The Design of Plantwide Control Systems; a Practical Approach · 2020. 9. 10. · we start with a...

The Design of Plantwide Control Systems;

a Practical Approach

A.E.M. Huesman

Delft Center for Systems and Control

Delft University of Technology

Mekelweg 2, 2628 CD Delft, The Netherlands

23 March 2004

Foreword

The best way to learn is to teach...

Since 1990 I have been involved intensively in the design of plantwide control systems. Firstas a part of my master’s assignment and during the period 1990 - 1999 as a control technolo-gist for Shell. In 1999 I returned to Delft as an assistant professor. My current research areacould be described as “economic optimal plantwide control”.

The text in front of you are the notes that go with the course ”Practical Approach to theDesign of Plantwide Control Systems” (st6121) 1. This course was developed especially forstudents of the TWAIO program “Process and Equipment Design”. Basically the coursetries to teach chemical engineers the competence of designing a control system for a completechemical plant.

In the past this course was developed by Giljam Bierman and updated by Sorin Bildea. Ihave taken the liberty to completely revise the course while trying to maintain all the valuableelements of the the original and updated course. I would like to thank Giljam and Sorin forthe opportunity to run this course.

The notes help to understand the subject. However in order to achieve the whole competencestudents should also do a few designs and discuss their designs with an experienced controltechnologist.

Although these notes have been written under somewhat hectic circumstances, I have enjoyedit. If you have suggestions how to improve these notes please do not hesitate to contact me(email: [email protected]).

Adrie Huesman, May 2003.

Chapters 3, 5 and 7 haven been corrected/extended.

Adrie Huesman, March 2004.

1All rights reserved by A.E.M. Huesman.

3

Contents

1 Introduction 7

1.1 Problem sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Purpose of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Outline of these notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 The objective of plantwide control 11

2.1 Purpose of chemical plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Design of chemical plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Objective of plantwide control . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 A summary of relevant theory 17

3.1 Process dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Process control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Basic process control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Decomposition of the plantwide control problem 27

4.1 Material balance and quality control . . . . . . . . . . . . . . . . . . . . . . . 274.2 Decomposition in the time domain . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Interaction in the frequency domain . . . . . . . . . . . . . . . . . . . . . . . 30

5 Material balance and quality control 33

5.1 Push, pull and other material balance schemes . . . . . . . . . . . . . . . . . 335.2 Standard quality schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 About recycles 37

6.1 The influence of recycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2 How to deal with recycles? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7 Plantwide control design 43

7.1 Procedure of Ponton and Laing . . . . . . . . . . . . . . . . . . . . . . . . . . 437.2 Procedure of Luyben, Tyréus and Luyben . . . . . . . . . . . . . . . . . . . . 447.3 Proposed procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.4 An example; sulfuric acid alkylation . . . . . . . . . . . . . . . . . . . . . . . 46

A Instrumentation details 59

5

6 CONTENTS

Chapter 1

Introduction

“...It is not usual to be personal or anecdotal in papers of this sort, but it seems an appropriatepoint for one of the authors (JWP) to tell a story at his own expense. Some years ago, whena fairly junior academic, but nonetheless with several years of teaching experience, he tookan ‘industrial sabbatical’ as a process engineer on a large project 1. The work in general hadnothing to do with control, computers or any of the author’s other main research interests.However, at one point he was asked what subjects he had taught, and mentioned that theseincluded process control. “Ah!” said the questioner, “I have a control problem for you”. Hewas then presented with a process flowsheet and asked to put the control loops on it.”

“The author was nonplussed and embarrassed. Despite having taught differential equa-tions, Laplace transforms, Bode diagrams, root locus plots, and other appurtenances of atraditional control course, he was at a loss even as to how to start this task. So must havebeen generations not just of author’s students, but graduates of most chemical engineeringdegree courses. And this is the control task which process engineers in industry are mostfrequently called upon to perform...”

Taken from J.W. Ponton and D.M.Laing [1], including footnote.

1.1 Problem sketch

Traditional process control courses typically teach subjects like differential equations, lin-earization, Laplace transforms, eigenvalues and the like (see for example Stephanopoulos [2]).However at first sight these subjects seem of little use when faced with the central questionof this course:

How to design a control system for a complete chemical plant?

As a matter a fact one wonders why these traditional subjects are taught at all. Actuallythere are good reasons for teaching the traditional process control subjects. A nice way toexplain this is by making use of analogy that was introduced by Montfoort [3]:

“...During the design an architect focuses on the purpose of the complete building and inte-grates parts like stairs, roof and walls into a complete whole fit for purpose. His integrating

1This incidentally is an activity to be highly recommended for young and indeed not so young academics.

7

8 CHAPTER 1. INTRODUCTION

task distinguishes him from the person who for example calculates the thickness of the walls......A technologist has a task similar to an architect. Just as an architect designs a buildingwith a certain purpose so does a technologist design a process with the purpose to makea specific product. In the case of an architect the focus is on the complete building, for atechnologist on the complete process. Also a technologist integrates various units (distilla-tion column, pump, reactor) into a complete whole fit for purpose. Just like an architect atechnologist has an integrating task that distinguishes him from the person who calculatesthe various units even though the latter also needs to be done and is often part of the work...” 2

So for the syntheses of a building and a process a designer uses a structural view that consistsof:

• Building blocks, these blocks represent the highest level of detail. In other words adesigner is concerned with walls and a reactor but not with bricks or catalyst.

• Knowledge how to put the building blocks together. In what way can the buildingblocks be connected to form a building or process?

It is fair to say that the level of detail of a typical process control course is actually at a higherlevel than the building blocks. Nevertheless such a course serves two purposes. First of all weneed the higher detail level to recognize/define adequate building blocks. And secondly oncethe design has been finished we still have to calculate/design the individual building blocks.

So in this course the focus will be on the structural aspects of process control but that theknowledge at a higher detail level is a prerequisite (Process Control, st3091).

1.2 Purpose of the course

The purpose of the course is to to teach chemical engineers the competence of designing aplantwide control system.

The contents is this course is limited by two considerations:

1. Current state of the art. At this point in time it is not possible to design an economicoptimal plantwide control system (it is in fact a research area). Therefore our aim willbe to design a workable plantwide control system. Something similar applies to thecontrol of batch processes in the sense that this field still needs a lot of research. Sowill limit ourselves to continuous processes.

2. Amount of time available. Currently the study load is set at 80 hours (all-in). Forthat reason we will limit ourselves to Basic Process Control (BPC) 3. So even thoughModel Predictive Control (MPC) and Real Time Optimization (RTO) can be part of aplantwide control system they are outside the scope of this course.

2The original text is in Dutch, this English translation tries to stay as close as possible to the original text.3In this course BPC means proportional integral derivative control with extensions like ratio, cascade,

override, selective, inferential and feedforward control.

1.3. OUTLINE OF THESE NOTES 9

1.3 Outline of these notes

Basically these notes consists out of three parts. The first part provides the starting point. Sowe start with a discussion on the objective of plantwide control (chapter 2). And in chapter3 the relevant (process) system and control theory is summarized.

The second part focuses on the structural aspects of plantwide control. Chapter 4 explainsthat in every plantwide control system we can recognize a material balance and a qualityaspect. Both aspect are investigated further in chapter 5. The second part finishes with adiscussion on recycles (chapter 6).

The third part (chapter 7) integrates the various structural aspects into a plantwide controldesign procedures. First some procedures are discussed and then a selected procedure is ap-plied on an example.

10 CHAPTER 1. INTRODUCTION

Chapter 2

The objective of plantwide control

The objective of plant-wide control should follow from the purpose of a chemical plant. There-fore we will first discuss the various aspects of the purpose of a plant. Since the processand control design are done separately we need to determine which aspects are relevant forthe control design. Finally these aspects will be used to formulate the objective of plantwidecontrol.

2.1 Purpose of chemical plants

From a technological point of view the purpose of a chemical plant is to convert feed intospecified product, see for example Douglas [4]. Figure 2.1 provides a general overview of thevarious feeds and products.

Raw materials (ca.10):oilmineralsairetc.

Intermediates (ca. 300):acectic acidethanalethene oxideetc.

Consumer products (ca. 30000):plasticsfertilizerspharmaceuticalsetc.

2 15 100

Base chemicals (ca. 20):etheneammoniasulfuric acidetc.

Fuels (ca.10):gasolinekerosinegas-oiletc.

Figure 2.1: From raw materials to consumer products, after Moulijn and others [5].

Normally conversion is described in terms of quantity and quality. However a plant shouldnot only be regarded in a technological sense, it should also be looked at from the angle of:

• Society; a plant should be safe and sustainable.

• Economics; a plant should be profitable.

So the purpose of a chemical plant is to convert feed into product (desired quantity andquality) in a safe, sustainable and profitable manner.

11

12 CHAPTER 2. THE OBJECTIVE OF PLANTWIDE CONTROL

2.2 Design of chemical plants

To make sure that the desired conversion is achieved in a safe, sustainable and profitable wayplants are designed in a systematic way (see figure 2.2). The first stage is the formulation.This stage specifies the required performance of the design (in terms of conversion, safety etc.)and limits the search space (what kind of operation, which unit operations etc.). The synthesisstage generates a number of alternatives. The behavior of the alternatives is determined inthe analysis stage (this typically involves simulation). Finally the evaluation compares theperformance of the design with the required performance as stated in the formulation. Incase that the performance is not acceptable one can adjust the formulation or return to thesynthesis stage and generate more alternatives.

formulation synthesis analysis evaluation

Figure 2.2: A systematic approach towards design, after Meeuse [6].

Figure 2.2 seems to suggest that chemical plants are designed “in one go”. However in actualpractice process and control are designed separately. Meeuse [6] argues convincingly that thisseparation has severe drawbacks. Since at this point in time no reliable alternative is availableyet we will assume that both designs are done separately. The consequence of this separationis that we need to determine which performance aspects are relevant for the process designand which are relevant for the control design:

Conversion As mentioned before conversion is described by quantity and quality. Bothaspects are relevant for the process as well as the control design. The process designmust ensure that the required quantity and quality range can be produced. It shouldbe noted that this flexibility allows for adjustment to market demand (more on thisin the last paragraph of this chapter). The control must be designed such that anycombination of quantity and quality in this range can be effected and maintained. Thesituation is shown in figure 2.3.

Quantity

Qualit

y

A

D C

Bxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xx

F

xxxx

E

Figure 2.3: The process design ensures that a plant can perform a flexible conversion (areaABCD). The control effects and maintains any conversion in this area (for example E or F).

2.2. DESIGN OF CHEMICAL PLANTS 13

Safety Safety in a process environment is normally translated in: “NO uncontrolled loss ofcontainment”. In other words we should avoid rupture of equipment. Safety is achievedby a combination of inherent safety, Safety Relief Valves (SRV’s) and Instrument Pro-tective Functions (IPF’s). Inherent safety means that the equipment has sufficientmechanical integrity to avoid rupture. SRV’s circumvent rupture by controlled release.IPF’s monitor situations that could lead to rupture and if necessary effect a (partial)shutdown to avoid it. Inherent safety, SRV’s and IPF’s are a part of the process design.However this does not mean that the aspect safety is completely irrelevant for the con-trol design. To be more precise; although the control design does not guarantee safetyit should support it in the sense that shutdowns effected by the IPF’s and the lifting ofSRV’s should be avoided. The last two events can have a large negative impact on theperformance aspects conversion, sustainability and profitability. Figure 2.4 depicts therelation between inherent safety, SRV’s, IPF’s and control in a graphic way.

Pressure

Tem

pera

ture

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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xxxxxxxxxxxxxxx

xxxxxxxxxx

limit inherent safety

protected by SRV and/or IPF

normal control range

Figure 2.4: The combination of SRV and IPF ensures that the plant stays well within theinherent safe area. The control operates with sufficient back-off from the SRV and IPF limit.

Sustainability This aspect has received increasing attention during the last decade. Andrecently inclusion of this performance aspect in the process design has been investigated,see Korevaar [7]. But since the amount and results of the research on this subject arestill limited, the author feels it would be premature to include this performance aspectin the control design at this point in time.

Profitability A good measure for the profitability of a plant, P , can be calculated as follows:

P = price products · quantity products︸︷︷︸

revenues

− fixed costs − variable costs︸︷︷︸

total costs

−taxes (2.1)

Fixed and variable costs are independent resp. dependent on the quantity of products.Smith [8] gives a breakdown of the fixed and variable costs (see figure 2.5).

It should be noted that only a part of the economic variables just mentioned are underthe influence of the process and control design. These variables are the capital costs(repayments) and a part of the variable costs; raw materials (quantity, not the price),consumed chemicals and catalysts and utilities. With some abuse of the term we willrefer to this part of the variable costs as operating costs. In other words for the process


raw materialsconsumed chemicals and catalystsutilitiesmaintenance due to operationquality controletc.

capital costs repaymentsroutine maintenanceoverheadslaborinsuranceetc.

Fixed costs include: Variable costs include:

Figure 2.5: A breakdown of the fixed and variable costs.

and control design only the capital and operating costs are relevant. It is even possibleto go one step further; for the process design both capital and operating costs matterbut for the control design only operating costs are relevant. This will be illustrated bymeans of an example; a two-cut splitter (simple distillation column).

Given the feed, the operating pressure and the required separation it is well-known thatthe process design boils down to choosing the right combination of number of traysand reflux ratio (provided the number of trays exceeds the minimum number of trays,see Huesman [9]). The link to costs is straightforward; the number of trays is directlyrelated to capital costs and the reflux ratio (mainly) to operating costs.

What about the control design? It is important to realize that the field instrumentation(transmitters and valves) including the cabling and the control system do not dependmuch on the control design. In the case of a two-cut splitter we typically need two leveltransmitters, one pressure transmitter, one or two quality or temperature transmitters,five valves and a system that enables the control of four or five variables. So the controldesign has little influence on the capital costs. A good control design however hasinfluence on the operating costs. The steam consumption by a two-cut splitter can beminimized by operating the column at minimum quality give-away.

2.3 Objective of plantwide control

We are now in the position to summarize the discussion in this chapter and formulate theobjective of plantwide control:

1. support (not guarantee) safety2. realize the required conversion3. minimize the operating costs

It should be noted that the three aspects of the objective are mentioned in the order of eco-nomic importance. The support of safety is essential to avoid (economic) loss, the requiredconversion is a necessary condition to make profit and minimizing the operating costs leadsto maximum profit.

There are at least two approaches to realize this objective. The first approach uses the

2.3. OBJECTIVE OF PLANTWIDE CONTROL 15

objective to formulate a dynamic optimization problem:

minDOF over time

(operating costs) (2.2)

s.t. support of safety, (2.3)

required conversion (2.4)

and plant behavior (2.5)

DOF stands for Degrees Of Freedom (valve positions, motor speeds etc.), the abbreviation s.t.means subjected to. We have already seen that the support of safety basically means keepingthe equipment in the allowed pressure/temperature range and that the required conversioncan be expressed in terms of quantity and quality. Plant behavior seems a new aspect butis described by a dynamic model of the plant. The optimization approach is attractive sinceit leads to economic optimal plantwide control. However the typical size of the optimizationproblems involved is considerable (the number of variables and equations is ≈ 103 − 105).And the effect of disturbances can only be taken into account by repeating the optimizationreal-time. Since real-time solution of large optimization problems is quite a challenge theoptimization approach is a research area.

The second approach could be called traditional. The traditional approach tries to realizethe required conversion by control. If necessary or possible the support of safety and theminimization of the operating costs is also handled by control (but the primary focus is onthe required conversion). In the rest of these notes the traditional approach will explained inmore detail. An advantage of this approach is that it leads to control problems that can besolved easily real-time. A disadvantage is that it most likely does not minimize the operatingcost to the lowest possible level, so potential profit is lost.

STORAGE STORAGEPLANT

aPLANT

bPLANT

c

PLANNING & SCHEDULING SYSTEM

MARKET

FEED PRODUCTS

UTILITYa

UTILITYb

PLANTWIDE

CONTROL

PLANTWIDE

CONTROL

PLANTWIDE

CONTROL

Figure 2.6: The relation between plantwide control and the scheduling & planning system.

It should be noted that the required conversion is determined outside the boundary of theplantwide control system. One may suspect that the required conversion is directly related


to the market demand. Although the market demand plays an important role in determiningthe required conversion it is normally not the only factor. This has to do with the typicalenvironment of a plant; a site. Basically a site is just a collection of a number of plants thatshare the organization (management, technical staff, maintenance etc.), but also things likestorage and utilities. Figure 2.6 shows a simple site. Note that the various plantwide controlsystems receive their production schedules (the required conversion for say the next 1 - 4weeks) from the the scheduling & planning system. The last system has a sidewide scope anddetermines the production schedules based on:

• An intensive interaction with the market (humans play an important role).

• Constraints related to the plants (throughput and quality).

• Constraints related to storage and utilities (capacity).

For example it might seem profitable to raise the throughput of a certain product (the marketreadily absorbs it and from a plant perspective there are no constraints). However raisingthroughput could be impossible due to lack of product storage or the fact that a utility isalready at maximum capacity.

Chapter 3

A summary of relevant theory

This chapter provides an overview of the relevant (process) systems and control theory. Italso discusses the building blocks of BPC. Please note that this chapter is just a summary; itis by no means complete or elaborate. The idea is to refresh the present knowledge (maybeadd a few things) and to organize it in a way that helps answering the central question of thiscourse.

3.1 Process dynamics

Given the fact that the plant of interest should have a certain controlled behavior, one mayask: “Why bother with process dynamics?”. The answer is that the controlled behavior isdetermined by both the control as well as the dynamics (to the point where certain controlledbehavior is impossible to achieve due to the dynamics).

There are two ways to determine the process dynamics:

1. Modelling. For a process system this boils down to the following procedure; 1) make andstate your assumptions, 2) choose one or more volume elements, 3) draw a schematic,introduce convenient notation, 4) apply the relevant laws of conservation and 5) addconstitutive equations (for transport of mass, heat and/or impulse). Since we are in-terested in the process dynamics the conservation laws should include accumulationterms.

2. Identification. Basically this involves choosing a base function (for example a first orderdifferential equation) and the design of an experiment (step, pseudo random binarynoise, etc.). After performing the experiment the parameters of the base function arecalculated such that the input-output behavior of the base function matches that of theexperiment as close as possible.

The modelling and identification approach represent the extremes of a complete spectrum ofpossibilities to determine a dynamic model. Therefore on one hand they are complementary(see table 3.1) and on the other hand related (the conservation laws play a similar role asbase functions).

The result of a modelling exercise is typically a set of dependent non-linear differential equa-tions. In this form the model is not of much use, in order to bring the model in a more usable

17

18 CHAPTER 3. A SUMMARY OF RELEVANT THEORY

Table 3.1: A comparison between modelling and identification.Approach Advantages Disadvantages

Modelling 1. large working range 1. labor intensive2. in agreement with balances 2. result often in non-usable form

Identification 1. fast for small systems 1. limited working range2. result in usable form 2. complex for non-linear systems

form we can either simulate 1 or linearize the model 2. Once the model is linearized, devia-tion variables can be introduced and the differential equation(s) can be written as algebraicequations using Laplace transforms. Modelling, linearization, the introduction of deviationvariables and Laplace transformation will be illustrated by means of an example. Figure 3.1shows an isothermal Continuous Stirred Tank Reactor (CSTR).

Fin, A

in

Fout

, Aout

V

Figure 3.1: An isothermal CSTR.

In the CSTR feed A is converted in product B via first order kinetics. Applying the conser-vation laws leads to (the density is assumed constant):

total mass:dV

dt= Fin − Fout (3.1)

component A:d(V Aout)

dt= FinAin − FoutAout − kAoutV (3.2)

Substitution of the first equation in the last one gives after rearranging:

dAoutdt

=FinV

(Ain − Aout) − kAout (3.3)

Note that this model has two state variables (V and Aout), two manipulated variables orinputs (Fin and Fout) and one disturbance variable (Ain). Suppose that we set the right handside of equation 3.1 equal to f1 and the right hand side of equation 3.3 equal to f2. Bysetting f1 = 0 and f2 = 0 we can find a steady state solution, for example; Fin,0 = Fout,0 = 1m3/hour, Ain,0 = 500 mol/m

3, k = 18 hour−1, V0 = 0.5 m3 and Aout,0 = 50 mol/m

3.

1For example with Matlab using a Ordinary Differential Equation (ODE) solver.2A linearized model will only be valid near the point of linearization. But if we effectively control the

process around this point then the process will always operate near this point. Please note that a linear modelallows for analytical results.

3.1. PROCESS DYNAMICS 19

Linear differential equations can be written in so-called state space format :

ẋ = Ax + Bu + B′d (3.4)

y = Cx (3.5)

In this format x, u and d are vectors and A, B, B′ and C matrices. In our case:

x =

[x1x2

]

=

[V − V0

Aout − Aout,0

]

(3.6)

u =

[u1u2

]

=

[Fin − Fin,0

Fout − Fout,0

]

(3.7)

d =[

d1]

=[

Ain − Ain,0]

(3.8)

So x, u and d contain respectively the states, inputs and disturbances as deviation variables.In this example the matrices are given by:

A =

[(δf1/δV )0 (δf1/δAout)0(δf2/δV )0 (δf2/δAout)0

]

=

[0 0

−1800 −20

]

(3.9)

B =

[(δf1/δFin)0 (δf1/δFout)0(δf2/δFin)0 (δf2/δFout)0

]

=

[1 −1

900 0

]

(3.10)

B′ =

[(δf1/δAin)0(δf2/δAin)0

]

=

[02

]

(3.11)

Since the output variables equal the state variables the matrix C equals the identity matrix.From the state space representation it is easy to determine the Laplace transforms via 3:

G(s) = C(sI − A)−1B + C(sI − A)−1B′ (3.12)

Applying this to the example gives:

y1(s)

u1(s)=

1

s(3.13)

y2(s)

u1(s)=

900s − 1800

s2 + 20s(3.14)

y1(s)

u2(s)=

−1

s(3.15)

y2(s)

u2(s)=

1800

s2 + 20s(3.16)

y1(s)

d1(s)= 0 (3.17)

y2(s)

d1(s)=

2s

s2 + 20s(3.18)

3For example with the Matlab command ss2tf


Laplace transforms can be used to obtain a solution of a differential equation (read responsein the time domain). However in most cases it is attractive to leave the equation in theLaplace domain and to rearrange it as a transfer function:

output(s) =nominator(s)

denominator(s)input(s) (3.19)

This is because transfer functions have nice properties:

• By examining the roots of the nominator and denominator we get considerable insightin the dynamic behavior. The location of the poles determine speed and stability 4, thelocation of the zeros indicate inverse response 5.

• The final-value theorem allows us to calculate directly the steady-state behavior:

limt→∞

f(t) = lims→0

[sf(s)] (3.20)

• The overall transfer function of a series of n transfer functions f1(s) to fn(s) is simplygiven by:

fn(s) · fn−1(s) . . . f2(s) · f1(s) (3.21)

Instead of determining a time domain response we can also calculate the frequency response.Both responses present the same information (in a different way) but the frequency responseis normally much easier to calculate. Basically in the transfer function the Laplace variables is replaced by jω and the result is written as the sum of a real and imaginary part saya(ω) + b(ω)j. The so-called amplitude ratio (AR) and the phase shift (φ) are calculated asfollows:

AR =√

(a(ω))2 + (b(ω))2 (3.22)

φ = arctan

[b(ω)

a(ω)

]

(3.23)

A frequency response shows the AR and the phase shift as a function of the frequency ω. Formore information on frequency responses see Stephanopoulos [2].

Table 3.2 shows the most common transfer functions in practice. Of course also combinationslike first order plus dead time frequently occur. Transfer functions can be extended by puttingthem in series with static non-linear functions. This can be an effective way to handle acommon “source” of non-linearity; saturation.

3.2 Process control

Figure 3.2 shows a general process (with u(s) being the input, d(s) a disturbance and y(s)the output).

4Poles further from the origin are faster and poles with a positive real part are unstable.5Right Half Plane (RHP) zeros invoke inverse response.

3.2. PROCESS CONTROL 21

Table 3.2: The most encountered transfer functions in practice.Name Laplace transform Remarks

1. mixing components

First order Kτs+1 2. mixing heat

1. β > 1 overdamped; process

Second order Kτ2s2+2βτs+1

2. β = 1 crit. damped; process

3. β < 1 underdamped; control

1. distributed equipmentdead time e−Ds 2. quality measurements

1. liquid inventory

integrator Ks

2. vapor inventory

d(s)

y(s)

+

+P(s)

Ds)

u(s)

Figure 3.2: A general process system.

It is important to realize that the input provides a Degree Of Freedom (DOF) which is anessential condition to do control. The number of DOF’s (#DOF’s) can be determined via:

#DOF’s = #variables - #equations (3.24)

A closer look at equation 3.24 reveals that it easily leads to errors; a small number is de-termined by the subtraction of two large numbers. Luyben [10] proposes to calculate the#DOF’s by just counting the number of external flows that can be manipulated. For exam-ple the number of external flows that can be manipulated in a two-cut splitter is five (seefigure 3.3, three material flows and two heat flows). So a two-cut splitter has five DOF’s. Bythe way, the #DOF’s for a unit operation is in the range 1 - 5. This implies that #DOF’sfor a complete plant is of the order 101 - 102.

If we want to control the output y(s) then the control problem could be stated as:

• Keep the output constant despite disturbances; regulator problem.

• Move the output to the desired value (yref (s)); servo problem.

In control terms the output y(s) is often referred to as the controlled variable. For controlpurposes two types of disturbances can be distinguished; measured and unmeasured.

There are different approaches to solve the control problem (also see figure 3.4):


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Heat


Materialxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Material


Material


Heat

Figure 3.3: The manipulable external flows of a two-cut splitter.

1. By feedforward control; u(s) is based on yref (s) and d(s).

2. By feedback control; u(s) is based on yref (s) and y(s).

u(s)

d(s)

y(s)

+

+y

ref(s) C(s) P(s)

D(s)

u(s)

d(s)

y(s)

+

+y

ref(s) C(s) P(s)

D(s)

-

+

Figure 3.4: Feedforward (left) and feedback (right) control.

The two control approaches have quite different characteristics:

Robustness The closed-loop transfer functions of the feedforward and feedback system arerespectively:

y(s) = D(s)d(s) + P (s)Cd(s)d(s) + P (s)Cref (s)yref (s) (3.25)

y(s) =D(s)

1 + P (s)C(s)d(s) +

P (s)C(s)

1 + P (s)C(s)yref(s) (3.26)

Lets focus on the servo problem. For good static performance (read no offset) of feedfor-ward control P (s)Cref (s) → 1.0 as s → 0 (in other words Cref (0) = P (0)

−1). Howeverfor feedback control it is sufficient that P (s)C(s) → ∞ as s → 0 (so C(0) = ∞). Theconclusion is that for good static performance feedforward control depends much more

3.2. PROCESS CONTROL 23

on a good process model than feedback control does. Actually this conclusion can beextended to dynamic servo as well as dynamic regulator performance.

Stability It is clear that in the case of feedforward control the poles of the controlled systemare just the poles of C(s) and P(s). However in the case of feedback control they aregiven by: 1 + P (s)C(s) = 0. In other words with feedforward control we do not changethe poles of the uncontrolled process but with feedback control we do. An importantconsequence is that with feedforward control we can not make a stable process unstableand we can not stabilize an unstable process. With feedback control it is completelythe opposite; a stable process can become unstable while an unstable process can bestabilized.

Speed Generally speaking feedforward control tends to be faster than feedback control. Notethat for feedback action (yref − y) 6= 0, so a disturbance first has to affect y before anycontrol action is taken. Another reason is that feedforward control can generate largeinput signals (to speed things up) without compromising stability.

Looking at the characteristics it could be said that feedforward and feedback control are toa large extend complementary. For that reason feedback control is often extended with feed-forward control. Feedback control is always necessary to achieve robustness.

Lets assume that the process is stable (so we don’t have to shift poles) and that a perfectmodel of the process is available. Can we now obtain perfect control? Perfect control meansthat the output tracks the reference tightly while disturbances do not affect the output.Given the assumptions (no stability or robustness issue) feedforward control will give thebest control possible. From equation 3.25 it is clear that for perfect control P (s)Cref (s) = 1and D(s) + P (s)Cd(s) = 0. So Cref (s) = P (s)

−1 and Cd(s) = −P (s)−1D(s) or in words the

controller should contain the inverse of the process. The latter is actually a very useful resultsince in certain cases this is difficult or even impossible:

1. Most process transfer functions are strictly proper, that is the degree of the nominatoris smaller than the degree of the denominator. After inversion such transfer functionsmight be far from proper and therefore difficult to realize (differentiation).

2. Dead times frequently occur in process transfer functions. Since the inversion of deadtime boils down to prediction dead time can not be inverted.

3. Right Half Plane (RHP) zeros do not often appear in process transfer functions 6. Afterinversion a RHP zero becomes a RHP pole which makes C(s) unstable, so a RHP zerocan not be inverted.

The conclusion is that control can not “remove” dead time or inverse response. So dead timeand inverse response pose limitations to the controlled behaviour.

Lets rearrange equation 3.26 somewhat:

y(s) =1

1 + P (s)C(s)︸︷︷︸

S(s)

d∗(s) +P (s)C(s)

1 + P (s)C(s)︸︷︷︸

T (s)

yref (s) (3.27)

6This is true for Single Input Single Output (SISO) systems, however in the case of Multiple Input MultipleOutput (MIMO) systems they occur more frequently.


In equation 3.27 D(s)d(s) has been replaced by d∗(s). Furthermore the part associated withdisturbance rejection (regulator problem) is indicated by S(s) while the part associated withreference tracking (servo problem) is indicated by T(s). S(s) is normally referred to as thesensitivity function, T(s) as the complementary sensitivity function. The dynamic closed-loopperformance can be assessed via a frequency response of the S and T function. By the waynote that S(s) + T (s) = 1.

3.3 Basic process control

BPC just uses one type of feedback control; Proportional Integral Differential (PID) control.Partly this is because of historic reasons, partly because given the most encountered transferfunctions (see table 3.2) it provides adequate control. The PID transfer function is given by:

u(s)

e(s)= Kc

[

1 +1

τis+ τds

]

(3.28)

Besides PID control BPC also uses extensions. Figure 3.5 shows a number of these extensions.The symbols that are used in this figure are explained in Appendix A: Instrumentation Details.

FT

FC

YXC b

a

a*b

Ratio

LC

FC

Cascade

PC

nitrogen

safe

location

Y0 - 45% 55 - 100%

Split range

PC

Y

SC

FC

Override

TT TT TT

reactant(s)

cat.

YTC

Selective

LC

steam

waterFC

Y

a+b

b

FT

a

Feedforward

Figure 3.5: The various extensions of BPC.

3.3. BASIC PROCESS CONTROL 25

Ratio This in fact a form of feedforward control. The objective is to keep the ratio betweentwo flows constant. In some case one flow is uncontrolled (“wild”), in other cases bothflows are controlled. Typical applications are reactor feed systems and fired heaters.

Cascade This scheme combines two controllers, the output of one controller (the master) isconnected to the reference of the other controller (the slave). The objective is just tocontrol the variable at the master level. The advantage of having a slave is in disturbancerejection; if the slave senses a deviation it is rejected immediately even if at the masterlevel no deviation is noticed. Typical examples include level/flow, temperature/flowand quality/flow combinations.

Split range This scheme uses two manipulated variables that typically have an oppositeeffect. Note that the split is made at the output level. Typical applications are nitrogenblanketing and priority control.

Override Also this scheme involves two controllers but in this case there are two objectives.Since there is only one manipulated variable only one objective can be satisfied (thechoice is made via a selector). Typical examples are flow/pressure control of compressorsand steam header control.

Selective This scheme uses various measurements that via some calculation are reducedto one “measurement”. Typical calculations are maximum, position of the maximum,average and weighted average.

Feedforward This represents a very diverse extension (that in fact includes ratio control).Often it is based on an inverse of the static process behavior. Typical examples includeboiler level control, flow/temperature decoupling and heat balance control.

Chapter 4

Decomposition of the plantwide

control problem

This chapter discusses what could be called the first structural aspect of a plantwide controlsystem; the decomposition of a plantwide control system in material balance and quality con-trol. Much of this chapter can be found in Buckley [11] but the material has been updated andcorrected. The paragraph ”Decomposition in the time domain” contains mainly new material.

4.1 Material balance and quality control

The plant-wide control problem was first described by Buckley although he uses the term“overall process control”. Buckley states that in the early 1950’s it was generally assumedthat the best way to control a complete plant was to keep all flows, pressures etc. constant sothat the product quality would be constant (“setpoint environmental control”). Besides thefact that a large number of loops were used, the loops were also tuned for the fastest responsepossible. By 1956 it was clear that this “blind” application of servomechanisms theory onindividual loops led to unstable and oscillating plants. Buckley tackled this problem by takingthe plant manager’s viewpoint and stated that a plant must fulfill two primary requirements:

a It must produce a product whose quality meets sales specification.

b It must maintain a material balance.

The first requirement is straightforward, Buckley extends the last requirement into threeseparate issues:

1. Inventories must be maintained between minimum and maximum limits.

2. Production rate must be adjusted to exactly equal, on a long-term basis, the sales rate(or rate of shipments).

3. The resulting adjustments in process flows must be sufficiently smooth and gradual toavoid upsetting process equipment.

The first issue recognizes the fact that a change in the production rate upsets the processinventories. The first requirement together with the second issue equals of course “the requiredconversion” (see chapter 2). The third issue refers to the interaction between the production

27

28 CHAPTER 4. DECOMPOSITION OF THE PLANTWIDE CONTROL PROBLEM

rate and the quality; the basic most important interaction existing in any plant and indicatesthat it can be handled by dynamic decoupling. In the rest of this chapter we will discuss thedecomposition in more detail; first in the time domain and then in the frequency domain.

4.2 Decomposition in the time domain

Figure 4.1 shows a simple process. The liquid feed enters a vessel, is passed through a pumpand heated with steam.

FT1

LC1

TT

1

Product

Steam

Feed

FCV1

TCV1

Fin

Fout

F, T

Figure 4.1: A simple process.

As a measure for the production rate we will take the flow of product F. The productionrate can be manipulated with valve FCV1. We assume that the liquid inventory in the vesselbehaves as an integrator (

Kps

). To stabilize the inventory it is controlled by LC1; a P-onlycontroller that manipulates the outlet flow (Fout). Using the theory of chapter 3 we get:

L(s) =−1/Kc1

−KpKcs + 1

Fin(s) +1

1−KpKc

s + 1Lref (s) (4.1)

Fout(s) =−1/Kps1

−KpKcs + 1

Lref (s) +1

1−KpKc

s + 1Fin(s) (4.2)

So for Kc < 0 we achieve stable closed-loop behavior (first order). The quality of the productis its temperature T. The temperature can be manipulated with valve TCV1. From thisexample it is clear that:

1. The open-loop dynamics of the material balance are determined by the transfers FCV 1 →Fin (open-loop flow control) and Fin → Fout (inventory control).

2. The open-loop dynamics of the quality is determined by the transfer TCV 1 → T (con-duction). In general the open-loop quality dynamics can also be determined by reactionkinetics and diffusion.

In other words the open-loop dynamics of the material balance are faster than the open-loopquality dynamics. This situation is reflected in figure 4.2. Note that an increase in the pro-duction rate leads to a lower temperature, so the material balance interacts with the quality!

4.2. DECOMPOSITION IN THE TIME DOMAIN 29

T

+

+

F1120

1

+s

1120

1

+

-

s

1360

180

+

-

s

e s

FCV1

TCV1

Fref

+

+

-

-s540

0.20.2 +Tref

s15

7.07.0 +

110

1

+s

Feed

FT1

Fin Fout

TT1

FC1

TC1

Process

Figure 4.2: Control of the process depicted in figure 4.1.

Figure 4.2 shows that the whole process can be controlled by two PI-controllers; one fromF → FCV 1 (FC1) and the other from T → TCV 1 (TC1).

Figure 4.3 shows the reaction of the controlled process during an increase of the productionrate (no filter). This figure clearly shows that a change in the production rate affects thequality. It should be said that this response can not be improved by the tuning of TC1. Thesituation can be improved by:

• Slowing down the interaction at a process level. Note that according equation 4.2 thetime constant is inversely proportional with Kp, so lowering Kp (read a larger vessel)gives a slower interaction. Buckley [11] advocates this approach, however it means anincrease in capital and operating costs and compromises safety.

• Extension with feedforward control. The speed of the controlled system can be increasedwith feedforward control. The typical starting point for this extension would be to setup a heat balance around the heat exchanger. The performance of feedforward controlhowever is limited by the presence of dead time.

• Slowing down the interaction at a control level (dynamic decoupling). This can either bedone by tuning FC1 more slowly or by filtering setpoint changes to FC1. A disadvantageof tuning FC1 more slowly is that also makes disturbance rejection slow. Filtering thesetpoint does not have this disadvantage. The effectiveness of setpoint filtering is shownin figure 4.3 (filter); the maximum quality deviation is reduced by a factor 4! Howeverthere is price to be paid, changes in the production rate are considerably slower!

An important difference between material balance and quality control is the fact that materialbalance control is really plantwide while quality control is localized. The explanation is thata process consists of a number of unit operations that support the same production rate buteach unit operation typically handles only one quality aspect. This is the second reason whythe decomposition of plantwide control in material balance and quality control is so useful.


0 500 1000 1500 2000 2500 3000 3500 4000−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time [seconds]

F a

nd T

[−]

filterno filter

no filter

filter

Figure 4.3: The response of the controlled process (see figure 4.2) when the setpoint of FC1is increased by 0.1. F is solid, T is dashed, the applied filter is 1900s+1 .

4.3 Interaction in the frequency domain

We will now discuss the interaction between the material balance and quality in the fre-quency domain. The material balance of most processes is not subjected to many or largedisturbances (there is an important exception but this will explained in the next chapter).So the closed-loop dynamics of the material balance control system are described by thecomplementary sensitivity function T. Often quality specifications change infrequently, so theclosed-loop dynamics of the quality control system are described by the sensitivity functionS. We have already seen that the open-loop dynamics of the material balance are faster thanthe open-loop dynamics of the quality (see figure 4.4).

Amplituderatio

Frequency

1.0

Quality Material balance

Figure 4.4: The typical open-loop frequency response of material balance and quality.

4.3. INTERACTION IN THE FREQUENCY DOMAIN 31

The material balance and the quality always interact. If the quality has to remain on spec-ification the interaction can be handled by slowing down the material balance control suchthat it does not interfere too much with the quality control (see figure 4.5). The fact that theprimary concern lies with the quality rather then with the material balance can be explainedas follows. Generally off specification product can be handled in three ways:

1. Mix it with product that has quality give-away. Note that this is only possible if thequality is mixable, so for low viscosity products like gasoline and solvents. Howeverthere are lots of products like polymers, glass and particulates for which this is notthe case. Since generally (see for example Shinskey [12]) the relation between cost andquality is typically hyperbolic or logarithmic, mixing on specification is more expensivethan producing on specification 1.

2. Reprocess it. Normally partly reprocessing is sufficient. There are cases like when toomuch of a stable side-product is formed that reprocessing is not possible. Effectivelyreprocessing means a reduction in production rate.

3. Sell it for a lower price. For a number of products like polymers there are markets forlower quality product. The price of lower quality product is of course less.

So although off specification product can be handled in various ways, economically it is moreattractive to avoid off specification production.

Amplituderatio

Frequency

1.0

Quality S-function

Material balance T-function

Figure 4.5: The typical closed-loop frequency response of material balance and quality.

1Douglas [13] points out that although this is true for steady state operation it is certainly not always truefor dynamic operation.

Chapter 5

Material balance and quality

control

In this chapter we will discuss material balance and quality control in more detail. So thefocus is no longer on the interaction between the two but rather on the individual aspects ofmaterial balance control and the individual aspects of quality control.

5.1 Push, pull and other material balance schemes

In the case of figure 4.2 material balance control system is in fact nothing more then produc-tion rate control followed by inventory control. Although this is the most common arrange-ment, there are other ways (see figure 5.1).

FC

1

LC

1

LC

2

FC

1

LC1

LC2

FC

1

LC

1

LC

2

Figure 5.1: Various possibilities to do material balance control, from top to bottom; a push,a pull and an intermediate scheme.

33

34 CHAPTER 5. MATERIAL BALANCE AND QUALITY CONTROL

The terms push and pull speak for themselves. Buckley [11] refers to push and pull as“control in the direction of flow” and “control in the direction opposite to flow” respectively.Buckley compares the stability of the push and pull strategy and concludes that the pullstrategy is better. However this conclusion is based on the assumption that the “managementor production supervision” is done by feedback while in practice it is based on planning(read feedforward). Luyben [14] also compares the push and pull strategy on three differentprocesses (a stripper, a CSTR/stripper and a ternary process consisting of a CSTR andtwo distillation columns) with the idea “to be more responsive to customer demands”. Heconcludes that pull schemes are workable but have a several inherent dynamic disadvantages:

1. Larger variability in the product quality loops.

2. Larger surge capacity requirements.

3. Control loops are more interacting and difficult to tune.

4. Stronger non-linear behavior.

5. More switching of loop pairing when flowsheet changes occur.

Luyben performs a detailed analysis what causes the disadvantages mentioned above. Thisanalysis explains convincingly that Luyben’s conclusion will be valid for most if not all pro-cesses. Also from the discussion on the open-loop and closed-loop dynamics of materialbalance and quality it is clear that a pull strategy cannot “bypass” the open-loop qualitydynamics. Luyben’s conclusion also provides a good explanation why in practice the pushscheme is the most popular one. There is however one important class of processes that ac-tually use the pull scheme intensively; utilities (steam, water, air, electricity, nitrogen etc.).This is because utilities should adjust their production rate to the plants they support. Sincestorage of utilities is expensive or even impossible a number of measures is taken to ensuregood dynamic performance. First of all utilities are designed to have certain open-loop dy-namics (this is not so difficult since they typically have the complexity of a unit operation).The second measure has to do with the way they are operated; often several utilities work inparallel to support a number of processes (see figure 5.2).

PLANT

a

PLANT

b

PLANT

c

UTILITY

a1

UTILITY

a2

25% 20% 25%

35% 35%

Figure 5.2: A typical utility configuration. If for example plant b raises its utility intake bya factor 2.00, then utility a1 and a2 only have to raise their production by a factor 1.29.

It should be noted that in figure 5.1 there is only one intermediate strategy, however if thereare more vessels then there are also more intermediate schemes possible. In their paper Price

5.2. STANDARD QUALITY SCHEMES 35

and others [15] discuss intermediate schemes. They state that these schemes have the ad-vantage of establishing faster a production rate change since a change is propagated in twodirections instead of one (as with push and pull). They test a large number of different ma-terial balance control systems on a CSTR/column example and the Tennessee Eastman testproblem and come to the conclusion that the intermediate strategy indeed performs some-what better than the other strategies. Given the fact that the material balance control hasto be slowed down to diminish interaction with quality control the argument of establishinga faster change of production rate seems to be irrelevant. A far more important conclusionof Price and others is that material balance control should be “self-consistent”; a productionrate change propagate itself up- and downstream via inventory control.

Finally a few words about production rate. Luyben, Tyréus and Luyben [16] clearly preferto set the production rate at the reactor. They mention the various alternatives for doingthis; by temperature, reactant concentration, reactor volume (liquid phase reaction) or re-actor pressure (gas phase reaction). Although other unit operations do not have that manyalternatives they do not give any reason why the production rate should be set at the reac-tor. Of course since normally the reactor is right at the beginning of the process setting theproduction rate at the reactor often leads to a push scheme. But the real reasons to prefer apush scheme are given by Luyben [14].

5.2 Standard quality schemes

It was already mentioned that material balance control is really plant-wide while qualitycontrol is localized. Already in the past this situation was recognized and utilized by theinvention and application of standard quality schemes. Such a scheme shows how the relevantquality aspect of a unit operation can be controlled. The production rate in these schemes istreated as an unmeasured or measured disturbance. Figure 5.3 shows two well-known stan-dard quality schemes.

REACTOR

FC

1

FT

2

Y

QC

1

b

a*b

FC

2

a

reactant A

reactant B

FC

3

QC

2

LC

1

FC

4

Figure 5.3: Two standard quality schemes; control of a stoichiometric reactor (left) andcontrol of the top of a distillation column (right).

36 CHAPTER 5. MATERIAL BALANCE AND QUALITY CONTROL

The reactor control scheme mixes the two reactants in ratio (feedforward). Since the accuracyof the flow measurements is limited, feedback is added; a quality controller that measures thecomposition of the reactor effluent adjusts the ratio in which both reactants are mixed. Thedistillation control scheme makes use of cascade; a quality controller that measures the topquality resets the reflux. Note that the level in the top accumulator is controlled by thedistillate flow. This only works well if R/D is say less than 3.0, otherwise it is better tocascade QC2 with FC4 and LC1 with FC3 (to make sure that the level can be controlled).So standard quality schemes can be quite case specific, therefore before applying such ascheme make sure you know its background. Although the process control literature does notmention these standard quality schemes often (only Van der Grinten [17], Shinskey [18], [12]and Luyben, Tyréus and Luyben [16]) give a number of these schemes) in practice they are ofconsiderable value. They allow a control engineer to do new designs as well as maintenancein an effective and efficient manner.

Chapter 6

About recycles

This chapter discusses the last structural aspect of plantwide control; the influence of recycleson plantwide behavior. First we will explain the influence in detail, then we will try to answerthe question how to deal with this influence.

6.1 The influence of recycles

Two different recycles exist; heat and material recycles. The recycle of heat is the result ofheat integration or in other words the desire to preserve energy and money. Typically thesame heat is used in various unit operations on decreasing temperature levels. But also heatintegration within one unit operation is possible like for example the coupling of the reboilerand condenser of a distillation column by means of a heat pump.

Materials that are normally recycled are unreacted reactants and auxiliary compoundslike solvents and catalysts. Economically the recycling of these materials is attractive (prod-uct is worth more than reactant) is or even a must (the value of catalyst can exceed manytimes that of product). In their book Luyben, Tyréus and Luyben [16] discuss a convincingexample that even the introduction of a material recycle itself is economically attractive.

Recycles can change the plantwide behavior in significant ways. Luyben, Tyréus and Luybenmention two effects:

1. Recycles can change the plantwide dynamics considerably. The dominant time constantmight easily be 10 times the sum of the time constants of the individual unit operations.

2. Recycles can introduce high sensitivity in the steady-state (so called “snowball” effect).In those cases a small change in for example the production rate results in a large changeof the recycle flow.

Both effects can be illustrated by means of an example (see figure 6.1).

Setting up the total mass balances (steady state) over the mixing point, the reactor and thesplitter results in:

ρF2 = ρF1 + ρF4 (6.1)

ρF3 = ρF2 (6.2)

ρF5 + ρF4 = ρF3 (6.3)

37

38 CHAPTER 6. ABOUT RECYCLES

21fresh feed A CSTR SPLITTER3 5

4

product B

Figure 6.1: An example of a process with recycle. In the CSTR the reactant A is convertedinto product B (according A → B, the reaction is first order in A). The splitter separates thereactor effluent and recycles A back to the reactor.

Note that the flows are volumetric flows and that the density is constant. The reactor isa CSTR, so the mole balances for component A (steady state) over the mixing point, thereactor and the splitter are given by:

F2[A2] = F1[A1] + F4[A4] (6.4)

F3[A3] = F2[A2] − kV [A3] (6.5)

F5[A5] + F4[A4] = F3[A3] (6.6)

The feed is pure, arbitrarily [A1] is assumed to be 10000. Furthermore the splitter behavesideally, so [A5] = 0 and [A4] = 10000. Finally the required production F5 is set at 100. Notethat the whole problem now consists of 10 variables (F1...F5 and [A1]...[A5]) and 10 equationsand is therefore fully defined. Since the production equals kV [A3] = F5[B5] = F5 · 10000(remember that the reactor is a CSTR) the term kV can be determined assuming a certainsingle-pass conversion (read [A3]). Table 6.1 shows the solution for a number of cases.

Table 6.1: The solution of the plantwide example for various cases.Case 1A Case 1B Case 1C Case 2A Case 2B Case 2C

Vr. F5 = 100 F5 = 101 F5 = 101 F5 = 100 F5 = 101 F5 = 101[A3] = 5000 [A3] = 5050 [A3] = 5000 [A3] = 1000 [A3] = 1010 [A3] = 1000kV = 200 kV = 200 kV = 202 kV = 1000 kV = 1000 kV = 1010

F1 100 101 101 100 101 101

F2 200 204.04 202 111.11 112.35 112.22

F3 200 204.04 202 111.11 112.35 112.22

F4 100 103.04 101 11.11 11.35 11.22

F5 100 101 101 100 101 101

[A1] 10000 10000 10000 10000 10000 10000

[A2] 10000 10000 10000 10000 10000 10000

[A3] 5000 5050 5000 1000 1010 1000

[A4] 10000 10000 10000 10000 10000 10000

[A5] 0 0 0 0 0 0

Note that the single-pass conversion for the 1 cases is low while for the 2 cases it is high. Alsonote that the B and C cases are small production rate increases of the A cases by respectivelyraising [A3] or the product kV . From table 6.1 several gains can be calculated:

6.1. THE INFLUENCE OF RECYCLES 39

Krecycle =∆F4

∆F1(6.7)

Kreactor =∆F3

∆F2(6.8)

Ksplitter =∆F4

∆F3(6.9)

For example for case 1A → 1B:

Krecycle =103.04 − 100

101 − 100= 3.04 (6.10)

All gains have been summarized in table 6.2.

Table 6.2: The gains for various production rate increases.Gain 1A → 1B 1A → 1C 2A → 2B 2A → 2C

Krecycle 3.04 1 0.24 0.11

Kreactor 1 1 1 1

Ksplitter 0.75 0.50 0.19 0.10

τfast 55.72 57.0 58.8 59.4

τslow 2584 1263 756 674

Table 6.2 clearly shows that for the production rate increase 1A → 1B there is a snowballeffect; a small increase of production requires a substantial increase of the recycle. This effectis absent for the other production rate increases.

Lets assume that the lumped dynamics of the reactor and the splitter are described by firstorder dynamics:

∆F3(s)

∆F2(s)=

Kreactorτreactors + 1

(6.11)

∆F4(s)

∆F3(s)=

Ksplitterτsplitters + 1

(6.12)

Using the theory of chapter 3 gives the overall transfer function for the recycle (keep in mindthat the mixing point introduces positive feedback !):

∆F4(s)

∆F1(s)=

KsplitterKreactor(τsplitters + 1)(τreactors + 1) − KsplitterKreactor

(6.13)

The characteristic equation of this second order system is:

τsplitterτreactors2 + (τsplitter + τreactor)s + (1 − KsplitterKreactor) (6.14)

It is trivial to find the roots of the characteristic equation and to convert them to timeconstants. Lets assume some values; τreactor = 60 and τsplitter = 600. This results in the timeconstants shown in table 6.2. Note that the production rate increase 1A → 1B has the largest


time constant; 2584 (almost four times the sum of the time constant of the reactor and thesplitter!). To understand what is happening lets combine equations 6.13 and 6.14:

∆F4(s)

∆F1(s)=

KsplitterKreactorτsplitterτreactors2 + (τsplitter + τreactor)s + (1 − KsplitterKreactor)

(6.15)

If KsplitterKreactor → 1 the term (1 − KsplitterKreactor) disappears so:

∆F4(s)

∆F1(s)=

(KsplitterKreactor

τsplitterτreactors + (τsplitter + τreactor)

)

·

(1

s

)

(6.16)

Note that an integrator has appeared in the overall transfer function! In other words ifKsplitterKreactor → 1 the plantwide dynamics start resembling an integrator. This is confirmedby figure 6.2.

0 1000 2000 3000 4000 5000 6000 70000

0.5

1

1.5

2

2.5

3

Time [s]

Del

ta F

4

1A −> 1B

1A −> 1C

2A −> 2B

2A −> 2C

Ksplitter Kreactor = 1

Figure 6.2: A step response of the recycle transfer function for the various production rateincreases.

From a process point of view it is not difficult to explain what is happening. For values ofKsplitter that approach 1 most of the extra feed introduced during a production rate increaseis recycled (see equation 6.7). So basically extra feed just accumulates in the process.

Generally speaking the influence of recycles on plantwide dynamics could be summarizedas; recycles might introduce integrator-like behavior. In process terms; recycles might lead toexcessive accumulation of material, certain components or heat.

6.2 How to deal with recycles?

Given the possible influence of recycles the question the question above is a fair one. There isone obvious answer; avoid recycles altogether or make them as small as possible. Although it

6.2. HOW TO DEAL WITH RECYCLES? 41

is clear that this answer will be of little use in many cases (equilibrium reaction, selectivity,exothermic reaction etc.) it is important to realize that there should be a good motivationfor the existence as well as the size of each recycle 1. To illustrate the importance of themotivation lets return to table 6.1, this table shows that for high single-pass conversion therecycle is small and that a production rate increase can be done by either raising the reactantconcentration, the reactor temperature or the reactor hold-up.

However even if we have to accept a recycle of a certain size normally it is still possible tolimit integrator-like behavior. For example from table 6.1 it is clear that in case of low single-pass conversion it is better to increase the production rate by raising the reactor temperatureor the reactor hold-up rather than by the reactant concentration. Now take a look at figure 6.3.

21fresh feed A CSTR SPLITTER 5

4

product B

FC

1

LC

1


4

product B

FC

1

LC1


4

product B

FC

1

LC

1

production rate

constant

Ya

production rate

k*a

production rate

3

3

3

constant

k = [V/F2] design

Figure 6.3: Different ways to control the production rate in the CSTR/splitter/recycle process.From top to bottom; constant reactor hold-up (push), constant space-time (balanced) andconstant effluent (Luyben).

Note that in the top scheme the reactor hold-up is kept constant. Since the reactor tempera-ture is assumed constant a production rate increase means raising the reactant concentration.So the top scheme corresponds with cases 1B and 2B. The top scheme should not be used forlow single-pass conversion (case 1B), however it can be used for high single-pass conversion(case 2B). Note that we already have already encountered this scheme in chapter 5 under thename push scheme. Production rate increases and disturbances are handled by the splitter.

1Occasionally one may encounter a recycle for which this is not the case.


The scheme in the middle adjusts the reactor hold-up such that during a productionrate increase the space-time ( V

F2) is constant2. This not only ensures a constant reactant

concentration it also makes all flows proportional to the feed! From table 6.1 it is clear thatthe scheme in the middle corresponds with cases 1C and 2C. This scheme can be used for bothlow as well as high single-pass conversion. In their paper Wu and Yu [19] refer to this schemeas “balanced” since a production rate increase and disturbances are handled partly by thereactor and partly by the splitter.

The bottom scheme was proposed by Luyben [20]. Since the reactor outlet flow is keptconstant the only way to change production is by changing the reactor hold-up. So productionrate increases and disturbances are handled by the reactor. The steady-state effects of thisparticular scheme are not included in table 6.1 but it is not difficult to establish that thisscheme should only be used for low single-pass conversion. It should be noted that this schemesuffers from a operational disadvantage; the production rate is set in an indirect way. So theoperator needs to know the relationship between the reactor hold-up and the production rate.

By the way the three control schemes mentioned above can also be discussed using theplant Damkohler number. For more information see Bildea, Dimian and Iedema [21].

Generally recycles could be dealt with as follows:

1. Challenge the existence and size of each recycle. As stated before existence and sizeshould be well motivated. And although one only encounters occasionally a badlymotivated recycle it also helps to understand the background of each recycle.

2. Make sure there are “sinks” for certain components and heat. There is always a clearsink for material and product in particular; the product flow. However this is not thecase for (unreacted) reactants, intermediates, byproducts, inerts and heat.

3. Investigate the influence of each recycle by steady state simulation. Typically a simpli-fied simulation is sufficient. During the simulation the available DOF’s are used to keepcertain variables constant while the production rate and/or the disturbance change.Integrator-like behavior will reveal itself as a high gain.

2This is in fact an application of ratio control.

Chapter 7

Plantwide control design

From the discussion so far it is clear that during the design of a plantwide control systemseveral aspects need to addressed like material balance control, quality control and recycles.However these aspects must be integrated into a plantwide control design procedure. In thischapter we will first discuss the design procedures from Ponton and Laing [1] and Luyben,Tyréus and Luyben [16] 1. Then a new design procedure will be proposed. Finally the newprocedure will be used to design a plantwide control system for a specific process.

7.1 Procedure of Ponton and Laing

In their paper Ponton and Laing refer directly to the process design procedure of Douglas [4].Douglas uses the engineering method or artist’s approach 2. This method or approach boilsdown to successive refinement; first develop a simple solution and then add successive layersof detail. In the case of process design one starts with a single block with just the feed andproduct flows and in various stages the block is expanded into a flowsheet. Ponton and Laingcouple each process design stage with a control design stage (see table A).

Table 7.1: Coupling process and control design stages, after Ponton and Laing [1].Stage Process design Control design

1 input-output structure feed and product rate control

2 recycle structure recycle rates and composition

3 separation sequence product and intermediate stream composition

4 energy integration temperature and energy balance control

5 - inventory regulation

Note that the inventory regulation (the fifth control stage) has no counterpart in (steadystate) process design. From table A it is clear that stage 1 and 5 handle the material balance

1Actually quite a number of procedures have been published see for example Buckley [11], Marlin [22] andLarsson and Skogestad [23].

2This term recognizes the fact that process design is actually a creative activity. The reason for this is thatdesign problems are underdefined; only a small fraction of the information needed to define the problem isavailable. The rest of the information must be selected, obtained etc. using creativity

43

44 CHAPTER 7. PLANTWIDE CONTROL DESIGN

while stage 2 and 3 take care of the quality control. Recycles are handled by stage 2 (material)and stage 4 (heat).

Ponton and Laing provide one (rather short) justification for the order of the stages; themore important control associated with product rate, overall conversion and product qualityinfluence the economic performance of the plant and should be done first.

7.2 Procedure of Luyben, Tyréus and Luyben

The procedure Luyben, of Tyréus and Luyben [16] consists of nine steps:

1. Establish control objectives.

2. Determine control degrees of freedom.

3. Establish energy management system.

4. Set production rate.

5. Control product quality and handle safety, environmental and operational constraints.

6. Fix a flow in every recycle loop and control inventories (pressures and liquid levels).

7. Check component balances.

8. Control individual unit operations.

9. Optimize economics and improve dynamic controllability.

The first step is an essential part of the formulation (see figure 2.2). In combination with step2 it becomes clear if the problem is underdefined, just defined or overdefined. Steps 4 and 6take care of the material balance control while quality control is addressed in step 5. Steps3 and 6 handle heat respectively material recycles. The authors explain the necessity of step7 by the fact that not only the overall conservation of energy and mass needs to be takeninto account but also the non-conserved entities like entropy and chemical components. Step8 completes the control for individual unit operations while the last step uses the remainingdegrees of freedom for optimization.

The authors justify the order of steps by the principle that the most important item comesfirst so that the designer has the largest set of degrees of freedom to choose from. For exam-ple step 3 comes first (steps 1 and 2 do not reduce the degrees of freedom) since the reactoris typically “the heart” of a chemical process and the ways to control the heat redrawn orsupplied to the reactor are limited. When dealing with an exothermic reaction together withheat integration care must be taken to ensure the final removal of heat to avoid recycling ofheat. The principle that the most important item comes first together with the fact that theprocedure begins with the reactor shows resemblance with the process design procedure ofSmith [8].

7.3. PROPOSED PROCEDURE 45

7.3 Proposed procedure

The procedure from Ponton and Laing is quite different from the procedure as proposed byLuyben, Tyréus and Luyben. That is because the starting point is also quite different; Pontonand Laing couple the process and control design while Luyben, Tyréus and Luyben assumethat the process design has been done. As explained before (chapter 2) in actual practiceprocess and control are designed separately. Basically the reason is that if the process hasnot been designed there is little one can say about the process dynamics. It would now beobvious to use the procedure from Luyben, Tyréus and Luyben. However several steps in thisprocedure are questionable. For example the first part of step 6 “Fix a flow in every recycleloop” is heavily debated. In paragraph 6.2 it was mentioned that the scheme proposed byLuyben suffers from a operational disadvantage and Larsson [24] points out that a “floatingreactor level” is normally not optimal from an economic point of view. Another example isstep 3 “Establish energy management system”; it seems to put too much emphasis on controlwhile it is mainly a design issue.

Therefore another procedure will be proposed. We will assume that the process design hasbeen done (so a process description and flowsheet are available) and we will base the procedureon the systematic design approach (see figure 2.2). The proposed procedure is summarizedin table 7.2.

Table 7.2: The proposed procedure for the design of a plantwide control system.Stage Step Remarks

Formulation 1. determine control objectives use • general objective (chapter 2)• process description/flowsheet• input technologist/operators

2. determine #DOF’s • count external manipulable flows

Synthesis 3. develop material balance control • start with a push scheme unlessthere is reason to use a pull scheme• make sure the production ratecontrol is effective• add inventory controls along theprimary path (from feed to product)

4. develop quality control schemes • use standard schemes if possible5. check the influence of recycles • by simplified steady state

nonlinear simulation6. minimize the operating costs • by extension of the quality control

Analysis 7. do some simple checks • check sinks for reactants, interme-diates, inerts, byproducts and heat

8. simulate by • rigorous simulation• steady state nonlinear simulation• dynamic linear simulation

Evaluation 9. evaluate • check if objectives are achieved

The stages and (the order of) the steps mentioned in the table above should not come as asurprise. In fact the only thing that needs explaining is the order of the steps in the synthesis


stage. Actually the order of the steps in this stage is somewhat irrelevant. The reason is thatthe different steps (material balance control, quality control etc.) normally do no compete forthe same DOF’s. So in other words the steps are largely independent. In the case that thesteps do compete for the same DOF’s then the best order is given in table 7.2. This order isbased on:

• The fact that the material balance control has a larger impact than quality control.Remember that only material balance control is plantwide while quality control is local-ized. Furthermore loss of inventory often leads to off specification product or (partial)shutdown.

• The fact that recycles and costs can be dealt with by extension of the material balanceand/or quality control. For example the balanced scheme (see figure 6.3) could beconsidered as a extension of the push scheme.

Finally a few words about the documentation of a plantwide control system. The documen-tation should enable the reader to answer the following questions: 1) What is the currentsituation? and 2) What is the background of the current situation? The first question reflectsthe minimal requirement and in most countries it is also a legal requirement. The answeron the second question facilitates maintenance. An easy and structured way to answer bothquestions is by documenting all the stages of the proposed procedure. So typically the firstchapter of the final documentation is called “Process and control objectives” and it containsa process description, the specific control objectives...

7.4 An example; sulfuric acid alkylation

The proposed procedure will now be used to design a plantwide control system for a processof average complexity; sulfuric acid alkylation. Alkylation is a general term for the reaction ofalkenes with isobutane (isoC4) to form larger branched alkanes; alkylate. The purpose of thisprocess is the production of components with a high octane number (for gasoline blending).In our case the alkene is propene (C3=), this means that alkylate consists of seven carbonatoms (C7). The reaction mechanism occurs via carbenium ions (see figure 7.1). Besides

C C C + H+ C C+ C

+ C C+ CC C C

C

C C+ C

C

+ C C C

+ C C CC C+ C

C

C C C

C

C C+ C

+ C C CC C C

C

C C C

C C C

C

C C+ C

C

+ C C+ C

C

Figure 7.1: The mechanism of the reaction of C3= with isoC4.

alkylation also polymerization of C3= can occur (C3= → C6= → C9=), to suppress this un-desired reaction a large excess of isoC4 (typical

isoC4C3=

= 10 [mole/mole]) is used. The reaction

7.4. AN EXAMPLE; SULFURIC ACID ALKYLATION 47

is catalyzed by acid, in our case sulfuric acid (H2SO4)3. The reaction is exothermic; per

mole of alkylate 80 kJ of heat is released. So thermodynamically alkylation is favored by lowtemperatures.

Figure 7.2 shows a flowsheet of the process as licensed by Exxon Research and Engineering;http://www.prod.exxonmobil.com/refiningtechnologies/fuels/mn_sulfuric.html.

C1

caustic

C3=

V1cooling

water

C2

C3

n-C4

C7+

spent

caustic

waste

water

E1

E2

R1

spent

acid

fresh

acid

M1CP1

V2 V3

isoC4

C3

water

Figure 7.2: Alkylation of C3= with isoC4 and H2SO4 as catalyst.

Note that this is a somewhat simplified flowsheet, for example the condensors, top accumu-lators and reboilers of the various distillation columns (C1, C2 and C3) are not shown. TheC3= feed is cooled and fed to the various stages of the reactor R1. Both isoC4 and acid enterthe first stage of R1. The amount of acid in the reactor is around 50% [vol/vol], the acidconcentration should be close to 90% [mass/mass]. Note that the isoC4 feed is first combinedwith the recycle. The reactor consists of four CSTR’s in series, it operates at a pressure of2 bar and a temperature of 5 oC. This low temperature is essential to avoid redox reactionsthat result in excessive acid consumption and the formation of tars and SO2. The residencetime in the reactor is around 30 minutes.

The reaction heat is absorbed by the evaporation of mainly propane (C3, present in

3In practice also hydrofluoric acid (HF) is used, for a comparison see Moulijn, Makkee and Van Diepen [5].


the C3= feed or formed during the reactions) and isoC4. The vapors are compressed andcondensed, a part of the liquid is used to cool the C3= feed, the rest is send to depropanizerC1. This column removes the C3 and as such prevents the accumulation of this component.The bottoms of C1 (isoC4) are recycled back to the reactor.

The last compartment of R1 acts as a settler. The heavy acid phase is recycled backto the reactor. The hydrocarbon phase is subsequently washed with a caustic solution (forexample a NaOH-solution) and water before it is fed to the de-isobutanizer C2. This columnseparates the isoC4 from the heavier components. The latter are send to the debutanizerC3, the isoC4 is recycled back to the reactor. Column C3 removes the normal butane (n-C4,present in the isoC4 feed) from the alkylate.

We will now start with the design of a plantwide control system according the proposed pro-cedure.

Step 1 Determine control objectives

The general objective of a plantwide control system could be stated as: 1) support (not guar-antee) safety, 2) realize the required conversion and 3) minimize the operating costs. Letsstart with safety, from the process description it is clear that besides the pressure in variousunit operations also the temperature in R1 is related to safety (a temperature increase leadsto faster reactions and higher pressures). So the control of the reactor temperature needsspecial attention. Note that this was already recognized during the design of the process; thereactions take place in an evaporating liquid phase.

For the required conversion we need to be able to set the production rate as well as the quality.The production rate boils down to setting the feed of C3=.

The best way to discuss the control objectives related to quality is by unit operation, seetable 7.3. Note that a number of objectives are taken directly from the process description.

In the settler of R1 we need to control the interface between acid and hydrocarbons. Sinceinterfaces are normally more difficult to measure than levels it is good to discuss this with theinstrumentation engineer. It important to realize that in R1 the temperature and pre

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