Basic Process Engineering Control 9783110289824, 9783110289817

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De Gruyter Graduate Agachi, Cristea ∙ Basic Process Engineering Control

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Paul S. Agachi, Mircea V. Cristea

Basic Process Engineering Control |

Authors Prof. Dr. Paul Serban Agachi Babes-Bolyai University Faculty of Chemistry and Chemical Engineering Arany Janos Str. 11 400028 Cluj-Napoca Romania

Assoc. Prof. Dr. Mircea Vasile Cristea Babes-Bolyai University Faculty of Chemistry and Chemical Engineering Arany Janos Str. 11 400028 Cluj-Napoca Romania

ISBN 978-3-11-028981-7 e-ISBN 978-3-11-028982-4 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://www.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Cover image: Janaka Dharmasena/Getty Images/Hemera Typesetting: PTP-Berlin, Protago TEX-Produktion GmbH, www.ptp-berlin.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ♾Printed on acid-free paper Printed in Germany www.degruyter.com

Foreword In the last decades, Chemical Engineering has evolved and developed towards Process Engineering which is a more comprehensive field also including other processes than those with the chemical product as the final target. The general principles developed over time in the field of Chemical Engineering are ripe to be used in other fields which might be seen as very different: biomedical processes in the human body, the management of utilities, of water resources, the analysis of the behavior of a society or administration, petroleum or natural gas extraction and transportation, production of textiles or other materials, or river and water table pollution. Process Control, referring initially to Chemical Engineering, is the theory and practice of the operation of the processes without human intervention and it is sometimes named Automatic Process Control. Historically, the control techniques were aimed mainly towards diminishing the deviation of a parameter from its desired value, a value imposed by the process technology. This concern represents only a part of the problems to be solved in the successful operation of a plant. It is very strange, but the control or process engineer has a much greater influence on the operation of the plant than it seems is imposed by its tasks. In fact, a control or a process engineer has the possibility of obtaining larger production in a technological process and can contribute substantially more to cost reduction and to increased economic efficiency than any other specialist of the plant, namely the technology engineer or the economist. If the control of the technological parameters is so restrictive, how can the process engineer contribute through automatic control towards the optimal operation of the process? The answer consists in the intelligent application of control theory: if the control is applied without a thorough understanding of the process, its results will be poor and useless. The most efficient control system or control scheme embeds in it the characteristics of the controlled process including its demands, its drawbacks and limitations. In a good cooperation between the process, the technological equipment and the means of monitoring and control, all parts are acting as a whole, inside the so called “Chemical System” or “Process System”. This cooperation is not easily obtained: to design a control system able to cope with expectations, an understanding of the steady state and dynamic behavior of the plant (process, technological and control equipment) is needed on the part of all contributors, starting from the researchers of the basic industrial process, continuing with the designers and ending with the plant operators. The book explains all the determinations in the system, starting from the intimacy of the processes, going on to the intricate interdependency of the process stages, analyzing the hardware components of a control system and ending with the design of an appropriate control system for a parameter or a whole process. The book is first addressed to students and graduates of departments of Chemical or Process Engineering.

VI | Foreword Second, to the chemical or process engineers in all industries or research and development centers, because they will notice the resemblance in approach from the system point of view, between different fields which might seem far apart from each other. As for the students in Chemical Engineering or Chemical Technology, as future specialists involved in the research, design or operation of a process, it is necessary to learn all aspects linked to the basics (physics, chemistry, mathematics), process and process equipment (reaction engineering and transport phenomena, technology, modeling, optimization), monitoring and control (measuring and control equipment, control). Because the control room is the interface between the operator and the control equipment is his/her “eye and hand” in the process, it is absolutely necessary to understand their functioning in the new conditions of increased technological complexity. It is also interesting and useful for the application of complex mathematical tools and methods to study the behavior of the process in a steady or dynamic state and in the diagnosis of the problems based on mathematical modeling. Usually, universities produce chemical and control engineers with different complementary competencies. The newer option for process engineers educated to understand not only the “chemistry” of the process, but also the way the process can be made operational in real time and economically efficient seems to be the future for their education. This book is a plead for process control as part of process engineering. The book is structured in three parts: Part I – Fundamentals of system theory and control, approach of the processes as systems, basics of mathematical modeling; Part II – The analysis of a control system treating all its hardware components (principles of operation, construction, dynamic and steady state behavior) all starting from the behavior of the controlled process; Part III – The synthesis of the control system, including the harmonization of the components’ behavior in the control system, the design and tuning of the controllers generally and specialized on the control of the main parameters controlled in process engineering. Finally, many thanks to our students with whom we cooperated in the elaboration of the book (Mihai Mogos, Botond Szilagyi, Abhilash Nair, Hoa Pham Thai), our professors who strived to educate us as citizens and specialists, to our former heads of departments where we have worked, who, each of them, left a mark on our present formation, to our university who nurtured us for many years in harder or easier times, to our colleagues from abroad who helped us especially after the changes in Romania in 1989, to our colleagues at home with whom we repeatedly debated the topics, and above all to God for giving us the opportunity and power to achieve. Last but not the least, all thanks to our families who supported us in our whole life endeavor. Cluj-Napoca, June 2014

Paul S. Agachi Cristea V. Mircea

Contents Foreword | V

Part I: Introduction 1

History of process control | 3

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

Basics of systems theory | 8 System concept | 8 System delimitation | 9 Input and output variables | 11 Classification of the systems | 12 The state concept | 18 Input-state-output relationship | 21 Stability of the system | 22 Types of elementary signals | 23 LTI systems described by input-output relationships | 26 Time response of the linear time-invariant systems | 27 Solution of the homogeneous differential equation | 27 Particular solutions of the nonhomogeneous differential equation | 28 General solution of the nonhomogeneous differential equation | 29 Stability of the system described by input-output relationships | 29 Stability of systems described by linear time-invariant differential equations | 30 Frequency response of the system described by input-output relationships | 31 Frequency response of the system initially at equilibrium | 33 Steady state and transient response to the harmonic input | 33 LTI systems described by input-state-output relationships | 34 Transformation of the input-output representation into the input-state-output representation | 35 Solutions of the state equations | 37 Solution of the nonhomogeneous state equation | 38 Laplace transform | 39 Definition of the one-sided and two-sided Laplace transform | 40 Properties of the Laplace transform | 41 Laplace transform of usual functions | 41 Inverse Laplace transform | 41

2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27

VIII | Contents 2.28 2.29 2.30 2.31 2.32 2.33 2.34

Use of Laplace transform in the analysis of linear time-invariant systems | 44 Use of Laplace transform for describing systems represented by input-output relationships | 45 Use of Laplace transform for describing systems represented by input-state-output relationships | 46 The transfer matrix | 48 Bode diagrams | 49 Nyquist diagrams | 57 Problems | 58

3 3.1 3.1.1 3.1.2 3.2 3.3 3.4 3.5

Mathematical modeling | 63 Analytical models | 64 The conservation laws | 64 Thermodynamics and kinetics of the process systems | 73 Statistical models | 76 Artificial neural network Models | 80 Examples of mathematical models | 82 Problems | 92

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Systems dynamics | 97 Proportional system | 97 Integral system | 99 Derivative system | 101 First order system | 104 Second order system | 106 Higher order system | 109 Pure delay system | 113 Equivalence to first order with time delay system | 114 Problems | 116

5 5.1 5.2 5.3 5.4

Manual and automatic control | 122 Manual control | 122 Automatic control | 124 Steady state and dynamics of the control systems | 125 Stability and instability of controlled process and control systems | 126 Performance of the control system | 126 Problems | 128

5.5 5.6

Contents

Part II: Analysis of the feedback control system 6 6.1 6.2 6.3

The controlled process | 133 Steady state behavior of the controlled process | 133 Dynamic behavior of the controlled process | 140 Problems | 147

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Transducers and measuring systems | 151 Introduction | 151 Measuring systems in process engineering | 155 General characteristics of the transducers | 156 Temperature transducers | 158 Pressure transducers | 166 Flow transducers | 173 Level transducers | 187 Composition transducers | 196 Problems | 224

8 8.1 8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5 8.3

Controllers | 227 Classification of controllers | 227 Classical control algorithms | 231 Proportional controller (P) | 231 Proportional-Integral controller (PI) | 238 Proportional-Integral-Derivative controller (PID) | 242 Controllers with special functions | 245 Distributed Control Systems | 250 Problems | 253

9 9.1 9.1.1 9.1.2 9.2 9.2.1 9.2.2 9.2.3 9.3 9.4 9.5 9.5.1 9.5.2

Final control elements (actuating devices) | 256 Types of final control elements | 256 Control valves | 256 Other types of final control elements | 262 Sizing the control valve | 263 The flow factor (Kv ) for incompressible fluids | 263 The flow factor (Kv ) for gases | 266 The flow factor (Kv ) for steam | 267 Inherent characteristics of control valves | 268 Installed characteristics of the control valves | 270 The dynamic characteristics of a control valve | 274 The gain of the control valve | 274 The dynamics of the control valve | 274

| IX

X | Contents 9.6 9.7 10 10.1 10.2 10.3 10.4 10.5

Sizing and choice of the control valves | 275 Problems | 278 Safety interlock systems | 281 Introduction | 281 Safety layers | 282 Alarm and monitoring system | 284 Safety instrumented systems | 285 Problems | 290

Part III: Synthesis of the automatic control systems 11 Design and tuning of the controllers | 295 11.1 Oscillations in the control loop | 295 11.2 Control quality criteria | 297 11.3 Parameter influence on the quality of the control loop | 300 11.4 Controller tuning methods | 300 11.4.1 Experimental methods of tuning controller parameters | 300 11.5 Tuning controllers for some “difficult to be controlled” processes | 323 11.6 Problems | 325 12 12.1 12.2 12.3 12.4 12.5 12.6

Basic control loops in process industries | 330 Flow automatic control systems | 330 Pressure automatic control systems | 335 Level automatic control systems | 338 Temperature automatic control systems | 344 Composition automatic control systems | 349 Problems | 355

Index | 357

| Part I: Introduction

1 History of process control As one may see, the development of process control is strongly related to manufacturing processes. These roots are traced to the ancient times of humanity, starting with metal, fabric and pottery production. Industrial manufacturing and actually engineering were innovations of the XVIIIth century, during the Industrial Revolution. The population increased during this century, being inclined to consume more and better. This consumerism led to an escalation in demand, both with regard to quantity and quality, for food, clothing, footwear, housing, transportation, which stimulated the production of construction materials, textiles, chemicals etc. Each global conflict, after its end (e.g. First and Second World Wars), induced the same behavior and the same reaction on behalf of society and production companies. Production became mass production with huge quantities of products delivered at deadlines and with a certain quality expectation. Mass production, under these circumstances, could no longer be controlled manually because of the expectations. In the meantime, in the second half of the XXth century, the environment became important, a fact which imposed environmental constraints on the manufacturers. On top on that, the globalization of the economy amplified the competition among world companies and only those capable of reducing costs and respecting the environment remained on the market. Together with these facts, an important impact on the development of technology and especially computing facilities was caused by the Cold War and the space race between the major world powers: the USA and the USSR. All these sequences led to a tremendous development of control equipment and techniques as a part of process control development. Process control was seen as a major tool for development and complying with the constraints. Milestones in the modern history of control are C. Drebbel’s contribution in inventing the first temperature control device for a furnace, around 1624, D. Papin’s invention of the first safety valve for his steam engine – a pressure regulator – in 1704, E. Lee’s first controlled positioning system for a windmill in 1745 [1, 2] (Fig. 1.1), T. Polzunov’s first level controller for his steam engine (1765), J. Watt’s fly ball governor in 1768 [3, 4] – a speed regulator for his rotary steam engine – (Fig. 1.2). The first obviously advanced combination between process engineering and process control was H. Jacquard’s loom in 1801 (Fig. 1.3) which stored the model of the silk fabric on punch cards [5, 6]. Actually, in that period in France there were several looms having similar control systems. The first publication in the field of control systems was elaborated by J. C. Maxwell in 1868 and approached a theoretical analysis of the stability of Watt’s fly ball governor (1868) [3]. The next papers on the subject of automatic control appeared only in the first half of the XXth century (1922–1934) and have to be noted the first in the field of control in chemical/process engineering [7–11]. A major innovation is represented by G. Philbrick’s “Automatic control analyzer – Polyphemus”, actually the first analogue computer (1937–1938) just before World

4 | Part I: Introduction

Fig. 1.1. Lee’s control positioning system for a windmill.

Fig. 1.2. The speed controller for Watt’s steam engine.

War II [12]. World War II brought extremely important innovations in the field of automatic control: automatic rudder steering, automatic gun positioning systems, automatic pilot of V1 and V2 etc. The innovative ideas of Ziegler and Nichols about how to tune the controllers in a loop stem from that time of progress! [13] Immediately after World War II, the field of process control exploded. It was helped by the construction of the megacomputers ENIAC (1946) [14] and UNIVAC (1951) [15], Shockley’s patent on the transistor (1950) [16] and Feynman’s premonition regarding nanospace expressed at the American Physics Society Meeting in Caltech

1 History of process control

| 5

Fig. 1.3. H. Jaquard’s loom with punch cards.

“There’s plenty of room at the bottom” (1959) [17]. The new frontier of challenging outer space launched by the US president John Fitzgerald Kennedy produced the portable computer which influenced thoroughly the more recent history of Process Control. The first industrial control computer system was built in 1959 at the Texaco Port Arthur, Texas, refinery with a RW-300 computer from the Ramo-Wooldridge Company [18] and the second one was installed in 1964 by Standard Oil California and IBM at El Segundo refinery in an FCC Unit, under the name 1710 Control System [19]. Thus, there were practically three stages in Process Control development: – the first stage: measurement and control devices (World War I–mid XXth century), dominated by manual control and operation of the industrial processes; sometimes field mechanical controllers were used; – the second stage: classical feedback control (1950s–1970s), when most of the processes were supervised and controlled automatically in the simplest way – feedback control; the control equipment was either electronic or pneumatic, rarely hydraulic; – the third stage: computer assisted control (1970s–present) dominated, as the main feature, by control systems involving a micro- or minicomputer; most processes are controllable and they need classical control systems, but around 20 % of them from process industry are less controllable and need Advanced Control techniques and equipment. The use of computers in process control can be detailed in several stages of complexity. In the first stage of complexity a microchip is embedded to perform some calculations

6 | Part I: Introduction inside the controller or even computer, allowing more complicated control algorithms: optimal, adaptive, fuzzy, ratio, inferential, feed-forward control algorithms. A second stage of complexity is the advanced process control (APC) characterized by a combination of hardware and software control tools used to solve complicated multivariable control problems or mixed integer-discrete control problems. It involves the computer techniques (hardware and software) applied to control, a good understanding of the process (process modeling and design), a good understanding of control techniques and optimization (model predictive control (MPC), distributed control systems (DCS) etc.). In the processes with multiple variables (hundreds or thousands) an efficient operation of the process, of the plant or of the industrial platform, can be done only through APC. Central control rooms looking like cosmic flight control centers are supervising and operating the whole plant. New skills of the operators are required, because the implications of their actions are on a large scale. With APC one can save energy, raw materials, time, reduce costs and make the processes more competitive in terms of quality and costs. The third stage of complexity is expressed by wise machinery (WM), a new concept which involves not only complying with the standards or setpoints, but also the intervention of the machine for its own functional efficiency and safety without human assistance [20]. For that, WM involves complex industrial equipment, the data acquisition system (DAS) and the ‘‘optimal parametrical control system (OPCS)”. They are intended for finding operative and optimal solutions of all possible management tasks arising in vital activities of process and the machine’s equipment in the limits of its life cycle. PCS is based on bio-cybernetic control systems (BCS) which are principally a novel and industry evaluated decision-making system (Fig. 1.4), which functions on the basis of formalization of the unconscious activity process of a brain in maintaining the functions of an organism. It is a new civilization in itself.

Equipment configuration

Parameters

BCS

Instructions

Decisions support system configuration

Fig. 1.4. Wise Machinery.

References [1] [2] [3]

Mayr, O., The origins of feedback control, Scientific American, 223, No. 4 (October), (1970), 110–118. Mayr, O., Feedback Mechanisms in the Historical Collections of the National Museum of History and Technology, Smithsonian Institution Press, 1971. Maxwell, J. C., On governors, Proceedings of the Royal Society of London, 16, (1868), 270–283 [This is presumably the first scientific article on feedback control].

1 History of process control

[4] [5] [6] [7]

[8]

[9] [10]

[11]

[12] [13]

[14]

[15]

[16] [17] [18] [19]

[20]

| 7

Centrifugal governor, http://en.wikipedia.org/wiki/Centrifugal_governor. Eclectica, Jacquard’s loom and the stored program concept, http://addiator.blogspot.ro/2011/10/jacquards-loom-and-stored-programme.html. Essinger, J., Jacquard’s Web: How a Hand-loom led to the Birth of the Information Age, New Ed., Oxford University Press, 2007. Minorsky, N., Directional stability of automatically steered bodies, J. Am. Soc. Nav. Eng., 34, (1922), 280. [The article came after WWI when serious attempts at the stabilization of the rudder of naval units had been made]. Nyquist, H., Regeneration theory, Bell Syst. Technol. J., 11, (1932), 126 [The article is related to papers by Black and Bode, representing another approach to a feedback system. Nyquist developed an original theory for feedback systems analysis and design]. Grebe I., Boundy R. H., Cermak R. W., The control of chemical processes, Trans. Am. Inst. Chem. Eng., 29, (1933), 211 [The article is the first to approach the control of chemical processes]. Hazen, H. L., Theory of servomechanisms, J. Franklin Inst., 218, (1934), 279 [Hazen is the first to describe how automatic equipment will replace human labor; he distinguished for the first time the “open cycle” (without feedback) from the “closed cycle” (with feedback) which are nowadays called open-loop and closed-loop control]. Ivanoff A., Theoretical foundation of the automatic regulation of temperature, Journal of the Institute of Fuel, 7, (1934), 117 [One of the first papers theoretically analyzing temperature control]. Holst, P., George A. Philbrick and Polyphemus: The first electronic training simulator, IEEE Annals, 4:2, (1982), 143. Ziegler J. G., Nichols N. B., Optimum settings for automatic controllers, Transactions ASME, 64, (1942), 759 [The first paper approaching the controller closed-loop tuning which was quite new at that time. The methods described are still used on a large scale for individual control loops] Goldstine H. H., Goldstine A., The Electronic Numerical Integrator and Computer (ENIAC),1946, reprinted in The Origins of Digital Computers: Selected Papers, pp. 359–373 Springer-Verlag, New York, 1982, [First digital computer created at the University of Pennsylvania for the US army; it had a calculation speed 1,000 times higher than the electromechanical machines]. UNIVAC Conference Oral History, 17–20 May 1990, Charles Babbage Institute, University of Minnesota [The second numerical computer created by the same team as ENIAC, delivered in 1951 to the US Census Bureau]. US 2502488(1950), Semiconductor Amplifier [Shockley W. submitted his first granted patent involving junction transistors]. Feynman R., There’s plenty of room at the bottom, Caltech Engineering and Science, 23:5, (1960), 22–36 [The conference presents the vision of nanospace and nanotechnology] Astrom, K., Wittenmark, B., Computer Controller Systems: Theory and Design, Prentice-Hall Inc., USA, 1997, p. 3. Strycker W. P., Use and application of control systems via a digital computer, SPE Production Automation Symposium, 16–17 April 1964, Hobbs, New Mexico, Conference paper. [The paper describes the concept and practice of installing the second computer control system in process industries, initially controlling offline, from a distant IBM computer center in San Francisco, via telephone lines with printer or punch card reader terminals, and afterwards online with a process computer installed at the refinery]. Bravy K., The compatibility of viability maintaining of machinery with viability maintaining of an animal’s organisms, New technologies for the development of civil and military organizations in the modern post-industrial world, November 30, 2004, Ashdod, Israel, Conference paper.

2 Basics of systems theory 2.1 System concept Definition. The system is a set of component entities, characterized by specific individual properties, being in interdependent relationships and coupled together in a certain way to give a well-defined purpose to the assembly [1]. The set of constituting entities reveals the structural aspect of the system, disclosing the existence of the individual elements that form it. There is a relative division of the system into its constituting components leading to the concept of subsystem, as a component entity. This relative evaluation of the system structure is directly tied to the a priori hypothesis of indivisibility assumed for its elements, named entities. The entities composing the system may be grouped in three categories: concepts, objects and subjects [1]. Concepts are abstract entities generating the abstract type system, such as: systems of mathematical equations, programming computer languages or numeration systems. The space delimited by the ensemble of these entities forms the main center of interest and development for the systems theory [2, 3]. Objects are material entities, without life, representing real physical systems, such as a chemical reactor, a mechanical vehicle or an electronic device. Subjects are natural material entities composed of living individuals, such as: the student community of a university, the operating personnel of a chemical unit or the microorganisms of a biological system. The system of a chemical plant is composed of the chemical equipment as object entities, the operating personnel as subject entities and the set of working rules as concept entities. Individual properties of the system entities are referred to as attributes. The nature of and interdependence between attributes provides identity to the entity. The example of a system composed of three entities and having a variable number of attributes aij is presented in Fig. 2.1. Attributes may be classified in two categories: variable attributes and structural attributes. This variability is reflected in mathematical functions that usually have time as the independent variable (and possibly one or several spatial variables). Physical or chemical quantities such as: reactor volume, molecular mass, type of chemical species involved in a chemical reaction or the reactants’ state of aggregation are all structural attributes. Physical quantities such as: mass or energy flows, reactants’ or products’ concentration and temperature, specific heat (possibly temperature dependent) of the chemical species, are examples of variable attributes. Some of the variable attributes describing the entity interact with other variable attributes of a different entity causing the interaction relationships that are inducing

2 Basics of systems theory

a11

a12

a21

| 9

a22

a13 Entity 2 Entity 1 a31

a32

a33

a34 Entity 3

System

Fig. 2.1. System formed by three interacting entities, each having interdependent attributes.

new properties to the ensemble (system). The connection mode between entities is decisive for the system’s behavior and performance as a whole. It becomes obvious that the whole is superior, as functionality, to the component parts and, once the entity properties and interactions are known, the problem of predicting the properties of the whole is not a trivial task. From the systems theory point of view the connection of entities, by certain dependence relationships, does not confer the statute of system as long as the purpose for which is was designed or analyzed has not yet been specified. For example, a system consisting in a chemical reactor may be analyzed from the mechanical point of view, in order to investigate aspects related to the reactor response to variable mechanical stress or it may be analyzed and designed in order to provide maximal conversion of reactants into the desired chemical products, from the chemical efficiency point of view.

2.2 System delimitation Delimitation of the systems is of primary importance being a direct consequence of the purpose specification and having an important effect on its way of representation. System delimitation is the operation of separating the entities composing the system from the rest of the surrounding environment. This delimitation consists in the determination of an interface (Σ ) represented by a real or imaginary surface that encompasses in its interior all entities that constitute the system [1]. Usually, for real physical systems, the delimiting surface corresponds to physical construction attributes of the entities in the system. A typical case is the continuous stirred tank reactor investigated from the mass, energy or momentum transfer processes. The reactor shell commonly represents the delimiting surface, as may be noticed from Fig. 2.2. There are situations when this delimitation assumes the specification of a more complex interface (surface) with an irregular form. As an example, the interface of a heat exchanger equipped with a heating coil may be presented, where the interface

10 | Part I: Introduction

Σ

Fig. 2.2. Typical interface delimitation of the system Σ , represented for the case of the stirred tank reactor.

consists of two surfaces Σ1 and Σ2 ; the system is delimited by the volume contained between these two surfaces, as presented in Fig. 2.3. In the behavior description for heterogeneous processes, the infinitesimal elementary volume is often put into evidence. For such elementary volume, considered as a system, the balance equations are first developed and then the space integral is considered in order to reveal macroscopic aspects, Fig. 2.4. The operation of delimiting the system interface has to take into account the definition of a minimal structure for the system, in accordance to the stated purpose. It is worthwhile to notice that the interface is settled according to the possibility of the mathematical description for its component entities and relationships involved. The surrounding environment is represented by all entities not included in the system, but significant are only those outside entities having (possible) direct interaction relationships with entities within the system.

Σ2

Σ1

Σ1–Σ2

Fig. 2.3. Interface delimitation Σ1 –Σ2 of the system represented by a heat exchanger with heating coil (textured represented volume).

2 Basics of systems theory

| 11

dΦ Ʃ

Ʃ dz dz dy

dr

dx

Fig. 2.4. Systems Σ represented by infinitesimal elementary volumes in rectangular and cylindrical coordinates.

2.3 Input and output variables For the analytical (both quantitative and qualitative) description of the entities belonging to the system, and consequently for the whole system, it is necessary to know the structural attributes, the variable attributes and the interactions between entities. Each entity can be described from the mathematical point of view by a system of (algebraic, differential or integral) equations representing the way attributes are connected by interdependence relationships. Consider a system described by a set of relations Ri , i = 1 ⋅ ⋅ ⋅ r between the variable attributes vj , j = 1 ⋅ ⋅ ⋅ n and the structural attributes sk , k = 1 ⋅ ⋅ ⋅ l: R1 (v1 , v2 , . . . , vn , s1 , s2 , . . . , sl ) = 0 R2 (v1 , v2 , . . . , vn , s1 , s2 , . . . , sl ) = 0 ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ Rr (v1 , v2 , . . . , vn , s1 , s2 , . . . , sl ) = 0

(2.1)

Relations between attributes of the entities forming the system originate from laws, postulates and theorems. Typical examples are the conservation laws for mass, energy or momentum, for state equations and for transport and transfer of property or reaction kinetics. Variable attributes are represented by mathematical functions having usually as independent variables time and one or more space variables. In the following, the structural attributes will not be explicitly represented. Variable attributes that intersect the delimiting interface of the system are simply named as terminal variables and the relations involving these variables as terminal relations [4]. Those variables that do not intersect the interface are named suppressed variables. The terminal variables of the system may be classified in two important categories, depending on the cause or effect nature they exhibit. This division is named the system orientation. An oriented system is a system for which the terminal variables are already

12 | Part I: Introduction separated in variables of type cause, named input variables (inputs) and variables of type effect, named output variables (outputs). The oriented system may be therefore described by the set of equations: R1 (u1 , u2 , . . . , uq , y1 , y2 , . . . , yp ) = 0 R2 (u1 , u2 , . . . , uq , y1 , y2 , . . . , yp ) = 0 or R(u, y) = 0 ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ Rr (u1 , u2 , . . . , uq , y1 , y2 , . . . yp ) = 0

(2.2)

In equation (2.2) may be noted the vector of the input variables u = [u1 u2 ⋅ ⋅ ⋅ uq ]T , the vector of the output variables y = [y1 y2 ⋅ ⋅ ⋅ yp ]T and the vector of the relations (functions connecting inputs and outputs) R = [R1 R2 ⋅ ⋅ ⋅ Rr ]T . Note: boldface font notation will be further used for vectors and matrices. Once the system orientation is performed the simplified, but very general, generic representation of the system may be obtained, as presented in Fig. 2.5.

Inputs (Causes)

u

y System

Outputs (Effects)

Fig. 2.5. Generic representation of the system.

2.4 Classification of the systems With the aim of bringing to attention the different properties and characteristics of the abstract or feasible systems, in the following a set of categories of systems are presented [1, 5, 6]. Classification of systems is performed on the basis of various criteria determined by both the nature of attributes (of variable or structure type) and by the nature of relations between attributes. This classification is also valid for the individual classification of entities composing the system.

Dynamic and static systems Classification of the systems into these two categories is performed considering or neglecting the aspect of time dependence for the variable attributes of the system. Time represents a major dimension of the surrounding reality. Irrespective of the physical nature of the system, almost all of its attributes have to undergo changes with respect to time. This time dependence may involve variations taking place at very different time scales, from the order of nanoseconds in the case of phenomena evolving at atomic scale to the order of years in the case of materials ageing phenomena. Systems for which the vector of outputs y(t), at every time moment t, depend on the values of the vector of inputs u(t) from the same time moment but also of the vec-

2 Basics of systems theory

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tor of inputs u(t ± Δt) and/or outputs y(t ± Δt), from neighboring moments ((t − Δt) for non-anticipative systems and (t + Δt) for anticipative systems) are named dynamic systems. Dynamic systems are also called systems with memory. This denomination derives, for feasible systems (those that can become physical), from the dependence of the outputs at the present moment of time also on values of variables from moments preceding the current moment. The dynamic character is a consequence of the inertia that characterizes all feasible systems. Inertia is the system property of making opposition to the change of the quantity of mass, energy or momentum contained in the system at a certain moment of time. Systems for which the vector of outputs y(t), at every time moment t, depend only on the values of the vector of inputs u(t) from the same time moment t, are named static systems. These systems are also named systems without memory. Static systems are ideal as they emerge from feasible systems when simplifying assumptions are considered. The system may be considered static when the aspect of time dependence of its attributes is not important for the stated purpose. The mathematical description of dynamic systems is usually performed by differential equations (relationships) and for static systems by algebraic equations. For static systems, the algebraic relationships (algebraic equations) describe in a simple and natural way the relationship between the non-temporal input and output variables. For dynamic systems, the differential relationships (differential equations) are the most common mathematical support for describing the dynamic character. Example 2.1. For illustrating the use of the differential relation in the description of dynamic system behavior consider the system consisting in an open tank continuously fed and evacuated with a liquid phase, Fig. 2.6. The inlet mass flow Fmi (t) and outlet mass flow Fmo (t) are considered to be time dependent. It is also assumed that the liquid level in the tank H is continuously measured with a transducer and the signal (level information) is sent as an electrical quantity to an observer placed at a distance from Fmi

Ʃ

H

HO Fmo

Fig. 2.6. Time-dependent level H(t) change in a tank.

14 | Part I: Introduction the tank (passes outside the system’s interface Σ ). For the initial time moment the level in the tank is known H(t0 ) = H0 . The aim is to describe the way that the liquid level in the tank changes with time for the change in time of the inlet and outlet flows. The system may be considered delimited by the interface Σ , with the terminal variable of input type u(t) = Fmi (t) − Fmo (t) (the cause) and the terminal variable of output type y(t) = H(t) (the effect). The known structural variables are the tank cross-section A and the liquid density ρ . For a finite time interval Δt the mass balance for the Σ system may be stated as: the net mass (difference between inlet and outlet mass) of liquid entered or evacuated in/from the tank during the time interval Δt is equal to the change of mass of liquid from the tank during the same time interval Δt. This equality is presented in the equation (Fmi − Fmo )Δt = (H − H0 ) ⋅ A ⋅ ρ .

(2.3)

Equation (2.3) may be reformulated under the form Fmi − Fmo = A ⋅ ρ

H − H0 . Δt

(2.4)

Equation (2.4) describes the “mean” behavior of the level in the tank over the considered finite time interval Δt. If we want to capture the change of the liquid level at every moment of time t, as close as possible to the initial moment t0 , it is necessary that the time interval Δt becomes infinitely small; this is equivalent to performing the limit Δt → 0 in equation (2.4): lim (Fmi − Fmo ) = A ⋅ ρ lim

Δt→0

Δt→0

H − H0 . Δt

(2.5)

It may be noticed that the right member of equation (2.5) becomes the derivative of the level function H(t) of the tank: Fmi (t) − Fmo (t) = A ⋅ ρ

dH . dt

(2.6)

The equation referring to the derivative of the level function may be reformulated using conventional notation used for the system input and output variables: u(t) = A ⋅ ρ

d(y(t)) . dt

(2.7)

The dynamic (with memory) characteristic of the system is sustained for the above example, by the dependence of the level function H(t0 + Δt), at the moment t = t0 + Δt, on the level value from a preceding moment of time, i.e. H(t0 ). In general, a single input single output (SISO) dynamic system may be described by the generic differential equation: qn

dn y dn−1 y dy dm u dm−1 u du + qn−1 n−1 + ⋅ ⋅ ⋅ + q1 + q0 ⋅ y = pm m + pm−1 m−1 + ⋅ ⋅ ⋅ + p1 + p0 ⋅ u. n dt dt dt dt dt dt

(2.8)

Depending on the ordering relationship between the maximal differentiation order of

2 Basics of systems theory

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the right m and left n members of equation (2.8) the systems may be classified in: (i) strictly proper systems, when n > m, (ii) proper systems, when n = m, (iii) improper (non-feasible) systems, when n < m.

Systems with lumped and distributed parameters The majority of physical systems are characterized by spatial dimensions. This fact determines that, besides time, the system attributes also depend on one or more spatial independent variables. Systems having attributes depending on one or more spatial coordinates are named systems with distributed parameters. Typical examples of system properties depending on space are: concentration of molecular species, density, pressure and temperature in a plug flow reactor where physical and chemical phenomena are taking place and depend on the position of the elementary volume in the reactor (for example with respect to the cylindrical system of coordinates r, θ , and z). There are systems for which the dependence of an attribute is important with respect to only two or just one of the spatial coordinates, so it is possible to consider the attribute constant with respect to the other spatial coordinates. Usually, this kind of approach represents a simplifying assumption serving the reduction of the mathematical model complexity. For example, in the case of the plug flow reactor the dependence of the properties with respect to the r and θ coordinates are often neglected taking into consideration only the dependence of the attribute with respect to the reactor length coordinate z (which is the moving direction of the mass flow in the system). Systems having attributes depending on neither of the spatial coordinates are named systems with lumped parameters. For a large majority of the systems in chemical engineering, the systems with lumped parameters are the result of simplifying assumptions. The typical example is the continuous stirred tank reactor (with total back-mixing regime) for which it is assumed that chemical and physical properties at each point of the system do not depend on any of the spatial coordinates, despite the fact that this dependence exists in the physical system (to a larger or smaller extent). The systems with distributed parameters are described by functions having at least one space independent variable. The relationships between the attributes of the dynamic and distributed parameters systems are of the partial differential type of equations. The system characteristic of having lumped or distributed parameters is determined by the quantitative relationship between the change with respect to time and the change with respect to the spatial coordinates of the system attributes. Whenever the propagation time of phenomena with respect to the spatial coordinates is small (reduced dimension or high speed), the system may be considered to have concentrated (lumped) parameters.

16 | Part I: Introduction From the mathematical point of view, obtaining the solution of a system of partial differential equations is in general a difficult task [6, 7]. This is the reason for making a compromise in the description of the dynamic systems with distributed parameters when only time behavior is taken into consideration. This compromise consists in introducing a pure time delay in the description of the phenomena evolution, usually named dead time. This way a relationship between attributes represented by functions only dependent on time may be obtained, mathematically represented by total (ordinary) differential equations. The dead time is represented, from the mathematical point of view, by functions having the time-independent variable shifted with the dead time value (translation in time). Example 2.2. A typical example for a dynamic system with distributed parameters consists in a conveyor belt for feeding a silo of solid material, Fig. 2.7.

Fmb(t)=u(t) Fm(t,x) Fmd(t)=Fmb(t–τ) v x L y(t)=Qm Fig. 2.7. Typical system presenting dead time τ .

The flow of solid material transported by the conveyor belt Fm (t, x) is both time t and space x dependent (the spatial coordinate stretches along the belt direction of movement). Consider the mass flow entering the conveyor belt at position x = 0 as the input variable u(t) and the quantity (mass) of solid material accumulated in the silo Qm (t) as the output variable y(t). Both input and output variables are considered dependent on time. The mass accumulation in the silo may be described by the equation: dQm (t) L = Fmb (t − τ ) τ = dt v

or

dy(t) = u(t − τ ). dt

(2.9)

The mass flow rate Fmd (t) discharged from the conveyor belt at position x = L, at a certain moment of time t, is identical to the mass flow rate Fmb (t − τ ) entering the conveyor belt at the preceding moment of time t − τ . The dead time τ describes in a global manner the delay produced by the conveyor belt in the propagation of the mass flow rate along the transport direction. The dead time is equal to the ratio between the length of the path of transport L and the belt constant velocity v. By the help of the dead time τ it is possible to describe the accumulation process of the solid in the silo

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using a total differential equation (with respect to time) having the argument of the input function shifted (translated) in time with the dead time value.

Deterministic and stochastic systems The deterministic or stochastic character of a system is related to the univocity with respect to time of the functions representing the terminal variables. If the vector of the output variable y(t) is uniquely determined (known) by the input vector u(t), along all of the time interval t > t0 , the system is considered deterministic. A deterministic system responds to a uniquely known (deterministic) input with a uniquely known (deterministic) output. If the vector of the output variable y(t, ξ ) is dependent both on time t and on a random variable ξ , therefore not being uniquely known over the time interval t > t0 , the system is considered stochastic. The stochastic system may have attributes represented by functions with random independent variables [8].

Linear and nonlinear systems A uniform system is a system described by a unique set of input-output relationships. Uniform systems having linear attributes and relations are denoted as linear systems. Linearity is characterized from the phenomenological point of view by the validity of the effects superposition principle and by the cause-effect proportionality principle. From the mathematical point of view, the properties of additivity and homogeneity should be fulfilled. Linearity of the relationships describing the linear systems consists in the existence of only constant coefficients (or only time-dependent coefficients for the non-stationary linear systems) in the underlying equations, over the whole definition domain of the independent variables. The presence of coefficients (structural attributes) depending on the terminal input or output variables (or depending on derivatives of the input or output variables) in the terminal relations denotes the nonlinear system. The non-uniform system is also nonlinear. A linear mathematical operator R is an application between two sets of functions presenting the following properties. Additivity. According to this property, for every two functions from the definition set, f1 and f2 , the following equality holds: R(f1 + f2 ) = R(f1 ) + R(f2 ); this relation sustains the equality between the effect of the sum of two causes, R(f1 + f2 ), and the sum of the individual effects of the two causes, R(f1 ) + R(f2 ), (principle of the effects superposition). Homogeneity. According to this property, for any real constant α and any function f from the definition set it holds the equality: R(α ⋅ f ) = α ⋅ R(f ); this relation sustains the equality between the effect of an individual cause f multiplied by α times, R(α ⋅ f ), and the individual effect R(f ) of the cause f , multiplied α times, α ⋅ R(f ), (principle of proportionality between the cause and the effect).

18 | Part I: Introduction Linear dynamic systems are mathematically described by differential equations with constant coefficients for the linear time-invariant (LTI) systems and by differential equations with time-dependent coefficients for linear non-stationary systems. The properties of effects superposition and of cause-effect proportionality are the basis for one of the most important consequences regarding the way of determining the linear system’s response to a free varying input. The response (output) of a linear system to a free varying input may be determined by the summation of the responses to the elementary functions with which the input function may be approximated by an infinite series (the most general case). When the system’s response to an elementary input function is known (such as Dirac impulse response), the response to any input may be determined by the convolution operation. A very useful characteristic in describing the system’s behavior is related to the way the system responds to the sine form of input function, when the frequency of the sine signal ranges from low to high values. The result of this analysis is denoted as the frequency response of the system. Linear systems own an essential property according to which the quasi-stationary response of the system to a sinusoidal input is also of sinusoidal form having the same frequency as the input signal. As the frequency of the input wave changes, both amplitude and argument (phase lag) of the quasi-stationary sinusoidal response change. Interconnecting linear subsystems also lead to systems with linear behavior. This property allows the decomposition and aggregation of complex systems having as a basic subsystem a system of the linear type.

2.5 The state concept For understanding intuitively the concept of state, consider the system presented in Fig. 2.8. It consists in a continuous stirred tank reactor with a heating jacket. The endothermic reaction A + B → C is performed in the continuous operated reactor. The system delimitation is represented by the Σ interface. Terminal variables passing over the interface Σ are of input type: F1 , F2 , Fsteam , CAi , CBi and of output type: F3 , Fcondensate , CA , CB , CC . But there is still a set of suppressed variables that do not pass over the delimiting interface, that can be important for describing the system’s behavior; such suppressed variables may be considered: the inventory of reactants and products in the reactor, the level of condensed vapors in the jacket or the pressure in the reactor vessel. The knowledge of these variables may reveal aspects related to the internal behavior of the system. The internal behavior may not be always determined by tracing the inputs’ and outputs’ evolution. This simple example points out the fact that, in general, knowing the relationships between the input and output variables may not always contribute to the exhaustive and intimate description of the aspects related to its behavior. A fundamental limitation of the system description using input-output relationships is the fact that

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C

F1

F2

CAi

CBi

Fsteam Ʃ

Fcondensate F3 CA , CB , CC Fig. 2.8. Terminal and suppressed variables for a continuous stirred tank reactor.

the output may not be uniquely determined for known inputs; it is necessary to specify a set of supplementary information, related to the internal behavior of the system, in order to accomplish the desired uniqueness. It is obvious that revealing these internal aspects may contribute to better understanding the system behavior, knowledge that may be later exploited for improved operation, design and control.

The state of an abstract oriented system Consider an abstract, oriented and deterministic system characterized by a set of input-output relationships and described by equation (2.2). A fundamental true statement is the following: if the input u(τ ) is known for every time moment τ , τ ∈ (−∞, t], then the output y(τ ) may be also uniquely determined (known) on the time interval τ ∈ (−∞, t]. But if the input u(τ ) is known only for time interval τ ∈ (t0 , t], then the output may not be uniquely determined on the same limited time interval τ ∈ (t0 , t]. For a dynamic system having a time evolution depending on values of the input or output variables from different (past and present) time moments, it is obvious that a unique behavior may be determined only if the “history” of the system’s behavior is known. Such a unique (and complete) specification of the output evolution assumes knowledge of this “history” from time moments situated as far as possible (infinitely) in the past.

20 | Part I: Introduction The main property of the state of the system at the moment t = t0 is to separate the past t < t0 from the future t > t0 by specifying that information necessary at moment t = t0 in order to determine (for known future input) the output in a unique way during the future time span t > t0 . Definition. The state of an abstract oriented system is a set of real numbers representing the minimal information necessary to be known at the initial moment of time t = t0 that will determine, together with the given input u[t0,t] , a unique output y(t) for the future time span t > t0 . For the abstract, oriented and dynamic system described by differential equations of equation (2.8) form, such a set of real numbers may consist in the output y(t0 ) and the first (n − 1) derivatives of the output function evaluated at the same initial moment of time y(1) (t0 ), . . . , y(n−1) (t0 ). This set of values characterize in a unique way the state of the system at the time moment t = t0 . If the output y(t) and its first (n − 1) derivatives are considered for all time moments t, then a set of functions representing the state variables will be obtained; these variables have the property of completely characterizing the system for each moment of time during the time interval t > t0 . The state variable is usually denoted by a vector function x(t) having the components xi (t), i = 1, 2, . . . , n, as presented in the following: x(t) = [x1 (t), x2 (t), . . . , xn (t)]T .

(2.10)

The space generated by the values of the state variable x(t) is referred to as the state space of the system and has the generic notation of Σ . The state variable concept from systems theory is the same as that of the Lagrange, Lyapunov, Poincare or Gibbs generalized variables from physics and chemistry. The selection of the state variables is not unique. The way of choosing the state variables presented before is just a simple and intuitive possibility of specifying them. For one representation of the system using the input-output relationship it corresponds a set of possible input-state-output representations, while only one unique input-output representation corresponds to one input-state-output representation. In conclusion, there are two main ways of representing the system: one uses the input-output relationship (I) and the other the input-state-output relationship (II): I II

R(u, y) = 0, y = RS(x0 , u).

(2.11)

To determine the variables fulfilling the requirements of becoming state variables it is necessary that they satisfy a set of conditions named as consistency conditions.

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2.6 Input-state-output relationship The input-state-output relation may be formulated by: {

x(t) = φ (t; t0 , x 0 , u[t0 ,t] ), y(t) = η (t; x(t), u(t)).

(2.12)

The functional dependence φ is denoted as the transition function and its equation as the state equation. Arguments of the function φ reveal the dependence of the state x(t) on the time t, in an explicit manner; on the initial moment t0 , on the initial state x(t0 ) and on the inlet segment u[t0,t] , in an implicit manner. The functional dependence η is denoted as the output function and its equation as the output equation [1]. Two, possibly neighboring, time moments t and t0 are involved in the state equation. At the limit, when t → t0 , the state equation actually becomes a differential equation. The output equation is an instantaneous equation involving values of the state x(t) and input u(t) from the same moment of time and being therefore an algebraic equation. Reformulating the equations (2.12), a practical form of the input-stateoutput relations may be obtained: {

ẋ = f (x, u), y = g(x, u),

(2.13)

where f and g are two vector functions (f = [f1 f2 ⋅ ⋅ ⋅ fn ]T , g = [g1 g2 ⋅ ⋅ ⋅ gp ]T ), of nonlinear type in the general case. As a particular but very useful case, the form of the input-state-output relations for linear systems is described by {

ẋ = A ⋅ x + B ⋅ u y =C⋅x+D⋅u

(2.14)

where A, B, C, D are matrices of appropriate dimensions (constant for linear timeinvariant systems and time dependent for linear time-dependent systems). An abstract, oriented and dynamic single-input-single-output (SISO) system has an input-output description of the form presented in equation (2.8) and is represented by a high order differential equation. For the general case of a multiple-input-multiple output (MIMO) dynamic system, the description consists in a system of high order differential equations. The system has also an input-state-output description of the form presented in equation (2.13), represented by a system of first order differential equations and an algebraic system of equations. The two ways of reflecting the behavior of the system are equivalent from the point of view of the input-output relationship. It may be concluded that the way of describing the behavior of the system using input-state-output relations is more complete than the input-output description because it reveals internal aspects of system behavior by the means of the state equations. Regarded from the perspective of controlling the system, the description using the input-state-output relationship offers the possibility of getting a better understanding of the system and consequently being in the position of maintaining control

22 | Part I: Introduction over those variables describing the system behavior. Obviously, this is an important incentive compared to the case of controlling only the output variables. In general, from the practical point of view, not all state variables have a physical sense because for many cases the state variables may not be measured, affecting the possibility of directly controlling the states of the system.

2.7 Stability of the system Stability is a very important property of the system, being given special attention in systems theory. The stability concept probably has its origin in the investigation of the mechanical system behavior. Three types of mechanical equilibrium may be distinguished according to the way the solid body behaves after the application of a small (force) disturbance: (i) stable equilibrium, if the mechanical body returns to its initial position after the application of a small disturbance, (ii) unstable equilibrium, if the mechanical body tends to get a different position after the application of a small disturbance, (iii) neutral equilibrium, if the mechanical body remains in a position arbitrary far from to its initial position, after the application of a small disturbance. Stability may be often interpreted by investigating the change in the energy of the system. If the energy is decreasing, the system tends to be more stable; if it is increasing it becomes more unstable and if it remains constant the equilibrium is neutral. In a larger sense, a system is denoted stable if, when having “small” and bounded inputs and initial conditions, it gets bounded outputs. If a small and bounded input or initial condition determines an unlimited increasing output, the system is denoted unstable. This mode of expressing stability leads to the concept of Bounded-Input BoundedOutput Stability (BIBO stability). Definition. A time-continuous system is denoted as stable in the bounded-input bounded-output sense (BIBO stable) if its response y to any finite amplitude input u remains of finite amplitude. A typical example of chemical systems instability is that of the reactor in which an exothermic reaction is taking place and for which the heat flow produced by the chemical reaction is superior to the evacuated heat flow of the cooling agent. It is assumed that at the initial time moment there is a balance between the heat flow produced by the mass of reaction and the heat flow evacuated with the cooling agent. If a small rise of the reactant temperature appears (as a small disturbance) the temperature of the reaction mass increases, being followed by a rise of the reaction rate (according to the Arrhenius equation) which, in the conditions of the limited cooling capacity, de-

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termines the further increase of the reaction mass temperature. This positive reaction (avalanche) finally determines the unlimited (unbounded) rise of the temperature in the reactor. Certainly, in practice, the reactor temperature rise may not be unlimited as the system is deteriorated (in the extreme case the system vanishes) before the unbounded values (close to infinity) are attained.

2.8 Types of elementary signals The signal can be defined as a terminal variable carrying information. In a more general approach, the signal is a phenomenon representing information [9]. The mathematical support for the signal is the function. Usually, the independent variable of the signal function is time, the signal being denoted as a time-dependent signal. In systems theory, signals depending on frequency-independent variable are also studied. A group of deterministic signals, named elementary signals or standard signals are defined by the following properties [1]: (i) signals are represented by functions having all derivatives continuous with respect to time, possibly excepting a single value of time, (ii) the functions that represent the signals may be obtained one from another by successive derivation and integration operations. The main elementary functions used for describing the elementary signals in systems theory are: (1) The unit step function u0 (t). It presents a discontinuity in the origin and has amplitude equal to 1. It is defined by: u0 (t) = {

0 for t < 0 1 for t ≥ 0

(2.15)

and has the representation shown in Fig. 2.9. The step function having a certain a amplitude is represented by u(t) = a u0 (t). (2) The unit ramp function v0 (t). It presents the discontinuity in the origin. It is defined by 0 for t < 0 v0 (t) = { (2.16) t for t ≥ 0 and has the representation shown in Fig. 2.10. u0(t) 1 t 0

Fig. 2.9. Unit step elementary signal.

24 | Part I: Introduction v0 (t)

t

0

Fig. 2.10. Unit ramp elementary signal.

The ramp function having a certain a slope is represented by v(t) = a v0 (t). It may be noticed that the unit step function may be obtained by the derivation of the unit ramp function and the unit ramp function may be obtained by the integration of the unit step function: u0 (t) =

t

dv0 (t) , dt

v0 (t) = ∫ u0 (τ )dτ .

(2.17)

−∞

(3) Other elementary functions may be defined by the general function form v(t) =

tn−1 u (t), (n − 1)! 0

n ≥ 1, n ∈ N.

(2.18)

(4) The unit impulse function (Dirac impulse) δ (t). The class of elementary functions may be completed with the unit impulse function, also named the Dirac function. For the definition of this function consider the v(t) function, shown in Fig. 2.11. v(t) 1 0

t

ε

Fig. 2.11. Signal leading to the unit step function u0 (t) when ε → 0.

When passing to the limit ε →0, the function v(t) gets identical to the unit step function u0 (t): lim v(t) = u0 (t), ε ≠ 0. (2.19) ε →0

By the differentiation of the v(t) function, with ε =0, ̸ the δ0 (t) function may be obtained: dv(t) δ0 (t) = (2.20) . dt Its representation is shown in Fig. 2.12. The integral of the δ0 (t) function (area under the function, proportional to the signal power) is equal to unity. It may be noticed that as ε →0 the v(t) function approaches the unit step function and the δ0 (t) function approaches the form of an impulse with an infinitely small

2 Basics of systems theory

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δ0 (t) 1/Ɛ 0

Fig. 2.12. Signal δ0 (t) having the form of a unit impulse with finite duration and amplitude.

t

Ɛ

duration (approaching zero) and an infinite amplitude, but having a finite and equal to unity area. The function obtained by this process of approaching the limit ε →0 is the unit impulse δ (t), denoted by δ (t) = lim δ0 (t)

(2.21)

ε →0

and having the representation shown in Fig. 2.13.

δ(t)

0

Fig. 2.13. The elementary unit impulse δ (t) also named the Dirac impulse.

t

The equation describing the unit impulse is δ (t) = {

0 ∞

for t ≠ 0 for t = 0.

(2.22)

Function δ (t) owns the following important property (t > 0): t

∫ δ (τ )dτ = 1 = u0 (t)

(2.23)

−∞

that justifies the consideration of the unit impulse function to be obtained by differentiating the unit step function δ (t) =

du0 (t) . dt

(2.24)

The Dirac function δ (t) may be understood as a generalized function or as a distribution. This function completes the class of the elementary functions. It features a set of properties having sense only under integration operation, such as ∞

t0 +ε

t0 +ε

∫ f (t) ⋅ δ (t − t0 )dt = ∫ f (t) ⋅ δ (t − t0 )dt = f (t0 ) ∫ δ (t − t0 )dt = f (t0 ). −∞

t0 −ε

t0 −ε

(2.25)

26 | Part I: Introduction The results of the (2.25) expression reveal the sampling property of the shifted unit impulse function δ (t − t0 ) consisting in extracting the value f (t0 ) of the function f (t) at the moment t0 when the unit impulse is applied. (5) The group of elementary signals may be extended with the sinusoidal signal, defined by u(t) = A sin(ω t), (2.26) where A is the amplitude and ω is the (angular) frequency. Its representation is shown in Fig. 2.14. u(t)=Asin(ωt)

A

0

t

T=2π/ω

Fig. 2.14. The sinusoidal signal.

The presented signals are used for describing and defining the behavior of the systems. In order to compare the behavior of two or more systems it is necessary to subject them to the same type of input signals and then to investigate their response that will reveal their dynamic and steady state characteristics. Elementary signals are simple but intuitive and represent a benchmark for studying the response of different systems for the same testing conditions.

2.9 LTI systems described by input-output relationships Definition: A system described by the input-output relationship is represented by the set of pairs of input and output functions (u(t), y(t)) satisfying the differential equation of the form f (y(t),

dy(t) du(t) dn y(t) dm u(t) , u(t), , t) = 0, ,..., , . . . , dt dtn dt dtm

(2.27)

for t ∈ T, T = [t0 , ∞], with n and m positive integer values and with f an application, f : Cn+m+2 × T → C.

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2.10 Time response of the linear time-invariant systems Linear time-invariant (LTI) systems are described by linear differential equations having constant coefficients. The general form of the differential equation is [14]: qn

dn y(t) dn−1 y(t) dy(t) + q + ⋅ ⋅ ⋅ + q1 + q0 ⋅ y(t) n−1 dtn dt dtn−1 m d u(t) dm−1 u(t) du(t) = pm + p + ⋅ ⋅ ⋅ + p1 + p0 ⋅ u(t), m−1 m m−1 dt dt dt

(2.28)

in which the qn and pm are nonzero coefficients. The domain of definition of the function u(t) and y(t) is the half-infinite T+ = [t0 , ∞) or infinite T∞ = R axis. The halfinfinite axis T+ is considered to include the moment t0− , leading to the interval T+ = [t0 −ε , ∞) with ε → 0. This approach allows the inclusion of the Dirac function δ (t−t0 ) from the time moment t0 . For each of the two members of the equation (2.28) a characteristic polynomial may be associated, further denoted with Q and P: Q(λ ) = q0 + q1 ⋅ λ + ⋅ ⋅ ⋅ + qn ⋅ λ n m

P(λ ) = p0 + p1 ⋅ λ + ⋅ ⋅ ⋅ + pm ⋅ λ .

(2.29) (2.30)

The roots of these polynomials play an important role in determining the solution of the differential equation (2.28), for a given input u(t). This solution is usually named the response of the system to the input u(t). Finding the response of the linear time-invariant system consists in determining the unique output y(t) corresponding to a given input u(t). First, this finding determines all outputs y(t) corresponding to a given input u(t), followed by the selection of one unique output on the basis of the initial conditions.

2.11 Solution of the homogeneous differential equation The homogeneous differential equation corresponding to the differential equation (2.28) is dn y(t) dn−1 y(t) dy(t) qn + qn−1 + ⋅ ⋅ ⋅ + q1 (2.31) + q0 ⋅ y(t) = 0. n dt dt dtn−1 This differential equation describes the behavior of the system in case of the input u(t) = 0. The solution of the homogeneous differential equation is related to determining the roots of the characteristic polynomial Q, denoted as the characteristic roots. Mathematics denotes as basic solutions (or as linear independent integrals) the set of the n linear independent solutions y1 , y2 , . . . , yn of the homogeneous differential equation (2.31). It can be demonstrated that any solution of the homogeneous differential equation is a linear combination of the basic solutions. It is therefore important to

28 | Part I: Introduction find a way for determining the basic solutions. One way of choosing a set of n basic solutions is defining the set of mi basic solutions, for each root λ of order of multiplicity mi , as mi yi = ti ⋅ eλ t , i = 0, 1, . . . , mi − 1; ∑ mk = n, t ∈ T+ or T∞ . (2.32) k=1

This set of basic solutions is not unique but is it determined in a convenient way. Taking into account that the solution of the homogeneous differential equation is a linear combination of the basic solutions, the set of all response functions to the zero input Yhomog is: Yhomog = {y|y = ∑ αi ⋅ yi , αi ∈ C}.

(2.33)

i

In general, the roots of the characteristic polynomial Q may also have complex roots, even if the coefficients qi are real. For this case, the roots are always complex conjugated pairs. Nevertheless, pairs of basic solutions having only real values may also be determined. If the roots of the characteristic polynomial are complex conjugated pairs, λ and λ ∗ , having the multiplicity order m, the basic solution contains 2m real linear independent integrals of the form yi = ti ⋅ eσ ⋅t ⋅ cos(ω ⋅ t), yi = ti ⋅ eσ ⋅t ⋅ sin(ω ⋅ t),

i = 0, 1, . . . , m − 1 i = 0, 1, . . . , m − 1,

(2.34)

where σ = Re(λ ) and ω = Im(λ ) are the real and imaginary parts of the root λ . It may be noticed that, in this case, the solution of the homogeneous differential equation is oscillating (harmonic), having decaying amplitude if σ = Re(λ ) is negative.

2.12 Particular solutions of the nonhomogeneous differential equation A particular solution of the differential equation (2.28) consists in any output function ypart verifying the differential equation for a given input u(t). It is often possible to guess the form of a particular solution. It is the case of the input functions having the form of a constant or of an exponential, both being particular solutions of the same type. For the case of feasible systems, the particular solution may be also computed by the convolution between the impulse response and the given input.

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2.13 General solution of the nonhomogeneous differential equation The general solution y(t) of the nonhomogeneous differential equation (2.28) is the sum of the solution of the homogeneous differential equation yhomog and a particular solution of the nonhomogeneous differential equation ypart : y(t) = ygen = yhomog + ypart .

(2.35)

It may be noticed that the general solution of the nonhomogeneous differential equation consists in a set of functions Ygen having as parameters the arbitrary constants αi : Ygen = {y | y = ∑ αi ⋅ yi + ypart , αi ∈ C}.

(2.36)

i

As previously mentioned, a unique solution y(t) of the equation (2.28), for a given input u(t), may only be obtained if the initial conditions are also specified. If the following initial conditions are given: y(t0 ) = y0 ,

y(1) (t0 ) = y1 , . . . , y(n−1) (t0 ) = yn−1 ,

(2.37)

the n arbitrary constants αi can be determined by solving a system of algebraic equations for the αi unknown.

2.14 Stability of the system described by input-output relationships Stability will be investigated in the sense of bounded-input bounded-output stability (BIBO stability). Consider the linear time-invariant system described by the differential equation (2.28). Definition. The roots of the characteristic polynomial Q are denoted as the characteristic roots of the system. The P󸀠 and Q󸀠 polynomials are the polynomials obtained by the simplification of the polynomials P and Q with the greatest common divisor. Definition. The roots of the Q󸀠 polynomial are denoted as the poles of the system and the roots of the P󸀠 polynomial as the zeros of the system. Due to the fact that the LTI system is completely characterized by its impulse response h(t), it is expected that the investigation of the BIBO stability to be performed on the basis of this response.

30 | Part I: Introduction Definition. The system described by a linear and time-invariant differential equation (LTI system) is BIBO stable if and only if its impulse response has a finite action (value). A stronger form of stability than the BIBO stability is the converging-input converging-output (CICO) stability. The condition for a system to be CICO stable is that for a bounded input approaching zero when time approaches infinity, i.e. u(t) → 0 when t → ∞, the corresponding output should also be bounded and should approach zero when time approaches infinity, i.e. y(t) → 0 when t → ∞.

2.15 Stability of systems described by linear time-invariant differential equations Taking into account that the response of the system described by the equation (2.28) is y = h ∗ u + ∑ αi ⋅ yi

(2.38)

i

and that BIBO stability implies that h is finite, it follows that for every pair of finite h and finite u, the first term in equation (2.38) is finite. The symbol “∗” denotes the convolution. Likewise, finite h implies the existence of poles having negative real values. Bounding conditions remain to be specified for the second term in equation (2.38), the part of the solution corresponding to the homogeneous differential equation. From this part, the conditions to be specified refer to those characteristic roots that are not poles. Definition. The necessary and sufficient conditions for the system, described by the LTI differential equation (2.28), to be BIBO stable are: (i) the degree of the P polynomial is less than or equal to the degree of the Q polynomial, (ii) the poles of the system have strictly negative real part, (iii) the system does not have any characteristic roots with strictly positive real part that cancel with zeros, (iv) every characteristic root cancelling with a zero and having a zero real part is only of first order of multiplicity. Definition. The necessary and sufficient conditions for CICO stability of the system described by the linear time-invariant differential equation (2.28) are (i) the degree of the P polynomial is less than or equal to the degree of the Q polynomial, (ii) the characteristic roots of the Q polynomial have strictly negative real part. CICO stability implies BIBO stability but the reversed implication in not true.

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2.16 Frequency response of the system described by input-output relationships In systems theory, a special role is played by the description of the way that the system responds to periodic input functions of the sine form or, more generally, to a harmonic function having complex values. The reason for this interest is related to the fact that input signals can be decomposed in finite or infinite linear combinations (series) of harmonic functions. Knowing the response of the system to the harmonic input and taking into account the linearity property, the response of the system to an arbitrary input may be determined as a linear combination of harmonic response functions. This description of the system is named the frequency response and reveals quantitative and qualitative aspects of both steady state and dynamic behavior. Definition. The harmonic signal is represented by a function having the time t as the independent variable and the real value ω , i.e. the frequency, as a parameter, and has values in the set of the complex numbers: x(t) = a ⋅ ejω t .

(2.39)

The constant a may have complex values in the general case and is denoted as the complex amplitude. If represented in the polar form, the complex constant a has the form a = υ ⋅ ejφ , where υ = |a| is the amplitude or module of the constant a and φ is the argument of the constant a. The complex harmonic signal has the form: x(t) = a ⋅ ejω t = υ ⋅ ejφ ⋅ ejω t = υ ⋅ cos(ω t + φ ) + j ⋅ υ ⋅ sin(ω t + φ ).

(2.40)

The real harmonic signal has the form c(t) = Re[x(t)] = υ ⋅ cos(ω t + φ ) = υ ⋅ sin (ω t + φ +

π ). 2

(2.41)

The frequency f is related to the (angular) frequency ω by the well-known relationship ω = 2 ⋅ π ⋅ f . Because the angular frequency ω is usually used (and not directly the frequency f ), the term frequency usually stands for the angular frequency. Consider the harmonic input signal with unity amplitude: η (t) = ejω t ,

t ∈ R.

(2.42)

The response of the LTI system to the harmonic input u = η (t), presented in equation (2.42), may be determined by convolution: +∞

+∞

y(t) = ∫ h(t − τ ) ⋅ u(τ )dτ = ∫ h(t − τ ) ⋅ ejω τ dτ , −∞

−∞

t ∈ R,

(2.43)

32 | Part I: Introduction provided that the integral exists. Making the variable change t − τ = θ , the following form for the output is obtained: +∞

+∞

y(t) = ∫ h(θ ) ⋅ e −∞

= hω (ω ) ⋅ e

jω (t−θ )

jω t

,

dθ = ( ∫ h(θ ) ⋅ e−jω θ dθ ) ⋅ ejω t

t ∈ R.

−∞

(2.44)

The notation hω (ω ) has been used for the function +∞

hω (ω ) = ∫ h(θ ) ⋅ ejω θ dθ ,

ω ∈ R.

(2.45)

−∞

This function is named the frequency response function of the system. It is a complex function. Comparing equation (2.42) and equation (2.44) it may be noticed that the system response to a harmonic input is an output signal of the harmonic form, too: y = hω (ω ) ⋅ η (t).

(2.46)

This is one of the remarkable properties of linear systems (property valid also for a =1). ̸ The way the harmonic response signal y changes its amplitude and argument, compared to the amplitude and argument of the harmonic input signal η , is specified by the frequency response function, namely by the amplitude and argument of the complex function hω (ω ). The following real harmonic input signal is considered: u(t) = υ ⋅ cos(ω t + φ ) = Re[a ⋅ ejω t ].

(2.47)

The response signal to the previously mentioned input signal u(t) has the form y(t) = υω ⋅ cos(ω t + φω ) = Re[aω ⋅ ejω t ],

(2.48)

where aω , υω and Φω have the explicit form aω = hω (ω ) ⋅ υ υω = |hω (ω )| ⋅ υ φω = φ + arg(hω (ω )).

(2.49)

The way the harmonic response signal y changes its amplitude and argument (phase) compared to the harmonic input signal η is specified by the frequency response function, namely by the module (amplitude) |hω (ω )| and argument (phase) of the complex function hω (ω ), as given in equations (2.49).

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2.17 Frequency response of the system initially at equilibrium The frequency response function of a system described by a differential equation may be directly obtained from the polynomials Q and P, where the coefficients of the differential equation are included. Knowing the frequency response function (shortly named frequency response), the response of the system to any input signal may be determined since the latter can be decomposed in a linear combination (series) of harmonic functions. Definition. The frequency response function hω (ω ) of an initially at equilibrium system (initially at rest or at steady state), described by the equation (2.28), exists if and only if all the poles of the system have strictly negative real part. The frequency response function hω (ω ) has the form hω (ω ) =

P(jω ) , Q(jω )

ω ∈ R.

(2.50)

Existence of the frequency response function is conditioned by the bounded or notbounded property of the expression under the integral defining it, as presented in equation (2.45). This request may be reduced to the condition that the impulse response h of the initially at rest system should have the basic solutions approaching zero when time approaches infinity. This condition may happen only when the poles of the system have negative real part: the situation when the basic solutions contain terms of the form tk ⋅ eλ ⋅ (with λ pole of the system). All these terms have exponential time decay for poles with negative real part. The form of the frequency response function given in equation (2.50) is obtained taking into account that the system’s response to a harmonic input u(t) = ej⋅ω ⋅t is also of harmonic form y(t) = hω (ω ) ⋅ ej⋅ω ⋅t . The pair of functions (u(t), y(t)) must satisfy the differential equation. Replacing them in the equation (2.28) and taking into account that the n-th derivative of the exponential function results in the multiplication of the function with the factor (jω ⋅ t)n , the equation (2.28) becomes Q(jω ) ⋅ hω (ω ) ⋅ ejω t = P(jω ) ⋅ ejω t ,

t ∈ R.

(2.51)

Thus, the form of the frequency response function given in equation (2.50) is straightforward.

2.18 Steady state and transient response to the harmonic input The frequency response hω (ω ) of the CICO stable LTI system, to a complex harmonic input u(t) = a0 ⋅ ejω t , t ∈ T, (2.52)

34 | Part I: Introduction or to a real harmonic input u(t) = a0 ⋅ cos(ω t + φ ),

t ∈ T,

(2.53)

where T is the infinite or half-infinite positive time axis, has the form y = ysteadystate + ytransient .

(2.54)

The steady state component is the response of the system in the (quasi) steady state regime, i.e. the situation when all input variables have transmitted their transient effects on the output variables. This regime is usually accomplished after a sufficiently long time interval (in fact when time approaches infinity). The steady state component of the response ysteadystate is given by ysteadystate (t) = hω (ω ) ⋅ a0 ⋅ ejω t ,

t ∈ T,

(2.55)

for the complex harmonic input, and by ysteadystate (t) = |hω (ω )| ⋅ a0 ⋅ cos(ω t + φ + arg(hω (ω ))),

t ∈ T,

(2.56)

for the real harmonic input. The transient component is the response of the system in transient regime, i.e. during the period of time immediately following the application of the input variable change. This is the period of time when the output variable has not yet suffered the entire effect produced by the input variable (the cause). The transient component of the response ytransient is the solution of the homogeneous differential equation, presenting the property ytransient (t) → 0 for

t → ∞.

(2.57)

As may be noticed, the frequency response of the system refers to the (quasi) steady state component of the response to a harmonic input.

2.19 LTI systems described by input-state-output relationships The description of the system’s behavior using input-state-output relations reveals not only the external behavior of the system (directly described by the input-output relationships) but also the internal behavior, which is the internal mechanism governing it. It is clear that the uncovering of these internal aspects contributes to a better comprehension of the system with direct benefits on system analysis, synthesis and control [10–13]. Following the theoretical elements related to the state concept, it was concluded that the basic property of the state at time moment t = t0 is to separate the past t < t0 and the future t > t0 by specifying the information necessary at moment t = t0 ,

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35

x(t0 ), in order to univocally determine the output over the future time t > t0 . The set of numerical values containing this information, for each time moment t, generates a vector function x(t) denoted as the state variable. The components of the state vector generate the state space Σ . Therefore, it is only necessary to know the state of the system x(t) (at a certain time moment t) and the given input u(t) (for t ≥ t0 ), in order to univocally determine the output y(t) for the time interval t > t0 .

2.20 Transformation of the input-output representation into the input-state-output representation Consider the linear time-invariant system described by the differential equation (2.28) where, without losing generality, qn = 1 has been assumed. For a representation of the system using an input-output relationship it may be assigned multiple input-state-output representations, but a single input-state-output representation has only a single (unique) associated input-output representation. From the set of the input-state-output representations of the same system, of first importance are those representations leading to particular (simple) forms of the matrices A, B, C, and D [1]. The way of choosing the state variables of a system is not unique. A simple but intuitive way of choosing the state variables is to consider as state variables the output y(t) and the first (n − 1) derivatives of the output. This way of choosing the states is suggested by the fact that when setting (specifying) the initial conditions y(t0 ), y󸀠 (t0 ), y󸀠󸀠 (t0 ), y(3) (t0 ), . . . , y(n−1) (t0 ) for a differential equation this implies depicting the minimal information necessary for uniquely determining the output for the future time moments t > t0 , when the input u(t) is given. This way of choosing the state variables, named phase (state) variables of canonical form, is leading to the following form of the input-state-output equations: ẋ1 = x2 ẋ2 = x3 .. . ẋn−1 = xn ẋn = −q0 ⋅ x1 − q1 ⋅ x2 − ⋅ ⋅ ⋅ − qn−2 ⋅ xn−1 − qn−1 ⋅ xn + u

(2.58)

y = (p0 − pn ⋅ q0 ) ⋅ x1 + (p1 − pn ⋅ q1 ) ⋅ x2 + (p2 − pn ⋅ q2 ) ⋅ x3 + ⋅ ⋅ ⋅ + (pn−1 − pn ⋅ qn−1 ) ⋅ xn + pn ⋅ u.

(2.59)

36 | Part I: Introduction These equations are equivalent to the set of linear first order differential equations: ẋ1 0 [ ẋ ] [ 0 [ 2 ] [ [ ] [ [ . ] [ . [ ] [ [ . ]=[ . [ ] [ [ . ] [ . [ ] [ [ ] [ [ ẋn−1 ] [ 0 [ ẋn ] [ −q0

1 0 . . . 0 −q1

0 1 . . . 0 −q2

. . . . . . .

. . . . . . .

. 0 x1 0 [ x ] [0] . 0 ] ] [ 2 ] [ ] ] [ ] [ ] . . ] [ . ] [.] ] [ ] [ ] [ ] [ ] . . ] ]⋅[ . ]+[ . ]⋅u ] [ [ ] . . ] [ . ] ] [.] ] [ ] [ ] . 1 ] [ xn−1 ] [ 0 ] . −qn−1 ] [ xn ] [ 1 ]

(2.60)

and an algebraic equation x1 ] [ [ x2 ] ] [ ] [ y = [(p0 −pn ⋅q0 ) (p1 −pn ⋅q1 ) . . . ( pn−2 −pn ⋅qn−2 ) (pn−1 −pn ⋅qn−1 )] [ ... ] +pn ⋅u, (2.61) ] [ [x ] [ n−1 ] [ xn ] where the matrices A, B, C, and D have the form 0 0 . . . 0 −q [ 0

[ [ [ [ [ A=[ [ [ [ [ [

1 0 . . . 0 −q1

0 1 . . . 0 −q2

. . . . . . .

. . . . . . .

. 0 . 0 ] ] ] . . ] ] . . ] ], . . ] ] ] . 1 ] . −qn−1 ]

0 [0] [ ] [ ] [.] [ ] ] B=[ [ . ], [.] [ ] [ ] [0] [1]

(2.62)

C = [(p0 − pn ⋅ q0 ) (p1 − pn ⋅ q1 )...(pn−2 − pn ⋅ qn−2 ) (pn−1 − pn ⋅ qn−1 )], D = pn . This form of representing the system, using input-state-output relations, is denoted as the standard canonical form. From the previously presented way of describing the system it has been shown that an LTI system has an input-output description of the form presented in equation (2.28) (i.e. an equation, or more general, a system of high order differential equations) but also has an input-state-output description of the form (2.14), (i.e. a set of first order state differential equations and a set of output algebraic equations). The two ways of representing the behavior of the system are equivalent from the input-output point of view. Compared to the input-output representation, the description based on the first order differential equation is possible by introducing a new variable that makes a connection between the input and output variables, i.e. the state variable. The direct consequence of this equivalence is the fact that any set of high order input-output differential equations may be transformed into a set of first order state

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differential equations associated with a set of output algebraic equations, on the basis of an appropriate definition of the state variables. The most appreciated and efficient representation form of the state equations, especially for the control system design, is the so called modal form where the matrix A has a pure diagonal form Λ (with nonzero values only on the main diagonal of matrix A). This way of choosing the state variables is only possible when the eigenvalues of the matrix A are of zero order of multiplicity. The linear transformation of a certain input-state-output representation to the modal one may be performed on the basis of the transformation matrix consisting in the eigenvectors of matrix A. The modal inputstate-space representation implies states that are decoupled (each state only depends on itself and on the input). When the eigenvalues of the matrix A are of nonzero order of multiplicity it is only possible to obtain a quasi-diagonal form of the matrix A, denoted as the Jordan form.

2.21 Solutions of the state equations Consider the general case of a nonlinear system described by the set of state equations: ̇ = f (t, x(t), u(t)), x(t)

(2.63)

y(t) = g(t, x(t), u(t)),

(2.64)

where the vector state variable belongs to an n-dimensional space of real or complex numbers Σ ∈ Rn or Σ ∈ Cn and the input and the output functions are also defined on real or complex spaces, Su ∈ Rk or Su ∈ Ck , respectively Sy ∈ Rm or Sy ∈ Cm . The existence of the solutions for the set of the state and output equations (2.63) and (2.64) may be guaranteed only for difficult formulated mathematical conditions and explicit solutions may be determined just for a few particular cases. For the linear systems described by the input-state-output relations ̇ = A(t) ⋅ x(t) + B ⋅ u(t), x(t)

(2.65)

y(t) = C(t) ⋅ x(t) + D ⋅ u(t),

(2.66)

with t ∈ T (where T = R or T = R+ ), it is possible to obtain explicit solutions. In the following it is considered that the input variables (vector of k dimension), the output variables (vector of m dimension) and the state variables (vector of n dimension) may have complex values, Su ∈ Ck , Σ = Cn , Sy ∈ Cm . Matrix A is time dependent and it is of n × n dimension, matrix B is time dependent and it is of n × k dimension, matrix C is time dependent and it is of m × n dimension and matrix D is also time dependent and it is of m × k dimension. Existence of the solutions for the state and output equations (2.65) and (2.66) is guaranteed if the matrices A(t), B(t), C(t), and D(t) are continuous and bounded functions.

38 | Part I: Introduction

2.22 Solution of the nonhomogeneous state equation The solution of the nonhomogeneous state equation for the nonzero input u(t): ̇ = A(t) ⋅ x(t) + B(t) ⋅ u(t), x(t)

(2.67)

with t ∈ T, is of the following form: t

x(t) = Φ (t, t0 ) ⋅ x(t0 ) + ∫ Φ (t, τ ) ⋅ B(τ ) ⋅ u(τ ) ⋅ dτ

(2.68)

t0

for every t and t0 from T. The vector function Φ (t, t0 ) is the transition matrix. Demonstration of the steps for obtaining this solution may be performed using the method of constants variation. If the solution of the state nonhomogeneous differential equation is known, the system output function y(t) may be determined by replacing the state expression from equation (2.68) into the output equation (2.66): t

y(t) = C(t) ⋅ Φ (t, t0 ) ⋅ x(t0 ) + ∫ C(t) ⋅ Φ (t, τ ) ⋅ B(τ ) ⋅ u(τ ) ⋅ dτ + D(t) ⋅ u(t).

(2.69)

t0

It may be noticed that, in general, the response of a linear system y(t) is a summation of two components y(t) = yzero-input + yzero-state , having the form yzero-input (t) = C(t) ⋅ Φ (t, t0 ) ⋅ x(t0 ),

(2.70)

t

yzero-state (t) = ∫ C(t) ⋅ Φ (t, τ ) ⋅ B(τ ) ⋅ u(τ ) ⋅ dτ + D(t) ⋅ u(t).

(2.71)

t0

The term yzero-input is named zero-input response and represents the response of the system having the input equal to zero but nonzero initial conditions. The term yzero-state is named zero-state response and represents the response of the system having the initial conditions equal to zero but nonzero input. This summation is a consequence of the linear character of the system that possesses the effect superposition property. Taking into account the form of the transition matrix, for the linear time-invariant systems the two components have the following simplified form: yzero-input (t) = C ⋅ eA⋅t x(0),

t0 = 0,

(2.72)

t

yzero-state (t) = ∫ C ⋅ eA(t−τ ) ⋅ B ⋅ u(τ ) ⋅ dτ + D ⋅ u(t).

(2.73)

0

According to the convolution operation, the response to the zero initial conditions (zero-state) may be formulated as t

yzero-state (t) = ∫ h(t − τ ) ⋅ u(τ )dτ , t0

(2.74)

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where h(t) is denoted as the impulse response matrix of the system. For the case of the LTI systems the impulse response matrix has the form h(t) = C ⋅ eAt ⋅ B ⋅ u0 (t) + D ⋅ δ (t).

(2.75)

For the case when both the input u(t) = u(t) and the output y(t) = y(t) are scalar functions (m = k = 1), the impulse response is also a scalar function h(t) = h(t), representing the response of the system with zero initial conditions to a Dirac input function δ (t). For the MIMO case (m, k > 1), the (i, j) element of the impulse response matrix hij is the response of the i-th component of the system output having zero initial conditions to the j-th component of the input function, considered as a Dirac function, and considering equal to zero all the other components of the input vector.

2.23 Laplace transform In the previous paragraphs the way of determining the time response of the systems described by input-output or input-state-output relationships, generally called the system behavior (analysis) in the time domain, has been presented. The methods for determining the response for the linear time-invariant systems are based on the convolution operation. Unfortunately convolution does not allow a direct evaluation of the system behavior. Additionally, the implied operation of integration is a relatively complicated and resource-consuming operation even for numerical computation. These considerations led to the development of other methods for investigating and describing the system’s behavior, based on the frequency response function, methods that are usually named the system analysis in the frequency domain. Frequency domain analysis is intuitive with respect to reflecting the system behavior and it is used for control system design [14]. The core of this analysis is based on two fundamental aspects: 1. every time-dependent signal may be decomposed in a linear combination of harmonic (complex) functions having increasing frequencies (Fourier series); 2. the response of the linear time-invariant system, to a harmonic (complex) input, is equal to that input multiplied by a factor that is described by the frequency response function. Consequently, the steps for determining the response of the linear time-invariant system are: 1. the input function is decomposed in a linear combination of harmonic functions having different frequencies; 2. the individual response for each harmonic function of the linear combination is determined (being also a harmonic combination), based on the principle of the cause-effect proportionality;

40 | Part I: Introduction 3.

the individual responses are added for obtaining the total response, the operation of summation being based on the effect superposition property characteristic for linear systems.

2.24 Definition of the one-sided and two-sided Laplace transform Although the one-sided Laplace transform is predominantly used for linear timeinvariant systems description, the two-sided Laplace transform will be introduced first in order to get a comprehensive overview of this mathematical instrument [14]: Definition. The two-sided Laplace transform is the application L that transforms the continuous signal x(t), defined on the entire time axis t ∈ R, into a complex function X(s) = L(x(t)) having the complex independent variable s, and being defined by the integral ∞

X(s) = ∫ x(t) ⋅ e−st dt,

s ∈ E ⊂ C.

(2.76)

−∞

The set E ⊂ C, is named the region (domain) of existence of the two-sided Laplace transform X(s) and consists in all complex values of the variable s for which the integral is convergent. It is usual to denote the Laplace transform of the signal x(t) by the capital character X(s) corresponding to the character used to denote the time signal x(t). X(s) is also named the image of the function x(t) and x(t) is named the original of the function X(s). Unlike the Fourier transform that may not be applied for a signal having exponential growth, the Laplace transform allows this application due to the exponential function included in the integral defining it. Definition. The one-sided Laplace transform is the application L+ that transforms the continuous signal x(t), defined on t ∈ R+ , into a complex function X+ (s) = L+ (x(t)) having the complex independent variable s, and being defined by the integral: ∞

X+ (s) = ∫ x(t) ⋅ e−st dt,

s ∈ E+ .

(2.77)

0

The set E+ ⊂ C, is named the region (domain) of existence of the one-sided Laplace transform X+ (s) and consists in all complex values of the variable s for which the integral is convergent [1, 14]. It is usual to denote the one-sided Laplace transform of the signal x(t) by the capital character X+ (s) corresponding to the character used to denote the time signal x(t). If no confusion is possible (the usual case), the subscript “+” may be omitted. X+ (s) is also named the image of the function x(t) and x(t) is named the original of the function

2 Basics of systems theory

| 41

X+ (s). There is a close relationship between the one-sided and the two sided Laplace transform, which may be formulated as L+ (x) = L(x ⋅ u0 (t)), where u0 (t) is the unit step elementary function. In particular, if x(t) is a signal defined such that x(t) = {

0 ≠ 0

for for

t 0}

Ex+

{s ∈ C|s − a ∈ Ex+ }

Ex+

e−s θ ⋅ X+ (s)

x(0+ ) = lim sX+ (s)

x(α t), α ∈ R, α ≠ 0 x(α t), α ∈ R, α > 0

Scaling

dX(s) ds

{s ∈ Ex |Re(s) > 0}

Ex

{s ∈ C|s − a ∈ Ex }

Ex+ ∩ Ey+

Ex+ ∩ Ey+

Region of existence E+

X(s)+ ⋅ Y(s)+

α X+ (s) + β Y+ (s)

One-sided Laplace transform s ∈ E+

Initial value theorem

−t ⋅ x(t)

s-differentiation

0−

∫ x(τ )dτ

−∞ t

)

s ⋅ X(s)

dx(t) dt X(s) s

X(s − a)

eat ⋅ x(t), a ∈ C

s-shifting

Integration

Ex Ex+

es θ ⋅ X(s) e−s θ ⋅ X+ (s)

x(t + θ ), θ ∈ R x(t − θ ) ⋅ u0 (t − θ ), θ ∈ R, θ ≥ 0

Time shifting

t

Ex ∩ Ey Ex+ ∩ Ey+

X(s) ⋅ Y(s) X(s)+ ⋅ Y(s)+

(x ∗ y)(t) x ⋅ u0 (t) ∗ y ⋅ u0 (t)

Convolution

Ex ∩ Ey

α X(s) + β Y(s)

(α x + β y)(t), α, β ∈ C

Linearity

Region of existence E

Two-sided Laplace transform s∈E

Time signal t∈R

Property

Table 2.1. Properties of the Laplace transform.

42 | Part I: Introduction

2 Basics of systems theory

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43

Table 2.2. Laplace transform of usual functions. Time signal t∈R

Two-sided Laplace transform

One-sided Laplace transform

Conditions

eat ⋅ u0 (t) −eat ⋅ u0 (−t) eat

1 s−a

Re(s) > Re(a)

1 s−a

Re(s) > Re(a)

1 s−a

Re(s) < Re(a)

0

a∈C

— —

—

1 s−a 1 s 1 s

Re(s) > Re(a)

a∈C a∈C a∈C

1 u0 (t) δ (t) δ (k) (t)

— Re(s) > 0 s∈C s∈C Re(s) > 0

1 s

1 sk

1 sk

Re(s) > 0 Re(s) > 0 s∈C s∈C Re(s) > 0

k∈N k∈N

tk−1 u (t) (k−1)! 0

1 sk

tk−1 ⋅eat (k−1)!

1 (s−a)k

Re(s) > Re(a)

1 (s−a)k

Re(s) > Re(a)

a ∈ C, k ∈ N

⋅e u0 (−t) − t(k−1)!

1 (s−a)k

Re(s) < Re(a)

0

s∈C

a ∈ C, k ∈ N

cos(ω t) ⋅ u0 (t) sin(ω t) ⋅ u0 (t)

s (s2 +ω 2 ) ω (s2 +ω 2 )

eat ⋅ cos(ω t) ⋅ u0 (t)

s−a (s−a)2 +ω 2

u0 (t)

k−1 at

at

e ⋅ sin(ω t) ⋅ u0 (t)

ω (s−a)2 +ω 2

1 sk

s (s2 +ω 2 ) ω (s2 +ω 2 )

Re(s) > 0 Re(s) > 0 Re(s) > a

s−a (s−a)2 +ω 2

Re(s) > a

ω (s−a)2 +ω 2

Re(s) > 0 Re(s) > 0

ω ∈R ω ∈R

Re(s) > a

a, ω ∈ R

Re(s) > a

a, ω ∈ R

(A) Definition. Consider X, the two-sided Laplace transform of the signal x, in the existence region E. The signal x(t) may be recovered from X(s) using the inverse Laplace transform x(t) = L−1 (X(s)) defined by σ +j∞

1 x(t) = ∫ X(s) ⋅ est ds, 2⋅π ⋅j

t ∈ R,

(2.79)

σ −j∞

where the integral is computed along a vertical line completely included in the existence region E and having the σ abscissa. (B) Definition. Application of the inverse Laplace transform to the one-sided Laplace image X+ (of the original x(t)) leads to the signal u0 (t) ⋅ x(t) defined by L−1 (X+ (s)) = {

x(t) 0

for for

t≥0 , t max Re(λi )},

(2.99)

where λi are the eigenvalues of the matrix A. The presented considerations represent the simple method for determining the transition matrix Φ (t, t0 ).

2.31 The transfer matrix The transfer matrix is the MIMO case generalization of the SISO case transfer function [1]. Consider the system described by the state equation (2.89) and the output equation (2.90); the system is represented by input-state-output relationships having zero initial conditions x(0) = 0. The Laplace transform of the system response Y(s), starting from zero initial conditions, may be obtained from equation (2.94) and accounting for the zero initial conditions: Y(s) = [C ⋅ (s ⋅ I − A)−1 ⋅ B + D] ⋅ U(s).

(2.100)

It may be noticed that equation (2.100) is of the form Y(s) = H(s) ⋅ U(s), where H(s) is the transfer matrix and is equal to: H(s) = C ⋅ (s ⋅ I − A)−1 ⋅ B + D = C

adj (s ⋅ I − A) B + D. det (s ⋅ I − A)

(2.101)

Existence of the relationship Y(s) = H(s) ⋅ U(s) is provided in the region of existence E = E0 ∩ E u . The transfer matrix is the element by element one-sided or two-sided Laplace transform, of the impulse response matrix: H(s) = L(h(t)) = L(C ⋅ eAt ⋅ B ⋅ u0 (t) + D ⋅ δ (t)).

(2.102)

Similar to the SISO case, it may be shown that the frequency response matrix hω (ω ) may be directly determined from the transfer matrix H(s) by the replacement of the s variable with the (jω ) product: 󵄨 hω (ω ) = H(s) 󵄨󵄨󵄨󵄨s=jω = H(jω ),

ω ∈ R.

(2.103)

2 Basics of systems theory

| 49

This relationship is possible if the region of existence for the transfer matrix E0 includes the imaginary axis. The transfer matrix is m × k dimensional for the MIMO case and becomes a scalar for the SISO case. The sense of the Hij element of the transfer matrix is the following: for the inlet vector u having all components equal to zero, excepting the j-th component that is an exponential signal uj = aj ⋅ est , the response of the i-th component of the output vector yi (t) is equal to the Hij element of the transfer matrix multiplied by the j-th component of the inlet vector, i.e. yi (t) = Hij ⋅ uj = Hij aj ⋅ est .

2.32 Bode diagrams Bode diagrams are graphical representations of the frequency response hω (ω ) function [1, 15]. As mentioned previously, the frequency response is a complex function that may be determined from the transfer function simply by replacing the variable s with s = jω and getting hω (ω ) = H(jω ). The Bode diagram is a pair of graphical representations. One shows the magnitude (module) of the frequency response function |hω (ω )| = |H(jω )| and the other shows the phase (argument) of the frequency response function arg(hω (ω )) = arg(H(jω )), as they change with respect to the frequency ω considered with logarithmic scale lg(ω ) [17]. For the phase (argument) the notation φ ((hω (ω )) = φ (H(jω )) = arg(H(jω )) is also used. The Bode diagram of magnitude commonly uses the decibel dB unit for module. The magnitude in decibel units is defined by |H(jω )|dB = 20 lg |H(jω )|.

(2.104)

The Bode diagram of phase usually uses the radian unit (the degree unit may be also used). The magnitude and phase Bode diagrams are usually placed one below another, thus having the same logarithmically spaced abscissa lg(ω ), as presented in Fig. 2.15. This positioning allows the correlation between values of the magnitude and the phase at the same frequency lg(ω ) . Due to the decibel unit, the values of the magnitude situated above the abscissa axis correspond to modules having values higher than unity (amplification) and the values of the magnitude situated below the abscissa axis correspond to values of magnitude less than unity (attenuation). Values of the phase above the abscissa axis correspond to positive phase (leading phase) and value below the abscissa axis corresponds to negative phase (phase lag). The logarithmically spaced frequency allows the representation of the Bode magnitude and phase with a good resolution both for very low and for very high frequency values. Bode diagrams representation of the frequency response usually starts with factorization of the frequency response function H(jω ) and its transformation into the

50 | Part I: Introduction

|H(j·ω)|dB 20 10

Amplification lg(ω) −4

−10 −20

−3 −2 −1 Attenuation

φ(H(j·ω)) π

Leading phase

π/2 lg(ω) −4

−π/2 −π

−3 −2 Phase lag

−1 Fig. 2.15. Form and axis significance of the Bode diagrams of magnitude and phase.

general form nm

nl

K ⋅ ∏ (1 + jω Tm ) ⋅ ∏ [ ω12 (jω )2 +

H(jω ) =

m=1 ni (jω )p ⋅ ∏

i=1

l=1 nk

(1 + jω Ti ) ⋅ ∏ k=1

l

[ ω12 (jω )2 k

2ξl (jω ) ωl

+ 1] ,

+

2ξk (jω ) ωk

(2.105)

+ 1]

where K, Ti , Tm , ωk , ωl , ζk , and ζl have real constant values. It may be noticed that binomials correspond to real zeros and poles. Trinomials correspond to complex conjugated poles or zeros, for ζk , ζl < 1, and to real and identical zeros or poles, for ζk , ζl = 1. The factor (jω )p corresponds to a pole situated in the origin and has the multiplicity order of p. In the following, the Bode diagrams of the magnitude and phase will be presented for each factor of the frequency response function presented in equation (2.105). (1) The first considered factor is H1 (jω ) = K/(jω )p , with the Bode magnitude: K = 20 lg K − 20 lg ω p = 20 lg K − 20p lg ω . (2.106) ωp Representation of the Bode magnitude for this factor is a straight line having the negative slope of −20p dB/decade. The decade is a distance on the abscissa between two frequencies having the 1 : 10 ratio. The family of lines, having p as parameter, have a common (fixed) point of coordinates (1, K dB ) and intersect the abscissa axis at the point lg(ω ) = lg(K 1/p ). The Bode diagram of magnitude for this factor is presented in Fig. 2.16. The phase of this factor is π φ (H1 (jω )) = −p . (2.107) 2 It is represented by a family of straight lines parallel to the abscissa axis and having the parameter p (independent of ω ). They are represented in Fig. 2.17. |H1 (jω )|dB = 20 lg

2 Basics of systems theory

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Bode diagram of magnitude for the factor H1

|H1|dB 200 100

p=0 p=1 p=2 p=3

20lg(K) 0 −100 −200 −2

lg(ω) −1

0

1

2

3

Frequency [log(rad/s)] Fig. 2.16. Bode diagram of the magnitude for the frequency response function H1 (j ⋅ ω ) = K/(jω )p , with p = 0, p = 1, p = 2 and p = 3.

φ(H1) [rad] −1 −π/2 −2

Bode diagram of phase for the factor H1 p=1 p=2

−π −3 −4 −3π/2

p=3

lg(ω)

−5 −2

−1

0 1 Frequency [log(rad/s)]

2

3

Fig. 2.17. Bode diagram of the phase for the frequency response function H1 (j ⋅ ω ) = K/(jω )p , with p = 1, p = 2 and p = 3.

(2) The second considered factor is H2 (jω ) = 1/(1 + jω Ti ), having the Bode magnitude determined as follows: H2 (jω ) =

1 − jω ⋅ Ti ω ⋅ Ti 1 1 = = −j 2 2 2 2 1 + jω ⋅ Ti 1 + ω ⋅ Ti 1 + ω ⋅ Ti 1 + ω 2 ⋅ Ti2

2 2 ω ⋅ Ti 1 1 ) +( ) = 2 2 2 2 1 + ω ⋅ Ti 1 + ω ⋅ Ti √1 + ω 2 ⋅ Ti2 1 = 20 lg = −10 lg(1 + ω 2 ⋅ Ti2 ). 2 2 √1 + ω ⋅ Ti

|H2 (jω )| = √ ( |H2 (jω )|dB

(2.108)

Representation of the Bode diagram of magnitude |H2 (jω )|dB is performed taking into account the asymptotes at the frequency extremities (frequency approaching zero and infinity values). The frequency domain is split in two regions, separated by the frequency value ω = 1/Ti denoted as the corner frequency. For the frequency values such as ω < 1/Ti , it is implied that ω 2 ≪ (1/Ti )2 and consequently, ω 2 Ti2 ≪ 1. As a result, the value of the Bode magnitude becomes |H2 (jω )|dB = −10 lg(1 + ω 2 ⋅ Ti2 ) ≈ −10 lg 1 = 0dB ,

(2.109)

52 | Part I: Introduction where the term ω 2 Ti2 has been neglected with respect to 1. This form of the Bode magnitude |H2 (jω )|dB is leading to a straight line graphical representation (named the low frequency asymptote). This asymptote may be approximated as being identical to the abscissa axis up to the corner frequency ω < 1/Ti . For the frequency values such as ω > 1/Ti , it is implied that ω 2 ≫ (1/Ti )2 and consequently, ω 2 Ti2 > 1. As a result, the value of the Bode magnitude becomes |H2 (jω )|dB = −10 lg(1 + ω 2 ⋅ Ti2 ) ≈ −10 lg(ω 2 ⋅ Ti2 ) = −20 lg ω − 20 lg Ti ,

(2.110)

where the term 1 has been neglected with respect to ω 2 Ti2 . This form of the Bode magnitude |H2 (jω )|dB leads to a straight line graphical representation (named the high frequency asymptote), having a negative slope of −20 dB/dec for ω > 1/Ti . The two asymptotes intersect on the abscissa, at the corner frequency ω = 1/Ti . The exact value of the Bode magnitude at the corner frequency is |H2 (jω )| dB = −10 lg 2 = −3dB . ω= 1

(2.111)

Ti

The value of −3dB corresponds to the value of |H2 (jω )| = 0.707. This is considered an acceptable deviation from the exact value when considering the case of the approximate asymptotic representation.

|H2|dB

Bode diagram of magnitude for the factor H2(jω) 0

−20 −40

Asymptotes

−60 −80 −2

lg(ω) 1

−1 lg(1/Ti ) 0

2

3

Frequency [log(rad/s)] Fig. 2.18. Bode diagram of magnitude for the frequency response function H2 (j ⋅ ω ) = 1/(Ti ⋅ jω + 1).

The Bode representation of the magnitude |H2 ( jω )|dB is shown in Fig. 2.18, where the dashed line has been used for representing the exact value of the magnitude and the plain line for the asymptote-based representation. The phase of the factor H2 ( jω ) is ω ⋅ Ti 1 + ω 2 ⋅ Ti2 = tan−1 (−ω Ti ) = − tan−1 (ω Ti ). 1 1 + ω 2 ⋅ Ti2

− φ (H2 (jω )) = tan−1

(2.112)

The Bode diagram representation of the phase φ (H2 ( jω )) is presented in Fig. 2.19.

2 Basics of systems theory

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53

Bode diagram of phase for the factor H2(jω)

φ(H2) [rad]

0

−0.5 −π/4 −1 −1.5 −π/2

−2 −2

lg(ω) −1 lg(1/Ti ) 0

1

2

3

Frequency [log(rad/s)] Fig. 2.19. Bode diagram of phase for the frequency response function H2 (j ⋅ ω ) = 1/(Ti ⋅ jω + 1).

It should be noted that the Bode diagram of phase has an inflexion point at lg ω = lg 1/T i . (3) The factor of the form H3 ( jω ) = 1 + jω Tm has the magnitude and phase representation similar to those of the factor H2 (jω ) but with changed sign of the magnitude (in dB units) and phase (symmetry with respect to abscissa). The corner frequency is the same ω = 1/Tm . This factor is represented in the Bode diagrams of magnitude and phase shown in Fig. 2.20 and Fig. 2.21.

|H3|dB

Bode diagram of magnitude for the factor H3 (jω) 80 +20 dB/dec

60 40

Asymptotes

20 lg(ω)

0 −2

−1 lg (1/Tm ) 0

1

2

3

Frequency [log(rad/s)]

φ(H3) [rad] π/2

π/4

Fig. 2.20. Bode diagram of magnitude for the frequency response function H3 (jω ) = (Tm ⋅ jω + 1).

Bode diagram of phase for the factor H3(jω) 2 1.5 1 0.5 0

lg(ω) −2

−1 lg(1/Tm ) 0 1 Frequency [log(rad/s)]

2

3

Fig. 2.21. Bode diagram of phase for the function H3 (j ⋅ω ) = (Tm ⋅jω +1).

54 | Part I: Introduction (4) The factor of the form H4 (jω ) = 1/[(jω )2 ⋅ 1/ωk2 + (jω ) ⋅ 2ζk /ωk + 1] has the Bode magnitude determined as follows: 1−(

H4 (jω ) =

ω 2 2ξk ) − (jω ) ωk ωk

1 = 2 2 2 2ξk 1 2 (jω ) + (jω ) + 1 (1 − ω ) + 4ξk ω 2 2 ωk 2 2 ωk ωk ωk 2

1

|H4 (jω )|dB = 20 lg

= −10 lg [(1 −

2

√ (1 −

(2.113)

4ξk2 2 ω2 ) + ω ωk2 ωk2

4ξk2 2 ω2 ) + ω ] . (2.114) ωk2 ωk2

Representation of the Bode magnitude |H4 (jω )|dB is performed taking into account the existence of asymptotes for the extremities of the frequency interval. The frequency domain is divided in two regions separated by the corner frequency ω = ωk . For the frequency values such as ω < ωk , it is implied that ω 2 ≪ ωk2 and consequently, ω 2 /ωk2 < 1. As a result, the value of the Bode magnitude |H4 (jω )|dB becomes |H4 (jω )|dB ≈ −10 lg 1 = 0dB ,

(2.115)

where the terms ω 2 /ωk2 and 4ζk2 ω 2 /ωk2 have been neglected with respect to 1 (ζk ≤ 1). For the frequency values such as ω > ωk , it is implied that ω 2 ≫ ωk2 and consequently, the highest term from the logarithm’s argument is ω 4 /ωk4 (the rest are neglected) and the value of the Bode magnitude |H4 (jω )|dB becomes: 2

|H4 (jω )| = −10 lg (

ω ω2 ) = −40 lg = −40 lg ω − 40 lg ωk . ωk ωk2

(2.116)

This form of the Bode magnitude |H4 (jω )|dB leads to a straight line representation (named high frequency asymptote) with the negative slope of −40 dB/dec for ω > ωk . The exact value of the Bode magnitude at the frequency corner is 󵄨󵄨 |H4 (jω )| 󵄨󵄨󵄨ω =ω = −10 lg (4ξk2 ) = −20 lg(2ξk ). 󵄨 k

(2.117)

The magnitude depends on the damping factor ζk . For damping factors of the values usually encountered in practice, i.e. ζk > 0.7, the differences between the approximate representation using asymptotes and the exact one are relatively small. But for damping factor ζk > 0.4 the differences increase. For the value ζk = 0 a discontinuity appears at ω = ωk . This value corresponds to the resonance (resonance frequency of the system). For ζk = 1 the two roots of the trinomial are real and equal being therefore equivalent to the product of two binomials. The Bode diagram of magnitude is presented in Fig. 2.22.

2 Basics of systems theory

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55

Bode diagram of magnitude for the factor H4(jω)

|H4|dB

ξk=0

20

ξk=0.1 ξk=0.7

0 Asymptotes

−20 −40 −60

−1

−0.5

0

0.5 lg(ωk ) 1

1.5

2

2.5

lg(ω)

Frequency [log(rad/s)] Fig. 2.22. Bode diagram of magnitude for the frequency response function H4 (j ⋅ ω).

The phase of the factor H4 (jω ) is

φ (H4 (jω )) = tan

−1

2ξk 2ξ − k ωk ωk −1 ω ω ( ) = − tan . ω2 ω2 1− 2 1− 2 ωk ωk −

(2.118)

The Bode diagram of phase φ (H4 (jω )) is presented in Fig. 2.23. Bode diagram of phase for the factor H4(jω)

φ(H4) [rad] 0

ξk=0.7 ξk=0.1 ξk=0

−1 −π/2 −2 −π

−3 −4

lg(ω) −1

−0.5

0

1.5 0.5 lg(ωk ) 1 Frequency [log(rad/s)]

2

2.5

Fig. 2.23. Bode diagram of phase for the frequency response function H4 (j ⋅ ω).

(5) The factor of the form H5 (jω ) = [(jω )2 ⋅1/ωl2 +(jω )⋅2ζl /ωl +1] has the magnitude and phase representations similar to those of the factor H4 (jω ) but with changed sign of the magnitude (in dB units) and phase (symmetry with respect to abscissa). The corner frequency is equal to ω = ωl . This factor is represented in the Bode diagrams of magnitude and phase shown in Fig. 2.24 and Fig. 2.25.

56 | Part I: Introduction Bode diagram of magnitude for the factor H5(jω)

|H5|dB 60 40

ξl=0.7 ξl=0.1 ξl=0

20 0

Asymptotes

−20 −1

−0.5

0.5 lg(ωl ) 1

0

1.5

lg(ω) 2.5

2

Frequency [log(rad/s)] Fig. 2.24. Bode diagram of magnitude for the frequency response function H5 (j ⋅ ω).

Bode diagram of phase for the factor H5(jω)

φ(H5) 4 [rad] π 3

ξl=0.7 ξl=0.1 ξl=0

2 π/2 1 0 −1

−0.5

0

0.5 lg(ωl ) 1

1.5

2

lg(ω) 2.5

Frequency [log(rad/s)] Fig. 2.25. Bode diagram of phase for the frequency response function H5 (j ⋅ ω).

Starting from the transfer function of the system, the steps for performing the Bode diagram representation of the frequency response function using asymptotes are: 1. building the frequency response function from the transfer function by replacing the Laplace variable s with s = jω ; 2. bringing the frequency response function H(jω ) to the factorized form presented in equation (2.105); 3. representing each factor of H(jω ) (with its magnitude and its phase) on the same Bode diagram for the magnitude and on the same Bode diagram for the phase of the function H(jω ); 4. obtaining the Bode diagram of magnitude and phase of the function H(jω ) by performing a graphical summation of the individual representations, both for the Bode diagram of magnitude and for the Bode diagram of phase. In particular cases, especially when the frequency domain is not very large, the Bode diagrams for the magnitude (module) of the frequency response function |hω (ω )| = |H(jω )| may be represented without the use of a logarithmic scale (dB) on the y-coordinate. As an example, Fig. 2.26 shows the Bode diagram without the logarithmic scale, for the magnitude of the factor H4 (jω ) previously presented with dB

2 Basics of systems theory

|H4(jω)| 1.5

|

57

Bode diagram of phase for the factor H4(jω)

1 0.5 lg(ω)

0 −2 −1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Frequency [log(rad/s)] Fig. 2.26. Bode diagram of magnitude for the frequency response function H4 (j ⋅ ω), with no logarithmic scale on the y-coordinate.

units in Fig. 2.22 (for ζk > 0.7). The present book will use both of these two equivalent Bode diagrams representations.

2.33 Nyquist diagrams Nyquist diagrams are also used for the graphical representation of the frequency response hω (ω ) = H(jω ) function [9]. The Nyquist plot has on the x-coordinate the real component of the frequency response, Re[H(jω )], and on the y-coordinate the imaginary component of the frequency response, Im[H(jω )]. Typical Nyquist plots are presented in Fig. 2.27. They show the Nyquist diagrams for the first order (plot 1), second order (plot 2) and third order (plot 3) systems. On the plot, the Nyquist diagram has a starting point, corresponding to the lowest value of the considered frequency interval and a final point corresponding to the highest value of the frequency interval of interest. An arrow usually shows the sense of growing frequency along the plot. Each point of the plot corresponds to a particular frequency value within the frequency interval. As presented in Fig. 2.27, for the generic ω ∗ value of the frequency it corresponds to one point on the plot. The length of the vector connecting the origin of the coordinate system with the particular point on the plot is equal to the magnitude (module) of the frequency response function |hω (ω ∗ )| = |H(jω ∗ )|, as |H(jω ∗ )| = [(Re[H(jω ∗ )])2 + (Im[H(jω ∗ )])2 ]1/2 . The angle this vector makes with respect to the x-coordinate φ (ω ∗ ) is equal to the phase (argument) of the frequency response function φ (ω ∗ ) = φ ((hω (ω ∗ )) = φ (H(jω ∗ )), as φ (H(jω ∗ )) = tan−1 (Im[H(jω ∗ )]/Re[H(jω ∗ )]). Between the form of the plot and its position on the graph, on one side, and the dynamic behavior of the system, on the other side, there is a close relationship that reveals patterns of frequency response of the systems. The Nyquist diagrams are equivalent to the Bode diagrams.

58 | Part I: Introduction Imaginary axis Im [H(jω)] 1

Nyquist diagram

0.8 0.6 0.4 0.2 ω=∞

0

ω=0

φ(H(jω*))

−0.2 3

−0.4 −0.6

2

1

|H(jω*)|

−0.8 ω*

−1 −1

−0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

0.8

1

Real axis Re [H(jω)] Fig. 2.27. Typical Nyquist diagrams of the frequency response for first (1), second (2) and third (3) order systems.

2.34 Problems (1) Please specify the delimitation and the most representative input, state and output variables of the following systems: (a) the natural gas transportation pipe, between the extraction site and an industrial user, investigated for the purpose of natural gas pressure loss reduction; (b) the air conditioning (heating and cooling) of a concert hall, investigated for the purpose of minimizing the energy consumption by temperature optimization; (c) the cell of a bacteria, investigated for the purpose of enhancing its living conditions and increasing its lifetime; (d) the operating personnel of the offshore oil-gas platform, investigated for the purpose of maximizing their contribution to the efficient and safe operation of the unit; (e) the market of sulfuric acid consumers, investigated for the purpose of improving the management of the production in an industrial plant. (2) Classify the systems discussed in the previous problem, form the following categorizing criteria: dynamic and static, with lumped and distributed parameters; deterministic and stochastic, linear and nonlinear.

2 Basics of systems theory

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59

(3) Consider an instantaneous discharge of a pollutant mass M at a given point of a river channel. Which of the elementary signals would best describe the time and point (space) mass discharge? (4) Make the graphical representation of the following signals: u0 (t) ⋅ v0 (t), u0 (t − t0 ) ⋅ sin(ω t) and u0 (t − t0 ) ⋅ e−10t . (5) Consider the system described in Example 2.2. Is this system linear or nonlinear? Prove your answer. (6) Consider a system described by the following input-output relationship: T

dy(t) + y(t) = K u(t), dt

(2.119)

where T and K are constant. (a) If the input signal is described by u(t) = u0 (t) and the initial value of the output is y(0)= y0 , compute and plot the response of the system y(t) for t ∈ [0, ∞]. (b) If the input signal is described by u(t) = u0 (t−t0 ) and the initial value of the output is y(t0 ) = y0 , compute the response of the system y(t) for t ∈ [t0 , ∞]. (c) If the input signal is described by u(t) = δ (t −t0 ) and the initial value of the output is y(t0 ) = 0, compute the response of the system y(t) for t ∈ [t0 , ∞]. (7) Let h1 (t), h2 (t), and h3 (t) be the impulse response of the three systems connected in series. What is the impulse response h(t) of the whole system (i.e. between the input of the first system and the output of the last one)? (8) Consider the system presented in Fig. 2.28, consisting in the process of liquid accumulation in series of two tanks. The two vessels operate at atmospheric pressure. Based on the mass balance of liquid in each of the tanks, the first order differential equations may be obtained: T1

dH1 + H1 = k1 ⋅ Fvi dt

(2.120)

dH2 (2.121) + H2 = k2 ⋅ Fv1 dt where T1 , T2 , k1 and k2 have constant values. Furthermore, the second order differential equation describing the relationship between the level in the second tank and the flow rate entering the first tank is: T2

dH d2 H2 + (T1 + T2 ) 2 + H2 = k2 ⋅ Fvi . (2.122) dt dt2 (a) Compute the general solution of equation (2.122), taking into account the constant initial conditions H2 (0) = a and (dH2 /dt)(0) = b. T1 T2

60 | Part I: Introduction Fvi

H1 Fv1 R1

H2

R2

Fv2

Fig. 2.28. Liquid accumulations in a series of two tanks.

(b) Verify whether the system is BIBO stable. (9) Consider the system consisting in a U shaped tube partially filled with liquid of density ρ , as presented Fig. 2.29. The branches of the tube are cylindrical and of equal cross-section S. p1

p2

h h S

Fig. 2.29. Oscillating behavior of the liquid level in the U tube.

When a differential pressure Δp = p1 −p2 is suddenly (step shaped) applied on the two branches of the U tube, the liquid level in the two branches will change, according to the simplified equation (momentum balance): p1 ⋅ S − p2 ⋅ S − ρ ⋅ g ⋅ (2 ⋅ h) ⋅ S −

λ ⋅L dh d2 h ⋅S⋅ρ ⋅ =ρ ⋅S⋅L⋅ 2 D dt dt

(2.123)

2 Basics of systems theory

| 61

where: λ is the friction coefficient, L is the length of the column occupied by the liquid in the U tube, and D is the U tube diameter (cross-section area of the tube S = π D2 /4). (a) Compute the general solution of equation (2.123), taking into account the constant initial conditions h(0) = 0 and (dh/dt)(0) = 0. (b) Verify whether the system is BIBO stable. (10) Describe the system presented in Fig. 2.29 by the input-state-output representation, choosing the state variables as phase variables. (11) Consider the transfer function of a dynamic system given by H(s) =

0.5 e−10s (s + 1) (2s + 1)(3s + 1) .

(2.124)

Plot the Bode diagrams of module and phase of the frequency response function of the system. The pure time delay and the time constants are considered in seconds as the time unit.

References [1]

Cristea, M. V, Agachi, S. P., Elemente de Teoria Sistemelor, Editura Risoprint, Cluj-Napoca, 2002. [2] Forrester J. W., Principles of Systems, Wright Allen Press, 1969. [3] Zadeh, L. A., Polak, E. System Theory, McGraw-Hill, New York, 1969. [4] Hăngănuţ, M., Noţiuni de Teoria Sistemelor, Cluj-Napoca, Atelierul de multiplicare al Institutului Politehnic, 1989. [5] Marquardt, W., Modellbildung und Simulation verfahrenstechnischer Prozesse, Vorlesungsmanuskript, Technische Hochscule Aachen, 1998. [6] Cristea, V. M., Agachi, S. P., Simulation and model predictive control of the soda ash calciner, Control Engineering and Applied Informatics, 3, No.4, (2001), 19–26. [7] Cristea, V. M., Bagiu, E. D., Agachi, P. S., Simulation and control of pollutant propagation in Somes River using comsol multiphysics, European Symposium on Computer Aided Process Engineering 20, Published in Computer Aided Chemical Engineering, pp. 985–991, Ischia, 2010. [8] Şerban, S., Şerbu, T., Corâci, I. C., Teoria Sistemelor, Bucureşti, Editura Matrix Rom, 2000. [9] Stephanopoulos, G., Chemical Process Control. An Introduction to Theory and Practice, Englewood Cliffs, New Jersey 07632, Prentice Hall, 1984. [10] Rolhing H., Systemtheorie II. Vorlesungsmanuskript, Hamburg Teschnische Universität, 1998. [11] Cruz, R. L., Linear Systems Fundamentals. Lecture Supplement. San Diego, University of California, 1999. [12] Ubenhauen, R., Systemtheorie. Grundlagen für Ingineure, München Wien, Oldenbourg Verlag, 1990. [13] Oppenheim, A. V., Willsky, A. S., Nawab, S. H., Nawad, H., Signal and Systems, Prentice Hall, 1996. [14] Kwakernaak, H., Sivan, R., Strijbos, R. C. W., Modern Signals and Systems, Englewood Cliffs, New Jersey 07632, Prentice Hall, 1991.

62 | Part I: Introduction [15] Hăngănuţ, M., Teoria Sistemelor, Cluj-Napoca, Universitatea Tehnică, 1996. [16] Cristea, V. M., Agachi, S. P., Reglarea evoluată a reactorului de carbonatare din instalaţia de producere a sodei amoniacale, Revista Română de Informatică şi Automatică, 7(4), (1997), 45–51. [17] Mihoc, D., Ceaprău, M., Iliescu, S. St., Bornagiu, I., Teoria şi elementele sistemelor de reglare automată, Bucureşti, Editura Didactică şi Pedagogică, 1980.

3 Mathematical modeling The mathematical model of a process represents the mathematical relationship between the output variables and the input and state variables. Generally, the mathematical model is the relationship y(t) = f (t; x(t), u(t)),

(2.12)

where u(t) and y(t) are the input and output vectors respectively, x(t)−t the state vector and wherever there is the discussion of the dynamic behavior, the variable (time) is present. Fag , T°iag

F, T°i

F, T°

Fag , T°ag (a) T°i T°ag F

Heat transfer process

T°

T°ext Fag (b)

Fig. 3.1. Heat transfer process in a heat exchanger.

In the case of the process from Fig. 3.1, the mathematical model is an expression in its most general form ∘ , t) , T ∘ = f (Ti∘ , F, Fag , Ti∘ag , Text (3.1) T ∘ is the output temperature, Ti∘ the input temperature of the fluid, F fluid flow, Fag ∘ temperature of the enheating agent flow, Ti∘ag heating agent input temperature, Text vironment. If the model describes the steady state, t is missing, the equation being an algebraic one. If the model describes the dynamic state, equation (3.1) is a differential one in a form similar to the differential equation (2.8). In case the process has several output variables, the model takes the form of a system of algebraic or differential equations or combined. The models can be theoretical, analytical ones, also denoted as mechanistic or first principle models, based on the conservation, thermodynamic and kinetic equations; or empirical ones in the form of mathematical regressions, based on experi-

64 | Part I: Introduction mental values, processed statistically, based on artificial neural networks. For more complex processes, the models are combined containing both analytical and empirical parts. It is extremely important for a model to be correctly determined and solvable that 1. the number of equations is equal with the number of variables; 2. all terms of the model are dimensionally consistent, that is, expressed in the same measurement unit system (e.g. SI (Système International d’Unités)). The use of mathematical models is extended from the activities of design and operation to those of simulation and control. The simulation is the representation of the reality based on the results from running a mathematical model of that reality. The process simulators can be used either for training the personnel in a plant, for designing the plant or its control system.

3.1 Analytical models Analytical models express most clearly the interdependence of the parameters of the process. The procedure of writing and solving a model is the following: 1. Elaborate a clear and correct flow sheet or drawing of the process described; if the process is too complex, divide it in lower complexity modules. 2. Identify all variables in the process and understand the correlation cause-effect between them: first inside the modules and second between the modules. 3. Elaborate the lists of variables and constants: output variables, input variables, design and construction parameters, thermodynamic and kinetic constants. 4. Elaborate the list with simplifying hypotheses. 5. Write the equations of the model if possible in the “natural” sequence of the development of the process; ensure that all unknowns in the equations are expressed in new “secondary” equations. 6. Solve the system of equations and interpret the results.

3.1.1 The conservation laws There are three fundamental conservation laws which are used in the mathematical modeling: mass, energy and momentum conservation laws. To be able to describe appropriately the processes, one has to add as tools of modeling the basic thermodynamics and kinetics laws [1–3]. It is important to define the nature of the systems from the point of view of the distribution of the parameters inside them: there are lumped parameter systems (where all parameters have the same value regardless of their measurement point) and dis-

3 Mathematical modeling

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65

tributed parameter systems (where the value of the parameter depends on its point of measurement) (see Chapter 2).

Mass conservation law The mass conservation law expresses the accumulation either of the total mass in a system (total mass conservation law) (equation (3.2)), or the composition variation inside the system (mass of components conservation law) (equation (3.3)). {mass input flow} − {mass output flow} = {

{

rate of mass }, accumulation

(3.2)

component i molar component i molar }−{ } input flow output flow ±{

rate of accumulation } { } { formation or consumption } = { of number of moles of } . } { rate of component i } { component i

(3.3)

Examples follow of applying total mass or mass of components conservation law at lumped and distributed parameter systems. Example 3.1. Total mass conservation law for lumped parameters system In a buffer tank for intermediate product in one chemical plant (Fig. 2.6), there is continuous input and output of the same material. Thus, the density is the same in the tank as in the input and output flows. The total mass conservation law states: Fvi ρ − Fvo ρ =

d (Vρ ) dt

(3.4)

where Fmi = Fvi ρ and Fmo = Fvo ρ are the volumetric input and output flows, ρ is the liquid density and V = H ⋅A the instant volume of liquid in the tank. The first two terms are those of mass convective flow and the term on the right-hand side of the equation is the mass accumulation term in the tank. The dimensional analysis is kg m3 kg m3 kg 1 − = ⋅ ⋅ ⋅ m3 ⋅ 3 s m3 s m3 s m which shows consistency, all terms being expressed by mass flow.

kg , s

the measurement unit for

Example 3.2. Component conservation law for lumped parameter system A continuous stirred tank reactor (CSTR) is represented in Fig. 3.2. The reaction is the k

simplest one, A 󳨀→ B,, first order with the rate constant k.

66 | Part I: Introduction Fi , CAi , Ti°

Fag , T°ag

Fag , T°agi CA , CB , V, T° F, CA ,CB , T °

Fig. 3.2. CSTR.

The mixing is considered perfect, thus all concentrations inside the reactor and at its output have the same values. The mass of component A conservation law for the mass of reaction inside CSTR is d Fvi CAi − Fvo CA − VkCA = (VCA ), (3.5) dt where CAi and CA are the input and inner/output concentrations respectively. The first two terms are the molar input and output flows of the reactor, the third term is that of reaction and the term on the right-hand side is that of accumulation. The dimensional analysis m3 kmol m3 kmol 1 kmol 1 kmol − − m3 ⋅ ⋅ = ⋅ m3 ⋅ ⋅ ⋅ s m3 s m3 s m3 s m3 shows all terms are expressed by

kmol s

, the measurement unit for molar flow.

Example 3.3. Total mass conservation law for distributed parameter system A long pipeline is used for the transportation of natural gas (Fig. 3.3). Both parameters, velocity and density (v and ρ ), are functions of position and time. This is why the volumetric flow is expressed decomposed by Av. It is natural that at the entrance of the pipeline both parameters have higher values than at the end of it, because of the friction along the road. Because there are different values for the parameters along the pipeline, one cannot write a single equation expressing the behavior of the whole system and we have to divide the system in small portions of dimension dz, “infinitely small elements” in which it can be assumed the values of the parameters are constant. The dimension of the small interval depends on the nature of the process: for a long pipeline of 100 km the dimension considered can be of 1 m, whereas for a particle of 10 cm of limestone in a process of carbonation dz could be of 5 mm. The material balance is written for the portion dz. The output variable at z + dz is calculated according to the Taylor expansion formula f (z + dz) = f (z) +

1 df 1 d2 f 2 1 d3 f 3 dz + dz + ⋅ ⋅ ⋅ ⋅ dz + ⋅ ⋅ 1! dz 2! dz2 3! dz3

(3.6)

3 Mathematical modeling

|

67

where dz is the infinitely small element of a dimension to be judged. Usually, since dz is very small, the dz2 , dz3 etc. terms become negligible and the expansion is reduced to the first two terms of the series expansion. ) Since the mass flow is Avρ at z, according to (3.7), it becomes Avρ + 𝜕(Avρ dz at 𝜕z z + dz. Thus, the mass balance becomes for the infinitely small element dz Avρ − [Avρ +

𝜕 (Avρ ) 𝜕 dz] = (A ⋅ dz ⋅ ρ ) , 𝜕z 𝜕t

(3.7)

where A is the cross-section area of the pipeline. The first two terms express the convective flows at the input and output of the dz element and the right-hand side term is that of accumulation in time. The partial derivatives, 𝜕z𝜕 and 𝜕t𝜕 show that the variables, v and ρ depend on both space and time. The dimension analysis is m2

kg m kg m kg m kg 1 1 − [m2 + m2 m] = m2 m 3 3 3 3 s m s m m s m s m

and shows that all terms are measured by

kg . s

dz

o

z z + dz

zf

Fig. 3.3. Long pipeline distributed parameter system.

Example 3.4. Component conservation law for distributed parameter system Let us consider a plug flow reactor (PFR) represented in Fig. 3.4, where the reaction is k

A 󳨀→ B, first order with the rate constant k. The profile of concentration of the reactant A in steady state is represented in Fig. 3.4. Obviously, the concentration depends on the position inside the reactor; a gradient of concentration exists between two neighboring points and the mass diffusion phenomenon is present. Thus, the total component molar flow at z is expressed by AvCA + A (−DA

𝜕CA ), 𝜕z

where the second term expresses the diffusion by Fick’s law and DA is the diffusion constant for the component A. The whole component A balance for the element dz is AvCA + A (−DA

𝜕CA 𝜕C 𝜕C 𝜕 ) − {AvCA + A (−DA A ) + [AvCA + A (−DA A )] dz} 𝜕z 𝜕z 𝜕z 𝜕z 𝜕 − A ⋅ dz ⋅ k ⋅ CA = (A ⋅ dz ⋅ CA ). (3.8) 𝜕t

68 | Part I: Introduction

L

d CA, CB

Concentration – C

CA0

1 Steady state concentration of CA in the PFR

0.8 0.6 0.4 0.2 0 0

1

2

3

4

5 6 Length – L

7

8

9

10

Fig. 3.4. PFR with the profile of concentration CA in the steady state along the reactor.

The dimensional analysis shows all terms are expressed in molar flow measurement units, kmol/s m2

m kmol m2 1 kmol + m2 3 s m s m m3 2 2 m kmol 1 2 m 1 kmol 2 m kmol 2 m 1 kmol − [m2 + m + + m ) m] (m s m3 s m m3 m s m3 s m m3 1 kmol 1 2 kmol − m2 m = mm 3 . s m3 s m

In its final form, after simplification, equation (3.8) becomes DA

𝜕C 𝜕2 CA 𝜕 (vCA ) − kCA = A . − 𝜕z 𝜕t 𝜕z2

(3.9)

The equation is with partial derivatives, meaning the parameters v and CA change both with time and distance. As a comment, everywhere a second derivative term appears in the equation, this is the sign of a diffusive process. Solving all the above constructed equations describing dynamically the mass balance in the system one can determine the changing of the property (mass or concentration) in time and in any place of the system. The partial differential equation is solved using, for example, a finite element method. Recently, it is not expected from the process or control engineers to elaborate the subroutines of solving the mathematical problems, since scientific software such as MatLab [1, 2] or Mathematica [3] exist and have incorporated special subroutines for solving ordinary differential equations (ODE).

3 Mathematical modeling

| 69

Energy conservation law The energy conservation law expresses the accumulation of heat in the system (equation (3.10)) input flow of kinetic, potential, and } { output flow of kinetic, potential, } { } } { { − and internal energy, (power) } } { internal energy (power) } { } { { } } { by convection and diffusion { by convection and diffusion volume or mechanical work flow of heat (power) formed } } { } { } { { ∓ { exercised in time (power) by the } ± { or transferred by } } } { { { reaction, radiation, or conduction } { system to the outerenvironment } ={

rate of energy }. accumulation (power)

(3.10)

In order to understand the true importance of the different terms in the energy balance, we have to compare the forms of energy. Approximation of the forms of energy (kinetic, potential and internal) and of the volume work/mechanical work is given below. For the kinetic term, let us consider one moves a kilogram of water with a normal velocity of 1 m/s in a pipeline (in industry the pipelines are dimensioned to let the liquids have an average velocity of 1–2 m/s and for the gases of 30–40 m/s from reasons of energy loss). 2

2

1 m2

v 1 kg 2s ΔEc m 2 Nm = = = 0.5 = 0.5 W. Δt Δt 1s s Transporting the same kilogram of water at a height of one meter, the potential energy per unit of time will be

ΔEp Δt

=

mgh 1 kg ⋅ 9.8 = Δt 1s

m 1m s2

= 9.8

Nm = 9.8 W. s

And heating the same kilogram of water with 1 K, the internal energy involved, U J ΔU mcp ΔT 1 kg ⋅ 4318 kg⋅K 1 K J = = = 4318 = 4318 W, Δt Δt 1s s

meaning that in the energy balance we may neglect the potential and kinetic energy, being too small in comparison with the heat involved. At the same time, we know that the volume work is expressed by L = ∫ p ⋅ dV or L = pΔV in isobaric conditions. The liquids being practically incompressible do not produce or are not subjected to volume work, which is also negligible. This term is to be taken into consideration only for processes in gaseous phase. Example 3.5. Energy conservation for a lumped parameter system k

Let us consider a CSTR (Fig. 3.2) in which an exothermic first order reaction A 󳨀→ B takes place, with the rate constant k and heat of reaction ΔHr < 0. The process being

70 | Part I: Introduction exothermic, the jacket has cooling agent and there is a heat transfer from the mass of reaction to the jacket through the wall. The heat transfer is defined by the thermal transfer coefficient KT and by the heat transfer area AT . ∘ ) − VkCA ΔHr = Fvi ρ cpA Ti∘ − Fvo ρ cp T ∘ − KT AT (T ∘ − Tag

d (Vρ cp T ∘ ), dt

(3.11)

where cp is generically the specific heat of the flows (cpA for raw material A and cp ∘ for the reactor inventory and the output flow), T ∘ and Tag are the temperature inside the reactor and that of the cooling agent respectively and V the volume of the reaction mass inside the reactor. The first and second terms are those of input and output convective energy flows, the third term is that of conductive heat transfer (there is no radiation), the fourth term is the reaction term, expressing the heat produced by the chemical reaction and the term on the right-hand side of the equation is the accumulative term of heat in the reactor. Dimensional analysis shows all terms are expressed in J/s = W meaning terms of energy flow: kg J m3 kg J 1 kmol J m3 kg J W 1 K − K − 2 m 2 K − m3 = m3 3 K. 3 3 3 s m kgK s m kgK mK s m kmol s m kgK Example 3.6. Energy conservation for a distributed parameter system k

Let us consider a PFR (Fig. 3.5) with an exothermic first order reaction A 󳨀→ B, with the rate constant k and heat of reaction ΔHr < 0. The profile of temperature in the reactor is depicted in Fig. 3.5.

L

TAgo°

D CA , CB , To°

CAO , Ti°

d TAgi°

T°

TAg°

T° Length – L Fig. 3.5. PFR with exothermic reaction and cooling agent and the profile of temperature with an exothermic reaction and counter-current cooling flow.

3 Mathematical modeling

| 71

Due to the variation of temperature along the reactor a gradient of temperature exists between the different points of the reactor. Thermal diffusion is thus present and it is expressed by Fourier’s law of thermal diffusion Q = Ar (−kT

𝜕T ∘ ), 𝜕z

where Q is the diffusive heat flow, Ar is the cross-section area of the reactor, kT is the Fourier diffusion coefficient and T ∘ is the temperature at the coordinate z. The heat transfer area of the reactor is that of the cylinder with the height dz being expressed by AT = π Ddz where D is the inner diameter of the reactor. The transfer coefficient through the wall is KT . The heat flow at the coordinate z + dz is given by the Taylor expansion formula. Thus the heat balance is Ar vρ cp T ∘ + Ar (−kT

𝜕T ∘ ) 𝜕z 𝜕T ∘ 𝜕T ∘ 𝜕 )+ [Ar vρ cp T ∘ + Ar (−kT )] dz} 𝜕z 𝜕z 𝜕z 𝜕 ∘ ) − Ar dzkCA ΔHr = (A dzρ cp T ∘ ) . − KT π Ddz (T ∘ − Tag 𝜕t r

− {Ar vρ cp T ∘ + Ar (−kT

(3.12)

The dimensional analysis shows the dimensional consistency, giving also a signal concerning the correctness of the formulae. All terms are expressed in W: m2

m kg J W 1 K + m2 K s m3 kgK mK m m kg J W 1 m kg J W 1 1 K + m2 K + (m2 K + m2 K) m − [m2 s m3 kgK mK m m s m3 kgK mK m kg J 1 kmol J 1 W ] = m2 m 3 K. − 2 mmK − m2 m mK s m3 kmol s m kgK

After simplification, the equation (3.12) becomes 4K T ∘ 𝜕T ∘ 𝜕 𝜕 𝜕 ∘ )= (ρ cp T ∘ ) + (vρ cp T ∘ ) + kCA ΔHr + (T − Tag (kT ). 𝜕t 𝜕z D 𝜕z 𝜕z

(3.13)

Momentum conservation law The momentum conservation law: the rate of change of the movement quantity of a system is equal to the total force applied to the system (equations (3.14) and (3.15)): { or

sum of forces } = {rate of change of movement quantity} applied to a system n

∑ Fij = j=1

d (mv) , dt

(3.14)

(3.15)

72 | Part I: Introduction Fj Fij

m

ν

o

Fig. 3.6. Application of the impulse conservation on the movement of one mechanical body.

i

where Fij is the projection of applied force j on the direction i (Fig. 3.6) and m and v are the mass and the velocity of movement of the mass m along the i direction. If there is = a, the acceleration, and only one force applied to the body and m is constant, dv dt equation (3.15) is reduced to the expression of Newton’s second law F = m ⋅ a. Example 3.7. Momentum conservation for a constant mass system Consider a tank with continuous outflow through a pipeline of length lp and crosssection area Ap (Fig. 3.7). The friction on the pipeline is expressed by Poisson’s law, 2

l

2

Δpp =∝ ρ2v +λ dp ρ2v where Δpp is the pressure loss on the hydraulic system (pipeline), p and ∝ and λ are the coefficients of local and linear hydraulic pressure loss and they are dimensionless. The pipeline is horizontal which makes the projection of the gravity force equal to 0 on the axis of liquid movement. The forces acting on the liquid column in the pipeline are: hydrostatic force, ρ ghAp , G, the weight of the liquid in the 2

l

2

pipeline and friction force against the movement, (∝ ρ2v +λ dp ρ2v )Ap . The atmospheric p pressure is equal on both sides of the pipeline. G, being vertical, has the projection on i axis Gij = 0. Thus, ρ ghAp − (∝

lp ρ v2 ρ v2 d +λ ) Ap = (Ap lp ρ v). 2 dp 2 dt

(3.16)

D

ρ lp

h

ν dp

Fig. 3.7. Tank with free outflow through a pipeline.

3 Mathematical modeling

| 73

The dimensional analysis kg m2 2 1 2 kg m kg m 2 ⋅ m ⋅ m − m = m ⋅m⋅ 3 m3 s2 m3 s2 s m s shows all terms are expressed in N = kg m . s2 From equation (3.15) one may calculate the velocity or the flow of the liquid in steady state when dtd (Ap lp ρ v) = 0. Usually, because the liquids are incompressible, the response of the liquid flow at an increase of the liquid height is instantaneous. Example 3.8. Momentum conservation law for variable mass systems One may consider a long pipeline for the transportation of petrochemical products from the refinery to the harbor (Fig. 3.8). In order to clean the pipeline after the transport of a product and to prepare it for the next transport of another product, one leather ball, a “pig”, is used to push the rest of the product out of the pipeline. The “pig” is pressured with inert gas at a constant pressure p0 . It is a typical example of a variable mass system. The equations describing the movement are p0 Ap − λ

(lp − z) ρ v2 dp

2

Ap =

d (A (l − z)ρ v) dt p p

(3.17)

and

dz , dt from which one may calculate the dynamics of emptying the pipeline. v=

lp Z dp p0

ρ, ν

Fig. 3.8. Pipeline with variable mass of liquid.

3.1.2 Thermodynamics and kinetics of the process systems The state and equilibrium equations indicate the way the physical properties change with process parameters such as temperature, pressure and composition. The most frequently used equations [7] are presented in the synthetic Tabs. 3.1 and 3.2.

74 | Part I: Introduction Table 3.1. Thermodynamic properties used in process modeling. Property/Thermodynamic laws

Formula

Notations

Liquid enthalpy Vapor enthalpy

ΔHl = cp T ∘ ΔHl = cp T ∘ + lv

cp – heat capacity; T ∘ – temperature lv – latent heat of vaporization

Enthalpy of a mixture of liquids and heat capacity of a mixture

ΔH lm =

cplm = ∑ xj cpj

Dependence of enthalpy with temperature

ΔHl = a0 + a1 T ∘ + 2 a2 T ∘ + ⋅ ⋅ ⋅

Boyle Mariotte law

pV = nRT ∘

∑n1 xj ΔH j Mj ∑n1 xj Mj

1

Gas density Molar concentration

or

n

pM RT ∘ ni Ci = ∑ m ρ j j

ρ =

xj – molar fraction; Mj – molar mass; ΔH j – enthalpy of j component in the mixture; cpj – heat capacity of component j a0 , a1 etc. are numerical coefficients; the dependence is to be considered for large (> 10∘ ) variations of temperature p – pressure of gas; V – gas volume; n – number of moles of gas; R – universal gas constant ρ – gas density

=

ρ xi󸀠 M i

Ci – molar concentration of component i, ni – number of moles of i, ∑j mj — mass of the mixture, ρ – density of the mixture, xi󸀠 – mass fraction of i, Mi – molar mass of component i

n

Liquid mixture density

ρ = ∑ Mj Cj or 1 n

ρ = ∑ xj ρ j 1

ρj – density of the component j in the mixture; Cj – molar concentration of component j in the mixture,

Dalton’s law (partial pressure of vapors in a vapor mixture)

pj = Pyj

pj – pressure of the component j in the mixture; P –total pressure of the mixture; yj – molar fraction of component j.

Raoult’s law (total vapor pressure of a liquid mixture)

P = ∑n1 xj pj

xj – molar fraction of the component j in the liquid mixture; pj – vapor pressure of the pure component j in the mixture; P – total vapor pressure of a liquid mixture

Antoine’s law (vapor pressure correlation with temperature)

lgP = a +

Fenske’s Law (liquid-vapor equilibrium)

yj =

Henry’s law (dependence of the gas concentration at gasliquid interface on gas pressure)

C∗ =

Repartition of a solvate between two solvents (diluted solutions)

y = kx

b T ∘ +c

∝xj 1−(∝−1)xj

p H

a, b, c –numerical coefficients; P – vapor pressure xj – molar fraction of component j in liquid phase; yj – molar fraction of component j in vapor phase; ∝ – relative volatility C∗ – gas concentration at the interface; p – gas pressure above the liquid; H – Henry’s constant y – molar concentration of the solvate in the principal solvent; x – molar concentration of the solvate in the secondary solvent; k – repartition constant

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Table 3.2. Kinetic properties used in process modeling. Property/Kinetic laws Rate of reactions

Formula

Notations k

Direct first order A 󳨀→ Br = kCA Direct second order k A + B 󳨀→ C + D, r = kCA CB Reversible first order A ⇐⇒ B, r = k1 CA − k−1 CB k1 , k−1

Parallel first order k1

k2

A 󳨀→ B, A 󳨀→ C, rA = (k1 + k2 )CA Parallel second order k1

ri – rate of reaction being understood as the number of moles produced/consumed in the reaction in a unit of time, per one unit of volume (kmol/m3 .s); ki – rate constants expressed in such a way as for the rate of reaction to be expressed in kmol/m3 .s;

k2

A + B 󳨀→ D, A + C 󳨀→ E rD = k1 CA CB , rE = k2 CA CC Consecutive reactions k1

k2

A 󳨀→ B 󳨀→ C, rB = k1 CA − k2 CB Arrhenius law (dependence of the rate constant on temperature) Relative conversion of the reactant

Relative conversion in a CSTR

k0 – pre-exponential constant; Ea – activation energy; R – universal constant; T ∘ – reaction temperature ξ – conversion of the reactant; CA0 – molar concentration of the reactant at the beginning of the reaction (or at the input of the reactor); CA – molar concentration of the reactant at the time t, or in the reactor

E − a∘

k = k0 e

ξ =

RT

CA0 −CA CA0

k

Direct first order A 󳨀→ B ξ =

k VF

1+k VF

Reversible first order aA ⇐⇒, k−1 bB k1

1 V

ξ = ξ∞ (1 − e−k1 (1+ K ) F ) K=

k1 k−1

E − 1a∘

k1 = k10 e

RT

E − −1,a ∘

k−1 = k−1,0 e Relative conversion in a PFR, with a first order direct reaction Excess factor for reversible reaction to assure maximum conversion in C Selectivity between the desired and by product in a CSTR

ξ =1−

RT

V – volume of the mass of reaction in the reactor; F – volumetric flow through the reactor; k –rate constant 𝛾 – excess factor; a, b – stoichiometric coefficients

V e−k F

aA + bB ⇐⇒ cC k1 , k−1

𝛾=

V− volume of the mass of reaction in the reactor; F – volumetric flow through the reactor; k – rate constant; K – equilibrium constant; ξ∞ – the final conversion at the end of the reaction

aCB0 bCA0

Parallel reactions k1

k2

A + B 󳨀→ P, A + B 󳨀→ R (desired) ΦR = k2 1n2 −n1 1+ k CAf 1

CAf = CA0 (1 − ξ )

ΦR – selectivity of product R relative n −n to the undesired P; CAf2 1 – final concentration of A with the orders of reaction n1 and n2

76 | Part I: Introduction As a conclusion, the analytical mathematical models are very general and their understanding gives an insight into the process described and offers solutions for operation, design or control. It is quite interesting that the general laws can be applied regardless of the field of interest. This makes analytical modeling a powerful tool.

3.2 Statistical models Wherever the analytical models are difficult to elaborate, because some of the kinetics or thermodynamics of the process are not known, there is a solution of elaborating a model based on direct measurements in the process. The process (Fig. 3.1) model is the general one (equation (3.1)) showing the relationship between the output, state and input variables. Analytical models are treated in Section 3.1. When the general analytical approach of a process is not possible, because of the many uncertainties, the statistical approach is the most appropriate one. Generally, the steady state mathematical models are in the form of a regression which can be linear, m

y = f (x1 , x2 , . . . ., xm ) = ∑ ai xi , ai ∈ R, i = 1, n

(3.18)

i=1

or nonlinear y = f (x1 , x2 , . . . , xm , x1 x2 , x1 x3 , . . . , x12 , x22 , . . .)

(3.19)

where xi are the variables of the process and index m represents the maximum number of variables. The statistical analysis [8, 9] is based on measurements made in a process and their statistical processing is aimed at the construction of a linear or nonlinear regression.

Regression analysis Let us consider the simplest regression y =∝0 + ∝1 x1 + ∝2 x2 + . . . + ∝m xm .

(3.20)

The coefficients ai to be determined estimate the “real” values αi and constitute the vector a0 [ ] [ a1 ] [ ] [⋅⋅⋅ ] ] Ā = [ (3.21) [a ]. [ i ] [ ] [⋅⋅⋅ ] [ am ]

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Table 3.3. Program of experiments in an industrial or laboratory plant for elaborating a statistical model.

Measurement no.

x1

x2

1 2 ... j ... n

x11 x12 ... x1j ... x1n

x21 x22 ... x2j ... x2n

Input variables ... xi ... ... ... ... ... ...

xi1 xi2 ... xij ... xin

... ... ... ... ...

xm

Output variables y

xm1 xm2 ... xmj ... xmn

y1 y2 ... yj ... yn

To estimate the coefficients a set of n observed values of x̄ and ȳ is taken into account. The constant input value 1 is allocated to the free term a0 . The measurements (n measurements) done in the process constitute the matrices x̄ (inputs) and ȳ (outputs) with the corresponding matrices 1 x11 x21 ⋅ ⋅ ⋅ xi1 ⋅ ⋅ ⋅ xm1 [ ] [ 1 x12 x22 ⋅ ⋅ ⋅ xi2 ⋅ ⋅ ⋅ xm2 ] [ ] [ ] ⋅⋅⋅ ] X̄ = [ [ 1x x ⋅⋅⋅x ⋅⋅⋅x ] [ 1j 2j ij mj ] [ ] ⋅⋅⋅ [ ] [ 1 x1n x2n ⋅ ⋅ ⋅ xin ⋅ ⋅ ⋅ xmn ]

y1 [y ] [ 2] [ ] Ȳ = [ ⋅ ⋅ ⋅ ] [ ] [⋅⋅⋅] [ yn ]

(3.22)

where x0j = 1 is a fictive variable to give uniformity to the regression form corresponding to the first term of the regression which is a free term, a0 . The estimation is based on the least squares principle which minimizes the difference between the measured and the calculated values of y: ε = min SSE = min

2 1 n ∑ (y − a0 x0j − a1 x1j − a2 x2j − ⋅ ⋅ ⋅ am xmj ) . n j=1 j

In order to obtain the minimum, one has to fulfill the conditions (3.24) 𝜕ε 𝜕ε 𝜕ε = 0; = 0; . . . . = 0. 𝜕a0 𝜕a1 𝜕am

(3.23)

(3.24)

Each derivative = 0 will have as a result an equation of the type n 2 𝜕 { ∑ [yj − (a0 x0j + a1 x1j + a2 x2j + ⋅ ⋅ ⋅ + am xmj )] } = 0, 𝜕ai j=1

(3.25)

and, in the end, n

n

n

n

a0 ∑ x0j xij + a1 ∑ x1j xij + ⋅ ⋅ ⋅ + am ∑ xmj xij = ∑ xij yj , j=1

where i = 0 ÷ m.

j=1

j=1

j=1

(3.26)

78 | Part I: Introduction The equation system with m + 1 equations in the matrix form is ̄ = X̄ T Y.̄ X̄ T XA

(3.27)

̄ −1 X̄ T Ȳ A = (X̄ T X)

(3.28)

The solution in the matrix form is

with the ai coefficients

m

ai = ∑ i=0

Aij Δ

n

( ∑ xij yj ),

(3.29)

j=1

where Δ is the determinant of the matrix X̄ T X̄ and Aij is the algebraic complement of the ∑nj=1 xij yj element in Δ. In order to determine if this is the case, the semilinear or quadratic terms coefficients, aij , or aii , are calculated by the same formula (3.29), but adding to the measurement matrix the corresponding terms. For example, for the nonlinear regression y = f (x1 x2 , . . . , xm x1 x2 , x1 x3 , . . . , xm−1 xm ) = a0 + a1 x1 + a2 x2 + ⋅ ⋅ ⋅ + am xm + a12 x1 x2 + a13 x1 x3 + ⋅ ⋅ ⋅ + am−1,m xm−1 xm the matrix of measurements is 1 x11 x21 ⋅ ⋅ ⋅ xi1 ⋅ ⋅ ⋅ xm1 x11 x21 x11 x31 ⋅ ⋅ ⋅ xm−1,1 xm1 [ [ X̄ = [ 1 x12 x22 ⋅ ⋅ ⋅ xi2 ⋅ ⋅ ⋅ xm2 x12 x22 x12 x32 ⋅ ⋅ ⋅ xm−1,2 xm2 [ ⋅⋅⋅ [ 1 x1n x2n ⋅ ⋅ ⋅ xin ⋅ ⋅ ⋅ xmn x1n x2n x1n x3n ⋅ ⋅ ⋅ xm−1,n xmn

] ] ], ]

(3.30)

]

which is different to that presented in the equation (3.22), being extended with the terms xi−1,j xij with i = 2 ÷ m and j = 1 ÷ n. It should be noted that the minimum number of measurements necessary is n = m + 1, in this situation the system has a unique solution. But the higher n > m + 1 is, the more chances to better estimate the coefficients there are. Once the form of the regression is established to be the most appropriate and the corresponding coefficients are calculated, one has to verify the adequacy of the model for the reality of: (a) the accuracy of the measurements; (b) the weight of the coefficients in the regression; (c) the adequacy of the form of the model to the reality of the described plant. (a) The accuracy of the measurements is judged using the Cochran test [10]. Because the measurements are randomly affected by disturbances, the measurement errors are transmitted ultimately to the values of the parameters. In order to minimize these errors, it is indicated to increase the number of measurements: both repeating

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79

the output measurement at the same values of the input parameters and using the average of the outputs and, on the other hand, increasing the number of measurements with different input parameters values (n ≫ m + 1). In order to visualize the experimental errors, one repeats the measurements with the same input parameters r times: y11 y12 y13 ⋅ ⋅ ⋅ y1r [ y y y ⋅⋅⋅y [ 21 22 23 2r Y=[ [ ⋅⋅⋅ [ yn1 yn2 yn3 ⋅ ⋅ ⋅ ynr and the average y1̄ =

∑rl=1 y1l r ∑rl=1 y2l r

[ [ ̄ [ y = ̄ Y=[ 2 [ ... [ ∑r ynl [ yn̄ = l=1r

] ] ] ] ]

] ] ] ] ] ] ]

where r is the number of repetitions of the same measurement. With these values, one calculates the dispersion for each measurement 2 Syj

=

∑rl=1 (yjl − yj̄ )

2

(3.31)

r−1

2 2 and the maximum value of these dispersions, Smax = max Syj . The total measurement dispersion of the whole experiment is 2 ∑nj=1 Syj

Sy2 = The Cochran variable Gmax , Gmax =

S2max S2y

n

.

(3.32)

is compared with the GT value from the Coch-

ran distribution in order to appreciate the precision of the measurements. If Gmax > GT , the precision is insufficient and the number of repetitions has to be increased until Gmax < GT . (b) The importance of the coefficients in a model is judged based on the Student distribution [11]; the calculated coefficients are compared with their dispersion of determination. If the values are comparable, it means the calculated coefficients are in the noise range and their values are not considered. S2 {ai } =

Sy2 n

=

2 ∑nj=1 Syj

n ⋅ (r − 1)

.

(3.33)

For that, ti has to be larger than one tT value taken from the Student distribution for a degree of confidence usually larger than 95 %: ti =

S2

ai > tT . {ai }

(3.34)

80 | Part I: Introduction (c) The adequacy of the model form refers to the linearity or nonlinearity chosen for the model in order to describe, most appropriately, the plant reality. There is often the situation when the distribution of the data is parabolic, for example, and, by mistake, a linear regression is chosen to describe the behavior of the plant. The adequacy 2 of the measurements is assessed by calculating the remnant dispersion (Srem ) and that 2 of determination of the model (Sy ) minimizing the difference between the measured and calculated output: Δy = ymeas − yc = [ymeas − (a0 + a1 x1 + . . . am xm )] = min 2 Srem =

and Sy2 = The model is adequate [12] when F =

∑nj=1 Δy2 n − (m + 1) 2 ∑nj=1 Syj

S2rem S2y

n

.

(3.35)

(3.36)

(3.37)

< FT (Fischer–Snedecor test), where FT is

a tabular value of the Fischer distribution, corresponding to a certain degree of confi2 dence (> 95 %). Ideally, if Srem = Sy2 the whole inaccuracy is given by the measurement error, the model form being perfect and not introducing any bias. This is not true, in reality, if the values of FT are larger than 1. If the condition is not fulfilled, another form is proposed, the new coefficients are calculated and again the Fischer–Snedecor condition is verified. Once the three tests have been done and passed, with the modifications required, the regression can be declared satisfactory.

3.3 Artificial neural network Models Artificial neural networks (ANNs) are computer programs having a form of processing information inspired from the simplified representation of the way the human brain operates. These programs do not attempt to thoroughly reproduce the human brain mode of working but only its logical operation, based on a collection of computing entities named neurons. The connections between neurons are stronger or weaker. These connections are the support for representing the relationship between the cause and the effect, by sending and weighting the input flux of data towards the output. On the basis of given input-output pairs of data examples, the weights are designed and computed, this process being denoted as learning or training the ANN. The input data is processed from the input towards the output of the ANN according to the ANN’s architecture, i.e. the particular structure of neurons and weight values settled in the training step. Following the training procedure, the ANN may be further used for predicting the output for new inputs, not given during the training process, and work as a black box model [13].

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The basic element of an ANN is the neuron, which is a very simple computing structure. A simplified representation of the neuron and its computing principle is presented in Fig. 3.9 [14]. Inputs Weights W1, 1 u1 W1, 2 u2

Net input

Output

n ∑

f

y

W1, R

uR

y=f(n)=f(Wu+b)

Bias b

W=[W1, 1 W1, 2 .... W1, R]

1

u=[u1u2 .... uR]T

Fig. 3.9. Typical structure of the artificial neuron.

The artificial neuron has R inputs u1 , u2 , . . . , uR , which are represented by the vector u = [u1 u2 ⋅ ⋅ ⋅ uR ]T . Each of these inputs is multiplied (weighted) with the factors (weights) w1,1 , w1,2 , . . . , w1,R . The weights may be represented by the line vector W = [w1,1 , w1,2 ⋅ ⋅ ⋅ w1,R ]. The neuron first computes the net input, n, i.e. the sum of products (input × weight) to which a bias (scalar), b, is added n = w1,1 ⋅ p1 + w1,2 ⋅ p2 + ⋅ ⋅ ⋅ + w1,R ⋅ ⋅ ⋅ pR + b. The net input is then sent to be processed by a function, f , named activation or transfer function, which generates the output of the neuron by a very simple calculus: y = f (n) = f (w1,1 ⋅ u1 + w1,2 ⋅ ⋅ ⋅ u2 + ⋅ ⋅ ⋅ + w1,R ⋅ uR + b) = f (W ⋅ u + b). The activation function may have different forms but the most commonly used forms are the signum, linear and sigmoidal functions. The modeling power of the ANNs consists in the association of several neurons arranged in different topologies, with the aim of processing the input data and generating the output data. The typical topology of a multilayer ANN is presented in Fig. 3.10. 1

Input layer

2

3

Hidden layers

n–1

n

Output layer

Fig. 3.10. Typical structure of the multilayer ANN.

82 | Part I: Introduction

– –

– – – –

The multilayer ANN architecture is built on the following rules. Artificial neurons are organized in layers, each layer consisting in a set of neurons having the same activation function. Each ANN has one input layer, which is the input gate for the data to be processed, one output layer, which is the gate of delivering the processed data, and several intermediate layers named hidden layers, where the input data is processed step by step and sent towards the ANN output. The connection paths (represented by arrows) are oriented from the input towards the output. Each neuron of a layer is only interconnected with all neurons of the neighboring (adjacent) layers. Each neuron receives data from all neurons of the previous layer, computes its output and sends data to all neurons of its next layer. The ANN may have a various number of hidden layers and neurons in each hidden layer, but the number of neurons in the input and output layer are fixed by the particular modeling problem (the dimension of the input and output pair examples of training data).

The ANNs may build complex models and are very efficient especially for cases when the laws governing input and output are not known or insufficiently formalized. They are trained on the statistical basis of the implicit information contained in the training examples. ANNs can also be used for one special case of modeling, namely for classification. ANNs may separate the elements of a set in classes, on the basis of a priori settled classification categories or by discovering these categories by themselves. The capacity of this structure of connected computing elements to process data using a very short computing time, compared to the numerical methods, is a very much appreciated feature for saving computer resources.

Simulation Simulation is the process of imitating reality using a model. The model is solved and its solutions represent the variations induced in the process by the inputs, or construction constants. Simulators are used in industry to train the operators before operating a process and to test different alternatives of design and control.

3.4 Examples of mathematical models Example 3.9. Non isothermal CSTR k

Considering the CSTR (Fig. 3.2) in which an exothermic first order reaction A 󳨀→ B takes place. The heat of reaction is ΔHr < 0. According to the procedures proposed at

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the beginning of the chapter, 1. elaborate a clear and correct flow sheet or drawing of the process described; if the process is too complex, divide it in lower complexity modules; 2. identify all variables in the process and understand the correlation cause-effect between them: first inside the modules and second between the modules. The variables of the process are specified on the drawing. 3.

Elaborate the lists of variables and constants: output variables, input variables, design and construction parameters, thermodynamic and kinetic constants.

Output variables are those of interest from the point of view of the reaction: F, CA , CB , T∘. ∘ . Input variables are those influencing the output ones: Fvi , CAi , Ti∘ , Fvagi , Tagi State variables: V, T ∘ , ρ , c , C , C , T ∘ , T ∘ . p

A

B

ag

w

Properties of the reactants/products, physical constants: ρA , ρB , MA , MB , cpA , cpB , k0 , Ea , ΔHr , R, g, ρag , cpag . Design and construction parameters: AR , Kv0 , Vmin , Vj , KT , AT , Mp , cpp , KTi , KTag , Ap , lp , dp , λ . Nomenclature: 3 F, Fvi – outflow and input flow of the reactor ( ms ) CAi , CA , CB – molar concentrations of the reactant in the input flow and of A and B in the reactor ( kmol ) m3 T ∘ – temperature inside the reactor and in the outflow (K) Ti∘ – temperature inside the reactor and in the outflow (K) 3 Fvagi – cooling agent flow in the jacket ( ms ) ∘ , T ∘ – input and jacket temperature of the cooling agent (K) Tagi ag T ∘ – temperature of the inner wall of the reactor (K) w

V, Vmin , Vj – volume of the reaction mass in the reactor, minimum volume allowed in the reactor for normal functioning of the level control and volume of the jacket (m3 ) ρA , ρB , ρ – density of A, B and of the mixture of A and B ( mkg3 ) MA , MB – molar mass of A and B ( kmol ) m3 cpA , cpB , cp – heat capacity of A, B and of the mixture ( kgJ K ) k, k0 – rate constant and pre-exponential factor ( 1s , 1s ) J Ea – activation energy ( kmol ) J R – universal gas constant (= 8,314 kmol ) K g – gravitational acceleration (= 9.8 sm2 ) J ΔHr – heat of reaction ( kmol ) ρag , cpag – density and heat capacity of the cooling agent ( mkg3 ,

J ) kg K 2

KT , AT – heat transfer coefficient and heat transfer area ( mW2 K , m )

84 | Part I: Introduction KTi , KTag – partial heat transfer coefficients inside the reactor and in the jacket ( mW2 K ) Mw , cpw – mass of the reactor wall and its heat capacity (kg, kgJ K ) Ap , lp , dp – cross area of the outflow pipeline, the length of the pipeline, diameter of the pipeline (m) λ – coefficient of pressure loss on the pipeline (−) 4. Elaborate the list with simplifying hypotheses: – the reactor is perfectly mixed (this shows we can consider the same concentrations, same density, same temperature at each point of the mass of reaction and also at the reactor’s output) and all reactions take place in liquid phase (no vapor phase to be considered); – the reaction does not continue in the output pipeline (it means that the concentrations in the reactor are the same as those at the end of the pipeline); – the reactor is cylindrical (simplifies the calculus of height of the liquid inside the reactor); – the reactor and pipelines are perfectly isolated thermally (this simplifies the heat balance because it does not consider the heat losses; usually these are around 5% of the heat load); – the fluctuations of temperature are not too large, meaning properties do not change with temperature; – the jacket is completely filled with cooling agent (this means Fvagi = Fvag ); – the wall of the reactor is considered “thin” from the heat transfer point of view (this means the transfer through the wall is instantaneous and does not incur any delay). 5.

Write the equations of the model if possible in the “natural” sequence of the development of the process; ensure that all unknowns in the equations are expressed in new “secondary” equations; Each variable on the lists is assigned one equation; if one variable is not on the lists, it should be assigned a new equation until all variables are “covered” by equations. Usually, for flows, momentum balance equations, for temperatures, energy balance equations and for mass and concentrations, mass balance equations are used.

Thus, for F: either ρ ghAp − λ

lp ρ v2 d A = (A l ρ v) dp 2 p dt p p

(3.38)

or the control equation F = F0 + K V0 (V − Vmin ).

(3.39)

is assigned. The second equation expressing the flow is the ‘‘level control” equation, where Kv0 is the gain factor of the level controller. The outflow is increasing directly

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85

with the increase of the volume of the mass of reaction over the minimum value required for a normal functioning of the control loop. The first equation gives the solution for v, needing an extra equation for F: F = vAp . Looking for the “new” variables we notice h is on no list, V, v, and ρ are on the list of state variables and we should link them to some “new” equations: V ; this equation introduces the need of another equation for V AR d for V: Fvi ρA − Fρ = (Vρ ) dt for ρ : ρ = MA CA + MB CB (Tab. 3.1)

for h: h =

(3.40) (3.41) (3.42)

for CA and CB component mass balance equations will be written Fvi CAi − FCA − VkCA = 0 − FCB + VkCA =

d (VCA ) dt

(3.43)

d (VCB ). dt

(3.44)

Whatever is transformed from A, contributes to B (+VkCA ). For T ∘ should be written the energy balance equation ∘ ) − VkCA ΔHr = Fvi ρA cpA Ti∘ − Fρ cp T ∘ − KT AT (T ∘ − Tag

d (Vρ cp T ∘ ) dt

(3.45)

for cp : cp = xA cpA + xB cpB (Tab. 3.1)

(3.46)

M M for xA : xA = CA A and xB = CB B ρ ρ

(3.47)

∘ ∘ ∘ ∘ )= for Tag : Fvag ρag cpag Tiag − Fvag ρag cpag Tag + KT AT (T ∘ − Tag

d (V ρ c T ∘ ) (3.48) dt j ag pag ag

the agent flow is the same since the jacket having the volume Vag is completely full; in this example, the inner wall of the reactor is considered “thin”, this having as a consequence the existence of only one KT . If the wall is not thermally “thin” (Fig. 3.11), the equation takes another form: ∘ ∘ ∘ Fvag ρag cpag Tiag − Fvag ρag cpag Tag + KTag AT (Tw∘ − Tag )= ∘ )= KTi AT (T ∘ − Tw∘ ) − KTag AT (Tw∘ − Tag

d (V ρ c T ∘ ) dt j ag pag ag

d (M c T ∘ ). dt w pw w

(3.49) (3.50)

A “thick” wall is either thick physically having an important dimension (e.g. the heat exchanger of the polyethylene produced in the high pressure process has a wall of stainless steel of 20 cm), or it is thermally “thick” being thin physically, but a very bad thermal conductor (e.g. an enameled steel wall). 6.

Solve the system of equations and interpret the results.

86 | Part I: Introduction D

KTag

KTi

T° T°w

T°i

T°ag L

Fig. 3.11. The wall is thermally “thick”.

Thus, the example of a CSTR involving practically only four variables describing the production quantity and its quality has a simplified model of 12–14 equations, 6–7 differential and 6–7 algebraic. The system is nonlinear and can be solved using a MatLab subroutine ODE15s, ODE23, ODE45. The interpretation of the results shows the behavior of all variables in time when one or several inputs change or when some of the characteristics of the process change (e.g. transfer coefficients due to deposits of limestone in the shell of the heat exchanger or in the cooling jacket). Example 3.10. Cascade of three CSTRs In some processes, one reactor is not enough and several reactors in cascade are used. The reactors in the example are isothermal, perfectly mixed, perfectly insulated and the reaction takes place only in the liquid phase. They function at the same temperature, have different holdups, same density and then the different outflows. The reack

tion is A 󳨀→ B, with ΔHr = 0. Fi , CAi V1

F1 , CA1 , CB1 Fig. 3.12. Cascade of three CSTR.

V2

F2 , CA2 , CB2

V3

F3 , CA3 , CB3

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The output variables are F3 , CA3 , CB3 . The input variables are Fi , CAi . The state variables are V1 , V2 , V3 , CA1 , CB1 , CA2 , CB2 , CA3 , CB3 , F1 , F2 , F3 . The model has 12 equations (first 9 independent), since the output variables are state variables as well: d Fi ρ − F1 ρ = (V1 ρ ). (3.51) dt d F1 ρ − F2 ρ = (V2 ρ ) (3.52) dt d (V ρ ) F 2 ρ − F3 ρ = (3.53) dt 3 d (V C ) Fi CAi − F1 CA1 − V1 kCA1 = (3.54) dt 1 A1 d F1 CA1 − F2 CA2 − V2 kCA2 = (V2 CA2 ) (3.55) dt d F2 CA2 − F3 CA3 − V3 kCA3 = (V3 CA3 ) (3.56) dt F1 = KV1 (V1 − Vmin1 ) (3.57) F2 = KV2 (V2 − Vmin2 )

(3.58)

F3 = KV3 (V3 − Vmin3 )

(3.59)

d 0 − F1 CB1 + V1 kCA1 = (V1 CB1 ) (3.60) dt d F1 CB1 − F2 CB2 + V2 kCA2 = (V2 CB2 ) (3.61) dt d (V C ) F2 CB2 − F3 CB3 + V3 kCA3 = (3.62) dt 3 B3 The system to be solved has thus 12 differential equations and we solve the first nine equations (3.51)–(3.59) of the equation system for F1 , F2 , F3 , CA1 , CA2 , CA3 , V1 , V2 , V3 . With these values, CB1 , CB2 , CB3 are calculated. The steady state values are required to be able to solve the system. Example 3.11. Binary distillation column [15] A binary distillation column is considered with N trays which separates two products with the same relative volatility, α , along the column (on each tray, the volatility is the same). List of variables: Output variables: FB , FD , xB , xD . Input variables: FF , xF . State variables: Mj , Lj , xj , yj , MB , xB , yB , MD , xD , FR , V, FB , FD . Each distillation unit (tray) is represented in Fig. 3.15 + section with the weir.

88 | Part I: Introduction

FD , xD

FF , xF

FB , xB Fig. 3.13. Binary distillation column.

Fig. 3.14. Distillation column Illudest in the Control Laboratory of the Faculty of Chemistry and Chemical Engineering, University BabesBolyai, Cluj-Napoca, Romania.

Simplifying hypotheses: 1. Each tray has the same efficiency (Et = 100%). 2. Vapors leaving the tray are in equilibrium with the liquid phase on the tray.

3 Mathematical modeling

|

89

Lj+1 , xj+1 V, yj

j+1th tray

Mj , xj

jth tray

V, yj–1

j–1th tray Lj , xj

Fig. 3.15. The unit of distillation + section with the weir.

3.

The column is fed on the tray NF with the feed flow FF at the boiling temperature corresponding to the concentration xF of the volatile component in the liquid phase. 4. Each tray is perfectly mixed and has a holdup Mj with the molar composition xj , of the liquid phase and yj of the vapor phase. The holdup of the vapors on the tray is negligible (the vapor mass is ca. 1000 times smaller than that of the liquid in an atmospheric column). 5. The holdups of the bottom are MB (molar composition xB and yB ) and that of the reflux tank MD (molar composition xD and 0 - the condensation is total). 6. The bottom molar flow is FB with its composition xB and the distillate molar flow is FD , with its composition xD. 7. The reflux molar flow is FR and the vapor molar flow is V.Supposing the values of the latent heat of vaporization substantially close, it can be approximated that one mole of most volatile component vaporizes at the cost of condensing of one mole of less volatile component, having the result that there is no need for an energy balance equation on the column and the molar vapor flow Vis constant along the column. 8. On each tray, the internal molar reflux depends on the holdup of the tray and is Lj . 9. The dynamic response of the condenser is much faster than that of the column. 10. There is no chemical reaction on the trays. Having 4 variables on each tray, the bottom with three variables, the distillate drum with two and 4 other flows to be determined (FR , V, FB , FD ) it means the model has to have 4N+9 variables. Tray equations: Lj+1 − Lj =

d M dt j

Lj+1 xj+1 − Lj xj + Vj−1 yj−1 − Vj yj = yj =

α xj 1 + (α − 1)xj

(3.63)

d (M x ) dt j j

(3.64) (3.65)

90 | Part I: Introduction Lj = f (Mj ) or Francis equation Lj = 1.837lj h3/2 j ,

(3.66)

where lj is the length of the weir and hj is the height of the liquid over the weir. Feed tray equations: FF + LNF +1 − LNF =

d M dt NF

(3.67)

FF xF + LNF +1 xNF +1 − LNF xF + VNF −1 yNF −1 − VNF yNF =

d (M x ). dt NF NF

(3.68)

The equilibrium equation of the feed tray is the same with all tray equilibrium equations. Distillate drum equations: V − FD − F R =

d M dt D

VyN − FD xD − FR xD =

(3.69)

d (M x ). dt D D

(3.70)

Bottom equations: L1 − FB − V =

d M dt B

L1 x1 − VyB − FB xB = yB =

(3.71)

d (M x ) dt B B

α xB . 1 + (α − 1)xB

(3.72) (3.73)

The control equations: FD = FD0 + KvD (MD − MDmin )

(3.74)

FR = FR0 −K vR (xD − xDset ).

(3.75)

The reflux has to decrease when xD > xDset the setpoint value for xD : V = V0 + KvV (xB − xBset ).

(3.76)

The vapor flow has to increase when xB >xBset . The total number of equations is 4N + 9 as is the number of variables. Example 3.12. The efficiency of an esterification reaction (in %) depends on the molar concentrations (mol/l) of the reactants (acid and alcohol). A factorial experiment at

3 Mathematical modeling

| 91

Table 3.4. Input variables. Input variables

x1

x2

Center level Variation step

1.0 0.2

2.4 0.4

Table 3.5. Results of experiments. Input variables (concentration mol/l) x1 x2

Output variable (efficiency %) yI yII

1.2 1.2 0.8 0.8

0.5 0.8 0.2 0.55

2.8 2.0 2.8 2.0

Output average (%) y

0.3 0.6 0.1 0.25

0.4 0.7 0.15 0.40

two levels [9] was developed, in order to determine the correlation between the output and input variables. The aim is to determine a linear regression y = a0 + a1 x1 + a2 x2 . There is a repetition of the measurements for each combination; the average is taken into consideration. 1 [1 [ X=[ [1 [1

1.2 1.2 0.8 0.8

2.8 1.0 2.0 ] ] T [ ] ; X = [ 1.2 2.8 ] [ 2.8 2.0 ]

1.0 1.2 2.0

1.0 0.8 2.8

1.0 ] 0.8 ] ; 2.0 ]

Δ = det(X T X) = 0.4096 T

−1

A = (X X)

4.0 [ X T X = [ 4.0 [ 3.6

4.0 4.16 9.6

9.6 ] 9.6 ] ; 23.7 ] (3.77)

0.68 ] [ (X Y) = [ 0.624 ] , giving a regression y = 0.68 + 0.624x1 − 0.38x2 . [ −0.38 ] T

Statistical processing [13]: (a) Weight of the coefficients in the model. S2 {ai } =

Sy2 n {

=

=

∑nj=1 ∑rl=1 (yjl − yj̄ )

2

n (r − 1)

(0.5 − 0.4)2 + (0.3 − 0.4)2 + (0.8 − 0.7)2 + (0.6 − 0.7)2 + (0.2 − 0.15)2 } + (0.1 − 0.15)2 + (0.55 − 0.4)2 + (0.25 − 0.4)2

= 0.0041,

4⋅1 where n = 4 and r = 2.

(3.78)

92 | Part I: Introduction The Student test (for α = 0.95, tT = 2.776): a0 0.68 = = 167 > 2.776 S2 {ai } 0.0041 a1 0.625 = = 153 > 2.776 2 S {ai } 0.0041 a2 0.38 = = 93.54 > 2.776, S2 {ai } 0.0041

(3.79)

meaning all coefficients have importance in the model. (b) The adequacy of the model is tested using Fischer’s test. For a degree of freedom of 1 and for a confidence coefficient α = 0.95, FT = 7.71: 2

2 Srem

= =

∑nj=1 [ymeas − (a0 + a1 x1 + ⋅ ⋅ ⋅ + am xm )] n − (m + 1)

(3.80)

(0.4 − 0.365)2 + (0.7 − 0.669)2 + (0.15 − 0.115)2 + (0.45 − 0.419)2 = 0.0044, 1

where n = 4 and m = 2. 2 Srem 0.0044 = = 0.271 < 7.71. 0.01625 Sy2

(3.81)

The model is adequate to the reality.

3.5 Problems (1) A mathematical model of a steam generator (Fig. 3.16) has to be constructed, taking into account the following: volume of the liquid (Vl ) is much larger than that of the vapors (Vv ); the liquid phase is in thermodynamic equilibrium with the vapor phase, meaning: Tl∘ = Tv∘ , the vapor pressure of the liquid at Tl∘ , and the pressure of the vapors at Tv∘ are equal (Pl = pv ), the flow of evaporated liquid is equal to the outflow of the vapors (Nv = Fv ρv ); the liquid level is a controlled function of the input flow and vapor pressure function of the heating agent flow; heat losses to the exterior are negligible. The geometric dimensions of the generator are known; all characteristics of the liquid (ρl , cpl , lv ) are known; the gains of the P controllers of the two control loops are KL and KP . k

(2) The chemical reaction A 󳨀→ B + C, first order, regarding A, takes place in a semicontinuous STR (Fig. 3.17). The desired product is B. C is very volatile product, its evacuation is necessary in order to maintain the pressure in the reactor. The evacuation is done via a condenser

3 Mathematical modeling

|

93

Fv

Vv , cpv , ρv , Tv° PC Nv

TC

VI , cpI , ρI , TI°

V0 , cp0 , ρ0 , T0°

Fig. 3.16. Steam generator.

FV

PC

TC

V, CA , CB , CC , T°

Fig. 3.17. Semi-continuous tank reactor (SCSTR).

in order to prevent the loss of A and B. The flow FvC of C is pure. The relative volatilities of A and C relative to B are ∝AB = 1.2 and ∝BC = 10. The gases are considered perfect and the process is isobaric. Develop the mathematical model of the SCSTR. (3) A double effect evaporator (Fig. 3.18) is fed at the solvent evaporation temperature corresponding to the feed mass concentrations xF for the first evaporator and x1 for the second. The volumes of the liquid phase of both evaporators are V. The concentrations

94 | Part I: Introduction V1 , IV1

V2 , IV2

P1

V0 , IV0

P2

F, xF

F1 , x1

F2 , x2

Fig. 3.18. A double effect evaporator.

of the solvate inside the evaporators are x1 and x2 . The evaporators have level control systems. The primary heating agent is steam with the latent heat lv0 and the secondary heating agent is solvent with the latent heat lv1 . The solid fraction is contained only in the liquid phase being absent in the vapor one. The heat losses to the exterior are negligible. The steam and solvent are totally condensing and lose only their latent heat of vaporization lv0 and lv1 . A steady state and dynamic model of the system is required. (4) In order to model a heat transfer process between a fluid and a plane wall, several experiments have been done, with the results presented in Tab. 3.6. Table 3.6. Results of experiments. Re

50

100

150

300

500

Nu

0.114

0.121

0.126

0.134

0.140

The regression to express the best dependence Nu = f (Re) is to be found. The statistical processing of data is expected. (5) A stirred tank is considered, operating at atmospheric pressure, fed with two fluxes having variable volumetric flows F1 (t) and F2 (t), temperatures T1o and T2∘ , specific heats cp1 = cp2 = cp and densities ρ1 = ρ2 = ρ (Fig. 3.19).

3 Mathematical modeling

F1 , C1 , T1°

|

95

F2 , C2 , T2°

V, C, T°

Fʼ, C, T°

H F, C, T° Fig. 3.19. CSTR with two influents.

Both fluxes contain a dissolved material having the molar concentrations C1 , and C2 respectively. The output flow is F(t), with concentration C(t), density ρ , and specific heatcp . It is supposed the tank is perfectly stirred and the mixing is done without additional heat and without heat exchange with the exterior. The values of densities, specific heat, of the inputs and outputs are considered equal. Considering the dependence of the output flow F(t) function of the tank liquid content (variable) F(t) = k ⋅ √V(t), one has to determine the following: (a) The dynamic mathematical model of the stirred tank taking care to evidence the unknowns: C(t), T ∘ (t)andV(t). There will be considered: – input variables: the volumetric flows F1 (t)siF2 (t). – output variables: output flow concentration C(t), output flow temperature T ∘ (t), and tank liquid volume V(t). (b) Which way does the dynamic mathematical model change when the heat losses are not negligible? (the heat transfer coefficient is KT and heat transfer area of the tank is AT ), the evacuation of the tank is done at the height H (dotted line in the figure). The characteristics of the hydraulic circuit (ξ , λ , lp , dp etc.) are assumed to be known. (6) During the PVC batch suspension process, the determinant characteristic of the PVC is given by Kw = f (T ∘ , min ) where Kw is the so called Kwert, T ∘ is the reaction temperature, and min is the total mass of initiator. The experimental data measured in the process are given in Tab. 3.7. Calculate the regression coefficients of the following types: (a) Kw = a + bT ∘ + cmin , (b) Kw = a(T ∘ )b mcin , and discuss which model is the most appropriate to describe the measurements.

96 | Part I: Introduction Table 3.7. The experimental data.

j

T∘ [∘ C]

min [kg]

I

Kw II

Kw, av

1 2 3 4 5 6

45 52 50 48 52 51

1.5 3 2 2.4 1 4

54.63 61.37 57.28 58.9 52 63.75

54.23 6.17 57.58 58.66 52.12 63.85

54.43 61.27 57.43 58.78 52.06 63.95

References [1] [2] [3] [4] [5] [6] [7]

[8] [9]

[10] [11] [12] [13] [14]

[15]

Smith, G. M., Van Ness, H. C., Abbot, M. M., Introduction to Chemical Engineering Thermodynamics, 7th Edition, McGraw Hill, 2011. Smith, J. M., Chemical Engineering Kinetics, McGraw-Hill Chemical Engineering Series, 1981. Franks, R. G. E., Modeling and Simulation in Chemical Engineering, Wiley Interscience, New York, Chapter 9, 1972. Myskis, A. D., Introductory Mathematics for Engineers – Lectures in Higher Mathematics, Mir Publishers, Moscow,1972, p. 497. MATLAB and Simulink for Technical Computing, TheMathWorks, Inc., Apple Hill Drive, Natick, Massachusetts 01760 USA. Wolfram Mathematica 8, Wolfram Research, Inc., 100 Trade Center Drive, Champaign, IL 618207237, USA. Floarea, O., Jinescu G., Balaban C., Dima, R., Operat.ii s.i utilaje în industria chimică – Probleme (Operations and equipment in the chemical industry – problems), Editura Didactică s.i Pedagogică, Bucures.ti, 1980, p. 4. Freund J. R., Wilson W. J., Regression Analysis: Statistical Modeling of a Response Variable, Academic Press, San Diego, 1998, p. 75. Penescu C., Ionescu G., Tertis.co M., Ceangă E., Identificarea experimentală a proceselor automatizate (Experimental identification of the controlled processes), Editura Tehnică, Bucures.ti, 1971, p. 228. Cochran, W. G., Cox, G. M., Experimental Designs, John Wiley and Sons, London, 1952, p. 77. Box, G. E., Multifactor design of the first order, Biometrika, 39, (1952), 49–57. Mehta, C. R., Patel, N. R., Tsiatis, A. A., Exact significance testing to establish treatment equivalence with ordered categorical data, Biometrics, 40 (3), (1984), 819–825. Freund J. R., Wilson W. J., Regression Analysis: Statistical Modeling of a Response Variable, Academic Press, San Diego, 1998, p. 407. Sipos, A., Pasat, G. D., Cristea, V. M., Mudura, E., Imre, L. A., Bratfalean, D., Modelarea, simularea si conducerea avansată a bioproceselor fermentative, Editura Universităţii “Lucian Blaga”, Sibiu, 2010. Luyben, W., Process Modeling, Simulation and Control for Chemical Engineers, McGraw Hill, 1990, p. 70.

4 Systems dynamics From the dynamic behavior point of view, systems may be classified in the following types [1, 3]: – proportional system, – integral (or pure capacitive) system, – derivative system, – first order (or first order capacitive) system, – second order system with underdamped, critically damped, oscillating and overdamped response (second order capacitive system), – higher order (n-th order multicapacitive) system, – pure delay (dead time) system.

4.1 Proportional system The proportional system is described by the input-output relationship: y(t) = K ⋅ u(t),

(4.1)

where K is denoted as the constant of proportionality between the output and the input variable [1, 3]. The proportional system implies an instantaneous relationship between the input variable (cause) change and output variable (effect) change. As all real systems have inertia (e.g. due to their mass), the time relationship between the input and output variables features a certain nonzero time lag or time delay. From this perspective, the proportional system is placed at the physically feasible limit. Nevertheless, when comparing two systems, one with a very large time lag or time delay and the other one with a very small inertia, the latter may be considered a proportional system from the dynamic point of view, because its dynamic response may be regarded as instantaneous in comparison to the slow one. The unit step input and the proportional system response to this input are presented in Fig. 4.1. The transfer function of the proportional system emerges from the Laplace transform of equation (4.1): Y(s) HP (s) = (4.2) =K U(s) and the frequency response function becomes HP (jω ) =

Y(jω ) = K. U(jω )

(4.3)

98 | Part I: Introduction u0(t)

y(t)

1

K

0

t

0

t

Fig. 4.1. Proportional system unit step response.

Bode diagram of magnitude for the proportional system

|Hp|dB [dB]

20lgK

lg(ω) –2 –1.5

–1

–0.5 0 0.5 1 1.5 Frequency [log(rad/s)]

2

2.5

3

Bode diagram of phase for the proportional system φp 1 [rad] 0.5 0 –0.5 lg(ω)

–1 –2 –1.5

–1

–0.5 0 0.5 1 1.5 Frequency [log(rad/s)]

2

2.5

3

Fig. 4.2. Bode diagrams of the proportional system

According to equation (4.3) the magnitude of the frequency response function |HP (jω )| equals K and its phase φP (jω ) is zero. The Bode diagrams of the proportional system are presented in Fig. 4.2. Example 4.1. Due to the fact that in electrical systems the transport rate of electrical charges may be very high, compared to usual mass, momentum or heat transfer processes, it may well be considered that in the resistive electrical circuit, such as the one presented in Fig. 4.3, the output voltage Uo is depending on the input voltage Ui

4 Systems dynamics

| 99

according to the proportional system behavior: Uo (t) =

R2 U (t), R1 + R 2 i

(4.4)

where the gain is K = R2 /(R1 + R2 ). R1

Ui

R2

Uo

Fig. 4.3. The resistive electrical circuit may be considered a proportional system.

The response of the electrical circuit to a step input voltage change is roughly instantaneous and attenuated with the gain factor K.

4.2 Integral system The integral system is described by the input-output relationship: t

1 y(t) = ∫ u(τ )dτ Ti

(4.5)

0

where the Ti is an integral constant [1, 3]. The integral system is also known as the pure capacitive system, because its output (effect) is proportional to the accumulation (positive or negative) of the input variable (cause). As a result, the integral system behaves as a pure capacity that is filling up or emptying due to the net difference between the incoming and outgoing fluxes of the extensive property, considered as input. This accumulation is proportional with the 1/Ti constant. The unit step input and the integral system response to this input are presented in Fig. 4.4. The transfer function of the integral system emerges from the Laplace transform of equation (4.5): Y(s) 1 HI (s) = (4.6) = , U(s) Ti s and the frequency response function becomes HI (jω ) =

Y(jω ) 1 j = =− . U(jω ) Ti jω Ti ω

(4.7)

100 | Part I: Introduction u0(t)

y(t)

1 1/Ti 0

t

0

t

Fig. 4.4. Integral system unit step response.

According to equation (4.7) the magnitude of the frequency response function |HI (jω )| equals 1/(Ti ω ) and its phase φI (jω ) is tan−1 (−1/(Ti ω )/0) = −π /2 (negative phase, i.e. phase lag). The Bode diagrams of the integral system are presented in Fig. 4.5. |HΙ|dB [dB] 60

Bode diagram of magnitude for the integral system

40 20 0 –20 –40 –2

lg(ω) –1.5

–1 –0.5 0 0.5 1 1.5 Frequency [log(rad/s)]

2

2.5

3

Bode diagram of phase for the integral system φΙ –0.5 [rad] –1 –1.5 –π/2 –2 –2.5 lg(ω)

–3 –2

–1.5

–1 –0.5 0 0.5 1 1.5 Frequency [log(rad/s)]

2

2.5

3

Fig. 4.5. Bode diagrams of the integral system.

Example 4.2. For illustrating the integral system behavior, consider the system consisting in a tank continuously fed and evacuated with a liquid phase, Fig. 4.6 [2]. The inlet mass flow Fmi (t) is considered time dependent and outlet mass flow Fmo (t) is constant. The system may be delimited by the interface Σ , with the terminal variable of

4 Systems dynamics

| 101

Fmi

∑

H

H0 Fmo=constant

Fig. 4.6. Integral system example: mass accumulation in a tank with constant outlet flow.

input type u(t) = Fmi (t) − Fmo (t) (the cause) and the terminal variable of output type y(t) = H(t) − H0 = ΔH (the effect). The change of the liquid level in the tank H − H0 , as output variable, is depending on the net inlet flow Fmi (t)−Fmo (t), according to an integral relationship emerged from the mass balance equation: Fmi (t) − Fmo = A ⋅ ρ

dH dt

(4.8)

which, by integration, becomes t

1 ∫ (Fmi (τ ) − Fmo )dτ H = H0 + Aρ 0

t

1 ∫ (Fmi (τ ) − Fmo )dτ . and ΔH = Aρ

(4.9)

0

The change of the liquid level in the tank shows an integral relationship with respect to the net difference between the inlet and outlet mass flow rates.

4.3 Derivative system The derivative system is described by the input-output relationship: y(t) = Td

du(t) , dt

(4.10)

where the Td is a derivative constant [1, 3]. The unit step input and the derivative system response to this input are presented in Fig. 4.7. The unit step response of the derivative system has a Dirac function (impulse) form.

102 | Part I: Introduction

y(t)=Tdδ(t)

u0(t)

Area=Td 1

0

t

0

t

Fig. 4.7. Derivative system unit step response.

The transfer function of the derivative system emerges from the Laplace transform of equation (4.10): Y(s) HD (s) = (4.11) = Td s U(s) and the frequency response function becomes HD (jω ) = j ω Td .

(4.12)

According to equation (4.12), the magnitude of the frequency response function |HD (jω )| equals Td ω and its phase φD (jω ) is tan−1 ((Td ω )/0) = π /2 (positive phase, i.e. phase lead). The Bode diagrams of the integral system are presented in Fig. 4.8. The (pure) derivative system is not physically feasible, but a system having a large derivative constant Td and a small time constant T may be approximated to a derivative system. Example 4.3. Consider the electrical circuit consisting in an Operational Amplifier with positive feedback, presented in Fig. 4.9. Applying the second Kirchoff law on the ABCDEG loop the following equation is obtained: Ui − Uc − RI − Uo = 0, (4.13) and from the conservation law of the electrical charges on the conductive plates of the capacitor Ce dU I = Ce c (4.14) dt the equation (4.13) becomes Uo = Ui − RCe

dUc − Uc . dt

(4.15)

Considering the second Kirchoff law on the ABFG loop Ui − Uc − Ri Ii = 0,

(4.16)

4 Systems dynamics

| 103

Bode diagram of magnitude for the derivative system

|HD|dB [dB] 60 40 20 0

lg(ω)

–20 –2

0 1 Frequency [log(rad/s)]

–1

2

3

Bode diagram of phase for the derivative system

φD [rad]

3 2.5 2 π/2 1.5 1 lg(ω)

0.5 –2

0

–1

1

2

2.5

3

Frequency [log(rad/s)] Fig. 4.8. Bode diagrams of the derivative system.

R I Ce A B Ui

Ii KA

Uc

D

Ri C

G

Uo

F E

Fig. 4.9. Operational Amplifier with a derivative system behavior.

and taking into account that the current Ii is very small (as input current in the Operational Amplifier), it may be approximated that Ui ≈ Uc . Consequently, equation (4.15)

104 | Part I: Introduction becomes

dUi (4.17) , dt which shows a derivative behavior, with the derivative time constant Td = RCe . The minus sign denotes the inverse polarity with respect to the Ui voltage. Due to the capacitor Ce , for a step input voltage Ui change, only the part of the rapid input voltage change of the step is sent to the output. Uo = −RCe

4.4 First order system The first order (or first order capacitive) system is described by the input-output relationship: dy(t) T (4.18) + y(t) = Ku(t), dt where the T is denoted as the time constant and K as the steady state gain of the first order system [1, 2, 4]. Described by a first order differential equation, the first order system is physically feasible and a large class of systems may be approximated to its dynamic behavior. The unit step input and the first order system response to this input are presented in Fig. 4.10. y(t) B

yst

< 0,95(yst–y0)

u0(t) 0,63(yst–y0) 1

y(0)=y0

A M

0

t

0

T

3T

t

Fig. 4.10. First order system unit step response.

The explicit solution of the first order equation (4.18), for the constant input u0 (t) and y(0) = 0, is t t y = K u0 (t) (1 − e− T ) = K(1 − e− T ), t ≥ 0. (4.19) The time constant T is defined as the period of time needed by the unit step response of the first order system to reach 63.2 % of its net change. The net change is the difference between the final steady state value and the initial value of the system output. This definition of the time constant is revealed by making t = T in equation (4.19). The output becomes T

y(T) = K(1 − e− T ) = K(1 − e−1 ) = 0.632 K.

(4.20)

4 Systems dynamics

| 105

The time constant can also be determined graphically, as shown in Fig. 4.10, by drawing the tangent AB in the origin A of the step response plot and considering the intersection of the tangent with the steady state asymptote, i.e. point B. The perpendicular drawn from point B to the abscissa determines the point M and the length of the segment OM is equal to the time constant T. The time constant is a measure for the rate by which the first order system changes its output as response to the input change (dynamic characteristic). The response time is the time it takes for the step response to reach about 95 % of its final change. It is considered to be of the order of 3T. The steady state (static) gain is defined as the ratio between the output change and the input step change (the change is considered as the difference between the final steady state value and the initial steady state value, for both the input and the output). For the step input change the gain is K=

y − y0 y(∞) − y(0) = st = yst − y0 . u0 (∞) − u0 (0) 1−0

(4.21)

When the step input change does not have a unit value, but the amplitude of the change is umax − umin , the first order steady state system gain may be computed as the ratio (yst − y0 )/(umax − umin ). The steady state gain shows the magnitude of the inlet variable influence on the output variable (steady state characteristic). The transfer function of the first order system emerges from the Laplace transform of equation (4.18): Y(s) K HD (s) = (4.22) = U(s) Ts + 1 and the frequency response function becomes HFO (jω ) =

K K KT ω K(1 − j T ω ) = −j . = 2 2 jTω +1 1 + (T ω ) 1 + (T ω ) 1 + (Tω )2

(4.23)

According to equation (4.23) the magnitude of the frequency response function |HFO (jω )| equals K/(1 + T 2 ω 2 )1/2 and its phase φFO (jω ) is − tan−1 (Tω ). The Bode diagrams of the first order system are presented in Fig. 4.11 [2]. The first order system is an accumulator of mass, energy or momentum. Example 4.4. An isothermal continuous stirred tank reactor (CSTR) in which a first order reaction is transforming the A reactant into B product can be considered as an accumulator of component A. Due to the change of the inlet concentration of A, CAi , entering the reactor (considered as the input variable) the output concentration of A, CA , (considered as the output variable) is changing. The mass balance of the component A in the reactor system, for constant volume and equal inlet/outlet volumetric flows F, is described by FCAi − FCA − VkCA =

d (VCA ). dt

(4.24)

106 | Part I: Introduction

|HFO|dB 0 [dB] –20

Bode diagram of magnitude for the first order system

–40 –60 lg(ω)

–80 –2 –1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

3

Frequency [log(rad/s)]

Bode diagram of phase for the first order system φFO [rad]

0 –0.5 –1

–1.5 –π/2 –2

lg(ω) –2 –1.5

–1

–0.5 0 0.5 1 1.5 Frequency [log(rad/s)]

2

2.5

3

Fig. 4.11. Bode diagrams of the first order system.

Following simple transformations, equation (4.24) becomes: V dCA F + CA = C , F + Vk dt F + Vk Ai

(4.25)

leading to the time constant and steady state gain parameters of the first order system: T=

V , F + Vk

K=

F . F + Vk

(4.26)

4.5 Second order system The second order system is described by the input-output relationship 1 d2 y(t) 2ζ dy(t) + + y = Ku(t), ωn dt ωn2 dt2

(4.27)

where the ωn is the natural (angular) frequency of oscillation, ζ is the damping factor of the oscillations, and K is the steady state gain of the second order system [1, 4, 5]. The second order system unit step response is presented in Fig. 4.12.

4 Systems dynamics

Overdamped Critically damped Underdamped Oscillating

y(t)

u0(t)

| 107

1

1 0

t

0

t

Fig. 4.12. Second order system unit step response.

According to the values of the damping factor ζ , the analytical (explicit) solution of equation (4.27), for u(t) = u0 (t), has different forms. These forms are y(t) = Ku0 (t)[1 − e−ζωn t ( cosh ωn √ζ 2 − 1t +

ζ √ζ 2

sinh ωn √ζ 2 − 1t)]

for

ζ >1

−1 (4.28)

y(t) = Ku0 (t) [1 − (1 + ωn t) e

− ωn t

y(t) = Ku0 (t)[1 − cos(ωn t)] for y(t) = Ku0 (t)[1 −

e−ζ ωn t √1 − ζ 2

]

for

ζ =1

(4.29)

ζ =0

sin (ωn √1 − ζ 2 t + arctan (

(4.30)

√1 − ζ 2 ζ

))] for

0 < ζ < 1. (4.31)

For values of the damping factor ζ exceeding unity, the second order system has an overdamped behavior, presented in equation (4.28), with a sluggish response. The shortest overdamped behavior, i.e. the critically damped response, is obtained for the damping factor ζ = 1, equation (4.29). When the damping factor ζ is less than unity the system response becomes underdamped and shows an oscillating behavior with diminishing amplitude, equation (4.31). For the case when the damping factor equals zero ζ = 0, the second order system exhibits an oscillating behavior with constant amplitude, equation (4.30). The transfer function of the second order system emerges from the Laplace transform of equation (4.27): HSO (s) =

Y(s) K = 1 2 2ζ U(s) s + s+1 ωn ωn2

(4.32)

108 | Part I: Introduction and the frequency response function becomes after simple transformations 1−( HSO (jω ) = (1 −

ω2 ωn2

ω 2 ) ωn 2

) +

−j

4ζ 2 ωn2

ω2

2ζ ⋅ω ωn 2

4ζ 2 ω2 (1 − 2 ) + 2 ω 2 ωn ωn

.

(4.33)

The Bode diagrams of the second order system are presented in Fig. 4.13 [2]. Bode diagram of magnitude for the second order system

|HSO|dB

ξ=0

20

ξ=0.1 0 ξ=0.7

–20 –40 –60 –1

–0.5

0

0.5 lg(ωn ) 1

1.5

2

lg(ω) 2.5

Frequency [log(rad/s)]

Bode diagram of phase for the second order system φSO [rad]

0 ξ=0.7 –1 ξ=0.1

–π/2 –2

ξ=0 –π

–3 –4 –1

–0.5

0

0.5 lg(ωn ) 1 1.5 Frequency [log(rad/s)]

2

lg(ω) 2.5

Fig. 4.13. Bode diagrams of the second order system.

For the case of the critically and overdamped behavior, the second order system may emerge from the coupling in series of two first order capacitive systems. Consequently, this second order system is denoted as the second order capacitive system. Example 4.5. Consider the case of the system composed of two isothermal CSTRs that are connected in series, as shown in Fig. 4.14 [1]. In both of them, a first order reaction is transforming the A reactant into the B product. Each of the CSTRs has as input the change of the inlet concentration of A component entering the reactor and as output the change of the output concentration of A component. The volume of the inventory in each reactor is kept constant.

4 Systems dynamics

CAi , F

A

B

A

| 109

B

CA1

CA2

CB1

CB2

V

V

CA1 , F

CA2 , F

Fig. 4.14. Second order system emerged from the series connection of two CSTRs.

The system of CSTRs has a second order critically damped behavior on the transfer path having CAi as input and CA2 as output. They are described by the mass balance equations for the A component dCA1 + (F + Vk)CA1 = FCAi dt dC V A2 + (F + Vk)CA2 = FCA1 dt

V

(4.34)

which emerge in the global equation d2 CA2 V dCA2 F2 V2 + 2 = C , + C A2 F + Vk dt (F + Vk)2 dt2 (F + Vk)2 Ai

(4.35)

where

F + Vk F2 . (4.36) , ζ = 1, K = V (F + Vk)2 The transfer function of the CSTRs system may be obtained by applying the Laplace transform to the equations (4.34) ωn =

T1 s CA1 (s) + CA1 (s) = K1 CAi (s)

(4.37)

T2 s CA2 (s) + CA2 (s) = K2 CA1 (s), where

V F , K 1 = K2 = F + Vk F + Vk and by following simple transformations becomes the form T1 = T2 =

HSO (s) =

CA2 (s) K 1 K2 K 1 K2 . = = CAi (s) (T1 s + 1)(T2 s + 1) [T1 T2 s2 + (T1 + T2 )s + 1]

(4.38)

(4.39)

4.6 Higher order system When connecting in series multiple first order capacitive systems, the input-output relationship features an increased time lag and presents the so called higher order

110 | Part I: Introduction capacitive system behavior [1, 4, 5]. The first order capacitive systems connected in series may or may not have interactions (cause-effect in both directions between systems). These two cases are presented in Fig. 4.15. (a) u=u1

FOC1

FOC2

FOC3

y1=u2

y2=u3

FOCn y3=u4

y=yn

yn–1=un

(b) u=u1

FOC1

FOC2

FOC3

FOCn

y=yn

Fig. 4.15. (a) Series connection of first order systems without interaction, (b) series connection of first order systems with interaction.

For the case of a noninteracting cascade of n first order systems, the differential equation describing the input u and output y relationship is of n-th order of differentiation with respect to the output y. The unit step response of the system is presented in Fig. 4.16. y1 0

t

y2 0

t

y3 u0(t)

0

t

yn

1 0

t

0

t

Fig. 4.16. n-th order system unit step response.

The transfer function of the noninteracting cascade of n first order systems has the following form: Kn K1 K2 HnO (s) = (4.40) ⋅⋅⋅ . T1 s + 1 T2 s + 1 Tn s + 1 Example 4.6. First, consider as an example the n-th order system, presented in Fig. 4.17, emerged from cascading n first order noninteracting systems [1, 4]. In each of them a first order reaction is transforming the A reactant into B product. Each of the CSTRs has as input the change of the inlet concentration of A component entering the

4 Systems dynamics

CAi , F

A

B

A

B

A

| 111

B

CA1

CA2

CAn

CB1

CB2

CBn

V

V

V

CA1 , F

CA2 , F

CAn , F

Fig. 4.17. n-th order system emerged from the series connection of n CSTRs.

reactor and as output the change of the output concentration of A component. The volume of the inventory in each of the reactors is kept constant. The system composed of multiple CSTRs has the concentration of A component entering the first reactor CAi as input and the concentration of A component leaving the n-th reactor CAn as output. It is described by the transfer function of the form presented in equation (4.40). Example 4.7. Consider as an example of a second order system, the pneumatic system emerged from cascading two interacting systems. Each of the interacting systems consists in a vessel (capacity) for air accumulation. Both inlet and outlet flow rate of air changes the pressure in the vessel. The input variable of the interacting system is the inlet pressure of the feeding flow and the output variable is the pressure in the vessel. The air accumulation is assumed to be isothermal and the process is considered homogeneous with respect to the space coordinates. The two vessel system is presented in Fig. 4.18.

pi

R1

C1 p1

R2

C2 p2

R3

p3

Fig. 4.18. Pneumatic systems with interaction.

The equations describing the pressure change in the two vessel system are dp1 pi − p1 p1 − p2 − = dt R1 R2 dp p − p2 p2 − p3 C2 2 = 1 − dt R2 R3 C1

(4.41)

112 | Part I: Introduction which becomes after the Laplace transform p p 1 1 + ) p1 = i + 2 R1 R2 R1 R2 p p 1 1 (C2 s + + ) p2 = 1 + 3 . R2 R3 R2 R3 (C1 s +

(4.42) (4.43)

After eliminating the p1 variable from equation (4.42), the following equation is obtained: p2 [

R1 C1 R2 C2 R3 s2 (R1 C1 R3 + R1 C1 R2 + R2 C2 R3 + R1 C2 R3 )s + + 1] R1 + R 2 + R 3 R1 + R 2 + R 3 R3 R1 + R 2 R C R s = p + ( 1 1 2 + 1) p3 R1 + R 2 + R 3 i R1 + R 2 + R 3 R1 + R 2

(4.44)

and the transfer function on the path pi to p2 becomes R3 R1 + R 2 + R 3 . H(s) = R1 C1 R2 C2 R3 2 (R1 C1 R3 + R1 C1 R2 + R2 C2 R3 + R1 C2 R3 ) s + s+1 R1 + R 2 + R 3 R1 + R 2 + R 3

(4.45)

The transfer function of the noninteracting cascaded capacities has the form R 2 R3 (R1 + R2 )(R2 + R3 ) . H(s) = R R C R1 R2 C1 R2 R3 C2 2 R R C s + ( 1 2 1 + 2 3 2)s + 1 (R1 + R2 )(R2 + R3 ) R1 + R 2 R2 + R 3

(4.46)

As may be noticed, the transfer functions of the cascaded interacting and noninteracting capacities are different. For the numerical case when C1 = C2 = 2 and R1 = R2 = R3 = 1 the transfer function (equation (4.45)) on the path pi to p2 becomes H(s) =

1 2 3(2s + 1) ( s + 1) 3

(4.47)

and the transfer function on the path p3 to p2 , obtained from equation (4.44), has the form 2(s + 1) H(s) = (4.48) . 2 3(2s + 1) ( s + 1) 3 The time constant for a single noninteracting capacity is T=

2 C = =1 1 1 2 + R1 R2

(4.49)

while the time constants for the interacting capacities have the 2 and the 2/3 values.

4 Systems dynamics

| 113

4.7 Pure delay system The pure delay (or dead time) system is described by the input-output relationship: y(t) = u(t − τ ),

(4.50)

where the τ is denoted the pure delay time or dead time [1, 4, 6]. The dead time is the time that passes from the moment of the input change until the moment the output starts to change, too. The unit step input and the pure delay system response to this input are presented in Fig. 4.19. u0(t)

y(t)

1

1

0

t

0

τ

t

Fig. 4.19. Pure delay system unit step response.

It may be noticed that the dead time system does not change the form of the input signal, only delays it. The transfer function of the pure delay system may be obtained by the series expansion of the output signal around the moment of time t: y(t) = u(t − τ ) = u(t) −

τ du(t) τ 2 d2 u(t) τ 3 d3 u(t) − + ⋅⋅⋅ . + 1! dt 2! dt2 3! dt3

(4.51)

Applying the Laplace transform to (4.51) the following equality is obtained: Y(s) = (1 −

τ s τ 2 s2 τ 3 s3 + − + ⋅ ⋅ ⋅) U(s), 1! 2! 3!

(4.52)

where in the parenthesis we find the Maclaurin series expansion of the function e−τ s , leading to the transfer function of the pure delay system: HPD (s) =

Y(s) = e−τ s . U(s)

(4.53)

The frequency response function of the pure delay system becomes: HPD (jω ) = e−jωτ = cos(ωτ ) − j sin(ωτ ).

(4.54)

According to equation (4.54), the magnitude of the frequency response function |HPD (jω )| equals 1 and its phase φPD (jω ) is tan−1 (tan(−ωτ )) = −ωτ . The Bode diagrams of the pure delay system are presented in Fig. 4.20.

114 | Part I: Introduction |HPD|dB [dB]

Bode diagram of magnitude for the pure delay system 1

0.5 0 –0.5 –1 –1

–0.5

0

0.5

1

1.5

log(ω) 2

Frequency [log(rad/s)]

Bode diagram of phase for the pure delay system φPD [rad]

0

–400

–800 –1

–0.5

0 0.5 1 Frequency [log(rad/s)]

1.5

log(ω) 2

Fig. 4.20. Bode diagrams of the pure delay system.

The pure time delay may directly emerge from transport processes or may come out as equivalent pure time delay, resulting from cascading several capacities. Example 4.8. The step change of the fluid flow rate (or temperature) produced at the inlet of a long transport pipe (considered as the input variable) produces at the end of the pipe the same step change of the fluid flow rate (or temperature), considered as the output variable, featuring a pure delay behavior. Considering only the convective transport, the pure delay time may be computed as the ratio between the length of the pipe L and the fluid velocity v, τ = L/v.

4.8 Equivalence to first order with time delay system Real systems may be approximated as having one of the previously presented dynamic behavior patterns. Nevertheless, they may exhibit dynamic behavior that emerges from combinations of the presented types of systems. One of the most usual cases of such combinations is the first order subsystem that has associated pure delay (dead time) behavior. The unit step response of the first order system with dead time has the form presented in Fig. 4.21. This system produces

4 Systems dynamics

| 115

y(t) yst < 0,95(yst–y0)

u0(t) 0,63(yst–y0) y(0)=y0

1

0

0

t

τ

Tp

3Tp+τ

t

Fig. 4.21. Unit step response of the first order system with dead time.

a combination of time delay lag on the input-output path that aggregates the pure time delay with the first order system time lag (time constant). The significance of the pure time delay and the time constant are illustrated in Fig. 4.21. The transfer function of the first order associated with the pure time delay system, has the following form [2]: HFOPD (s) =

Y(s) K e−τ s = U(s) Ts + 1

(4.55)

and its frequency function is HFOPD (jω ) =

K e−τ jω . 1 + jω T

(4.56)

For the case of second order overdamped systems or the case of higher order systems emerged from connecting in series of several first order capacitive systems, it is usual to approximate the dynamic behavior to an equivalent first order system with dead time (having an equivalent time constant Te and an equivalent dead time τe ). The transfer function of the equivalent system has the transfer function and the frequency function of the forms presented in equations (4.55) and (4.56). The equivalent gain Ke , equivalent time constant Te and equivalent dead time τe may be computed either analytically or graphically. Consider the case of connecting in series n first order capacitive systems (of known gains and time constants, Ki and Ti , i = 1 ⋅ ⋅ ⋅ n). The analytical computation of the equivalent gain Ke , equivalent time constant Te and equivalent dead time τe emerges from the equalities of the gain factor, magnitude and phase lag of the frequency response between the equivalent first order system with dead time and the n first order series of capacitive systems. They are presented in the following equations: n

K e = ∏ Ki i=1

(4.57)

116 | Part I: Introduction n

Ki

∏ i=1

√1 +

Ti2

Ke

= ω2

(4.58)

√1 + Te2 ω 2

n

∑ (− arctan(Ti ω )) = −τe ω − arctan(Te ω ).

(4.59)

i=1

Figure 4.22 presents the step response of n-th first order system and the equivalent step response of the first order with pure time delay system [2]. y(t)

y(t) B

yst

yst

I y(0)=y0

y0

A O 0

M τe

N Te

t

0

τe

Te

t

Fig. 4.22. (a) Step response of the n-th order system (b) Equivalent (first order with dead time) step response.

The equivalence may also be accomplished graphically by drawing the tangent line to the step response plot of the n-th order system, in the inflexion point I, and obtaining the segments OM = τe and MN = Te . These segments are determined by the projections on the abscissa, M and N, of the tangent line intersections with the asymptotes of the initial and final steady state values, i.e. points A and B of the tangent line.

4.9 Problems (1) The chemical reaction of transforming the A reactant into the B product has first order kinetics and is performed in a CSTR. Build the mathematical model of the system, o the Bode plots and the transfer function for the transfer path Tag to T o (i.e. heating agent temperature to reactor temperature). Determine also the type of the dynamic behavior and parameters (gain, time constant, pure time delay) characterizing the dynamic behavior of the heat transfer process. The mass of the reactor wall may be considered negligible. The following data are known: volume of reactor V = 1 m3 , heat transfer area AT = 4 m2 , heat transfer coefficient KT = 1000 kcal/m2 h ∘ C, flow rates of the equal inlet and outlet streams F = 5 m3 /h, specific heat of the reactant and products cp = 1 kcal/kg ∘ C, density of the reactant and products ρ = 1000 kg/m3 , temperature of the

| 117

4 Systems dynamics

∘ = 20 ∘ C, heat of reaction ΔHr = inlet stream Tio = 40 ∘ C, cooling agent temperature Tag 3

−20 kcal/mol, nominal concentration of A component CA = 0.5 kmol/m , reaction rate constant at steady-state temperature T ∗∘ , k(T ∗∘ ) = 0.002 s−1 , activation energy E = 15 kcal/mol, gas constant R = 1.989 kcal/kmol K. Note: the reaction rate constant can be expressed as k = a exp(−E/(RT ∘ )). This expression can be linearized around the steady state temperature T ∗o by the first two terms of the Taylor series expansion: dk 󵄨󵄨󵄨󵄨 ∗∘ ∘ k(T o ) ≅ k(T ∗∘ ) + (4.60) 󵄨 (T − T ). dT 󵄨󵄨󵄨T ∗ (2) Consider two CSTRs coupled in series. They are connected by a very long pipe, where only the transport phenomenon occurs. Determine the transfer function for the transfer path CAi to CA1 and to CA2 . The following data are given: reactors’ inlet and outlet equal flow rates Fi = F1 = F2 = 2 m3 /h, interior diameter of the connecting pipe dc = 20 mm, length of pipe lc = 150 m, volumes of the reactors V1 = V2 = 0.3 m3 , reaction rate constant in both reactors k = 0.005 s−1 . Plot the Bode diagrams of the system. (3) Compute the global transfer function of the control system presented in Fig. 4.23, with the time constants and the pure time delay given in seconds. Yref(s) + 5 –

0.3 8s+1

0.15 e–2s 150s+1

Y(s)

2 10s+1 Fig. 4.23. Feedback control system with proportional controller.

(4) Consider a tank equipped with a stirrer and a heating jacket, as presented in Figure 4.24. It is assumed that the mixing of the fluid is perfect and the wall has negligible heat capacity. Determine the transfer function of the system for the transfer path Ti∘ to T2∘ and put it in the following form: T ∘ (s) K1 = (4.61) H(s) = 2∘ . Ti (s) (Ta s + 1)(Tb s + 1) Find the mathematical expressions of the parameters K1 , Ta , and Tb . (5) The data below in Tab. 4.1 were obtained after an inlet step change of the steam flow rate entering a heat exchanger. The heat exchanger outlet temperature measurements were performed with a glass thermometer. What is the time constant of the heat exchanger?

118 | Part I: Introduction FV , TA°

Fag , T1°

M2 M1 Fag , T°i ag

FV , T2°

Fig. 4.24. Perfectly mixed tank with heating jacket.

Table 4.1. Step response of the heat exchanger outlet temperature. t [min]

0

2

4

6

8

10

12

14

16

20

30

40

50

T [∘ C]

50

51

54

58

62

65

68

71

73

76

79

80

80.5

o

(6) Consider a batch reactor with diameter D = 2 m and height H = 3.5 m. The reactor temperature is controlled by the (cooling) water circulating in the jacket. The overall heat transfer coefficient between the jacket and the reactor is KT = 1000 kcal/m2 hK and the mean residence time of the water agent in the jacket is t = 2 min. Neglecting the heat capacity of the reactor wall and considering a chemical reaction with zero enthalpy of formation, determine the time constant of the system. What error is introduced if the capacities’ interactions are neglected? (7) A tank has the cross-sectional area A = 1 m2 , the nominal liquid level H = 4 m and the nominal outlet flow rate F = 2 m3 /h. How will the liquid level change in time if the outlet flow rate suddenly (stepwise) increases to the flow rate value F ∗ = 2.5 m3 /h? Plot the exact solution of the process equation and the solution obtained by linearization. (8) Solve numerically (using the Euler and Runge–Kutta methods) the differential equations of the processes describing: (a) the tank from Problem (7), (b) the cascade (series) of three isothermal CSTRs where the first order reaction transformation of the A reactant into the B product occurs. The residence time in all reactors is TR = V/F = 2 minutes, the reaction rate constant is k = 0.5 min−1 , initial concentration in the reactors are CA1 (0) = 0.4 kmol/m3 , CA2 (0) = 0.2 kmol/m3 and CA3 (0) = 0.1 kmol/m3 , and the step change of the inlet flowrate is from F = 0.8 kmol/m3 to F = 0.6 kmol/m3 .

4 Systems dynamics

| 119

(9) Develop a software application which solves, based on the Euler method, the system of ordinary differential equations which describes the gas accumulation process in a series of three tanks, as presented in Fig. 4.25. The following parameters are known: V1 = V2 = V3 = 20 L, R1 = R2 = R3 = R4 = 108 Pa/(m3 /s), gas constant R = 8314 J/kmolK, p4 = 0 (atmospheric pressure), the molecular mass of gas M = 29 kg/kmol and the temperature T o = 20 ∘ C. Compute the steady state pressures if the pressure p0 has the initial value p0 (0) = 2 bar and it changes stepwise to p0 = 3 bar. Plot the change of the pressures in the tanks. p0

V1 , p1

V2 , p2

R1

R2

V3 , p3 R3

p4 R4

Fig. 4.25. Cascade of tanks with gas accumulation.

(10) The dynamic behavior of the liquid level h in the tank R1 is investigated when the inlet volumetric flow rate F1 is changing. Both tanks, R1 and R2 , are operated at atmospheric pressure. The liquid emerging from tank R1 is recycled through the tank R2 with the pump P .The system of the two tanks is shown in Fig. 4.26. F1 C1 R1

LAS

h C2 h1

V2

F2

LRC

VR FI

R2

V1 PI P

Fig. 4.26. Liquid accumulation in the tank R1 .

(a) Based on the mass balance of the liquid in the tank R1 and the momentum balance for the liquid in the output pipe of the same tank, develop the dynamic model

120 | Part I: Introduction describing the relationship between the level h in the tank (output) of the R1 and the inlet flow rate F1 (input). (b) Linearize the obtained model and compute the transfer function of the process with the same input-output variables, considering the system as a capacitive first order system. Use the following notation for the variables and their associated numerical values: A – cross sectional area of cylindrical tank R1 (0.07065 m2 ), C1 – inlet piping of the tank R1 , C2 – outlet piping of the tank R1 , Dp – inner diameter of the outlet pipe C2 (0.015 m), F1 – inlet volumetric flow in the tank R1 , F10 – inlet volumetric flow in the tank R1 at steady state conditions (1.111⋅10−4 m3 /s), F2 – outlet volumetric flow evacuated from the tank R1 , h – level of liquid in the tank R1 , h0 – level of liquid in the tank R1 at steady state conditions (0.25 m), h1 – elevation of the tank R1 , relative to the tank R2 (0.65 m), k – steady state (static) gain of the system, k1 , k2 – real constants, Lp – length of the outlet pipe C2 (2 m), n – number of flow restrictions (4 = 3 elbows + 1 tap partially open), patm – atmospheric pressure, Sp – cross-sectional area of the outlet pipe C2 (1.76625 ⋅ 10−4 m2 ), T – time constant of the first order system, w – velocity of the liquid flowing in the outlet pipe C2 , ξi – pressure loss coefficients on different local flow restrictions of the outlet pipe C2 , λ – friction coefficient between fluid and wall of the pipe, μ – fluid dynamic viscosity (10−3 Ns/m2 ), ρ1 = ρ2 = ρ – liquid density at the inlet, outlet and inside the tank R1 (1000 kg/m3 ), Numerical values: Mean velocity of the fluid in the outlet pipe (in steady state conditions, when F10 = F20 ), vm = F10 /Sp = 1.111 ⋅ 10−4 /1.76625 ⋅ 10−4 = 0.629 m/s, Reynolds number: Re = ρ ⋅ vm ⋅ Dp /μ = 9435 ≈ 9500 (turbulent flow) Friction coefficient between fluid and wall of the pipe λ = 0.3164/Re0.25 = 0.03205 (smooth pipe), Coefficient of pressure loss on the 90∘ elbow local flow restriction: ξ1 = ξ2 = ξ3 = 0.125, Coefficient of pressure loss on the partially open tap V2 local flow-restriction ξ4 = 40.

4 Systems dynamics

| 121

References [1] Agachi, S., Automatizarea Proceselor Chimice, Editura Casa Cărţii de Ştiinţă, Cluj-Napoca, 1994. [2] Cristea, M. V., Agachi, S. P., Elemente de Teoria Sistemelor, Editura Risoprint, Cluj-Napoca, 2002. [3] Agachi, S., Cristea, M. V., Lucrări Practice de Automatizarea Proceselor Chimice, Tipografia Univ., Cluj-Napoca, 1996. [4] Stephanopoulos, G., Chemical Process Control. An Introduction to Theory and Practice, Englewood Cliffs, New-Jersey 07632, Prentice Hall, 1984. [5] Marinoiu, V., Paraschiv, N., Automatizarea Proceselor Chimice, Vol. I, Vol II, Editura Tehnică, Bucureşti, 1992. [6] Marlin, T. E., Process Control, Designing Processes and Control Systems for Dynamic Performance, 2nd edition, Mc-Graw Hill, 2000.

5 Manual and automatic control [1] As mentioned before, process control deals with the process operation without human intervention. Manual control operations are the basis of automatic control.

5.1 Manual control Manual control of a plant is the operation of the plant, manually, by the human operator. Taking into consideration a simple process of heat transfer in a shell and tube heat exchanger, from Fig. 5.1, one may consider the way a process operator is trying to maintain a stable temperature at a value given by the technological process (Fig. 5.2). Since a disturbance occurs, say a decrease in input temperature during the winter (Ti∘ ), or the decrease of the input temperature of the heating agent, the operator watching the process observes on a thermometer the change and manipulates the valve, opening it stepwise, increasing the heating agent flow and the heat flow to the process. The temperature will increase after a long while, the delay being given by the time constant of the process (Fig. 5.2a). An experienced operator, knowing the dynamics of the process, opens the valve abruptly (Fig. 5.2b), gives a thermal shock to the process, hurrying in this way the change of temperature (Fig. 5.2c). After a while, the operator decreases the agent flow to an intermediate value and smoothly reaches the desired output temperature. This is a way of controlling the temperature, similar to the automatic control (PID). The drawbacks of the manual control are obvious: – it needs continuous attention in supervising the parameters whether on night or day shift; if there are hundreds of parameters to be supervised, it is very difficult to operate the process appropriately; – for a good operation of the process, it needs a high level of understanding of the process (including its steady state and dynamics) on behalf of the operator. These considerations support the introduction, wherever possible, of automatic control systems.

Fig. 5.1. Heat exchanger.

5 Manual and automatic control

|

Fag , T°iag

T°

T°i

Fag , T°ag

T°i

t T°

t1

(a) t0

t

Fag

t T°

t1

(b) t0

t

Fag

t T°

(c)

t0

t1

t Fig. 5.2. Operation of a heat exchanger.

123

124 | Part I: Introduction

5.2 Automatic control Automatic control of a plant means the operation of the plant without human intervention, by an “automaton” which is the Automatic Control System (ACS). Figure 5.3 shows the usual control solution for the heat exchanger. Actually, the control system copies the actions of the operator: (a) the observation of the temperature; (b) the calculation of the deviation from the prescribed value; (c) the estimation of the change needed in the manipulated variable (in operator’s mind); (d) the actual change of the heating flow until the setpoint is reached. T°set TC

TT

Fag Ti°

T° Fig. 5.3. The temperature control system for the heat exchanger.

Fag

The devices performing all the above mentioned tasks are: the transducer for action (a); the controller for actions (b) and (c); the control valve (final control element) for action (d). The block diagram for a general feedback control system is given in Fig. 5.4. S r

+

e –

Controller C xr

c

Actuating device AD

Transducer T

m

Process

y

Fig. 5.4. Block diagram of a general feedback control system.

The significance of the notations is the following: Process - the controlled process, namely the transfer path manipulating the variable/controlled variable; T (transducer) – the device which measures the controlled parameter and transforms it into a signal compatible with the other signals in the loop; S (summing point or comparator) – the device which compares the setpoint with the signal of the transducer and calculates the error; C (controller) – the device which calculates the control signal, according to an embedded control algorithm and the sign and magnitude of the er-

5 Manual and automatic control

|

125

ror; AD (actuating device or final control element) – the device which modifies the manipulating variable, to change the process output. The system is named feedback control system, because it measures the controlled variable (y) with T (xr – the reaction variable), feeds it back to the entrance of the system, where it is compared with the desired value (r – reference, input or setpoint) inside the S; S calculates the deviation of the controlled variable from the setpoint (e – error; e = r − xr ); the error is supplied to C and the controller calculates according to its algorithm (P, PI, PID or others) the control variable c; the AD transforms c in the manipulated variable m which is a mass or energy flow which, modified, is supposed to bring the deviation/error e to 0. The subtraction of the controlled variable (xr ) from the setpoint (r) produces the so called negative feedback (negative reaction) and the addition of the two variables, positive feedback (positive reaction). Sometimes, the positive reaction exists naturally, embedded in the process (e.g. exothermic reactions). Of course, automatic control systems require a better preparation of the personnel, increased initial costs (about 10 %–15 % additional costs of the equipment), a good organization of the plant including metrology, maintenance, a good understanding of the economics of additional areas like optimization, heat integration, control.

5.3 Steady state and dynamics of the control systems The time functioning of a system in steady state implies the time invariability of its parameters. Thus, the period of functioning of the heat exchanger outside the interval t0 –t1 (Fig. 5.2b) represents a steady state. From the moment t0 , the process modifies its working parameters entering into the so called dynamic state (until t1 ). Starting with t1 , the process calms down to a steady state. Usually the dynamic state of a process is not desirable, because the process can become uncontrollable. But sometimes, especially in the batch processes case, dynamic behavior is desired and conducted in such a way as to obtain the maximum production, for example, in the minimum time. Unfortunately, the ever appearing disturbances offer short periods of stationarity for a process, seriously affecting its technical and economic performance. For example, in the functioning of the distillation column (Fig. 3.14), the stability of the composition at the top of the column is disturbed almost permanently by: changes of the composition of the feed flow (changed raw material, changes in the functioning conditions of the previous stages), variations of temperature at the bottom of the column (due to the changes in the steam quality), variations of the feed flow (clogging of the pipelines, ageing of the centrifugal pump), the outside pressure change (variable weather fronts). The permanent presence of disturbances imposes the automatic control of a process.

126 | Part I: Introduction

5.4 Stability and instability of controlled process and control systems From the point of view of stability, the process can be self-regulating or non-selfregulating. A process is self-regulated (or BIBO stable) if, at the occurrence of a bounded disturbance, it goes from a steady state to another steady state via a dynamic transitory period (Fig. 5.5a). If the process, at the occurrence of a bounded disturbance, goes from a steady state to an unstable regime (Fig. 5.5b), it is nonself-regulated. This is the case of the exothermic reactors operated at their unstable equilibrium point. y

y

t

Fig. 5.5. Self-regulating and non-self-regulating process.

t (b)

(a)

There are cases when a stable by nature process becomes unstable due to a badly designed control system (Fig. 5.6). This is a cause which sometimes makes the control loops to be set on “manual” operation. y

y

t (a)

t (b)

Fig. 5.6. The stable process (a) becomes unstable due to a bad control (b).

5.5 Performance of the control system The performance of the control system is judged both in a steady state and in a dynamic one. Thus, if a control system is well designed and tuned, responds to a step disturbance with a damped oscillation with the following distinguishable elements (Fig. 5.7): σ1 – overshoot; σ2 and σ3 are the amplitudes of the second and third oscilla-

5 Manual and automatic control

|

127

tion; est – the steady state error; tr – transient time/settling time (duration of the transient process – time needed by the controlled parameter to arrive and remain around the final steady state with a deviation of maximum 5 % or 2 % of the final value). 1.5 Setpoint Actual σ1

ε st

Parameter

1 σ2

σ3

tr

0.5

0 0

20

40

60

80

100

Time Fig. 5.7. The transient response of a control system.

One can say a process control system behaves properly if during the dynamic state σ1 ≤ σ1imp ,

(5.1)

meaning that the overshoot does not exceed a technologically imposed value – e.g. 10 % of the setpoint value; tr ≤ trimp , (5.2) meaning that the transient time is shorter than a technologically imposed value – e.g. for a drying process it is important that the deviations from the setpoint are shorter than 10 min; σ ζ ≥ ζimp (ζ = 1 − 3 ) , (5.3) σ1 where ζ is the damping ratio and it is important that it is smaller than an imposed value; it is usually considered good behavior if the ζ = 0.75, meaning that the oscilσ lations respect the rule of quarter decay ratio, σ3 = 14 (Fig. 5.8). 1 The performance in the steady state is the steady state error (SSE), or offset, that is the difference between the settled controlled variable and its setpoint. est ≤ estimp.

(5.4)

Meaning the steady state error should be smaller than an imposed value; technologically, est is acceptable if it is under 2 % of the setpoint value. This statement is very

128 | Part I: Introduction 1.5 Actual Setpoint σ1

σ3

Parameter

1

σ1 1 σ3 = 4

0.5

0 0

20

40

60

80

100

Time Fig. 5.8. Good response of a control system with a quarter decay ratio.

general and it is seen from the technology point of view. We shall demonstrate that even a 2 % error can cause important economic losses.

5.6 Problems (1) Describe the advantages and disadvantages of automatic control compared with the manual one. Each situation should have at least three advantages and disadvantages. Why do you think automatic control is to be preferred in the complex situation of an industrial process? (2) Explain what happens with the controlled temperature when, in the temperature control loop described in Section 5.2, the summer does not subtract xr from r, (e = r−xr ) but adds them (e = r + xr ). (3) Explain why the heating process of a classroom is a self-regulated process and the change of the flow rate at the entrance of a tank is either self-regulated (free outflow through a valve) or a non-self-regulated one (forced outflow through a pump)? (4) Suppose the tiles in an oven are burnt normally at 1200 ∘ C. The technological constraint is that the temperature of 1320 ∘ C should not be exceeded for more than 10 minutes. How can we describe the functioning of a temperature control loop if σ1 = 15 % of the setpoint and tr = 25 min?

5 Manual and automatic control

|

129

(5) Which of the following processes are “naturally” stable? Exothermic reaction, nuclear reaction, filling a glass with water under the tap, pressure increase in an air buffer compressor tank with inflow and outflow pipes.

References [1] Agachi, S., Automatizarea Proceselor Chimice, Editura Casa Cărţii de Ştiinţă, Cluj-Napoca, 1994, p. 4.

| Part II: Analysis of the feedback control system

6 The controlled process Figure 5.4 depicts all elements of a feedback control system. Its analysis is the description of the steady state and dynamic behavior of all its parts: the controlled process, the transducer, the controller and the actuator.

6.1 Steady state behavior of the controlled process The steady state behavior of the controlled process is important to be known because it determines first the design of the equipment. The whole design is done in steady state at the so called “normal” or “nominal” values of the operating parameters. Second, the steady state characteristics are important in order to design the control solution and the configuration of the control loop. A mathematical model of the steady state of a process can give important indications on the relationship between the input and output variables. Further, studies of sensitivity or relative gain array (RGA) [1–3] give important indications on the pairing of output and manipulating variables especially when it is about MIMO systems. Let us take the example of a simple process of heating in a steam heat exchanger (Fig. 3.1) with the steady state mathematical model given in the equations (6.1) and (6.2): F ρ c T ∘ − F ρ c T ∘ − K A (T ∘ − T ∘ ) = 0 (6.1) vi

p i

vo

p

T

T

ag

∘ − F ρ c T ∘ + K A (T ∘ − T ∘ ) = 0. Fvag ρag cpag Tiag vag ag pag ag T T ag

(6.2)

∘ between the two equations, the output temperature is expressed in Eliminating Tag the general form as ∘ ) T ∘ = f (Ti∘ , Fvi , Fvag , Ti∘ag , Text (3.1) and, more detailed, T∘ =

∘ K A Fvag ρag cpag F vi ρ cp Ti∘ +Fvi ρ cp Ti∘ KT AT + Fvag ρag cpag Tiag T T Fvo ρ cp KT AT + Fvag ρag cpag Fvo ρ cp + Fvag ρag cpag KT AT

.

(6.3)

Fvi is responsible mainly for the production of the heat exchanger, the only solution for pairing in a temperature control loop is Fag → T ∘ (Fvi → Fvo is the pairing for the flow control loop) so that the natural structure of the feedback temperature control system is that from Fig. 5.3. This more than intuitive method is used to define grossly the solution of automation of more complex processes, mostly reaction and separation processes. Another simple example is that of a CSTR with a first order reaction taking place. The conversion (Tab. 3.2) is expressed by the relation ξ =

k(T ∘ ) VF 1 + k(T ∘ ) VF

,

(6.4)

134 | Part II: Analysis of the feedback control system where k is the rate constant, V is the mass reaction volume and F is the flow. Obviously, when the conversion is the objective of the reaction process ξ = f (T ∘ , V, F),

(6.5)

the solution for automation is of three control loops, controlling temperature, level and flow. The RGA analysis [4] is a rather simple method used to determine the structure of automation of a MIMO process. RGA is useful for MIMO systems that can be decoupled in several SISO systems without or with weaker interaction. RGA is a normalized form of the gain matrix that describes the impact of each control variable on the output, relative to each control variable’s impact on other variables. The process interaction of open-loop and closed-loop control systems is measured for all possible input-output variable pairings. A ratio of this open-loop “gain” to this closed-loop “gain” is determined and the results are displayed in a matrix λ11 [λ [ 21 RGA = Λ = [ [ ⋅⋅⋅ [ λn1

λ12 λ22 ⋅⋅⋅ λn2

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

λ1n λ2n ] ] ]. ⋅⋅⋅ ] λnn ]

(6.6)

The array is a matrix with one column for each input variable and one row for each output variable in the MIMO system. This format allows a process engineer to match the input and output variables that have the strongest effect on each other minimizing the undesired side effects. The relative gain for a selected ij pair of variables is defined as the ratio of the open-loop gain for that pair with all other loops open to its openloop gain when all other loops in the process are closed, with their controlled variables held at setpoint by their controller. That is, 󵄨 𝜕yi 󵄨󵄨󵄨 󵄨󵄨 m = ct 𝜕mj 󵄨󵄨󵄨 λij = (6.7) . 󵄨 𝜕yi 󵄨󵄨󵄨 󵄨󵄨 y = ct 𝜕mj 󵄨󵄨󵄨 The numerator is the open-loop gain determined with all other manipulated variables constant, and the denominator is the open-loop gain determined with all other controlled variables kept constant, which, in fact, applies to the steady state. If the RGA matrix is analyzed one should observe the following: – the closer the values in the RGA are to 1, the more decoupled the system is; – the value closest to 1 in each row of the RGA determines which variables should be coupled or linked; – each row and each column should sum to 1. The calculation of the RGA elements can be done experimentally, and this is preferable because it gives the most accurate results based on real data, or theoretically based

6 The controlled process

|

135

on the steady state mathematical model. In the present work, the theoretical model is described. If a process model is available, the steady state gain matrix relates the manipulated variables to the controlled variables according to the following equation: ȳ = Gm̄

(6.8)

where ȳ and m̄ are the output and manipulating vectors respectively and G is the steady state gain matrix g11 g12 ⋅ ⋅ ⋅ g1n [ ] G = [ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ] (6.9) [ gn1 gn2 ⋅ ⋅ ⋅ gnn ] 𝜕y

where the values gij = 𝜕mi . j Since the G matrix is computed, one can calculate if the system can be decoupled, using the Singular Value Decomposition (SVD) method (condition number of the SVD is CN < 50)¹ [5]. If not, a multivariable controller approach can be used. If yes, the transposed matrix of the inverse of the gain matrix is calculated R = (G−1 )T .

(6.10)

The elements of RGA matrix are computed by λij = gij rij .

(6.11)

The interpretation of the λij values is the following: each of the rows in the RGA represents one of the outputs. Each of the columns represents one manipulated variable: – if λij = 0, the manipulated variable (mj ) will have no effect on the output or the controlled variable (yi ); – if λij = 1, the manipulated variable mj affects the output yi without any interaction from the other control loops in the system; – if λij < 0, the system will be unstable whenever mj is paired with yi , and the opposite response in the actual system may occur if other loops are opened in the system; – if 0 < λij < 1, other control loops (mj − yi ) are interacting with the manipulated and controlled variable control loop. In these cases, the possible advice is given in Table 6.1. Example 6.1. [5] A blending unit is used to dilute and cool down the product stream of a reactor. Three streams are combined in the mixer: the hot, concentrated stream from

1 Usually CN > 10 shows a system difficult to be controlled.

136 | Part II: Analysis of the feedback control system λij λij λij λij λij λij

Possible pairing =0 =1 ≪0 ≤ 0.5 >1

Avoid pairing mj with yi Pair mj with yi (best pairing) Avoid pairing mj with yi Avoid pairing mj with yi Pair mj with yi

Table 6.1. Possible pairing of controlled and manipulated variables relative to RGA elements’ values.

the reactor, a room temperature stream containing none of the product A, and a second room temperature stream containing some A produced elsewhere in the process, with concentration c2 . It is desired to control the flowrate, temperature, and concentration of A in the product stream by manipulating the flowrates of the three input streams. The process is presented in Fig. 6.1, and the steady state values of flowrate, temperature, and concentration are indicated. m indicates the manipulated variables and y the controlled variables.

m1 = F1 = 20 L/min T1 = 423 K c1 = 10 g/L

y3 m3 = F3 = 70 L/min T3 = 293 K c3 = 0 g/L

m2 = F2 = 40 L/min T2 = 293 K c2 = 4 g/L

y1 = F = 130 L/min y2 = T = 313 K y3 = c = 2.77 g/L

Fig. 6.1. A blending process for three streams, two of them containing the product A.

Temperatures are different. Total flow, temperature and concentration have to be controlled. The equations used to model the system are the heat and mass balances: y1 = m1 + m2 + m3

(6.12)

T m + T2 m2 + T3 m3 T1 m1 + T2 m2 + T3 m3 y2 = 1 1 = y1 m1 + m2 + m3

(6.13)

y3 =

c1 m1 + c2 m2 + c3 m3 c1 m1 + c2 m2 + c3 m3 = , y1 m1 + m2 + m3

(6.14)

6 The controlled process

| 137

the partial derivatives of the control variable equations in order to determine the elements of the steady state gain matrix: 𝜕y1 =1 𝜕m1 (T − T2 ) m2 + (T1 − T3 ) m3 𝜕y2 = = 1 2 𝜕m1 (m1 + m2 + m3 )

g11 =

(6.15)

g21

(6.16)

g31 =

(c − c2 ) m2 + (c1 − c3 ) m3 𝜕y3 = 1 2 𝜕m1 (m1 + m2 + m3 )

(6.17)

𝜕y1 =1 𝜕m2 (T − T1 ) m1 + (T2 − T3 ) m3 𝜕y2 = = 2 2 𝜕m2 (m1 + m2 + m3 )

g12 =

(6.18)

g22

(6.19)

g32 =

(c − c1 ) m1 + (c2 − c3 ) m3 𝜕y3 = 2 2 𝜕m2 (m1 + m2 + m3 )

(6.20)

𝜕y1 =1 𝜕m3 (T − T1 ) m1 + (T3 − T2 ) m2 𝜕y2 = = 3 2 𝜕m3 (m1 + m2 + m3 )

g13 =

(6.21)

g23

(6.22)

g33 =

(c − c1 ) m1 + (c3 − c2 ) m2 𝜕y3 = 3 . 2 𝜕m3 (m1 + m2 + m3 )

(6.23)

Substituting the values in the picture, [ G=[ [ [

1

1

11 13 47 845

2 − 13 8 845

2 13

1

[ ] −1 T 2 ] [ − 13 ] R = (G ) = [ 1 18 − 845 [ 0 ]

and thus [ RGA = [ [

2 13 11 13

[ 0

4 13 5 13 4 13

7 13 3 − 13 9 13

] ]. ]

4 13 − 52 65 2

7 13 3 2 − 65 2

] ] ]

(6.24)

]

(6.25)

]

m1 , the flow rate of stream 1, should be used to control y2 , the temperature of the output stream, since the value of λ21 is the closest to 1 in the matrix ( 11 ). The element 13 9 second closest to 1 is λ33 , with its value of 13 . This indicates that m3 , the flow rate of stream 3, should control y3 , the concentration of A in the product stream. In the case of controlling y1 , the best option from the point of view of RGA value would be m3 7 (λ13 = 13 ), but this is already controlling the concentration of A in the product, so the next best option is m2 , the flow rate of stream 2. This is not the best choice because the relative gain is less than 0.5, but it is the best option available.

138 | Part II: Analysis of the feedback control system

30

T˚C

25

20

Process characteristics N1 N2

15 0

2

4

6 Fag(m3/h)

8

10

Fig. 6.2. The nonlinear dependence of the output heat exchanger temperature on the steam flow.

Besides the pairing of the variables, it is important to also configure the inner structure of the control loop. The equation (6.3) gives the nonlinear dependence T ∘ = f (Fvag ) presented in Fig. 6.2. It makes a difference to the efficiency of the control if the nominal point of operation is N1 or N2 . This is the difference between a heat exchanger having a reserve for heating and another one designed at the limit. In normal/nominal situations (operating point N1 in Fig. 6.2) the control system designed as in Fig. 5.3 is efficient enough. But, in the situation of a much stronger disturbance (curve P1 in Fig. 6.3), the controller is “ordering” the control valve to open at maximum, but does not compensate the caloric deficit due to the disturbance. In this case, another inner structure of the control loop should be sought after. Thirdly, the steady state characteristic of the process is needed at the choice and sizing of the actuator (Fig. 6.3). T° P1 N T1°

P2 P3

Tn° T3° T2° Fag min

Fag n

Fag max Fag n + 0.5m3/h

Fag

Fig. 6.3. The sizing of the control valve in the conditions of the nonlinear characteristic of the process.

6 The controlled process

|

139

Usually (80 % of the situations in process industry), the actuator is a control valve. In the following section, we demonstrate the role of the steady state behavior of the process at choosing and dimensioning of the valve. In the case of one “positive” disturbance (curve P1 ), the output temperature of the exchanger moves to the value T1∘ corresponding to the nominal steam flow Fag n . The controller, noticing it has an error, pushes the system to move on curve P1 towards the nominal value Tn∘ . In the opposite case, when a negative disturbance occurs (curve P2 ), the controller determines the system to move from T2∘ to the nominal one Tn∘ . In both cases, the controller determines the closing or opening of the valve at the flow values of Fag min and Fag max , respectively. These are the values of flow with which the valve will be sized (see Part II, Chapter 9). But choosing the valve does not only mean dimensioning the valve orifice, but also the design of the installed valve characteristics. As seen in Fig. 6.3, the characteristics are highly nonlinear. Usually all process characteristics are nonlinear. Nonlinearities cause inaccuracy in controlling the processes due to the variable gain factor along the characteristic. One may observe that in N1 and N2 (Fig. 6.2) the slopes and thus the gains are very different. If the control system operates around N1 , when determining a positive deviation (say +10 ∘ C), it manipulates the steam flow decreasing it with a certain quantum (say equal to 0.5 m3 /h). If the deviation is negative (−10 ∘ C), manipulating it with the same increase (absolute value of 0.5 m3 /h) which is normal for a controller with a fixed proportional gain Kc (see Part II, Chapter 8), the result of the action will not be the nominal temperature Tn∘ , but a lower temperature. There is a steady state error (SSE) totally due to the nonlinearity and to the changing slope in the two opposite actions on the characteristic (to the left, the curve is more abrupt than to the right of the nominal point). One must say that nonlinearity is causing problems to controlling the processes. The solution is to choose a compensating characteristic of the valve (which has a small slope where the slope of the process characteristic is large and vice versa). That is, in the discussed case, to choose a logarithmic installed characteristic of the valve (Part II, Chapter 9, Fig. 9.21). The choice being done, the total characteristic of the process + valve will be linear (Fig. 6.4). Fourth, the steady state characteristic of the process is used to calculate the gain factor of the process at the operating point, Kpr , which is actually the slope of the curve in the operating point (Fig. 6.5). The gain is expressed in the units resulting from the ratio of the two derived variables. The gain factor of the process is used at the controller tuning (Part III, Chapter 11). As it is the slope in the operating point, Kpr =

Δy 󵄨󵄨󵄨 dy 󵄨󵄨󵄨󵄨 󵄨󵄨 = 󵄨 . Δm 󵄨󵄨N dm 󵄨󵄨󵄨N

For Example 6.2, where T∘ =

Ti∘ag 4Ti∘ + (T ∘i ag )Fag 4Ti∘ + = 4 + 2Fag 2 + 4 4 + 2Fag F ag

(6.26)

140 | Part II: Analysis of the feedback control system 1 0.8

y

0.6 0.4 1

0.2 Process characteristic

0 0

0.2

0.4

0.6

0.8

0.8 1

0.6 y

m

0.4 1

0.2

0.8

0

Process+valve characteristic 0

0.2

0.6

0.4

0.6

0.8

1

m

xc

0.4 0.2 Valve characteristic

0 0

0.2

0.4

0.6

0.8

1

xc

Fig. 6.4. Linearizing the process characteristic through a compensating valve characteristic.

Kpr |N =

󵄨 dT ∘ 󵄨󵄨󵄨 󵄨󵄨 dFag 󵄨󵄨󵄨N

= (Ti∘ag )

−2 1 + [4Ti∘ + (T i∘ag )FagN ] , 4 + 2FagN (4 + 2FagN )2

(6.27)

with the agent flow of 2 m3 /h, the gain is 7 ∘ C/(m3 /h).

6.2 Dynamic behavior of the controlled process The dynamic behavior is important showing the speed of response of the process to the action of the manipulating variables or disturbances. The capacity of a process to be appropriately controlled is given by its controllability [6]. The concept of controllability refers to the capability of the controller to change the state of the controlled plant. A system ẋ = Ax + Bu is said to be controllable if for all initial conditions x(0) = x0 , terminal conditions x1 , and t1 < ∞ there exists an input u(t), 0 ≤ t ≤ t1 such that

6 The controlled process

141

|

30

25 T0

Kpr = tg α 20 Process characteristics Operating point 15 0

2

4

6

8

10

Fag Fig. 6.5. The gain of the process is the slope of the characteristic of the process at the operating point.

x(t1 ) = x1 . That is, given x0 , x1 , and t1 < ∞, we wish to find u(t), 0 < t < t1 , such that t1 At1

x1 = x(t1 ) = e

x0 + ∫ eA(t1 −t) Bu (t) dt

(6.28)

0

for a system to be controllable. Note that this equation can be solved for all xo and x1 if and only if it can be solved for all x1 with x0 = 0. So we will now just consider the zero initial condition response. What does this mean? It means that if a plant is controllable, its steady state can be changed without great trouble, within a finite time horizon, into another steady state, by the control system, or that a system is kept at setpoint by the control system without difficulty in the presence of a disturbance. Usually the dynamics of a process are given by a combination of first order (T and dead time (τ ) systems corresponding to an equivalent first order (Te ) and equivalent dead time (τe ). This is because most of the processes can be described by a sequence of capacities (e.g. multiple effects evaporators, cascade of CSTRs, distillation columns with sequence of distillation units etc.). The practice of the process control determines simply the range of controllability of the processes using the values of the ratio Te /τe (Tab. 6.2). Table 6.2 also gives suggestions for the complexity of the control loop in all cases of controllability. The poorer the controllability, the more complex the control system must be in order to stabilize the process. At a fair controllability, the system can have the structure of a cascade. At a poor controllability, the complexity of the cascade control is not enough and a more complex loop such as feed forward control or model predictive control is necessary. Shinskey [7] even gives some indications of the values of the ratio depending on the capacities in series (Figs. 6.6 and 6.7).

142 | Part II: Analysis of the feedback control system Table 6.2. The controllability of the process measured by the ratio Te /τe . Te /τe

Controllability

Response of the control system to a step input change

Type of adequate control system

6–10 3–6

good fair

feedback control cascade control

0 Fm = 0 ΔT°

I Fig. 7.28. Measuring principle of the thermal mass flowmeter.

From equations (7.39) and (7.40) the voltage-mass flow rate relationship can be obtained as U = 0.5α Ek1 cp Fm = KFm (7.41) thus, indicating a linear dependence. This type of direct mass flow meter is very appropriate for measuring low gas flow rates, in laboratory and small or medium size pilot applications. It may be also used for viscous and low electrically conductive liquids. Corrections have to be done if the temperature, composition and viscosity of the fluid changes considerably. Its accuracy class is 1 %. The inferential mass flow meter computes the mass flow rate based on the volumetric flow rate and the fluid density, as Fm = Fv ρ [4]. For gases, the most common mass flow rate measuring approach is presented in Fig. 7.29. Special temperature and pressure transducers, placed in the neighborhood of the differential pressure transducer, are used to obtain the necessary pressure, temperature and differential pressure signals. These signals are further used to accurately compute the volumetric flow rate, fluid density and consequently, the mass flow rate of the gas stream.

186 | Part II: Analysis of the feedback control system F Δp

P

T°

ρ

Δp

ρΔρ Fm

Fig. 7.29. Inferential mass flowmeter.

Flow integrators Besides the flow rate measurements, in process engineering applications information is sometimes needed that reveals the total quantity of delivered volume or mass, for a certain period of time. If the mass or volumetric flow rates are measured and transformed into a standard signal, the total quantity may be obtained by time integrating these signals. This operation may be performed by the special devices named integrators. A particular class of flow meters measures and integrates the volumetric flow rate by dividing the flow into precisely measured volumes and counting the number of such volumes passed through the flow meter. The number of volumes multiplied by the value of a single sample volume provides the total volume (integrated flow rate) that was passed through the flow meter in a certain period of time. The flow rate may be computed if the time is also measured. The ratio between the total volume passed through the volumetric meter and the corresponding period of time for its accumulation, equals the mean flow rate for the corresponding period of time. From this category of flow integrators, the most common are the Woltman turbine meter and the oval shaped gear meter [3]. The Woltman turbine meter is frequently used for integrating water streams. It is schematically represented in Fig. 7.30. The turbine, having helical blades, is rotated by the fluid flow and the number of rotations is transmitted by a mechanical gear coupling system to a mechanical gear integrator. An alternative construction transforms the rotations of the turbine in a pulsed electrical signal that is generated by the magnetic coupling of the turbine with the exterior electrical circuit or by electromagnetic induction. An electronic counter totalizes the number of electrical impulses and hence, the total volume. Its accuracy class is 2 % to 3 %. The oval shaped gear meter is presented in Fig. 7.31. It is composed of two oval gears that rotate and separate fixed volumes of fluid V which are passed from the upstream to the downstream of the meter (four volumes for one full rotation). The number of rotations is transmitted to mechanical or electrical

7 Transducers and measuring systems | 187

Fig. 7.30. Woltman turbine flow integrator.

V

V Fig. 7.31. Oval shaped gear flow integrator.

counters that show the integrated flow rate. The typical accuracy class of the meter is higher than 0.5 %.

7.7 Level transducers Level measurement is of fundamental importance in process engineering because of its direct relationship to the material inventory of the system, the information on the latter being considered to be most significant for process operation. The level of both liquid and solid materials may be measured. The level measurement can be continuous with respect to the level values or discrete, i.e. only for detecting threshold values (for alarming, interlocking or on-off control). The measuring methods may imply direct or lack of contact with the measured medium. The measuring methods can directly or indirectly measure the level. For the indirect methods, either the environment or the measured medium characteristics may affect level measurement, such as: density, composition, pressure, temperature, agitation, dust, vapors, electrical conductivity, dielectric properties, moisture and material deposition. A large variety of level measuring instrumentation is offered by the manufacturers.

188 | Part II: Analysis of the feedback control system Level transducers based on floats The optical reading of the level in a tank or in an associated communicating vessel and the float-based level measurement are direct level measuring methods. The float is a solid body whose density is lower compared to the liquid density. Due to Archimedes’ force the body floats and shows the liquid level. It may be mechanically coupled by a pulley system to a counterweight in order to transmit the float movement to an indicator outside the tank or to a displacement electrical transmitter for standard signal generation. The displacement transmitter may use magnetic (permanent magnet) or inductive coupling to the float. The float can be also mounted in an associated communicating vessel. The level measuring principle using the float coupled directly to a spring is presented in Fig. 7.32. (1)

(3)

(2)

2 1

y y0

H +y –y H0 0 0

H

Fig. 7.32. Float based level measurement.

The float 1 is placed in a communicating vessel 2. As the level in the tank changes, the float changes its position [3]. Three forces acting on the float are in equilibrium. They are the gravity force G, Archimedes’ force FA , and the elastic force of the spring Fs , and at equilibrium: G = F A + FS (7.42) G − Aρ [H − (H0 + y0 − y)]g − KS y = 0, where y is the spring elongation corresponding to the partially immersed float (position (3) in Fig. 7.32), y0 is the spring maximum elongation corresponding to the nonimmersed float (position (2) in Fig. 7.32), i.e. when H = H0 and G = Ks y0 , and A is the cross-sectional area of the float. The current elongation of the spring y is linearly dependent on the level H: y = y0 −

Aρ g (H − H0 ). KS + Aρ g

(7.43)

The elongation can be transformed into an electrical signal by magnetic or inductive coupling.

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189

The dynamic behavior of the float system is described by the equation G − Aρ [H − (H0 + y0 − y)]g − KS y = M

d2 y , dt2

(7.44)

where M is the mass of the float and spring. The second order differential equation (7.44) reveals the possibility of getting into an oscillatory regime. This system’s intrinsic behavior is occurring beside the oscillations induced by the liquid circulation aimed to equalize the level in the communication vessels. Choosing the appropriate values for the mass M, the float cross-sectional area A and the elastic constant Ks , may render to the system an aperiodic second order dynamic behavior. The accuracy class of this type of level transducers is higher than 1 %. Fouling and building deposits on the float may affect the accuracy of the measurements.

Level transducers based on the pressure measurement This is the most commonly used level measuring method [3, 5]. Actually, these level transducers are pressure or differential pressure transducers. For a tank opened to atmospheric pressure, the level measuring principle is based on the measurement of the hydrostatic pressure p (relative pressure) produced by a column of liquid having the height H, as their direct relationship shows: p = ρ g H.

(7.45)

There is a linear relationship between the level and the hydrostatic pressure provided that density is constant. If the tank operates under pressure, the differential pressure transducer is required to measure the pressure exerted by the column of the liquid, as revealed by the equation Δp = PH − PL = Pgas + ρ g H − Pgas = ρ g H. (7.46) The measuring principle is presented in Fig. 7.33. In order to make the level measurement independent of the atmospheric pressure it is possible to use the differential pressure method for both cases of open and closed (under pressure) tanks. If the gas vapors may condense, the gas connection line should be filled with the condensate of the liquid with the help of a condensing vessel mounted in the neighborhood of the gas tap. PL becomes the high pressure connection and PH becomes the low pressure connection of the transducer. As a result, the differential pressure received by the transducer and hence, the standard signal, becomes inversely proportional to the liquid level. For the case when the liquid and vapors are corrosive and produce deposits, the connection lines of the transducer may be filled with a neutral liquid or separated by elastic and corrosive-resistant metallic membranes.

190 | Part II: Analysis of the feedback control system

Gas connection line Pgas

Standard signal H

Differential pressure transducer PH

PL

Fig. 7.33. Differential pressure based level measurement.

The accuracy class of this level transducers category is that of the pressure transducer. The time response of the level transducer is less than 2 seconds. In order to overcome the direct contact between the pressure transducer and the corrosive medium or the medium creating deposits (polymerization, crystallization) the bubble tube level measuring principle may be used, as presented in Fig. 7.34. Standard signal FC Air PL

PH p

H

Fig. 7.34. Bubbler level measurement principle.

Air or neutral gas is injected into a tube immersed in the liquid so as to create a bubbling flow [3, 5]. The pressure created in the tube is proportional to the liquid level, provided that liquid density is constant. The pressure is converted into a standard sig-

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nal by a pressure transducer. It is important that the injected flow rate of the neutral gas to be kept constant. This may be achieved by a flow rate control loop, such as by the constant flow rate rotameter. For pressurized tanks the differential pressure transducer can be used in this case where the low pressure connection of the differential transducer PL is connected to the pressure of the gas acting on the liquid surface.

Capacitance level transducers The measuring principle of the capacitance level transducer is based on the electrical capacitance of a capacitor that depends on the dielectric material between the plates, the distance and the area of the plates, as presented by equation (7.16) [5]. For a given geometry, the overall capacitance depends on the dielectric coefficient (permittivity). The dielectric coefficient is formed by two components, the one corresponding to the liquid and the other one corresponding to the gas, as shown in Fig. 7.35. Standard signal

Standard signal

Transmitter

Transmitter Metal conductor wall

Electrical conductor

Electrical insulator Electrical conducator core

C=f(ɛ)

Level in a non–conductive liquid

Nonconductive wall

Metal

C=f(ɛ)

Level in a conductive liquid

Fig. 7.35. Capacitance level measurement in conductive and nonconductive liquids.

For a nonconductive liquid the two plates of the capacitor are: an immersed electrical conductor and the conductive wall of the tank. In the case of a conductive liquid the two plates are: an immersed electrical conductor coated with an electrical insulator material and a metal rod (cable probe) also immersed in the liquid. In this second case the tank should have nonconductive walls. The dielectric coefficient of the air (and most of the gases) is about tens of times smaller compared to that of the liquids. As a consequence, when the level in the tank is small, the overall capacitance is small and when the level is high, the capacitance is also high.

192 | Part II: Analysis of the feedback control system If the liquid in the tank has high viscosity or produces buildup, the capacitive level measurement may show a large time constant or even fail in performing the measurement. The accuracy class of the capacitive level transducer is 0.5 %. The time response is less than 1 s. Special compensation is needed when the liquid temperature and composition are changing.

Ultrasonic and radar level transducers The measuring principle of the ultrasonic level transducer is based on the reflection of the sonic wave sent by an ultrasound transmitter. The reflection of the wave occurs when it meets the surface of the liquid or solid in the tank [5], as presented in Fig. 7.36.

Transmitter – Receiver

T–R

Standard signal

Δt

H

HT

Δt

h

T–R

Fig. 7.36. Ultrasonic level measurement principle.

An ultrasonic receiver detects the reflected wave and the transmitter-receiver ensemble measures the time interval between the moment of emitting the ultrasonic pulse and the moment of receiving the echo pulse 2Δt. Correlated with the known ultrasound propagation velocity in the gas vW , the level of the liquid may be directly calculated by H = HT − h = HT − vw Δt, (7.47) where HT is the height position of the transmitter-receiver, measured from the bottom of the tank. This level measuring method is very appreciated when the direct contact between the transducer and liquid or solid is undesirable (corrosion, buildup, high viscosity) or the properties of the medium are changing (temperature, composition). Nev-

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193

ertheless, it is necessary to have a good reflectivity of the surface (e.g. heavy agitation or foam may disturb the measurement). Smart transducers can filter undesired false echoes generated by tank interior obstacles, such as baffles, ladders and tank braces. As the ultrasounds propagation speed depends on the temperature and the composition of the ultrasound propagation medium, special compensation should be provided if these measuring conditions change significantly. The use of ultrasound level transducers is limited to medium temperature up to 150 ∘ C and to medium pressure up to 5 bar. Its accuracy class is 0.5 % and time response usually less than 3 s. The measuring principle of the radar level transducer is similar to the ultrasound level transducer (i.e. measuring the flight time), except that the reflection of the frequency modulated electromagnetic wave sent by the transmitter is reflected on the surface of the liquid or solid in the tank. It is a noncontact measuring method that can be used in difficult operating conditions, as the measurement is neither depending on the temperature, composition or buildup of the solid or liquid medium, nor on the temperature, pressure, dust or composition of the gaseous propagation medium. A single requirement is still needed and this implies that the dielectric constant of the gas has to exceed a minimum limit (i.e. over 1.9). The measuring error is very small, i.e. of about ± 1 mm. The time response is almost instantaneous. Laser-based measurement of the level uses the same flight time principle. It is recommended for very accurate measurements, for long measuring ranges and for severe conditions, such as melting furnaces. The method may be affected by the presence of dust. The accuracy class of the measurement may reach 0.02 % and the time response is almost instantaneous.

Radiometric level transducers Gamma rays are radiation electromagnetic waves, of very short wavelength, produced by energy state transitions in the nucleus of some elements, such as Cobalt 60 and Cesium 137 isotopes [5]. Their power of penetration is very high, making possible the installation of the gamma sources outside of the tank wall and making the level measurement independent of the gas properties. The radiometric level measuring principle is presented in Fig. 7.37. A gamma ray source sends a beam towards the gamma radiation detector. The detector is usually a Geiger–Müller counter for point level detection and a rod scintillation counter for the continuous level measurement. The amplitude of the gamma radiation received by the counter depends on the thickness of the liquid or solid to be penetrated, and hence on the level in the tank. For point level detection, the gamma radiation source is unidirectional while for the continuous level measurement it is multidirectional. When the level of the liquid or solid in the tank reaches the preset level of the detection point there is a drastic decrease in the signal detected by the counter, thus triggering the state of the alarm relay or the predefined operation. The continuous level detector consists in a rod having scintillation crystals distributed in-

194 | Part II: Analysis of the feedback control system

Detection of a point level Threshold detector

Gamma sources

Analog detector Continuous level measurement

Photo-electronic transmitter Standard signal Fig. 7.37. Gamma radiometric level measurement principle.

side it. When radiated, the crystals produce light that is integrated and processed by a photo-electronic transmitter and converted into a standard signal, proportional to the level. The radiometric level transducer operation is independent of the gas, liquid and solid properties (temperature, pressure, viscosity, dust) inside the tank, but is biologically hazardous and consequently special attention should be paid to operation personnel protection and to tank inventory of biological nature. The measuring accuracy class is 1 % and the time response is less than 2 seconds.

Conductive level transducers The measuring principle of this level measurement transducer is based on measuring the electrical resistance of the non-shunted part of an electrical resistance immersed in the conductive liquid of the tank [5], as presented in Fig. 7.38. The electronic transmitter senses only the non-immersed resistance R1 created in its input electrical circuit due to the fact that part of the total resistance, i.e. the R2 resistance part, is shunted by the conductive liquid. As the level increases, the resistance R1 reduces its value. The electrical resistance is made of a tape mounted on a metal bar, the latter being connected to the always immersed end of the tape. This simple method for level measurement may present errors of about ± 3 cm, but the time response is of less than 2 s. For low cost applications, electrodes may be mounted at specified threshold detection levels and the liquid be used for acting as an electrical switch.

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195

Standard signal Electronic transmitter Electrical resistance

Metal base R1 R2

Fig. 7.38. Conductive level measurement principle.

Level transducers based on weighing This is an indirect measuring method for the level in tanks of known weight, geometry and of the liquid or solid density of the tank inventory. The measuring principle is based on weighing the overall mass of the tank. This is performed by measuring the deformation of the strain gauges, mounted on metallic elements that support the tank [5], as presented in Fig. 7.39. E

Transmitter 4–20 mA Indicator Wheatstone bridge

Fig. 7.39. Weighting based level measurement.

As the level of the liquid or solid in the tank changes, its weight changes and the strain gauge resistance as well. A Wheatstone bridge and associated electronic circuits generate a standard signal proportional to the weight, and hence to the level. Compensation with changes of the strain gauge temperature is necessary and special mechanical measures must be applied to avoid undesired influences on the system’s weight that may be introduced by the connections with other equipment, such

196 | Part II: Analysis of the feedback control system as piping. This method has many applications for solids level and weight measurements. Accuracy is the one specific to the strain gauges and the time response is less than 5 s.

7.8 Composition transducers Composition is the most important process parameter in industrial process engineering applications because the quality of raw materials and products needs to be assessed in order to provide safe and efficient operation. Composition transducers may be classified in two main categories: selective and non-selective instruments. The selective composition transducers are able to provide composition information on the individual components forming the mixtures, on the basis of the components’ particular physical and chemical properties. The non-selective transducers indirectly measure the composition by inferring the components’ concentration on the basis of physical and chemical properties of the mixture (density, viscosity, pH, conductivity).

Density transducers There are different methods for finding the composition based on density measurements. They are typical for the case of binary mixtures of miscible components, such as: ethanol-water, acid-water or salt-water. Even for a multiple component mixture, in which one component is dominant, density may provide useful information on the key component composition. Continuous density measurement can be performed by measuring the hydrostatic pressure exerted by a column of liquid, by measuring the displacement of a hydrometer, by weighing and by measuring the oscillating frequency of a mechanical probe immersed in the liquid. A simple and noncontact method for hydrostatic density measurement is presented in Fig. 7.40. Inert gas (e.g. nitrogen) is introduced with constant flow rate in each of the tubes 3, so as to produce a bubbling flowing regime [3]. The tubes are immersed in two tanks: one tank 1 containing the measuring liquid of unknown density ρmes and the other one, tank 2, containing a reference liquid with known density ρref . Liquid levels in the two tanks are the same, due to the same overflow weir position. The difference in the hydrostatic pressure in the two tubes is Δp = pmes − pref = ρmes g H − ρref gH.

(7.48)

This differential pressure is linearly dependent on the density of the measuring liquid and is converted into a standard signal by the differential pressure transducer. In order

7 Transducers and measuring systems |

197

Standard signal FC PH

PL 3 1

+ p–

FC

3

Inert gas

2

H ρref ρmes

h

Measuring liquid density ρmes

Liquid of reference density ρref Fig. 7.40. Bubbling system density measurement.

to change the zero point of the emerged differential pressure, for example when ρref > ρmes , the high pressure tube can be moved down with an appropriate displacement h. The influence of the liquid temperature on the density measurement is prevented by the design of the instrument. This is due to the fact that the reference tank 2 is completely immersed in the measuring tank 1 and consequently, the temperature in both tanks is the same. This means that density changes produced by temperature show up in both tanks and are mutually compensated by the emerged pressure changes subtraction. The dynamic behavior of the instrument depends on the time needed for liquid homogenization in the measuring tank, which is the rate-controlling subsystem of the measuring device. This is described by the equation Vmes

dρmes = F(ρmes-in − ρmes ), dt

(7.49)

where Vmes is the measuring tank volume, while F and ρmes-in are the flow rate and density entering the measuring tank. According to equation (7.49) the time constant of the instrument is equal to the residence time of the liquid in the measuring tank Tp = Vmes /F. The instrument accuracy class is higher than 1.5 %. The density measurement based on the displacement of a hydrometer floating in the liquid has an inductive transmitter, like the one presented for the inductive pressure transducer, or may have a differential transformer. The density measurement that

198 | Part II: Analysis of the feedback control system relies on weighing has as measuring principle that takes the same approach as the level transducer based on weighing, i.e. using the strain gauge. The density measurement based on bringing a mechanical probe (bar, fork, tube) immersed in the liquid to oscillation can be performed by changing the mechanical excitation frequency until resonance is obtained. This principle was presented with the Coriolis flow meter.

Viscosity transducers Viscosity measurement is important not only for the study of the fluid’s flowing behavior but also as in indicator for the evolution of reactions, specifically in the polymers industry. There are several methods for measuring cinematic viscosity η , such as the measurement of the flow rate through a capillary or between two rotating cylinders and measuring the free falling time of a solid body in the medium of unknown viscosity. Most of the methods are specific to laboratory measuring conditions and do not provide continuous viscosity measurement (e.g. U-tube, falling sphere, falling piston, vibrational or bubble viscometers). For continuous measurement, rotational viscometers, such as Searle, Couette, electromagnetically spinning sphere or rotameter type viscometers are used [3]. Figure 7.41 presents the Searle viscometer. 7 7

6 5

5 φ 6

4

1 3

2

Fig. 7.41. Searle viscometer.

The liquid to be measured 2 is placed in a tank where the cylinder 3 is rotating. This inner tank is immersed in a second tank in which the fluid 1 is used to keep constant the temperature of the liquid 2. Cylinder 3 rotates with constant angular speed ω . Due to the medium’s viscosity, cylinder 3 is dragged back in its rotation compared to the driving shaft. Its movement, with the same ω , will show a phase difference φ . This phase-shift is possible due to the elastic coupling provided by the spring 4.

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At constant angular speed there is equilibrium between the cylinder 3 moment of rotation MR = k1 ηω and the counteracting moment of rotation produced by the spring 4, Ms = k2 φ , leading to a linear dependence between the phase-shift φ and the viscosity: k ω φ = 1 η = Kη . (7.50) k2 The cursor 5 of the rheostat is fastened on the rotating cylinder shaft and changes its angular position with respect to the rheostat resistance 6 according to the same phase-shift φ . The rheostat electrical resistance 6 is fastened on the driving shaft system. The position of the cursor 5 is proportional to the phase-shift and, hence to the viscosity. The rheostat electrical resistance, depending on viscosity, is processed and transformed into a standard electrical signal. The dynamic behavior of the Searle viscometer is of first order and it is controlled by the viscosity homogenization in the measuring tank [9]. The accuracy class of the measuring system is 2 %. Another type of continuous measuring viscometer has a similar construction with a rotameter. The difference is that the float is designed so that its position is dependent on the viscosity, while the flow rate entering the viscometer is kept constant by a pump. The position of the float is transformed into a standard signal in a similar way to the case of the rotameter. Its accuracy class for measuring cinematic viscosity is 3 %.

Electrical conductivity transducers Electrical conductivity σ is the property of a medium to pass an electric current through it [3]. It is the inverse of the medium electrical resistivity ρ . The electrical resistance R of a cell containing the conductive medium and its conductance G depend on the geometry of the cell, i.e. the area S and distance between electrodes l, according to the relationships R =ρ

l S

S 1 1S = =σ R ρ l l l σ = G = KG, S G =

(7.51)

where K is a constant of the cell. The SI unit for conductivity is Ω−1 ⋅ cm−1 or S ⋅ cm−1 . The concentration-conductivity relationship, which is the basis for this concentration measurement method, is presented in Fig. 7.42 for diluted, and in Fig. 7.43 for concentrated solutions of electrolytes (acids, salts and bases). As it is revealed, the conductivity is linearly dependent on the concentration, at low conductivity values, and is nonlinear (presenting negative slope), at high conductivity values. The conductivity measuring transducer is presented in Fig. 7.44.

HS 2 O Ca(O 4 H) NaO 2 H

HCl

200 | Part II: Analysis of the feedback control system

MgCl2

180

CaCl2

Electrical conductivity [μS/cm]

160

NaCl

140

KCl

120

CaSO4 Na2SO4

100

KNO3

80 60 40 20 Fig. 7.42. Electrical conductivity of diluted solutions of electrolytes at the temperature of 18 ∘ C.

0 0

10

20

30

40

50

60

70

80

Concentration [mg/L]

0.7

HF (–15°C) HCI

Electrical conductivity [S/cm]

0.6 0.5

KOH

0.4

HNO3

0.3 H2SO4

NaOH

0.2 0.1 0

0

10

20

30

40

50

60

70

80

90

100

Concentration [%mass] Fig. 7.43. Electrical conductivity of concentrated solutions of electrolytes at the temperature of 18 ∘ C.

7 Transducers and measuring systems |

201

RT R1

P1

S

I

R3

R2

Conductivity measuring cell

Fig. 7.44. Electrical conductivity measurement principle.

Relative electrical conductivity [σ/σ20]

The conductivity measuring cell consists in two parallel plate electrodes placed in the flowing conductive solution. The cell resistance, depending on liquid conductivity, is introduced in a branch of the electrical measuring bridge. The measuring bridge is supplied with alternating current (1000 Hz) in order to avoid electrical polarization, which, for direct current, consists in the accumulation of ions around the plates and the formation of a barrier for the current passage. The electrical conductivity depends on temperature, as shown in Fig. 7.45.

2.5

NaOH 50% NaOH 20%

2

H2SO4 96%

1.5

KCI – 1N 1

H2SO4 30%

0.5 0

0

10

20

30

40

50

Temperature° [C] Fig. 7.45. Electrical conductivity dependence on temperature.

202 | Part II: Analysis of the feedback control system As a consequence, compensation of the conductivity change due to the temperature changes is performed with the RTD, RT , also immersed in the measured solution. The RTD brings a change of the resistance in the measuring branch of the bridge that compensates the change of the cell resistance produced by temperature. The dynamic behavior is close to a proportional system due to the rapid response of the electric phenomena. The accuracy class is higher than 0.5 %. The direct contact conductivity cell, presented in Fig. 7.44, can be used for the conductivity measurements ranging from 0.05 μ S ⋅ cm−1 to 200 μ S ⋅ cm−1 . For measuring high corrosive concentrated solutions (acids and bases) or when deposition may appear on the electrodes of the cell, the conductivity can be measured by the conductivity meter with no electrodes [4, 5], presented in Fig. 7.46. Transmitter Standard signal Oscillator Primary coil

Detector Secondary coil

Conductive Solution

Fig. 7.46. Electrical conductivity transducer without electrodes.

The measuring principle relies on Faraday’s law of electromagnetic induction. Two toroidal coils are used, both situated around the pipe passed by the conductive solution and encapsulated in an epoxy resin. One coil is supplied with an oscillating voltage. The second coil is coupled with the first one by the magnetic field and due to the presence of the conductive solution. The voltage induced in the second coil is dependent on the conductivity of the solution and hence, to the concentration. The induced voltage is processed by a detector and transformed into a standard signal. This electrodeless transducer can measure conductivities starting from 30 μ S ⋅ cm−1 . Usually, the calibration of the instrument is performed using KCl etalon solutions, at controlled temperature. The accuracy class of the instrument is higher than 1 % and its time response is almost instantaneous.

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pH transducers The measurement of the acidity or basicity of a reaction medium (e.g. of wastewaters) is given by the concentration of the hydrogen ions H+ or of hydroxyl ions OH− [3]. The product of the two ion concentrations is the ionization constant. For the case of water this product is CH+ COH− = KH2 O = 10−14 . (7.52) At the neutral point, where the concentration of the hydrogen ions H+ or of the hydroxyl ions OH− is equal, their concentration is: CH+ = COH− = √KH2 O = 10−7 .

(7.53)

In order to simplify the representation of the concentration exponential values, the following logarithmic quantities have been defined: pH = − log10 CH+ ,

(7.54)

pOH = − log10 COH− ,

(7.55)

pKH2 O = − log10 KH2 O .

(7.56)

According to these definitions, equation (7.52) becomes: pH + pOH = pKH2 O .

(7.57)

It may be stated that, depending on the acidity and basicity, the pH of aqueous solutions takes the following typical values: pH = 1 for strong acid solution (e.g. N/10 HCl), pH = 2 to pH = 4 for weak acid solution (e.g. N/102 to N/104 HCl), pH = 7 for neutral solution (e.g. distilled water), pH = 10 to pH = 12 for diluted basic solution (e.g. N/104 to N/102 NaOH) and pH = 14 for strong basic solution (e.g. N/1 NaOH). The pH measurement is based on the formation of a galvanic cell between a measurement electrode, ME, and a reference electrode, RE. The electromotive force of the galvanic cell depends on the pH of the solution to be measured. Both the measurement and the reference electrode may consist of a metal wire immersed in a solution of chloride ions. The reference electrode can use a potassium chloride solution. The measurement electrode uses a pH 7 chloride buffer. In the measurement electrode, a special glass produces an electrical differential potential on its internal and external faces. This is due to the different concentration of hydronium ions on the faces of the glass membrane, produced by the migration of the protons (hydrogen ions). The measurement electrode (galvanic half-cell) may have different types: hydrogen, quinhydrone, metal-metallic oxide, glass, etc. The most commonly used one is the glass electrode, presented in Fig. 7.47. The glass electrode consists of the glass membrane 1, the buffer solution 2, the platinum electrode 3 and an isolated connection wire 4. As reference electrode, the calomel electrode is the most common. It is presented in Fig. 7.48.

204 | Part II: Analysis of the feedback control system 4

3

2 1

Fig. 7.47. Glass electrode for pH measurement.

4 3 2 1 Fig. 7.48. Calomel reference electrode.

The calomel electrode consists of the platinum Pt wire 1, mercury Hg inventory 2, calomel Hg2 Cl2 inventory 3, and saturated potassium chloride solution 4. The galvanic cell formed by the two half-cell electrodes is: Pt, Hg | Hg2 Cl2 , saturated KCl || pHi | glass membrane | pHm electrode || H2 , Pt. pHm is the pH of the measured medium and pHi is the pH of the glass electrode internal solution. The newer implementations use AgCl and Au wire.

7 Transducers and measuring systems | 205

The e.m.f. of the galvanic cell is u = uas − uN (pHm − pHi )

(7.58)

where the asymmetry voltage uas is the voltage drop on the membrane of the glass electrode and uN is the Nernst potential term, defined by uN =

2.303 R T ∘ , F

(7.59)

where R is the universal gas constant, T ∘ is the absolute temperature, and F is the Faraday’s constant. Due to the fact that the e.m.f. of the galvanic cell depends both on pH and temperature, as shown in Fig. 7.49, the industrial pH meter is a voltmeter provided with the compensation of the e.m.f. change with the temperature change. The industrial pH meter is schematically represented in Fig. 7.50. It provides the compensation of the pH changes with temperature. u [mV]

400 300 200 Δu

100 0 2

4

6

8

10 12 14

–100

80ᵒC

–200

40ᵒC

–300 0ᵒC

–400

pH

20ᵒC

Fig. 7.49. pH dependence on temperature.

The input voltage of the electronic amplifier ui is a summation between the e.m.f. voltage of the galvanic cell ue and the temperature compensation voltage uBA : ui = ue − uBA .

(7.60)

The temperature compensation voltage uBA is obtained from the diagonal of the Wheatstone bridge which has an RTD arm of resistance RT . When the temperature of the medium changes from the reference temperature of 20 ∘ C, the e.m.f. of the galvanic cell changes: ue = ue + Δu. (7.61) 20 ∘ C

206 | Part II: Analysis of the feedback control system

uBA Electronic amplifier Standard signal

ui A

ME

B

RE

RT

ue

Fig. 7.50. pH transducer.

The change of the e.m.f. voltage produced by the temperature change Δu is compensated by the uBA voltage, uBA = Δu. The Wheatstone bridge is designed in such a way that at the reference temperature of 20 ∘ C the compensation voltage to be uBA = 0. Due to the glass membrane of the measuring electrode, the ME-RE galvanic cell has a high internal electrical resistance, Rint = 0.1–1 MΩ. As presented in Fig. 7.51, the input resistance of the electronic amplifier Ri must be considerably higher than Rint (Ri > 100 MΩ) in order to provide an accurate measurement: ui = ue

Rint

Ri

1 . R 1 + int Ri

(7.62)

Ui

Ue

Fig. 7.51. Electrical scheme of the pH transducer.

7 Transducers and measuring systems

| 207

In order to prevent building material from being deposited, the electrodes’ construction provides different self-cleaning methods, such as: ultrasonic, brush, water and chemical cleaners. New electrode designs include the two (double junction) or all three electrodes in a single case and the microprocessor-based transmitter makes the required compensations (temperature, aging, operating conditions). The pH measurement may be performed with absolute errors higher than ± 0.02 pH. The time constant of the pH meter is depending on the electrodes’ particular geometry and can reach the order of seconds.

Dissolved oxygen transducers Continuous Dissolved Oxygen (DO) concentration measurement is of very high interest in wastewater treatment, as it reveals one of the most important requirements for microorganisms’ growth. Two types of electrochemical cells are frequently used: the galvanic cell and the polarographic cell. The galvanic cell consists of two electrodes, the anode made of silver Ag and the cathode made of lead Pb [3]. The electrolyte to be analyzed is passed through the cell where the two electrodes are immersed, as presented in Fig. 7.52. Electronic transmitter Standard signal

+

–

Ag

Pb

Fig. 7.52. Dissolved oxygen transducer.

The redox reactions involving oxygen are O2 + 2H2 O + 4e− → 4OH−

(7.63)

Pb + 2OH− → 2e− + Pb(OH)2

(7.64)

at the anode, and at the cathode.

208 | Part II: Analysis of the feedback control system A galvanic cell is formed and its e.m.f. u depends on the DO concentration: a RT ∘ (7.65) ln 0 εAg = εAg0 + zO F aO2− εPb = εPb0 +

a + RT ∘ ln Pb zPb F aPb

(7.66)

u = εAg − εPb

(7.67)

where ε is the notation for the potential developed at the electrodes, a is the ion activity, F is the Faraday’s constant, and z is the number of electrons transferred in the balance equation. The galvanic cell is connected to an electronic transmitter that transforms the voltage in a standard current signal. Using the same measuring approach, the oxygen concentration in a gaseous mixture can be determined if the gas is bubbled in an alkaline solution of the instrument. By means of electrochemical methods it is possible to determine the composition of species or substances that participate in redox reactions (e.g. Na in amalgam for the NaCl electrolysis process or hydrazine for hydrazine production). The polarographic cell has two electrodes made of gold, Au, for the cathode, and of silver, Ag, for the anode [5]. The DO cell is presented in Fig. 7.53. Electronic transmitter Standard signal

RTD Anode

Cathode Electrolyte Selective membrane

Fig. 7.53. Polarographic dissolved oxygen transducer.

The electrochemical reactions are O2 + 2H2 O + 4e− → 4OH−

(7.68)

4Ag + 4Cl− → 4AgCl + 4e−

(7.69)

at the cathode, and at the anode.

7 Transducers and measuring systems | 209

Oxygen diffuses through the membrane made of Teflon and it is reduced at the polarized cathode. The electrical current generated depends on the DO concentration. Compensation with temperature changes is provided by an immersed RTD. The response time is about 1 minute or more. The instrument’s accuracy class is higher than 3 %.

Gas analyzers The most common gas analyzing methods and their measurement domains are presented in Tab. 7.1. Table 7.1. Measuring principles, measuring domains and measured components of typical gas analysers. Measuring principle

Measuring domain

Measured component

Paramagnetism

0–5 % 0–100 % 0–5 ppm 0–1000 ppm

O2 H2 O H2 +CO CO2

Thermal conductivity

0–5 % 0–100 %

O2 , O3 , CH4 , SO2 , H2 S, Cl2 , H2 O, H2 , Cl, NH3 , NO2 , CO, CO2 , CO + H2

Electric conductivity

Specific domain for each gas

O2 , CH4 , SO2 , H2 S, H2 , Cl2 , NH3 , NO2 , CO, CO2 , CO + H2

Infrared absorption

Specific domain for each gas

O2 , O3 , CH4 , SO2 , Cl2 , H2 O, H2 , Cl, NH3 , NO2 , CO, CO2 , long chain hydrocarbons

UV absorption

Specific domain for each gas

Cl2 in H2 , benzene in alcohol, phenol in steam

Gas chromatography

Specific domain for each gas

O2 , SO2 , H2 S, H2 , Cl2 , NH3 , NO2 , CO, CO + H2

Chemical absorption

0–0.013 %

O2 , SO2 , H2 S, H2 O, CO2

Catalytic combustion

CO, CO + H2

Infrared gas analyzers The measurement principle of the IR analyzers is based on the absorption of the infrared radiation when passed through the measurement chamber filled with gas mixtures [3]. It is a selective measuring method, as it relies on the fact that each gas has particular IR wavelengths it absorbs. This selective IR wavelength absorption is presented in Fig. 7.54, for a set of common gases.

210 | Part II: Analysis of the feedback control system

Absorption ratio [%]

CH4

CO2 CO

C2H4

C2H4

CH4

C2H4

100 CO2

0

5

10

15

λ [μm]

Fig. 7.54. IR absorption spectra of some gases.

The magnitude of the IR radiation that suffered absorption by one of the gas mixture components is described by the Lambert–Beer law: I = I0 e−α (λ )Ck l ,

(7.70)

where I is the magnitude (intensity) of transmitted IR radiation, I0 is the intensity of the incident IR radiation, Ck is the concentration of the absorbing gas, l is the distance the IR radiation travels through the gas, and α (λ ) is an absorption coefficient that depends on the IR wavelength λ . The measuring principle of the IR analyzer in presented in Fig. 7.55. The main parts of the analyzer are: L1 and L2 are filament lamps that produce the IR radiation, Mo is the motor driving a helix for creating an intermittent IR radiation beam, F1 and F2 are radiation IR filters, 1 and 2 are the measuring and reference gas chambers, 3 and 4 are detecting chambers filled with an IR absorbing gas, M is an elastic metallic membrane of the electrical condenser C, and d is the distance between the condenser plates. A gas mixture consisting of three gas components C1 , C2 and C3 is considered, of which only the C1 component absorbs the IR radiation. The concentration of C1 is the target of the measurement. The gas to be analyzed is passed into the measuring chamber 1. In the second chamber 2 a gas is introduced that does not absorb the IR radiation (e.g. N2 ). The intensity of the radiation received in the detecting chamber 3 is higher compared to that received in the detecting chamber 4, due to the IR absorption of the gas analyzed in the measuring chamber 1. The difference between these two radiation intensities is proportional to the concentration of the C1 gas component contained in the gas mixture. As a consequence, the temperature in chambers 3 and 4 is different. The pressure is also different, p3 > p4 , producing a differential pressure Δp = p3 –p4 . The differential pressure Δp changes the distance d by displacing the mobile plate of the electric condenser. As a result, the electrical capacitance of the condenser C = ε Sc /d changes proportional to the component C1 concentration in the

7 Transducers and measuring systems | 211

+ – L2

L1

Mo

F2

F1

2

1

I1

I2 3

M C

4 d Transmitter Standard signal

~

Fig. 7.55. IR analyzer.

measured gas mixture. The relationship between the condenser electric capacitance C and the concentration of C1 gas component may be obtained from the following equations: p3 SM − p4 SM − Km x = 0

(7.71)

x = dmax − d

(7.72)

C=

ε Sc ε Sc ε Sc , = = d dmax − x dmax − (p3 −p4 )SM K

(7.73)

m

but from the ideal gas law the pressure is related to the temperature p V = n R T∘ 3 3

3

3

p4 V4 = n4 R T4∘ ,

(7.74) (7.75)

and equation (7.73) becomes ε Sc T∘

C=

R (n3 V − 3

dmax −

3

Km

T∘ n4 V4 ) 4

.

(7.76)

SM

The p, V, T ∘ , and n notations have been used for pressure, volume, temperature, and the number of moles in the chambers 3 and 4 (the latter denoted by the subscript

212 | Part II: Analysis of the feedback control system index); Km is the elastic constant of the metallic membrane having the area SM ; Sc is the area of the condenser plates; ε is the permittivity of the gas; and d is the distance between the condenser plates. From the expression of the gas kinetic energy in chambers 3 and 4, the temperatures may be obtained as 2E0 T3∘ = (7.77) 3k and 2E0 e−α CC1 l (7.78) T4∘ = , 3k where E0 is the energy of the light emitted by the lamp and k is the Boltzmann constant. Equation (7.76) becomes C=

ε Sc . 2E0 R SM n3 n4 −α C1 l dmax − ( − e ) 3k Km V3 V4

(7.79)

The concentration-dependent capacitance of the condenser is a branch of the electrical bridge supplied by alternative voltage. The voltage obtained from the bridge is processed by an electronic transmitter and the concentration dependent standard current signal is generated. For the case when there are two or more gaseous components that absorb the IR radiation in the gas mixture, but only one is of interest, the chambers of the filters will be filled with the unmeaning gas components. In this way, the component of the emitted IR radiation having the wavelength corresponding to the parasite gases are removed by absorption in the filters. The dynamic behavior of the IR analyzer is characterized by a time constant less than 3 s due to the small volumes of the chambers. The accuracy class of the instrument is 1.5 %.

Thermal conductivity gas analyzers The thermal conductivity λ is related to the heat transfer as presented in the equation Q = −λ A

𝜕T ∘ t, 𝜕x

(7.80)

where Q is the quantity of heat passing through the surface A in a defined period of time t and due to a spatial gradient of the temperature 𝜕T ∘ /𝜕x, with the heat flux being normal to the surface [3]. Different gases have different thermal conductivities, as presented in Tab. 7.2. The thermal conductivity of two gas mixtures depends on the molar fraction of the gas components, x1 and x2 , according to the additive relationship: λ = λ 1 x1 + λ 2 x2 .

(7.81)

7 Transducers and measuring systems

| 213

Table 7.2. Thermal conductivity of different gases λ , [cal/(cm s grd)]. Gas

λ ⋅ 105 , 0∘ C

Gas

λ ⋅ 105 , 0∘ C

Gas

λ ⋅ 105 , 0∘ C

Gas

λ ⋅ 105 , 0∘ C

He Ne Ar H2 N2 O2 Cl2

34 11 3.9 41.6 5.81 5.89 1.88

CO CO2 HCl SO2 H2 S NH3 Air

5.6 3.4 2.7 2.0 3.1 5.2 5.83

CH4 C2 H6 C3 H8 C4 H10 C2 H4 C2 H2 CHCl3

7.2 4.3 3.6 3.2 4.2 4.5 1.6

CH2 Cl2 CH3 Cl CH3 OH Ag Invar Glass H2 O (l)

1.6 2.2 3.45 105 2600 200 130

This is only valid for particular gas mixtures, such as air-CO or air-CH4 . Other mixtures, such as air-NH3 , CO–NH3 , or air-steam, show nonlinear dependence between the thermal conductivity and the fraction of the components, with thermal conductivity maxima. For these cases, this measuring method based on the dependence of the components’ concentration on thermal conductivity is not applicable without special nonlinearity compensation. Nevertheless, this measuring method is pertinent when, in a mixture of two gases, one component has a high thermal conductivity (e.g. H2 ) and the other one has a low thermal conductivity (e.g. N2 ), as presented in Fig. 7.56.

λ 104 [cal/(cm s deg)]

4 3 2 1 0

0

20

40

60

H2 in N2 at 0°C [%]

80

100

Fig. 7.56. Dependence of the N2 –H2 gas mixture thermal conductivity on the H2 fraction.

The gas concentration measuring principle, based on thermal conductivity, is presented in Fig. 7.57. The measuring cell MC and the reference cell RC are placed in two branches of the Wheatstone bridge. Each of these cells contains an electrical resistance. The measuring gas mixture flows over the measurement cell resistance RMC and the reference gas over the reference cell resistance RRC . The Wheatstone bridge is balanced when the reference gas flows through both cells, due to the fact that the resistances are equally cooled by the gas having the same thermal conductivity. In this situation R1 RRC =

214 | Part II: Analysis of the feedback control system Measured gas

Reference gas

MC

RC

RMC

RRC u

R2

R1

E Fig. 7.57. Wheatstone bridge for measuring the composition of a gas mixture, based on thermal conductivity.

R2 RMC and the voltage u = 0. The output voltage of the Wheatstone bridge changes u ≠ 0 when a different gas mixture or the same gas mixture with different mole fraction passes through the measuring chamber. This is due to the fact that the RMC and RRC resistances are unequally cooled. The output voltage is proportional to the thermal conductivity change and thus, the voltage is proportional to the gas composition. This analyzer may be successfully used for measuring gas mixtures such as: N2 – H2 , H2 -hydrocarbons but also for gas mixtures containing SO2 , He, NH3 , and Cl2 . The response time is of about 5 s, due to the small volume of the measuring cell. The accuracy class is 2.5 %.

Gas analyzers based on paramagnetism Paramagnetism is the property of materials to be attracted by an externally applied magnetic field. Diamagnetic materials are repelled by the magnetic field. Magnetic susceptibility χ describes the degree of magnetization due to the externally applied magnetic field. It is a dimensionless proportionality constant of the material. Magnetic

7 Transducers and measuring systems

| 215

susceptibility χ and magnetic permeability μ are related by the relationship μ = μ0 (1 + χ ).

(7.82)

Paramagnetic materials have positive magnetic susceptibility χ > 0, while diamagnetic materials have negative magnetic susceptibility χ < 0. The large majority of gases are diamagnetic. Some gases are paramagnetic, such as O2 and NO. They can be measured quantitatively from mixtures of gases, based on their paramagnetic property. Magnetic susceptibility χ and temperature T ∘ are related by Curie’s law: χ T ∘ = C,

(7.83)

where C is the Curie constant. Magnetic susceptibility depends on pressure and temperature for paramagnetic gases according to the relationship χ=

ρ0 T0∘ 1 pC ∘ 2 . p0 (T )

(7.84)

Magnetic susceptibility of a mixture of gases is the weighted sum of the components’ magnetic susceptibility. As oxygen is the most common paramagnetic gas, the oxygen analyzer based on paramagnetism is presented in Fig. 7.58.

R2

R1 A

N

B Z

S Transmitter Standard signal

R3

R4

Fig. 7.58. Oxygen gas analyzer based on paramagnetism.

Its measuring principle relies on the attraction of oxygen in an external magnetic field, followed by its release due to heating. As a result of the generated “magnetic wind”, the electrical resistances in a Wheatstone bridge will be changed [3]. The gas to be analyzed enters a toroidal tube. A glass tube connects the two pathways of the gas in the toroidal tube. A permanent magnet (N–S) is asymmetrically

216 | Part II: Analysis of the feedback control system mounted on this glass connecting tube and its magnetic field attracts the oxygen inside it. An oxygen “magnetic wind” is generated along the z direction. Two electrical resistances, R1 and R2 , are mounted on the same glass tube. They are supplied with electrical current. Resistance R1 heats the oxygen attracted in the permanent magnet zone. Once heated, the oxygen loses its paramagnetic properties and it is released from the permanent magnet zone. As a result, the “magnetic wind” is generated and heat is transported in the zone of the resistance R2 , heating it. The resistance R1 is cooled and resistance R2 is heated. They unbalance the Wheatstone bridge formed with R3 and R4 resistances and produce an electrical voltage transformed into a standard signal by the transmitter. The generated voltage is proportional to the oxygen content in the measured gas. The dynamic behavior of the paramagnetic gas analyzer is characterized by the time constant is of about 1 second. The accuracy class is 3 %.

Zirconia oxygen gas analyzer The zirconia oxygen gas analyzer consists in a high temperature ceramic sensor [6, 7]. The measuring cell has two platinum electrodes and zirconia ceramic between them. At high temperature (over 500 ∘ C) zirconia ceramic has the property of allowing oxygen ions to pass through it. One electrode is placed in a gas with constant oxygen content (reference gas, which is usually air) and the other one is in contact with the measured gas. Oxygen ions move through zirconia ceramic in the direction of descending gradient of oxygen concentration and generate an electromotive force proportional to the oxygen concentration in the measured gas. Figure 7.59 schematically represents the zirconia oxygen measuring cell. Zirconia ceramic

Electrodes Heating oven

Reference gas Measured gas

Reference gas Zirconia ceramic

O2O2-

Measured gas

Transmitter +

O2

O2

Fig. 7.59. Zirconia oxygen gas cell.

Standard signal

E −

7 Transducers and measuring systems

| 217

At the high oxygen concentration zone of the cell the electrochemical reaction is O2 + 4e− → 2O2−

(7.85)

and at the low oxygen concentration zone it is 2O2− → O2 + 4e− .

(7.86)

The generated e.m.f. is given by the Nernst relationship E=

P RT ∘ ln REF , nF PMES

(7.87)

where PREF and PMES are the partial pressure of oxygen on the reference gas side of the zirconia ceramic electrolyte and, respectively, on the measured gas side. Zirconia analyzers may be used for measurement of oxygen concentration over the range 0.01 ppm to 100 % in gases or gas mixtures. It is important to note that the measured gas must not contain combustible gases, such as: CO, H2 , hydrocarbons (CH4 ) due to the high temperature of the oven. Contact with the halogens, halogenated hydrocarbons, sulfur and lead containing compounds is also restricted, because they may poison the cell. The time response is short (less than 2 s) and is of the instrument accuracy class 1.5 %.

Humidity gas analyzers Humidity measurement is important for a large set of applications, ranging from pharmaceuticals, food, environment, textile, pulp and paper, warehouse storage, laboratory and medicine fields. Some of the most common humidity terms are as follows [4, 5]: 1. Humidity is the quantity of water vapors contained in the gas mixture (air). 2. Absolute humidity is the mass of water vapors contained in the unit of volume of a gas mixture. 3. Dew point temperature is the temperature at which water saturation occurs or the temperature at which water starts to condense on a surface. 4. Relative humidity is the ratio between the mass of water vapors and the maximum mass of water vapors in the gas mixture, at given temperature (pressure). As a result, relative humidity may also be defined as the ratio between the partial pressure of water vapors in the measurement gas pp (or partial pressure at dew point) and the saturation vapor pressure of water ps (or partial pressure if the dew point were equal to the gas mixture temperature): φ= 5.

pp ps

.

Dry-bulb temperature is the temperature of the measuring gas mixture.

(7.88)

218 | Part II: Analysis of the feedback control system 6.

Wet-bulb temperature is the temperature of a wetted thermometer which is cooled due to evaporation of the gas mixture stream.

Dew point temperature measurement is an indirect measurement of the absolute humidity and vapor pressure. Transformation of relative humidity and dew point can be performed if the dry-bulb temperature and total pressure of the gas mixture are known. Psychrometric diagrams allow the conversion of dew point, relative humidity, absolute humidity, dry- and wet-bulb temperature to each other. They may be implemented in computer applications. Some of the measuring principles of humidity gas analyzers are [4]: 1. measurement of the difference between the dry-bulb and wet-bulb temperature produced by evaporation of a wetted thermometer, and converting the temperature difference into relative humidity or dew point; 2. chilled surface temperature measurement that reveals the dew point obtained on a cooled mirror; 3. mechanical expansion of materials due to the influence water content has on their volume; 4. electrical capacitance and electrical resistance property change of materials (polymers, aluminum oxides, silicon oxides, hygroscopic salts) with the relative humidity or dew point. The humidity measurement based on the psychrometer is presented in Fig. 7.60. Dry-bulb thermometer

Wet-bulb thermometer

Water tank T °, pp

Twb° , ps

Wet wick

Fig. 7.60. Psychrometer based on dryand wet-bulb thermometers.

Two thermometers are placed in the stream of the measurement gas mixture [3]. The gas mixture flowing velocity should exceed a minimum velocity (e.g. higher than 2 m/s). The wet-bulb is continuously wetted by a wick supplied with distilled wa-

7 Transducers and measuring systems | 219

ter which causes a temperature drop due to evaporation. The dry-bulb and wet-bulb ∘ , the dry-bulb temperature and the relative humidity temperature difference T ∘ –Twb relationship is presented in the plots of Fig. 7.61.

50

0

5

10

45

20

hu m id ity [% ]

35 30

30

Re la tiv e

Psychrometric difference [°C]

40

25

40

20

50

15 60 10

70

5

90

0 0

10

20

30

40

50

60

70

80

90

100

Dry-bulb thermometer [°C] Fig. 7.61. Psychrometric chart for relative humidity measurement.

As presented in Fig. 7.61, a curve describing the dry-bulb temperature and psychrometric difference relationship corresponds to each given air relative humidity. For practical calculations, the Sprung formula may be used for computing the partial pressure of water from the dry-bulb and wet-bulb temperature difference: pp = pwb − 0.5

p ∘ ), (T ∘ − Twb 760

(7.89)

where the 0.5 coefficient is the Assmann psychrometer constant, pwb is the saturation ∘ and p is the air pressure. partial pressure at the wet-bulb temperature Twb The relative humidity becomes φ =

p ∘ ) pwb − 0.5 760 (T ∘ − Twb

ps

.

(7.90)

Replacing the thermometers with RTDs, for the measurement of the dry-bulb and wetbulb temperature, the psychrometric transducer is obtained. Its schematic representation is shown in Fig. 7.62.

220 | Part II: Analysis of the feedback control system Transmitter Standard signal

Water tank

RTD

RTD

Wet wick

Fig. 7.62. Psychrometric transducer using RTDs.

The RTD’s temperature measurements are sent to the transmitter. The latter computes the temperature difference and converts it, according to the equation (7.90), into a standard signal proportional to the relative humidity. The time constant of the instrument is up to 10 s and its accuracy class is 3 %. The chilled surface humidity measuring method (dew point method) is based on determining the condensation temperature of water vapors from the gaseous mixture. This is the temperature of saturation that corresponds to the saturation partial pressure of the water vapors. Condensation appears on a thermo-electrically cooled metallic surface. The principle of the chilled mirror humidity measuring method is presented in Fig. 7.63.

Electrical source

Light source Phototransistor Gas to be analyzed

Phototransistor Transmitter Chilled mirror Thermo–electrical cooling

Fig. 7.63. Chilled mirror hygrometer.

i [mA]

t [s]

7 Transducers and measuring systems |

221

The hygrometer works on a semi-continuous base, with repeated measuring cycles [5]. The light source sends a beam towards the chilled mirror which has a reflecting surface. The mirror is placed in the stream of the gas to be analyzed. The phototransistors receive the reflected beam having high amplitude. The thermo-electrical cooling system, controlled by an electrical source, diminishes the temperature of the mirror up to the moment (temperature) when fine particles of water (condensate) appear on its surface. This moment is detected by the electro-optical system of the phototransistors, as the reflected beam amplitude changes to low amplitude. This final temperature of the mirror is the dew temperature. The temperature is converted into a standard signal by the transmitter. The measuring cycle repeats after the condensate is removed. The measuring cycle lasts for about 30 s and the instrument accuracy class is 0.5 %.

Output signal

Process chromatographs The operation principle of gas chromatographs consists in the selective retention of the components from a gaseous mixture on the stationary (solid) phase [3]. The retention depends on the molecular mass of the components, favoring the retention of the components with smaller molecular mass. The successive separation is accomplished by the support of a carrying gas that entrains the retained gas components, but due to its high molecular mass it is not retained on the stationary phase. Desorption of the retained gas components is performed in the increasing order of the retention time. The output gas from the column enters a gas analyzer (IR, thermal conductivity, etc.) and the result is presented in a chromatogram, of the generic form shown in Fig. 7.64.

Injection time

D A C B

trA

time

Fig. 7.64. Chromatogram showing each of the A, B, C, and D components removed by desorption at fixed moments of time.

The time elapsed from the injection of the sample and the appearance of the i-th peak maximum is denoted as the retention time of the component i (e.g. trA is the retention time for component A). The retention time is a qualitative indication of the i-th component’s presence in the sample. If the operation parameters (such as temperature, flow rate of the injected gas mixture and carrying gas) do not change, a certain component will leave the chromatographic column after the same period of time during

222 | Part II: Analysis of the feedback control system the measuring cycle. These moments of time allow the identification of the components. By the integration of the peak it is possible to obtain quantitative information about the component, as the concentration of the component is proportional to the area of the peak. Numerical integration is performed by the process chromatographs. The components’ concentration of the sample may be computed by Ci =

A i fi n

,

(7.91)

∑ A i fi i=1

where Ai is the area of component i and fi is the detector specific calibration coefficient of correction. The computation of the correction coefficients is carried out following two steps. First, exact quantities of each of the gas mixture components are injected into the column and the areas of the obtained peaks S1 , S2 , . . . , Sn are determined. Second, one of the components is selected as reference (e.g. the n-th component) and the correction coefficients are computed by fi =

Si , Sn

i = 1, . . . , n.

(7.92)

These correction coefficients account for the different sensitivities of the detector with respect to each of the components. The simplified representation of the chromatograph is shown in Fig. 7.65. Flux 1 Flux 2 5

Flux 3

6

7

3

8

4 1

2

Fig. 7.65. Simplified scheme of the process chromatograph.

The operation of the chromatograph is discontinuous and a measuring cycle consists in a sequence of operations. First, the sample is extracted from the process using the sampler 1. The sample is introduced into the conditioning unit 2, which brings it to the operating conditions specific to the analyzing unit 4. For the case of the single-flux chromatograph, the conditioned sample enters directly in the analyzing unit 4. For the multi-flux chromatograph, the sample enters in the switching unit 3 which has the role to sequentially connect the conditioned samples to analyzing unit 4. The switching moments for the different fluxes are received from the control unit. The fluxes are switched so that while the sample from a flux is analyzed, the sample from the next

7 Transducers and measuring systems | 223

flux is circulated to prepare a fresh sample. The injection unit 5 sends the sample to the system of the chromatographic columns. Subsequently, it introduces the carrying gas for the desorption step. The distribution of the sample-carrying gas mixture towards one of the chromatographic columns is performed by the switching unit 6. The role of the switching unit 6 is to reduce the time of the chromatographic measurement by overlapping part of the operations. While in one of the chromatographic columns the components are removed by desorption, in another column the sample is adsorbed and the third one is cleaned for receiving the next sample. The analysis is effectively performed in the detector unit 7 and the computation of each component concentration is accomplished by integration in the computation unit 8. The sequence of the operations developed by the control unit is: sampling, conditioning, switching the injection unit on the flux to be analyzed, adsorbing the components on the column, switching the injection unit on the carrying gas, removing by desorption the components from one column, analyzing the flux, computing the concentration of components and cleaning the column. In the meantime, switching between the entering fluxes and between the chromatographic columns are carried on in order to prepare the measurement of the next flux. The signal obtained from the integrator is stored in memory units until the next measuring cycle (sample and hold action) [10]. From the dynamic point of view, the chromatograph is a pure time delay system (time delay spanning from minutes to tens of minutes). High costs and complexity of the chromatograph design and operation impede their large scale application in industry [11]. Nevertheless, process control and process management might also benefit from the advent and new developments of analytical measurements [4]. Some of these systems cross over from the laboratory environment to online process employment. Their sophisticated design, complexity of operation, qualified maintenance and large cost may be surpassed by the valuable added information aimed to optimize process design, operation and control. The measuring principles of these instruments usually rely on electromagnetic radiation or ultrasonic wave emission which is reflected, refracted, absorbed, scattered, produces ionization or induces electromagnetic responses from the sample. They use the gamma ray, x-ray, ultraviolet, visible, near-infrared, mid-infrared, far-infrared, microwave, sound and ultrasound wave domains to investigate the presence and concentration of different compounds or molecular species in the solid, liquid and gas sample. Some of these analyzers are: IR Fourier Transform based spectrometers, quadrupole mass spectrometer, ion cyclotrone resonance mass analyzer, ultraviolet and visible spectrometers, Raman scattering emission spectrophotometers, nuclear magnetic resonance spectrometers, X-ray fluorescence spectrometers, microwave spectrometers, neutron activation analyzers and atomic emission spectrometers.

224 | Part II: Analysis of the feedback control system

7.9 Problems (1) What is the absolute error of a thermocouple temperature measuring system, having the measuring domain ranging from −20 ∘ C to 180 ∘ C if the true measured temperature is of 80 ∘ C and its accuracy class of 1.5 % is defined on the basis of the maximum permissible related error? What is the absolute error of the same temperature measurement system if its accuracy class of 1.5 % is defined on the basis of the maximum permissible relative error? (2) The relative error of a magnetic flow meter, defined by equation (7.2), is 2.5 %. What would the relative error of the same flow meter be when computed by the equation er =

xm − xt 100. xm

(7.93)

(3) Both the volume and the density measurement meters have an accuracy class of 2 %, defined on the basis of the maximum permissible relative error. What is the relative error for inferring the mass, computed by multiplication between the volume and the density measured values? (4) A pressure transducer having the measuring domain between 0 and 50 bar and the accuracy class of 1 % (based on the maximum permissible related error) is used for measuring a pressure of p = 2 bar. (a) From the practical point of view, does the measurement meet the accuracy requirements? (b) What solution should be proposed for obtaining a measurement with an absolute error situated in the interval of ± 5 % of the pressure p to be measured? (5) Consider a volumetric flow rate measuring system based on the orifice plate, differential pressure transducer and square root extractor, presented in Fig. 7.22. For the flow rate spanning from 0 to 10 m3 /h, the differential pressure domain of the orifice plate ranges from 0 to 3200 mm H2 O. The relationship between the flow rate and the differential pressure is F = 0.18 (Δp)1/2 . The root extractor equation is i󸀠 = 2 + √8 √i − 2.

(7.94)

Compute the static gain of the flow rate measuring system (orifice plate, differential pressure transducer and square root extractor) for the particular flow rate of F ∗ = 5 m3 /h. (6) What is the static gain of the following transducers? (a) RTD Pt100 detector and electrical resistance transmitter having the measuring range between 20 ∘ C and 60 ∘ C and the output signal in the range of 2 ⋅ ⋅ ⋅ 10 mA.

7 Transducers and measuring systems | 225

(b) pH transducer at the temperatures of 20 ∘ C and 80 ∘ C. (c) Electrical conductivity meter measuring the conductivity of the KOH solution with mass concentration of 20 % and the HNO3 solution with mass concentration of 70 %, coupled with a conductivity-electrical signal transmitter working in the range of 2 ⋅ ⋅ ⋅ 10 mA and the conductivity measurement in the domain of 0 S/cm ⋅ ⋅ ⋅ 0.8 S/cm. (7) A rotameter is calibrated to measure water flow rates between 60 l/h and 600 l/h. The diameter of the rotameter conic tube at the inlet of the liquid is of Dn = 20 mm and the density of the float is of ρf = 800 kg/m3 . How does the scale of the rotameter change when it is used for measuring the flow rates of sulfuric acid with concentration of C = 530 g/l? The sulfuric acid concentration-density is presented in Tab. 7.3. It is assumed that both liquids have the same viscosity. Table 7.3. Sulphuric acid concentration-density relationship. C [g/l] 3

ρ [kg/m ]

129.2

162

195.2

515.8

553.9

591.9

630.1

707

1080

1100

1120

1300

1320

1340

1360

1400

(8) Consider the water flow rate measurement in a pipe with the interior diameter of Dp = 60 mm, using the orifice plate. The flow rates to be measured range from 0 m3 /h to 3.6 m3 /h. The available differential pressure transmitter has the measuring domain ranging from 0 mmH2 O to 2000 mmH2 O. Design the diameter of the orifice plate so as to be properly used for the given domains of the flow rate and the transmitter differential pressure. The pressure drop coefficients are given in Table 7.4. Table 7.4. Pressure drop coefficients. (D0 /Dp )2

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

ξ

7000

1670

730

400

245

165

117

96

65.6

51.5

(9) Consider a level transducer based on float with the following characteristics: cross-sectional area of the float A = 18 cm2 , fluid density ρ = 1000 kg/m3 , spring’s constant Ks = 12.6 N/m and the measuring domain ranging from 0.2 m to 1 m. Compute the static gain of the transducer and make a graphic representation of its steady state characteristic. Compute also the static gain of the level transducer coupled with a transmitter having the following parameters: float displacement range of 100 mm and output signal of the transmitter ranging from 4 mA to 20 mA.

226 | Part II: Analysis of the feedback control system (10) Ethanol of 50 % purity is used in an esterification column. The quality of the ethanol presents mass concentration fluctuations of ± 10 %. Compute the measuring domain of the bubbling differential pressure transducer Δp, used for the concentration measurement. The following parameters are given: immersion difference between the tubes is Δh = 0.1 m, the reference liquid is pure ethanol at the temperature of 20 ∘ C. The differential pressure is converted in a standard signal ranging from 4 mA to 20 mA by an electronic transmitter. Compute the errors caused by the summer-winter temperature changes of ΔT ∘ = 30 K. The density-concentration relationship is presented in Tab. 7.5, the volumetric thermal expansion is 𝛾 = 10−3 K−1 and the density at the temperature of T ∘ = 303 K is ρet = 0.781 kg/dm3 for the pure ethanol. Table 7.5. Density-concentration relationship. C [%] ρ [kg/m3 ]

0

10

20

30

40

50

60

70

80

90

100

0.99

0.982

0.968

0.954

0.935

0.914

0.891

0.868

0.843

0.818

0.789

(11) Consider a pH meter using the reference calomel electrode and the glass measurement electrode for measuring pH between 2 pH and 12 pH and working at the temperature of 20 ∘ C. The calomel electrode becomes polarized at currents higher than i = 0.5 nA. The internal electrical resistance of the glass electrode is Rint-glass = 100 MΩ and of the calomel electrode is Rint-cal = 100 KΩ. Compute the necessary input resistance of the electronic amplifier. What is the value of this input resistance such that the error of the measurement is less than 1 %?

References [1] [2] [3]

Millea, A., Cartea metrologului Metrologie generală, EdituraTehnică, Bucureşti, 1985. http://www.french-metrology.com/en/more/glossary.asp. Agachi, S., Automatizarea Proceselor Chimice, Editura Casa Cărţii de Ştiinţă, Cluj-Napoca, 1994. [4] McMillan, G. K., (editor-in-chief), Considine, D. M., (late editor-in-chief), Process/Industrial Instruments and Controls Handbook, 5th Edition, McGraw-Hill, 1999. [5] Bsata, A., Instrumentation et Automatisation dans le Controle des Procedes, Deuxieme edition, Les Editions Le Griffon d’argile, Quebec, 1994. [6] http://www.systechillinois.com/en/zirconia-oxygen-analysis_51.html. [7] http://www.toray-eng.com/measuring/tec/zirconia.html. [8] Agachi, S., Cristea, M. V., Lucrări Practice de Automatizarea Proceselor Chimice, Tipografia Univ., Cluj-Napoca, 1996. [9] Cristea, M. V., Agachi, S. P. Elemente de Teoria Sistemelor, EdituraRisoprint, Cluj-Napoca, 2002. [10] Stephanopoulos, G., Chemical Process Control. An Introduction to Theory and Practice, Englewood Cliffs, New-Jersey 07632, Prentice Hall, 1984. [11] Marinoiu, V., Paraschiv, N., Automatizarea Proceselor Chimice, Vol. I, Vol. II, Editura Tehnică, Bucureşti, 1992.

8 Controllers 8.1 Classification of controllers The controller is the element of the control system (Fig. 5.3) which receives the signal from the transducer, compares it with the desired value for the controlled parameter (setpoint), calculates the error e and elaborates a control signal c with the goal of cancelling (e = 0) or, at least decreasing the error. A classification of controllers can be made from different points of view [1]: – depending on the way the control action is calculated and emitted: continuous or discrete controllers (continuously or discretely at certain moments); – depending on the computing algorithm of the controller output (c = f (e)): P (Proportional), PI (Proportional-Integral), PID (Proportional-Integral-Derivative), other algorithms; – depending on the specialization of the controller: specialized on different processes or universally applied controllers to all types of processes; – depending on the type of auxiliary energy used for functioning: electric, pneumatic (indirect action) and without auxiliary energy (direct action) controllers. According to the first type of classification, if the control function of the controller f (e) is continuous, the controller is a continuous-action controller (Fig. 8.1).

Variable value

Actual

Setpoint

% signal to final control element

100

0 time

Fig. 8.1. Continuous action of the controller.

If the control action is discontinuous (meaning that the signal quantization occurs in the controller), then the controller is a discrete-action one (Fig. 8.2). The discretization is of many types, producing different types of discrete controllers: pulse controllers with time quantization, relay controllers with level quantization and digital controllers with time and level quantization. Where the process allows (the controlled variable can float between two limits without any restriction), two-position or three-position controllers can be used. This

228 | Part II: Analysis of the feedback control system

Variable value

Actual

Setpoint

% signal to final control element

100

0 time

(a)

Variable value

Actual

Setpoint

% signal to final control element

100

0

Fig. 8.2. Two types of discrete actions of the controller: (a) on-off, two-position controller; (b) digital controller.

time

(b)

Fe

Y

Xdg

Xbot Xtop Xmax X

Magnet

Ts

time

time

Fig. 8.3. Bimetallic controller of the flat iron (see SAMSON http://www.docentes.unal.edu.co/ sorregoo/docs/sistemas%20de%20control.pdf) [2].

is the case with domestic applications where the thermostats of the central heating, of the flat iron or of the refrigerator are usually on-off controllers (Figs. 8.2a and 8.3).

8 Controllers

|

229

If the process is stricter, a digital controller is usually used and the sampling time (Ts ) is as small as possible to assure an accurate control (Fig. 8.2b). The second classification depending on the control algorithm is the mostly used one, the P, PI, PID algorithm controllers being used in about 80 % of the control loops in industry. One can mention at this point the advanced controllers (adaptive, optimal, predictive controllers etc.) which have other functions for the control algorithm than P, PI, PID. These controllers are more sophisticated and more expensive, being used in the control of the processes with weak controllability. The proportional (P) controller has the control action expressed by equation (8.1). c = c0 + Kc e,

(8.1)

where c is is the control signal addressed to the actuator, c0 is the value of the control action at steady state which assures controlled variable at setpoint value (e = 0), Kc is gain factor, e is error, and Kc e is the proportional component of the signal. The proportional integral (PI) controller has the control action expressed by equation (8.2): t

1 c = c0 + Kc (e + ∫ e (τ ) dτ ), Ti

(8.2)

0

t

where Ti is the integral time constant and T1 ∫0 e(τ )dτ is the integral component of the i control signal. The proportional integral derivative (PID) controller has the control action expressed by equation (8.3): t

c = c0 + Kc (e +

de (t) 1 ∫ e (τ ) dτ + Td ), Ti dt

(8.3)

0

where Td is the derivative time constant and Td de(t) is the derivative component of the dt control signal. Concerning the third classification, in practice either universal controllers exist, used for the control of any process variable, or specialized controllers used just for specific parameters: temperature, pressure, level, flow controllers. The first category implies that the controller is an electronic or a pneumatic one and operates on the process through an intermediate actuator (Fig. 8.4).

230 | Part II: Analysis of the feedback control system

(a)

(b)

(c)

Fig. 8.4. Universal controller; (a) pneumatic controller (FEPA Barlad, Romania); (b) transistor based electronic controller (FEA Bucharest, Romania); (c) microprocessor based electronic controller (ABB Switzerland, through the courtesy of ABB).

The specialized controllers have the actuator embedded in the controller (e.g. the gas pressure controller on the methane gas pipelines, or the water-level controller in the reservoir of the flush toilet, see Fig. 8.5).

Load

Fig. 8.5. The level controller in the flush toilet reservoir.

From the point of view of the energy used, there are the so called controllers using auxiliary energy (electrical or pneumatic) or those using the energy of the controlled process (temperature, pressure, level etc.). The level controller in Fig. 8.5 is an example of a controller using the energy of the controlled process: accumulation of mass in a reservoir.

8 Controllers

| 231

8.2 Classical control algorithms In the following sequences, we will discuss in more detail the behavior of the “classical controllers”, P, PI, PID, and only at introductory level, the advanced controllers such as adaptive, optimal, predictive controllers.

8.2.1 Proportional controller (P) [3] The equation describing the P algorithm is (8.1). The P component shows that the controller takes a proportional action with the dimension of the error. The result is a quick increase or decrease of c, from the value of c0 , depending on the sign of the error and magnitude of Kc , the proportional controller gain. The response of the controller at step input signal is given in Fig. 8.6. e (%) 20

10

0

5

10

15

20

t (s)

Fig. 8.6. P controller response at a step input.

The result of such a behavior on the heater (Fig. 5.2) is the abrupt opening or closing of the control valve up to one intermediate position, changing the agent flow and thus influencing the outflow temperature. The magnitude of the control action change (Δc) which has a direct relationship with the opening/closing of the valve depends on the value of Kc . Kc is the gain of the proportional controller and can be calculated as Kc =

Δc Δe

or

Kc =

Δc . Δxr

(8.4)

The steady state characteristic The Proportional Band (PB), which in some industrial controllers replaces Kc , is the portion of the measuring range for which the controller has a linear/ proportional action. Thus, for a PB = 100 %, the measured variable is possible to change from 0 to maximum (0 %–100 %), for which the controller has the corresponding proportional

232 | Part II: Analysis of the feedback control system action from 0 to maximum (2 mA–10 mA or 0 %–100 %). For PB = 50 % the measurement range for which the corresponding controller action is 0 %–100 % (2 mA–10 mA), is only half of the measurement range of the transducer. Similarly, for a PB = 200 %, for the whole measurement range (0 %–100 %), the controller limits its action only to half of its maximum domain (25 %–75 % or 4 mA–8 mA). The steady state characteristic is given in Fig. 8.7. c (%) 100

Kc=2

75 c0

Kc=0.5

25

0

25

r

75

100

xr (%)

Fig. 8.7. Steady state characteristic of the proportional controller.

The relationship between the Proportional Band and the controller gain is equation (8.5) 100 PB = . (8.5) Kc An industrial controller has a range of the PB between 0.5 % and 500 % allowing the choice of an appropriate controller gain for a certain process controlled. How do we choose the controller gain? The tuning of the controller’s parameters is a special subject to be treated in a separate book. This section only gives an orientation, a rule of the thumb, about choosing the controller gain relative to the process gain factor. Example 8.1. If the pH of two different neutralization processes is to be controlled, one strong acid-strong base and the other one weak acid-weak base (Fig. 8.8), the titration curve is different, presenting different Kpr , much larger in the first case (the slope is about 1000 : 1 at the neutralization point). This shows that the process in the first case is much more sensitive than in the second situation, meaning that to correct the deviation from pH 7, the first neutralization system needs a very fine intervention and the second one a much more severe intervention in the dosage of the base. In order to properly adjust the pH to the value 7, the system in Fig. 8.8a needs a very small opening or closing of the valve, which is determined by a very “small” Kc , whereas in the second case Kc is “much larger” allowing the dosage of a larger amount of base. This “small” or “large” is a very relative notion, meaning “small” or “large” in comparison with what the process is demanding. The “rule of the thumb” is that

8 Controllers

|

233

pH (a) Equivalence point

(b)

SA – SB WA – WB Fig. 8.8. Two neutralization processes at which the pH is controlled: (a) strong acid-strong base; (b) weak acid-weak base.

Volume of base (%)

where the process is very sensitive and is characterized by a high Kpr , the controller has to have a small Kc and vice versa. The transfer and frequency functions of the P controller are those of the P element (equations (4.2) and (4.3)); the Bode diagrams are the same as in Fig. 4.2.

The P control with steady-state error (offset) The proportional controller has the advantage that its intervention is very fast, but at the end of the controlling process, the control accuracy is given by the steady state error est . Example 8.2. Let us discuss the behavior of a pressure control system (Fig. 8.9), which controls the pressure produced by an air-blower in a pipeline. c

Xr

i P

po

pi Δp1

P

Δpv

Fig. 8.9. Pressure control system with a P controller.

In the normal steady state condition, the values of the pressure are p0 = 200 mmH2 O;

Δp1 = 50 mmH2 O;

Δpv = 50 mmH2 O;

pi = 150 mmH2 O;

p = 100 mmH2 O.

The steady state characteristics of the elements in the control system are

234 | Part II: Analysis of the feedback control system Transducer: Measurement range of the pressure transducer: 0 ⋅ ⋅ ⋅ 200 mmH2 O Output range of the transducer: 2 ⋅ ⋅ ⋅ 10 mA Measuring 100 mmH2 O, the output signal xr = 6 mA The steady state characteristic is given in Fig. 8.10. Xr (mA) 10

6

2

0

100

200

P (mmH2O)

Fig. 8.10. Steady state characteristic of the pressure transducer in the example.

Controller: Input range of the controller: 2 ⋅ ⋅ ⋅ 10 mA (0 ⋅ ⋅ ⋅ 100 %) Output range of the controller: 2 ⋅ ⋅ ⋅ 10 mA (0 ⋅ ⋅ ⋅ 100 %) PB = 100 % (Kc = 1) c0 = 6 mA Setpoint value: 6 mA (corresponding to p = 100 mmH2 O to be maintained) At p = 100 mmH2 O, xr = xr0 = 6 mA, e = 0, and c = c0 = 6 mA The steady state characteristic of the controller is given in Fig. 8.11. c (mA) 10

6

2 2

6

10

xr (mA)

Fig. 8.11. Steady state characteristic of the controller in the example.

8 Controllers

|

235

Actuator (control valve) + electro-pneumatic convertor: Input range of signal: 2 ⋅ ⋅ ⋅ 10 mA (0 ⋅ ⋅ ⋅ 100 %) Stem variation range: 0 ⋅ ⋅ ⋅ 100 % Pressure drop range: 0 ⋅ ⋅ ⋅ 100 mmH2 O Stem position at steady normal state: h = 50 % Pressure drop value at normal steady state: Δpv = 50 mmH2 O When p = 100 mmH2 O, xr = 6 mA, e = 0 and c = c0 = 6 mA, Δpv = 50 mmH2 O The steady state characteristic of the control valve is given in Fig. 8.12. h (%)

ΔPV (mmH20)

100

100

50

50

0

0 2

6

10

c (mA)

0

50

100

h (%)

Fig. 8.12. Steady state characteristic of the control valve in the example (h = h(xc )) and (Δpv = Δpv (h)).

The control system aims to keep 100 mmH2 O in the pipeline in the presence of disturbances. Let us analyze the behavior of the system. Suppose a disturbance of +20mmH2 O at the tap after the blower, meaning that Δp1 = 70 mmH2 O. This can be caused by the manipulation of the tap during an ordinary maintenance operation. The consequences are: pi = 130 mmH2 O, the valve does not react yet and Δpv = 50 mmH2 O, the pressure in the pipeline is p = 80 mmH2 O, giving a deviation of −20 mmH2 O (−10 % of the measurement range of the transducer). Δxr = −10 % = −0.8 mA; xr = 6 mA–0.8 mA = 5.2 mA Kc =1; Δc = −10 % = −0.8 mA; c = 6 mA–0.8 mA = 5.2 mA Δh = −10 %; h = 50 %–10 % = 40 % Δpv = 40 mmH2 O. The result after the first intervention of the controller is thus p = 130 mmH2 O– 40 mmH2 O = 90 mmH2 O, an increase of the pressure with 10 mmH2 O, or a relative increase from the former situation of +5 %. The control system continues to act and to correct the error step by step. The functioning of the control system is synthesized in Tab. 8.1 and at different controller gain factors. The dynamic behavior of the control system is illustrated in Fig. 8.13.

236 | Part II: Analysis of the feedback control system Table 8.1. Functioning of the pressure control system in time at different controller gain factors.

t p [s] [mmH2 0]

Δp [%]

ΔXr [%]

Xr [mA]

ΔXc [%]

Xc [mA]

Δh [%]

h

Δpr [mmH2 0]

Kp = 0.5 0 2 4 6 8 10

100 80 85 83.75 84.37 84

0 0 6 0 6 0 50 −10 −10 5.2 −5 5.6 −5 45 2.5 2.5 5.4 1.25 5.7 1.25 46.25 −1.25 −1.25 5.3 −0.125 5.65 −0.625 46.62 0.625 0.625 5.35 0.625 5.67 0.31 45.93

50 45 46.25 46.62 45.93

est1 = 100 − 84 = 16 mmH2 0 = 8%

0 −10 5 −2.5 1.25

0 −10 5 −2.5 1.25

6 5.2 5.6 5.4 5.2

0 −10 5 −2.5 1.25

6 5.2 5.6 5.4 5.2

0 −10 5 −2.5 1.25

50 40 45 42.5 43.75

50 40 45 42.5 43.75

est2 = 100 − 86.7 = 16 mmH2 0 = 6%

0 −10 0 −10 0

0 −10 0 −10 0

6 5.2 6 5.2 6

0 −20 0 −20 0

6 4.4 6 4.4 6

0 −20 0 -20 0

50 30 50 30 50

50 30 50 30 50

Oscillating operation mode

Kp = 1 0 2 4 6 8 10

100 80 90 85 87.5 86.25

Kp = 2 0 2 4 6 8 10

100 80 100 80 100 80

p (mmH2O) 100 K C =2

e st2 e st1

90

K C= 1 K C= 0 . 5

80 0

2

4

6

8

10

time (s)

Fig. 8.13. Dynamic behavior of the pressure control system in Fig. 8.9.

8 Controllers

|

237

One can observe that by increasing the controller gain, the steady state error decreases, but at a certain point the system begins to oscillate. The steady state error can be calculated with the final value theorem (Tab. 2.1) which is enounced in the following sentence: If the time function f (t) and its first derivative admit Laplace transform and the function sF(s) is holomorph on the imaginary axis and in the right half plan s, then lim sF(s) = lim Δf (t).

(8.6)

t→∞

s→0

For the pressure control system formerly discussed, the block diagram is given in Fig. 8.14. Pi

KprD

+

P(S)

+ Kv TvS+1

Kprm

–

KC

+

KT TTS+1

r Fig. 8.14. Block diagram of the pressure control system.

For this example mA mA mmH2 O 100 Kv = = 12.5 ; 8 mA

Hc (s) = Kc = 1 Hv (s) =

Kv ; Tv s + 1

Hpr d (s) = Kpr d = 1

Tv = 3 s

mmH2 O mmH2 O

(the transfer function of the process with the input of the disturbance; there is no ramification of the pipeline to cause any pressure loss) HT (s) =

KT 8 mA ;K = = 0.04 ; TT s + 1 T 200 mmH2 O Hpr m (s) = Kpr m = 1

TT = 1 s

mmH2 O mmH2 O

(the transfer function of the process with the input being the manipulating variable; there is no ramification of the pipeline to cause any pressure loss and any decrease of the process gain factor) H(s) =

Kpr d 1 = . 12.5 0.04 Kv KT 1 1 1+ Kc Kpr m 1 + 3s + 1 s + 1 Tv s + 1 TT s + 1

238 | Part II: Analysis of the feedback control system Then lim sH(s)D(s) = lim s

s→0

s→0

(3s + 1)(s + 1) −20 ⋅ = lim Δp(t) = −13.5 mmH2 O t→∞ (3s + 1)(s + 1) + 0.5 ⋅ 1 s

where −20 is the input disturbance of −20 mm H2 O pressure drop through closing the s tap positioned in front of the control valve. When Kc changes to 0.5, the offset changes to 16 mm H2 O. From this example one can see the steady state error disappears only when the disturbance disappears. From the practical point of view, one can raise the question: is it so important not to have steady state error? The answer is that it depends on the situation. If, for example, the kinetics of a reaction shows that the reaction takes place between 55∘ and 65∘ it is not so important form the chemical point of view to keep the temperature tight at 60∘ . But from the economic point of view it is important to keep the lowest temperature tight in the range for the sake of energy saving. In order to understand what an error only of +2∘ means, we can give the following example. Let us consider a heat exchanger (Fig. 3.1). The heated fluid flow is of F = 5 m3 /h, having the density of ρ = 1000 kg/m3 and specific heat cp = 1 kcal/kg∘ . Suppose the fluid has to be heated at 60∘ and instead, it is heated at 62∘ . This is not very much from the point of view of the reaction. But the heat loss is calculated as ΔQ = F ρ cp ΔT ∘ = 5 ⋅ 1000 ⋅ 1 ⋅ 2 = 10−2 Gcal/h. In one year, a plant is working 7200 hours and with a cost of a Gcal of around 70 Euros, the total money loss is Economic loss = ΔQ ⋅ 7200 ⋅ 70 = 5040 Euros/year. This amount may be considered insignificant for an industry, but on an industrial platform where there are around hundreds or thousands of heat exchangers, the total loss becomes quite important; and it is due to only a very common functioning of a plant. This is one of the reasons why it is important to eliminate the steady state error, which is done using the PI controller.

8.2.2 Proportional-Integral controller (PI) [4] The equation describing the PI algorithm is (8.2). It is observed that, different to the P controller, the integral term occurs, meaning that as long as an error exists, c increases or decreases, depending on its sign, until the error is cancelled. There is no offset in steady state under PI control. The step response of the controller is given in Fig. 8.15. Ti , the integral time, representing the time until the initial proportional step doubles. The demonstration of this

8 Controllers

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239

statement is in the case of a forced constant error e = A produced by the step input Δr = A: t KA 1 Δc = Kc (A + ∫ Adτ ) = Kc A + c t = 2Kc A (8.7) Ti Ti 0

when t = Ti . e

time c 2Kce Kce 0 t=Ti1

t=Ti2

time

Fig. 8.15. The step response of the PI controller.

It can be observed that by increasing the integral time, the control action becomes slower, a fact adequate to a slower dynamics process. The choice of the Ti is important in order to make possible the harmonization of the I action of the controller with the delay in the controlled process; it is useless to impose a control signal with a small Ti to a process with a large time constant, T, since, due to the inertia of the process, the controlled variable will never be able to follow the action of the actuator “inspired” by the controller action. Fixing a much smaller Ti than the time constant of the process, T, involves the quick opening or closing of the control valve, actions at which the slow process will react as in Fig. 8.16. The quick integral action opens the valve very fast to the maximum, the consequence is that, very slowly, the process arrives to the setpoint value, but, because of the great inertia, this value is exceeded and the process reverses, going to an oscillatory behavior. How do we choose the integral time? Example 8.3. Let us consider two pressure control loops for two recipients with different volumes, V1 > V2 (Fig. 8.17). Supposing a decrease of pressure, the same −Δp in both recipients, one has to choose the controllers’ parameters, but specifically to have a relative perception on the values of the two integral time constants, Ti1 and Ti2 . Using the mass balance equations

240 | Part II: Analysis of the feedback control system I [mA] 10

2 t m [Flow] max

0

t

y y1 y0

t

Fig. 8.16. The response of a slow controlled process to a too fast control action imposed by a small integral time.

PC1

V1 P1=P0 –ΔP

Fi

Fe1=Fe2

PC2

Fi

V2 P2=P0 –ΔP

Fe1=Fe2 Fig. 8.17. Pressure control in recipients with different volumes.

and the linear approximation of the mass flow in a pipeline, d(V1 ρ1 ) dt d(V2 ρ2 ) Fi2 − Fe2 = dt p − po pi − p1 Fi1 = and Fe1 = 1 Rp1 Rp2 p2 − po pi − p2 Fi2 = and Fe2 = , Rp1 Rp2

Fi1 − Fe1 =

(8.8) (8.9) (8.10) (8.11)

8 Controllers

and considering the expression of density ρ = (8.12) and (8.13)

pM , RT ∘

Rp1 + Rp2

241

the equations (8.8)–(8.11) become

Rp1 Rp2 V1 M dp1 Rp2 Rp1 p + p + p1 = Rp1 + Rp2 RT ∘ dt Rp1 + Rp2 i Rp1 + Rp2 0 Rp1 Rp2

|

Rp2 Rp1 V2 M dp2 pi + p . + p2 = ∘ Rp1 + Rp2 Rp1 + Rp2 0 RT dt

(8.12)

(8.13)

The only difference between the processes described is the volume of the two recipients. Since V1 > V2 , Tpr1 > Tpr2 results in Ti1 > Ti2 . The transfer function of the controller is obtained by applying the Laplace transform to equation (8.2): 1 ), HPI (s) = Kc (1 + (8.14) Ti s the frequency function HPI (jω ) = Kc (1 +

1 1 ) = Kc (1 − j ), Ti jω Ti ω

(8.15)

and the module and phase angle M(ω ) = Kc √1 +

1 Ti2 ω 2

and φ (ω ) = − tan−1

1 . Ti ω

(8.16)

The Bode plots are given in Fig. 8.18. It is observed that the own frequency of oscillation of the closed control system with PI controller, at which the loop phase angle becomes −180∘ , called the crossover frequency, decreases proportionally with the decrease of the Ti (ωosc 2 < ωosc 1 ). If the crossover frequency of oscillation is lower, the probability of a disturbance inducing resonance in the control loop is higher, since the typical frequencies characterizing the oscillations of the chemical/process systems are low, being comprised in the range of 10−5 Hz (one oscillation in 24 hours for e.g. the daily change of outside temperature) to 1 Hz. This means that by decreasing the value of Ti, the loop becomes more unstable. One frequent phenomenon linked to the PI control is the windup. Integral windup refers to the situation in a PI controller (but it may happen to the PID as well), where a change in setpoint, especially when it is large, produces a large increase or decrease (function of the error sign) of the integral term (windup), thus overshooting and continuing to increase or decrease as the accumulated error is unwound. This causes unpleasant repeated deviations in both directions (negative or positive) of the controlled variable, producing thus oscillations and unstable control. There are several anti-windup solutions to be applied, but the most reliable is to disconnect the I component of the controller until the controlled variable enters in a controllable range (say, ± 10 % of the setpoint). Modern controllers have the anti-windup setup [5].

242 | Part II: Analysis of the feedback control system Bode plots of the PI controller 350 300 250 200 M 150

Ti2

100 50

Ti1

0 –4 –3.5 –3 –2.5 –2 –1.5 –1 –0.5 0 0.5 1 Frequency [log(rad/s)]

Phase angle 0°

Bode plots of the PI controller ωosc2 ωosc1 φpr φTi1

φTi2

–50° –100° –150° –180° –200°

φt1 φt2

–4 –3.5 –3 –2.5 –2 –1.5 –1 –0.5 Frequency [log(rad/s)]

Fig. 8.18. Bode plots for the PI controller.

8.2.3 Proportional-Integral-Derivative controller (PID) [6] The equation describing the PID algorithm is (8.3). The step response of the controller is given in Fig. 8.19. It can be demonstrated that the higher Td is, the higher the impulse characterizing the step response is; the role of the D component is to abruptly open or close the actuator of the control system, forcing the process to come back to the setpoint. In the case of the temperature control of a heater, the derivative part of the control action opens the steam valve almost completely for a short while, giving the process a thermal shock, which forces it to recover much faster the decrease of temperature than in the case of P or PI control. Usually, the PID control algorithm is used to control better and faster, the slow processes rather than the heat transfer ones. How do we choose the derivative time? Example 8.4. Let us consider two heat control loops for two drying chambers with different volumes, V1 > V2 (Fig. 8.20).

8 Controllers

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243

e

A

time c D part KpA

I part

P part c0

time F, ρ, cp , T°i

Fig. 8.19. Step response of the PID controller.

F, ρ, cp , T°i TC2

TC1 V1 , T°1

V2 , T°2 KT , AT , T°ag

KT , AT , T°ag Fag

–ΔT°

Fag

F, ρ, cp , T°

–ΔT°

F, ρ, cp , T°

Fig. 8.20. Choosing comparatively the values of Td for the temperature controllers in two different CSTRs.

One can observe the volumes are different, V1 > V2 . According to the heat balances (3.11) where the term of reaction is 0, ∘ ) = d (Vρ c T ∘ ), Fvi ρ cpA Ti∘ − Fvo ρ cp T ∘ − KT AT (T ∘ − Tag p dt

(8.17)

where Fvi = Fvo = F is the air flow in the drying chambers and cpA = cp its heat capacity, calculating Tpr1 and Tpr2 for the two drying chambers, Tpr1 =

V1 ρ cp Fρ cp + KT AT

and Tpr2 =

V2 ρ cp Fρ cp + KT AT

,

Tpr1 > Tpr2 .

(8.18)

When the temperature changes in the two chambers, the process of change is slower in the first one (with larger Tpr1 ) and faster in the second one. Considering a decrease in temperature of the input air flows of both chambers, the temperatures decreases in both of them. In order to bring back the temperatures to the setpoint, both controllers have to act in similar ways, but differently: the first one stronger (to recover

244 | Part II: Analysis of the feedback control system the larger delay in action given by the larger volume of the first reactor) and the second one weaker. Thus results Td1 > Td2 . The first controller opens in this way the steam control valve more than the second one in order to give a stronger temperature shock for the slower system. The transfer function of the PID controller is obtained by applying the Laplace transform to the equation (8.3). 1 E(s) + Td sE(s)] Ti s 1 HPID (s) = Kc (1 + + Td s). Ti s Xc (s) = Kc [E(s) +

and thus

(8.19) (8.20)

The frequency function is HPID (jω ) = Kc (1 +

1 1 + Td jω ) = Kc [1 + j (Td ω − )] Ti jω Ti ω

(8.21)

with module and phase angle M(ω ) = Kc √ 1 + (Td ω −

1 2 ) Ti ω

and φ (ω ) = tan−1 (Td ω −

1 ). Ti ω

(8.22)

The Bode diagram is presented in Fig. 8.21. It may be observed that when the Td has a smaller value (Td2 ), the integral effect prevails and the crossover frequency ωosc becomes ωosc2 in the lower frequencies range; thus, the control system becomes more unstable since the majority of the disturbances in process engineering are in the low frequency range. When Td is larger (Td1 ), ωosc becomes ωosc1 placed in a higher frequency range. Thus, the control system becomes more stable. One may observe too, that at ωosc1 , the module of the controller has a very high value, meaning that any disturbance with a frequency close to ωosc1 will be severely amplified (the actuator strongly opened and closed) and the loop strongly destabilized. As a result the choice of the two main parameters Kc and Td should be done in accordance with these considerations (see Chapter 11). Recently, the classical controllers have been evaluated for more comprehensive functions so as to adapt their algorithms to the unknown parts of the controlled processes [7–9]. These controllers evolved towards the so called class of intelligent (i-PID) controllers. They take into account the changing parts of the processes without any modeling needed, just fast estimation and identification techniques [10]. This is a solution to help those control/electrical engineers who do not have enough knowledge of the process they are in charge of controlling. The authors are somewhat cautious to recommend any application of a control solution without having the knowledge of the process controlled.

8 Controllers

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245

Bode plots of the PID controller 50 45 40 35

Td1

30 MdB 25 20

Td2

15 10 5 0 –3.5

ω1=(Ti·Td1 )–1/2 ω2=(Ti·Td2 )–1/2 0

–3

–0.5

Frequency [log(rad/s)] Phase angle

Bode plots of the PID controller

100°

φTd1

50°

φpr

φTd2

0° –50°

φt1

–100° φt2

–150° –200°

ωosc2

ωosc1

–250° –300° –3.5

–3

–2.5

–2

–1.5

Frequency [log(rad/s)]

–1

–0.5

Fig. 8.21. Bode plots for the PID controller.

8.2.4 Controllers with special functions The PID controller is the one with the most complicated algorithm used most frequently in industry. There are situations where the performances which could be obtained with the PID algorithm are not sufficient for the process controlled. There are more performing control algorithms which will be described briefly in this section, but in detail they will be the object of the second volume of the book, entitled Advanced Process Control.

246 | Part II: Analysis of the feedback control system Adaptive controllers The Adaptive controllers have algorithms which “adapt” themselves to the changes in the dynamics and steady state behavior of the process. In the paper [11] the authors present a situation involving the temperature control of the hydrazine hydrate manufacturing process using as manipulating variable the NH3 liquid flow (Fig. 8.22). T Tn

Fn

FNH

3

Fig. 8.22. Steady state characteristic of the reactor producing aqueous chloramine solution in the hydrazine hydrate process [12].

The operating point for the temperature is at the point of inversion of the characteristic slope; because of the exothermic behavior of the process, the increase of the liquid ammonia flow, through its expansion, compensates the exothermic character of the reaction between NaOCl and NH3 exactly at the equilibrium point (Fn NH3 , Tn∘ ); over this flow value at equilibrium, the endothermic character of the expansion dominates the exothermic character of the reaction. In order to keep the temperature at the operating point, the controller has to memorize the process steady state characteristic and to adapt its action to the characteristic. This can be obtained through a scheme of the type presented in Fig. 8.23 [13].

Transducer – r +

Control algorithm Correction of the control algorithm

Final control element

Controlled process

Model of the process

y

Fig. 8.23. The structure of an adaptive controller.

There are several surveys and comprehensive scientific contributions to the theory and practice of self-adaptive controllers [14–16]. The authors preferred to mention a practical self-adaptive controller produced by the company ASEA Novatune and used later

8 Controllers

V3

|

247

Temp. control

V2

Flowrate control

Fig. 8.24. Adaptive control of an ethylene oxide reactor [17].

on to control an ethylene oxide reactor in the presence of a catalyst [17]. The control scheme is given in Fig. 8.24. The reactor has a production of 30,000 tons/year of ethylene oxide from ethylene and oxygen. The reaction is extremely exothermic, the constraints of deviation of temperature being ± 0.1∘ in normal conditions and ± 0.5∘ in extreme conditions of strong load disturbances. These disturbances usually occur when an almost empty raw material tank is replaced with another full one, producing an extremely brutal change of parameter (pressure or flow). The PID controllers usually used are decoupled in these situations and work only in “manual” operation mode. Bengtsson and Egart proposed an adaptive control scheme, presented in Fig. 8.24. The temperature controller uses two control valves, V2 and V3 , the latter for normal operation and the first for large disturbances which have to be processed. Both valves are controlled in “automatic” mode and the change of production is done without decoupling the controllers, the change of the setpoint being done according to one preprogrammed scheme. The general scheme of the adaptive controller, considering a function of the setpoint, is different from that in Fig. 8.23, and is presented in Fig. 8.25. Adaptive control can be used in many processes such as PVC batch polymerization or ε -caprolactam, the so called nylon 6,6, continuous manufacturing process, where the properties of the reaction mass change continuously during the batch time (case of PVC), or along the process (ε -caprolactam) influencing the initial tuning of the controller.

248 | Part II: Analysis of the feedback control system Measuring, filtration r

Algorithm design

Control algorithm

Final control element

Controlled process

y

Transducer Fig. 8.25. The structure of a self-adaptive controller taking into consideration the set-point value.

Optimal controllers The optimal control implies minimizing or maximizing a global performance index: N

J(k, x(k)) = ∑ f (x(k), u(k)).

(8.23)

k=1

This performance index can be the integral of the squared error to assure the minimum of deviation from the setpoint but also, for example, the shortest route from one point to another in a town [19]. For a process in process industry, the performance can be the batch distillation time for a batch distillation, preserving a quality performance index; in the case presented in Fig. 8.26, x(k) is the distillate average concentration, and the control action is the optimal reflux ratio time function R(k) = u(k) (Fig. 8.26). The general structure of such an optimal control scheme is given in Fig. 8.27. All optimal control problems have at their base either Pontryagyn’s maximum principle or Bellman’s dynamic programming formulation [18, 20]. These problems will be treated in detail in the second volume of Advanced Process Control. In a study made for the methanol industry [21], the authors succeeded in demonstrating that through finding an optimum temperature profile along the catalytic methanol reactor, the efficiency can be increased by 30 %.

Predictive controllers The authors’ volume [22] is one work describing in detail the principle of predictive control. Predictive control is the most advanced one at the moment, because it “foresees” what it will happen in the future of the process when a control action is taken. The principle of model predictive control is presented in Fig. 8.28. The controller is a discrete one; at each step (k), the algorithm calculates one action based on the minimization of the error between the predicted trajectory y(t) and the setpoint r(t): p

∑ ‖ y(k + l | k) − r(k + l) ‖2 . l=1

(8.24)

8 Controllers

|

249

D, xD = ct

R

Optimal reflux computing algorithm

FC

R

Optimal reflux

R(k)

t(k)

time

Fig. 8.26. Optimal operation of a batch distillation.

Transducer

Control algorithm

Final control element

Computing algorithm of Xi

Controlled process

y

Model of the process “J” index

Model parameter estimation

Fig. 8.27. The structure of an optimal control system using the change of the set-point value.

The minimization assures the fastest way to the setpoint which allows the process control system to have minimum deviations from the setpoint trajectory. An example is the temperature control of the batch PVC reaction [23] which, using the MPC technique, allows the close match of the real temperature in the reactor with the optimal profile calculated (Fig. 8.29).

250 | Part II: Analysis of the feedback control system past

future Predicted outputs

Setpoint y(k+1 | k) y(k)

Manipulated inputs

u(k+1) k k+1 k+2...... k+l ............k+m ..................k+p Input horizon Output horizon Fig. 8.28. Principle of model predictive control.

T(°C) 60 55 Optimal profile PID NMPC

50 45 0

50

Cooling flowrate

100 150 200 250 300 350 400 450 500

time (min)

NMPC PID

2 1.5 1 0.5 0 0

50

100 150 200 250 300 350 400 450 500

time (min)

Fig. 8.29. PID and NMPC control of the PVC reactor for the optimal temperature profile.

8.2.5 Distributed Control Systems The development of Distributed Control Systems (DCS) is directly related to the huge potential offered by the microprocessor and data communication technologies that

8 Controllers

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251

emerged in the 1970s, with the most representative product at that time the Honeywell TDC-2000. The DCS [24] principally consists of integrated hardware and software systems accomplishing two basic functions for the real-time management of the enterprise: monitoring and control (Fig. 8.30). Its architecture is designed so that the computed-based control and monitoring levels are distributed both physically and functionally. The monitoring and control parts of DCS use communication networks for linking their units and for sharing the collected and analyzed data or for distributing the control or decision information. DCS development is continuously expanding with new tasks such as: decision support capabilities, expert system facilities or management and business abilities. The jobs accomplished by the DCS are distributed into partially independent subsystems organized on multiple levels and having a hierarchical structure, able to offer a high degree of reliability. This is due to the fact that functions are not concentrated in a centralized structure, but in different computer systems, and the failure of one unit does not imply the collapse of the whole system. This modularity allows the quick and prompt intervention for maintenance and for restoring of either hardware or software affected capabilities. The distributed architecture of the DCS demonstrates very great flexibility due to the straight reconfiguration of its software. Built-in libraries for the control algorithms, monitoring charts, alarm functions or displays can be either directly chosen or adapted to fit the particular application needs. The communication network linking the distributed units allows the connection and separation from the DCS, for rapid implementation of changes, upgrades or even improving the structure of the design. They offer the user the opportunity to focus on the process management and control tasks and relieve the effort aimed at solving the implementation solution for the instrumentation layer. The bottom layer of the DCS performs the data acquisition and the regulatory control but also interacts with the higher monitoring and control layers for sending and receiving data. The input signals are conditioned, analyzed and data reconciliation or statistical processing may be performed. Smart transducers or control valves are easily integrated onto the DCS platform and their incentives are fully exploited. DCS controllers offer the facility of also providing, besides the traditional proportional, integral and derivative algorithms, combinatorial, sequential logic and supervisory control. They are integrated with the feed forward, robust and model-based control and with advanced control algorithms implementing artificial intelligence tools such as fuzzy logic, neural networks or genetic algorithms. The new DCS systems enlarge the control functions with production planning and scheduling associated with the management of resources. One of the most appreciated features of the DCS is its system concept-based design, implemented in a harmonious architecture that integrates different subsystems communicating which each other, having distributed functions and responsibilities

252 | Part II: Analysis of the feedback control system

Computer

Operating center

Interface

Interface

Traffic directory

Data BUS

Operator console

Interface data acquisition

Interface

Basic control units

Displays

Power supply

Terminal

Protection barrier

Field equipment

Fig. 8.30. Block diagram of a distributed control system.

in such a way as to make the DCS work as a whole and offer the means for finding the globally optimal control solutions.

8 Controllers

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253

8.3 Problems (1) The control system in Fig. 8.31 has the following characteristics: Hc (s) = Kc ;

HAD (s) =

10 ; 2s + 1

Hpr (s) =

0.5 ; 30s + 1

HT (s) =

1 . 10s + 1

y

Xset +

HC

Hpr

HAD

– HT

Fig. 8.31. Feedback control system.

The controller has the PB = 50 %. Draw the Bode and Nyquist plots for the whole control system. How are these diagrams modified if the PB = 10 %? (2) The automatic control system in Fig. 8.31 has a PI controller with PB = 25 % and Ti = 30 s. Draw the Bode plots for the ACS. Draw the Bode plots in the situation of Ti = 5 min. (3) The phase margin of the PD controller of the ACS in Fig. 8.31 is +45∘ . What is the Td of the controller in this case? What is the open loop gain for this system with the calculated value of the derivative time and a PB = 100 %? (4) Consider the heat exchanger temperature ACS from Fig. 8.32.

KT , AT FN

ρm ,T°

T°ag , Fag Tr

TC

Fig. 8.32. Temperature feedback control system with P controller.

The temperature is controlled with a P controller with Kc = 1. Being given a disturbance of the input temperature from 50 to 60 ∘ C, calculate the extra energy consump-

254 | Part II: Analysis of the feedback control system tion due to the steady state offset. The characteristics of the equipment in the control loop are: 3 final control element – KAD = 2 mmA/h ; transducer – KTR = 0.4 mA ∘ ; 3

process – nominal flow through the heat exchanger – FN = 1 mh ; density of the fluid kcal ρ = 1000 mkg3 ; specific heat cp = 1 kg ; heat transfer coefficient KT = 1000 mkcal 2 hK ; heat K transfer area AT = 1 m2 . What is the offset and the extra heat consumption if the setpoint is changed with +10o C? (5) Two CSTRs have the following characteristics: kg kcal 2 Reactor 1: V1 = 1 m3 ; KT1 = 1000 mkcal 2 hK ; AT1 = 3 m ; ρ1 = 1000 m3 ; cp1 = 1 kg K ;

F1 = 1

m3 . h

2 Reactor 2: V2 = 2 m3 ; KT2 = 800 mkcal 2 hK ; AT2 = 5 m ; ρ2 = 1000

F2 = 1

kg ; m3

cp2 = 1

kcal ; kg K

m3 . h

The reactions are neither producing nor consuming heat. Choose comparatively PB, Ti , and Td for the temperature controllers of both reactors. Explain the differences. (6) Design the behavior of the auto-adaptive controller for the process described in Fig. 8.22.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

Agachi P. S., Automatizarea proceselor chimice (Automation of chemical processes), Casa Cartii de Stiinta, Cluj-Napoca, 1994, p. 108. *** SAMSON (http://www.docentes.unal.edu.co/sorregoo/docs/sistemas%20de%20control. pdf). Agachi P. S., Automatizarea proceselor chimice (Automation of chemical processes), Casa Cartii de Stiinta, Cluj-Napoca, 1994, pp. 110–116. Ibidem, pp. 117–120. Packard A., Saturation and antiwindup strategies, ME 132, Chap.15, Spring 2005, UC. Berkley, p.134–143. Agachi P. S., Automatizarea proceselor chimice (Automation of chemical processes), Casa Cartii de Stiinta, Cluj-Napoca, 1994, pp. 120–124. Ang, K. H., Chong, G., Li, Y., PID control system analysis, design, and technology, IEEE Trans. Control Systems Techn., 13, (2005), 559–576. Aström, K. J., Hägglund, T., Advanced PID Control, Instrument Soc. Amer., 2006. Li, Y., Ang, K. H., Chong, G. C. Y., PID control system analysis and design, IEEE Control Systems Magaz., 26, (2006), 32–41.

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[10] Fliess, M., Join C., Intelligent PID Controllers, Proc. of 16th Mediterranean Conf. on Control and Automation, Ajaccio, France, 2008. [11] Agachi S., Topan V., Automatizarea unei reactii chimice cu comportare neliniara (Automation of a chemical reaction with nonlinear behavior), Rev. Chim. 1, (1980), 63– 67. [12] Silberg A. I., Topan V. A., Agachi P. S., et al. Procedure and reactor for manufacturing chloramine, Romanian Patent no.120893/25.11.1985 [13] Calin S., Dumitrache I., Regulatoare automate (Automatic controllers), Ed. Didactica si Pedagogica, Bucuresti, 1985, Chap. 8,11. [14] Åström, K. J., Theory and applications of adaptive control — a survey, Automatica, 19, (1983), 471–486. [15] Åström, K. J., Wittenmark, B., On self-tuning regulators, Automatica, 9, (1973), 185–199. [16] Åström, K. J., Adaptive Control, Dover, 2008, pp. 25–26. [17] Bengtsson G., Egart B., Experience with Self-tuning Control in the Process Industry, Proc. IX IFAC Congress, Budapest, 1984, Section Case Studies. [18] Pontryagin, L. S., The Mathematical Theory of Optimal Processes, Mir Publishers, Moscow, 1962. [19] Doya, K., How can we learn efficiently to act optimally and flexibly, Proc. of the National Academy of Sciences of USA,106, No. 28, (2009), 11429–11430. [20] Betts, J. T., Practical Methods for Optimal Control Using Nonlinear Programming, SIAM Press, Philadelphia, Pennsylvania, 2001. [21] Agachi S., Vass E., Optimizarea reactorului de fabricare a metanolului, utilizand Principiul Maximului lui Pontriaghin (Optimization of the methanol reactor using Pontriagyn’s Maximum Principle), Rev. Chim., 6, (1995), 513–521. [22] Agachi, P. Ş., Nagy, Z. K., Cristea, M. V., Imre-Lucaci, A., Model Based Control, Case Studies in Process Engineering, Wiley-VCH, ISBN 3-527-31545-4, Weinheim, 2006, pp. 17–20. [23] Nagy Z., Agachi S., Model predictive control of a PVC batch reactor, Computers & Chemical Engineering, 6, (1997), 571–591. [24] http://www.automation.com/library/articles-white-papers/fieldbus-serial-bus-ionetworks/ieee-1394-and-industrial-automation-a-perfect-blend.

9 Final control elements (actuating devices) The actuating device is the element of the control loop which receives the control signal from the controller (c) and transforms it in a mass or energy flow (m) which modifies the controlled variable (y) in the controlled process. m can be the steam flow at the entrance of a heater or a column reboiler, a combustible flow at the entrance of a burner, or an NaOH flow at the entrance of a neutralizing plant.

9.1 Types of final control elements There are several ways of manipulating the material or energy flow in or out of a process: varying the speed of a conveyor belt, the rotation of a screw pump or of a centrifugal pump, the frequency of switching on/off a switch device (relay) [1]. But by far the most popular way of manipulating the flow is using the control valves, due to their simplicity and reliability.

9.1.1 Control valves The most frequently used final control elements are the control valves (Fig. 9.1). Such a control valve assembly has two main parts: the actuator (drive) which can be pneumatic, electric or hydraulic and the valve which manipulates the flows with a plug, ball, diaphragm or baffle. Figure 9.1a depicts the control valve with its drive and Fig. 9.1b only the pneumatic drive/actuator. The manipulation of the valve is done pneumatically by the pressured air, usually in a range of 0.2–1 bar. The pneumatic actuator is formed from a diaphragm case of cast iron separated in two chambers by a rubber membrane (M). The membrane is held in a certain position by a spring (SP) which also has the role of bringing back the membrane when the pressure signal disappears. Fixed on the membrane it is the stem (S) which has at one end the plug (P), which obliterates to a certain proportion the seat ring/orifice (SR) through which the fluid passes from the input of the valve cage to its output. Because the position of the plug is variable, the opening of the orifice is also variable and so the fluid flow passing through the valve is variable as well. Nowadays, the majority of the controllers are electronic ones, having as control variable c, an electric current in the range 2–10 mA or 4–20 mA (see Chapter 7). The conversion from the electric control signal ic to the pneumatic signal addressed to the control valve (pc ) is done by the electro-pneumatic convertor (EPC) (Fig. 9.2b and c). In the EPC, the instrumental air of pressure 6 bar is first conditioned (filtered and dried) in the reducer and then the resulting feed air at 1.4 bar is varied proportionally with the value of the ic in the above mentioned range. The result is the stem travel, which – through the

9 Final control elements (actuating devices)

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257

Ac

SP S

Pc

M

P

S

SR

Fig. 9.1. Pneumatic control valve: P - plug; pc – control pressure; Ac – actuator; Sp – spring; M – membrane; SR – seat ring.

position of the plug attached to it – opens, to a certain percentage, the orifice of the valve. Because of the friction in the seal rings of the stem, the valve functions with hysteresis; in order to control exactly the position of the plug to correspond to that of the control signal, a positioner (Fig. 9.2c) is sometimes added. The positioner is an internal stem position control loop which mechanically measures the stem travel and ensures it is exactly that given by the control signal. We may find different types of control valves on free sources (Fig. 9.2): http://www. google.ro/search?q=control+valve&tbm=isch&tbo=u&source=univ&sa=X&ei=9rXUf2qGImYPc6sgYAM&ved=0CEcQsAQ&biw=1280&bih=615 The actual valve can be of different types [2]: – globe valve with direct flow single or double ported (Fig. 9.3), three-way (Fig. 9.4); – angle valves in which one port is co-linear with the valve stem, and the other port is at a right angle to the valve stem (Fig. 9.5); – butterfly valve (Fig. 9.6) used for large pipelines and low pressure fluids (gas usually); – ball or V-notch control valve (Fig. 9.7) is used for partially solid flows or viscous flows or suspensions which are not able to be manipulated by the other types of valves. The V-notch of V-shape in the ball allows a greater range ability of the manipulation of the flow.

258 | Part II: Analysis of the feedback control system –

slide/gate control valves (Fig. 9.8) are used usually for very large cross area sections of the pipelines through which either water (e.g. flood water from reservoirs) or catalyst (e.g. Fluid Catalytic Cracking Unit) are transported.

(a)

(b)

M Pp 0,2 – 1 bar

Positioner S

Electro-pneumatic Pc c converter 0,2 – 1 bar 4 – 20 mA

Ps, air supply 1,4 bar (c) Fig. 9.2. Different types of industrial control valves with electro-pneumatic convertors aside: (a) regular industrial valve; (b) control valve with electropneumatic convertor; (c) actuator with electropneumatic convertor and positioner.

9 Final control elements (actuating devices)

|

Fig. 9.3. Globe valve two-way single-ported.

Fig. 9.4. Globe valve three-way single-ported.

259

260 | Part II: Analysis of the feedback control system

Fig. 9.5. Angle valve.

Fig. 9.6. Butterfly valve.

Fig. 9.7. Ball control valve.

9 Final control elements (actuating devices)

A

|

261

A

A’ A’ Fig. 9.8. Slide/gate control valve.

There are other types of control valves specifically for non-usual fluids manipulation: – membrane control valves (Fig. 9.9) for very corrosive, viscous and high density fluids (e.g. slurry). Its principle of operation is that putting pressure on the elastic membrane tube, the section passing the fluid diminishes proportionally with the control pressure exercised. There are construction versions with the diaphragm closing the valve on a weir or saddle.

Fig. 9.9. Membrane valve.

262 | Part II: Analysis of the feedback control system 9.1.2 Other types of final control elements –

Conveyor belts with variable speed (Fig. 9.10);

Electric motor with frequency converter for variable speed drive

–

Fig. 9.10. Variable speed conveyor belt.

screw pumps or screw conveyors (Fig. 9.11);

Fig. 9.11. Screw drive/ conveyor used for solids or viscous fluids.

–

pumps with variable speed (centrifugal pumps, gear pumps, piston pumps, peristaltic pumps). The rotation speed or the piston movement frequency is varied using a motor equipped with a variable frequency drive (VFD) [2] (Fig. 9.12);

Fig. 9.12. Left: gear pump and right: screw drive pump.

–

reflux distributors (Fig. 9.13) are used for very small reflux ratio flows. These distributors operate by varying the ratio between the reflux time and evacuation of

9 Final control elements (actuating devices)

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263

Condenser

C. B.

R

D

Fig. 9.13. Reflux distributor at the top of a distillation column.

the distillate time. The distribution funnel is commuted by an electromagnetic relay outside the column and a ferromagnetic piece embedded in the funnel, either on “total reflux” position (R), or “distillate collection” position (D). The ratio of the two mentioned times represents the reflux ratio.

9.2 Sizing the control valve There are many technical publications of excellent quality which deserve to be mentioned here: Emerson’s Control Valve Handbook [1], FNW material about flow coefficients [5], Parcol’s Handbook for Control Valve Sizing [6], or Samson’s Application Notes, Kv coefficient, valve sizing [7].

9.2.1 The flow factor ((Kv ) for incompressible fluids [3] The flow factor (Kv ) is the most important parameter which characterizes the flow through a valve. Let us consider a simple hydraulic circuit in which a valve is placed (Fig. 9.14). It is important because it synthesizes, as seen in Fig. 9.14, the simultaneous changes of several factors due to the stem travel position: pressure drop coefficient, cross area section through the control valve, the pressure drop in the control valve. The valve manufacturers give the Kv as the main value for sizing the control valve. If one expresses the volumetric flow Fv function of √p0 –p1 (Fig. 9.15), the relationship is linear (9.1); pc is the pressure at which the liquid vaporizes at the operating temperature. For 0 < p0 < pc , Fv = Kv √p0 –p1 .

(9.1)

264 | Part II: Analysis of the feedback control system

P0

P1 FV

Fig. 9.14. Hydraulic circuit including a control valve.

FV Non-critical flowrate

Critical flowrate

FC

PC–P1

P0–P1

Fig. 9.15. The dependence of the volumetric flow on the pressure difference (p0 –p1 )1/2 .

Kv is the slope of the characteristic in its first portion, the straight line, and it is defined as the water flow expressed in m3 /h, at 15 ∘ C, passing at a pressure difference of 1 bar. In the American scientific literature, the flow coefficient is Cv = 1.16 Kv , being the flow in US gallon/minute, passing through the completely open valve at 60 ∘ F at a pressure difference of 1 lb/in2 . Applying the Hagen–Poiseuille relationship, ρ v2 (9.2) , 2 where Δpv is the pressure drop on the valve, ξv is the pressure drop coefficient on the valve, and v, the velocity of the fluid in the pipeline, the flow rate through the valve is Δpv = ξv

Fv = v ⋅ Av = √

2 1 A √ Δp , ξv v ρ v

(9.3)

and at etalon conditions where ρe and Δpve are the density (1 kg/dm3 ) and pressure difference of 1 bar, Av being the cross-section area through the control valve, this flow is exactly the flow factor Kv : Kv = √

ρ 2 1 A √ Δp = Fv √ . ξv v ρe ve Δpv

(9.4)

9 Final control elements (actuating devices)

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265

This relationship of the flow factor Kv is valid for the turbulent flow, noncritical, in the absence of cavitation and vaporization. When Kv is calculated, the density is always expressed in kg/dm3 , Δpv in bar, and the flow rate in m3 /h. Example 9.1. The data for the calculation of a control valve are the following: maximum alcohol flow rate through the valve is 6.8 m3 /h, its density ρa = 735 kg/m3 and the pressure drop on the valve at the corresponding flow rate is Δpv = 0.8 bar. Kv has to be calculated. The relationship (9.4) becomes

Kv = Fv [

ρ [ mkg3 ] 735 m3 √ ] = 6.8√ = 6.5. h 1000Δpv [bar] 1000 ⋅ 0.8

For laminar flows in the pipeline, Bernoulli’s law is applied and Reynold’s number can be calculated with (9.5): vD (9.5) Re = ρ , ν where D is the pipeline diameter [m] and ν the viscosity [Ns/m2 ]. The profile of the flow tube is presented in Fig. 9.16.

P0

P1 Fig. 9.16. The profile of the fluid vein in the pipeline when the section is strangled.

At the level of the plug, Re is not the same as in (9.5) because the area of the passing section is much smaller and the velocity of the fluid is much larger. Re inside the valve can be recalculated [4]: 1 D ReR = Re . (9.6) 2.59 √K v When ReR < 500 the flow is laminar and in these conditions (9.4) can be used. Observing that the flow regime is changing inside the valve, the flow factor has to be √Re recalculated using the corrected ReR with the factor C = 20 R relationship (9.4) becoming F ρ Kv = v √ . (9.7) C 1000Δpv

266 | Part II: Analysis of the feedback control system Cavitation It has to be noted that at the minimum section of the strangled vein of the fluid (vena contracta), the fluid pressure drops abruptly (Fig. 9.17) at values which can be equal to the saturation vapor pressure (pvs ) thus forming vapor bubbles. These bubbles implode when the pressure goes back to the regular pressure value in the pipeline. The implosions create strong shock waves eroding severely the interior of the valve. P0

Critical Turbulent Laminar

PC

PVK

L

Fig. 9.17. The pressure profile along the pipeline in the section of the valve: (a) laminar flow pressure profile; (b) turbulent flow pressure profile; (c) critical flow pressure profile (pc – critical pressure).

In order to avoid cavitation, Δpv ≤ p0 –pv ,

(9.8)

and because only a part of the pressure is recovered, a critical flow coefficient is defined [4], Cf < 1, which depends on the control valve inner form and equation (9.8) becomes Δpv ≤ Cf2 (p0 –pv ). (9.9) This equation imposes that, for avoiding the cavitation, special control valves with several expansion stages have to be built and for which Cf becomes close to 1.

9.2.2 The flow factor (Kv ) for gases [3, 5, 7] The simplifying assumptions, at the basis of the computational relationships for Kv , are: the gas is considered ideal, the density is that after the valve, the gas temperature does not vary at passing through the valve. Applying the general law of gases, pN FvN p F = 2∘v , ∘ T TN

(9.10)

where the pN , FvN , TN∘ are the pressure, volumetric flow and temperature in normal conditions (1 atm = 101.325 kPa, 20∘ = 293.15 K); p2 , T ∘ , Fv are the pressure, temperature and flow after the valve.

267

9 Final control elements (actuating devices) |

Expressing the volumetric flow as the ratio between the mass flow and its density, F

F

pN ρm p2 ρm N = . T∘ TN∘

(9.11)

From both equations (9.10) and (9.11), Fv = FvN

pN T ∘ p T∘

and

(9.12)

2 N

ρ = ρN

p2 TN∘ , p T∘

(9.13)

N

and with (9.4), the expression of Kv becomes Kv =

FvN p T∘ √ N , 514 p2 Δpv

(9.14)

where p2 is the absolute pressure and is expressed as p2 = 1 + p2 rel .

9.2.3 The flow factor (Kv ) for steam [3, 8] For superheated steam the specific volume is used, v = 1/ρ [m3 /kg] and equation (9.4) becomes 1 Kv = Fv √ , 1000Δpv v or using the mass flow Fv = Fm v, Kv =

Fm v , √ 31.6 Δpv

(9.15)

where Fm is measured in [kg/h] – for saturated dry steam in which p2 > 100 bar, formula (9.15) is used; – for saturated dry steam with p2 < 100 bar, the approximation p2 v2 ≈ 2 is done and (9.15) becomes Kv = –

Fm 1 √ 22.4 Δpv p2

(9.16)

for wet steam with p2 > 100 bar, the relationship (9.16) becomes Kv =

Fm x √ 22.4 Δpv p2

(9.17)

with x the proportion of saturated steam in the mixture wet-saturated steam.

268 | Part II: Analysis of the feedback control system Example 9.2. A control valve is mounted on a superheated steam pipeline, and the flowing process has the following characteristics: Fm max = 100 kg/h, the specific volume after the valve v2 = 0.188 m3 /kg, the pressure before the valve p1 = 20 bar and after the valve, p2 = 12 bar. The flow factor Kv has to be calculated. Kv =

100 0.188 √ = 0.486. 31.6 8

There are numerous other situations when the control valves are in operation as in the case of compressible fluids, or two phase flows. Sizing procedures are given in [3, 4, 6].

9.3 Inherent characteristics of control valves The second element, after the flow coefficient (Kv ), characterizing the flow through a control valve, is the flow variation function of the valve’s stem travel (h). Knowing that the Kv is actually the flow in certain etalon conditions, ξv (h) and Av (h) are functions of h, and 1 Kv = √ (9.18) A (h) = Kv (h), ξv (h) v the result is that Kv = Kv (h) represents the inherent characteristic of the control valve. The inherent characteristic is determined by measuring the flow rates at different valve openings and at a constant pressure drop. There are several types of inherent characteristics (Fig. 9.18): ΔK (a) The linear characteristic, where Δhv = ct for all values of Kv and with the analytical expression K K Kv h = v0 + (1 − v0 ) (9.19) Kv100 Kv100 Kv100 h100 5

Flow [%]

4

Linear Equal Percentage Quick Opening Square Root Modified Parabolic Hyperbolic

3 2 1 0 0

0.2 0.4 0.6 0.8 Stem opening [%]

1

Fig. 9.18. Typical inherent characteristics of the control valves [9].

9 Final control elements (actuating devices)

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269

where h = 0 is the stem travel of the valve with the valve completely closed; h = 100 is the stem travel of the valve with the valve completely open; Kv0 is the flow factor for the completely closed valve (h = 0); Kv100 is the flow factor for the completely open valve (h = 100). (b) The logarithmic (equal percentage) characteristic, where Kv K K h = v0 exp [ln ( v100 ) ]. Kv100 Kv100 Kv0 h100

(9.20)

Using the differentiated expression of (9.20), dKv K dh = ln ( v100 ) Kv Kv0 h100 which shows that when the stem travel h h varies from 50–60 %, the Kv varies 100 from its value to Kv + 10% Kv which explains the equal percentage characteristic name. (c) The modified parabolic characteristic with the analytical expression (9.21) is approximately in-between the linear and the equal percentage characteristic, providing a fine control at low flow values and approximately linear characteristic at higher flow capacity: Kv K = v0 + Kv100 Kv100

h h100

3 − 2 ( hh )

2

.

(9.21)

100

(d) The quick/fast opening control valve designed for quick flooding or evacuation of the reaction mass in case of emergency. In addition to these frequently used control valves characteristics, there are two others: hyperbolic, which is the reverse of quick opening and we may name it “slow opening”; and square root, placed in-between modified and hyperbolic to better control the flows in the middle part of the flow range. The profile of the characteristic is either resolved through a certain adequate form of the plug and seat (Fig. 9.19), or via a special characteristic of the electro-pneumatic converter.

270 | Part II: Analysis of the feedback control system

Valve plug

Valve spindle

Spindle movement

Fast opening

Valve seat

Linear

Equal percentage

Fig. 9.19. Special form of the plugs and seats for corresponding valve characteristics.

In the other case of constructing the form of the characteristic using special characteristics of the actuator and control valve, x h = f1 ( c ) , h100 xc100 x Kv = f2 ( c ) , Kv100 xc100

(9.22)

and

Kv = f1 ∘ f 2 . Kv100

(9.23)

(9.24)

9.4 Installed characteristics of the control valves Once included in the hydraulic circuit (Fig. 9.20), the control valve changes its inherent characteristics depending on the structure of the system and the total pressure drop on the system. The total pressure drop on the system is formed out of the sum of the local pressure drops (9.25), the sum of the linear pressure drops (9.26), the sum of the pressure decrease due to the level difference (9.27), the pressure drop on the control valve itself (Δpv ). n

Δploc = ∑ ξi i=1 m

Δplin = ∑ λj j=1

ρ v2 2

(9.25)

lj ρ v2 Dj 2

(9.26)

l

Δpz = ∑ ρ ghk k=1

(9.27)

9 Final control elements (actuating devices)

|

271

P2 CV H HE PC P1

Fig. 9.20. The control valve does not work independently but is a part of a hydraulic system.

with ξi – the local pressure drop coefficient, λj – the linear pressure drop coefficient, lj and Dj the lengths and diameters of the different pipeline sections, hk – different level differences which contribute to the total pressure drop. The total pressure drop on the system is thus: n

Δps = ∑ ξi i=1

l ρ v2 m lj ρ v2 + ∑ λj + ∑ ρ ghk + Δpv + Δppump = ΔpL + Δpv , 2 dj 2 j=1 k=1

(9.28)

with ΔpL being the sum of all pressure drops except the valve. The flow rates in the pipeline and valve are equal and according to (9.3) and (9.4), FL = KL √ Thus, Δps = ρ F 2 ( K12 + v

1 ), KL2

Δp ΔpL = Fv = √ v = F. ρ ρ

(9.29)

and from this results

F=

Kv 2

√

√ 1 + ( Kv )

Δps ρ

or

(9.30)

KL

Kv

F= √1 +

K (K v ) v100

2

K ( Kv100 ) L

2

√

Δps . ρ

(9.31)

At h = h100 , all variables depending on stem travel get the index 100 and one can write instead of (9.29), F100 = Kv100 √ (

Δpv100 Δp = KL √ L100 ρ ρ

Kv100 2 ΔpL100 ) = KL Δpv100

and ΔpL100 = Δps − Δpv100 .

and (9.32)

272 | Part II: Analysis of the feedback control system Supposing that the flow does not change its regime and the fluid is incompressible, KL stays constant and replacing (9.32) in (9.31), Kv

F= √1 +

2 Δp −Δp K s v100 (K v ) Δpv100 v100

√

Δps ρ

and using (9.32) F F100

Kv Kv100

= √1 + (

Kv ) Kv100

2

√ Δps

( Δp

v100

− 1)

Δps . Δpv100

(9.33)

Δpv100 Δps

If we denote the parameter m = where Δps = p1 max − p2 (p1 max is sometimes called maximum pumping height), the relation (9.32) becomes (9.34), representing the installed characteristic of the valve: Kv F = F100 Kv100

1 √m + (

2 Kv ) Kv100

.

(9.34)

(1 − m)

Kv Kv100

represents the inherent characteristics which are given in (9.19)–(9.21), the ratio depending on the control valve’s stem travel h: K (h) F = v F100 Kv100

1 2

.

(9.35)

√ m + ( Kv (h) ) (1 − m) K v100

The installed characteristics of the control valves are plotted in Fig. 9.21 for the three most utilized control valves. The variable parameter in the figure is m, which takes values between 0.04 and 1. We may easily observe that being installed in a hydraulic circuit, the inherent characteristics are deformed due to the nonlinearities induced by the square dependence of the pressure drop on the flow. These deformations can be used in compensating the undesired nonlinearities of the process characteristics (Chapter 6, Figs. 6.2–6.4). Generally, how do we choose a control valve characteristic? There are some practical recommendations [10] which are very general and helpful. Summarizing the paper: for pressure control valves the recommendation is equal-percentage; for flow, if the setpoint varies – linear; if load varies – equal-percentage; for temperature – equalpercentage.

9 Final control elements (actuating devices)

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1 0.8

0.4

Linear

F F100

0.04 0.1 0.25 0.33 0.5 1

0.6

0.2 0 0

0.2

0.4

0.6

0.8

1

h h100 1

F F100

Modified parabolic

0.8 0.6 0.1 0.25 0.33 0.5 1

0.4 0.2 0 0

0.2

0.4

0.6 h h100

0.8

1

1

F F100

Logarithmic

0.8 0.6 0.04 0.1 0.25 0.33 0.5 1

0.4 0.2 0 0

0.2

0.4

0.6 h h100

0.8

1 Fig. 9.21. The installed characteristics of the control valve.

273

274 | Part II: Analysis of the feedback control system

9.5 The dynamic characteristics of a control valve 9.5.1 The gain of the control valve The final control element is composed of the electro-pneumatic convertor (EPC), the actuator (AC) and the valve (CV). Each of them have their own gain factor, the electropneumatic convertor – KEPC , the actuator – KAC , the valve – KCV and the total gain is the product of all three: KFCE = KEPC KAC KCV . (9.36) Usually, the EPC has linear characteristic, having as input the control signal 2–10 or 4–20 mA and as output the pressure addressed to the actuator, 0.2–1 bar. Thus, the gain of the EPC is ΔpAC KEPC = Δic and for the first case, 0.8 bar bar KEPC = = 0.1 . 8 mA mA The actuator, AC has as input the pressure in the actuator, 0.2–1 bar and as output the whole stem travel 0–100 %. Thus, its gain is KAC =

Δh 100% % = = 125 . ΔpAC 0.8 bar bar

The gain of the control valve depends on the type of the characteristic, being for the linear valve dF (9.37) =a dh and for the equal percentage (logarithmic) dF (9.38) = bF. dh Because of the fact that these characteristics change in the hydraulic circuit, the gain becomes dependent on the stem travel and the variation is given in Fig. 9.22 on the next page [11].

9.5.2 The dynamics of the control valve The dynamic behavior of control valves is largely treated in [11] Chapter 3 and orientations are given in [12]. Concluding, because of the series pneumatic capacities, of the transport line delay, the behavior is in the most cases described by a transfer function of second order a3 Hv (s) = , (9.39) a2 s2 + a1 s + 1 where a3 is KFCE , a2 has values of around 10−1 s2 , and a1 values of around 1–20 s. In the most cases, the behavior is treated as a first order dynamics [12].

9 Final control elements (actuating devices)

Linear 5

4

4

0.05 0.1 0.2 0.5 1

3 2

dF dh

1 0

275

Logarithmic

5

dF dh

|

3 0.2 0.5 1

2 1

0

0.2 0.4 0.6 0.8 h h100

1

0

0

0.2 0.4 0.6 0.8 h h100

1

Fig. 9.22. The change of the control valve gain, function of travel and m coefficient in the case of: (a) linear valves; (b) logarithmic valves.

9.6 Sizing and choice of the control valves In choosing and dimensioning the control valve (Kv and the inherent characteristic), one has to take into consideration the two following situations: (a) the pressure drop on the hydraulic system is not given and the source pressure has to be calculated; (b) the pressure at source is given and also the pressure drop on the system. In both cases, the following steps for sizing the valve are given. Case (a) (a.1) establish the steady state process characteristic on the transfer path manipulated variable → controlled variable (Section 6.1); (a.2) based on the maximum and minimum disturbance, the minimum and maximum respectively flow through the valve are calculated (Section 6.1, Fig. 6.3); (a.3) at the maximum flow value, ΔpL100 is computed, using the relations (9.25)– (9.28); (a.4) depending on the form of the steady state characteristic of the process (a.1) and other steady state characteristics of the transducer (e.g. orifice plate flow meter), the compensating characteristic of the valve is chosen (Section 6.1, Fig. 6.4) in order to linearize the behavior of the whole ensemble valve and process. This Δp means the most appropriate m value is chosen. m = Δpv100 ; s (a.5) the value of Δpv100 (from m) and then the Kv100 value are calculated using the relations (9.4), (9.14), (9.15)–(9.17), depending on the valve sized; (a.6) the superior value Kvs > Kv100 is chosen from the manufacturer’s tables;

276 | Part II: Analysis of the feedback control system (a.7) with Kvs chosen, the new Δpv100 value is calculated and consequently, the new installed characteristic chosen, to observe if the linearization suffers. If the linearization suffers, the whole sizing procedure is resumed from the beginning; (a.8) the pumping pressure is determined using p1 = p2 + ΔpL100 + Δpv100 Case (b) (b.1) establish the steady state process characteristic on the transfer path manipulated variable → controlled variable (Section 6.1); (b.2) based on the maximum and minimum disturbance, the minimum and maximum respectively flow through the valve is calculated (Section 6.1, Fig. 6.3); (b.3) at the maximum flow value, ΔpL100 is computed using the relations (9.25)– ((9.27); (b.4) Δpv100 = p1 − p2 − ΔpL100 is calculated; Δp (b.5) m = Δpv100 is calculated and the adequate (for linearization) installed characters istic is chosen; (b.6) Kv100 is determined; (b.7) the superior value Kvs > Kv100 is chosen from the manufacturer’s tables; (b.8) with Kvs chosen, the new Δpv100 value is calculated and consequently, the new installed characteristic chosen, to observe if the linearization suffers. If the linearization suffers, the whole sizing procedure is resumed from the beginning. Example 9.3. A flow control valve is mounted on the feed pipeline of a reactor working at the pressure of 20 bar. The nominal operating flow is Fn = 120 m3 /h, the minimum flow is Fmin = 70 m3 /h, and the maximum Fmax = 147 m3 /h. The density of the fluid is ρ = 875 kg/m3 . The pressure drop on the pipeline is Δpp100 = 12.6 bar. The pressure drop inside the pump is supposed to be 0. Sizing and characteristic choice of the control valve is requested. The problem is included in case (a). The steady state characteristic (flow control on a pipeline), F = f (F) is a straight line with the slope 45∘ . Consequently, the characteristic of the valve is linear in the range Fmin /Fmax = 70/147 = 0.47 and Fmax /Fmax = 147/147 = 1.0 and especially around Fn /Fmax = 0.81. A modified parabolic control valve with m = 0.33 (Fig. 9.21) should be chosen, but also a logarithmic characteristic with m = 0.2 fits. Choosing a logarithmic valve, Δpv100 = 0.2 Δps

(9.40)

Δpv100 = 0.2Δps = 0.2(Δpv100 + Δpp100 )

(9.41)

Δpv100 =

0.2Δpp100 1 − 0.2

=

0.2 ⋅ 12.6 = 3.15 bar. 0.8

(9.42)

9 Final control elements (actuating devices)

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277

According to (9.4) and from Example 9.1, Kvmax = F100 √

ρ 875 = 147√ = 74. 1000Δpv100 1000 ⋅ 3.15

(9.43)

We choose from the catalogue Kvs = 90 We recalculate the pressure drop Δpv100 = and m=

2 F100 ρ 1472 875 = = 2.2 bar 2 1000 Kvs 902 1000

Δpv100 2.2 = = 0.15, Δps 12.6 + 2.2

(9.44)

(9.45)

and the valve characteristic remains the same. The source has the calculated pressure p1 100 = p2 + Δps = 20 + 12.6 + 2.2 = 94.8 bar.

(9.46)

Example 9.4. Calculate and choose the control valve of a temperature control system of a heat exchanger through which water is circulated (Fig. 5.3). The characteristics of 3 the heat exchanger are: nominal flow through the heat exchanger – FN = 4 mh ; density kg kcal of the fluid and heating agent ρ = ρag = 1000 m3 ; specific heat cp = cp ag = 1 kg ; heat K 2 transfer coefficient KT = 1000 mkcal 2 hK ; heat transfer area AT = 4 m ; Δpp100 = 1 bar; ∘ ∘ ∘ ∘ Ti = 20 ; Tiag = 50 . The temperature which has to be kept constant is Tn∘ = 30∘ . The temperature fluctuations around the normal T ∘ = 20∘ are of ± 5∘ . The pressure is i

given by a centrifugal pump with the operation characteristic given in Fig. 12.2 and its pressure at Fag max = F2 in the figure has to be calculated. The agent is evacuated at atmospheric pressure. The steady state characteristic of the heat transfer process is given by the equations (6.1) and (6.2): ∘ ) − VkC ΔH = 0 Fvi ρ cp Ti∘ − Fvo ρ cp T ∘ − KT AT (T ∘ − Tag A r ∘ ∘ ∘ ∘ F ρ c T − F ρ c T + K A (T − T ) = 0, vag ag pag iag

vag ag pag ag

T

T

ag

(6.1) (6.2)

where VkCA ΔHr = 0, Fvi = Fvo = FN . ∘ , and replacing the values in the problem, one obtains the steady Eliminating Tag state characteristic of the process: T∘ =

Ti∘ag 4Ti∘ + . 4 + 2Fag 2 + 4 F

(9.47)

ag

Data calculated for the disturbances of ± 5∘ in the input temperature and for different values of the agent flow are given in Tab. 9.1. Figure 6.9 is the graphic representation of the temperature variations in this case.

278 | Part II: Analysis of the feedback control system

15

Ti∘ [ ∘ C] 20 T ∘ [ ∘ C]

25

23.75 23.66 28.12 29 29.6 30

27.5 30 31.25 32 32.5 32.8

31.25 33.33 34.37 35 35.4 35.7

3

Fag [m /h] 2 4 6 8 10 12

Table 9.1. Variation of the temperature of the heated fluid in the heat exchanger function of the disturbances.

(a.2) The minimum and maximum disturbances are Ti∘min = 15∘ and Ti∘max = 25∘ and the nominal operating input temperature is Ti∘n = 20∘ . At these values, for an outflow temperature Tn∘ = 30∘ , the heating agent flows are 3 3 3 Fag max = 12 mh , Fag min = 1.33 mh , Fag n = 4 mh The operating range of the valve is between Fmin /Fmax = 1.33/12 = 0.11 and Fmax /Fmax = 12/12 = 1.0 with special attention to be given to the nominal point Fn /Fmax = 4/12 = 0.33. (a.3) Δpp100 = 1 bar from the problem data. (a.4) From Figs. 9.23 and 9.21, a logarithmic characteristic of the valve with m = Δpv100 = 0.9 is chosen. Δps (a.5) Δpv100 = 0.9Δps = 0.9(Δpv100 + Δpp100 ) which gives the result Δpv100 = 9 bar. ρ Thus, Kvmax = F100 √ 1000Δp

v100

1000 = 12√ 1000∙9 = 4.

(a.6) The catalogue value for Kvs is just 4, so we do not have to recalculate the characteristics. (a.7) – (a.8) p1 100 = 9 + 1 = 10 bar so the head of the pump has to be bigger than this value.

9.7 Problems (1) Consider a composition ACS from Example 6.3, with the following characteristics: the molar fraction of the concentrate can vary between 0.9 and 1, but at the nominal point xcN = 0.95; the concentration of the active component in the diluting flux is xD = 0; the target composition after the dilution is xN = 0.5; the diluting flow D = 1 m3 /h constant. The dilution is done at atmospheric pressure and the pump pumps the concentrate at 2 bar. The pipeline pressure drop is Δpp100 = 1 bar and the density kg of the concentrate is ρ = 800 m3 . The control valve has to be sized and chosen and is manipulating the concentrate (Fig. 12.13). (2) The height of the liquid in the bottom of a distillation column is h = 1.5 m and the pressure drop between the column and the following column in the sequence is

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279

4 bar. The density of the product which passes through the control valve of the level kg ACS is ρ = 700 m3 . What is the influence of the liquid level in the column bottom upon the flow between the two pieces of equipment? How much does an increase of 20 % change the Δps ? How is the sizing of the control valve influenced? (3) Consider a flow ACS with the transducer an orifice plate without square root ex3 3 tractor. The data of the piping system are: F100 = 20 mh and FN = 15 mh ; pressure at the source p0 = 9.5 bar; pressure at the end of the pipeline p2 = 4.1 bar; the pressure drop in the pump and on the line ΔpL100 = 4.2 bar; density of the fluid circulated kg through the valve ρ = 700 m3 . The control valve has to be sized and the linearizing characteristic to be chosen. (4) Calculate the gain factors of the final control elements, including the control valves for the cases in the Problems (1) and (3) at nominal and maximum values of the flow. The electro-pneumatic convertors have the input signal 4–20mA and the output 0.2–1 bar; the drive has the input 0.2–1 bar and the output stem travel 0–100 %. (5) What types of final control elements should we choose for manipulating the following products: 1. steam at 25 bar; 2. flow of acetic acid at 10 l/h nominal flow; 3. air flow at 100 Nm3 /h and pressure 0.5 bar; 4. sulfuric acid 60 % produced through the contact chamber process; 5. water emulsion of polyvinyl acetate flow; 6. air conditioning air flow at 300 mmH2 O pressure; 7. slurry flow at the bottom of a distillation column; 8. sand flow for the glass production process.

References [1]

[2] [3] [4] [5] [6]

Emerson Process Management Handbook, Fisher, Control Valve Handbook, Fourth Edition, Cernay, France, 2005, p. 4–17. http://www.documentation.emersonprocess.com/groups/public/documents/book/cvh99.pdf Campbell, S. J., Solid-State AC Motor Controls. New York: Marcel Dekker, Inc., pp. 79–189. ISBN 0-8247-7728-X. Agachi, S., Batiu, I., Automation of a volatile oils distillation plant, Contract with Plafar Orastie, no. 112/1984. Marioniu, V., Automatizarea proceselor chimice si petrochimice, Editura Didactica si Pedagogica,Bucuresti, 1979, chap. 11. Singh, M., Elloy, J. P., Mezencev, R., Munro, N., Applied Industrial Control, Pergamon Press, 1980, Chap. 7. FNW, About Cv flow coefficients), 2012, http://www.fnwvalve.com/FNWValve/assets/images/ PDFs/FNW/tech_AboutCv.pd

280 | Part II: Analysis of the feedback control system [7] [8] [9] [10]

[11] [12] [13]

Handbook for control valve sizing, Parcol Bulletin 1-I, 2010, http://www.parcol.com/docs/1i_gb.pdf, Samson’s Application Notes, Kv Coefficient, Valve Sizing, Samson AG. Mess- und Regeltechnik, Frankfurt am Main, 2012, http://www.samson.de/pdf_en/t00050en.pdf SpiraxSarco Ltd., Steam Engineering Tutorials, Block 6, 2013 The Engineering Toolbox, Tools and Basic Information for Design, Engineering and Construction of Technical Applications, Control Valves and Flow Characteristics, http://www.engineeringtoolbox.com/control-valves-flow-characteristics-d_485.html. Headley, M., Guidelines for selecting proper valve characteristics, Valve Magazine, 15 (2), (2003), 28–33. Marioniu, V., Poschina I., Stoica M., Robinete de reglare, Ed. Tehnica Bucuresti, 1980, Section 2.5.4. Emerson Process Management Handbook, Fisher, Control Valve Handbook, Fourth Edition, Cernay, France, 2005, p. 32.

10 Safety interlock systems 10.1 Introduction Safety systems play a very important role in the operation of a plant, as the secure operation is the first task to be fulfilled by any management and control system. Interlocking is performed by a set of instruments intended to prevent the plant from reaching states that have negative impact on the operators, equipment, environment or products. The design of interlocking systems should consider different levels of risk that may affect the process and can only be implemented after a deep and comprehensive understanding of the process. Although different standards are known, they are not generalized and custom-tailored solutions are commonly encountered in practice. The safety system type of technology logic to be used (e.g. relays, integrated electronics or software systems) and the frequency of testing the system are application dependent. The safety standards follow the same approach. Nevertheless, designing an efficient and reliable safety system implies a set of steps to be performed as a general procedure for fulfilling the objective of providing secure operation. These steps are also known as the model of the safety life-cycle [1]. The prerequisite for designing any safety system is the thorough knowledge of the plant, process, equipment and their interaction with human operators and the environment, in order to be operated for preventing any damaging and harming effects. A multidisciplinary approach and team may be needed (e.g. process, biochemical, control, mechanical and electrical engineers, safety and management specialists) to cover the diversity and complexity of the system under study. Interactions between the subunits should be comprehensively revealed. The hazard analysis is the first step to be performed. The ample identification of the hazards is important for defining the potential tasks the safety system must cope with. The hazard analysis implies a Hazard and Operability (HAZOP) study. The ranking of the hazards with respect to the probability of their occurrence and the magnitude of their associated effects is also necessary. This task is accomplished by the risk assessment analysis, preceded by the hazard one. As a result, based on the risk assessment, the hazards are managed either at the level of the conceptual plant design or at the level of the safety system under design. It is worth mentioning that risk can only be reduced and not completely eliminated and absolute safety may never be attained. The next step is to establish the measure of importance for different safety goals. Accordingly, the safety system performance is established for each of the safety goals to be achieved. Several levels of safety may be defined at this step. They will be considered with specific safety equipment, in a hierarchy designed according to the emerged levels. The actions implied for satisfying each safety goal and the underlying logic is the following step of the safety system design. The logic of these actions is the core of

282 | Part II: Analysis of the feedback control system the design methodology and special attention should be given to considering both normal operation conditions and start-up or shutdown procedures. If any information from practical experience of operation in similar cases is available, it would be very valuable for using it to evaluate (pre-test) the solutions of the designed safety system. The design should consider a conservative margin for fulfilling its safety goals in order to cover the variability and the possible lack of exact quantitative assessment of the risk and associated impact [2]. According to the design, the safety instrumentation is built, commissioned and pre-startup tested. A solid state construction is desired and usually thorough verification is made to ensure long-life operation. Procedures for scheduled tests and maintenance of the safety system must be clearly specified, and stipulated in rigorous time program diagrams. Any changes to the design and the on-site implementation adjustments of the safety system have to be performed only after a competent analysis involving the team participating in the design work. Decommissioning and disposal of the safety system is the last step of the safety life-cycle model. It is intended to develop procedures to be followed at the end of the system’s life. As a general rule of thumb the practitioners recommend the safety system be as simple as possible.

10.2 Safety layers The safety system is commonly designed with a hierarchical structure having the control system at the bottom of the safety multilayer hierarchy. The hierarchy mainly addresses the extent or gravity of the hazards to be treated by the safety system. Nevertheless, it is desired that safety layers work independently in order to maximize the probability of responding to the possible malfunction of the other layers. A typical multilayer structured safety system is presented in Fig. 10.1 [6]. Some of the layers act as prevention measures, i.e. to block the incidence of hazards, while other layers mitigate for diminishing the effect of the already started incidents. The safety multilayer system may have additional layers as the risk analysis proceeds and considering that safety is increased if the number of layers is also increased. However, each of the safety layers must be designed as simply as possible. Part of the layers may be situated inside the plant while others outside it. In order to minimize the complexity of the safety system, the intrinsic safety provided by the conceptual design of the plant must be maximized. Achieving this fundamental objective will bring additional benefits directly reflected in the costs of the safety system implementation. The plant design should decide on solutions to operate at mild conditions, i.e. at low temperatures, pressures and inventories, with mechanical equipment working exposed to reduced wear, resistant to corrosion and abrasion, at low rotating speed, and in-

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Community emergency measures Plant emergency measures Safety interlock system Alarm and monitoring system Regulatory and advanced control system Plant & process Fig. 10.1. Typical configuration of the safety multilayer system.

volving raw materials and products that exhibit reduced hazard for explosion or for harming human health [3]. By accomplishing its designed role, the regulatory or advanced control system may be considered part of the safety system. Although this layer’s main task is the efficient operation of the plant, by keeping the controlled parameters of the process at the desired setpoints, the safety needs would be inherently satisfied in normal automatic operation mode. In control system design there is a tendency to integrate parts of the safety functions in the control equipment (e.g. Distributed Control Systems) or duplicate them for redundancy reasons. The latter is the preferable approach. The alarm and monitoring system is the first one to inform the operating personnel that a process is about to reach unsafe states and that opportune action must be taken by the operator for preventing the appearance and development of potential hazards. Light, sound, moving parts of the alarm equipment or animation on the monitors (flashing light, changing color, lists) are simple but effective ways of drawing the operator’s attention to the emerging hazardous state. A quick answer is needed for this equipment and its operation should be simple, robust and capable to work autonomously (e.g. have its own energy supply, operate during plant start-up or shutdown). The operators should have competence for interpreting the alarm system information and for taking appropriate (manual) measures to bring the operation safely back to normal or to shut down the plant. These procedures must be stated in special regulation documents and operators be previously trained to cope with all (predicted) potential situations. The safety interlock system (also denoted as Safety Instrumented System, SIS) is usually designed to shut down the plant in a failsafe way, without the operator’s intervention. Its operation is directly related to and dependent on the success of the inherently safety design of the plant. The safety interlock system must have its logic of operation designed according to the previously analyzed and anticipated scenarios and possibly transposed into Boolean logic mathematical form. The physical implementa-

284 | Part II: Analysis of the feedback control system tion of this safety layer benefits from different options. The most common is based on the traditional relays. Nonetheless, hardwired electronic implementation using combined digital and analog electronic devices and software-based systems are increasing in importance and their frequency of application is continuously growing. The choice of the physical implementation of the safety interlock system has to consider several aspects, such as: level of security, cost, promptitude of operation, operators’ acceptance or training, frequency of testing and cost of maintenance. During the last years, the software-based interlock systems have gained increasing acceptance under the implementation of the Programmable Logic Controllers, although they are not yet considered as reliable and low-cost as the relay or hardwired solutions. But some of their appreciated features make them competitive and capable to extend and become dominant in the future, especially for large applications. Such incentives consist in versatility, capability of self-testing or diagnosis and compatibility with the computerbased control systems. The plant emergency measures safety level may consider physical protection systems and fire or gas systems. From the first category, the relief devices and containment dikes (bunds) may be used to protect equipment working at hazardous pressure and to prevent the spread of the hazardous reliefs (relief valves, rupture disks, flares, scrubbers). The fire and gas systems are intended to act against the fire by stopping the fire in an automatic way (e.g. by activating fire sprinklers) or by alarming (e.g. by the sirens) the community and the fire-fighter service. Evacuation plans for the people in the neighborhood are part of the mitigation actions in case of severe accidents. The community emergency measures extend the actions for limiting the consequences of accidents to the outside of the plant.

10.3 Alarm and monitoring system The alarm and monitoring system is aimed to draw an operator’s attention to the deviation of plant operation parameters from normal limits and appearance of potentially hazardous events, therefore asking for opportune intervention. This is due to the fact that the control system is not designed to solve by itself all problems during plant operation and the operators have to take the responsibility of running the process safely and efficiently in such special situations. The most common alarm function consists in a lamp (light source) that flashes inside a box having a colored glass or plastic cover on which (or under which) a label is printed for denoting the alarm source. The alarm may be associated with a continuous or intermittent sound generated by a horn or buzzer. By the operator’s acknowledgment, performed when pressing a button, the lamp stops flashing (and usually gets into continuous lighting) and the sound is stopped. Process alarms emerging from the control system use the same sensors with the control system and point out problems with the equipment. The safety alarms originate from the safety instrumentation layer and indicate the development of critical operation states.

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The shutdown alarms are generated by the automatic shutdown system of the safety instrumented layer and inform the operator on the shutdown trip [7, 8]. The alarm display has to be easily visible by the operator and color coded alarm information may be assigned to different levels of alarms, according to their importance: pre-alarms, typical alarms, bypass alarms and shutdown alarms. The red color is usually assigned to the highest priority alarm (i.e. emergency). According to the order in which the operator should be informed, the alarms may be ranked as having: low, medium and high priority. The alarms will provide focused information on the circumstances of appearance, equipment involved, straight measure to be applied and priority. The sounds generated by multiple alarms must have the amplitude and frequency such as to be easily recognized by the operator and differentiated according to the level of priority. The alarm system has to cope with the circumstances of multiple alarms acting together, during the same period of time. It is important for the operator’s analysis and for taking mitigating decisions, to be able to discriminate between alarms and to indicate which of the alarms acted first. This task becomes complicated if the time responses of the alarms are different. In the case of an automatic shutdown of the unit, the first acting alarm can be directly discriminated by the operator who acknowledges the trip and, as a consequence of the acknowledgment, the first alarm may become flashing while the others will have a continuous illumination. The standards for the alarm systems consider the limited capacity of an operator to treat a large number of active alarms. The alarm system should process the flux of multiple and simultaneous alarms and provide the operator the filtered flux of information that reveals the most important ones, according to their priority. Operator acceptance and capacity for processing information from alarms, under difficult working conditions and high workload, have to be supervised and appropriate management regulations should be developed.

10.4 Safety instrumented systems The safety instrumented system (SIS) consists of measuring devices (sensors) and actuators, working together according to the logic solvers, with the aim of bringing the plant to a safe state when predetermined setpoint conditions are not met or safe operation is jeopardized [5]. The typical form of the SIS, having the actuator of a control valve as final element, is presented in Fig. 10.2. The main parts of the SIS are the sensors, the input and outputs interfaces, the logic solver and the final control elements [8].

286 | Part II: Analysis of the feedback control system

Power supplies

Sensors

Input Interfaces

Logic solver E/E/EP devices

Output Interfaces

Actuator

Communications Fig. 10.2. Typical configuration of an electrical Safety Instrumented System.

Sensors The sensors are of two main categories. They may have the role of measuring the process parameter and providing a continuous signal, proportional to the parameter, or they may only have the duty of signalizing the violation of particular low or high limits of the process parameter. For the first case, sensors are associated with analog transmitters and for the second one, they consist in switches (implemented in either pneumatic or electric technologies). The sensor switch is a passive device that provides a binary on-off signal. This signal is changed when the process parameter exceeds the predefined upper-lower limits [8]. The sensors used in the SIS should have a robust design and work in a failsafe way for protecting the plant. As they are complex by construction and are exposed to the interaction with process parameters and the environment, the sensors are sometimes prone to fail in fulfilling their tasks. Exposure to high heat or pressure sources, mechanical stress (vibrations), aggressive or corrosive media and clogging the connecting pipes are some of the typical causes of sensor failures. The most common malfunction behaviors shown by the sensors are: providing changes of the output signal without process parameter change, providing improper output signal change when the process parameter changes, erratic behavior and very long response time. The implications of the sensor response in the case of power loss should be carefully analyzed. For instance, senor switches should have normally closed contacts. The analog transmitters should have either upscale or downscale signal response in case of failure, depending on the effect they should have through SIS on the process. A trade-off must be made when choosing between sensor switches and analog sensors. The first category is simple, reliable and cost effective but it is limited with respect to the quantity of information provided and self-diagnosis. The second one offers more information on the process variable and the sensor’s state of operation but it is more expensive and requires qualified maintenance. The SIS may have functions to perform diagnostics on both the transmitter and the field-wiring. The choice of using analog sensors is preferable when the risk analysis recommends it. An important measure for dealing with the malfunction or failure of the SIS sensors is doubling the sensors. The sensor redundancy will significantly reduce the probability that SIS acts wrong due to sensor fail. The redundant sensor should be sepa-

10 Safety interlock systems | 287

rated from the first one with respect to the power supply, the mounting and connection to the process and to the lines for transmitting the signal. Periodically testing the sensors’ state of operation is good practice for keeping the protection potential of the SIS.

Logic solver The logic solver is the part of the SIS which automatically takes the necessary decisions for activating the final control elements, based on the information received from the sensor switches and analog transmitters and according to an incorporated logic [8]. The safety logic was traditionally performed by relays or, later, by hardwired electronics. Although still present in many applications, the logic solvers based on these technologies have today been replaced and mostly implemented by safety Programmable Logic Controllers (PLCs), especially for large applications. An example of the logic for the activation of the final control element is presented in Fig. 10.3, using the logical AND and OR gates and Boolean logic.

High temperature switch AND Low cooling flow switch OR High pressure switch

Output port

Final control element

AND High level switch Fig. 10.3. Logic solver example of the SIS.

The logic of the solver consists in the following statement: if either the simultaneous condition of high temperature and low cooling flow or the simultaneous condition of high pressure and high level are fulfilled, then the final control element should act and protect the system. In order to prevent the malfunction of the ports and logic gates, special diagnostic electronic circuits may be added to the logic solver. These diagnostic elements detect, in a switching operation mode, if special generated square wave signals sent to the gates and ports are transmitted through them. If these square sig-

288 | Part II: Analysis of the feedback control system nals are also present at the outputs of the gates and ports, they confirm the integrity of the gates and ports and the fact that they are in a ready state of operation. The PLC implemented logic solver offers flexibility, reduced costs for large applications and good compatibility with all digital equipment. The logic solvers are preferably implemented on safety PLCs which, compared to traditional PLCs, are designed to be fail tolerant and failsafe. The PLCs have input and output ports and are capable of performing logical and mathematical operations. They also have special communication electronics. Fault tolerance is the ability of a functional unit to continue to perform the required function in the presence of faults and errors [7]. The fault tolerant device will also provide an alarm and possibly indicate the fault origin. The diagnosis of the fault is a feature safety PLC should have implemented in its software. Continuation of the safety PLC operation in the presence of the faults may be obtained by the use of redundant hardware and software units, thus making possible the online restoring of the no-fault working regime. Figure 10.4 presents the 1oo1 (one out of one) architecture.

Sensors

Input circuit

Logic solver

Output circuit

Actuator

Fig. 10.4. 1oo1 architecture.

This architecture has only one channel for executing the safety function on the final element (actuator). The output is in energized state, i.e. closed circuit, when there is no violation of the safety conditions. The output will be de-energized (i.e. became open circuit) when a safety function of the logic solver demands it. This is the situation of the failsafe scenario. But, if the output remains stuck in the energized state, i.e. closed circuit, the fail will be dangerous, as any triggering of the actuator will not be able to become executed. This second scenario makes the 1oo1 architecture fail unsafe (dangerously). The 1oo2 (one out of two) architecture is presented in Fig. 10.5. Input circuit

Logic solver

Output circuit

Logic solver

Output circuit

Sensors

Actuator Input circuit

Fig. 10.5. 1oo2 architecture.

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This architecture has two channels for executing the safety function on the final element. The channels work in parallel and their de-energized to trip outputs are connected in series. This architecture provides a failsafe operation if any of the channels demands it. The safety system only fails dangerously if both of the channels fail dangerously, i.e. if both outputs become stuck in the energized state. In the case when only one of the channels remains in an energized stuck state, i.e. in closed circuit, the other channel will continue to offer shutdown protection. This makes the safety system show a lower chance of failing dangerously. The 1oo1D architecture presented in Fig. 10.6 has one safety channel associated to the diagnostic one. The outputs of the two channels are again connected in series. The diagnostic channel makes possible the detection of a dangerous failure and transforms it into a safe failure. Input circuit

Logic solver

Sensors

Output circuit Actuator

Diagnostic circuit

Fig. 10.6. 1oo1D architecture.

The 2oo2 architecture has a two channel configuration and it is used for the case of energize to trip safety systems. In this case the outputs are connected in parallel. The system remains protected even in the case when one of the outputs is stuck in the de-energized state, as the other channel will still offer protection. The 2oo2D architecture consists of a double 1oo1D architecture setup, with four channels (two of them being offered by the diagnostic paths) [8].

Actuator and final element The actuator is driving the final element by executing the automatic shutdown of the unit. There are different types of actuators coupled with the final elements. Typical implementations of the actuator-final element devices are: electrical relays that act as on or off control in the powering (starter) circuit of electrical motors, solenoid activated emergency valves and solenoid activated pilot valves acting on the air supply of the pneumatic actuators (diaphragms or pistons) of the emergency shut-off valves [8]. As the actuator and final element have to work in the field, they are exposed to tough operating circumstances and environmental conditions. Consequently, they have a high potential to exhibit malfunction or failure. The most common actuator and final element misfunction situations are, for the solenoid valves: electrical coil shortcircuit or interruption, blocking of the valve stem, plugging of the vents, corrosion

290 | Part II: Analysis of the feedback control system and abrasion of the valve internal parts, and for the trip valves: leaking, interruption or constriction of the air supply pipe, blocking of the valve stem. Limit switches or positioners are used for feedback, in order to get information on the actual state of operation of the final element.

Safety integrity level The safety integrity of a system is its quality of be fault tolerant. Integrity means to preserve the system’s performance and operating features, despite the faults and errors affecting its hardware and software. The integrity of a safety system is the probability of the system to maintain its safety functions. The safety integrity level (SIL) of the safety system is directly related to the hazard and risk analysis of the system to be protected. The risk estimation or risk assessment of the potential hazards showing up in the protected system implicitly determines the need for keeping it safe and quantifies the necessary risk reduction. Several levels of safety (three or four) are used to measure the tolerable failure rate of a specific safety function. The safety integrity levels are proportionally related to the safety availability they provide, such that the higher the availability offered, the higher the safety integrity level. For example, the SIL 1 will provide an availability of about 0.99 and may consist in a non-redundant structure having one piece for each of its sensor, logic solver and actuator elements. The SIL 2 will provide an availability of 0.999 and may have redundant elements, while the SIL 3 will offer an availability higher than 0.999 and should have only redundant elements [8]. One of the trends in the design and development of the safety interlocking systems consists in the expansion of the software implementations over the hardwarebased solutions. This tendency is sustained by the flexibility of the software-based safety system that allows the simple change of configuration and multiplication of the input-output paths together with the logic solver new design. The cost implied by the software solution is considerably lower, compared to the hardwired solution, and the difference increases radically with the number of the safety elements. Nevertheless, the system becomes more centralized and the failure of the software may affect the protected plant to a dramatic extent. Redundant software-based safety systems are recommended. Another trend is the advent of smart instruments that may have implemented safety functions, such as self-state diagnosis, comprehensive measurement analysis and communication with the safety instrumented system.

10.5 Problems (1) Please draw the 2oo2 architecture with four channel configuration and describe the operation of the safety system.

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(2) Explain if the safety interlock system must accomplish its designed tasks in automatic mode, manual mode or in abnormal operation conditions. (3) Please explain when the interlock system may be disabled (bypassed): in automatic mode, manual mode or in abnormal operation conditions? (4) Common process taps for redundant transducers are not desired. Please explain the motivation of this design recommendation. (5) Consider the system of five tanks that are supplied with liquid by a single feed pump. Each of the tanks has a level transducer and an on/off inlet valve. The interlock system is designed to work according to the following logic. If the level in the tank exceeds a predefined upper limit value, then its inlet valve is locked to a closed position. The feed pump is locked closed if all the inlet valves are closed (feedback provided by closing a limit switch) and any of the predefined level upper limit values of the tanks is reached. 1. Please make the logic solver diagram using the logical gates. 2. Please describe the operation of the interlock system. 3. Please describe the operation of the interlock system when one of the level upper limit transducer fails due to a low signal. 4. Please describe the operation of the interlock system when one inlet valve limit switch fails (i.e. remains in an opened switch position). 5. Provide a safer design for the interlock system.

References [1] Overton, T., Berge,r S., Process safety: How are you doing?, Chemical Engineering Progress, 104, (5), (2008), 40–43. [2] Overton, T. A., and King, G., Inherently safer technology: An evolutionary approach, Process Safety Progress, 25 (2), (2006), 116–119. [3] Gentile, M., Summers, A., Cookbook versus performance SIS practices, Process Safety Progress, 27 (3), (2008), 260–264. [4] Summers, A. E., Hearn, W. S., Quality assurance in safe automation, Process Safety Progress, 27 (4), (2008), 323–327. [5] Marszal, E. M., Scharpf, E. W., MIPENZ, Safety Integrity Level Selection – Systematic Methods Including Layer of Protection Analysis, ISA, 2002. [6] McMillan, G. K. (editor-in-chief), Considine, D. M. (late editor-in-chief), Process/Industrial Instruments and Controls Handbook, 5th ed., McGraw-Hill, 1999. [7] http://www.aiche.org/ccps/topics/elements-process-safety. [8] http://www.processoperations.com/SafeInstrSy/SS_Main.htm.

| Part III: Synthesis of the automatic control systems

11 Design and tuning of the controllers Obviously it is not enough that the process engineer or the process control engineer puts together and interconnects the elements of the control loop (process, transducer, controller and final control element) in order to obtain a good quality (see Chapter 5) control. These elements have to be finely “harmonized”, to make them able to cooperate properly with each other. Of course there are several ways of “harmonization”, starting with the appropriate choice of equipment relative to the process controlled. The quality of the control depends very much on the process characteristics. If these are properly understood, the control system has the chance to work appropriately. Then, because the characteristics of the process, transducer, control valve are somehow fixed (they are not always fixed because in time their properties are often changing), the only way of “harmonizing” the functioning of the elements of the control loop is the controller tuning, which means the appropriate choice of the values of the controller parameters. Up to now, the majority of the controllers have a P, PI, or PID structure, meaning that the tuning parameters are Kc , Ti , or Td . In this chapter, we describe the optimal choice of these parameters.

11.1 Oscillations in the control loop [1] Let us consider a basketball player who dribbles on the court and we analyze the movement of the ball (Fig. 11.1).

F

T

ymax

t ymin F Fig. 11.1. A basketball ball dribbled by a player.

In order to have an undamped oscillation of the ball (meaning the same period T and the same amplitude H), needed by the player to advance on the court, there are two conditions to be fulfilled: the player has to apply the same force F, of the same magnitude, at the same time equal to the oscillation period of the ball, T. Note that the court

296 | Part III: Synthesis of the automatic control systems “applies” the same reverse force, F, at mid-period. Similar examples can be given for a pendulum, for spring-mass systems, for playground swings etc. A similar phenomenon takes place in a control loop (Fig. 11.2).

D S +

C

FCE

Process

–

T

D

t

c

C C

C

r

C

C t1

t2

t3

t4

t5

t

Fig. 11.2. The closed control loop oscillates when a disturbance D occurs.

When a disturbance D occurs, the output y is pushed away from the setpoint value and begins to increase, for example; the deviation is “immediately” perceived by the controller which gives a command in order to bring the process back to the setpoint; due to the process inertia, the controlled variable follows its increase for a while and

11 Design and tuning of the controllers

|

297

then, “feeling” the order, begins to go back to the setpoint value; since the controller is not necessarily properly tuned, the controlled variable goes beyond the setpoint: the moment in which the controller intervenes and tries to restrict the decrease. The inertia provokes the decrease for a while and afterwards the controlled variable goes back towards the setpoint. The oscillation continues infinitely if the controller does not buffer the amplitude. It is observed in both cases (basketball player and control loop) that: – intervention of the control action in the same direction is at a full period (1 period is equivalent with −360∘ phase delay), meaning that the total summed phase angle for an oscillation to appear is −360∘ . Because the summing element represents a phase delay itself of −180∘ (the vectors of the setpoint and reaction variable are opposite e = r − xr ), the condition phase reduces to −180∘ ; – the control action upon the process and the other elements in the control loop is applied with the same magnitude. The conditions for having undamped oscillations are then (series elements in the control system): 5

∑ φi = φS + φC + φAD + φPr + φT = −360∘

(11.1)

i=1

and

5

∏ Mi = MS MC MAD MPr MT |ωosc = 1.

(11.2)

i=1

Since φS = −180∘ (Fig. 5.4) and MS = 1, the equations (11.1) and (11.2) become φC + φAD + φPr + φT = −180∘

(11.3)

MC MAD MPr MT |ωosc = 1.

(11.4)

11.2 Control quality criteria [2, 3, 5] The quality of the control action depends on many factors, among which some are more important: 1. controllability of the process (see Section 6.2); 2. structure of the control loop; 3. magnitude and form of the disturbance; 4. the place in the process where the disturbance occurs; 5. tuning of the controller parameters. To appreciate the quality of the control action, there are quality criteria on which the tuning of controller parameters is based. The most frequently used criteria of quality are:

298 | Part III: Synthesis of the automatic control systems (1) Criterion of the integral of the error (IE) (Fig. 11.3) expressed through (11.5): ∞

IIE = ∫ e(t)dt = ∑ Si = min

(11.5)

0

meaning that the closer the control loop answer is to the setpoint, the better the quality of the control action is; because the integral of a curve has the geometrical significance of the sum of the areas under the curve, the equation has the form in (11.5) min takes a numerical value, e.g. 2. y

r

+

+ –

–

t Fig. 11.3. The IE criterion.

This criterion has a problem: when the oscillations are symmetrical, the sum of the areas is 0 and the quality of the control is bad. This observation produced more advanced criteria. (2) Criterion of the integral of the absolute error (IAE) (Fig. 11.4) expressed through (11.6): ∞

󵄨 󵄨 IIAE = ∫ |e(t)|dt= ∑ 󵄨󵄨󵄨Si 󵄨󵄨󵄨 = min,

(11.6)

0

which eliminates the disadvantage of the symmetrical areas with opposite sign (+/−). y

r

+

+

+

+ t Fig. 11.4. The IAE criterion.

(3) Criterion of the integral squared error (ISE) (Fig. 11.5) which is a stronger criterion since if min has the same value as at IE or IAE, the conditions to be fulfilled are harsher

11 Design and tuning of the controllers

in this case:

|

299

∞

IISE = ∫ e2 (t)dt = ∑ Si2 = min.

(11.7)

0

y

s12

s22

s 32

s3

s1

s42

t

s4

s2

Fig. 11.5. The ISE criterion.

(4) Criterion of integral time-weighted absolute error (ITAE): ∞

IITAE = ∫ t|e(t)|dt = min

(11.8)

0

which penalizes the errors at long time more heavily. (5) Criterion of the area of the step response area (Fig. 11.6): SSR = min .

(11.9)

y

y r si t

Fig. 11.6. The step response area criterion.

The smaller the area of the step response is, the better the quality of the control action. (6) Quarter decay response criterion (Fig. 11.7): σ3 =

1 σ 4 1

(11.10)

The quality of the control action is good if the third oscillation is a quarter of the first one (Chapter 5). The two last criteria are very practical and are frequently used during the real operation of a plant.

300 | Part III: Synthesis of the automatic control systems y σ1 σ3 r

t Fig. 11.7. Quarter decay ratio response criterion.

11.3 Parameter influence on the quality of the control loop [3, 6] In Chapter 8, we described the separate influence of each of the controller parameters, Kc (PB), Ti , Td on the quality of the control loop response to disturbances. In the present chapter, the combined influence of the above mentioned parameters in the most frequently used controller structures are illustrated in the Tabs. 11.1 and 11.2. The statements “too small”, “good” and “too large” are relative to what the process asks for. For example, controlling the concentration in a CSTR a PB = 1000% is good and when the temperature is controlled, PB = 15% is good as well. Increasing both Ti on the x axis and Kc on the y axis, a combined effect is obtained as presented synthetically in Fig. 11.8. The situation of the PID controller involves three parameters with 27 combinations (C33 = 27). It is known that the D and P components of the controller have the strongest influence in this order. The Derivative part of the controller will always force the control action even at a small error. This is why, if Td is larger than “needed” by the process, an oscillatory regime will be imposed, even if the P or I are attenuated. If Td is smaller than “needed” by the process, the beneficial effect of the D component, which is meant to speed up the response of the slow process dynamics, is not felt enough and the process together with the control loop remains at the sluggish behavior.

11.4 Controller tuning methods [4, 7, 9–14] 11.4.1 Experimental methods of tuning controller parameters Probably about 90 % of the control loops are tuned experimentally and in about 75 % of cases the operating personnel can “guess” closely the appropriate controller parameters based on the knowledge they have about the process and the practical experience they have with other similar control systems. The most frequently used methods are those of trial and error [4].

11 Design and tuning of the controllers

| 301

Table 11.1. The influence of the PB upon the response to disturbance of a feedback control with P controller. PB

Step response to disturbance, of the classic (feedback) ACS

Observations

too large

y

Practically, the influence of the disturbance is not attenuated (large overshoot and large offset)

σ1

est

t good

Small overshoot, quick dampening and small offset

y est t

too small

Undamped oscillations

y

t

The method of successive trials in the process has successive steps described in the following sequence. 1. Both Integral and Derivative components of the PID controller are eliminated (Ti = max; Td = 0); the controller is in the “manual” (M) operation mode. 2. The PB is fixed at its maximum value (e.g. PB = 200 %). 3. The controller is switched from “manual” (M) to “automate” (A) functioning mode. 4. A small disturbance is given either at the setpoint or load; the very small gain of the controller does not importantly influence the response of the control system, allowing a large offset.

302 | Part III: Synthesis of the automatic control systems Table 11.2. The influence of the PB and Ti upon the response to disturbance of a feedback control with PI controller. There are nine combinations of parameters (C23 ). The influence of the P component is always stronger than that of the I component. PB

Ti

Step response to disturbance, of the classic (feedback) ACS

Observations

large

large

y

There is no influence of the I component and a very weak one of the P component

t

good

large

The behavior is of a P controller with “good” PB

y est t

small

large

The behavior is of a P controller with “small” PB

y

t

large

good

The P behavior is dominant; although the Ti is good and the overshoot smaller, cancelling the steady state error is not practically obtained

y

t

(continued on next page)

11 Design and tuning of the controllers

|

303

Table 11.2. (continued) PB

Ti

Step response to disturbance, of the classic (feedback) ACS

Observations

good

good

y

Small overshoot, quick dampening and 0 offset

t

small

Because of the dominant P action, oscillations appear

good y

t

large

small

Large overshoot, slow attenuation and noise with small oscillations

y

t

good

small

Small overshoot, quick dampening and small repeated oscillations around the setpoint

y

t

small

small

Undamped, amplified oscillations

y

t

304 | Part III: Synthesis of the automatic control systems Kc

y

y

y

r

r

r

t

t

t

y

y

y

r

r

r

t

t

t

y

y

y

r

r

r

t

t

t Ti

Fig. 11.8. Combined effect of changing both Ti on the x axis and Kc on the y axis; the response is given at a setpoint step change [10].

5.

6. 7. 8. 9. 10. 11. 12.

With the controller again on M mode, the PB is reduced at half value and the experiment from 4 is repeated in A mode of the controller; the action of the controller is observed as being stronger than before. PB is reduced continuously repeating the actions from 3 and 4 until undamped oscillations are obtained; this is the ultimate or oscillation value of the PBu . PB is fixed at PB = 2PBu value. Ti is reduced at half value from the maximum; the operation mode is M. The experiment regarding Ti is continued in a similar procedure as expressed in the sequence of steps 3–6, until undamped oscillations are obtained (Ti = Tiu ); Ti is fixed at = 2Tiu value. Td is increased step by step until the same oscillatory regime is obtained (Tdu ); The derivative component is fixed at Td = 0.5 Tdu .

11 Design and tuning of the controllers

|

305

13. PB is reduced in steps of 10 % until the overshoot is the desired one and the damping is adequate (usually 1/4 decay). In some cases, this method does not give the expected results and not necessarily because of its 13 “bad-lucky” number of steps; there are systems of “conditioned” stability that become unstable both at too small or too large PBs.

The method of limit of stability The most popular method of tuning the controller parameters bringing the process to instability is Ziegler–Nichols [5]. With this method, the process is brought to instability as before (steps 2–7). The testing of the maintained, undamped oscillations is done as in Fig. 11.9 and the ultimate period Pu is measured on the diagram.

y

Changed set point increase Kc to Amplified provoke oscillations oscillations. Permanent oscillations with the same amplitude

r

Kc = 4 (not enough)

Kc increase again Kc = 5 (too much)

Kc decrease Kc = 4.5

Pu = 40 sec

Ultimate period

Ku = 4.5 t(sec)

t0

t1

t2

30sec 40sec

Fig. 11.9. Testing Ku in the neighborhood of the limit of instability.

With the values of Ku and Pu experimentally determined, Ziegler and Nichols proposed the optimal controller parameters given in Tab. 11.3. Table 11.3. Optimal controller parameters proposed by Ziegler and Nichols.

Optimal controller parameter PB Ti Td

P

Controller structure PI PID

2PBu — —

2.2PBu 0.83Pu —

1.7PBu 0.5Pu 0.12Pu

306 | Part III: Synthesis of the automatic control systems

–

–

–

A short discussion can be made here. It is normal that the proportional band be increased relative to that corresponding to the oscillation one; based on experiments, Ziegler and Nichols determined as optimum PB = 2PBu . When the integral component is introduced, a plus of instability is added; this is the reason for increasing the optimum PB by 10 % (to calm a little bit the instability added by the I component); at the same time, in order to give I more importance, Ti is decreased to 0.83 from its previously established value (Pu ). Considering a PID structure, all controller parameters are a little enhanced to counterbalance the “laziness” of the process; the importance of the D component is much stronger than the I component; that is why the Ti is 4–5 times larger than Td .

Example 11.1. For a cascade of 3 CSTRs (Fig. 3.12) [6], at the input concentration CA0 = 0.98 kmol/m3 , the steady state values of concentration of A in the three reactors are: CA1 (0) = 0.4 kmol/m3 , CA2 (0) = 0.2 kmol/m3 , CA3 (0) = 0.1 kmol/m3 ; following the step change in input from 0.98 kmol/m3 to 1.8 kmol/m3 , the steady state values of CA in the reactors are changing. The reaction is a first order one in all three reactors, the retention time in each is τ = 2 min, the ultimate period of oscillation is Pu = 3.63 min and the PBu = 1.56 %. The cascade of reactors has a concentration control loop measuring the output concentration of A. The optimal settings of the controller, according to Ziegler and Nichols are given in Tab. 11.4. Table 11.4. The optimal controller settings in the case of the three CSTR control loop.

Optimal controller parameter PB [%] Ti [min] Td [min]

P

Controller structure PI PID

3.12 — —

3.6 3.03 —

2.8 1.82 0.453

The behavior of the control loop in the three cases of optimal setting is given in Fig. 11.10. The method of bringing the process to instability has two major disadvantages: – it is a slow method, especially when the process has delays of hours or even days; such an experiment as those described before may last weeks, during which the process does not work properly; – it is a dangerous method, since the process cannot be controlled once at instability; the nonlinear behavior of the process can cause severe deviations from the tolerable range of operation.

11 Design and tuning of the controllers

|

307

CA3[kmol/m3] P 0,11

0,10

PID PI

0,09 5

10

15

t [min]

Fig. 11.10. Behavior of the concentration control system in the three CSTR cascade.

This is the reason why in the case of slow processes, other more rapid methods are used; at the same time, the disturbance of the process is a minimum one.

The methods of process response curve and Cohen–Coon [7] In this case, the control loop is on manual mode and the step input is given by the control signal. The “process” response is in fact the response of the chain control valve, process and transducer. The process response curve of a level ACS is given in Fig. 11.11. The control signal c is changed by 10 % and the result is the increase of the measured level L with 1 m. On the record of the measured variable L, there are identified several constants: the dead time τ , measured from the step application time and finished when the level begins to increase. The time constant T is delimited by the right extreme of the dead time and the cotangent. The gain is obtained by dividing the change r in the measured variable (ME) by the change of the setpoint r ( Δx ). Δr Δxr The “process” gain is given by the ratio Kpr = Δc . The values recommended by Ziegler and Nichols for optimal settings of the controller are given in Tab. 11.5. Table 11.5. Optimal settings of the controller based on the process response curve.

Optimal controller parameter PB [%] Ti [min] Td [min]

P Kpr ⋅ τ /T ⋅ 100 − −

Controller structure PI 1.1Kpr ⋅ τ /T ⋅ 100 2τ −

PID 0.7Kpr ⋅ τ /T ⋅ 100 2.5τ 0.4τ

308 | Part III: Synthesis of the automatic control systems y(t) (m)

y(t) (m) B

2

2

I 1

Δy 1

A O 0

N 2

M 1 τ

t [min] 3

4

t [min] 0 τ

T

1 T

2

3

4

r[%]

60 50 1

2

3

4

t [min]

Fig. 11.11. The step response evidences the dead time, time constant and the gain. Left: practical curve showing the behavior of the level in the tank when the setpoint is changed 10 %. Right: the common way τ and T are identified on the “process” curve.

The observations made are related to the controllability of the process: if the ratio τ /T increases, its controllability is worse and the instability of the feedback control loop increases (Section 6.2). Also, if the process gain is high, the instability increases. In order to stabilize the process, the PB has to increase together with Kpr ⋅ τ /T. The ratio between Ti and Td is in the range 4–6. If instead of PB, Kc is used, Kc has to decrease. The Cohen–Coon method [7] is similar (Fig. 11.12). With the controller on “Manual” mode, a small Δc is given. On the diagram of the process, all elements are identified: t0 – the time when Δc was initiated; t2 – the time where the half point of the final settlement occurs; t3 – time when 63.2 % point of the final process settlement occurs. The difference is the way the “process” parameters are calculated: the equivalent dead time is t1 –t0 , the time constant is t3 –t1 , and the gain is Δy/Δc. t1 = (t2 − ln(2)t3 )/(1 − ln(2)). According to [7], the optimal settings are given in Tab. 11.6. Comments can be made in order to explain the form of the calculations formulae. It is obvious that the controller gain is inversely proportional to the process gain, which is quite normal; at the same time, the process controllability (expressed in the formulae by the ratio T/τ ) directly influences the controller parameters. If the control-

11 Design and tuning of the controllers

y

|

309

Δy 0.632 Δy

Δy/2

2

1 t2

t0

3

4

t3

t (min)

c Δc

1

2

3

4 t (min)

Fig. 11.12. The process step response gives details about the steady state and dynamics of the process.

lability is worse (T/τ is small), the controller gain is smaller; as a general rule of the thumb, Ti and Td are inverse proportional with T/τ which is normal if we take into consideration the influence of both parameters on the stability of the loop.

Lambda method The third frequently used tuning method which is worth mentioning is the Lambda method of tuning [2, 7] which takes care of the properties of the process as well. “Lambda tuning” refers to all tuning methods where the control loop speed of response is a selectable tuning parameter; the closed loop time constant is referred to as “Lambda”. The method has as a result a choice of parameters which gives a robust stability to the control loop. Figure 11.13 gives the way to determine the process parameters, which is very similar to the Cohen–Coon method. Smuts [7] gives the following advice: Pick a desired closed loop time constant (λ ) for the control loop. A large value for λ will result in a slow control loop, and a small λ value will result in a faster control loop. Generally, the value for λ should be set between one and three times the value of T. Use λ = 3 ⋅ T to obtain a very stable control loop. If you set λ to be shorter than T, the advantages of Lambda tuning listed above soon disappear. Calculate PID controller settings using the equations below:

310 | Part III: Synthesis of the automatic control systems Table 11.6. Optimal stings calculated through the Cohen–Coon method. www.opticontrols.com

Controller Gain

Integral Time

PB Controller

Kc = (

1.03 T ) ( + 0.34) gp τ

PI Controller

Kc = (

0.9 T ) ( + 0.092) gp τ

PD Controller

Kc = (

1.24 T ) ( + 0.129) gp τ

TD = 0.27τ (

T − 0.324τ ) T − 0.129τ

PID Controller (Noninteracting)

Kc = (

1.35 T T + 0.185τ ) ) ( + 0.185) TI = 2.5τ ( gp τ T + 0.611τ

TD = 0.37τ (

T ) T + 0.185τ

TI = 3.33τ (

Derivative Time

T + 0.092τ ) T + 2.22τ

y(t) yst c(t)[%] Δy

0,63(yst–y0)

60

y(0)=y0

Δc 50 t

0 τ

T

Fig. 11.13. Step response experiment for determining the controller parameters using Lambda method; c is the controller output and y is the process variable.

Controller gain (Kc ): Kc = T/(Kpr ⋅ (λ + τ ));

Kpr =

Δxr [%] . Δc[%]

(11.11)

Integral time (Ti ): Ti = T.

(11.12)

Td = zero.

(11.13)

Derivative time (Td ): Olsen and Bialkowski [2] give the same values for the calculation of the controller parameters. The specialists from VisSim [13] give some practical advice which we found appropriate to be mentioned in this book. The aforementioned methods (e.g. Cohen–Coon, Ziegler–Nichols), calculate values which are approximately in the right range and can be used with success. The authors experimented several times with these methods and the tuning was quite appropriate. But usually, additional manual tuning is required. Most tuning is usually done by trial and error. The authors mentioned above proposed a method of approaching trial and error tuning which of course can be improved by any engineer in the field.

11 Design and tuning of the controllers

|

311

1.

At the beginning, a small step in the manipulated variable should be done and the effect on the controlled variable response should be observed: the direction and size of the change (to be able to calculate the sign and process gain); the stabilizing time of the response when a step change in load or setpoint is made (to calculate the process time constant – one third to one fifth of the time to settle after the change). 2. With the controller on Manual mode, the “set-up” parameters, e.g. the limits on the manipulation, and measurement, the direction of control action (a reverse acting controller is required when the process gain is negative) should be established. 3. Switch off the integral and derivative action in the controller by setting the Ti to maximum and the Td to zero. Set the controller gain to a relatively small value (say Kc at 0.5 or PB at 200 %). This initial value can be higher if the process gain is low and smaller if the process gain is high. 4. With the controller on Automatic mode, we should make a small change in the setpoint and observe the controlled variable’s response: if the response is oscillatory one has to reduce the controller gain, if it is slow and weak, one has to increase it. Changes in the controller gain by a factor of between 2 and 10 are to be done, depending on the knowledge about the process and the limitations we have concerning the safety. The action should be repeated (with the controller on Manual when changing the gain or PB) until a response with a decay ratio from one quarter to one half is obtained. 5. At this point, one should reduce the controller gain by around 25 % to better stabilize the control loop and to prepare the introduction of the I and D components. Now set the integral time constant (if integral action is needed) to a value of the order of magnitude or even to the value of the process time constant. 6. Modify the Ti until a one quarter decay ratio is obtained. If the response is too oscillatory, then increase Ti , if it is too slow, reduce the Ti . Recommended changes in the Ti are with a factor of 2. 7. If derivative action is needed (slow processes), the Td should be chosen at around one quarter the process time constant. The action from point 4 is repeated and we should observe the response adjusting the constant as desired. Derivative action is very sensitive and sometimes unpredictable. Examples of “difficult to tune processes” will be given afterwards. 8. At the end, the controller gain should be changed by small amounts until we obtain the desired controlled response.

Theoretical methods of tuning controller parameters [13–15] Sometimes, especially during the design process of an industrial plant, there is no physically present installation on which we can experiment. In other situations, especially in potentially dangerous processes, one cannot simply experiment and the

312 | Part III: Synthesis of the automatic control systems controller parameters should be calculated otherwise. There are theoretical methods close at hand.

Method based on Routh stability criterion Considering the control system from Fig. 11.14, Y(s) =

HprD (s) 1 + HT (s)HC (s)HAD (s)Hpr m (s)

D(s) +

HprD (s) Y(s) = D(s) 1 + HT (s)HC (s)HAD (s)Hpr m (s)

HC (s)HAD (s)Hpr m (s) 1 + HT (s)HC (s)HAD (s)Hpr m (s)

R(s) (11.14)

and

(11.15)

HC (s)HAD (s)Hpr m (s) Y(s) = . D(s) 1 + HT (s)HC (s)HAD (s)Hpr m (s)

(11.16)

D(s)

R(s) +

E

HC(s)

C

+ HAD(s)

–

Xr(s)

HprD(s)

Hprm(s)

+

Σ

Y(s)

HT(s)

Fig. 11.14. Control system subjected to a disturbance.

The roots of the polynomial 1 + HT (s)HC (s)HAD (s)Hpr m (s) define the stability of the control system (Chapter 2, Section 2.15). The characteristic equation, 1 + HT (s)HC (s)HAD (s)Hpr m (s) = 0

(11.17)

and considering HT (s) = HAD (s) = 1 in order to simplify the calculation, is 1 + HC (s)Hpr m (s) = 0.

(11.18)

If the equation (11.17) has at least one root in the right half-plane s, the closed system is unstable.

11 Design and tuning of the controllers

|

313

Routh stability criterion Consider a closed-loop transfer function H(s) =

b0 sm + b1 sm−1 + ⋅ ⋅ ⋅ + bm−1 s + bm B(s) = , A(s) a0 sn + a1 sn−1 + ⋅ ⋅ ⋅ + an−1 s + an

(11.19)

where the ai ’s and bi ’s are real constants and m ≤ n. An alternative to factoring the denominator polynomial, Routh’s stability criterion, determines the number of closed loop poles in the right half-plane s. Algorithm for applying Routh’s stability criterion: 1. Calculate the roots of A(s) at the origin from the polynomial a0 sn + a1 sn−1 + . . . + an−1 s + an = 0,

2.

(11.20)

where a0 ≠ 0 and an >0. If the order of the resulting polynomial is at least two and any coefficient ai is zero or negative, the polynomial has at least one root with nonnegative real part. To obtain the precise number of roots with nonnegative real part, proceed as follows. Arrange the coeficients of the polynomial, and values subsequently calculated from them as shown below: sn sn−1 sn−2 sn−3 sn−4 ... s2 s1 s0

a0 a1 b1 c1 d1 ... e1 f1 g0

a2 a3 b2 c2 d2 ... e2

a4 a5 b3 c3 d3 ...

a6 a7 b4 c4 d4 ...

... ... ... ... ... ...

where b1 =

a1 a2 − a0 a3 a1

b2 =

a1 a4 − a0 a5 a1

b3 =

a1 a6 − a0 a7 a1

......

314 | Part III: Synthesis of the automatic control systems generated until subsequent coefficients are 0. Similarly, b1 a3 − a1 b2 b1 b a − a1 b3 c2 = 1 5 b1 b1 a7 − a1 b4 c3 = b1

c1 =

⋅⋅⋅⋅⋅⋅ c1 b2 − b1 c2 c1 c b − b1 c3 d2 = 1 3 c1 d1 =

until the n-th row of the array has been completed. Missing coefficients are replaced by zeros. The resulting array is called the Routh array. The powers of s are not considered to be part of the array. We can think of them as labels. The column beginning with a0 is considered to be the first column of the array. The Routh array is seen to be triangular. It can be shown that multiplying a row by a positive number to simplify the calculation of the next row does not affect the outcome of the application of the Routh criterion. (3) Count the number of sign changes in the first column of the array. It can be shown that a necessary and sufficient condition for all roots of (11.19) to be located in the left half-plane is that all the ai are positive and all of the coefficients in the first column are positive. Example 11.2. Considering the 3 CSTRs cascade system (Fig. 3.12) [6], we want to determine the condition of stability of a concentration ACS with a P controller. The transfer function of the process is Hpr m (s) =

( 12 )

3

(s + 1)3

=

CA3 (s) CA0 (s)

(11.21)

and that of the controller is Hc (s) = Kc .

(11.22)

s3 + 3s2 + 3s + 1 = 0.

(11.23)

The characteristic equation is

Routh’s matrix is [ [ [ [ [ [ [

1 3 9−1 3 8 3 8 3

=

8 3

3 ] 1 ] ] ] 0 ] ] ]

(11.24)

=1 [ ] and does not present any sign change, meaning the system is stable in open loop.

11 Design and tuning of the controllers

|

315

Considering (11.18), the stability of the closed loop is given by the condition 3

1 + Kc

( 12 )

(s + 1)3

which becomes s3 + 3s2 + 3s + (1 +

= 0,

(11.25)

Kc ) = 0. 8

(11.26)

] ] ] ]. ] ] ]

(11.27)

Routh’s matrix is [ [ [ [ [ [ [

1

3

3

1+

9−(1+ 3

Kc 8

)

Kc 8

0

Kc

] [ 1+ 8 The only term capable of becoming negative and thus bringing instability to the closed loop is K 9 − (1 + 8c ) (11.28) < 0, 3 and the condition means (8 − 3

Kc ) 8

< 0 and then Kc > 64.

(11.29)

Then, for a controller gain Kc < 64, the ACS is stable. In order to have a smaller offset, we may choose Kc = 64.

The method of direct substitution The technique consists in replacing s = jω in the characteristic equation and to determine ω and the other controller parameters which satisfy the characteristic equation. Example 11.3. Consider the same system of 3 CSTRs with the concentration control loop and a P controller as in the previous examples. The characteristic equation is s3 + 3s2 + 3s + (1 +

Kc ) = 0. 8

(11.30)

Substituting s with jω , the characteristic equation becomes − jω 3 − 3ω 2 + 3jω + 1 + (1 +

Kc =0 8

Kc − 3ω 2 ) + j(3ω − ω 3 ) = 0. 8

(11.31) (11.32)

316 | Part III: Synthesis of the automatic control systems This means 3ω − ω 3 = 0

from which ω = ±√3

Kc − 3ω 2 = 0 8 which gives Kc = 64.

1+

(11.33)

The value of the gain at the limit of stability is 64, the same value as that obtained with the previous method. Regarding s, considering that the real part is = 0, the roots of the equation are situated on the imaginary axis and the ω obtained is the crossover frequency, that of oscillation, ωosc . The oscillation period, Posc = ω2π = 3.64 min, is osc the same as obtained in Example 11.1.

Method based on Nyquist stability criterion [13, 14] The Nyquist stability criterion mentions: the condition for a control system to be stable is that the number of rotations of the vector’s H(s) hodograph around the point (−1, j0) (Fig. 11.15) is equal with the number of poles of the transfer function H(s) placed in the right half-side of the plane s. H(s) is the transfer function of the open loop. Im[H(jω)]

ω=∞ (–1, 0j)

ω=0 Re[H(jω)]

Fig. 11.15. Nyquist plot.

The algorithm is as follows. 1. The open loop transfer function H(s) is constructed on the path r → xr . 2. The number of poles of the function 1+H(s) in the right half-plane is determined. 3. The hodograph of H(jω ) with values ω ∈ (0, ∞) and ω ∈ (0, −∞) is plotted. 4. The number of encirclements P of (−1, j0) is counted to see if the system is stable. Example 11.4. The pressure in the system of two vessels in series (Fig. 11.16) is controlled with a P controller. The maximum Kc to assure the stability of the loop is to be determined.

11 Design and tuning of the controllers

|

317

PC

ΔF F

Patm=0 P2

P1

H4(s) ΔF

+ –

Kp

H1(s)

H2(s)

2.2 2s+1

3 15s+1

2 20s+1

0.8 20s+1

ΔP2

H3(s)

H5(s) 0.16 Fig. 11.16. Pressure ACS with (a) two vessels in a series; (b) block scheme of the ACS.

– –

The data are as follows. The control valve is a first order element with Tcv = 2 s and a gain Kcv = 0.022 Nm3 /min/bar. The first vessel has a time constant Tv1 = 15 s and an increase of 10 % of the nominal flow (Fn = 2 Nm3 /min) produces an increase of pressure of 0.6 bar. The second vessel has a time constant Tv2 = 20 s and an increase of 1 bar in the first vessel produces a plus of pressure in the second one of 0.8 bar. On the channel ΔF → p2 , an increase by 10 % of the outflow produces a decrease of pressure of 0.4 bar and with a time constant of 20 s. The pressure transducer is pneumatic, a proportional element with the output range 0.2 ⋅ ⋅ ⋅ 1 bar and measurement range 0 ⋅ ⋅ ⋅ 5 bar.

318 | Part III: Synthesis of the automatic control systems The transfer functions are Kcv = Hcv (s) = Tcv s + 1 Hv1 (s) =

Kv1 = Tv1 s + 1

0.022 Nm3 /min 0.01 bar

2s + 1 0.6 bar 0.1⋅2 Nm3 /min

15s + 1

=

2.2 2s + 1

(11.34)

=

3 15s + 1

(11.35)

0.8 bar

Kv2 0.8 Hv2 (s) = = 1 bar = Tv2 s + 1 20s + 1 20s + 1 HD (s) = HT (s) =

0.4bar 0.1⋅2 Nm3 /min

20s + 1

=

2 20s + 1

1–0.2 bar = 0.16. 5–0 bar

(11.36)

(11.37)

(11.38)

Applying the algorithm: 1. The open loop transfer function is H(s) = Kc 2. 3.

3 0.8 2.2 0.16. 2s + 1 15s + 1 20s + 1

(11.39)

The function has no pole in the right half-plane and thus P = 0. The hodograph of the function is presented in Fig. 11.17.

Im[H(jω)]

ω=∞ (–1, 0j)

ωk

ω=0 ω=0 Re[H(jω)] K p1 K p1 Fig. 11.17. The hodograph of the open loop transfer function H(s).

For stability, it is demanded that there is no encirclement of the point (−1, j0). The conditions for that are Im[H(jω )]|ωosc = 0 and Re[H(jω )]|ωosc > −1

(11.40)

11 Design and tuning of the controllers

Re[H(jω )] =

319

0.844(1 − 370ω 2 )Kc (1 − 370ω 2 )2 + (37 − 600ω 2 )2 ω 2

and Im [H (jω )] =

|

0.844(37–600ω 2 )ω Kc 2

(11.41)

2

(1 − 370ω 2 ) + (37 − 600ω 2 ) ω 2

37 From Im[H(jω )]|ωosc = 0 results ω1 = 0 and ω2 = ±√ 600 ; from these values the one verifying Re[H(jω )]|ωosc > −1 is chosen:

Re[H(jω )] =

0.844(1–370ω 2 )Kc (1 −

2 370ω 2 )

=

0.844Kc 1- −

37 370 600

> −1 (ω2 = √

37 ). 600

The result is Kc < 25.8; the period of oscillation is Posc = Pu =

(11.42)

1 37 2π √ 600

= 0.64 min.

(11.43)

Method based on the criterion of quarter amplitude decay ratio (QADR) The conditions for undamped permanent oscillations are (11.3) and (11.4). In order to have quarter decay ratio oscillations, the control loop behaves in such a way that at every circulation of the information inside the loop (Fig. 11.2), its steady state open loop gain is 12 . Condition (11.3) stays the same: φC + φAD + φPr + φT = −180∘ , and from it, the crossover frequency ωosc is calculated. Equation (11.4) becomes MC MAD MPr MT |ωosc = 0.5.

(11.44)

The algorithm for calculation of the controller’s parameters is: (a) The controller has P structure: 1. ωosc is calculated from (11.3); 2. ωosc is replaced in the formula (11.44) and Kc is deduced. (b) The controller has PI structure: 1. Ti is chosen for the magnitude of the maximum value of time constants of the elements in the control loop (the Cohen–Coon method allocates Ti = T); 2. ωosc is calculated from (11.3); 3. Kc is deduced from (11.44). (c) The controller has PID structure: 1. Ti is chosen for the magnitude of the maximum value of time constants of the elements in the control loop (the Cohen–Coon method allocates Ti = T);

320 | Part III: Synthesis of the automatic control systems 2.

Td is chosen from the phase margin of +30∘ or +45∘ to assure the stability of the ACS 1 ) = +30∘ φc = tan−1 (Td ωosc − Ti ωosc

3.

Kc is deduced from (11.44).

Example 11.5. Consider a flow ACS, in a pipeline (Fig. 11.18). The characteristics of the elements of the control loop are l/min l/min mA TT = 0.2 s and KT = 2.5 l/min l/min TAD = 3 s and KAD = 1.3 . mA Tpr = 0.5 s and Kpr = 1

Fset

i F

F Fig. 11.18. Flow ACS on a pipeline.

The controller parameters are to be determined using the QADR method. All elements in the control loop are first order capacities. The expressions defining the first order capacity element are Mj (ω ) =

Kj √1 +

Tj2 ω 2

and φj (ω ) = −tan−1 (Tj ω )

(11.45)

(a) Consider a P controller with Mc (ω ) = Kc and φc (ω ) = 0. Applying the expressions above in (11.3) −1 360 (− tan Tpr ω − tan−1 TT ω + 0 − tan−1 TAD ω ) = −180∘ , 2π

and replacing the values one obtains a transcendental equation which can be numerically solved (halving interval, Newton–Raphson etc.). Other very simple solution is

11 Design and tuning of the controllers

| 321

that of substituting guessed values directly into the equation. The range in which we choose the guessed values is one including the corner frequency at which the module of a capacity decreases importantly. The corner frequency, ωc , for the system described is 2π ωc = , where Tmax = TAD = 3 s. Tmax The equation (11.3) becomes 360 (− tan−1 0.5ω − tan−1 0.2ω + 0 − tan−1 3ω ) = −180∘ 2π

(11.46)

and the solution has to be searched for in the range 0.01 ⋅ ⋅ ⋅ 10 rad ⋅ s−1 . Applying the trial values one obtains ω [rad ⋅ s−1 ]

φpr [grd]

φT [grd]

φc [grd]

φAD [grd]

∑ φi

0.1 1.0 10.0 3.5

−2.9 −26.56 −78.69 −60.25

−1.1 −11.3 −63.43 −34.99

0 0 0 0

−16.7 −71.56 −88.09 −84.55

−20.7 −109.44 −230.21 −179.8

ωosc = 3.5 rad ⋅ s−1 meaning that the loop has an oscillation at every 1.8 s. From (11.44) 1

2.5

√1 + 0.5 3.5 √1 + 0.2 3.5 2

2

Kc opt = 4.0 or

2

2

Kc

1.3 √1 + 32 3.52

= 0.5,

PBopt = 25 %.

(b) Consider a PI controller at which MPI (ω ) = Kc √1 +

1 Ti2 ω 2

and φPI (ω ) = − tan−1

1 . Ti ω

Choosing Ti = 5s, we can find the crossover frequency ωosc ω [rad s−1 ] 0.1 1.0 10.0 3.3

φpr [grd]

φT [grd]

φc [grd]

φAD [grd]

∑ φi

−2.9 −26.56 −78.69 −58.78

−1.1 −11.3 −63.43 −33.42

−63.4 −11.3 −1.14 −3.46

−16.7 −71.56 −88.09 −84.23

−84.1 −20.75 −231.4 −179.9

and we can approximate ωosc = 3.3 rad s−1 .

(11.47)

322 | Part III: Synthesis of the automatic control systems From (11.21) 1 2.5 1.3 1 Kc √ 1 + 2 2 = 0.5 √1 + 0.52 2.12 √1 + 0.22 2.12 5 2.1 √1 + 32 2.12 results Kc opt = 3.52

or

PBopt ≈ 28.3 %

and Ti opt = 5 s. (c) Consider a PID controller at which MPID (ω ) = Kc √ 1 + (Td ω −

1 2 ) Ti ω

and φPID (ω ) = − tan−1 (Td ω −

1 ) . (11.48) Ti ω

Choosing Ti = 3 s and calculating Td from the phase margin condition φc = tan−1 (Td ωosc −

1 ) = +30∘ , Ti ωosc

and in the first approximation considering ωosc = 3.5s−1 without the influence of I and D components of the controller. tan 30∘ + Td =

3.5

1 3 ⋅ 3.5 = 0.148 s.

󸀠 > Taking into consideration the influence of I and D, the real crossover frequency ωosc ωosc above approximated. This induces an equivalent module condition for (11.3):

∏ Mi (ω )|ω 󸀠 = 0.5 is equivalent with i

(11.49)

osc

∏ Mi (ω )|ωosc = 1.0.

(11.50)

i

Thus, 1

2.5

√1 + 0.52 3.52 √1 + 0.22 3.52

Kc √ 1 + (0.148 ⋅ 3.5 −

2 1 1.3 ) = 1.0 3 ⋅ 3.5 √1 + 32 3.52

gives a set Kc opt = 6.5 or PBopt ≈ 15 % Ti opt = 3 s Td opt = 0.15 s. If we examine Tab. 11.3, we observe these values are confirmed orientatively by the Ziegler–Nichols method as well.

11 Design and tuning of the controllers

|

323

Actually, the small value of Td opt , which cannot be practically fixed in the controller, shows that its value should be 0 instead. As we shall see in the next chapter, and as was already shown in Chapter 8, “fast” processes such as the flow in the pipeline do not need a D component of the controller. At the end of this section, we want to mention that there are processes that are difficult to tune. For these, the rules detailed previously do not give appropriate results.

11.5 Tuning controllers for some “difficult to be controlled” processes [17, 18] An “easy to be controlled” process control loop is one which satisfies the following characteristics imposed to the controlled process when a disturbance on the process occurs: 1. the process responds quickly without a significant delay; 2. it goes directly towards the new steady state without going first in the opposite direction; 3. it settles to the new steady state. In this case, the controllability is a good one (see Section 6.2) and the ACS easily controls the process (Fig. 11.19). y y

r

0

0

t

r

t

(a)

(b)

Fig. 11.19. Step responses for “easy to be controlled” processes.

1. 2.

3.

There are three classes of processes which are difficult to be controlled: processes with dead time (see Sections 4.7 and 4.8) for which the transfer function has a e−τ s term; processes with inverse response where the transfer function has a Right HalfPlane (RHP) zero (with two capacities in parallel, each with inverse action upon the process output, one faster than the other – see Section 4.4); processes presenting open-loop instability where the transfer function has an RHP pole.

324 | Part III: Synthesis of the automatic control systems These types of behavior are depicted in Fig. 11.20. y

y r

y

r

t

t (a)

(b)

t (c)

Fig. 11.20. Step response for “not easy to be controlled” processes.

For processes with a dead time, the problem can be solved using the experimental or theoretical methods introduced before. If the process is not controllable, some other ACSs (cascade, feed forward etc.) are used. 1. For processes with inverse response the solution can be given by a properly tuned PID controller. There are other methods too to deal with this problem, either to approximate the inverse-response with a dead time and capacity behavior, or to compensate the inverse response; the last method gives the best results and it is largely described in [17, 18]. 2. In the case of open-loop unstable systems, the transfer function has at least one RHP pole. It is obvious that when disturbing the process in any direction the system goes to its limit (e.g. an exothermic thermally unstable reactor [19]). The system can be stabilized (details are given in [19]) using a properly tuned controller. Example 11.6. Consider the transfer function of the open loop without controller H(s) =

K Ts − 1

(11.51)

and a proportional controller in the loop. Obtain the range of Kc for which the closedloop system is stable. The characteristic equation is KKc = 0 which gives Ts − 1 Ts − 1 + KKc = 0 and

1+

s=

1 − KKc T

is the root.

If we want the root to be negative (and the closed loop stable), the condition is Kc >

1 . K

(11.52) (11.53) (11.54)

11 Design and tuning of the controllers

|

325

When it is noticed that with all proper tuning of the controller parameters, the ACS is still unstable, one has to pass to another level of complexity, that of unconventional or advanced control which is the subject of the next book in the series.

11.6 Problems (1) A non-isothermal CSTR has a temperature control system. Determine the structure and optimal values of a temperature controller. Knowing dynamic and steady state properties – of the jacket: time constant Tag = 4 min, transport dead time in the jacket τag = ∘ 0.5 min, Kag = 0.63 m3 C/h ; ∘ – of the CSTR wall: Tw = 4 min, Kw = 0.45 ∘ C ; C ∘ – of the reaction mass in the interior, Tr = 26 min; Kr = 0.8 ∘ C ; C 3 – of the actuator, KAD = 3.5 mmA/h ; –

of the transducer, KT = 2 mA ∘ C , where the delays of the actuator and transducer are considered 0.

(2) A distillation column (Fig. 11.21) has 8 trays in its concentration section. As is shown, a concentration control system collects the sample from the tray 3 and manipulates the reflux flow. Each tray behaves as a capacity and has a time constant of Tt = 0.1 min and a gain % mol mA factor Kt = 1.1 kmol/min . The actuator and transducer have a total gain of 1 kmol/min mA % mol and negligible delay. The crossover frequency, ωosc has to be determined. How is this frequency modified when the concentration sample is collected on tray 8? What is the optimal value of the P controller gain of the control loop in both situations? 4m (3) The speed of the conveyor belt (Fig. 11.22) is v = min . The weight transducer is placed at 8 m distance from the slide valve of the ACS. The crossover frequency in the conditions of an integral control and the integral time of the controller are to be calculated for the QDAR criterion.

(4) Having a two CSTR cascade with a concentration ACS (Fig. 11.23), using Nyquist criterion, determine the maximum controller gain for which the loop is stable. The data are the following: V1 = V2 = 1 m3 ,

F 1 = F 2 = Fi = 4

m3 ; h

the reaction is isothermal with first order kinetics and the rate constant is k = 2 h−1 . The concentration transducer has the measurement range 11.8 kmol and the output m3

326 | Part III: Synthesis of the automatic control systems

R D 3 AC

8 F

B Fig. 11.21. Distillation column with a concentration ACS.

l=8m C

Fig. 11.22. Conveyor belt with a flow ACS.

signal in the range 4–20 mA; the delay of the transducer is given by the time constant TT = 2 min. The control valve is linear and works with unified signal 4–20 mA and 3 controls a flow between 0 and 4 mh . (5) The components of a heater control system are shown in Fig. 11.24. The time constants of the actuator, process and transducer are 5, 20, and 100 s. The improvement of the performance of the ACS is proposed introducing a supplementary sheath of the resistor which doubles its time constant, allowing a larger controller gain.

11 Design and tuning of the controllers

Fi

|

327

F1

Vi

V2

k

k

F2 AC Fig. 11.23. Two CSTR cascade with concentration ACS.

r +

KAD 5s+1

Kc –

+

D +

KPR 20s+1

y

1 100s+1 Fig. 11.24. Temperature ACS of a heater.

Draw the diagram of the ACS response to a reference temperature change of 10 % in both cases. (6) The maximum value of a P controller using Routh’s criterion for a control loop with the following characteristics has to be determined. One first order capacity with T1 = 1 min and K1 = 2; six capacities in series each with T2 = 1 min, K2 = 1; one control valve behaving capacitive with Tcv = 2 s, Kcv = 1.5. (7) A step response experiment on a heating process gives the following data: Time (min)

0

1

2

3

4

5

6

7

8

9

20

30

Response ( ∘ C)

0

0

0

4

10

19

27

35

41

45

48

50

The input flow variation is of 2 flow units. Determine, using all methods available, the crossover frequency and the maximum P controller gain. The optimal parameters of a PID controller using the process control curve, Cohen–Coon and Lambda methods have to be computed. Discussion is required.

328 | Part III: Synthesis of the automatic control systems (8) The temperature in a continuous CSTR is controlled by an ACS with the following characteristics: The transducer has the measuring range 100–200 ∘ C and an output standard signal 2–10 mA. The controller is PI and has a PB = 25 % and Ti = 3 min, having both input and output signals in the range 2–10 mA The control valve has a Kv = 4, a linear characteristic, an input signal in the range 2–10 mA and s constant pressure drop Δpv = 0.25 bar. Cooling water passes through the valve. If the control signal is 6 mA at error e = 0, what flow passes through the valve? If a step change of the reactor temperature of 5 ∘ C occurs, what will be the immediate effect on the output of the controller and on the water flow value? What happens if the tuning is done with PB = 100 % and Ti = 10 min? (9) The ACS of a heat exchanger has the following transfer functions: 0.047 m/bar m3 /min ⋅ 112 0.083s + 1 m ∘ 3 2 C/(m /min) Hpr (s) = (0.017s + 1)(0.432s + 1) 0.12 bar/ ∘ C . HT (s) = 0.024s + 1

HCV (s) =

Using the direct substitution method, show that the limit gain factor of the P controller is Kc = 12 and the period of oscillation is Tu = 0.36 min. (10) Determine the same parameters as in Problem (9), using the Nyquist method.

References [1] [2]

[3] [4] [5] [6] [7] [8]

Agachi, S., Automatizarea proceselor chimice, Ed. Casa Cartii de Stiinta, Cluj-Napoca, 1994, p. 162. Olsen, T., Bialkowski., B., Emerson Process Management, Lambda Tuning as a Promising Controller Tuning, Method for the Refinery, prepared for Presentation at 2002 AIChE Spring National Meeting in New Orleans, March 2002, Session No. 42 – Applications of Control to Refining. Agachi, S., Automatizarea proceselor chimice, Ed. Casa Cartii de Stiinta, Cluj-Napoca, 1994, p. 163. ibidem, p. 166. Ziegler, J. G., Nichols, N. B., Optimum settings for automatic controllers, Trans. ASME, (1942), 759–768. Luyben, W., Process Modeling, Simulation and Control for Chemical Engineers, McGraw Hill, N.Y., 1996, Chap. 10. Smuts, J., Control Notes, Reflections of Process Control Practitioner, Cohen Coon Tuning Rules, http://blog.opticontrols.com/archives/383, 2010–2013, March 24, 2011. Ogunnaike, B., Harmon Ray, W., Process Dynamics, Modeling and Control, Oxford University Press, 1994, Chapter 15, p. 520.

11 Design and tuning of the controllers

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

| 329

Skogestad, S., Probably the best simple PID tuning rules in the world, Annual AIChE Meeting, Reno, NV, paper 276h, 6 Nov., 2001. Rivera, D. E., Morari, M., Skogestad, S., Internal model control. 4. PID controller design, Ind. Eng. Chem. Res. 25 (1), (1986), 252–265. Buckbee, G., Best Practices for Controller Tuning, ExperTune Inc., 2009. Rice, R., PID Tuning Guide, A Best – Practices Approach for Tuning and Understanding PID Controllers, Second Edition, Control Station Inc., 2013. http://www.online-courses.vissim.us/Strathclyde/tuning_pid_controllers.htm Agachi, S., Automatizarea proceselor chimice, Ed. Casa Cartii de Stiinta, Cluj-Napoca, 1994, p. 170. Ahmad, A., Somanathi, M., PID controller tuning for integrating processes, ISA Transactions 49, (2010), 70–78. O’Dwyer A., Handbook of PI and PID Controller Tuning Rules, 2nd ed. London, Imperial College Press, 2006. Ogunnaike, B., Harmon Ray, W., Process Dynamics, Modeling and Control, Oxford University Press, 1994, Chapter 17. Waller, K., Nygardas, C., On inverse response in process control, I&EC Fund, 14, (1975), p. 221. Agachi, P., Automatizarea proceselor chimice, Ed. Casa Cartii de Stiinta, Cluj-Napoca, 1994, p. 241.

12 Basic control loops in process industries Chemical processes are generally controlled through five parameters: flow, pressure, level, temperature and concentration. There are also other parameters which can be controlled, but they are not very frequently met: distance between electrodes, current and inter-electrode voltage (all in electrochemical processes), rotation speed, torque, weight or turbidity etc. The present chapter focuses only on the main parameters mentioned above: flow, pressure, level, temperature and concentration. Each of the controlled parameters has particularities which impose a certain control solution.

12.1 Flow automatic control systems Measuring and controlling the flow is important due to the importance of the mass balance in process industries. It is extremely important to have control of the material consumption/production in all processes of reaction or separation. It is important not only for the stoichiometry of a reaction or the appropriate separation of compounds, but also for the economy of the process which is very often determining. Different ways of controlling the flow are depicted in Fig. 12.1. The flow can be controlled either by choking (strangling) the material vein (Fig. 12.1a), modifying the centrifugal or dosage pump motor speed by a Variable Frequency Drive (VFD) (Fig. 12.1b, c, d), modifying the number of strokes of the metering (piston) pump (Fig. 12.1e), or modifying the rotation speed of the motor of a conveyor belt or of a screw pump (Fig. 12.1f, g).

Process characteristics Flow is a momentum transfer process. In most cases, the liquid flow control system manipulates the material flow modifying the pressure drop on the transport line, using a control valve. The characteristic of the centrifugal pump is given in Fig. 12.2. The characteristic of a centrifugal pump and of the pipeline can be deduced from [1] and are given by equations (12.1). pcp = K1 n2 + K2 nF + K3 F 2 pp = p0 + KF 2 ,

(12.1)

where pcp is the centrifugal pump head pressure [N/m2 ], pp – the pressure characteristic of the pipeline [N/m2 ], p0 – the output pressure of the pump [N/m2 ], n – the rotational speed of the motor [rpm], F – the fluid flow rate [m3 /s]. K, Ki are constants. When the pressure drop on the pipeline increases, the outflow decreases from the nominal value FN to F1 . To return to the nominal flow rate, it is necessary to decrease

12 Basic control loops in process industries

M

FC

(a)

|

331

FC

(b) FC TB

FC

(d)

(c)

FC

M

(e)

FC

(f)

M FC (g) Fig. 12.1. Automatic control systems for solid or liquid flow.

P 1 N P1

2

PN P2 Pe F1

FN F2

F

Fig. 12.2. Steady state characteristics of the centrifugal pump and of attached pipeline.

the pressure drop on the pipeline by opening the control valve. Thus, the pressure drop on the pipe decreases and the curves 1 and N overlap. When the flow rate control is done by modifying the rotational speed n, (Fig. 12.3), the characteristics of the pump change with ni .

332 | Part III: Synthesis of the automatic control systems P 1 N 2

n2 nN F1

FN

n1

F2

Fig. 12.3. Steady state characteristics of the centrifugal pump and of attached pipeline when the flow is controlled by the rotational speed.

F

With the same pipeline characteristic, modifying n in the range n1 ⋅ ⋅ ⋅ n2 , one may change the flow in the range F1 ⋅ ⋅ ⋅ F2 . The momentum transfer process is a special case in which the flow is both the controlled and the manipulated variable. As a consequence, apparently, the steady state characteristic is a straight line with a slope of 45∘ and the steady state gain of the process is 1. The situation is a little bit more complicated since the flow is not manipulated by itself (Fig. 12.1a) but through the pressure drop on the pipeline (Fig. 12.2). The dynamic characteristic is given by the law of momentum conservation (3.16): ρ ghAp − (∝

lp ρ v2 ρ v2 d ) Ap = (Ap lp ρ v) +λ 2 dp 2 dt

and if on the left-hand side of the equation we replace the first term with Ap Δp, and then v =

F , Ap

l

m = Ap lp ρ , ∝ +λ dp = C, the equation becomes p

Ap Δp − C

Ap lp ρ dF ρ F2 Ap = . 2 Ap dt 2Ap

(12.2)

Deriving from (12.2), one can deduce the steady state characteristic Ap Δp − C F = Ap √

ρ F2 A = 0, 2A2p p

2 Δp. Aρ

(12.3)

(12.4)

The steady state characteristic F = f (Δp) is nonlinear and is shown in Fig. 12.4. Equation (12.2) is first order nonlinear due to the F 2 term and in order to describe the dynamics of the flow using a capacitive behavior one has to linearize around the nominal point of operation. It is quite acceptable to do this approximation because the control system, if it is properly tuned, should not allow deviations from the nominal flow larger than ±10 %.

12 Basic control loops in process industries

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F

Fn

Δpn

Δp

Fig. 12.4. Steady state characteristic of the flow.

The linearization is done using Taylor’s expansion formula (equation (3.6)) around the steady state point of operation (nominal point), x0 : f (x0 + Δx) = f (x0 ) +

1 df 1 d2 f | Δx2 + ⋅ ⋅ ⋅ . |x0 Δx + 1! dx 2! dx2 x0

(12.5)

Because Δx is a very small quantity, the terms with superior powers to 1 are negligible, so we can approximate the function with f (x0 + Δx) ≈ f (x0 ) +

1 df | Δx. 1! dx x0

(12.6)

One cannot say the Δx has to be chosen 0.01 m, 1 m3 /h, or 10 ∘ C and these values are small enough to allow the linearization. All depends on the system under discussion: if we need to approximate an infinitely small element in an electrolysis DeNora amalgam cell of 22 m, the Δx infinitely small element can be the dimension of an anodic frame of 0.8 m; if the temperature in a CSTR is of 80 ∘ C, Δx can be of 8 ∘ C. Thus, F 2 from equation (12.2), can be expressed as F 2 = Fn2 +

d(F 2 ) | (F − Fn ) = Fn2 + 2Fn (F − Fn ) = 2Fn F − Fn2 , dF Fn

(12.7)

where Fn is the nominal operating flow desired as controlled variable value. Thus, (12.2) becomes Ap Δp − C

ρ F 2 Ap lp ρ dF ρ 2Fn F + C n = . 2Ap 2Ap Ap dt

(12.8)

Arranged in another form, lp ρ Fn dF Fn Δp, +F = 2Ap Δpn dt 2Δpn

(12.9)

it depicts a capacitive behavior with a time constant and gain factor of the process Tpr =

lp ρ Fn 2Ap Δpn

and Kpr =

Fn . 2Δpn

(12.10)

334 | Part III: Synthesis of the automatic control systems Characteristics of the other elements in the control loop Transducer The flow transducer usually has a steady state linear characteristic (Chapter 7). If the flow is measured with an orifice plate sensor + differential pressure transmitter, the characteristic can be nonlinear if the transducer does not have a square root extractor. The nonlinearity of the transducer has to be compensated by the inverse nonlinear characteristic of the control valve. The dynamic behavior is that of a first order capacitive element with a time constant of seconds. Controller The structure of the controller is either P or PI with usual values of PB = 50–200 % and Ti ≈ 130 s depending on the time constant of the control valve which is the slowest element in the loop [2]. The D component of the controller is eliminated due to the instability of the process. The same influence has a too small Ti , destabilizing the process. Control valve The steady state characteristic has to compensate the other nonlinearities in the control loop: process (equation (12.4)) and transducer. Thus, usually it has an equal percentage or modified parabolic characteristic. The Kv is calculated as in Chapter 9. The control valve, although theoretically from the dynamic point of view is a second order element, it is considered a first order one with largest time constant in the control loop, of the order of seconds for the regular control systems. Example 12.1. Considering a flow control loop in a pipeline of L = 50 m and Dn = 20 on which in the normal operating point, Fn = 2.4 m3 /h, the pressure drop is Δpn = 1 bar. The fluid is water with ρ = 1000 kg/m3 . What are the constants defining the behavior of the process? From equation (12.10), kg

3

2.4 m 50 ⋅ 1000 m3 ⋅ 3600 Lρ Fn s Tpr = = = 1.08 s, 2Ap Δpn 2 ⋅ 3.14 ⋅ 0.0004 m2 ⋅ 105 N2 m 2.4 m3

Fn Kpr = = 3600 s = 0.33 ⋅ 10−7 2Δpn 2 ⋅ 105 N2 m

m3 s N m2

.

From this example we can observe that the process is a fast one with time constants of the order of seconds. It is well known that due to the turbulence in the pipelines, the process is noisy, very unstable around the average values of the flow rate, inducing instability in the control loop.

12 Basic control loops in process industries

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Considering the Example 11.5 of a flow ACS, with the data from the example, Tpr = 1 s

and Kpr = 0.33 ⋅ 10−7

TT = 0.2 s

and KT = 2.5

TAD = 3 s

and KAD

l/min m3 /s = 20 , N/m2 bar

mA , l/min bar = 0.05 . mA

The optimal values for a PI control are the same or very close to those in the example mentioned: ωosc = 2.1 s−1 ,

PBopt ≈ 65 %,

Ti opt = 5 s.

12.2 Pressure automatic control systems Pressure is another important parameter for controlling the mass balance of a process. Pressure is important for the chemical equilibria in the catalytic gaseous reactors, the total pressure imposing the adsorption, absorption and kinetic rates. Measuring the pressure of the liquid is irrelevant for the mass content, but it is very relevant for gaseous systems. When a gas is transformed isothermally, the pressure can be changed either by volume variation or by flow rate variation. When the thermodynamic system implies vapor/liquid equilibrium, the pressure can be changed, changing the temperature. As an example, the pressure at the top of a column (Fig. 12.5) can be modified either through the outflow of the non-condensable gases when these exist, or through the cooling agent flow rate of the condenser at the top of the column. PC

PC R

D R

(a)

D

(b)

Fig. 12.5. Controlling the pressure in a distillation column.

Another solution for controlling the pressure in the same system is to control the level of condensate in the condenser. In this way we modify the heat transfer area and, implicitly, the quantity of vapors condensed. In a simple pressure control, pressure is kept constant either through changing the inflow or the outflow (Fig. 12.6).

336 | Part III: Synthesis of the automatic control systems

PC

PC

P P0 R1

P R2 P1

P0 R 1

R2 P1

Fig. 12.6. The pressure can be controlled through the recipient inflow (a) or outflow (b).

Process characteristics Considering the first situation in Fig. 12.6, the mass balance written for a recipient is Fmi − Fmo =

dm , dt

(12.11)

where Fmi and Fmo are the input and output mass flows. The mass in the recipient is in an isothermal transformation expressed by Tab. 3.1: m = Vρ = V

pM , RT ∘

(12.12)

where: V – the volume of the recipient, ρ – the density of the gas in the recipient, p – the pressure in the recipient, M – the molar mass of the gas, and R – the universal gas constant. The mass flow is expressed through the electric-pneumatic equivalence (valid for laminar flow): u ≡ Δp and i ≡ Fm , by p −p p − p1 Fmi = 0 and Fm2 = , R1 R2 for pneumatic circuits, equivalent with Ohm’s law i=

u V1 − V2 = . R R

Equation (12.11) becomes p0 − p p − p1 M dp − =V ∘ , R1 R2 RT dt

(12.13)

which is a first order differential equation and brought up to the canonic form (4.18), R2 R1 R1 R2 VM dp p + p +p= R1 + R2 RT ∘ dt R1 + R 2 1 R1 + R 2 0 where Tpr =

R1 R2 VM R1 + R2 RT ∘

(if the control is done as in case (a)).

and Kpr =

R2 R1 + R 2

(12.14)

(12.15)

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Beside the capacitive behavior, with time constants of seconds or hundreds of seconds, one may mention that the pressure is an unstable, noisy process due to the turbulence induced by the source of pressure, usually compressors. If there are piston compressors, vibrations and periodic increases and decreases of pressure are characteristic. A pressure diagram is characterized by small quick variations around an average which does not change as fast. The steady state characteristic is linear in the approximation of (12.13).

Characteristics of the other elements in the control loop Transducer The pressure transducer is a first order capacitive element and the steady state characteristic is linear. Control valve The steady state installed characteristic should be linear but they depend on the operating point. The dynamic behavior is considered first order capacitive. Controller The structure of the controller is either P or PI with usual values of PB = 2–50 % and Ti ≈ 10 s–10 min, depending on the time constants in the loop [3]. The derivative action is absent due to the white noise present in the process. Example 12.2. In a recipient (Fig. 12.6) the pressure is controlled using a pressure ACS. The volume of the recipient is V = 200 l, the steady state values of the pressure are: p0 = 2 bar, p = 1 bar, p1 = patm = 0 bar. The temperature in the recipient is T ∘ = 27 ∘ C. The pneumatic resistors have the values of R1 = 1.2108 N/m2 /kg/s and R2 = 1.6108 N/m2 /kg/s. What are the dynamic characteristics of the process? Using equation (12.15), 2

2

−3 3 ⋅ 1.6 ⋅ 108 N/m 1.2 ⋅ 108 N/m kg/s kg/s 200 ⋅ 10 m ⋅ 29 kg/kmol Tpr = = 159 s, 2 2 8314 J/kmolK ⋅ 300K 1.2 ⋅ 108 N/m + 1.6 ⋅ 108 N/m kg/s

1.6 ⋅ Kpr =

kg/s

2 108 N/m kg/s 2

2

1.2 ⋅ 108 N/m + 1.6 ⋅ 108 N/m kg/s kg/s

= 0.57.

The process has a capacitive dynamic behavior.

338 | Part III: Synthesis of the automatic control systems If the control loop contains a transducer and a control valve with the following characteristics, TT = 1 s, TAD = 1 s,

KT = 8 mA/bar KAD = 5 bar/mA,

what are the parameters of a PI controller? Using the QADR method, 360 ( − tan−1 (1 ⋅ ω ) − tan−1 (1 ⋅ ω ) − 10 − tan−1 (159 ⋅ ω )) = −180∘ , 2π the crossover frequency ωosc = 1 rad s−1 meaning that the ACS has an oscillation every 6.3 s. We choose Ti = 120 s: 8 1 5 0.57 K √1 + = 0.5, √1 + 12 ⋅ 12 √1 + 12 ⋅ 12 √1 + 1592 ⋅12 c 1202 ⋅ 12 which results in Kc = 7 or PB = 14 %.

12.3 Level automatic control systems The level is also a significant process variable, mainly because it indicates the mass content of a liquid in a recipient. Usually, the level in the buffer raw material tank of a plant is maintained between certain limits which are designed to assure the autonomy of operation of the plant. But in some other cases, the level has an increased importance. In the CSTRs, the level determines the residence time which is calculated as τres = VF , where V is the volume of the liquid in the reactor and F the flow passing through the reactor. The residence time determines the chemical conversion, e.g. for a first order reaction ξ = (k VF /1 + k VF ), where k is the reaction rate constant (Tab. 3.2). The level (h) is in direct relationship with V, e.g. for cylindrical tank V = Ar h, where Ar is the cross area section of the reservoir. In the case of a distillation column, the bottom and reflux drum content is given by their volumes and it has enhanced importance because it has to enssure the presence of the liquid phase in the two pieces of equipment [4]. The level control (Fig. 12.7) can be assured through the manipulation either of the inflow or the outflow in a recipient.

Process characteristics In the first situation (Fig. 12.7a) the process has integral dynamics if the outflow is kept constant by a pump. If the outflow is free and dependent on the level, the process has capacitive behavior as in the second case (12.7b)

12 Basic control loops in process industries

Fi

| 339

Fi LC

LC Fe=ct

Fe

Fig. 12.7. Level control in a recipient.

Applying the mass conservation law Ar ρ

dh = F i ρ − Fo ρ . dt

(12.16)

The outflow can be expressed from equation (3.16) where the dynamics of the pipeline flow is negligible compared with that of mass accumulation in the tank ρ ghAp − (∝

lp ρ v2 lp ρ v2 ρ v2 +λ ) Ap = ρ ghAp − (∝ +λ ) A = 0, 2 dp 2 dp 2 p

(12.17)

from which v=√

2gh α +λ

and Fo = Ap √

lp dp

2gh l

α + λ dp

.

p

Equation (12.16) becomes Ar

dh 2gh . = Fi − A p √ l dt α + λ dp

(12.18)

p

If we denote = Ap √

2g α +λ

lp dp

, equation (12.18) becomes Ar

dh = Fi − C√h, dt

(12.19)

a differential nonlinear first order equation. The steady state characteristic, h = f (Fi ), h= A2p

1 2g

Fi2

(12.20)

l

α + λ dp

p

is represented in Fig. 12.8. In order to express the dynamic behavior as a capacitive one, which is closer to reality, one has to linearize (12.19), expanding √h using Taylor’s expansion formula, √ √h = √hn + d h |h (h − hn ) = √hn + 1 (h − hn ), dh n 2√hn

(12.21)

340 | Part III: Synthesis of the automatic control systems h [m]

Fi[m3/s]

Fig. 12.8. Nonlinear behavior of the level in the tank.

where hn is the level in the tank at steady state; around this value, the control loop operates the system. Replacing (12.21) in (12.19), the equation becomes 2√hn Ar dh 2√hn Fi − hn , +h= Ap √ 2g lp dt Ap √ 2g lp α +λ α +λ dp

(12.22)

dp

where Tpr =

2√hn Ar Ap √

2g α +λ

[s] and Kpr =

lp dp

2√hn Ap √

2g α +λ

lp dp

[ m3 ] . m [ s ]

(12.23)

There are some other important characteristics of the level as controlled process. The process is characterized by high instability given by stirrer (inside the CSTR), by a boiling process in a steam boiler, or at the bottom of a separation column due to the boiling process in the reboiler. These processes (stirring, boiling) permanently disturb the surface measured. If the transducer is one with a side measuring chamber (Fig. 7.32), the equivalent scheme of functioning is given in Fig. 12.9. Gas

Gas A1 h1

(a)

L1

h2

A2

L2

(b)

Fig. 12.9. Placing the transducer in a lateral chamber induces oscillations.

12 Basic control loops in process industries

|

341

If a pressure disturbance occurs, for example accidentally at the surface of the liquid in the recipient, a mass of liquid is pushed down to the lateral chamber; because of the difference of areas between the two communicating vessels, the level in the chamber increases to a larger extent. After the disappearance of the disturbance, due to the gravity force, the supplementary liquid in the chamber (over the normal point of operation), goes down, provoking another increase of the liquid level in the recipient and the process is alternatively repeated. The level oscillates. Mathematically, this process is expressed in the equations (12.24)–(12.30) (momentum balance for two mechanical systems bound together): ρ gh1 A1 − ρ gh2 A2 − R1 A1 v1 − R2 A2 v2 = M1

dv1 dv + M2 2 , dt dt

(12.24)

where: A1 A2 – the cross area sections of the recipient and chamber; h1 , h2 – the levels with which the normal level value changes in both recipient and chamber; R1 A1 v1 , R2 A2 v2 – the friction forces; 32L μ 32L μ R1 = d21 and R2 = d22 – the hydraulic resistance from the Hagen–Poiseuille law; 1

2

μ – the dynamic viscosity: M1 = L1 A1 ρ ,

M2 = L2 A2 ρ .

(12.25)

Applying the continuity law v 1 A 1 ρ = v2 A 2 ρ

(12.26)

and the equality between the quantities displaced in both recipient and chamber, (h − h1 ) A1 = (h2 − h)A2 ,

(12.27)

equation (12.24) becomes ρ gh (A1 + A2 ) − 2ρ gh2 − (R1 + R2 ) A2 or

d2 h dh2 = (L1 + L2 )A2 ρ 22 dt dt

A + A2 (L1 + L2 ) d2 h2 R1 + R2 dh2 + h, + h2 = 1 2g ρg dt A2 dt2

(12.28)

(12.29)

which describes a damped oscillations behavior (equations (4.27) and (4.28) and Fig. 4.12) with R1 + R 2 2g ωn = √ and ζ = (12.30) . L1 + L2 ρ √2g(L1 + L2 ) Any liquid surface is very unstable even if it is not the case of two communicating vessels. Remember the instability of the water in a tray when we deice the refrigerator and we transport the melted ice (in water form) to the sink.

342 | Part III: Synthesis of the automatic control systems Characteristics of the other elements in the control loop Transducer The level transducer has linear steady state characteristic and the dynamics was discussed in the present section. Control valve The control valve has to have a compensating characteristic, inverse to that in Fig. 12.28. The installed characteristic should be linear. In fact, the “linear” intrinsic characteristic is not linear and looks like a logarithmic one. Because of that when calculating the gain, the formula should be Kcv = aF, which is closer to the reality. Controller The controller should in most cases be a P controller since the level is not such a constrictive variable; in most cases it represents the inventory in a tank [2]. The controller gives smooth changes in flow rates addressed to the downstream units of the process. The usual values for PB are in the range 1 %–100 %. Example 12.3. What are the optimal values of a level controller in a level control loop (Fig. 12.10) with the following characteristics? Fi Gas

LC

Fo

Fig. 12.10. Tank with level control loop.

The tank has Ar = 1 m 2 ,

hn = 0.5 m,

Ap = 3 cm2 ,

α +λ

lp dp

= 1;

from (12.20), we can calculate the nominal flow passing through the tank at steady 3 state, Fn = 3.4 mh .

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The transducer has measuring range: 0–2 m; output signal range: 2–10 mA; capacitive behavior: TT = 5 s. The control valve has electro-pneumatic converter input signal range: 2–10 mA; electro-pneumatic converter output signal range: 0.2–1 bar; the actuator input range: 0.2–1 bar; the actuator output range: 0–100 %; the control valve operates linearly in the flow range 0–8 m3 /h; the control valve stem range is 0–100 %; hn = 50 %; the time constant of the actuating device is TAD = 5 s. The controller has P structure. The time constant and gain of the process, according to (12.23) is Tpr =

2√0.5 ⋅ 1 3 ⋅ 10−4 √ 2⋅9.8 1 2√0.5

Kpr = 3⋅

10−4 √ 2⋅9.8 1

= 0.104 ⋅ 104 s = 0.28 h = 17.4 min

m m m = 1064 m3 = 0.295 m3 = 17.73 m3 . s

h

min

The time constant and gain of the transducer: TT = 10 s = 0.17 min; 10 − 2 mA KT = =4 . 2−0 m The time constant and gain of the actuating device: Tcv = 30 s = 0.5 min m3 h % 1 − 0.2 bar 100 − 0 % m3 /h m3 /min = ⋅ ⋅ 0.68 = 0.14 . 10 − 2 mA 1 − 0.2 bar mA mA

Kcv = aFn = 0.2 ⋅ 3.4 = 0.68 KAD = KEPC ⋅ KAC ⋅ Kcv

The crossover frequency condition 360 (− tan−1 (17.4 ⋅ ωosc ) − tan−1 (0.17 ⋅ ωosc ) + 0 − tan−1 (0.5⋅ωosc )) = −180∘ 2π gives as the result ωosc = 3.2 rad min−1 meaning that the ACS has about 1 oscillation every 2 min.

344 | Part III: Synthesis of the automatic control systems The module condition: 4 0.14 17.73 Kc = 0.5 2 2 2 2 √1 + 17.4 ⋅ 3.2 √1 + 0.08 ⋅ 3.2 √1 + 0.082 ⋅ 3.22 Kc = 2.8;

PB = 35 %.

12.4 Temperature automatic control systems The temperature is the parameter showing the energy content of a system. Temperature control is extremely important, since a “negligence” of +2 ∘ C may have as a consequence a waste of more than 100,000 Euros/year on an industrial platform through the energy spent in vain (Chapter 8). But this is only the economic point of view. Very often, the quality of a product depends severely on the temperature at which it is produced (e.g. a certain sort of PVC, with specified characteristics, needs a reaction temperature of 52 ± 0.5∘ ). Usually, the temperature is controlled through the manipulation of an agent (heating or cooling) flow (Fig. 12.11). Fvi , ρA , cρA , T°i TC T°

T° KTi T°W

TC KTag Fag

T°ag

Fvag

F, ρ, cρ , T° (a)

T°iag

B (b)

Fig. 12.11. Temperature control of a CSTR or at the bottom of a distillation column.

Process characteristics The steady state and dynamic characteristics of the heating process are obtained from the heat conservation law applied to the CSTR system for example. The transfer path from the manipulated variable to the output one (Fag → T ∘ ) (Fig. 12.12) can be decomposed in three capacitive elements (jacket, reactor wall, mass of reaction). Supposing the simplifying assumptions: thermal agent is homogenous in the jacket – lumped parameter system; the temperature of the wall is the average one and the wall is considered a lumped parameter system as well; the wall is considered

12 Basic control loops in process industries

Fag

Jacket

T°ag

T°P

Wall

Mass of reaction

| 345

T°

Fig. 12.12. The capacitive elements in series describing the process behavior in the temperature control loop.

“thermally thin”; CSTR is perfectly mixed – lumped parameter system; the reaction is neither exothermic nor endothermic; there is no heat loss outside the reactor, we can write the equations describing the heat transfer in the reactor for each of the elements in Fig. 12.12. The equations are those are from the Example 3.1: ∘ − F ρ c T ∘ + K A (T ∘ − T ∘ ) = d (V ρ c T ∘ ) Fvag ρag cpag Tiag (12.31) vag ag pag ag Tag T w ag dt j ag pag ag ∘ ) = d (M c T ∘ ) KTi AT (T ∘ − Tw∘ ) − KTag AT (Tw∘ − Tag (12.32) dt w pw w d Fvi ρA cpA Ti∘ − Fρ cp T ∘ − KTi AT (T ∘ − Tw∘ )−K TT ATT (T ∘ − TT∘ ) − VkCA ΔHr = (Vρ cp T ∘ ), dt (12.33)

where ΔHr = 0. Besides these equations, we should write the equation describing the functioning of the thermocouple: d KTT ATT (T ∘ − TT∘ ) − Qel = (MT cpT TT∘ ), dt

(12.34)

where TT∘ , KTT , ATT are the temperature, heat transfer coefficient and heat transfer area of the transducer; Qel is the quantity of thermal energy transformed into electrical energy needed to change the electrical properties of the transducer. Qel is much smaller than the thermal energy involved and thus is negligeble. For the steady state characteristic we notice that it is nonlinear, the nonlinearity being given by ∘ − F ρ c T ∘ + K A (T ∘ − T ∘ ) = 0. Fvag ρag cpag Tiag vag ag pag ag Tag T w ag

(12.35)

The other characteristics, give a linear behavior Fvi ρA cpA Ti∘ − Fρ cp T ∘ − KTi AT (T ∘ − Tw∘ )−K TT ATT (T ∘ − TT∘ ) = 0

(12.36)

(when the heat of reaction is neglected) ∘ ) = 0. KTi AT (T ∘ − Tw∘ ) − KTag AT (Tw∘ − Tag The combined three equations can be assembled in T ∘ = f (Fv ag ),

(12.37)

346 | Part III: Synthesis of the automatic control systems which has a form of the type T∘ =

Ti∘ag aTi∘ + a + 2Fv ag b + F a

v ag

and is given in (9.47). All equations ((12.31)–(12.34)) are first order differential and they could describe a capacitive behavior. With one exception: equation (12.31), where there is one non∘ . The nonlinearity is linearized around the steady state point, (F linearity Fvag Tag vag n , ∘ T ) ag n

∘ =F ∘ ∘ ∘ ∘ Fvag Tag vag n Tag n + Fvag (Tag − Tag n ) + Tag n (Fvag − Fvag n ) Q = F T∘ + T∘ F −F T∘ . vag ag n

ag vag n

(12.38)

vag n ag n

In this way, the two variables are separated and the nonlinearity linearized. Replacing (12.38) in (12.31), we obtain d ∘ (M T ∘ ) + (Fvag n ρag cpag + KTag AT )Tag dt j ag ∘ − T∘ ) + F ∘ ∘ = Fvag ρag cpag (Tiag vag n ρag cpag Tag n + KTag AT Tw . ag n

(12.39)

Trying to obtain the canonic form of the capacity, we obtain ∘ ∘ ∘ = ρag cpag (Tiag − Tag n ) F + Tag + KTag AT dt Fvag n ρag cpag + KTag AT vag ∘ Fvag n ρag cpag Tag KTag AT n + + T∘ . Fva?? n ρag cpag + KTag AT Fvag n ρag cpag + KTag AT w ∘ dTag

Mj cpag Fvag n ρag cpag

(12.40)

The jacket time constant and jacket gain are Tj =

Mj cpag Fvag n ρag cpag + KTag AT

and Kj =

∘ − T∘ ) ρag cpag (Tiag ag n Fvag n ρag cpag + KTag AT

.

(12.41)

Similarly, for the wall, for the inner part of the reactor (mass of reaction) and for the temperature transducer (mainly the sheath) Mw cpw

Tw = Tr =

KTag AT + KTi AT Mr cpr KTi AT + Fρ cp + KTT ATT

KTag AT

and Kw = and Kr =

KTag AT + KTi AT KT i A T KTi AT + Fρ cp + KTT ATT

(12.42)

(12.43)

where Mr = Vρ and cpr = cp , TTw =

MT cpT KTT ATT

and KTw =

KTT ATT = 1. KTT ATT

(12.44)

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Characteristics of the other elements in the control loop Transducer The temperature transducer has linear steady state characteristic and the dynamics was discussed already in the present section. The dynamics depend essentially on the diameter, material and mass of the protecting well. The transducer additionally has the adaptor, which transforms the variation of voltage/resistance into unified signal. The characteristic of the adaptor is linear. The dynamics of these parts of the transducer are negligible. Control valve The control valve has to have a compensating characteristic, inverse to that in Fig. 6.9. The installed characteristic should be logarithmic. Because of that, when calculating the gain the formula should be Kcv = ah. Controller The controller should be PID controller since the heat transfer is a very slow process; the reset (integral) time is of the same order as the maximum time constant in the loop and the derivative time, one fourth of the reset time [2]. The usual values for PB are in the range 5 %–50 %. Example 12.4. Consider a heat transfer process in a CSTR with the following characteristics: Process: reactor volume and heat transfer area: Vr = 1.5 m3 , AT = 6 m2 ; volume of the jacket and heat transfer area: Vj = 0.3 m3 , AT = 6 m2 ; mass and heat capacity of the inner wall of the reactor: Mw = 350 kg, cpw = 0.15 kcal/kg.K; partial heat transfer coefficient heating agentwall: KTag = 500 kcal/m2 K; partial heat transfer coefficient wall-reaction mass: KTr = 1500 kcal/m2 K; ∘ = 90 ∘ C; T ∘ = 78 ∘ C; characteristics of the fluids: ρ = ρag = 1000 kg/m3 ; Tiag agn cpag = cpr = 1 kcal/kg.K; nominal flow rate of the heating agent: Fag n = 3 m3 /h; nominal flow rate of the reactant/product: Fi = Fo = F = 2 m3 /h. Transducer: mass, heat transfer area and heat capacity of the thermowell: MT = 0.5 kg, ATT = 0.02 m2 , cpT = 0.15 kcal/kg.K; partial heat transfer coefficient reaction mass-thermowell: KTT = 1500 kcal/m2 K; measurement range of the adaptor: 40 ∘ C–80 ∘ C; output signal range of the adaptor: 2–10 mA. Control valve: electro-pneumatic converter input signal range: 2–10 mA;

348 | Part III: Synthesis of the automatic control systems electro-pneumatic converter output signal range: 0.2–1 bar; the actuator input range: 0.2–1 bar; the actuator output range: 0–100 %; the control valve operates linearly in the flow range 0–6 m3 /h; Fag n = 3 m3 /h; the control valve stem range is 0–100 %; hn = 50 %; the time constant of the actuating device is TAD = 5 s; the gain of the control valve: Kcv = aF; a = 0.2. The process constants are found by applying equations (12.41)–(12.44): Tj =

0.3 ⋅ 1000 ⋅ 1 = 0.05 h = 3 min 3 ⋅ 1000 ⋅ 1 + 500 ⋅ 6 ∘C 1000 ⋅ 1 ⋅ (90 − 78) and Kj = = 2 m3 3 ⋅ 1000 ⋅ 1 + 500 ⋅ 6 h

350 ⋅ 0.15 Tw = = 0.0044 h = 0.27 min 500 ⋅ 6 + 1500 ⋅ 6 ∘C 500 ⋅ 6 and Kw = = 0.25 ∘ 500 ⋅ 6 + 1500 ⋅ 6 C Tr =

1.5 ⋅ 1000 ⋅ 1 = 0.13 h = 8.2 min 1500 ⋅ 6 + 2 ⋅ 1000 ⋅ 1 + 1500 ⋅ 0.02 and Kr =

∘C 1500 ⋅ 6 = 0.82 ∘ . 1500 ⋅ 6 + 2 ⋅ 1000 ⋅ 1 + 1500 ⋅ 0.02 C

The temperature transducer: 0.5 ⋅ 0.15 = 0.025 h = 1.5 min and KTw = 1 1500 ⋅ 0.02 ∘C mA 8 mA = 0.2 ∘ . KT = KTw Kad = 1 ∘ ⋅ C 80 − 40 ∘ C C

TTw =

∘C ∘C

The control valve: TAD = 5 s KAD = KEPC ⋅ KAC ⋅ Kcv =

1 − 0.2 bar 100 − 0 % m3 /h m3 /h ⋅ ⋅ 0.6 = 7.5 . 10 − 2 mA 1 − 0.2 bar % mA

Controller: The PID controller has the reset time chosen to be the same value as the maximum time constant, that is Tr = 8.2 min. Then, Ti = 8 min. The crossover frequency condition 360 (− tan−1 (3 ⋅ ωosc ) − tan−1 (0.27 ⋅ ωosc ) 2π + 0 − tan−1 (8.2⋅ωosc ) − tan−1 (1.5⋅ωosc ) − tan−1 (0.08⋅ωosc )) = −180∘

12 Basic control loops in process industries

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gives as the result ωosc = 0.5 rad min−1 , meaning that the ACS has about 1 oscillation in 12.5 min. The derivative time constant is calculated from the phase margin condition φc = tan−1 (Td ωosc −

1 ) = +30∘ , Ti ωosc

which, applied to this case, becomes φc = tan−1 (Td 0.5 −

1 ) = +30∘ . 8 ⋅ 0.5

This gives the value for Td = 1.65 min; we choose Td = 2 min. The module condition: 0.25 0.82 0.2 2 √1 + 32 ⋅ 0.52 √1 + 0.272 ⋅ 0.52 √1 + 8.22 ⋅ 0.52 √1 + 1.52 ⋅ 0.52 ⋅

7.5 √1 + 0.082 ⋅ 0.52

Kc √ 1 + (2 ⋅ 0.5 −

2 1 ) = 1.0. 8 ⋅ 0.5

The solution is Kc = 11.2

or PB = 9 %

Ti = 8 min Td = 2 min .

12.5 Composition automatic control systems Since chemical products are the main goal of the chemical industry, composition control seems to be one of the most important of all control systems. As it was demonstrated in the previous sections of this chapter that all controls are equally important because they address different issues of such complex processes as those in process industries (quality, economic, environmental issues). In what follows, the basic problems raised by the composition control systems are presented. One of the most relevant works in this respect is Shinskey’s “Quality control” chapter in [5]. According to our knowledge, but it is also evidenced in [5], one of the crucial issues in composition control is that of mixing [6, 7]. Mixing is not very well approached in the education of chemical engineers, being considered a process of secondary importance, obviously after that of chemical synthesis. But if we look at the huge progress today in enhancing the efficiency of processes, using microreactors for example, we may understand the importance of the process.

350 | Part III: Synthesis of the automatic control systems Process characteristics When we treat chemical reaction control theoretically, we consider that CSTR is a perfectly mixed system, which, in the case of practical quality control is not true: no mixing impeller has the capacity of perfect mixing, transporting one particle from the entrance to the exit in a time equal to 0. The dynamics of the process includes a time constant (Tpr ) and a dead time (τpr ), the ratio between them being decided by the efficiency of the impeller. The process of mixing is schematically described in Fig. 12.13. The considered process is one of diluting a concentrate C with a solvent S.

S

C, x

AC Fs+F

Fs

F, x

Fig. 12.13. Physical mixing process in a CSTR.

The steady state characteristic is obtained by writing the component conservation law under the assumption that there is no solute in the diluting solvent: Cxc + S ⋅ 0 = (S + C)x,

(12.45)

where x is the solute mass fraction in the diluted solution and xc the solute mass fraction in the concentrate. The steady state characteristic is x = f (C), and, further, x=

xc C . x = S+C c 1+ S C

(12.46)

The steady state characteristic is presented in Fig. 12.14. The gain of the process in the nominal operating point (nominal concentrate flow rate Cn ) is Kpr n =

Sxcn Sx Cn dx 1 ]= − = 2cn . | =x [ dC Cn cn S + Cn (S + Cn )2 F (S + Cn )2

(12.47)

The dynamic characteristic, as mentioned before, depends on the agitator’s efficiency The efficiency of the mixing agitator is defined by [5] ηs =

Fa , Fa + F

(12.48)

12 Basic control loops in process industries

|

351

x

xc xn

c

cn

Fig. 12.14. Nonlinear characteristic of a diluting process.

where Fa is the flow entrained by the agitator from the bottom of the reactor back to the top and F the total flow passing through the reactor. A rotating agitator generates high speed streams of liquid, which in turn entrain stagnant or slower moving regions of liquid resulting in uniform mixing [7]. Thus, the higher the efficiency of the agitator is, the higher the value Fs has and ηa has a value closer to 1. If the reactor could have perfect mixing, the process behaves dynamically capacitive, with no dead time: Tpr =

V V = ; F S+C

τpr = 0,

(12.49)

where V is the volume of the reactor, S is the solvent flow for dilution, C is the concentrate flow. If the reactor has no mixing, the process behaves dynamically as an element with dead time: V V τpr = (12.50) = ; Tpr = 0. F S+C The real situation is among the two extremes: the process has both time constant Tpr =

Fa V V ⋅ = ηa ⋅ F Fa + F F

τpr =

V V F ⋅ = (1 − ηa ) . F Fa + F F

(12.51)

Characteristics of the other elements in the control loop Transducer The transducer, if it is a gas chromatograph, presents a dead time equal to the measured component residence time, multiplied with the number of fluxes measured in the process. output signal The gain of the transducer is obviously standard ; because the accuracy of measurement range the measurement is essential in the case of the composition, usually, the measurement range is very narrow and, as a consequence, the gain is very high.

352 | Part III: Synthesis of the automatic control systems Control valve Due to the nonlinear characteristic of the process (Fig. 12.14), the installed characteristic of the valve is logarithmic with the adequate gain KCV = bF (equation (9.38)). The other components of the final control element are quite the same for all control valves. Controller The controller is PID, due to the fact that both mass transfer and mixing in a reactor or vessel are slow processes and the analyzer has a dead time of the order of minutes minimum. Due to the very high gain of the transducer, the controller gain should be very small. The usual values for the controller parameters are: PB = 100–1000 %, Ti = minutes to tens of minutes, Td = minutes.

pH control problem pH is a special variable especially due to its steady state characteristic (Fig. 12.15). The very steep slope of the characteristic induces a cycle of oscillations of the control system. 14 pH 12 10 8 ONa

6

O CH 3C

4 CH3COOH

2

HCI

0 0

0.5

1.0

1.5 mol NaOH mol acid

2.0

Fig. 12.15. pH steady state characteristics for different combinations: acid neutralized with NaOH.

But this is not the only reason why pH control is a really difficult problem to be solved in industry: especially in neutralization plants, the large range of flow rates (ratios from 1 : 1 to 5 : 1) and different composition of the inflows (the effluents from industrial plants are either acid or basic and require two neutralization agents; or, the acid content can vary with 7 orders of magnitude; the titration curve changes with the nature of the system neutralized) has as a result a low quality composition control. These are the reasons why the pH control problem is not treated in this chapter which approaches the most common controls in the process industries.

12 Basic control loops in process industries

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A usual pH control system in a CSTR use the cascade control, the inner loop being that of a ratio flow control (Fig. 12.16). The pH control in a wastewater treatment plant is more complicated and is approached in the second volume of this book.

rFC

Raw material to neutralize

Neutralizing reactant

pHc Linear controller

pH Fig. 12.16. pH control system in a CSTR reactor.

Example 12.5. Consider a dilution process control system in a CSTR (Fig. 12.13), with the following characteristics: Process: reactor volume: Vr = 350 l; nominal flow rate of the dilute: Fi = Fo = F = 70 l/min; nominal concentrate flow rate: Cn = 35 l/min; maximum concentrate flow rate: Cmax = 70 l/min; mass fraction of the concentrate: xcn = 1; efficiency of the mixing agitator: ηa = 95 %. Transducer: measurement range of the analyzer: 0.45–0.55 mass fractions; output signal range of the adaptor: 4–20 mA; dead time of the gas chromatograph: τT = 0.25 min; the time constant of the gas chromatograph: TT = 0.05 min. Control valve: electro-pneumatic converter input signal range: 4–20 mA; electro-pneumatic converter output signal range: 0.2–1 bar;

354 | Part III: Synthesis of the automatic control systems the actuator input range: 0.2–1 bar; the actuator output range: 0–100 %; the control valve operates equally percentage in the flow range 0–70 l/min; Cn = 35 l/min; the time constant of the actuating device is TAD = 3 s; coefficient of the installed characteristic: b = 3. According to the relations (12.51) and (12.47), the process has the characteristics Tpr =

Fa V V 350 ⋅ = ηa ⋅ = 0.95 = 4.75 min F Fa + F F 70

τpr =

V V F 350 ⋅ = (1 − ηa ) = 0.05 = 0.25 min F Fa + F F 70

Kpr n =

Sxcn (S + Cn )2

=

(F − Cn )xcn 35 ⋅ 1 − = = 0.0071 . l/min F2 702

The transducer is a gas chromatograph with τT = 0.25 min TT = 0.05 min 20 − 4 mA = 160 . KT = 0.55–0.45 − The final control element is a valve with electro-pneumatic convertor and actuator with the characteristics: KAD = KEPC ⋅ KAC ⋅ Kcv =

1 − 0.2 bar 100 − 0 % l/min l/min ⋅ ⋅ 3 ⋅ 0.7 = 13.12 and 20 − 4 mA 1 − 0.2 bar % mA

TAD = 3 s = 0.05 min . The crossover frequency condition: 360 (− tan−1 (4.75 ⋅ ωosc ) − (0.25 ⋅ ωosc ) 2π − tan−1 (0.05 ⋅ ωosc ) − (0.25 ⋅ ωosc ) − tan−1 (0.05⋅ωosc )) = −180∘ . The calculation gives as the result ωosc ≈ 2.75 rad min−1 , meaning that the ACS has about 1 oscillation every 2 minutes. We choose Ti = 5 min and Td results from the phase margin condition (equation (11.20)): tan−1 (Td ⋅ 0.6 − Td =

tan 30∘ + 0.6

1 ) = +30∘ 5 ⋅ 0.6

1 5⋅0.6

= 1.5 min .

12 Basic control loops in process industries

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From the module condition, 0.0071 160 1 √1 + 4.752 ⋅ 2.752 √1 + 0.052 ⋅ 2.752 ⋅1

13.12 √1 + 0.052 ⋅ 2.752

Kc √ 1 + (1.5 ⋅ 2.75 −

1 2 ) = 1.0 2.75

results in Kc = 0.21

PB = 476 %

Ti = 5 min Td = 1.5 min .

12.6 Problems (1) The normal liquid flow rate towards a process is 1.5 m3 /h, its limits being 0.5 m3 /h and 3 m3 /h. The flow is subjected to step changes up to 100 l/h. What are the dimensions of the buffer tank placed before the process in order to limit the variations at 10 l/h? The pressure upstream of the buffer is 4 bar and downstream 2 bar. Which dimensions will the buffer have if the pressure difference is 0.5 bar only? The controller gain is chosen in such a way that at a flow rate of 0.5 m3 /h the tank is empty and at 3 m3 /h is full. (2) Calculate the optimal controller parameters for a composition control loop controlling the concentration in a cascade of three CSTR described in the Examples 3.2 and 6.5 (Fig. 12.17). Fs , Cs

Fc , CAc V

V

F

CA1

V

F

CA2

F

CA3

F

AC Fig. 12.17. Composition ACS for a cascade of 3 CSTR.

The pumping flow of the mixing impeller is Fa = 80 l/min, each reactor volume is V = V1 = V2 = V3 = 200 l. The residence time in each reactor is 10 min. The correc-

356 | Part III: Synthesis of the automatic control systems tion of concentration is done adding reactant A (concentrate) with the concentration CAc = 5 kmol/m3 , with the flow rate nominal value of 5 l/min, the nominal input concentration being CAi = 0.8 kmol/m3 . The transducer is a densimeter with the measuring range 0–2.5 kmol/m3 and an output signal 2–10 mA having 0 time lag. The gain of the final control element is KAD = 4.66 l/min/mA and the time constant of the control valve is considered 0. (3) Repeat the calculation for one CSTR and 2 CSTRs in series. (4) Elaborate a MatLab simulation program for Problems (2) and (3).

References [1] Couzinet, A., Gros L., Pierrat, D., Characteristics of centrifugal pumps working in direct or reverse mode: Focus on the unsteady radial thrust, International Journal of Rotating Machinery, 2013, (2013), Article ID 279049. [2] Luyben, W. L., Luyben, M. L., Essentials of Process Control, McGraw-Hill, 1997, ISBN 0-07-114193-6. [3] Seborg, D., Mellichamp, D., Edgar, T., Doyle III, F., Process Dynamics and Control, John Wiley and Sons, 2011, Chapter 12. [4] Szabo, L., Nemeth, S., Szeifert, F., Three-level control of a distillation column, Engineering, 4, (2012), 675–681.

Index abstract oriented system, 19, 20 accuracy, 78, 152, 153, 157, 163, 164, 177–179, 187, 189, 193, 194, 196, 197, 199, 202, 209, 212, 214, 233, 351 accuracy class, 154, 157, 166, 169–172, 181–186, 189, 190, 192, 193, 216, 217, 220, 221 activation energy, 83 actuator, 285, 289 adaptive controller, 246, 247 advanced process control (APC), 6 advanced process control (book), 245, 248 alarm and monitoring system, 283 analytical model, 64, 76 artificial neural network (ANN), 64, 80 automatic control, 3, 122, 124, 125 – system, 122, 125 automatic control system, 293, 330, 331, 335, 338, 344, 349 Barton cell, 167, 168 bio-cybernetic control systems (BCS), 6 Bode diagram, 49, 50, 55, 100, 113 bottom molar flow, 89 Bourdon tube, 163, 164, 166, 167 capacitive level transducer, 192 capacitive pressure transducer, 170 capacitive system, 97, 99, 104, 108 cavitation, 265, 266 centrifugal pump, 125, 262, 330 centrifugal pump head pressure, 330 chemical engineering, 15 chromatograph, 221 Cohen–Coon method, 307–310, 319 cold junction, 161–163 composition control, 349, 352 condenser, 89, 92, 170, 210, 212, 335 conductive level transducers, 194 continuous stirred tank reactor (CSTR), 65, 105 control quality criteria, 297 control valve, 247, 251, 256–258, 352 – ball or V-notch, 257 – butterfly valve, 257 – centrifugal pump, 262 – conveyor belt, 256, 262, 330

– gear pump, 125, 262, 330 – globe valve, 257, 259 – membrane control valve, 261 – notch, 257 – piston pump, 262 – screw pump, 262 – slide/gate control valve, 258, 261 controllability, 140, 141, 148, 229, 297, 308, 323 controller, 227 – digital controller, 227, 229 – on-off controller, 228 – tuning, 139, 295, 300 converging-Input converging-output stability (CICO), 30 conveyor belt, 256, 330 Coriolis mass flow meter, 183 criterion of integral time-weighted absolute error (ITAE), 299 crossover frequency, 241, 244, 343 crystallization, 190 CSTR cascade, 325 damping ratio, 127 decay ratio, 127, 128, 300, 311, 319 delimiting surface, 9 density transducer, 196 derivative system, 97, 101–103 derivative time, 104, 229, 242 deterministic system, 17, 19 differential equation, 21, 36, 87 dissolved oxygen transducer, 207 distillate drum, 89, 90 distillate molar flow, 89 distillation column, 87, 125, 141, 279, 338 distributed control system (DCS), 6, 250, 283 distributed parameter system, 65, 66, 70 Doppler effect, 181, 182 dynamic system, 13, 14, 18 electrical conductivity transducer, 199 electro-pneumatic convertor, 235, 256, 274, 354 elementary signal, 23, 26 energy, 83 energy conservation law, 69, 148

358 | Index error, 80, 124, 125, 128, 139, 143, 152–154, 157, 193, 227, 229, 231, 235, 238, 239, 241, 248, 298, 300, 310 feed flow, 89, 125 feed-forward control, 6, 141 feedback control system, 124, 125, 131, 133, 143 final control element, 256, 262 first order system, 104–106, 115 flow control, 330, 353 flow controller, 229 flow factor (Kv ), 263, 265 – for incompressible fluids, 263 – for gases, 266 – for steam, 267 flow sheet, 64 frequency response, 18, 31–34, 45, 57 gas analyzer, 209 – humidity gas analyzer, 217 – infrared gas analyzer, 209 – paramagnetism-based gas analyzer, 214 – thermal conductivity gas analyzer, 2 – zyrconia oxygen gas analyzer, 216 harmonic signal, 31 heat capacity, 83, 243 heat of reaction, 69, 83, 117 heat transfer area, 70, 160, 254, 335, 345 higher order capacitive system, 110 higher order system, 109 holdup, 89 homogeneous differential equation, 27, 28, 30 hot junction, 161, 162 impulse response matrix, 39, 48 inductive pressure transducer, 171 inferential mass flow meter, 183, 185 infra-red thermometer, 165 input variable, 12, 16, 34 input-output path with logic solver – configuration, 290 – multiplication, 290 input-output relationship, 45, 59, 97, 101 input-output representation, 20, 35, 36 input-state-output relationship, 20, 21, 39, 46, 47 installed characteristic, 270, 272 integral of the absolute error (IAE), 298

integral of the error (IE), 297 integral of the squared error, 248 integral system, 99, 100, 102 integral windup, 241 interlocking system, 281, 290 internal molar reflux, 89 intrinsic characteristic – linear characteristic, 268, 269, 274 – logarithmic (equal percentage) characteristic, 269 – modified parabolic characteristic, 269, 334 inverse response, 323, 324 ionization gauge, 172, 173 kinetic equation, 63 Lambda method, 309 laminar flow, 265 Laplace transform, 39 – of usual functions, 41, 43 – one-sided, 40 – properties, 41, 42 – two-sided, 40 large overshoot, 301 latent heat of vaporization, 74, 89 level control, 84, 230, 338 level controller, 342 level transducers based on floats, 188 linear time-invariant system, 27, 29, 35, 39, 44 linearity, 17, 31, 47, 80 290 looms, 3 lumped parameters system, 65 magnetic flow meter, 180, 181 manual control, 5, 122 mass conservation law, 65, 66, 339 mathematical model, 64, 76, 82 measurand, 151–154 measuring device, 155, 177 method of limit of stability, 305 MIMO system, 133 model predictive control (MPC), 6, 141, 248 molar mass, 83 momentum conservation law, 64, 71, 73 multiple effects evaporator, 141 nonhomogeneous differential equation, 29, 38 nonlinearity, 139, 175

Index | 359

Nyquist diagram, 57 Nyquist stability criterion, 316 object, 8 optimal parametrical control system (OPCS), 6 orifice plate, 174–176, 275, 334 oscillation, 106, 127, 143, 168, 189, 241, 295, 297, 298, 304, 305, 319, 341, 352 oval shaped gear meter, 186 overshoot, 126, 127, 303, 305 Peltier effect, 161, 162 pH meter, 205 PI controller, 238, 241 piezoelectric effect, 171 Pirani gauge, 172, 173 Pitot tube, 177 plug, 256 plug flow reactor (PFR), 67 polarographic cell, 207, 208 positioner, 257, 290 predictive controller, 248 pressure control, 233, 237, 272, 335 pressure controller, 230 pressure drop, 174–176 process control, 3–5, 122, 127, 155, 223, 249, 295 process engineering, 3, 158, 166, 173, 186 proportional band (PB), 231 proportional controller, 231 – gain, 231 proportional system, 97, 99, 202 psychrometric transducer, 219 pure delay system, 113 PVC, 157, 179, 247 quarter decay response criterion, 299 radar level transducer, 193 radiometric level transducers, 193 reaction rate, 22 – constant, 117, 338 reboiler, 256, 340 reflux flow, 325 regression analysis, 76 relative gain array (RGA), 133 relative volatility, 74, 87 resistance temperature detector (RTD), 158, 184 response curve, 307

Reynold’s number, 176, 183, 265 rotameter, 178 routh stability criterion, 312, 313 safety instrumented system, 283 – final control element, 256, 262, 285, 287 – logic solver, 285, 287, 288, 290 – sensors, 284 safety integrity, 290 safety layer, 284 seat ring, 256 second order system, 97, 106–108, 111 – critically damped, 107 – damping factor, 106 – natural period of oscillation, 106 – overdamped, 107 – steady state gain, 106 – underdamped, 107 – unit step response, 106, 107 Seebeck effect, 160–162 semi-xontinuous STR, 92 sensor, 158, 166, 168, 183, 285–287 sinusoidal signal, 26 SISO system, 134 solvate, 74 solvent, 74, 350, 351 stability, 29, 30 state variable, 20, 22, 35, 36 static system, 13 statistical model, 76 steady state, 26, 33 – error (SSE), 127, 139, 233, 237, 238, 302 – gain, 104–106, 135, 137, 332 stem, 4, 235, 256, 257, 289, 290, 343, 348 stem travel, 256, 257, 263, 269, 271, 272, 274 stochastic system, 17 subject, 8 successive trial in the process, 301 system concept, 8 temperature control, 3, 128, 133, 242, 246, 249, 344 thermistor, 158, 164, 183 thermocouple, 160, 162, 163, 165 thermodynamic and kinetic equations, 63 thermodynamic equation, 63 Thomson effect, 161 transducer, 13, 124, 133, 142, 155–158, 163, 167, 172, 174, 175, 177, 179, 185, 189, 191, 194,

360 | Index 196, 197, 199, 202, 206–208, 227, 232, 234, 235, 275, 295, 307, 317, 334, 337, 340, 342, 345–348, 351–354 transfer function, 44, 45 transfer matrix, 48, 49 transfer path, 109, 116, 124, 275, 276, 344 transient response, 33 transient time, 127 transition matrix, 38 transmitter, 155, 157, 159, 160, 164, 165, 168, 170, 177, 179, 181, 188, 192–194, 197, 207, 208, 212, 216, 220, 221, 286, 334 tray, 87, 88, 90 turbulent flow, 181, 265

unit step function, 23

ultrasonic level transducer, 192 unified signal, 156, 347 unit impulse function, 24, 26 unit ramp function, 23

Ziegler–Nichols, 310

vapor flow, 89, 90 variable frequency drive, 262, 330 Venturi tube, 173 viscosity transducer, 198 vortex flow meter, 180 vortex shedding flow, 183 weir, 87, 89, 90, 196, 261 Wheatstone bridge, 159, 169, 172, 184, 195, 205, 206, 213, 214, 216 wise machinery (WM), 6 Woltman turbine meter, 186