be used in connection with developing the advanced control strategies described in this section. The process model defines the distillation process by the use of dynamic and steady-state equations that describe the material and energy balance equations. As shown in Figure 8.21f, binary distillation has 14 apparent variables, but only 11 independent variables. As the feed properties are usually fixed, the available independent variables are seven. Because there are two defining equations (the conservation of material and energy), this process has 7 − 2 = 5 degrees of freedom (Section 2.1). Therefore, the maximum number of control loops that we can place on this process is five. Therefore, one would usually close the energy balance of the column by pressure control and close the material balance around the column by controlling the level in the bottom of the column and in the reflux drum. The remaining two degrees of freedom are used up by the bottom and overhead composition control loops.

Overhead product (D)

Feed Steam (V)

Apparent variables: c1 c2 c3 c4 u1 u2 u3 u4 u5 u6 u7 u8 u9 m

= = = = = = = = = = = = = =

Overhead temperature Overhead pressure Overhead composition Overhead flow rate Bottom temperature Bottom pressure Bottom composition Bottom flow rate Feed temperature Feed pressure Feed composition Feed percent vapor Feed flow rate Steam flow rate (heat input)

Bottom product (B) Independent variables 2 1 2 1 2 1 1 1 11

FIG. 8.21f A binary distillation process has five degrees of freedom, and therefore five of its process variables can be independently controlled: one pressure, two levels, and two compositions.

© 2006 by Béla Lipták

8.21(12)

If the feed flow is uncontrolled, B is dependent upon F and D: B=F−D

8.21(13)

or if the bottoms product is the manipulated variable: D=F−B

8.21(14)

where F = feed rate (the inflow) D = overhead rate (an outflow) B = bottoms rate (an outflow) If the compositions of the feed, distillate product, and bottoms product are all known, then the component materials balance can be solved: B/F =

L

1873

(100 − %HK D − %LLK F − %LK F ) (100 − %HK D − %LK B )

8.21(15)

For a given feed composition and desired product compositions, only one bottoms-to-feed ratio, B/F (product split), will satisfy both the overall and the component material balances. A series of energy balances produce additional equations. The vapor boil-up rate VB equals the heat QB added by the reboiler divided by the heat of vaporization (∆H) of the bottoms product: VB = QB /∆H

8.21(16)

The vapor rate V above the feed tray equals the vapor boil-up rate plus the vapor entering with the feed (feed rate F times vapor fraction VF, provided the feed is neither subcooled nor superheated): V = VB + ( F )(VF )

8.21(17)

The internal reflux rate (Li), or the liquid at the top tray, of the column is derived by a heat balance around the top of the tower. If a total condenser is employed, this gives the equations: L=

Li [1 + K1 (To − Tr )]

8.21(18)

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Control and Optimization of Unit Operations

or Li = ( L ) ⋅ [1 + ( Kr )( ∆T )]

8.21(19)

The liquid rate, Lf, below the feed tray equals the internal reflux plus the liquid in the feed: L f = Li + (1 − VF )( F )

8.21(20)

The distillate rate, D, equals the vapor rate, V, minus the internal reflux: D = V − Li

8.21(21)

The bottoms rate, B, equals the liquid rate, Lf, minus the boilup, VB: B = L f − VB

Inversely, the percentage transmitter signal (scaled equivalent) corresponding to an engineering measurement can be obtained as % signal =

measurement in engineering units − zero range 8.21(25)

Most DCS and other electronic controllers require that all signals conform to 0–100%. As mentioned previously, scaling is done to convert engineering unit inputs and outputs into normalized values that these DCS and electronic systems can use. Figure 8.21g shows how the internal reflux rate of a distillation column is calculated. The calculation for internal reflux is given by the equation below: LI = L.[1 + (CP/∆H)∆T ]

− t2

GB = ta (1 − e )(1 − e ) GT = tb (1 − e − t3 )(1 − e − t4 )

8.21(26)

The subtracter is scaled first. Assuming ∆Tmax of 50°F (27.8°C), the span of To between 150°F and 250°F (65.6°C and 121°C), and the span of Tr between 125°F and 225°F (51.7°C and 107°C), the equation for the subtracter is written first in engineering units as

8.21 (23)

where − t1

measurement in engineering units = zero + range (% signal) 8.21(24)

8.21(22)

The criterion for separation is the ratio of reflux (L) to distillate (D) vs. the ratio of boil-up (V) to bottoms (B). Manipulating reflux affects separation equally as well as manipulating boil-up, albeit in opposite directions. Thus, for a twoproduct tower, two equations define the process: One is an equation describing separation and the other is an equation for material balance. During the unsteady state of upsets, the process model must account for the dynamics of the process. This extends the steady-state internal flow model and requires additional consideration. For this reason two dynamic terms, GT and GB, are included, which provides a dynamic model for the tower based on its dead time and second-order lag, giving L = GB [GT Li + (1 − VF ) ⋅ F ]

central control system. A simple example of this type of conversion has already been given for zero-based signals in connection with Figure 8.19l in Section 8.19. In this section scaling will be illustrated on more complex systems, involving several nonzero-based transmitter signals. The value of a transmitted signal in engineering units can be obtained from the normalized (scaled) transmitter signal and from the zero and range of the transmitter as follows:

∆T = To − Tr

8.21(27)

Now, converting from engineering to scaled units and denoting the scaled transmitter signal values as To′ and Tr′, the scaled equivalent of Equation 8.21(27) is 0 + 50 ∆T ′ = (150 + 100 To′) − (125 + 100 Tr′)

8.21(28)

where ta, tb = dead times Scaling When using process models, it is very important that the measurements be correctly represented, that all I/O values be properly scaled. This was less of a challenge in the analog age, when a 9 PSIG or a 12 mA signal always meant 50%, no matter which supplier, industry, or continent was involved. This is not necessarily the case in the present digital age, with its multiple protocols and the need for interfacing translators when connecting them. Scaling (the conversion from engineering units to fractions or percentages) is required in order to make the various transmitter signals meaningful to the DCS, PLC, or other

© 2006 by Béla Lipták

150−250°F = 0−100% T TT o 125−225°F = 0−100% TT Tr 0−10,000 GPM 2 FT L

∆

To−Tr

X Li = L (1 + CP ∆T) ∆H

TY UY [Li′ = 0.754 L′ (0.885 0−50°F = 0−100% + 0.115∆T′)] FY

L

FIG. 8.21g The steps in the calculation required to determine the internal reflux flow rate of a distillation column.

8.21 Distillation: Optimization and Advanced Controls

This reduces to the scaled equations:

1875

To TT

∆T ′ = 2(To′ − Tr′ + 0.25)

8.21(29) X

If the following assumptions are made:

FRC

UY

TY

FT

FY

TT

LI max = 15,000 gpm (0.95 m /s) 3 Lmax = 10,000 gpm (0.63 m /s) Cp = 0.65 BTU/lb°F (0.65 kcal/kg°C) ∆H = 250 BTU/lb (450 kcal/kg) 3

the equation for the multiplier then becomes  10, 000 L ′   0.65  50 ∆T ′  1 +   15, 000   250  

Vap

Equation 8.21(30) then reduces to Li′ = 0.667 L ′(1 + 0.13 ∆T ′)

When ∆T′ is zero, the internal reflux equals 0.667 times the external reflux. The number 1 within the parentheses, therefore, sets the minimum internal reflux. When ∆T ′ is 100%, the ratio of internal reflux to external reflux is at a maximum. The expression within the parentheses must be normalized. This is done by dividing both terms by the total numerical value, that is 1.13. To preserve the equality, the coefficient of L ′ is multiplied by 1.13. The scaled equation becomes Li′ = 0.754 L ′(0.885 + 0.115 ∆T ′)

8.21(32)

Internal reflux systems are designed to compensate for changes outside the column, such as reflux temperature that is affected by ambient conditions. It should be understood that a change within the column can introduce positive feedback. Figure 8.21h shows a typical internal reflux application and its response to an upset within the column. The control system reacts in the same way to an increase in overhead vapor temperature and to a decrease in reflux liquid temperature, but the required control actions are in the opposite direction.

FEEDFORWARD SYSTEMS Feedforward controls represented the first steps on the road towards multivariable model-based process control. They were first applied in well-understood processes such as heat transfer and distillation, where the material and heat balance equations made it possible to predict and anticipate the consequences of the process outputs to changes in the inputs, before they had time to evolve.

© 2006 by Béla Lipták

flux l re

d ute mp o C Extern

8.21(31)

Tr

ture pera m e t or

8.21(30) Variables

Li′ =

∆

rna inte

al reflu

x flow

Time

FIG. 8.21h Response of internal reflux control system to an increase in the concentration of heavy components in the overhead vapors.

Feedforward control techniques react to variations in disturbance variables, predict the disturbance’s effects, and take corrective action before the tower is significantly affected. Feedback control attempts to maintain the set point of a controlled variable by measuring its value at the outlet of the tower. In most cases, a combination of feedforward and feedback techniques can correct process deviations in the shortest time. This correction is accomplished by considering process dynamics (dead times and time lags), the nonlinearities between separation efficiency and column loading, loop interactions, and process measurements. The types of disturbances that feedforward control is most often used to compensate for include (1) feed flow rate, (2) ambient temperature (top and reflux temperatures), and (3) reflux flow rate. Other variables that can be compensated for, but to a lesser degree include the disturbances are (4) tower pressure, (5) feed composition, (6) feed temperature or enthalpy, and (7) reboiler heat. The application of feedforward techniques involves the use of the models and equations described in Section 8.19, but dynamically tuned to approximate the response of the distillation tower. Literally dozens of different feedforward control strategies have been proposed for distillation column control, and many of the more successful ones will be described and analyzed in this section. While feedforward control is common, it cannot be considered to be a universal solution for all columns.

1876

Control and Optimization of Unit Operations

V

L

V V FIC

Σ −KD FY

L = m − KD

SET

+m L

LT

FY

D

D V=L+D

FY

FT

FIG. 8.21i Reflux accumulator material balance.

FIG. 8.21j Reflux rate control system for overcoming accumulator lag.

8.21(33)

where V = boil-up (vapor rate) L = reflux rate D = distillate rate To overcome the accumulator lag, the reflux rate, L, must be manipulated in direct response to a change in distillate rate, D, rather than by waiting for the response of a level controller. If V is constant (k), Equation 8.21(33) can be solved for L, which is the manipulated variable in this part of the system.

© 2006 by Béla Lipták

where m is the output of the level controller and K is an adjustable coefficient. The resulting control system is shown in Figure 8.21j. The range of coefficient K should be broad enough to allow scaling and adjustment to be done during commissioning. The level controller trims the computation, the scaling and the value of K but do not alter the steady-state value of external reflux, L, because these factors affect the transient response only. The response of the reflux flow to changes in distillate for several values of K is given in Figure 8.21k. The full-scale values of reflux, Lmax, and distillate, Dmax, flows in 3 this case are 1000 gpm (3.79 m /min) and 500 gpm 3 (1.89 m /min), respectively. When K = 0, the reflux is adjusted by the level controller. In other cases, the reflux flow is immediately altered by some percentage for a change in distillate, and the level controller forces the balance of the change. The response is a firstorder lag. If K = 0.5, the reflux flow is changed to the exact new steady-state value, because K equals the ratio of Dmax/Lmax, and therefore the computation is exact; the lead equals the lag and the net effect is no dynamic contribution. If K = 1.0,

8.21(34)

8.21(35)

L1

50 GPM (0.19 m3/m)

K=0 K = 0.25

Flow

For this equation to be satisfied, L must be decreased one unit for every unit D is increased, and vice versa. If V is indeed constant and both the computations and the flow manipulations are perfectly accurate, no level controller is needed. If these conditions cannot be met, a trimming function is introduced. The system equation becomes L = m − KD

−K

FIC

D

The column interactions that otherwise might necessitate the use of an internal reflux control system can be eliminated in some cases when the flow of distillate product draw-off is controlled and reflux is put under accumulator level control. This is a slower system than one in which flow controls the reflux, and its response is not always adequate. If necessary, the response can be speeded up by reduction of the accumu4 lator lag. The steady-state material balance around the accumulator (Figure 8.21i) is expressed by

L=k−D

X FY

FT

Flow Control of Distillate

V =L+D

LIC

K = 0.5 50 GPM (0.19 m3/m) D1 Time

FIG. 8.21k Reflux response as function of K.

K = 1.0

8.21 Distillation: Optimization and Advanced Controls

the initial response is a first-order lead-lag function. In this case, the reflux is greater than required for the new steady state, and the level controller eventually corrects the flow. The value of K does not change the steady-state flow. It affects the transient response only, and therefore it can be used to adjust the dynamics of the loop. The greater the value of K, the faster the response. Care must be taken to prevent increasing the response to the point of instability. A rule of thumb is K max = 1.5( Dmax /Lmax )

8.21(36)

Therefore, in this example, K should not be set greater than 0.75. In some implementations, the range of adjustability of K is limited, and scaling is necessary. For the values used in the illustration (reflux full scale value equals 1000 gpm and distillate full scale equals 500 gpm), Equation 8.21(35) becomes 1000 L ′ = 1000 m′ − 500 KD′

8.21(37)

where L′ and D′ are the normalized values of L and D. The maximum value of m is equal to the maximum value of L, because the level controller by itself can cause the level control valve to open fully. The scaled equation is L ′ = m′ − 0.5KD′

8.21(38)

1877

(lagged) feed rate, FL, is then multiplied by the desired bottomsto-feed ratio to obtain the target for bottoms flow rate. B = ( B/F )( FL )

8.21(40)

V = m − K ( B/F )( FL )

8.21(41)

so that

Because these models are only approximations of the real process, inaccuracies do exist. For this reason, the bottomsto-feed ratio target obtained by the feedforward calculation should be trimmed by analysis-based feedback control. In the majority of feedforward applications, their purpose is not to replace feedback but to minimize the amount of work that the feedback part of the loop has to do. This requires that the advanced control must be able to measure and quantify the disturbance, then react before the fractionator separation can be upset in the first place. Constant Separation A distillation column operating under constant separation conditions has one fewer degree of freedom than others, because its energy-to-feed ratio is constant. At a given separation, for each concentration of the key component in the distillate, a corresponding concentration exists in the bottoms. In other words, for a constant-feed composition, holding the concentration of a component constant in one product 4 stream fixes it in the other. Figure 8.21l shows an example of a constant separation feedforward system in which distillate is the manipulated variable.

K must be adjustable over a range of ±10% for satisfactory tuning flexibility. V, y

Flow Control of Bottoms Similarly to the feedforward systems described for distillate flow control, a similar system can be used on control the column bottom flows, if the bottom product is flow controlled and the bottoms level is maintained by manipulation of the heat input or boil-up (V). The equation for that system is V = m − KB

Dynamics FY F2

ARC

AT

LC

(D ( F

2

LT

2 FRC D

L FL2

FT

8.21(39)

where V is the boil-up, B is the bottom product flow, m is the output of the bottoms level controller, and K is the same kind of adjustable coefficient as in Equation 8.21(38). Another commonly used model for feedforward compensation involves the bottoms-to-feed ratio (B/F). The bottoms product draw, ratioed with the feed rate, is a function of the overhead and bottoms composition targets at a given feed composition. The implementation of the bottoms-to-feed ratio control usually requires dynamic (dead time and lag) compensation of the feed rate. The dynamically compensated

© 2006 by Béla Lipták

FL2

Feed (F, z)

X Q 2 (F( 2 Q

FFY

FT Distillation (D)

FIC

FT FY

Q

LC

X

F2 FL – Lagged feed flow rate B, x

FIG. 8.21l Feedforward distillation control system with constant separation.

1878

Control and Optimization of Unit Operations

A material balance on the light key component gives Fz = Dy + Bx = Dy + ( F − D) x  z − x  D D=F =F   y − x  F

Lead/lag

Lag – lead/lag

8.21(42)

Step change in load input

8.21(43) Impulse

If the flow measurements are of the differential pressure type, then:

Lag

Multiple lag Time

2

 D  z− x = F2   D2 = F 2   F  y − x 

Load input

2

8.21(44)

Because boil-up must change in proportion to feed rate, a second feedforward loop is obtained for setting heat input: Q = F (Q/F )

or

Q 2 = F 2 [Q/F ]2

8.21(45)

where z, y, x = mole fraction of the key light component in feed, overheads, and bottoms, respectively D/F = required distillate-to-feed ratio Q/F = required energy-to-feed ratio No scaling is required of this equation if an adjustable ratio is used for both D/F and Q/F. Normal design practice for scaled systems calls for the output of the trim analyzer controller ARC to be at 50% when the design or normal distillate-to-feed ratio is required. If the gain of the multiplier is set at 2, the output tracks the load when this normal distillate-to-feed ratio occurs. In a linear system, the gain of the multiplier equals the scaling factor. In this system, however, the gain of the multiplier equals the square root of the scaling factor. When this rule is applied to the example, the scaled form of Equation 8.21(44) is D 2′ = 4.0( F 2 )′[( D/F )2 ]′

8.21(46)

where D ′, F ′, and [(D/F) ]′ are the normalized values of the respective terms in Equation 8.21(44). The block labeled “dynamics” in Figure 8.21l is a special module designed to influence the transient response. This is because the time response of the distillate to a feed rate change must be dynamically matched. The dynamic block is generally a dead time module and a lead-lag module in series. In the steady state, its output equals its input. Figure 8.21m illustrates the temporary modifications that various dynamic compensators can introduce to match the “dynamic personality” of the process. For a discussion of dead time compensation, refer to Section 2.9 in Chapter 2. 2

2

2

Maximum Recovery In many distillations, one product is worth much more than the other, and the control system is designed to maximize the

© 2006 by Béla Lipták

FIG. 8.21m Dynamic compensators introduce temporary modifications into the value of their output signals which match the dynamic “personality” of the controlled process.

recovery of the more valuable stream. One such equation for 4 this type of system is D = m( KF + K 2 F 2 )

8.21(47)

where D = distillate rate F = feed rate K = adjustable coefficient K2 = 1 – K m = feedback trim This equation assumes that energy is free and that the distillate product is worth more than the bottoms. Distillate product flow is not linear with feed rate when boil-up is held constant. The control diagram for this maximum recovery system is shown in Figure 8.21n. Note that the distillate-toreflux loop for accelerated response is also used. The sum2 ming block (FY-1) used to compute (KF + K2F ) needs no special scaling. The values of m can be computed from the feed composition. A typical range for m is 0.35–0.65. This is the output signal range of ARC-2, the feedback controller. Although the coefficients can be calculated in advance with reasonable accuracy, on-line adjustment is quite easy (these coefficients are accessible in most DCS and PLC systems), and the rigor of the calculations can be avoided. If energy is not free and only one product composition needs to be controlled, then a linear relationship can be assumed. In this case, product flows will be directly proportional to feed rate when separation is fixed. D = m1 ( K3F )

8.21(48)

B = m2 ( K 4 F )

8.21(49)

or

8.21 Distillation: Optimization and Advanced Controls

KF + K2F2

control the compositions of both products. One benefit of dual composition control is minimized energy consumption. However, it is difficult to implement dual composition control on many columns because of the severe interaction problems that may exist. Also, with a given feed and tower design, it may not be possible to achieve two arbitrarily chosen product compositions. An example of a feedforward dual-composition control model will be described here, after which a method for determining the degree of interaction, based on actual process data, will be discussed. The control of distillate composition can still be done by manipulating distillate flow as required by

D = m (KF + K2F2)

D LIC FY

X m FY

ARC 2 Σ

F2

∆ LT

FY 1 AT F FY

SP V

FIC

FY

FY

FT L

FT

1879

FRC

FT D

 z− x D=F  y − x 

F

8.22(52)

Q

LC B

FIG. 8.21n Maximum recovery system: instrumentation solves quadratic equation for distillate rate.

However, in order to also enforce composition control of the bottom product, an additional manipulated variable is needed. Another product stream cannot be independently manipulated without changing the accumulation in the column, which is not practical. The energy balance must, therefore, be adjusted to control bottoms composition x. The relationship between x and the energy balance was 5 developed by Shinskey, for binary mixtures, as a function of separation S: S=

and Q = m3 ( K5 F )

8.21(50)

y(1 − x ) x (1 − y)

8.21(53)

For multicomponent mixtures separation is defined as:

or L = m4 ( K 6 F )

8.21(51)

where D = distillate rate B = bottoms flow rate F = feed rate Q = heat input rate L = reflux rate K3, K4, K5, K6 = adjustable coefficients m1, m2, m3, m4 = feedback trim signals Composition Control of Two Products Because of the many variables that affect product composition, which are difficult to anticipate or control (e.g., feed composition), and because composition specifications for both products may be tight, some columns require better control than can be achieved by the previous constant separation strategy. One method that can be used on some columns for achieving the required product specifications is to directly

© 2006 by Béla Lipták

S=

yL / x L y /x = L H x H / yH x L / yH

8.21(54)

where the separation factor is the ratio of light to heavy key in the distillate divided by the same ratio in the bottoms product. The relationship between separation (S) and the ratio of boil-up to feed (V/F) over a reasonable operating range is V /F = a + bS

8.21(55)

where a and b are functions of the relative volatility, the number of trays, the feed composition, and the minimum V/F. The control system therefore computes V based on the equation for a binary mixture as  y(1 − x )  V = F a + b x (1 − y)  

8.21(56)

Because y is held constant, the bottom composition controller adjusts the value of the parenthetical expression if an

1880

Control and Optimization of Unit Operations

D = F(m) = F FY F

X

m

it is. The severity is a function of feed composition, product specification, and the pairing of manipulated and controlled variables. Severe interactions frequency occur when the energy balance is manipulated by two independent composition controllers. A column in which reflux flow and steam flow are the manipulated variables is an example of a severely interacting column. The control system equations are

z−x y− x

ARC

AT

F(t) FY Dynamics

F FY LC F m2

F, z SP

V, y

FY 2

X FY

V = F(m2) = FT = F a+b V F V

D, y LC

FY B, x FT

Q ARC

AT B, x

FIG. 8.21o Feedforward control system provides closed-loop composition control of two product streams.

error should appear in x. Let V/F = y(1 − x)/(1 − y), and the control equation becomes V = F (a + b[V /F ])

8.21(57)

where [V/F] equals the desired ratio of boil-up to feed. The system implementing Equation 8.21(52) is shown in Figure 8.21o. FY-1 and FY-2 are multipliers. The FY block labeled “dynamics” is a special block for dynamic compensation similar to the one described in Figure 8.21l and 8.21m. Included in a and b are the relationship between boil-up (vapor rate), and energy flow (Q), and the minimum ratio of boil-up to feed. Equation 8.21(57) can therefore be written Q = kF ([V /F ]min + [V /F ])

8.21(58)

where k represents the proportionally constant. Two Products with Interaction Interaction always exists between the material and energy balances in a distillation column. In some columns, this interaction is not severe enough to impede closed-loop composition position control of the two product streams, but in others

© 2006 by Béla Lipták

8.21(59)

L = F ([ L/F ])

8.21(60)

FIC

FT

FIC

Q = kF ([V /F ]min + [V /F ])

where L is the reflux rate and [L/F] is the desired reflux-tofeed ratio. Note that in the control system described by these two equations, the rates of products leaving the column are dependent on two energy balance terms. Increasing heat input at the reboiler forces the composition controller that is resetting reflux flow to increase heat withdrawal, and the top and bottom composition controllers, therefore, “fight” each other. The only way to avoid this fighting is by preventing a change at one end of the column from upsetting the other end. The heat input is changed when the bottom composition controller is upset. If the upset is because of a high concentration of light ends in the bottom product, heat is increased to adjust the separation being performed and to drive the extra light ends up and out the top. The top composition controller does not know how to split the increased vapor load, but it sees a measurement indicating an upset and responds to an increase in heat input by increasing the reflux flow. Theoretically, if the reflux rate is compensated for the change in heat input, the top composition controller upset can be avoided. One can find the relationship between reflux L and heat input Q by solving Equations 8.21(59) and 8.21(60) for L in terms of Q. The resultant equation is of the form L = k1Q − k2 F

8.21(61)

The values of k1 and k2 are found by deriving Equation 8.21(61) using actual process values of [L/F], [V/F]min, and [V/F]. The decoupling equation, Equation 8.21(61), replaces Equation 8.21(60) in the control model. The resulting system is shown in Figure 8.21p. The system is now half-decoupled: A change in heat input will not upset the top temperature, because the decoupling loop adjusts the reflux independently of the top temperature (analysis) controller. However, the heat input is still coupled to reflux, because a change in reflux will still cause the bottom temperature controller to adjust steam flow. This degree of decoupling is enough to reduce the interaction approximately 20-fold. The two multipliers are scaled as described previously (under the paragraph Scaling), and the adder is tuned on-line.

8.21 Distillation: Optimization and Advanced Controls

L = K1Q − K2F

1881

Q = kF [(V/F)]min+ (V/F)]

Σ

FY +K1 −

Q

Q X

F

L

FY

FY

FRC

LC

Dynamics K2 FY

ARC

AT

FT L

F

FY Dynamics k

X

FY

FT

SP FIC

(V/F)min + (V/F)

LT

ARC

FY

FT

Q

Q

AT LIC

FIG. 8.21p The decoupling of a distillation tower when the compositions of both products are being controlled.

6

Classical decoupling schemes, however, often do not provide a solution to the problem of interaction because of prac7,8 tical problems encountered on real columns. Decoupling systems that include overrides can drive to saturation when constraints are encountered. Most seriously, decouplers applied to systems with negative interaction (defined later) may have very little tolerance for errors in decoupler gains. For this group, which always includes the interaction encountered in reflux and boil-up controls, small errors can transform a system that provides complete decoupling into one that provides no control at all. Because the proper decoupler gains depend on the process gains, which inevitably change with variations in feed rate, product specifications, and column characteristics, these systems require constant attention and adjustment beyond the ordinary capability of plant operating personnel. The difficulties associated with the application of decoupling systems have prompted a re-examination of interaction itself. The problem may be postulated in two ways: 1. For a given column, is the interaction equally strong in each of the control structures available to the designer? 2. For a given control structure, will the interaction be equally strong in every column in which it is applied?

© 2006 by Béla Lipták

The stumbling block of loop assignment may in this way be converted into a stepping stone by providing the opportunity to select a control structure that will exhibit minimum interaction in any particular application. Shinskey and Ryskamp have given consistent guidelines 9 for assigning loops to minimize loop interaction. Shinskey suggests that the controller assigned to the more pure product 10 should manipulate separation. Ryskamp suggests that the controller for the component with the shorter residence time should adjust vapor flow, and the controller for the component with the longer residence time should adjust the liquid/ vapor ratio. Feed Composition Compensation Occasionally, changes in feed composition occur too fast to be handled by feedback control, and feedforward compensation for these changes is necessary (Figure 8.21q). The basic material balance equation, Equation 8.21(62), already has a term, z, representing concentration of the key component in the feed:  z− x D=F   y − x 

8.21(62)

1882

Control and Optimization of Unit Operations

SUPERVISORY CONTROL

D = zF/m z

FY

×&÷ m

F

ARC 1

FY

LT

Dynamics FY

HIC

L AT

AT

LIC

FY

FRC

FT FT

F, z

SP FY

D

Ratio RIC controller LT

FT LIC

Q

FIG. 8.21q Feed composition measured and used to compute distillate flow.

When z is measured, the equation for distillate can be simplified to D = zF/m

8.21(63)

where m is the output of the overhead analyzer feedback trim controller (ARC-1). The auto/manual station (HIC) is used in the event of analyzer failure. Dynamic compensation is placed on the flow feed signal only. The control of the bottoms flow in Figure 8.21q is indirectly provided by the feedforward control of the reboiler heat input based on the (dynamically compensated) feed flow rate. If, instead of this approach, feedforward analyzer control of the bottoms flow is desired, Equation 8.21(64) can be utilized. V = m − K ( B/F )( FL )

8.21(64)

where FL is the dynamically compensated feed rate. Substituting the B/F ratio: B/F =

(100 − %HK D − %LLK F − %LK F ) (100 − %HK D − %LK B )

8.21(65)

gives the feedforward expression for the vapor rate up the tower:  (100 − %HK D − %LLK F − %LK F )  V = m−K  FL (100 − %HK D − %LK B )   8.21(66)

© 2006 by Béla Lipták

On-line computer control can greatly enhance the profitability of the distillation process and data collection improvements, and increased flexibility can often justify computer control even if rigorous on-line optimization is not implemented. The optimization strategy can be implemented in the supervisory mode (recommendations to the operator) or in the automatic mode and can involve the whole plant or only particular subsystems of the total process. The main computer control functions applied to distillation include engineering calculations, operating assistance, 11 quality controls, and heat balance controls. The primary engineering calculations are made from material and heat balances around column sections and include tray loadings, internal vapor flows, internal liquid flows, and heat duties. These calculations are helpful as operating guidelines and as inputs for on-line control. However, they are usually based on steady-state conditions, and therefore the input signals must be averaged to make the calculations. Response, although normally fast enough for on-line control, may not be adequate if frequent, short-term disturbances must be handled. However, information gained from these types of calculations can often justify the computer system by providing better operating guidelines, even if it is not used for on-line control. An example of a computerized control system that makes the operator’s job easier is the balancing of the heater coil outlet temperature on furnaces (Figure 8.21r). Other examples of computer controls include (1) the feedforward adjustments of products and pump-arounds on the basis of feed rates, (2) the control of the column’s bottom level by throttling the feed preheater bypass flows, and (3) the control of overhead receiver level. Such control systems will be described in more detail later. Because many of these calculations result in the need for changing the set points of several controllers simultaneously, supervisory control can reduce the workload and the potential for human errors by the operator. Product quality controls are enhanced if the computer adjusts the column temperature and side-draw flow rates to control product specification. Often computers are used to infer product specifications from local flows, temperatures, and pressures. Examples of these inferred calculations include true boiling point (TBP) cut points, ASTM 95% boiling points, Reid vapor pressure (RVP), octane, viscosity, freeze points, cloud points, and pour points. These calculated measurements can be used in feedback controllers themselves or as a fast inner loop with an analyzer trim. The advantage of such model-based controls is that they can anticipate future events, because their outputs are not delayed by the process dead time and time lag. This results in tighter control. Adjustment of pump-around reflux flows, as shown in Figure 8.21s, is an application example where the computer

8.21 Distillation: Optimization and Advanced Controls

Computer raw crude rate control

+

A careful analysis of the limits and operating constraints is essential, because if the system is not designed to provide limit checks and overrides to handle operating limits, frequent operator intervention will be required during upsets. This can cause a lack of confidence in the computer system, which can result in the column’s being off computer control more than necessary.

Computer pass balancing control

+ + + FC

TT

FC

TT

1883

The Total Model To crude column

FI

Raw crude

TT

FC

TC

TT

FC

Fuel

FIG. 8.21r The balancing of heater coil charge rates in crude oil furnaces can be under computer control or supervision.

assists in heat balance control. The goal of such systems usually is to maximize the exchange of heat to feed, subject 19 to certain limits, which will be discussed in the subsection ‘‘Optimization.” SP FC

FT Sidecut

SP Heat balance logic

FC

FT Sidecut

Limits

Feed

Internal flows

FIG. 8.21s Computer adjustment of pumparound refluxes.

© 2006 by Béla Lipták

Bottoms

It is possible to design a system to compensate for all load variables: feed rate, composition, enthalpy, reflux, and bottoms enthalpy. The goal of these systems is to overcome the problems associated with unfavorable interactions and to isolate the column from changes in ambient conditions. These problems can usually be solved by careful system analysis and variable pairing, thus avoiding complicated total energy and material balance control systems. The complexity of the total material and energy balance systems is made apparent by the list of equations required in the model: Feed enthalpy balance Bottoms enthalpy balance Internal reflux computation Reboiler heat balance Overall material balance Suboptimization The following derivations will provide insight into the derivation and modeling of optimization equations and the handling of constraints. However, the reader is advised that every distillation column is unique; the examples given here are for illustrative purposes only and should not be considered to be suboptimal or optimal solutions for every column. Optimization of a single distillation column normally implies a maximum profit operation, but to achieve maximum profit, the price of the column’s products must be known. It is impossible to control every column in a system based on this criteria, because the prices of products for many columns are unknown. Product prices are often unknown because the products are feed streams to other units, whose operations would need to be taken into account to establish the column’s product prices. When product prices are unknown, it is possible to carry optimization only to the stage at which specified products can be produced for the least operating cost. This can be called an optimum with respect to the column involved, but only a suboptimum with respect to the system of which the column is a part. When column product prices are known, complete economic optimization can be achieved. However, a number of different situations may still exist. If there is a limited market for the products, then the control problem is to establish the separation that results in a maximum profit rate. Such an

Control and Optimization of Unit Operations

optimum separation will be a function of all independent inputs to the column involved. When an unlimited market exists for the products, and sufficient feedstock is available, the optimization problem becomes more difficult. Not only must the optimum separation be established, but also the value of feed must be determined. Optimization for this case results in operating the column at maximum loading or at maximum energy efficiency. One of three possible constraints will be involved: Throughput will be limited by the overhead vapor condenser, the reboiler, or the column itself. In some cases, the constraint will change from time to time, depending upon product prices and other independent variables of the system. The design of optimal automatic control systems to single columns should follow three logical steps:

A single column that is automated in this manner is called a suboptimized system. This suboptimum is defined as an operation that will produce close to the specified separation, whether or not that separation is ideal with regard to the total system of which the column is a part. If product purities are higher than specified, the operation cannot be considered suboptimum. When a single column is automated through the suboptimum operation stage, it will still exhibit up to five degrees of freedom. As a basis for proceeding into the optimization phase, Figure 8.21t is presented as one example of a column automated through the suboptimization stage. As shown in Figure 8.21t, the system used to regulate the separation is a predictive control system, similar to that described earlier. The function of the predictive control system is to manipulate the energy balance (reflux flow rate) and the material balance (bottom product flow rate) to give the specified separation. The equations derived for these manipulations are called the operating control equations. Figure 8.21u is another example of suboptimization. Both figures achieve their goal of operating at suboptimum; the difference is mainly in their the basic controls. In the paragraphs below, the steady-state operating equations and the dynamics for bottom product and reflux will be described. Bottoms Product Operating Equation The equation for predicting bottom product flow rate was derived in Section 8.19 as B/F =

© 2006 by Béla Lipták

(100 − %HK D − %LLK F − %LK F ) (100 − %HK D − %LK B )

8.21(67)

TDT

Specify set(PC)

PRC To FRC

Hot vapor by pass

PRC LRC

Tr

Set

FRC

Steam

Set (D)

Reflux Feed (F)

FT

AT

TRC

LLKF LKF HKF F Computer inputs

Set (Tf)



1. Design the basic controls to regulate basic functions, such as pressures, temperatures, levels, and flows 2. Configure the controls to regulate the main sources of heat inputs, including regulation of internal reflux flow rate, feed enthalpy, and reboiler heat flow rate 3. Apply controls to regulate the specified separation

Computer input (To − Tr)

 

1884

Bottom product Specified inputs LKB HKB FT ∆HF P E

Set (L) FRC

Top product

LRC Set FRC

FRC

Steam

Internal Set reboiler (B) Computer set point outputs Measurements Computer Bottom product LLKF LKF operating equation m = Measurements HKF and dynamics required to F compute feed Reflux operating T − Tr temperature for equation and dynamics m o specified value Feed enthalpy of feed enthalpy regulation

FIG. 8.21t Distillation column automated through the suboptimization stage (to produce close to the specified separation.)

where %LLKF = lighter than light key in the feed (mol%) %LKF = light key in the feed (mol%) = z %LLKD = lighter than light key in the distillate product (mol%) %LKD = light key in the distillate product (mol%) = y %HKD = heavy key in the distillate product (mol%) %LKB = light key in the bottoms product (mol%) = x Assuming that the composition in both ends of the tower are to be held at specification: B/F =

(100 − %HK D − %LLK F − %LK F ) (100 − %HK D − %LK B )

8.21(68)

where %HK D = specification of the heavy key in the distillate product (mol%) %LK B = specification of the light key in the bottoms product (mol%) = x

8.21 Distillation: Optimization and Advanced Controls

where

Computer input (To − Tr)

PRC FRC

Set

TDT Hot To vapor bypass Tr

E FT ∆H F P PRC LRC

Steam

Top product

Reflux Feed (F)

FT

AT



 

LLKF LKF HKF F Computer inputs

FRC

TRC

Set

Set (L) Set (Q) FRC

Set (Tf ) LRC

FRC Set (D) Steam

FRC

Bottom product

Internal reboiler

Specified Measurements inputs Computer LKB m Top product operating HKB LLKF equation and dynamics LKF FT ∆HF HKF Reflux operating F P equation and dynamics To − Tr E Feed enthalpy regulation

Computer set-point outputs

Derivation of the internal reflux operating equation is more difficult. Typically, this equation is developed in two parts:

constant average efficiency value of feed tray location value of feed enthalpy value of column pressure

The theoretical part of Equation 8.21(69) is normally developed by tray-to-tray runs of calculations performed by an off-line digital computer. A statistically designed set of runs is made, and the information thus obtained is curvefitted to an assumed equation form. Once the steady-state theoretical equation is developed and placed in service, the experimental part is determined by online tests. These tests involve operating the column at different loads to determine the correction required to (Li/F)t for the separation to be equal to that specified. Average overall efficiency E is set to make (Li/F)t required to equal the actual Li/F that exists. The loading tests are carried out under this condition. Often plant tests are performed to determine Li/F without consideration to the theoretical term. For a given distillate composition, the calculated internal reflux in Equation 8.21(72) is found for several different feed rates.

The experimental part of this equation is necessary because the effect of loading on the overall separating efficiency (E) is normally unpredictable. Both parts of the reflux operating equation are functions of all independent inputs to the system. However, simplifications are normally considered for the experimental part, as follows: ( Li /F )t = f1[( LLK F ),( LK F ),( HK F ),( E ),( FT ),

8.21(72)

The result often gives the relationship as shown in Figure 8.21v. Implementation of these curves is generally via a polynomial equation or segmented function curve. Reflux Operating Equation Internal reflux controls were described in Figure 8.21h. One can approximate internal reflux flow rate of a distillation column by making a heat balance around the top tray. If that is done, the following equation is obtained: ( Li ) = LK 2 [1 + K1 (To − Tr )]

8.21(69)

where (Li/F) = internal reflux to feed flow rate ratio required to give a specified separation (Li/F)t = theoretical part of reflux operating equation (Li/F)e = experimental part of reflux operating equation

© 2006 by Béla Lipták

specified specified specified specified

 C pL Li ∆H L  = ⋅  1.0 + ⋅ (To − Tr )  L ∆H Li  ∆H L   

FIG. 8.21u Alternative method of column automation through the suboptimization control stage.

( Li /F ) = ( Li /F )t + ( Li /F )e

= = = =

Internal reflux flow rate, (GPM)

Specify set (PC)

1885

8.21(73)

Concentration of %HKD

Feed flow rate, (GPM)

( ∆H F ),( P ),(%LK B ),(%HK D)]

8.21(70)

( Li /F )e = f2 ( Li )

8.21(71)

FIG. 8.21v Relationship of internal reflux to feed at several distillate compositions.

1886

Control and Optimization of Unit Operations

Substitute this equation into Equation 8.21(69) to eliminate Li: ( L/ F ) =

[( Li /F )t + ( Li /F )e ] K 2 [1 + K1 (To − Tr )]

8.21(74)

where L = external reflux flow rate K1 = ratio of specific heat to heat of vaporization of the external reflux K2 = ratio of heat of vaporization of external reflux to heat of vaporization of internal reflux To = overhead vapor temperature Tr = external reflux temperature Dynamics Equations Equations 8.21(68) and 8.21(69) are steady-state equations. Applied without alteration, undesirable column response will result, especially for sudden feed flow rate changes. Feed composition changes are less severe than are feed flow rate changes and seldom require dynamic compensation. Dynamic elements should compensate for feed flow rate changes in such a way that when the feed flow rate changes, the column’s terminal stream flows should respond at the proper time, in the correct direction, and without overshoot. The simplest form of dynamics to meet these criteria involves dead time plus a second-order exponential lag response. The feed flow rate signal is passed through this dynamic element before being used in the operating equations to obtain the bottom product flow rate (B) and the reflux flow rate (L) set points. The transfer function for the dynamic element is ( FL ) Ke −ts = ( F ) (T1s + 1)(T2s + 1) where (FL) (F) t T1, T2

= = = =

8.21(75)

feed flow rate lagged feed flow rate measured dead time time constant

Using Equation 8.21(75), F is eliminated from the left side of Equations 8.21(68) and 8.21(69) to obtain the complete set of operating equations as used in Figure 8.21v. These equations become   Ke −ts B=F  T s + T s + ( 1 ) ( 1 ) 2  1   100 − (%HK D ) − (%LLK ) − (%LK )  F F   100 − (%HK D ) − (%LK B )  

8.21(76)

  ( Li /F )t + ( Li /F )e   Ke −ts L=F    (T1s + 1) (T2s + 1)   K 2 [1 + K1 (To − Tt )] 

8.21(77)

© 2006 by Béla Lipták

where in functional form: ( Li /F )t + ( Li /F )e = f1[( LLK F ),( LK F ),( HK F ),( E ), (FT), ( ∆H F ), (P),(%LKB),(%HKD)] + f2(Li)

8.21(78)

Note that the bottoms and overhead dynamic constants, t, T1, and T2 in Equations 8.21(76) and 8.21(77), are not necessarily the same values. Application of Equations 8.21(76), 8.21(77), and 8.21(78) will result in a suboptimized operation. This is an operation producing a performance close to the specified one. The control block diagram of Figure 8.21w illustrates the application of these equations. Inspection of Equations 8.21(76) and 8.21(78) shows that the system still has five degrees of freedom. Therefore, feed tray location (FT), feed enthalpy ( ∆H F ), column pressure ( P ), concentration of heavy key component in the top product (%HK D ), and concentration of light key component in the bottom product (%LK B ), must all be specified. Although tray efficiency is also included, it remains a fixed value, as explained earlier. Local Optimum Variables Feed enthalpy and column pressure are local optimization variables that can be manipulated to achieve two different objectives to increase profitability: minimizing utilities costs or maximizing throughput. Often, this type of optimization is implemented via valve position controllers. The purpose of valve position controllers is to drive the column to a constraint condition on either reboiler heat, condenser duty, or column loading. Strategies to reduce utility costs include a valve position controller cascaded to pressure control. This is commonly referred to as “floating-pressure control.” Strategies to increase throughput include valve position controller cascaded to feed flow rate. In both cases, the valve position controller will drive the manipulated variable to an equipment constraint. Minimum Pressure Control Floating-pressure operation can often reduce energy consumption by providing minimum pressure operation within the constraints of the system. It is possible to operate a total condensing distillation column 8 with no pressure control. Although this provides for optimum operation at steady state, major problems could occur (e.g., flooding the column) during transient upsets. Flooding is caused by high vapor rates and can result in entrainment or foaming, or in preventing the liquid from flowing down the column. To prevent this, pressure control should be provided. Most distillation columns are operated with constant pressure control. However, several advantages can be achieved

8.21 Distillation: Optimization and Advanced Controls

Bottoms Valve analyzer Temperature position setpoint setpoint setpoint

1887

Distillate analyzer setpoint

Measurements: Bottoms analysis Overhead analysis Bottoms temperature Steam valve position Reflux valve position

Feed tray temperature

× FY

Reflux flow rate

FY

Top temperature

UY

• A •• M

A•• • M

A ••• M

> a1 + a2F + a3F2

×

Li

× FY

K2(1 + K1∆T )

A - Auto L - Local M - Manual R - Remote

Li

Σ FY

V

UY λb λs

× UY

FY Setpoint R• L

Setpoint R• L

FIC

FIC

••

Steam flow rate

AIC PID A •• • M

UY
PB

FIG. 8.21hh Control hierarchy to maximize profit rate when operating constraints are involved.

For this case, L will equal Li. Also, eliminate D by substituting (F − B). Therefore, by substituting L  L = Li =  i  F F

8.21(104)

B B= F F

8.21(105)

and

the following is obtained:   L B a1 + a3 ( P ) + (a2 − 1) i + − 1 F = 0 8.21(106) F F   In this equation a1, a2, and a3 are known from the experimental loading equation. P is specified, F is measured, and Li /F and B/F are obtained from the operating control equations for reflux and bottom product flow rate. It is now possible to find the optimum value for HK D . As HK D is lowered from its maximum allowable value as given

© 2006 by Béla Lipták

by Equation 8.21(93), the values for Li /F and B/F will change. HK D can be lowered until loading Equation 8.21(102) is equal to zero. Also, column pressure P can be raised to allow a greater loading that will result from a lower HK D . A point will be found at which maximum profit rate will exist. The maximum limit for P will be determined by several factors. As P is increased, operating costs will also increase, because of the resulting smaller differential temperature at the reboiler. Therefore, one possible limit would be a reboiler (or operating cost) limit. Another pressure limit will be set by the column pressure rating. Another limit for maximum P may be determined by requirements of upstream processing equipment. Yet another limit for maximum P could be loading of the downcomers between each tray. This comes about because at higher vapor densities, disengagements of vapor from the liquid becomes more difficult. Therefore, at some maximum pressure, density of the vapor can approach the point at which vapor will not have sufficient time to disengage from the liquid in the downcomers, and a condition known as downcomer flooding will occur. Assume for the purpose of illustration that the maximum for P is set at the pressure rating of the column. Therefore, P will be set and left at this value.

1896

Control and Optimization of Unit Operations

B=F

Ke−ts (T1s +1)(T2 s +1)

100 − (%HKD) − (%LLKF) − (%LKF)

L=F

Ke-ts (T1s +1)(T2s +1)

(Li/F )t + (Li/F )o K2[1 + K1(To − Tr)]

Set (P)

100 − (%HKD) − (%LKB)

8.21(77)

(Li/F )t + (Li/F )e = f1[(%LLKF), (%LKF), (%HKF), (E), (∆HF), (P), (FT), (%HKD)(%LKB)] + f2(Li)

8.21(78)

(FT )o = f3[(%LLKF), (%LKF), (%HKF), (F), (%HKD), (%LKB)]

8.21(82)

(∆HF)o = f4[(%LLKF), (%LKF), (%HKF), (%HKD), (%LKB)]

8.21(83)

V

PRC

8.21(76)

L

Set (Tf ) TRC F (FT)o

(P)o = (P)max, set by column pressure rating

FRC

Set (L)

Set (B) FRC

B

Plus equation for feed temperature to give ∆ HF As determined by Equation 8.21(84) (P)max, set by column pressure rating Control equations through the optimization stage without product prices

(P)

(Li/F)

a1 + a3(P) + (a2 − 1)

(B/F )

Li B + −1 F F

8.21(106)

F=0

Loading equation (%LKB) = (%LKB)o (LKB)max =

[1 − (HKD)o][(LLKF) + (LKF) + (HKF) − (HKB)ss] − [(LLKF) + (LKF)][1 − (HKB)ss] (LKF) − (HKD)o

(%HKD) lower from maximum value until loading equation = 0, this will give (%HKD)o 8.21(94)

Optimum for (%LKB) when bottom product price is higher than distillate product price

FIG. 8.21ii System for maximizing profit rate when bottom product price is higher than top product price and when entrainment above the feed tray limits loading.

Figure 8.21ii shows the optimizing control system connected to a typical distillation column. Only part of the basic controls are shown for purpose of clarity. The basic controls are not show in this figure and can be assumed to be the same as shown in Figure 8.21t. Unlimited Market and Feedstock When an unlimited market exists for the products, the feed flow rate and separation resulting in maximum profit rate must be determined. Operation for a column under this condition will always be against an operating constraint. The following example assumes that the operating constraint is the overhead vapor condenser capacity. In general, the overall optimization problem for this case is illustrated in Figure 8.21jj. Values must be determined for the optimum separation, ( HK D )o , ( LK B )o ; feed flow rate, (F )o; and column pressure, ( P )o , that will give maximum profit. One of the key component specifications, HK D or LK B , can be easily determined from the general optimizing policies. Often, the incremental gain in recovery of the most valuable product will not exceed the incremental gain resulting from increasing feed flow rate. This means that both products operating at minimum purity will allow the largest quantity of feed to be charged for maximum profit rate.

© 2006 by Béla Lipták

Optimum Concentrations Both products must be produced at minimum purity to achieve the most profitable operation for this case. Therefore, LK D = ( LK D )ss

8.21(107)

HK B = ( HK B )ss

8.21(108)

The concentration for each control component (LK B and HK D ) must be as follows to satisfy Equations 8.21(107) and 8.21(108): ( LK B )o = ( LK B )max

8.21(109)

( HK D )o = ( HK D )max

8.21(110)

( HK D )max and ( LK B )max are given by Equations 8.21(93) and 8.21(94). The separation is optimized independently by column pressure ( P ) and feed rate (F).

Loading Constraint As was previously explained, the overhead vapor condenser limits loading for this example. Column

8.21 Distillation: Optimization and Advanced Controls

Column loading limited by condenser Set (P) PRC

1897

Maximum coolant flow D (PD), (%LLKD) (%HKD), (%LKD) ≥ (%LKD)ss

Set (TF)

L

TRC

FRC F Set (L) FRC Set (F)

(FT)o

Set (B) FRC B (PB), (%HHKB) (%LKB), (%HKB) ≥ (%HKB)ss

Set points obtained from control equations described in Figure 8.21ee (F)

(%HKD)o

(%LKB)o

(P)o

1. Determine optimum separation (%HKD)o, (%LKB)o 2. Determine optimum column pressure, (P)o 3. Determine feed flow rate set point, (F), that will load overhead vapor condenser.

FIG. 8.21jj When an unlimited market exists for the products and the product prices (PD, PB) are known, the optimum separation, the optimum column pressure, and the value for feed flow rate that will give maximum loading must be found.

pressure must, therefore, be operated at maximum in order to obtain maximum condensing capacity. (∆T across the condenser tubes will be the largest at maximum column pressure, and therefore maximum condenser capacity will result.) The maximum pressure that can be used will be determined by one of the following five constraints, assuming operating costs do not become prohibitive before a physical constraint is reached: 1. 2. 3. 4. 5.

Column downcomer capacity Reboiler capacity Upstream equipment pressure specifications Pressure rating of the column shell Fouling of the reboiler or condenser tubes

As column pressure is increased, the capacities of the reboiler and tower trays will be approached. Also, the pressure rating of the shell and of other processing equipment will be approached. The one of the five constraints that is approached first, as column pressure is raised, will set the operating pressure. Assume for the purpose of illustration that the shell pressure rating is the limit on column pressure.

© 2006 by Béla Lipták

Pressure can, therefore, be set constant at the pressure rating of the column shell. Optimum Feed Flow Rate After the optimum separation ( LK B )o and ( HK D )o and the optimum column operating pressure ( P )o have been determined, the feed rate can be increased until the condenser capacity is approached. There are several ways that feed rate can be manipulated to maintain condenser loading. One very useful method involves a predictive control technique. Overhead vapor flow rate can be expressed in terms of various independent inputs and terms obtained from the operating control equations. Development of the predictive control equation proceeds as follows: Vo = D + L where Vo = vapor flow rate overhead D = top product flow rate L = external reflux flow rate

8.21(111)

1898

Control and Optimization of Unit Operations

First, eliminate D by D = F − B. Now, the maximum overhead vapor flow rate will be given by Equation 8.21(85) and Equation 8.21(111) can be set equal. a1 + a2 ( ∆T ) + a3 ( ∆T )2 = L + F − B

8.21(112)

or F = a1 + a2 (To − Tc ) + a3 (To − Tc )2 + B − L

8.21(113)

where a1, a2, a3 = coefficients for condenser-loading equation that are determined by column tests F = feed flow rate To = overhead vapor temperature Tc = temperature of coolant to overhead vapor condenser B = bottom product flow rates from the output of operating control equation L = reflux flow rates from the outputs of operating control equation

the feed rate required to load the condenser for the particular values of ( HK D )o and ( LK B )o . Figure 8.21kk shows the overall optimizing control system. Only the necessary basic controls are shown. The other controls can be assumed to be as shown in Figure 8.21t. Reboiler Limiting Let us now assume that loading is limited by the reboiler instead of the condenser. Optimum separation remains the same. However, now column pressure must be operated at a minimum value in order to gain maximum reboiler capacity. For this example, assume that minimum column pressure is set by the pressure requirements of downstream equipment. Therefore, column pressure is set at a constant value and will not be changed unless the pressure requirements of downstream equipment are changed. Having achieved the optimum separation (minimum purity of products) and optimum column pressure, the feed rate can now be increased up to the maximum capacity of the reboiler. This, then, will represent the most profitable operation. Again, manipulation of the feed flow rate can be handled by a predictive control technique. Liquid flow rate below the feed tray (Lf) is given by L f = Li + Fi

Temperature of the overhead vapor will be a function of all independent inputs to the system. However, column pressure is usually the main variable of concern. For this example let To = f ( P )

where Fi is the internal feed flow rate. Fi = F[1 + ( K F )(Tv − Tt )]

8.21(118)

8.21(114)

where f ( P ) is some function of column pressure. In many cases f ( P ) can be considered a linear function such as f ( p) = d1 + d 2 ( p)

8.21(115)

This equation can be determined off-line from correlation of data obtained from flash calculations at the average composition of the existing overhead vapor. If changes in composition of the overhead vapor affect temperature of the overhead vapor by a significant amount, then composition also has to be taken into account. Composition for the overhead vapor can be easily approximated from feed composition analysis. If Equation 8.21(113) is carried to this extent, then the feed flow rate can be predicted to keep the condenser against its maximum capacity. For the purpose of illustration here, To is assumed to be a function of column pressure only. Eliminate To from Equation 8.21(113) by Equations 8.21(114) and 8.21(115) to obtain

where KF = a constant equal to the specific heat of the feed divided by the heat of vaporization Tv = temperature of vapor above the feed tray Tf = temperature of feed at column entry The vapor flow rate out of the reboiler is given by VB = L f − B

VB = Li + F[1 + ( K F )(Tv − T f )] − B

P and Tc are measured, and B and L are obtained from the operating equations’ set point calculations. Fmax will be

8.21(120)

Next, substitute (Li /F)F for Li, and (B/F)F for B. Then solve for F to obtain F=

a1 + a2 [d1 + d 2 ( P ) − Tc ] + a3[d1 + d 2 ( P ) − Tc ] B − L = Fmax 8.21(116)

8.21(119)

Now, substitute Equation 8.21(118) into Equation 8.21(117) to eliminate Fi. Then, substitute Equation 8.21(117) into Equation 8.21(119) to eliminate Lf. The following is obtained:

2

© 2006 by Béla Lipták

8.21(117)

(VB )max ( Li /F ) + 1 + ( K F )(Tv − T f ) − ( B/F )

where F = set point of feed flow controller (VB)max = maximum reboiler heat input rate

8.21(121)

8.21 Distillation: Optimization and Advanced Controls

1899

Optimum Separation equations (%HKD)o = (%HKD)max From Equation 8.21(93) (%LKB)o = (%LKB)max From Equation 8.21(94) (%HKD)o

(%LKB)o

Maximum coolant flow

Optimization without product prices

Set (P) Set (TF) F

(∆HF)o, from Equation 8.21(83) (FT)o, from Equation 8.21(82) (P)o = (P)max, for this case set by column pressure rating

TT D

TRC

(B), from Equation 8.21(76) (L), from Equation 8.21(77–78)

PRC

FRC Set (F)

R L

FRC

(FT)o

Set (L) Set (B) FRC B

Plus equation for TF to give (∆HF)o as determined by Equation 8.21(83)

(P)o

(L)

(B)

(F)

Loading function Equation 8.21(116) for optimum feed rate (F ) = a1 + a2[d1 + d2(P) − (Tc)] + a3[d1 + d2(P) − (Tc)]2 + B − L

Tc

FIG. 8.21kk Optimizing for maximum profit rate when unlimited market exists for the products and loading is limited by condenser capacity.

Equation 8.21(121) calculated that feed flow rate which is required to cause the vapor rate (VB)max to exist for all separations specified. Li/F and B/F are obtained from the operating control equations used to achieve a suboptimum operation. Tv and Tf are measured. Equation 8.21(121) is used by specifying (VB)max and then evaluating the column operation. After sufficient time for the column to stabilize, the reboiler valve position (output of reboiler heat flow controller) is observed. If, for example, the reboiler valve is 85% open, (VB)max can then be increased until the reboiler heat control valve is near its maximum opening, say, 95% open. Enough room must be left to maintain control. (VB)max can be adjusted by the plant operator to maintain the reboiler valve near open or can be handled automatically by a valve-positionbased feedback controller. Once (VB)max is established by experience, few adjustments will be required to maintain the column in a fully loaded condition. Adjustments to (VB)max will be

© 2006 by Béla Lipták

required only as the heat transfer capability of the reboiler varies. The control scheme is illustrated in Figure 8.21ll.

CONCLUSIONS Example solutions to some of the common distillation column optimizing problems have been given. Although many different situations can exist, they usually are combinations of those presented. Optimization by feedback control methods cannot approach the quality of control obtained by predictive (feedforward) techniques. This is true even though the predictive control equations may require updating by feedback. In effect, predictive optimization control greatly attenuates any error that must be handled by feedback (updating). The application of feedforward optimizing control forces development of mathematical models of the component parts

1900

Control and Optimization of Unit Operations

Optimum separation equations (%HKD)o = (%HKD)max from Equation 8.21(93) (%LKB)o = (%LKB)max from Equation 8.21(94) (%LKB)o

(%HKD)o

Optimization without product prices

Set (P)

PRC

Set (Tf) TRC (B), from Equation 8.21(76) (L), from Equation 8.21(77–78) (∆HF)o, from Equation 8.21(83) (FT)o, from Equation 8.21(82)

F

(P)o = (P)max, for this case set by downstream requirements

FRC Set (F) (FT)o

(B/F)

FRC Set (L) Set (B) FRC B

Plus equation for TF to give (∆HF)o as determined by Equation 8.21(83)

(Li/F)

D Tf Tv L

Tf

Tv

(F)

Loading function for optimum feed rate (VB)max 8.21(121) (F) = (Li/F) + 1 + (KF)(Tv − Tf) − (B/F)

Tf Tv

FIG. 8.21ll Optimization of a distillation column when unlimited market exists for the products, prices of the products are known, and loading is limited by the reboiler.

of a process. The mathematical models developed for optimizing unit operations will eventually be required to extend optimization to include an entire plant complex. The APC products most applicable to distillation modeling are the white box models, where the theoretical dynamic models are derived on the basis of the mass, energy, and momentum balances of the process. Fuzzy logic and black box models are used less often, as they are more applicable to processes that are poorly understood or when it is acceptable to use a complete mechanistic empirical model constructed solely from a priori knowledge.

5. 6. 7.

8.

9.

10.

References

11. 12.

1. 2. 3. 4.

Lipták, B. G., “Process Control Trends,” Control, January, March, and April 2004. Jensen, B. A., and Collins, P. L., “Incentives for Tighter Fractionator Control,” Control, November 1990. Smith, D. E., Stewart, W. S., and Griffin, D. E., “Distill with Composition Control,” Hydrocarbon Processing, February 1978. Van Kampen, J. A., “Automatic Control by Chromatograph of a Distillation Column,” Convention on Advances in Automatic Control, Nottingham, England, April 1961.

© 2006 by Béla Lipták

13.

14.

Shinskey, F. G., Process-Control Systems, Application, Design, Adjustment, 3rd edition, New York: McGraw-Hill Book Company, 1988. Luyben, W. L., “Distillation Decoupling,” AIChE Journal, Vol. 16, No. 2, pp. 198–203, March 1970. Shinsky, F. G., “The Stability of Interacting Loops with and without Decoupling,” presented at the IFAC Symposium on Multivariable Control, Fredericton, New Brunswick, July 4–8, 1977. Gordon, L. M., “Practical Evaluation of Relative Gains: The Key to Designing Dual Composition Controls,” Hydrocarbon Processing, December 1982. Shinskey, F. G., Distillation Control for Productivity and Energy Conservation, 2nd edition, New York: McGraw-Hill Book Company, 1984. Ryskamp, C. J., “New Control Strategy Improves Dual Composition Control,” Hydrocarbon Processing, June 1980. Van Horn, L. D., “Crude Unit Computer Control: How Good Is It?” Hydrocarbon Processing, April 1980. Garcia, C. E., and Morari, M., “Internal Model Control: 1. A Unifying Review and Some New Results,” Industrial & Engineering Chemistry Process Design & Development, Vol. 21, pp. 308–323, 1982. Richalet, J. A., Rault, A., Testud, J. L., and Papon, J., “Model Predictive Heuristic Control: Applications to an Industrial Process,” Automatica, Vol. 14, pp. 413–428, 1978. Cutler, C. R., and Ramaker, B. L., “Dynamic Matrix Control: A Computer Control Algorithm,” presented at the AIChE 86th National Meeting, Houston, TX, 1979.

8.21 Distillation: Optimization and Advanced Controls

15.

16.

17. 18.

19.

Singh, D. K., “Neural Network Model-Based Control of Chemical Processes,” Proceedings of the ISA/90 International Conference and Exhibit, Part 2, New Orleans, LA, October 1990, pp. 965–968. Douglas, J. M., Jafarey, A., and McAvoy, T. J., “Short-Cut Techniques for Distillation Column Design and Control, Part 1: Column Design,” I&EC Process Design and Development, Vol. 18, pp. 197–202, April 1979. Lee, P. L., and Sullivan, G. R., “Generic Model Control,” Computers and Chemical Engineering, Vol. 12, p. 573, 1988. Tai, H. M., Wang, J., and Ashenayi, K., “A Neural Network Tracking Controller for High-Performance Applications,” Proceedings of the Oklahoma Symposium on Artificial Intelligence, November 1991, p. 225. Bannon, R. et al., “Heat Recovery in Hydrocarbon Distillation,” Chemical Engineering Progress, July 1978.

Bibliography For pre-1990 literature, refer to Section 8.19 1990 Cingara, A., Jovanovic, M., and Mitrovic, M., “Analytical First-Order Dynamic Model of Binary Distillation Column,” Chemical Engineering Science, Vol. 45, No. 12, pp. 3585–3592, 1990. Ding, S. S. and Luyben, W. L., “Control of a Heat-Integrated Complex Distillation Configuration,” Industrial & Engineering Chemistry Research, No. 29, p. 1240, 1990. Farrel, R. J. and Polli, A., “Comparison of Unconstrained Dynamic Matrix Control to Conventional Feedback Control for a First Order System,” Proceedings of the ISA/90 International Conference and Exhibit, Part 2, New Orleans, LA, October 1990, pp. 1037–1046. Gerstle, J. G., Hokanson, D. A., and Anderson, B. O., “Multivariable Control of a C2 Splitter,” Proceedings of the AIChE Annual Meeting, Chicago, IL, November 1990. Kister, H. Z., Distillation Operation, New York: McGraw-Hill Book Company, 1990. Li, R., Olson, J. H., and Chester, D. L., “Dynamic Fault Detection and Diagnosis Using Neural Networks,” Proceedings of the Fifth IEEE International Symposium on Intelligent Control, Philadelphia, PA, September 1990, pp. 1169–1174. Papastathopoulou, H. S. and Luyben, W. L., “Potential Pitfalls in Ratio Control Schemes,” Industrial & Engineering Chemistry Research, October 1990, pp. 2044–2053. Pitt, M. J., Instrumentation and Automation in Process Control, New York: E. Horwood, 1990. Riggs, J. B., “Advanced Model-Based Control of a Sidestream Draw Column,” Proceedings of the ISA/90 International Conference and Exhibit, Part 2, New Orleans, LA, October 1990, pp. 1023–1032. Riggs, J. B., Watts, J., and Beauford, M., “Industrial Experience with Applying Nonlinear Process Model-Based Control to Distillation Columns,” Proceedings of the ISA/90 International Conference and Exhibit, Part 2, New Orleans, LA, October 1990, pp. 1047–1054. Skogestad, S., Jacobsen, E. W., and Morari, M., “Inadequacy of SteadyState Analysis for Feedback Control, Distillate, Bottom Control of Distillation Columns,” Industrial & Engineering Chemistry Research, December 1990, pp. 2339–2346.

1991 Coughanowr, D. R., Process Systems Analysis and Control, 2nd edition, New York: McGraw-Hill Book Company, 1991. Farhat, S., Piouleau, L., Domenech, S., and Czernicki, M., “Optimal Control of Batch Distillation via Nonlinear Programming,” Chemical Engineering and Processing, Vol. 29, No. 1, pp. 33–38, January 1991.

© 2006 by Béla Lipták

1901

Figuerosa, J. L., Desages, A. C., Romagnoli, J. A., and Palazoglu, “Highly Structured Stability Margins for Process Control Systems. A Case Study of Decoupling Control in Distillation,” Computers & Chemical Engineering, Vol. 15, No. 7, pp. 493–502, July 1991. Gerstle, J. G. and Hokanson, D. A., “Experiences with Applying Dynamic Matrix Control in Olefins Plants,” Proceedings of the ISA/91 International Conference and Exhibit, Part 1, Anaheim, CA, October 1991. Houk, B. G., Snowden, D. L., and Stevenson, T. E., “Improved Control of Ethylene Recovery Train Using Dynamic Matrix Control,” Proceedings of the ISA/91 International Conference and Exhibit, Part 1, Anaheim, CA, October 1991. Jacobsen, E. W., and Skogestad, S., “Control of Unstable Distillation Columns,” Proceedings of the American Control Conference, Green Valley, AZ, Vol. 1, pp. 773–778, 1991. Jensen, B. A., “Improve Control of Cryogenic Gas Plants,” Hydrocarbon Processing, May 1991. Kettinger, J. G., Gaines, L. D., and McGee, N. F., “Multivariable Control a Depropanizer,” Proceedings of the ISA/91 International Conference and Exhibit, Part 1, Anaheim, CA, October 1991. O’Connor, D. L., Grimstad, K., and McKay, J., “Application of a Single Multivariable Controller to Two Hydrocracker Distillation Columns in Series,” Proceedings of the ISA/91 International Conference and Exhibit, Part 1, Anaheim, CA, October 1991. Papadopoulos, M. N. and Berkowitz, P. N., “Multivariable (MVC) Process Control. A User-Based On-Line Optimal Control System,” Proceedings of the Industrial Computing Conference, Anaheim, CA, October 1991. Papastathopoulou, H. S. and Luyben, W. L., “Control of a Binary SideStream Distillation Column,” Industrial & Engineering Chemistry Research, April 1991, pp. 705–713. Patwardhan, A. A. and Edgar, T. F., “Nonlinear Model-Predictive Control of a Packed Distillation Column,” Proceedings of the American Control Conference, Green Valley, AZ, Vol. 1, 1991, pp. 767–772. Radhakrishnan, T. K., and Gangiah, K., “Critical Study of Multivariable Self-Tuning Algorithms for Distillation Control,” Chemical Engineering & Technology, Vol. 14, No. 6, pp. 399–405, December 1991. Sandelin, P. M., Haeggblom, K. E., and Waller, K. V., “Disturbance Rejection Properties of Control Structures at One-Point Control of a Two-Product Distillation Column,” Industrial & Engineering Chemistry Research, June 1991, pp. 1182–1186. Sandelin, P. M., Haeggblom, K. E., and Waller, K. V., “Disturbance Sensitivity Parameter and Its Application to Distillation Control,” Industrial & Engineering Chemistry Research, June 1991, pp. 1187–1993. Tinetti, G., Ghassan, A., and Cheung, J. Y., “Comparing Intelligent and Classical Control Systems in a Nonlinear Environment,” Proceedings of the Oklahoma Symposium on Artificial Intelligence, November 1991, p. 199. Yang, D. R., Seborg, D. E., and Mellichamp, D. A., “Combined Balance Control Structure for Distillation Columns,” Industrial & Engineering Chemistry Research, September 1991, pp. 2159–2168. Yang, D. R., Waller, K. V., Seborg, D. E., and Mellichamp, D. A., “Dynamic Structural Transformations for Distillation Control Configurations,” AIChE Journal, Vol. 36, No. 9, pp. 1391–1402, September 1991.

1992 Brambilla, A. and D’Elia, L., “Multivariable Controller for Distillation Columns in the Presence of Strong Directionality and Model Errors,” Industrial & Engineering Chemistry Research, February 1992, pp. 537–543. Luyben, W. L. (ed.), Practical Distillation Control, New York: Van Nostrand Reinhold, 1992. Rovaglio, M., Raravelli, T., Biardi, G., Gaffuri, P., and Soccol, S., “Precise Composition Control of Heterogeneous Azeotrophic Distillation Towers,” Computers & Chemical Engineering, Vol. 16, 1992.

1902

Control and Optimization of Unit Operations

Yiu, Y., and Papadopoulos, M. N., “On-Line Optimal Advanced Control of a Ryan Holmes Gas Processing Process,” Advances in Instrumentation and Control, Vol. 47, Part 1, Research Triangle Park, NC: Instrument Society of America, 1992, pp. 967–975.

1993 Anderson, B., and Mejdell, T., “Using Temperature Profile for Product Quality Estimation on a Distillation Column,” Advances in Instrumentation and Control, Vol. 48, Part 1, Research Triangle Park, NC: Instrument Society of America, 1993, pp. 39–46. Freitas, M. S., Campos, M. C. M. M., and Lima, E. L., “Dual Composition Control of a Debutanizer Column,” Advances in Instrumentation and Control, Vol. 48, Part 1, Research Triangle Park, NC: Instrument Society of America, 1993, pp. 501–509. Ganguly, S., “Model Predictive Control of Distillation,” ISA/93 Technical Conference, Chicago, IL, September 19–24, 1993. McKetta, J. J. (ed.), Unit Operations Handbook, Vol. 1: Mass Transfer, New York: Marcel Dekker, 1993.

1994 Gokhale, V., Shukla, N., and Munsif, H., “Analysis of Advanced Distillation Control on a C3 Splitter and a Depropanizer,” 1994 AIChE National Annual Meeting, San Francisco, CA, November 1994.

1995 Banerjee, A., and Arkun, Y., ‘‘Control Configuration Design Applied to the Tennessee Eastman Plantwide Control Problem,” Computers. Chem. Engng., 19(4), 453–480, 1995. Diwekar, U. M., Batch Distillation: Simulation, Optimal Design and Control (Series in Chemical and Mechanical Engineering), Taylor & Francis, September 1995. Fleming, B., and Sloley, A.W., “Feeding and Drawing Products: The Forgotten Part of Distillation,” Proceedings of the ChemShow and Exposition, New York, December 1995. Hurowitz, S. E., and Gokhale, V., “A Dynamic Model of a Superfractionator: A Test Case for Comparing Distillation Control Techniques,” DYCORD ’95, 4th IFAC Symposium, Helsingor, Denmark, June 1995. Lundstrom, P., and Skogestad, S., “Opportunities and Difficulties with 5 × 5 Distillation Control,” J. Process Control, 1995, 5, 249–261. Musch, H. E., and Steiner, M., “Robust PID Control for an Industrial Distillation Column,” Control System Magazine, 1995, 15, 4, 46–55. Rawlings, J. B., “Dynamics and Control of Chemical Reactors, Distillation Columns, and Batch Processes (Dycord ’95),” a postprint volume from the 4th IFAC Symposium on Dynamics and Control of Chemical Reactors, Distillation Columns, and Batch Processes (DYCORD ’95).

1996 Koggersbøl, A., Andersen, B. R., Nielsen, J. S., and Jørgensen, S. B.,“Control Configuration for Energy Integrated Distillation,” Computers & Chem Eng., 20 (supplement), pp. S853–S858, 1996. th Shinskey, F. G., Process Control Systems, 4 ed., New York: McGraw-Hill, 1996, pp. 340–347.

1997 Anderson, N. A., Instrumentation for Process Measurement and Control, 3rd edition, Boca Raton, FL: CRC Press, October 1997.

© 2006 by Béla Lipták

Hurowitz, S. E., and Anderson, J. J., “Distillation Configuration Selection for Dual Composition Control,” AIChE Spring National Meeting, Houston, TX, April 1997. Hurowitz, S. E., and Anderson, J. J., “Control of High-Purity Distillation Columns,” Control 97 Conference, Sydney, Australia, October 1997. Linsley, J., “New, Simpler Equations Calculate Pressure-Compensated Temperatures,” Oil & Gas Journal, May 24, 1997, pp. 58–64. Mahoney, D. P., and Fruehauf, P. S., “An Integrated Approach for Distillation Column Control Design Using Steady State and Dynamic Simulation,” Aspentech technical articles, March 1997. Tham, M. T., “Distillation,” Base Document URL: http://lorien.ncl.ac.uk/ ming/distil/distil0.htm, Date: October 1997. Skogestad, S., ‘‘Dynamics and Control of Distillation Columns: A Tutorial Introduction,” Trans. IChemE, Vol. 75, Part A, pp. 539–562, 1997.

1998 Betlem, B. H. L., Krijnsen, H.C., and Huijnen, H., “Optimal Batch Distillation Control Band on Specific Measures,” Chemical Engineering Journal, 71, pp. 111–126, 1998. Luyben, W. L., Tyreus, B. D., and Luyben, M. L., Plantwide Process Control, New York: McGraw-Hill, 1998. Ochiai, S., “Calculating Process Control Parameters from Steady State Operating Data,” ISA Transactions, Vol. 36, No. 4, pp. 313–320, 1998. Riggs, J. R., “Improve Distillation Column Control,” Chemical Engineering Progress, October 1998, pp. 31–47. Stichlmair, J. G., and Fair, J. R., Distillation: Principles and Practices, New York: John Wiley & Sons, 1998.

1999 Hurowitz, S. E., Anderson, J., Duvall, M., and Riggs, J. B., “An Analysis of Controllability Statistics for Distillation Configuration Selection,” Presented at the AIChE Annual Meeting, Dallas, TX, November 1999. Eker, I., and Sakthivel, K., “Automation & Lube Oil Additives Blending Plant Using an S88.01-Consistent Batch Software: A Case Study,” Proceedings of the World Batch Forum, San Diego, CA, April 1999.

2000 Sloley, A. W., “Steady under Pressure: Distillation Pressure Control,” presented at the American Institute of Chemical Engineers Spring Meeting, March 6–9, 2000. Betlem, B. H. L., “Batch Distillation Column Low-Order Models for Quality Control Program,” Chemical Engineering Science, 55, pp. 3187– 3194, 2000. Hugo, A., “Limitations of Model Predictive Controllers,” Hydrocarbon Proc., January 2000, p. 86. Roffel, B., Betlem, B. H. L., and De Ruijter, J. A., “Modeling and Control of a Cryogenic Distillation Column,” Computers and Chemical Engineering, 24, pp. 111–123, 2000. Roffel, B., “Distillation: Instrumentation and Control Systems,” in Encyclopedia of Separation Science, Academic Press, 2000. Willis, M. J., “Selecting a Distillation Column Control Strategy (A Basic Guide),” University of Newcastle, UK: 2000.

2002 Florez, M., “Batch Distillation: Practical Aspects of Design and Control,” Proceedings of the World Batch Forum, Woodbridge Lake, NJ, April 2002. Cook, B., Engel, M., Landis, C., Tedeschi, S., and Zehnder, A., “Synthesis of Optimal Batch Distillation Sequences,” Proceedings of the World Batch Forum, Woodbridge Lake, NJ, April 2002.

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2003

2004

Blevins, T. L., McMillan, G. K., Wojsznis, W. K., and Brown, M. W., Advanced Control Unleashed, Research Triangle Park, NC: ISA, 2003. Kralj, F., “Application of the S88 Model in the Control of Continuous Distillation Facilities,” Proceedings of the World Batch Forum, Woodbridge Lake, NJ, April 2003.

Hurowitz, S., Anderson, J., Duvall, M., and Riggs, J. B., “Distillation Control Configuration Selection,” submitted to J. Process Control, March 2004. Lipták, B. G., “Process Control Trends,” Control, January, March, and May, 2004.

© 2006 by Béla Lipták