This book provides the methods, problems and tools necessary for process control engineering. This comprises process kno
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English Pages 416 [410] Year 2020
Paul Serban Agachi, Mircea Vasile Cristea, and Emmanuel Pax Makhura Basic Process Engineering Control
Also of Interest Process Engineering. Addressing the Gap between Study and Chemical Industry Kleiber, 2020 ISBN 9783110657647, eISBN 9783110657685
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Process Intensification. Breakthrough in Design, Industrial Innovation Practices, and Education Harmsen, Verkerk, 2020 ISBN 9783110657340, eISBN 9783110535365
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Paul Serban Agachi, Mircea Vasile Cristea, and Emmanuel Pax Makhura
Basic Process Engineering Control 
2nd edition
Authors Professor Paul Serban Agachi Botswana International University of Science and Technology Plot no. 10071, Private Bag 16 Palapye Botswana Professor Emeritus at University BabeșBolyai No. 1 Kogălniceanu Str. 400084 ClujNapoca Romania [email protected]
Prof. Dr. Mircea Vasile Cristea University BabesBolyai Faculty of Chemistry and Chemical Engineering Department of Chemical Engineering Str. Arany Janos No. 11 400084 ClujNapoca Romania [email protected] Emmanuel Pax Makhura Private Bag BR364 Gaborone Botswana [email protected]
ISBN 9783110647891 eISBN (PDF) 9783110647938 eISBN (EPUB) 9783110648539 Library of Congress Control Number: 2020934817 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Cover image: Janaka Dharmasena/Getty Images/Hemera Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck 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, https://doi.org/10.1515/9783110647938201
VI  Foreword or, as the case is in Botswana, where the notion of Chemical/Process Engineer is quite new, to those from Ore Processing or Metallurgical Engineering. 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. We may mention here, the food and beverage industry, mining and ore processing, coal and gas industry, materials manufacturing, water treatment industries. 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. The second edition of the book was suggested by our Editor. It is reasonable to add some new information, since the development of the control systems is so fast. Since this decade, driven by social demand and the rapid development of ICT technologies, e. g., big data, machine learning, cloud computing, and the Internet of Things (IoT), industrial automation systems are shifting from centralized systems to CyberPhysical Systems (CPS). This wave of technical innovation is now recognized as the 4th industrial revolution. We now touch these new developments in the chapters of the book. Additionally, we updated the information on new types of transducers, controllers (especially Programmable Logic Controllers), final control elements, introducing SCADA to the readers. Each chapter will benefit now from some problems solved by MATLAB or by LabView.
Foreword  VII
Finally, many thanks to: the editors (Karin Sora, Ria Sengbusch, Anna Bernhard, Ina Talandiene, Frauke Schafft), who, during these hard times of pandemy had the most professional attitude, approach and kindly guided us; our students with whom we cooperated in the elaboration of the first and second edition of the book (Mihai Mogos, Botond Szilagyi, Abhilash Nair, Hoa Pham Thai, for the first edition, and again Mihai MogoșKirner plus Daniel Erich Botha and Emmanuel Pax Makhura, who, in the end, became the third author, for the second edition); our professors who strived to educate us as citizens and specialists; our former or present heads of departments where we have worked or where we work now, for their support; our universities who nurtured us for many years in harder or easier times; our colleagues from abroad who helped us especially after the changes in Romania in 1989; our colleagues at our different homes 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. Palapye, May 2020
Paul S. Agachi Cristea V. Mircea Emmanuel P. Makhura
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  9 System concept  9 System delimitation  10 Input and output variables  12 Classification of the systems  13 The state concept  19 Inputstateoutput relationship  21 Stability of the system  23 Types of elementary signals  24 LTI systems described by inputoutput relationships  27 Time response of the linear timeinvariant systems  28 Solution of the homogeneous differential equation  28 Particular solutions of the nonhomogeneous differential equation  29 General solution of the nonhomogeneous differential equation  30 Stability of the system described by inputoutput relationships  30 Stability of systems described by linear timeinvariant differential equations  31 Frequency response of the system described by inputoutput relationships  32 Frequency response of the system initially at equilibrium  34 Steady state and transient response to the harmonic input  34 LTI systems described by inputstateoutput relationships  35 Transformation of the inputoutput representation into the inputstateoutput representation  36 Solutions of the state equations  38 Solution of the nonhomogeneous state equation  39 Laplace transform  40 Definition of the onesided and twosided Laplace transform  41 Properties of the Laplace transform  42 Laplace transform of usual functions  44
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
X  Contents 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34
Inverse Laplace transform  44 Use of Laplace transform in the analysis of linear timeinvariant systems  45 Use of Laplace transform for describing systems represented by inputoutput relationships  47 Use of Laplace transform for describing systems represented by inputstateoutput relationships  48 The transfer matrix  49 Bode diagrams  50 Nyquist diagrams  59 Problems  63
3 3.1 3.1.1 3.1.2 3.2 3.3 3.4 3.5
Mathematical modeling  67 Analytical models  68 The conservation laws  68 Thermodynamics and kinetics of the process systems  78 Statistical models  78 Artificial neural network Models  85 Examples of mathematical models  87 Problems  99
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9
Systems dynamics  105 Proportional system  105 Integral system  107 Derivative system  109 First order system  112 Second order system  114 Higher order system  117 Pure delay system  121 Equivalence to first order with time delay system  122 Problems  130
5 5.1 5.2 5.3 5.4
Manual and automatic control  135 Manual control  135 Automatic control  137 Steady state and dynamics of the control systems  138 Stability and instability of controlled process and control systems  139 Performance of the control system  140 Problems  141
5.5 5.6
Contents  XI
Part II: Analysis of the feedback control system 6 6.1 6.2 6.3
The controlled process  145 Steady state behavior of the controlled process  145 Dynamic behavior of the controlled process  153 Problems  161
7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9
Transducers and measuring systems  165 Introduction  165 Measuring systems in process engineering  169 General characteristics of the transducers  170 Temperature transducers  172 Pressure transducers  180 Flow transducers  187 Level transducers  204 Composition transducers  213 Problems  243
8 8.1 8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5 8.2.6 8.3
Controllers  247 Classification of controllers  247 Classical control algorithms  250 Proportional controller (P)  250 ProportionalIntegral controller (PI)  258 ProportionalIntegralDerivative controller (PID)  262 Controllers with special functions  265 Distributed Control Systems and Supervisory Control and Data Acquisition Systems  270 Programmable Logic Controllers (PLC)  277 Problems  287
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
Final control elements (actuating devices)  291 Types of final control elements  291 Control valves  291 Other types of final control elements  298 Sizing the control valve  299 The flow factor (Kv ) for incompressible fluids  300 The flow factor (Kv ) for gases  303 The flow factor (Kv ) for steam  304 Inherent characteristics of control valves  305 Installed characteristics of the control valves  307 The dynamic characteristics of a control valve  311 The gain of the control valve  311
XII  Contents 9.5.2 9.6 9.7 10 10.1 10.2 10.3 10.4 10.5
The dynamics of the control valve  311 Sizing and choice of the control valves  312 Problems  317 Safety interlock systems  319 Introduction  319 Safety layers  320 Alarm and monitoring system  322 Safety instrumented systems  323 Problems  328
Part III: Synthesis of the automatic control systems 11 Design and tuning of the controllers  333 11.1 Oscillations in the control loop  333 11.2 Control quality criteria  335 11.3 Parameter influence on the quality of the control loop  338 11.4 Controller tuning methods  342 11.4.1 Experimental methods of tuning controller parameters  342 11.5 Tuning controllers for some “difficult to be controlled” processes  361 11.6 Problems  366 12 12.1 12.2 12.3 12.4 12.5 12.6
Basic control loops in process industries  371 Flow automatic control systems  371 Pressure automatic control systems  376 Level automatic control systems  379 Temperature automatic control systems  385 Composition automatic control systems  391 Problems  397
Index  401
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 First 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. But, before controlling one has to measure. The instrumentation, at that time, was confined to measuring steam pressure, water flows, water level in steam drum [1]. 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 of 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 [2, 3] (Figure 1.1), T. Polzunov’s first level controller for his steam engine (1765), J. Watt’s fly ball governor in 1768 [4, 5] – a speed regulator for his rotary steam engine – (Figure 1.2). The first obviously advanced combination between process engineering and process control was H. Jacquard’s loom in 1801 (Figure 1.3) which stored the model of the silk fabric on punch cards [6, 7]. 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) [4]. The Second Industrial Revolution started with mass production in the automotive industry (Beginning of the 20th century). It still required mostly manual https://doi.org/10.1515/9783110647938001
4  1 History of process control
Figure 1.1: Lee’s control positioning system for a windmill.
Figure 1.2: The speed controller for Watt’s steam engine.
labour and few processes were mechanized. 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 [8–12]. A major innovation is represented by G. Philbrick’s “Automatic control analyzer – Polyphemus”, actually the first analogue computer (1937–1938) just before World War II [13]. World War II brought extremely important innovations in the field of automatic control: automatic rudder steering, automatic gun positioning systems,
1 History of process control  5
Figure 1.3: H. Jaquard’s loom with punch cards.
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! [14] Immediately after World War II, the field of process control exploded. It was helped by the construction of the megacomputers ENIAC (1946) [15] and UNIVAC (1951) [16], Shockley’s patent on the transistor (1950) [17] and Feynman’s premonition regarding nanospace expressed at the American Physics Society Meeting in Caltech “There’s plenty of room at the bottom” (1959) [18]. The Third Industrial Revolution started in the ’60s with the development of electronics. 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 RW300 computer from the RamoWooldridge Company [19] 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 [20]. 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 (’50s–’70s), 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 (’70s–present) dominated, as the main feature, by control systems involving a micro or minicomputer; most processes
6  1 History of process control 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. With this third stage mentioned, the Process Instrumentation and Control entered in the Fourth Industrial Revolution, the “digitalization” revolution. “As digital has taken over from analogue, computer power has increased and reduced in size and cost, signal processing and conditioning has become more effective. Universities continue to develop new ways to process signals and remove noise. Oxford University is in the process of licensing its PRISM technology that is a major leap in signal processing capabilities only possible with the huge processing power now available.” [1, 21] 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 inside the controller or even computer, allowing more complicated control algorithms: optimal, adaptive, fuzzy, ratio, inferential, feedforward 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 integerdiscrete 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 [22]. 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 biocybernetic control systems (BCS) which are principally a novel and industry evaluated decisionmaking system (Figure 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, the civilization of the Artificial Intelligence (AI). As it was pointed out, Artificial Intelligence (AI) is unstoppable at now. ABI Research, a global tech market advisory firm, forecasts that the total installed base of AIenabled devices in industrial manufacturing will reach the figure of 15.4 million
References  7
Figure 1.4: Wise Machinery.
in 2024, with a Compound Annual Growth Rate (CAGR) of 64.8 % from 2019 to 2024 [23]. As we stated before, it is a new civilization in itself, and, at this point, we should not forget the opinion in this regard of one of the most brilliant minds of the humanity, Stephen Hawkins: “The development of full artificial intelligence could spell the end of the human race. . . It would take off on its own, and redesign itself at an everincreasing rate. Humans, who are limited by slow biological evolution, couldn’t compete and would be superseded”. (Interview with the BBC, December 2014). Nowadays, the global process automation and instrumentation market is estimated at USD 71.4 billion in 2019 and is anticipated to register a CAGR of 6 % to reach USD 95.5 billion by 2024 and . . . the lion’s share (almost 50 %) is the Chemical/Materials Industries [24]. We doubt that the slow down of the economy because of the unexpected COVID19 will stop the trend. Maybe it will delay it for a while only.
References [1]
S. Amos, A brief history & future of process instrumentation, Process Industry Informer, September 23, 2017. [2] Mayr, O., The origins of feedback control, Sci. Am., 223 (4), (October), (1970), 110–118. [3] Mayr, O., Feedback Mechanisms in the Historical Collections of the National Museum of History and Technology, Smithsonian Institution Press, 1971. [4] Maxwell, J. C., On governors, Proc. R. Soc. Lond., 16, (1868), 270–283 [This is presumably the first scientific article on feedback control]. [5] Centrifugal governor, http://en.wikipedia.org/wiki/Centrifugal_governor. [6] Eclectica, Jacquard’s loom and the stored program concept, http://addiator.blogspot.ro/2011/ 10/jacquardsloomandstoredprogramme.html. [7] Essinger, J., Jacquard’s Web: How a Handloom led to the Birth of the Information Age, New Ed., Oxford University Press, 2007. [8] 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]. [9] Nyquist, H., Regeneration theory, Bell Syst. Tech. 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]. [10] 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]. [11] 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
8  1 History of process control
[12] [13] [14]
[15]
[16]
[17] [18] [19] [20]
[21]
[22]
[23]
[24]
time the “open cycle” (without feedback) from the “closed cycle” (with feedback) which are nowadays called openloop and closedloop control]. Ivanoff A., Theoretical foundation of the automatic regulation of temperature, J. Inst. 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 Ann., 4:2, (1982), 143. Ziegler J. G., Nichols N. B., Optimum settings for automatic controllers, Trans. Am. Soc. Mech. Eng., 64, (1942), 759 [The first paper approaching the controller closedloop 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 SpringerVerlag, 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 Eng. Sci., 23:5, (1960), 22–36 [The conference presents the vision of nanospace and nanotechnology]. Astrom, K., Wittenmark, B., Computer Controller Systems: Theory and Design, PrenticeHall 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]. M. Henry, O. Yu. Bushuev, Olga L. Ibryaeva, Prism Signal Processing for Sensor Condition Monitoring https://ora.ox.ac.uk/objects/uuid:2465f6beabd84e9a9e79d5f30084872e, IEEE International Symposium on Industrial Electronics, Edinburgh, 17–21 June, 2017, pp. 1404–1411. 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 postindustrial world, November 30, 2004, Ashdod, Israel, Conference paper. ABI Research, Industrial manufacturing sector will have an installed base of over 15 Million AIEnabled Devices in 2024, ABI Research’s Industrial AI Market Tracker market data report, 2017. “Process Automation & Instrumentation Market by Instrument (Field Instrument, Control Valve, Analytical Instrument), Solution (APC, DCS, HMI, MES, PLC, Safety Automation, SCADA), Industry, and Geography – Global Forecast to 2024”, ResearchAndMarkets.com, 01 August, 2019.
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 welldefined 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 system type, 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 Figure 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 new properties to the ensemble (system). The connection mode between entities is dehttps://doi.org/10.1515/9783110647938002
10  2 Basics of systems theory
Figure 2.1: System formed by three interacting entities, each having interdependent attributes.
cisive 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 it 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 [18, 19].
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 Figure 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
2.2 System delimitation
 11
Figure 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 Figure 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, Figure 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.
Figure 2.3: Interface delimitation Σ1 –Σ2 of the system represented by a heat exchanger with heating coil (textured represented volume).
12  2 Basics of systems theory
Figure 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
2.4 Classification of the systems  13
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 Figure 2.5.
Figure 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 vector
14  2 Basics of systems theory of inputs u(t ±Δt) and/or outputs y(t ±Δt), from neighboring moments ((t −Δt) for nonanticipative 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 physically real), 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 nontemporal 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, Figure 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
Figure 2.6: Timedependent level H(t) change in a tank.
2.4 Classification of the systems  15
distance from 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 crosssection area 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−1 u du dm u +q +⋅ ⋅ ⋅+q +q ⋅y = p +p +⋅ ⋅ ⋅+p1 +p0 ⋅u. n−1 1 0 m m−1 dt n dt dt m dt dt n−1 dt m−1
(2.8)
16  2 Basics of systems theory Depending on the ordering relationship between the maximal differentiation order of 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 (nonfeasible) 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 backmixing 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.
2.4 Classification of the systems  17
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. They are mathematically represented by total (ordinary) differential equations. The dead time is represented, from the mathematical point of view, by functions having the timeindependent 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, Figure 2.7.
Figure 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) = Fmb (t − τ) dt
τ=
L 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
18  2 Basics of systems theory 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 inputoutput 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 causeeffect 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 timedependent coefficients for the nonstationary 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 nonuniform 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).
2.5 The state concept 
19
Linear dynamic systems are mathematically described by differential equations with constant coefficients for the linear timeinvariant (LTI) systems and by differential equations with timedependent coefficients for linear nonstationary systems. The properties of effects superposition and of causeeffect 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 quasistationary 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 quasistationary 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 Figure 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 the system behavior. A fundamental limitation of the system description using inputoutput relationships is the
20  2 Basics of systems theory
Figure 2.8: Terminal and suppressed variables for a continuous stirred tank reactor.
fact that 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 inputoutput 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 far) in the past. 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
2.6 Inputstateoutput relationship
 21
t = t0 in order to be able 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 inputoutput relationship it corresponds a set of possible inputstateoutput representations, while only one unique inputoutput representation corresponds to one inputstateoutput representation. In conclusion, there are two main ways of representing the system: one uses the inputoutput relationship (I) and the other the inputstateoutput relationship (II): I II
R(u, y) = 0, y = RS(x 0 , 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.
2.6 Inputstateoutput relationship The inputstateoutput 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)
22  2 Basics of systems theory 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 inputstateoutput 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 inputstateoutput 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 timedependent systems). An abstract, oriented and dynamic singleinputsingleoutput (SISO) system has an inputoutput description of the form presented in equation (2.8) and is represented by a high order differential equation. For the general case of a multipleinputmultiple output (MIMO) dynamic system, the description consists in a system of high order differential equations. The system has also an inputstateoutput 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 inputoutput relationship. It may be concluded that the way of describing the behavior of the system using inputstateoutput relations is more complete than the inputoutput 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 inputstateoutput relationship offers the possibility of getting a better understanding of the system and consequently, being in the position of maintaining control 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
 23
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 and then disappearance of a small (force) disturbance: (i) stable equilibrium, if the mechanical body returns to its initial position after the application and then disappearance of a small disturbance, (ii) unstable equilibrium, if the mechanical body tends to get a different position after the application and then disappearance 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 and then disappearance 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 BoundedInput BoundedOutput Stability (BIBO stability). Definition. A timecontinuous system is denoted as stable in the boundedinput boundedoutput 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 by 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, determines 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.
24  2 Basics of systems theory
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 timedependent signal. In systems theory, signals depending on frequency, as 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 at 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 Figure 2.9.
Figure 2.9: Unit step elementary signal.
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 v0 (t) = {
0 for t < 0 t for t ≥ 0
(2.16)
and has the representation shown in Figure 2.10. 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
2.8 Types of elementary signals  25
Figure 2.10: Unit ramp elementary signal.
of the unit step function: dv (t) u0 (t) = 0 , dt
t
v0 (t) = ∫ u0 (τ)dτ.
(2.17)
−∞
(3) Other elementary functions may be defined by the general function form v(t) =
t n−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 Figure 2.11.
Figure 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.
ε→0
(2.19)
By the differentiation of the v(t) function, with ε =0, ̸ the δ0 (t) function may be obtained: δ0 (t) =
dv(t) . dt
(2.20)
Its representation is shown in Figure 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 duration (approaching zero) and an infinite amplitude, but having a finite and equal to
26  2 Basics of systems theory
Figure 2.12: Signal δ0 (t) having the form of a unit impulse with finite duration and amplitude.
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 Figure 2.13.
Figure 2.13: The elementary unit impulse δ(t) also named the Dirac impulse.
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
−∞
t0 −ε
t0 −ε
∫ f (t) ⋅ δ(t − t0 )dt = ∫ f (t) ⋅ δ(t − t0 )dt = f (t0 ) ∫ δ (t − t0 )dt = f (t0 ).
(2.25)
2.9 LTI systems described by inputoutput relationships  27
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 Figure 2.14.
Figure 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 inputoutput relationships Definition. A system described by the inputoutput 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) dn y(t) du(t) dm u(t) ,..., , u(t), , . . . , , t) = 0, dt dt n dt dt m
(2.27)
for t ∈ T, T = [t0 , ∞], with n and m positive integer values and with f an application, f : C n+m+2 × T → C.
28  2 Basics of systems theory
2.10 Time response of the linear timeinvariant systems Linear timeinvariant (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 + q0 ⋅ y(t) + ⋅ ⋅ ⋅ + q1 n−1 dt n dt dt n−1 du(t) dm−1 u(t) dm u(t) + ⋅ ⋅ ⋅ + p1 + pm−1 + p0 ⋅ u(t), = pm m dt dt dt m−1
(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 halfinfinite 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 ⋅ λn
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 timeinvariant 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 qn
dn y(t) dn−1 y(t) dy(t) + qn−1 + q0 ⋅ y(t) = 0. + ⋅ ⋅ ⋅ + q1 n dt dt dt n−1
(2.31)
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
2.12 Particular solutions of the nonhomogeneous differential equation
 29
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 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 yi = t i ⋅ eλt ,
mi
i = 0, 1, . . . , mi − 1;
∑ mk = n,
k=1
t ∈ T+ or T∞ .
(2.32)
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}. i
(2.33)
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 = t i ⋅ eσ ⋅t ⋅ cos(ω ⋅ t), yi = t i ⋅ 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 having 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.
30  2 Basics of systems theory
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}. i
(2.36)
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 inputoutput relationships Stability will be investigated in the sense of boundedinput boundedoutput stability (BIBO stability). Consider the linear timeinvariant 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.
2.15 Stability of systems described by linear timeinvariant differential equations  31
Definition. The system described by a linear and timeinvariant 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 converginginput convergingoutput (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 timeinvariant differential equations Taking into account that the response of the system described by the equation (2.28) is y = h ∗ u + ∑ αi ⋅ yi i
(2.38)
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 timeinvariant 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.
32  2 Basics of systems theory
2.16 Frequency response of the system described by inputoutput 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 ⋅ ejωt .
(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 ⋅ ejωt = υ ⋅ ejφ ⋅ ejωt = υ ⋅ cos(ωt + φ) + j ⋅ υ ⋅ sin(ωt + φ).
(2.40)
The real harmonic signal has the form c(t) = Re[x(t)] = υ ⋅ cos(ωt + φ) = υ ⋅ sin (ωt + φ +
π ). 2
(2.41)
The frequency f is related to the (angular) frequency ω by the wellknown 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) = ejωt ,
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)
2.16 Frequency response of the system described by inputoutput relationships  33
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) ⋅ ejωt
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(ωt + φ) = Re[a ⋅ ejωt ].
(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 ⋅ ejωt ],
(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).
34  2 Basics of systems theory
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 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 t k ⋅ 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 nth 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) ⋅ ejωt = P(jω) ⋅ ejωt ,
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 ⋅ ejωt ,
t ∈ T,
(2.52)
2.19 LTI systems described by inputstateoutput relationships  35
or to a real harmonic input u(t) = a0 ⋅ cos(ωt + φ),
t ∈ T,
(2.53)
where T is the infinite or halfinfinite 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 ⋅ ejωt ,
t ∈ T,
(2.55)
for the complex harmonic input, and by ysteadystate (t) = hω (ω) ⋅ a0 ⋅ cos(ωt + 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 inputstateoutput relationships The description of the system’s behavior using inputstateoutput relations reveals not only the external behavior of the system (directly described by the inputoutput 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
36  2 Basics of systems theory and the future t > t0 by specifying the information necessary at moment t = t0 , 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 inputoutput representation into the inputstateoutput representation Consider the linear timeinvariant 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 inputoutput relationship it may be assigned multiple inputstateoutput representations, but a single inputstateoutput representation has only a single (unique) associated inputoutput representation. From the set of the inputstateoutput 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 inputstateoutput equations: ẋ1 = x2 ẋ2 = x3 .. . ẋn−1 = xn ẋn = −q0 ⋅ x1 − q1 ⋅ x2 − ⋅ ⋅ ⋅ − qn−2 ⋅ xn−1 − qn−1 ⋅ xn + u y=
(p0 − pn ⋅ q0 ) ⋅ x1 + (p1 − pn ⋅ q1 ) ⋅ x2 + (p2 − pn ⋅ q2 ) ⋅ x3 + ⋅ ⋅ ⋅ + (pn−1 − pn ⋅ qn−1 ) ⋅ xn + pn ⋅ u.
(2.58)
(2.59)
2.20 Transformation of the inputoutput into the inputstateoutput representation
 37
These equations are equivalent to the set of linear first order differential equations: 0 ẋ1 [ ẋ ] [ 0 [ 2 ] [ ] [ [ [ . ] [ . ] [ [ [ . ]=[ . ] [ [ [ . ] [ . ] [ [ ] [ [ [ ẋn−1 ] [ 0 [ ẋn ] [ −q0
0 x1 1 0 . . . 0 ] [ ] [ 0 1 . . . 0 ] ] [ x2 ] [ 0 ] ] [ ] ] [ . . . . . . ] [ . ] [.] ] [ ] ] [ ] [ ] [ . . . . . . ] ]⋅[ . ]+[ . ]⋅u [ ] [ ] . . . . . . ] [ . ] ] [.] ] [ ] ] [ 0 0 . . . 1 ] [ xn−1 ] [ 0 ] −q1 −q2 . . . −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, ] [ [x ] [ n−1 ] [ xn ] (2.61) where the matrices A, B, C, and D have the form 0 0 . . . 0 [ −q0
[ [ [ [ [ A=[ [ [ [ [ [
1 0 . . . 0 0 1 . . . 0 ] ] ] . . . . . . ] ] . . . . . . ] ], . . . . . . ] ] ] 0 0 . . . 1 ] −q1 −q2 . . . −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 inputstateoutput 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 inputoutput 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 inputstateoutput 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 inputoutput point of view. Compared to the inputoutput 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.
38  2 Basics of systems theory The direct consequence of this equivalence is the fact that any set of high order inputoutput differential equations may be transformed into a set of first order state 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 inputstateoutput representation to the modal one may be performed on the basis of the transformation matrix consisting in the eigenvectors of matrix A. The modal inputstatespace 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 quasidiagonal 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 ndimensional space of real or complex numbers Σ ∈ Rn or Σ ∈ C n and the input and the output functions are also defined on real or complex spaces, Su ∈ Rk or Su ∈ C k , respectively Sy ∈ Rm or Sy ∈ C m . 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 inputstateoutput 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 ∈ C k , Σ = C n , Sy ∈ C m . 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.
2.22 Solution of the nonhomogeneous state equation
 39
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) = y zeroinput + y zerostate , having the form y zeroinput (t) = C(t) ⋅ Φ(t, t0 ) ⋅ x(t0 ),
(2.70)
y zerostate (t) = ∫ C(t) ⋅ Φ(t, τ) ⋅ B(τ) ⋅ u(τ) ⋅ dτ + D(t) ⋅ u(t).
(2.71)
t
t0
The term y zeroinput is named zeroinput response and represents the response of the system having the input equal to zero but nonzero initial conditions. The term y zerostate is named zerostate 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 timeinvariant systems the two components have the following simplified form: y zeroinput (t) = C ⋅ eA⋅t x(0), t
t0 = 0,
y zerostate (t) = ∫ C ⋅ eA(t−τ) ⋅ B ⋅ u(τ) ⋅ dτ + D ⋅ u(t). 0
(2.72) (2.73)
40  2 Basics of systems theory According to the convolution operation, the response to the zero initial conditions (zerostate) may be formulated as t
y zerostate (t) = ∫ h(t − τ) ⋅ u(τ)dτ,
(2.74)
t0
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 ith component of the system output having zero initial conditions to the jth 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 inputoutput or inputstateoutput relationships, generally called the system behavior (analysis) in the time domain, has been presented. The methods for determining the response for the linear timeinvariant 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 resourceconsuming 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 timedependent signal may be decomposed in a linear combination of harmonic (complex) functions having increasing frequencies (Fourier series); 2. the response of the linear timeinvariant system, to a harmonic (complex) input, is equal to that input multiplied by a factor that is described by the frequency response function.
2.24 Definition of the onesided and twosided Laplace transform
 41
Consequently, the steps for determining the response of the linear timeinvariant 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 causeeffect proportionality; 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 onesided and twosided Laplace transform Although the onesided Laplace transform is predominantly used for linear timeinvariant systems description, the twosided Laplace transform will be introduced first in order to get a comprehensive overview of this mathematical instrument [14]: Definition. The twosided 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 twosided 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 onesided 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, 0
s ∈ E+ .
(2.77)
42  2 Basics of systems theory The set E+ ⊂ C, is named the region (domain) of existence of the onesided 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 onesided 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 X+ (s). There is a close relationship between the onesided 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 t < 0 for t ≥ 0,
(2.78)
then the onesided and twosided Laplace transforms are identical L+ (x) = L(x), having identical domains of definition E+ = E. This is the case in the majority applications using the Laplace transform. The definition of the onesided Laplace transform refers to the integration domain between 0 and +∞, provided that the function x does not include a singularity at the moment of time t = 0. If the function x has a singularity at the initial moment of time, for example the case of a signal x(t) containing Dirac functions δ(t), it is assumed that integration is stared from the moment t = 0− . Making a parallel between the onesided and the twosided Laplace transform the following remarks may be performed: 1. the twosided Laplace transform is a generalization of the onesided Laplace transform, the latter being extensively used due to historical reasons; 2. the onesided Laplace transform is suitable for causal systems having zero initial conditions, the causality being introduced for the twosided Laplace transform by the multiplication with the unit step function u0 (t); 3. when using the onesided Laplace transform no specifications are made for the time interval t < 0, while the twosided Laplace transform is applied for t ∈ R; 4. left time shifting may be problematic for the onesided Laplace transform, while for the twosided Laplace transform this shifting is straightforward.
2.25 Properties of the Laplace transform A summary of the Laplace transform properties is presented in Table 2.1 [14].
{s ∈ Ex  Re(s) > 0} Ex
X (s − a) s ⋅ X (s) X (s) s dX (s) ds 1 X ( αs ) α 1 X ( αs ) α
x(t + θ), θ ∈ R x(t − θ) ⋅ u0 (t − θ), θ ∈ R, θ ≥ 0
eat ⋅ x(t), a ∈ C
dx(t) dt t ∫−∞ x(τ)dτ t ∫0− x(τ)dτ
−t ⋅ x(t)
x(αt), α ∈ R, α ≠ 0 x(αt), α ∈ R, α > 0
Time shifting
sshifting
Time differentiation
sdifferentiation
Scaling
s→∞
{s ∈ C  s/α ∈ Ex+ }
Ex+
{s ∈ Ex+  Re(s) > 0}
Ex+
{s ∈ C  s − a ∈ Ex+ }
Ex+
Ex+ ∩ Ey+
t→∞
s→0
lim x(t) = lim sX+ (s)
dX+ (s) ds 1 X (s) α + α
X+ (s) s
−
s ⋅ X+ (s) − x(0 )
X+ (s − a)
e−s θ ⋅ X+ (s)
X (s)+ ⋅ Y (s)+
Final value theorem
{s ∈ Cs/α ∈ Ex } {s ∈ C  s/α ∈ Ex }
{s ∈ C  s − a ∈ Ex }
Ex Ex+
Ex ∩ Ey Ex+ ∩ Ey+
Ex+ ∩ Ey+
Region of existence E+
x(0+ ) = lim sX+ (s)
X (s) ⋅ Y (s) X (s)+ ⋅ Y (s)+
αX+ (s) + βY+ (s)
Onesided Laplace transform s ∈ E+
Initial value theorem
Integration
Ex
es θ ⋅ X (s) e−s θ ⋅ X+ (s)
(x ∗ y)(t) x ⋅ u0 (t) ∗ y ⋅ u0 (t)
Ex ∩ Ey
Convolution
αX (s) + βY (s)
(αx + βy)(t), α, β ∈ C
Region of existence E
Linearity
Twosided Laplace transform s∈E
Time signal t ∈R
Property
Table 2.1: Properties of the Laplace transform.
2.25 Properties of the Laplace transform  43
44  2 Basics of systems theory
2.26 Laplace transform of usual functions The Laplace transforms of a set of usual functions are presented in Table 2.2 [14].
Table 2.2: Laplace transform of usual functions. Time signal t ∈R
Twosided Laplace transform
Onesided Laplace transform
Conditions
eat ⋅ u0 (t) −eat ⋅ u0 (−t) eat
1 s−a 1 s−a
1 s−a
a∈C a∈C a∈C
1 u0 (t) δ(t) δ(k) (t)
t k−1 u (t) (k−1)! 0 t k−1 ⋅eat u (t) (k−1)! 0 t k−1 ⋅eat − (k−1)! u0 (−t)
cos(ωt) ⋅ u0 (t) sin(ωt) ⋅ u0 (t)
eat ⋅ cos(ωt) ⋅ u0 (t) eat ⋅ sin(ωt) ⋅ u0 (t)
Re(s) > Re(a) Re(s) < Re(a) —
— — 1 s
1 sk 1 sk
1 s−a 1 s 1 s
— Re(s) > 0 s∈C s∈C Re(s) > 0
1 (s−a)k 1 (s−a)k
1 sk 1 sk
Re(s) > Re(a) Re(s) < Re(a)
s (s2 +ω2 ) ω (s2 +ω2 ) s−a (s−a)2 +ω2 ω (s−a)2 +ω2
Re(s) > Re(a) a∈C Re(s) > Re(a)
0
Re(s) > 0 Re(s) > 0 Re(s) > a Re(s) > a
Re(s) > 0 Re(s) > 0 s∈C s∈C Re(s) > 0
1 (s−a)k
0
Re(s) > Re(a) s∈C
s (s2 +ω2 ) ω (s2 +ω2 ) s−a (s−a)2 +ω2 ω (s−a)2 +ω2
Re(s) > 0 Re(s) > 0 Re(s) > a Re(s) > a
k∈N k∈N
a ∈ C, k ∈ N a ∈ C, k ∈ N ω∈R ω∈R a, ω ∈ R a, ω ∈ R
2.27 Inverse Laplace transform As mentioned in the definition of the Laplace transform, its independent variable is complex and may be formulated by s = σ + jω, where σ, ω ∈ R. (A) Definition. Consider X, the twosided 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∞
x(t) =
1 ∫ 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.
2.28 Use of Laplace transform in the analysis of linear timeinvariant systems  45
(B) Definition. Application of the inverse Laplace transform to the onesided 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 inputstateoutput relationships having zero initial conditions x(0) = 0. The Laplace transform of the system response Y (s), starting
50  2 Basics of systems theory 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 ∩ Eu . The transfer matrix is the element by element onesided or twosided 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)
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 jth component that is an exponential signal uj = aj ⋅ est , the response of the ith component of the output vector yi (t) is equal to the Hij element of the transfer matrix multiplied by the jth 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.
2.32 Bode diagrams 
51
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 Figure 2.15. This positioning allows the correlation between values of the magnitude and the phase at the same frequency lg(ω).
Figure 2.15: Form and axis significance of the Bode diagrams of magnitude and phase.
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 general form n
H(jω) =
n
m l K ⋅ ∏m=1 (1 + jωTm ) ⋅ ∏l=1 [ ω12 (jω)2 +
ni (jω)p ⋅ ∏i=1 (1
+ jωTi ) ⋅
nk ∏k=1
l
2ξl (jω) ωl
[ ω12 (jω)2 + k
+ 1]
2ξk (jω) ωk
+ 1]
,
(2.105)
52  2 Basics of systems theory 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: H1 (jω)dB = 20 lg
K = 20 lg K − 20 lg ωp = 20 lg K − 20p lg ω. ωp
(2.106)
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 Figure 2.16.
Figure 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.
The phase of this factor is π φ(H1 (jω)) = −p . 2
(2.107)
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 Figure 2.17.
2.32 Bode diagrams 
53
Figure 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 + jωTi ), having the Bode magnitude determined as follows: H2 (jω) =
1 − jω ⋅ Ti ω ⋅ Ti 1 1 = = −j 1 + jω ⋅ Ti 1 + ω2 ⋅ Ti2 1 + ω2 ⋅ Ti2 1 + ω2 ⋅ Ti2 2
2
ω ⋅ Ti 1 1 ) +( ) = 2 2 1 + ω ⋅ Ti 1 + ω2 ⋅ Ti2 √1 + ω2 ⋅ Ti2 1 = 20 lg = −10 lg(1 + ω2 ⋅ 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 ω2 ≪ (1/Ti )2 and consequently, ω2 Ti2 ≪ 1. As a result, the value of the Bode magnitude becomes H2 (jω)dB = −10 lg(1 + ω2 ⋅ Ti2 ) ≈ −10 lg 1 = 0dB ,
(2.109)
where the term ω2 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 ω2 ≫ (1/Ti )2 and consequently, ω2 Ti2 > 1. As a result, the value of the Bode magnitude becomes H2 (jω)dB = −10 lg(1 + ω2 ⋅ Ti2 ) ≈ −10 lg(ω2 ⋅ Ti2 ) = −20 lg ω − 20 lg Ti ,
(2.110)
where the term 1 has been neglected with respect to ω2 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 .
54  2 Basics of systems theory 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 Ti
(2.111)
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. The Bode representation of the magnitude H2 ( jω)dB is shown in Figure 2.18, where the dashed line has been used for representing the exact value of the magnitude and the plain line for the asymptotebased representation.
Figure 2.18: Bode diagram of magnitude for the frequency response function H2 (j ⋅ ω) = 1/(Ti ⋅ jω + 1).
The phase of the factor H2 ( jω) is ω⋅T
φ(H2 (jω)) = tan
−1
− 1+ω2 ⋅Ti 2 1 1+ω2 ⋅Ti2
i
= tan−1 (−ωTi ) = − tan−1 (ωTi ).
(2.112)
The Bode diagram representation of the phase φ(H2 ( jω)) is presented in Figure 2.19.
Figure 2.19: Bode diagram of phase for the frequency response function H2 (j ⋅ ω) = 1/(Ti ⋅ jω + 1).
2.32 Bode diagrams 
55
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 + jωTm 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 Figure 2.20 and Figure 2.21.
Figure 2.20: Bode diagram of magnitude for the frequency response function H3 (jω) = (Tm ⋅ jω + 1).
Figure 2.21: Bode diagram of phase for the function H3 (j ⋅ω) = (Tm ⋅ jω + 1).
(4) The factor of the form H4 (jω) = 1/[(jω)2 ⋅ 1/ω2k + (jω) ⋅ 2ζk /ωk + 1] has the Bode magnitude determined as follows: H4 (jω) =
1 (jω)2 ω2k
+
2ξk (jω) ωk
1
H4 (jω)dB = 20 lg √(1 −
2
ω2 ) ω2k
2
1
+
4ξk2 2 ω ω2k
+1
=
1 − ( ωω ) − k
(1 −
2
ω2 ) ω2k
= −10 lg [(1 −
+
2ξk (jω) ωk 4ξk2 2 ω ω2k 2
4ξk2 2 ω2 ) + ω ]. ω2k ω2k
(2.113)
(2.114)
56  2 Basics of systems theory 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 ω2 ≪ ω2k and consequently, ω2 /ω2k < 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 ω2 /ω2k and 4ζk2 ω2 /ω2k have been neglected with respect to 1 (ζk ≤ 1). For the frequency values such as ω > ωk , it is implied that ω2 ≫ ω2k and consequently, the highest term from the logarithm’s argument is ω4 /ω4k (the rest are neglected) and the value of the Bode magnitude H4 (jω)dB becomes: H4 (jω) = −10 lg (
2
ω2 ω ) = −40 lg = −40 lg ω − 40 lg ωk . ωk ω2k
(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=ωk = −10 lg (4ξk2 ) = −20 lg(2ξk ).
(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 Figure 2.22.
Figure 2.22: Bode diagram of magnitude for the frequency response function H4 (j ⋅ ω).
2.32 Bode diagrams 
57
The phase of the factor H4 (jω) is φ(H4 (jω)) = tan−1 (
2ξk ω ω k ) ω2 1− ω 2 k
−
= − tan−1
2ξk ω ω k . ω2 1− ω 2 k
−
(2.118)
The Bode diagram of phase φ(H4 (jω)) is presented in Figure 2.23.
Figure 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/ω2l + (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 Figure 2.24 and Figure 2.25.
Figure 2.24: Bode diagram of magnitude 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;
58  2 Basics of systems theory
Figure 2.25: Bode diagram of phase for the frequency response function H5 (j ⋅ ω).
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 ycoordinate. As an example, Figure 2.26 shows the Bode diagram without the logarithmic scale, for the magnitude of the factor H4 (jω) previously presented with dB units in Figure 2.22 (for ζk > 0.7). The present book will use both of these two equivalent Bode diagrams representations.
Figure 2.26: Bode diagram of magnitude for the frequency response function H4 (j ⋅ ω), with no logarithmic scale on the ycoordinate.
2.33 Nyquist diagrams 
59
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 xcoordinate the real component of the frequency response, Re[H(jω)], and on the ycoordinate the imaginary component of the frequency response, Im[H(jω)]. Typical Nyquist plots are presented in Figure 2.27. They show the Nyquist diagrams for the first order (plot 1), second order (plot 2) and third order (plot 3) systems.
Figure 2.27: Typical Nyquist diagrams of the frequency response for first (1), second (2) and third (3) order 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 Figure 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 xcoordinate 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∗ )]).
60  2 Basics of systems theory 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. Example 2.3. Consider the system presented in Figure 2.28, consisting in the process of liquid accumulation in series of two tanks. The two vessels operate at atmospheric pressure.
Figure 2.28: Liquid accumulations in a series of two tanks.
Based on the mass balance of liquid in each of the tanks, the first order differential equations may be obtained: dH1 + H1 = k1 Fvi (2.119) dt dH T2 2 + H2 = k2 Fv1 (2.120) 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: T1
dH2 d2 H2 + (T1 + T2 ) + H2 = k2 Fvi (2.121) dt dt 2 Compute the general solution of equation (2.121), taking into account the first con2 stant initial conditions H2 (0) = a and ( dH )(0) = b. dt T1 T2
a.
Equation 2.121 is a second order differential equation of the form: a
d2 y dy +b + cy = 0 dx dx 2
(2.122)
2.33 Nyquist diagrams  61
The solution to the equation of this form is given by y = C.F + P.I, where C.F → Complementary function P.I → Particular integral The procedure and solution to equation 2.122 is as follows: i. Rewrite the given differential equation as (aD2 + bD + c)y = f (x) ii. Substitute m for D, and solve the auxiliary equation am2 + bm + c = 0 for m iii. Obtaining the complementary function, u, is given next: a) Rewrite the differential equation 2.122 as (aD2 + bD + c)y = 0 b) Substitute m for D and solve the auxiliary equation am2 + bm + c = 0 ∴ [T1 T2 D2 + (T1 + T2 ) D + 1] H2 = 0 where: D =
d dt
P = T1 T2 ( Product ) {T1 , T2 } Sum = T1 + T2
[T1 T2 m2 + (T1 + T2 ) m + 1] = 0 T1 T2 m2 + T1 m + T2 m + 1 = 0
T1 m (T2 m + 1) + (T2 m + 1) = 0 (T2 m + 1) (T1 m + 1) = 0 1 1 or m = − m=− T2 T1
1
t
1
t
Since the roots are real and different, the general solution is y = Ae T 2 + Be T 1 iv. To determine the particular integral, v, firstly assume a particular integral which is suggested by f (x), but which contains undetermined coefficients. Since the term on the right hand side of equation (2.121) is a constant, i. e. f (x) = K2 Fvi , let P.I also be a constant, say v = k. v. Substituting v = k into [T1 T2 D2 + (T1 + T2 )D + 1]v = k2 Fvi gives −
[T1 T2 D2 + (T1 + T2 ) D + 1] k = k2 Fvi
−
(2.123)
2
Since D(k) = 0 and D (k) = 0 then k = k2 Fvii , v = k2 Fvi vi. The general solution is given by y = u + v, i. e. y = Ae
− T1 t 2
+ Be
− T1 t 1
+ k 2 F vi
(2.124)
Since we are given boundary conditions, arbitrary constants A and B in the C.F may be determined and particular solution to the differential equation obtained. vii. When t = 0, y = a a = Ae0 + Be0 + k2 Fvi a = A + B + k2 Fvi
A = a − B − k2 Fvi
dy A −1 t B − 1 t = − e T2 − e T1 dt T2 T1
62  2 Basics of systems theory When t = 0,
dy dt
=b B A − T2 T1 (a − B − k2 Fvi ) B − b=− T2 T1 −T1 a + T1 b + T1 k2 Fvi − T2 B b= T2 T1
b=−
T2 T1 b + T1 a − T1 k2 Fvi = B (T1 − T2 ) T2 T1 b + T1 a − T1 k2 Fvi T1 − T2 T (−a + T1 b + k2 Fvi ) A= 2 T1 − T2
B=
Finally: H 2 = k 2 F vi +
(T 1 a + T 2 T 1 b − T 1 k 2 F vi ) e
t T1
(T 1 − T 2 )
−
(T 2 a + T 1 T 2 b − k 2 T 2 F vi ) t
e T 2 (T 1 − T 2 )
(2.125)
The solution above is the analytical solution to the problem which applies basic mathematical principles, using MATLAB the same solution can be obtained through the code as shown below (Figure 2.29).
Figure 2.29: The MATLAB program solving numerically the problem from Example 2.3.
2.34 Problems  63
Example 2.4. Consider the transfer function of a dynamic system given by: H (s) =
0.5e−10s (s + 1) (2s + 1)(3s + 1)
(2.126)
Plot the Bode diagrams of the 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. This problem can be easily solved in MATLAB (Figure 2.30), the program can either be written in the editor or MATLAB workspace.
Figure 2.30: The MATLAB code and graphic solution for the transfer function in Example 2.4.
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;
64  2 Basics of systems theory (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 oilgas 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. (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(ωt) 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 inputoutput relationship: T
dy(t) + y(t) = K u(t), dt
(2.127)
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 consisting in a U shaped tube partially filled with liquid of density ρ, as presented Figure 2.31. The branches of the tube are cylindrical and of equal crosssection S. 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
References  65
Figure 2.31: Oscillating behavior of the liquid level in the U tube.
to the simplified equation (momentum balance): p1 ⋅ S − p2 ⋅ S − ρ ⋅ g ⋅ (2 ⋅ h) ⋅ S −
dh d2 h λ⋅L ⋅S⋅ρ⋅ =ρ⋅S⋅L⋅ 2 D dt dt
(2.128)
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 (crosssection area of the tube S = π D2 /4). (a) Compute the general solution of equation (2.128), taking into account the constant initial conditions h(0) = 0 and (dh/dt)(0) = 0. (b) Verify whether the system is BIBO stable. (9) Describe the system presented in Figure 2.31 by the inputstateoutput representation, choosing the state variables as phase variables.
References [1] [2] [3] [4] [5] [6] [7]
[8] [9]
Cristea, M. V., Agachi, S. P., Elemente de Teoria Sistemelor, Editura Risoprint, ClujNapoca, 2002. Forrester J. W., Principles of Systems, Wright Allen Press, 1969. Zadeh, L. A., Polak, E. System Theory, McGrawHill, New York, 1969. Hăngănuţ, M., Noţiuni de Teoria Sistemelor, ClujNapoca, Atelierul de multiplicare al Institutului Politehnic, 1989. Marquardt, W., Modellbildung und Simulation verfahrenstechnischer Prozesse, Vorlesungsmanuskript, Technische Hochscule Aachen, 1998. Cristea, V. M., Agachi, S. P., Simulation and model predictive control of the soda ash calciner, Control Eng. Appl. Inform., 3 (4), (2001), 19–26. 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. Şerban, S., Şerbu, T., Corâci, I. C., Teoria Sistemelor, Bucureşti, Editura Matrix Rom, 2000. Stephanopoulos, G., Chemical Process Control. An Introduction to Theory and Practice, Englewood Cliffs, New Jersey 07632, Prentice Hall, 1984.
66  2 Basics of systems theory
[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. [15] Hăngănuţ, M., Teoria Sistemelor, ClujNapoca, Universitatea Tehnică, 1996. [16] Cristea, V. M., Agachi, S. P., Reglarea evoluată a reactorului de carbonatare din instalaţia de producere a sodei amoniacale, Rev. Rom. Inform. Autom., 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. [18] Whitchurch, G. G., Constantine, L. L., Systems Theory, In: Boss P., Doherty W. J., LaRossa R., Schumm W. R., Steinmetz S. K. (eds) Sourcebook of Family Theories and Methods, Boston, MA, Springer, 2009. [19] Adams, K. M., Hester, P. T., Bradley, J. M., Meyers, Thomas J., Keating, C. B., Systems Theory as the Foundation for Understanding Systems, Syst. Eng., 17 (1), (2014), 112–123.
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.
Figure 3.1: Heat transfer process in a heat exchanger.
In the case of the process from Figure 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. https://doi.org/10.1515/9783110647938003
68  3 Mathematical modeling 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 experimental 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 causeeffect 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].
3.1 Analytical models  69
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 distributed 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 (Figure 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 righthand side of the equation is the mass accumulation term in the tank. The dimensional analysis is 1 kg m3 kg m3 kg ⋅ − ⋅ = ⋅ m3 ⋅ 3 s m3 s m3 s m which shows consistency, all terms being expressed by mass flow.
kg , s
the measurement unit for
70  3 Mathematical modeling
Figure 3.2: CSTR.
Example 3.2 (Component conservation law for lumped parameter system). A continuous stirred tank reactor (CSTR) is represented in Figure 3.2. The reaction is the simk
plest one, A → B„ first order with the rate constant k. 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 Fvi CAi − Fvo CA − VkCA =
d (VCA ), dt
(3.5)
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 righthand side is that of accumulation. The dimensional analysis m3 kmol m3 kmol 1 kmol 1 kmol − − m3 ⋅ ⋅ = ⋅ m3 ⋅ ⋅ ⋅ s s s m3 s m3 m3 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 (Figure 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
3.1 Analytical models 
71
Figure 3.3: Long pipeline distributed parameter system.
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 d2 f 1 df 1 d3 f ⋅ dz + ⋅ 2 dz 2 + ⋅ 3 dz 3 + ⋅ ⋅ ⋅ 1! dz 2! dz 3! dz
(3.6)
where dz is the infinitely small element of a dimension to be judged. Usually, since dz is very small, the dz 2 , dz 3 etc. terms become negligible and the expansion is reduced to the first two terms of the series expansion. 𝜕(Avρ) Since the mass flow is Avρ at z, according to (3.7), it becomes Avρ + 𝜕z dz at 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 crosssection area of the pipeline. The first two terms express the convective flows at the input and output of the dz element and the righthand 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
m kg kg m kg 1 1 m kg − [m2 + m2 m] = m2 m 3 s m3 s m3 m s m3 s m
and shows that all terms are measured by
kg . s
Example 3.4 (Component conservation law for distributed parameter system). Let us consider a plug flow reactor (PFR) represented in Figure 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 Figure 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
72  3 Mathematical modeling
Figure 3.4: PFR with the profile of concentration CA in the steady state along the reactor.
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
The dimensional analysis shows all terms are expressed in molar flow measurement units, kmol/s m2
2 m kmol 2 m 1 kmol + m s m3 s m m3 2 2 m kmol 1 2 m 1 kmol 2 m 1 kmol 2 m 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 𝜕z 2
(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
3.1 Analytical models 
73
using, for example, a finite element method [4]. 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 [5] or Mathematica [6] exist and have incorporated special subroutines for solving ordinary differential equations (ODE). 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 } flow of heat (power) formed volume or mechanical work { } } { } { { } ± { or transferred by ∓ exercised in time (power) by the } { } { } { } { reaction, radiation, or conduction } { system to the outer environment } ={
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
m v2
ΔEc = = Δt Δt
1 kg
2
1 m2
1s
s
2
= 0.5
Nm = 0.5 W. 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 ⋅ 4 318 kg⋅K 1 K J = = = 4 318 = 4 318 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.
74  3 Mathematical modeling 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). Let us consider a k
CSTR (Figure 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 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 . ∘ Fvi ρcpA Ti∘ − Fvo ρcp T ∘ − KT AT (T ∘ − Tag ) − VkCA ΔHr =
d (Vρcp 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 righthand 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: m3 kg J m3 kg J W 1 kg J 1 kmol J K − K − 2 m2 K − m3 = m3 3 K. 3 3 3 s m kgK s m kgK s m kmol s m kgK mK Example 3.6 (Energy conservation for a distributed parameter system). Let us conk
sider a PFR (Figure 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 Figure 3.5. 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 crosssection 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 = πDdz where D is the inner diameter of the reactor. The transfer coefficient
3.1 Analytical models 
75
Figure 3.5: PFR with exothermic reaction and cooling agent and the profile of temperature with an exothermic reaction and countercurrent cooling flow.
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 vρcp T ∘ + Ar (−kT
𝜕T ∘ ) 𝜕z
𝜕T ∘ 𝜕 𝜕T ∘ )+ [Ar vρcp T ∘ + Ar (−kT )] dz} 𝜕z 𝜕z 𝜕z 𝜕 ∘ − KT πDdz (T ∘ − Tag ) − Ar dzkCA ΔHr = (A dzρcp T ∘ ) . 𝜕t r
− {Ar vρcp 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
W 1 m kg J K + m2 K 3 s m kgK mK m m kg J W 1 1 m kg J W 1 − [m2 K + m2 K + (m2 K + m2 K) m s m3 kgK mK m m s m3 kgK mK m W 1 kmol J 1 kg J − 2 mmK − m2 m ] = m2 m 3 K. s m3 kmol s m kgK mK
After simplification, the equation (3.12) becomes 4K T ∘ 𝜕 𝜕 𝜕T ∘ 𝜕 ∘ (ρcp T ∘ ) + (vρcp T ∘ ) + kCA ΔHr + (T − Tag )= (kT ). 𝜕t 𝜕z D 𝜕z 𝜕z
(3.13)
76  3 Mathematical modeling 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)): {
sum of forces } = {rate of change of movement quantity} applied to a system
(3.14)
or n
∑ Fij = j=1
d (mv) , dt
(3.15)
where Fij is the projection of applied force j on the direction i (Figure 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 only one force applied to the body and m is constant, dv = a, the acceleration, and dt equation (3.15) is reduced to the expression of Newton’s second law F = m ⋅ a.
Figure 3.6: Application of the impulse conservation on the movement of one mechanical body.
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 (Figure 3.7). The friction on the pipeline is expressed by Poisson’s law, Δpp = ∝
ρv2 2
l ρv2 p 2
+λ dp
Figure 3.7: Tank with free outflow through a pipeline.
3.1 Analytical models 
77
where Δpp is the pressure loss on the hydraulic system (pipeline), 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, ρghAp , G, the weight of the liquid in the pipeline and friction ρv2
l ρv2
force against the movement, (∝ 2 + λ dp 2 )Ap . The atmospheric pressure is equal on p both sides of the pipeline. G, being vertical, has the projection on i axis Gij = 0. Thus, ρghAp − (∝
lp ρv2 ρv2 d +λ ) Ap = (Ap lp ρv). 2 dp 2 dt
(3.16)
The dimensional analysis kg m kg m kg m2 2 1 2 2 ⋅ m ⋅ m − m = m ⋅m⋅ 3 3 3 2 2 s m s m 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 d (Ap lp ρv) = 0. Usually, because the liquids are incompressible, steady state when dt 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 (Figure 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.
Figure 3.8: Pipeline with variable mass of liquid.
The equations describing the movement are p0 Ap − λ
(lp − z) ρv2 dp
2
Ap =
d (A (l − z)ρv) dt p p
(3.17)
78  3 Mathematical modeling and v=
dz , dt
from which one may calculate the dynamics of emptying the pipeline.
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 Tables 3.1 and 3.2. 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 (Figure 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 , i=1
ai ∈ R, i = 1, n
(3.18)
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.
3.2 Statistical models 
79
Table 3.1: Thermodynamic properties used in process modeling. Property/Thermodynamic laws
Formula
Notations
Liquid enthalpy
ΔHl = cp T ∘
cp – heat capacity; T ∘ – temperature
Vapor enthalpy
ΔHl = cp T ∘ + lv
lv – latent heat of vaporization
Enthalpy of a mixture of liquids and heat capacity of a mixture
ΔHlm =
Dependence of enthalpy with temperature
ΔHl = a0 + 2 a1 T ∘ + a2 T ∘ + ⋅ ⋅ ⋅
a0 , a1 etc. are numerical coefficients; the dependence is to be considered for large (> 10°) variations of temperature
Boyle Mariotte law
pV = nRT ∘
p – pressure of gas; V – gas volume; n – number of moles of gas; R – universal gas constant
Gas density
ρ=
pM RT ∘ ni ρ Ci = ∑ m ρ = xi M i j j ∑ Mi xi = ρ ⋅ Ci
ρ – gas density
Molar concentration and mole fraction
∑n1 xj ΔHj Mj ∑n1 xj Mj
or
cplm = ∑n1 xj cpj
xj – molar fraction; Mj – molar mass; ΔHj – enthalpy of j component in the mixture; cpj – heat capacity of component j
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
Liquid mixture density
ρ = ∑n1 Mj Cj or ρ = ∑n1 xj ρj
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 (liquidvapor 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
80  3 Mathematical modeling 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 ⇐⇒k1 , k−1 B, r = k1 CA − k−1 CB Parallel first order k2
k1
A → B, A → C, rA = (k1 + k2 )CA Parallel second order k1
k2
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;
A + B → D, A + C → E rD = k1 CA CB , rE = k2 CA CC Consecutive reactions k2
k1
A → B → C, rB = k1 CA − k2 CB E − a∘
Arrhenius law k = k0 e (dependence of the rate constant on temperature) Relative conversion of the ξ = reactant
Relative conversion in a CSTR
k0 – preexponential constant; Ea – activation energy; R – universal constant; T ∘ – reaction temperature
RT
CA0 −CA CA0
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
k
Direct first order A → B ξ=
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
k VF
1+k VF
Reversible first order aA ⇐⇒k1 , k−1 bB
1 V
ξ = ξ∞ (1 − e−k1 (1+ K ) F ) k K = k1 −1
E − 1a∘
k1 = k10 e Relative conversion in a PFR, with a first order direct reaction Excess factor for reversible reaction to assure maximum conversion in C
RT
E − −1,a ∘
k−1 = k−1,0 e
RT
V
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
ξ = 1 − e−k F
aA + bB ⇐⇒k1 , k−1 cC γ=
aCB0 bCA0
k1
Selectivity between the Parallel reactions A + B → P, k2 desired and by product in A + B → R (desired) a CSTR 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
3.2 Statistical models 
81
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 ] ] [ [⋅⋅⋅ ] ̄ ]. [ A=[ ] [ ai ] ] [ [⋅⋅⋅ ] [ am ]
(3.21)
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 = [ [ 1 x 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 . A general form of one program of experiments is given in Table 3.3. Table 3.3: Program of experiments in an industrial or laboratory plant for elaborating a statistical model. Measurement no. 1 2 ... j ... n
Input variables x1 x2 ...
xi
...
xm
Output variables y
x11 x12 ... x1j ... x1n
xi1 xi2 ... xij ... xin
... ... ... ... ...
xm1 xm2 ... xmj ... xmn
y1 y2 ... yj ... yn
x21 x22 ... x2j ... x2n
... ... ... ... ...
82  3 Mathematical modeling The estimation is based on the least squares principle which minimizes the difference between the measured and the calculated values of y (sum of square residuals SSR ): ε = min SSR = min
1 n 2 ∑ (y − a0 x0j − a1 x1j − a2 x2j − ⋅ ⋅ ⋅ am xmj ) . n j=1 j
(3.23)
In order to obtain the minimum, one has to fulfill the conditions (3.24) 𝜕ε 𝜕ε 𝜕ε = 0; = 0; . . . . = 0. 𝜕a0 𝜕a1 𝜕am
(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
j=1
j=1
j=1
j=1
a0 ∑ x0j xij + a1 ∑ x1j xij + ⋅ ⋅ ⋅ + am ∑ xmj xij = ∑ xij yj ,
(3.26)
where i = 0 ÷ m. 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 ), j=1
(3.29)
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
3.2 Statistical models 
83
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 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
[ [ [ y2̄ = Ȳ = [ [ ... [ [ ∑rl=1 ynl [ yn̄ = r
] ] ] ] ] ] ] ]
where r is the number of repetitions of the same measurement. With these values, one calculates the dispersion for each measurement 2 Syj
2
=
∑rl=1 (yjl − yj̄ ) r−1
(3.31)
84  3 Mathematical modeling 2 2 and the maximum value of these dispersions, Smax = max Syj . The total measurement dispersion of the whole experiment is
Sy2 = The Cochran variable Gmax , Gmax =
2 Smax Sy2
2 ∑nj=1 Syj
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 =
ai 2 S {a
i}
> tT .
(3.34)
(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 =
∑nj=1 Δy2
n − (m + 1)
(3.35) (3.36)
and Sy2 The model is adequate [12] when F =
=
2 ∑nj=1 Syj
2 Srem Sy2
n
.
(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
3.3 Artificial neural network Models 
85
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 inputoutput 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]. 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 Figure 3.9 [14].
Figure 3.9: Typical structure of the artificial neuron.
86  3 Mathematical modeling 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 Figure 3.10.
Figure 3.10: Typical structure of the multilayer ANN.
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 par
3.4 Examples of mathematical models  87
ticular 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. Considering the dynamic simulation, using it, one can observe almost instantaneously the dynamic behavior of very slow processes with time changes of hours, days or years; the steady state simulation is extremely useful to quantify the effect of the disturbances on the final characteristics of a process (e.g. feed molar ratio on the steady state molar ratio of the products in a distillation tower), or to quantify how much, changes in construction of the equipment affect the performance of it (e.g. increasing the heat exchange area affects the output temperature of the heat exchanger).
3.4 Examples of mathematical models Example 3.9 (Non isothermal CSTR). Considering the CSTR (Figure 3.2) in which an k
exothermic first order reaction A → B takes place. The heat of reaction is ΔHr < 0. According to the procedures proposed at 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 causeeffect 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 ∘ .
88  3 Mathematical modeling ∘ . 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) T ∘ – temperature inside the reactor and in the outflow (K) i
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 ) kg ρA , ρB , ρ – density of A, B and of the mixture of A and B ( m 3)
) 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 preexponential factor ( s1 , s1 ) J Ea – activation energy ( kmol ) J ) R – universal gas constant (= 8,314 kmol K m g – gravitational acceleration (= 9.8 s2 ) J ΔHr – heat of reaction ( kmol )
kg ρag , cpag – density and heat capacity of the cooling agent ( m 3,
J ) kg K 2
KT , AT – heat transfer coefficient and heat transfer area ( mW2 K , m ) 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);
3.4 Examples of mathematical models  89
– –
– – –
5.
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).
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 ρghAp − λ
lp ρv2 d A = (A l ρv) 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 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 (3.40) AR d for V: Fvi ρA − Fρ = (Vρ) (3.41) dt for ρ: ρ = MA CA + MB CB (Table 3.1) (3.42)
for h: h =
90  3 Mathematical modeling 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 ∘ Fvi ρA cpA Ti∘ − Fρcp T ∘ − KT AT (T ∘ − Tag ) − VkCA ΔHr =
for cp : cp = xA cpA + xB cpB (Table 3.1)
d (Vρcp T ∘ ) dt
M M for xA : xA = CA A and xB = CB B ρ ρ
(3.45) (3.46) (3.47)
∘ ∘ ∘ ∘ for Tag : Fvag ρag cpag Tiag − Fvag ρag cpag Tag + KT AT (T ∘ − Tag )=
d (V ρ c T ∘ ) dt j ag pag ag (3.48)
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” (Figure 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 (M c T ∘ ). dt w pw w
d (V ρ c T ∘ ) dt j ag pag ag
Figure 3.11: The wall is thermally “thick”.
(3.49) (3.50)
3.4 Examples of mathematical models  91
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. 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). Based on the model presented above, one simulator (Figure 3.12) was created in the Laboratory of Process Control from the University BabesBolyai, Romania [15]. The advantages of such a simulator are obvious: – Behaviour of the CSTR at different disturbances (e. g. Fi ) or manipulating variable (e. g. Fag ) can be observed very easily and in a short time. – The combined effect of the disturbances can be observed without difficulty and the implicit effect of the parameters not obviously related to the outputs are easy to correlate and explain (e. g. Fi effect on the concentrations or temperatures of the mass of reaction or jacket in the reactor). – Input parameters can be changed easily and their influence observed.
Figure 3.12: Simulator for a CSTR. Process Control Laboratory. University BabeșBolyai, Romania.
92  3 Mathematical modeling The simulator suggests what happens in the reactor, when the input reactant flow A increases stepwise from 5 to 6 l/s: – the level in the reactor increases with a capacitive behavior with a time constant of cca. 7 minutes and a gain of 0.5 m/l/s; this may suggest a certain tuning of the level controller parameters; – the concentration of the reactant A increases in the reactor, presenting a peak at the beginning (excess of unconsumed A), but decreases afterwards due to the increase of the rate of reaction (kCA ); at the same time, the concentration of B (product) follows A, but stabilizes at a higher value. All this change lasts around 20 minutes; – the temperature of the reactor increases due to the intensification of the exothermic reaction and influences the increase of the temperature in the jacket if no action is taken to to absorb the excess of heat (a control action to increase the coolant flow). The dynamic behavior imposes some structure of the parameters’ control loops and the tuning of the respective controllers’ parameters (Chapters 6 and 12). 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 k
and then the different outflows. The reaction is A → B, with ΔHr = 0.
Figure 3.13: Cascade of three CSTRs.
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: Fi ρ − F1 ρ =
d (V ρ). dt 1
(3.51)
3.4 Examples of mathematical models  93
d (V ρ) dt 2 d F2 ρ − F3 ρ = (V ρ) dt 3 F1 ρ − F2 ρ =
d (V C ) dt 1 A1 d F1 CA1 − F2 CA2 − V2 kCA2 = (V2 CA2 ) dt d F2 CA2 − F3 CA3 − V3 kCA3 = (V3 CA3 ) dt F1 = KV1 (V1 − Vmin 1 ) Fi CAi − F1 CA1 − V1 kCA1 =
F2 = KV2 (V2 − Vmin 2 )
(3.52) (3.53) (3.54) (3.55) (3.56) (3.57) (3.58)
F3 = KV3 (V3 − Vmin 3 )
(3.59)
0 − F1 CB1 + V1 kCA1 =
(3.60)
d (V C ) dt 1 B1 d F1 CB1 − F2 CB2 + V2 kCA2 = (V2 CB2 ) dt d F2 CB2 − F3 CB3 + V3 kCA3 = (V C ) dt 3 B3
(3.61) (3.62)
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 [16]). 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 .
Figure 3.14: Binary distillation column.
94  3 Mathematical modeling
Figure 3.15: Distillation column Illudest in the Control Laboratory of the Faculty of Chemistry and Chemical Engineering, University BabesBolyai, ClujNapoca, Romania.
Figure 3.16: The unit of distillation + section with the weir.
State variables: Mj , Lj , xj , yj , MB , xB , yB , MD , xD , FR , V, FB , FD . Each distillation unit (tray) is represented in Figure 3.16 + section with the weir. 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.
3.4 Examples of mathematical models  95
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. 1 000 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 more 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 V is 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 =
αxj
(3.63) d (M x ) dt j j
(3.64) (3.65)
1 + (α − 1)xj
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
FF xF + LNF +1 xNF +1 − LNF xF + VNF −1 yNF −1 − VNF yNF =
(3.67) d (M x ). dt NF NF
(3.68)
96  3 Mathematical modeling The equilibrium equation of the feed tray is the same with all tray equilibrium equations. Distillate drum equations: V − FD − FR =
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 = αxB . yB = 1 + (α − 1)xB
(3.71) d (M x ) dt B B
(3.72) (3.73)
The control equations: FD = FD0 + KvD (MD − MDmin )
FR = FR0 −K vR (xD − xDset ).
(3.74) (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. Figure 3.17 presents the result of a dynamic simulation of the behaviour of a binary distillation column (Figure 3.15), elaborated in the Process Control Laboratory of the University BabesBolyai [17]. It can be seen, in case the feed does not have the same designed concentration, how the profile of molar fractions along the column changes dramatically and the specifications of quality (100 % purity at the top) might not be met. From the simulation, we may observe: at a given construction of the column, when disturbances occur, the equipment does not deliver the quality at the specifications (if the input keeps the disturbed value, we may need more separation trays to obtain the product at the required specifications); since the construction can not be changed during the operation, the objective of purity of the distillate is not anymore achieved; this may suggest a change of the operational parameters (e. g. the Reflux Rate), an automatic intervention, a certain structure of the ACS etc.
3.4 Examples of mathematical models  97
Figure 3.17: Simulation of the behaviour of a binary distillation column at feed molar fraction change. The simulator belongs to the Process Control Laboratory, University BabeșBolyai, Romania.
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 two levels [9] was developed (Table 3.4 and Table 3.5), 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 2.0 ] ] ]; 2.8 ] 2.0 ]
1.0 [ X = [ 1.2 [ 2.8 T
Table 3.4: Input variables. Input variables
x1
x2
Center level Variation step
1.0 0.2
2.4 0.4
1.0 1.2 2.0
1.0 0.8 2.8
1.0 ] 0.8 ] ; 2.0 ]
4.0 [ X X = [ 4.0 [ 3.6 T
4.0 4.16 9.6
9.6 ] 9.6 ] ; 23.7 ]
98  3 Mathematical modeling Table 3.5: Results of experiments. Input variables (concentration mol/l) x1 x2
Output variable (efficiency %) yI
1.2 1.2 0.8 0.8
0.5 0.8 0.2 0.55
2.8 2.0 2.8 2.0
yII
Output average (%) y
0.3 0.6 0.1 0.25
0.4 0.7 0.15 0.40
Δ = det(X T X) = 0.4096
(3.77)
0.68 −1 [ ] A = (X T X) (X T Y) = [ 0.624 ] , giving a regression y = 0.68 + 0.624x1 − 0.38x2 . [ −0.38 ] Statistical processing [13]: (a) Weight of the coefficients in the model. S2 {ai } =
Sy2 n
{ =
2
=
∑nj=1 ∑rl=1 (yjl − yj̄ ) 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,
where n = 4 and r = 2.
4⋅1
(3.78)
The Student test (for α = 0.95, tT = 2.776): a0 2 S {a
0.68 = 167 > 2.776 0.0041 i} a1 0.625 = = 153 > 2.776 S2 {ai } 0.0041 a2 0.38 = 93.54 > 2.776, = 2 S {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 Srem
= =
∑nj=1 [ymeas − (a0 + a1 x1 + ⋅ ⋅ ⋅ + am xm )]2 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
3.5 Problems  99
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 (Figure 3.18) 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 .
Figure 3.18: Steam generator.
k
(2) The chemical reaction A → B + C, first order, regarding A, takes place in a semicontinuous STR (Figure 3.19).
100  3 Mathematical modeling
Figure 3.19: Semicontinuous tank reactor (SCSTR).
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 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 (Figure 3.20) is fed at the solvent evaporation temperature corresponding to the feed mass concentrations xF for the first evaporator and x1
Figure 3.20: A double effect evaporator.
3.5 Problems  101
for the second. The volumes of the liquid phase of both evaporators are V. The concentrations 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 Table 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 (Figure 3.21).
Figure 3.21: CSTR with two influents.
102  3 Mathematical modeling 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 heat cp . 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) and V(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 Table 3.7.
Table 3.7: The experimental data. j
T° [ °C]
min [kg]
1 2 3 4 5 6
45 52 50 48 52 51
1.5 3 2 2.4 1 4
Kw
I
II
Kw, av
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
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.
References  103
References [1] [2] [3] [4] [5] [6] [7]
[8] [9]
[10] [11] [12] [13] [14]
[15] [16] [17]
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, McGrawHill 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., Operaṭii ṣ i utilaje în industria chimică – Probleme (Operations and equipment in the chemical industry – problems), Editura Didactică ṣ i Pedagogică, Bucureṣ 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., Tertiṣ co M., Ceangă E., Identificarea experimentală a proceselor automatizate (Experimental identification of the controlled processes), Editura Tehnică, Bucureṣ 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. ImreLucaci, A., LabView Simulator for a CSTR, Laboratory of Process Control, University BabesBolyai, Romania, 2010. Luyben, W., Process Modeling, Simulation and Control for Chemical Engineers, McGraw Hill, 1990, p. 70. Nagy, Z. K., Simulator of a binary distillation column, Laboratory of Process Control, University BabesBolyai, Romania, 2000.
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 (nth order multicapacitive) system, – pure delay (dead time) system. The dynamic behavior of any system can be expressed as one of the particular types mentioned above, or a combination of two or several types.
4.1 Proportional system The proportional system is described by the inputoutput 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 Figure 4.1. The transfer function of the proportional system emerges from the Laplace transform of equation (4.1): HP (s) =
Y(s) =K U(s)
(4.2)
and the frequency response function becomes HP (jω) = https://doi.org/10.1515/9783110647938004
Y(jω) = K. U(jω)
(4.3)
106  4 Systems dynamics
Figure 4.1: Proportional system unit step response.
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 Figure 4.2.
Figure 4.2: Bode diagrams of the proportional system.
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 Figure 4.3, the output voltage Uo is depending on the input voltage Ui
4.2 Integral system
 107
Figure 4.3: The resistive electrical circuit may be considered a proportional system.
according to the proportional system behavior: Uo (t) =
R2 U (t), R1 + R2 i
(4.4)
where the gain is K = R2 /(R1 + R2 ). 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 inputoutput 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 Figure 4.4.
Figure 4.4: Integral system unit step response.
108  4 Systems dynamics The transfer function of the integral system emerges from the Laplace transform of equation (4.5): HI (s) =
Y(s) 1 = , U(s) Ti s
(4.6)
and the frequency response function becomes HI (jω) =
Y(jω) 1 j = =− . U(jω) Ti jω Ti ω
(4.7)
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 Figure 4.5.
Figure 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, Figure 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
4.3 Derivative system
 109
Figure 4.6: Integral system example: mass accumulation in a tank with constant outlet flow.
of 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
H = H0 +
1 ∫ (Fmi (τ) − Fmo )dτ Aρ
t
and
ΔH =
0
1 ∫ (Fmi (τ) − Fmo )dτ. 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 inputoutput 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 Figure 4.7. The unit step response of the derivative system has a Dirac function (impulse) form.
110  4 Systems dynamics
Figure 4.7: Derivative system unit step response.
The transfer function of the derivative system emerges from the Laplace transform of equation (4.10): HD (s) =
Y(s) = Td s U(s)
(4.11)
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 Figure 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 Figure 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 I = Ce
dUc dt
(4.14)
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)
4.3 Derivative system 
111
Figure 4.8: Bode diagrams of the derivative system.
Figure 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) becomes Uo = −RCe
dUi , dt
(4.17)
112  4 Systems dynamics 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.
4.4 First order system The first order (or first order capacitive) system is described by the inputoutput relationship: T
dy(t) + y(t) = Ku(t), dt
(4.18)
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 Figure 4.10.
Figure 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)
4.4 First order system
 113
The time constant can also be determined graphically, as shown in Figure 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): HD (s) =
K Y(s) = U(s) Ts + 1
(4.22)
and the frequency response function becomes HFO (jω) =
K K(1 − j T ω) K KT ω = = −j . 2 2 jT ω+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 ω2 )1/2 and its phase φFO (jω) is − tan−1 (Tω). The Bode diagrams of the first order system are presented in Figure 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)
114  4 Systems dynamics
Figure 4.11: Bode diagrams of the first order system.
Following simple transformations, equation (4.24) becomes: F V dCA + 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 inputoutput relationship 1 d2 y(t) 2ζ dy(t) + + y = Ku(t), ωn dt ω2n dt 2
(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].
4.5 Second order system
 115
Figure 4.12: Second order system unit step response.
The second order system unit step response is presented in Figure 4.12. 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 +
y(t) = Ku0 (t) [1 − (1 + ωn t) e
− ωn t
]
ζ √ζ 2 − 1
sinh ωn √ζ 2 − 1t)] for ζ > 1 (4.28)
for ζ = 1
(4.29)
y(t) = Ku0 (t)[1 − cos(ωn t)] for ζ = 0 y(t) = Ku0 (t)[1 −
e−ζ ωn t √1 − ζ 2
(4.30)
sin (ωn √1 − ζ 2 t + arctan (
√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) = U(s)
1 2 s ω2n
K +
2ζ s ωn
+1
(4.32)
116  4 Systems dynamics and the frequency response function becomes after simple transformations 2
HSO (jω) =
1 − ( ωω ) (1 −
ω2
ω2n
2
n
) +
4ζ 2 ω2n
ω2
−j
2ζ ωn
(1 −
ω2
ω2n
2
⋅ω
) +
4ζ 2 2 ω ω2n
.
(4.33)
The Bode diagrams of the second order system are presented in Figure 4.13 [2].
Figure 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 Figure 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. 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
4.6 Higher order system
 117
Figure 4.14: Second order system emerged from the series connection of two CSTRs.
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 V2 V dCA2 F2 +2 C , + CA2 = 2 2 F + Vk dt (F + Vk) dt (F + Vk)2 Ai
(4.35)
where ωn =
F + Vk , V
ζ = 1,
K=
F2 . (F + Vk)2
(4.36)
The transfer function of the CSTRs system may be obtained by applying the Laplace transform to the equations (4.34) T1 s CA1 (s) + CA1 (s) = K1 CAi (s) T2 s CA2 (s) + CA2 (s) = K2 CA1 (s), where T1 = T2 =
V , F + Vk
K1 = K2 =
F F + Vk
(4.37)
(4.38)
and by following simple transformations becomes the form HSO (s) =
K1 K2 K1 K2 CA2 (s) . = = CAi (s) (T1 s + 1)(T2 s + 1) [T1 T2 s2 + (T1 + T2 )s + 1]
(4.39)
4.6 Higher order system When connecting in series multiple first order capacitive systems, the inputoutput relationship features an increased time lag and presents the so called higher order
118  4 Systems dynamics capacitive system behavior [1, 4, 5]. The first order capacitive systems connected in series may or may not have interactions (causeeffect in both directions between systems). These two cases are presented in Figure 4.15.
Figure 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 nth order of differentiation with respect to the output y. The unit step response of the system is presented in Figure 4.16.
Figure 4.16: nth order system unit step response.
The transfer function of the noninteracting cascade of n first order systems has the following form: HnO (s) =
Kn K2 K1 ⋅⋅⋅ . T1 s + 1 T2 s + 1 Tn s + 1
(4.40)
Example 4.6. First, consider as an example the nth order system, presented in Figure 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
4.6 Higher order system
 119
Figure 4.17: nth 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 nth reactor CAn as output. It is described by the transfer function of the form presented in equation (4.40) [7]. 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 Figure 4.18.
Figure 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 p − p3 dp p − p2 C2 2 = 1 − 2 dt R2 R3
C1
(4.41)
120  4 Systems dynamics which becomes after the Laplace transform p p 1 1 + )p = i + 2 R1 R2 1 R1 R2 p p 1 1 + )p = 1 + 3. (C2 s + R2 R3 2 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 + R2 + R3 R1 + R2 + R3 R3 R1 + R2 RCRs p + ( 1 1 2 + 1) p3 = R1 + R2 + R3 i R1 + R2 + R3 R1 + R2
(4.44)
and the transfer function on the path pi to p2 becomes H(s) =
R1 C1 R2 C2 R3 2 s R1 +R2 +R3
+
R3 R1 +R2 +R3 (R1 C1 R3 +R1 C1 R2 +R2 C2 R3 +R1 C2 R3 ) s R1 +R2 +R3
+1
.
(4.45)
The transfer function of the noninteracting cascaded capacities has the form H(s) =
R2 R3 (R1 +R2 )(R2 +R3 ) R1 R2 C1 R2 R3 C2 2 R R3 C2 2 C1 s + ( RR1 R+R + R2 +R )s (R1 +R2 )(R2 +R3 ) 1 2 2 3
+1
.
(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
(4.47)
3(2s + 1) ( 32 s + 1)
and the transfer function on the path p3 to p2 , obtained from equation (4.44), has the form H(s) =
2(s + 1)
3(2s + 1) ( 32 s + 1)
.
(4.48)
The time constant for a single noninteracting capacity is T=
1 R1
C +
1 R2
=
2 =1 2
(4.49)
while the time constants for the interacting capacities have the 2 and the 2/3 values.
4.7 Pure delay system
 121
4.7 Pure delay system The pure delay (or dead time) system is described by the inputoutput 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 Figure 4.19.
Figure 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) τ2 d2 u(t) τ3 d3 u(t) − + ⋅⋅⋅ . + 1! dt 2! dt 2 3! dt 3
(4.51)
Applying the Laplace transform to (4.51) the following equality is obtained: Y(s) = (1 −
τs τ2 s2 τ3 s3 + − + ⋅ ⋅ ⋅) U(s), 1! 2! 3!
(4.52)
where in the parenthesis we find the Maclaurin series expansion of the function e−τs , leading to the transfer function of the pure delay system: HPD (s) =
Y(s) = e−τs . 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)
122  4 Systems dynamics
Figure 4.20: Bode diagrams of the pure delay system.
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 Figure 4.20. 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.
4.8 Equivalence to first order with time delay system
 123
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 Figure 4.21. This system produces a combination of time delay lag on the inputoutput 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 Figure 4.21.
Figure 4.21: Unit step response of the first order system with dead time.
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−τs = U(s) Ts + 1
(4.55)
and its frequency function is HFOPD (jω) =
K e−τjω . 1 + jωT
(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
124  4 Systems dynamics 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
Ke = ∏ Ki n
∏ i=1 n
i=1
Ki
√1 + Ti2 ω2
(4.57) =
Ke √1 + Te2 ω2
∑ (− arctan(Ti ω)) = −τe ω − arctan(Te ω). i=1
(4.58) (4.59)
Figure 4.22 presents the step response of nth first order system and the equivalent step response of the first order with pure time delay system [2].
Figure 4.22: (a) Step response of the nth 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 nth 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. Example 4.9. The chemical reaction of transforming the A reactant into the B product has first order kinetics and is performed in a CSTR (Figure 4.23). Build the mathematical model of the system, the Bode plots and the transfer function for the transfer path Ti∘ to T ∘ (i. e. input reactor temperature to reactor temperature). Determine also the type of the dynamic behavior and its characteristics (gain, time constant, pure time delay if any) characterizing the dynamic behavior of the heat transfer process. The mass of the reactor wall may be considered negligible. Plot also the dynamic response at 10 % change in manipulating variable (Fag ) and 10 % change in disturbance (Ti∘ ); plot the steady state behavior of the process in normal conditions and at a disturbance of 10 %.
4.8 Equivalence to first order with time delay system
 125
Figure 4.23: CSTR.
The following data are known: volume of reactor V = 1 m3 , heat transfer area AT = 4 m2 , heat transfer coefficient KT = 1 000 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 = 1 000 kg/m3 , temperature ∘ of the inlet stream Ti∘ = 40 °C, cooling agent temperature Tag = 20 °C, heat of reac
tion ΔHr = −20 kcal/mol, nominal concentration of A component CA = 0.5 kmol/m3 , reaction rate constant at steadystate temperature T ∘∗ = 30 °C, k(T ∘∗ ) = 0.002 h−1 , activation energy E = 15 kcal/mol, gas constant R = 1.989 kcal/kmol K, volume of the jacket Vj = 0.2 m3 . Note: The reaction rate constant can be expressed as k = a exp(−E/(RT o )). This expression can be linearized around the steady state temperature T ∘∗ by the first two terms of the Taylor series expansion: dk k (T ∘ ) = k (T ∘∗ ) + ( (T ∘ − T ∘∗ )) (4.60) dT T ∘
The mathematical model to describe the thermal behavior of the CSTR is made of the heat balances of the system, including that of the reaction mass and that of the cooling (exothermic reaction) jacket. Heat balance for the mass of reaction ∘ FρCp Ti∘ − FρCp T ∘ − KT AT (T ∘ − Tag ) − vkCA ΔHr =
d(VρCp T ∘ )
dt Because the wall is “thin”, its heat transfer equation is neglected, so we do not have an extra equation for the wall; if it were “thick” (meaning important from the delay point of view), the equation would have been: ∘ KTag AT (Tag − Tw∘ ) − KTi AT (Tw∘ − T ∘ ) =
d(Mw Cpw Tw∘ )
Heat balance for the agent ∘ ∘ ∘ Fag ρag Cpag Tiag − Fag ρag Cpag Tag + KT AT (T ∘ − Tag )=
dt
∘ d(Vj ρag Cpag Tag )
dt
126  4 Systems dynamics The steady state of the process is given by the following equations: ∘ FρCp Ti∘ − FρCp T ∘ − KT AT (T ∘ − Tag ) − vkCA ΔHr = 0 ∘ ∘ ∘ Fag ρag Cpag Tiag − Fag ρag Cpag Tag + KT AT (T ∘ − Tag )=0
Combining the mass of reaction and agent balance equations we obtain the sim∘ plified expression of output reaction temperature (eliminate Tag ): T∘ =
∘ KT AT − vkCA ΔHr Fag ρag Cpag − vkCA ΔHr KT AT Fag ρag Cpag FρCp Ti∘ + FρCp Ti∘ KT AT + Fag ρag Cpag Tiag
FρCp KT AT + Fag ρag Cpag FρCp + Fag ρag Cpag KT AT
dk (T ∘ ) ] (T ∘ − T ∗ ) dT ∘ T ∗ −E E (T ∘ − T ∗ ) k (T ∘ ) ≅ k (T ∘∗ ) + ae RT ∘∗ ∗ RT ∘∗2 E k (T ∘ ) ≅ k (T ∘∗ ) + k (T ∘∗ ) ∗ (T ∘ − T ∗ ) Linearized formula RT ∘∗2 k (T ∘ ) ≅ k (T ∘∗ ) + [
The MATLAB program solving the system of energy equations for the reactor and the jacket (Problem 4.9), with the linearized dependence of the rate constant function of temperature k (T ∘ ), is given in Figure 4.24. As the derivative functions on time are 0, the steady state curves are given in Figure 4.25.
Figure 4.24: MATLAB program solving the system of the steady state equations from Problem 4.9. 3
The nominal operation point is at Fag = 5 mh and T o = 33.51 °C. From these characteristics we can calculate the steady state gain of the process in the point of operation: ΔT ∘ Kpr = ΔF = −0.5m°C ≈ −0.33 m°C3 . The ‘minus’ sign of the slope shows that the tempera3 ag
1.5
h
h
ture decreases with the increase of the coolant flow.
4.8 Equivalence to first order with time delay system
 127
Figure 4.25: The steady state characteristics of the cooling process in the nominal regime (Ti∘ = 40 °C) and disturbed state (±10 %).
The equations are nonlinear since the term containing the differentiable variable, k (T) , is exponential. Linearizing, we are able to approximate the capacitive behavior k (T ∘ ) ≅ 0.001495 + 0.0000168265T ∘ ∘ ) − V [0.0015 + 0.0000168T ∘ ] CA ΔHr = FρCp Ti∘ − FρCp T ∘ − KT AT (T ∘ − Tag
d(VρCp T ∘ ) dt
The dynamic response given by the linearized k (T ∘ ) was calculated with the MATLAB program presented in Figure 4.26. The results of the runs of the program at different disturbances and manipulated variables are presented in Figures 4.27a, b. If Ti∘ increases from 40 °C to 44 °C, naturally, the output temperature has to increase from the steady state which is 33.5 °C to a value corresponding to the gain of 2.5 °C ΔT ∘ = 0.625 °C ). This value is 36 °C (Figure 4.27a). The bethe process (Kpr = ΔT ∘ = 4 °C °C i
havior is obviously capacitive as the form of the differential equations indicate. Since the reactor is quite voluminous, V = 1 m3 , one can expect that the rate of increase will be slow: in cca. 0.7 hours (settling time) it reaches the plateau and has a time constant = 0.11 h, (time in which the process reaches 63.2 % of the change of the variable to the plateau). Usually, in practical terms, the time constant is 4–6 times smaller than the settling time.
128  4 Systems dynamics
Figure 4.26: The MATLAB program calculating the dynamic behavior of the CSTR in Problem 4.9.
In Figure 4.27b, one may observe the temperature increase when the coolant flow de3 3 creases with 10 % from 5 mh to 4.5 mh . The dynamic characteristics are almost the same (settling time 0.7 h and time con°C ΔT ∘ stant T = 0.11 h). The gain of the process is Kpr = ΔF = +0.16m°C 3 = −0.32 m3 . ag
−0.5
h
h
For a first order system, defining the behavior on the transfer path Ti∘ → T ∘ (see Figure 4.27a) the transfer function is derived from Equation (4.22). Therefore: HD (s) =
0.625 K = Ts + 1 420s + 1
The Bode plots are presented in Figure 4.28. T is measured in s, since the frequency is measured in rad . s It may be observed that the dynamics are slow, implying oscillation frequencies between 0.06 osc/h and 60 osc/h (1 rad/s = 0.159 Hz). This is a reasonable frequency of oscillation for CSTR of volumes of the order of m3 and small heat transfer coefficients of around 1000 mkCal 2 h °C . This is happening in conditions of poor stirring. The cutting or angular frequency where the phase angle is φ = − π4 = −45∘ (see Figure 2.18), ω = 1 T
1 = 420 = 0.00238 rad . At this s 2π = 2639.9 s = 0.7333 h. 0.00238
value, the corresponding period of oscillation is Tosc =
If, for example, the input temperature of the reactant from the outdoor tank is oscillating in a nightday cycle (1 osc/24 hours), it will disturb the temperature control loop in a sinusoidal way, with the above frequency, in the range of 10−3 rad/s and 10−4 rad/s. The crossover frequency, appearing because of the other components of the loop coming with their own delays, will be placed closer to 10−3 rad/s meaning the temperature control loop is quite stable (see Chapters 8 and 9), its own period of os2π cillation being quite far from that of the disturbance, ωosc dist = 24⋅3600 = 7.26 ⋅ 10−5 rad . s
4.8 Equivalence to first order with time delay system
 129
Figure 4.27: (a) The profile for the output temperature change in the CSTR when the input temperature Ti∘ changes from 40 to 44 °C (+10 %). (b) The change of the temperature of the CSTR when the coolant flow decreases with 10 %.
130  4 Systems dynamics
Figure 4.28: Bode plots for the capacitive behavior of the path Ti∘ → T ∘ (see Figure 4.27a).
4.9 Problems (1) 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. (2) Compute the global transfer function of the control system presented in Figure 4.29, with the time constants and the pure time delay given in seconds.
Figure 4.29: Feedback control system with the proportional controller.
4.9 Problems  131
(3) Consider a tank equipped with a stirrer and a heating jacket, as presented in Figure 4.30.
Figure 4.30: Perfectly mixed tank with the heating jacket.
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: H(s) =
T2∘ (s) K1 = . Ti∘ (s) (Ta s + 1)(Tb s + 1)
(4.61)
Find the mathematical expressions of the parameters K1 , Ta , and Tb . (4) The data below in Table 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? 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 o [°C]
50
51
54
58
62
65
68
71
73
76
79
80
80.5
(5) 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 = 1 000 kcal/m2 hK and the mean residence time of the water agent in the jacket is t = 2 min. Neglecting
132  4 Systems dynamics 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? (6) A tank has the crosssectional 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. (7) 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 . (8) 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 Figure 4.31. The following parameters are known: V1 = V2 = V3 = 20 L, R1 = R2 = R3 = R4 = 108 Pa/(m3 /s), gas constant R = 8 314 J/kmolK, p4 = 0 (atmospheric pressure), the molecular mass of gas M = 29 kg/kmol and the temperature T o = 20 °C.
Figure 4.31: Cascade of tanks with gas accumulation.
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. (9) 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 Figure 4.32. (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 describing the relationship between the level h in the tank (output) of the R1 and the inlet flow rate F1 (input).
4.9 Problems  133
Figure 4.32: Liquid accumulation in the tank R1 .
(b) Linearize the obtained model and compute the transfer function of the process with the same inputoutput 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 – crosssectional 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 ,
134  4 Systems dynamics 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 (1 000 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 = 9 435 ≈ 9 500 (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 flowrestriction ξ4 = 40.
References [1] Agachi, S., Automatizarea Proceselor Chimice, Editura Casa Cărţii de Ştiinţă, ClujNapoca, 1994. [2] Cristea, M. V., Agachi, S. P., Elemente de Teoria Sistemelor, Editura Risoprint, ClujNapoca, 2002. [3] Agachi, S., Cristea, M. V., Lucrări Practice de Automatizarea Proceselor Chimice, Tipografia Univ., ClujNapoca, 1996. [4] Stephanopoulos, G., Chemical Process Control. An Introduction to Theory and Practice, Englewood Cliffs, NewJersey 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, McGraw Hill, 2000. [7] Baldea, M., Daoutidis, P., Dynamics and Nonlinear Control of Integrated Process Systems, Cambridge, Cambridge University Press, 2012.
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 Figure 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 (Figure 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 (Figure 5.2a). An experienced operator, knowing the dynamics of the process, opens the valve abruptly (Figure 5.2b), gives a thermal shock to the process, hurrying in this way the change of temperature (Figure 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.
Figure 5.1: Heat exchanger.
https://doi.org/10.1515/9783110647938005
136  5 Manual and automatic control
Figure 5.2: Operation of a heat exchanger.
5.2 Automatic control  137
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.
Figure 5.3: The temperature control system for the heat exchanger.
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 Figure 5.4.
Figure 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,
138  5 Manual and automatic control according to an embedded control algorithm and the sign and magnitude of the error; 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), transforms it in the reaction variable r with the “help” of the transducer T and feeds it back to the entrance of the system, where it is compared with the desired value (sp  set point reference, input or setpoint) inside the S; S calculates the deviation of the controlled variable from the setpoint (e – error; e = sp − r); 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 or Final Control Element (FCE) 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 y transformed in the reaction variable (r) from the setpoint (sp) 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). There are obvious advantages of the ACS: permanent monitoring and correction of the deviations of the parameters, without fatigue; one operator is able to supervise up to 100 parameters during the operation of the plant, making the supervision efficient and cost effective. But it has also disadvantages. ACSs require a better preparation of the personnel, increased initial costs (about 10 %–15 % additional costs to the total technological investment cost), 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 (Figure 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 (Figure 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
5.4 Stability and instability of controlled process and control systems  139
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.
5.4 Stability and instability of controlled process and control systems From the point of view of stability, the process can be selfregulating or nonselfregulating. A process is selfregulated (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 (Figure 5.5a). If the process, at the occurrence of a bounded disturbance, goes from a steady state to an unstable regime (Figure 5.5b), it is nonselfregulated. This is the case of the exothermic reactors operated at their unstable equilibrium point.
Figure 5.5: Selfregulating and nonselfregulating process.
There are cases when a stable by nature process becomes unstable due to a badly designed control system (Figure 5.6). This is a cause which sometimes makes the control loops to be set on “manual” operation.
Figure 5.6: The stable process (a) becomes unstable due to a bad control (b).
140  5 Manual and automatic control
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 (Figure 5.7): σ1 – overshoot; σ2 and σ3 are the amplitudes of the second and third oscillation; 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).
Figure 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; ζ ≥ ζimp (ζ = 1 −
σ3 ), σ1
(5.3)
5.6 Problems  141
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 oscillaσ tions respect the rule of quarter decay ratio, σ3 = 41 (Figure 5.8). 1
Figure 5.8: Good response of a control system with a quarter decay ratio.
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 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 ).
142  5 Manual and automatic control (3) Explain why the heating process of a classroom is a selfregulated process and the change of the flow rate at the entrance of a tank is either selfregulated (free outflow through a valve) or a nonselfregulated 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) 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, ClujNapoca, 1994, p. 4.
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 (Figure 3.1) with the steady state mathematical model given in the equations (6.1) and (6.2): ∘ Fvi ρcp Ti∘ − Fvo ρcp T ∘ − KT AT (T ∘ − Tag )=0
(6.1)
∘ ∘ ∘ Fvag ρag cpag Tiag − Fvag ρag cpag Tag + KT AT (T ∘ − Tag ) = 0.
(6.2)
∘ Tag
Eliminating between the two equations, the output temperature is expressed in the general form as ∘ T ∘ = f (Ti∘ , Fvi , Fvag , Ti∘ag , Text )
(3.1)
and, more detailed, T∘ =
∘ Fvag ρag cpag F vi ρcp Ti∘ +Fvi ρcp Ti∘ KT AT + Fvag ρag cpag Tiag KT AT
Fvo ρcp KT AT + Fvag ρag cpag Fvo ρcp + 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 Figure 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 (Table 3.2) is expressed by the relation ξ = https://doi.org/10.1515/9783110647938006
k(T ∘ ) VF
1 + k(T ∘ ) VF
,
(6.4)
146  6 The controlled process 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 openloop and closedloop control systems is measured for all possible inputoutput variable pairings. A ratio of this openloop “gain” to this closedloop “gain” is determined and the results are displayed in a matrix λ11 [λ [ RGA = Λ = [ 21 [ ⋅⋅⋅ [ 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 openloop 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, λij =
𝜕yi m 𝜕mj
= ct . 𝜕yi y = ct 𝜕mj
(6.7)
The numerator is the openloop gain determined with all other manipulated variables constant, and the denominator is the openloop 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.
6.1 Steady state behavior of the controlled process  147
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 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 [ G = [ ⋅⋅⋅ [ gn1 where the values gij =
g12 ⋅⋅⋅ gn2
⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅
g1n ] ⋅⋅⋅ ] gnn ]
(6.9)
𝜕yi . 𝜕mj
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. 1 Usually CN > 10 shows a system difficult to be controlled.
148  6 The controlled process 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.
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 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 Figure 6.1, and the steady state values of flowrate, temperature, and concentration are indicated. m indicates the manipulated variables and y the controlled variables.
Figure 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
T m + T2 m2 + T3 m3 T1 m1 + T2 m2 + T3 m3 = y2 = 1 1 y1 m1 + m2 + m3
(6.12) (6.13)
6.1 Steady state behavior of the controlled process  149
y3 =
c1 m1 + c2 m2 + c3 m3 c1 m1 + c2 m2 + c3 m3 = , y1 m1 + m2 + m3
(6.14)
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 g21 = 𝜕m1 (m1 + m2 + m3 )2 g11 =
g31 =
(6.15) (6.16)
𝜕y3 (c − c2 ) m2 + (c1 − c3 ) m3 = 1 𝜕m1 (m1 + m2 + m3 )2
(6.17)
𝜕y1 =1 𝜕m2 (T − T1 ) m1 + (T2 − T3 ) m3 𝜕y2 g22 = = 2 𝜕m2 (m1 + m2 + m3 )2
(6.18)
g32 =
(6.20)
g12 =
(6.19)
𝜕y3 (c − c1 ) m1 + (c2 − c3 ) m3 = 2 𝜕m2 (m1 + m2 + m3 )2
𝜕y1 =1 𝜕m3 (T − T1 ) m1 + (T3 − T2 ) m2 𝜕y2 = 3 = 𝜕m3 (m1 + m2 + m3 )2
g13 =
(6.21)
g23
(6.22)
g33 =
𝜕y3 (c − c1 ) m1 + (c3 − c2 ) m2 . = 3 𝜕m3 (m1 + m2 + m3 )2
(6.23)
Substituting the values in the picture, [ G=[ [ [
1
1
1
11 13 47 845
− 132
− 132
8 845
18 − 845
2 13
] [ ] R = (G−1 )T = [ 1 ] [ ] [ 0
4 13 − 52 65 2
7 13 3 2 − 65 2
] ] ]
(6.24)
]
and thus [ RGA = [ [
2 13 11 13
[ 0
4 13 5 13 4 13
7 13 3 − 13 9 13
] ]. ]
(6.25)
]
m1 , the flow rate of stream 1, should be used to control y2 , the temperature of the out11 put stream, since the value of λ21 is the closest to 1 in the matrix ( 13 ). The element 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
150  6 The controlled process 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. 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 Figure 6.2.
Figure 6.2: The nonlinear dependence of the output heat exchanger temperature on the steam flow.
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 Figure 6.2) the control system designed as in Figure 5.3 is efficient enough. But, in the situation of a much stronger disturbance (curve P1 in Figure 6.3), the controller is “ordering” the control valve to open at maximum, but does not compensate the caloric
Figure 6.3: The sizing of the control valve in the conditions of the nonlinear characteristic of the process.
6.1 Steady state behavior of the controlled process  151
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 (Figure 6.3). 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 Figure 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 (Figure 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, Figure 9.24). The choice being done, the total characteristic of the process + valve will be linear (Figure 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 (Figure 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 =
dy Δy = . Δm N dm N
(6.26)
152  6 The controlled process
Figure 6.4: Linearizing the process characteristic through a compensating valve characteristic.
Figure 6.5: The gain of the process is the slope of the characteristic of the process at the operating point.
6.2 Dynamic behavior of the controlled process 
For Example 6.2, where T∘ =
153
Ti∘ag 4Ti∘ + (T ∘i ag )Fag 4Ti∘ + = 4 + 2Fag 2 + 4 4 + 2Fag F ag
1 −2 dT ∘ ∘ + [4Ti∘ + (T ∘i ag )FagN ] , Kpr N = = (Ti ag ) dFag N 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 x(t1 ) = x1 . That is, given x0 , x1 , and t1 < ∞, we wish to find u(t), 0 < t < t1 , such that x1 = x(t1 ) = e
At1
t1
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 (Table 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 (Figures 6.6 and 6.7).
154  6 The controlled process 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
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 ratecontrolling subsystem of the measuring device. This is described by the equation Vmes
dρmes = F(ρmesin − ρmes ), dt
(7.49)
where Vmes is the measuring tank volume, while F and ρmesin 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, similar to the one presented for the inductive
7.8 Composition transducers  215
pressure transducer, or may have a differential transformer. The density measurement that 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. Example 7.4. 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 summerwinter temperature changes of ΔT ∘ = 30 K. The densityconcentration relationship is presented in Table 7.2, the volumetric thermal expansion is γ = 10−3 K−1 and the density at the temperature of T ∘ = 303 K is ρet = 0.7881 kg/dm3 for the pure ethanol. Table 7.2: Densityconcentration relationship. C [%] 3
ρ [kg/m ]
–
–
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
Considering the fact that the density of the measured liquid is bigger than that of the etalon liquid, immersing of the bubbling flow tube in the etalon liquid is larger than that in the measured liquid (the difference of the immersing depth is h = 100 mm). The reason is that at the minimum measured density ρmes_min , the difference of pressure Δp = (p+ − p− ) = 0; and at the maximum measured density, ρmes_max , the difference of pressure will be maximum (equal to the Δp measurement range). First, the etalon density is calculated at 20 °C: ρet20 °C = ρet30 °C /(1 + γΔT),
–
where the values from the right side of the equation are known (20 °Cis a considered reference). From condition: Δp = 0; H is determined: Δp = ρmas_min gH − ρet20 °C g (H + h) = 0 H=
ρet20 °C h
ρmes_min − ρet20 °C
216  7 Transducers and measuring systems
where: ρmes_min = 0.891 kg/dm3 , according to Table 7.2 at the highest liquid concentration 50 % + 10 % = 60 %. H=
(788.81 kg/m3 )(0.1 m) (891 − 788.81)kg/m3
= 0.771905274 m = 0.77 m –
Maximum pressure drop is obtained at the maximum value of the measured liquid density: Δpmax = ρmas_max gH − ρet20 °C g (H + h) where: ρmes_max = 0.935 kg/dm3 , according to data in Table 7.2 at the lowest concentration of the measured liquid, 50 %–10 % = 40 %. Δpmax = (935 kg/m3 )(9.81 m/s2 ) (0.77 m)
− (788.81 kg/m3 )(9.81 m/s2 ) (0.77 m + 0.1 m) .
= 7 062.7095 − 6 732.256707
= 330 Pa
Domain (The measurement range of the differential pressure transducer): Δpmin to Δpmax ↔ 0–330.45 Pa
7.8 Composition transducers  217
–
–
To calculate the errors due to the variation of temperature summerwinter (ΔT = 30 °C), precise calculations for the differential pressure Δp can be done, for example, for Δpmax , using the densities, measured and etalon, at 20 °C and 50 °C. It is expected that the pressure differences will be small, because the temperature affects both densities (measured and etalon). Observation: The construction of the device is done in such a way that the etalon and measured liquid have practically the same temperature (the box for the etalon liquid is immersed in the measurement liquid box). Variation of the transducer signal function of concentration is inverse proportional in the measured range (there are differential pressure transducers which can reverse the characteristic of dependence, making it direct proportional). ρet20 °C = 788.81 kg/m3
ρet50 °C = ρet20 °C / (1 + γΔT)
= (788.81 kg/m3 )/(1 + 10−3 .30 K) = 788.021978 kg/m3
Δpmax = ρmasm 50 °C gH − ρet50 °C g (H + h)
Δpmax = (935 kg/m3 )(9.81 m/s2 ) (0.77 m) /1.001
− (788.022 kg/m3 )(9.81 m/s2 ) (0.77 m + 0.1 m) .
= 7 055.653846 − 6 725.531175 = 330.12 Pa
Viscosity transducers Viscosity measurement is important not only for the study of the fluid’s flowing behavior but also as an 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. Utube, 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 7.44 presents the 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 phaseshift is possible due to the elastic coupling provided by the spring 4.
218  7 Transducers and measuring systems
Figure 7.44: Searle viscometer.
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 phaseshift φ and the viscosity: φ=
k1 ω η = Kη. k2
(7.50)
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 phaseshift φ. The rheostat electrical resistance 6 is fastened on the driving shaft system. The position of the cursor 5 is proportional to the phaseshift 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
7.8 Composition transducers  219
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
1 1S S = =σ R ρl 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 concentrationconductivity relationship, which is the basis for this concentration measurement method, is presented in Figure 7.45 for diluted, and in Figure 7.46 for concentrated solutions of electrolytes (acids, salts and bases).
Figure 7.45: Electrical conductivity of diluted solutions of electrolytes at the temperature of 18 °C.
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 Figure 7.47. The conductivity measuring cell consists in two parallel plate electrodes placed in the flowing conductive solution. The cell resistance, depending on liquid conductivity,
220  7 Transducers and measuring systems
Figure 7.46: Electrical conductivity of concentrated solutions of electrolytes at the temperature of 18 °C.
Figure 7.47: Electrical conductivity measurement principle.
7.8 Composition transducers  221
is introduced in a branch of the electrical measuring bridge. The measuring bridge is supplied with alternating current (1 000 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 electrical current flow. The electrical conductivity depends on temperature, as shown in Figure 7.48.
Figure 7.48: Electrical conductivity dependence on temperature.
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 Figure 7.47, can be used for the conductivity measurements ranging from 0.05 μS ⋅ cm−1 to 200 μS ⋅ 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 Figure 7.49. 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 μS ⋅ cm−1 . Usually, the calibration of the instrument is performed using KCl etalon solutions, at
222  7 Transducers and measuring systems
Figure 7.49: Electrical conductivity transducer without electrodes.
controlled temperature. The accuracy class of the instrument is higher than 1 % and its time response is almost instantaneous. 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+ and of the hydroxyl ions OH− are 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+ ,
pOH = − log10 COH− ,
pKH2 O = − log10 KH2 O .
(7.54) (7.55) (7.56)
According to these definitions, equation (7.52) becomes: pH + pOH = pKH2 O .
(7.57)
7.8 Composition transducers  223
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 halfcell) may have different types: hydrogen, quinhydrone, metalmetallic oxide, glass, etc. The most commonly used one is the glass electrode, presented in Figure 7.50. The glass electrode consists of the glass membrane 1, the buffer solution 2, the platinum electrode 3 and an insulated connection wire 4. As reference electrode, the calomel electrode is the most common. It is presented in Figure 7.51.
Figure 7.50: Glass electrode for pH measurement.
224  7 Transducers and measuring systems
Figure 7.51: 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 halfcell 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. 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 Figure 7.52, 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 Figure 7.53. It provides the compensation of the pH changes with 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)
7.8 Composition transducers  225
Figure 7.52: pH dependence on temperature.
Figure 7.53: pH transducer.
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 = ue20 °C + Δu.
(7.61)
226  7 Transducers and measuring systems 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 MERE galvanic cell has a high internal electrical resistance, Rint = 0.1–1 MΩ. As presented in Figure 7.54, 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
1
1+
Rint Ri
.
(7.62)
Figure 7.54: Electrical scheme of the pH transducer.
In order to prevent building material from being deposited, the electrodes’ construction provides different selfcleaning 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 microprocessorbased 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 Figure 7.55. The redox reactions involving oxygen are O2 + 2H2 O + 4e− → 4OH−
(7.63)
7.8 Composition transducers  227
Figure 7.55: Dissolved oxygen transducer.
at the anode, and Pb + 2OH− → 2e− + Pb(OH)2
(7.64)
at the cathode. A galvanic cell is formed and its e. m. f. u depends on the DO concentration: 0 εAg = εAg +
a RT ∘ ln 0 zO F aO2−
(7.65)
0 εPb = εPb +
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 Figure 7.56. The electrochemical reactions are O2 + 2H2 O + 4e− → 4OH−
(7.68)
4Ag + 4Cl− → 4AgCl + 4e−
(7.69)
at the cathode, and
228  7 Transducers and measuring systems
Figure 7.56: Polarographic dissolved oxygen transducer.
at the anode. 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 Table 7.3. 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 Figure 7.57, for a set of common 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.
7.8 Composition transducers  229 Table 7.3: 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 0–1 000 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
Figure 7.57: IR absorption spectra of some gases.
The measuring principle of the IR analyzer in presented in Figure 7.58. 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.
230  7 Transducers and measuring systems
Figure 7.58: IR analyzer.
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 = εSc /d changes proportional to the component C1 concentration in the 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)
7.8 Composition transducers  231
C=
εSc εSc εSc = = , d dmax − x dmax − (p3 −p4 )SM K
(7.73)
m
but from the ideal gas law the pressure is related to the temperature p3 V3 = n3 R T3∘
p4 V4 = n4 R
(7.74)
T4∘ ,
(7.75)
and equation (7.73) becomes εSc
C= dmax −
T∘
T∘
3
4
R(n3 V3 −n4 V4 ) Km
(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 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 3k
(7.77)
2E0 e−αCC1 l , 3k
(7.78)
T3∘ = and T4∘ =
where E0 is the energy of the light emitted by the lamp and k is the Boltzmann constant. Equation (7.76) becomes C=
dmax −
εSc 2E0 R SM n3 ( 3k Km V3
−
n4 −α C1 l e ) V4
.
(7.79)
The concentrationdependent capacitance of the condenser is a branch of the electrical bridge supplied by alternating 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 components of the emitted IR radiation having the wavelengths 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 %.
232  7 Transducers and measuring systems 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 Table 7.4. Table 7.4: Thermal conductivity of different gases λ, [cal/(cm s grd)]. Gas λ ⋅ 105 , 0 °C
Gas λ ⋅ 105 , 0 °C
Gas
He Ne Ar H2 N2 O2 Cl2
CO CO2 HCl SO2 H2 S NH3 Air
CH4 C2 H6 C3 H8 C4 H10 C2 H4 C2 H2 CHCl3
34 11 3.9 41.6 5.81 5.89 1.88
5.6 3.4 2.7 2.0 3.1 5.2 5.83
λ ⋅ 105 , 0 °C 7.2 4.3 3.6 3.2 4.2 4.5 1.6
Gas CH2 Cl2 CH3 Cl CH3 OH Ag Invar Glass H2 O (l)
λ ⋅ 105 , 0 °C 1.6 2.2 3.45 105 2 600 200 130
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 = λ1 x1 + λ2 x2 .
(7.81)
This is only valid for particular gas mixtures, such as airCO or airCH4 . Other mixtures, such as airNH3 , CO–NH3 , or airsteam, 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 Figure 7.59. The gas concentration measuring principle, based on thermal conductivity, is presented in Figure 7.60. 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
7.8 Composition transducers  233
Figure 7.59: Dependence of the N2 –H2 gas mixture thermal conductivity on the H2 fraction.
Figure 7.60: Wheatstone bridge for measuring the composition of a gas mixture, based on thermal conductivity.
are equally cooled by the gas having the same thermal conductivity. In this situation R1 RRC = 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
234  7 Transducers and measuring systems 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 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: χT ∘ = 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 Figure 7.61. 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 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
7.8 Composition transducers  235
Figure 7.61: Oxygen gas analyzer based on paramagnetism.
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 7.62 schematically represents the zirconia oxygen measuring cell. 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)
236  7 Transducers and measuring systems
Figure 7.62: Zirconia oxygen gas cell.
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.
7.8 Composition transducers  237
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. 6.
pp ps
.
(7.88)
Drybulb temperature is the temperature of the measuring gas mixture. Wetbulb 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 drybulb temperature and total pressure of the gas mixture are known. Psychrometric diagrams allow the conversion of dew point, relative humidity, absolute humidity, dry and wetbulb 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 drybulb and wetbulb 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 Figure 7.63. 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 wetbulb is continuously wetted by a wick supplied with distilled water which causes a temperature drop due to evaporation. The drybulb and wetbulb tem∘ , the drybulb temperature and the relative humidity relaperature difference T ∘ –Twb tionship is presented in the plots of Figure 7.64. As presented in Figure 7.64, a curve describing the drybulb temperature and psychrometric difference relationship corresponds to each given air relative humidity. For
238  7 Transducers and measuring systems
Figure 7.63: Psychrometer based on dryand wetbulb thermometers.
Figure 7.64: Psychrometric chart for relative humidity measurement.
practical calculations, the Sprung formula may be used for computing the partial pressure of water from the drybulb and wetbulb temperature difference: pp = pwb − 0.5
p ∘ (T ∘ − Twb ), 760
(7.89)
7.8 Composition transducers  239
where the 0.5 coefficient is the Assmann psychrometer constant, pwb is the saturation ∘ partial pressure at the wetbulb temperature Twb and p is the air pressure. 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 drybulb and wetbulb temperature, the psychrometric transducer is obtained. Its schematic representation is shown in Figure 7.65.
Figure 7.65: 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 thermoelectrically cooled metallic surface. The principle of the chilled mirror humidity measuring method is presented in Figure 7.66. The hygrometer works on a semicontinuous 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 photo
240  7 Transducers and measuring systems
Figure 7.66: Chilled mirror hygrometer.
transistors receive the reflected beam having high amplitude. The thermoelectrical 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 electrooptical 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 %. 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 Figure 7.67. The time elapsed from the injection of the sample and the appearance of the ith 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 ith 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 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
7.8 Composition transducers  241
Figure 7.67: Chromatogram showing each of the A, B, C, and D components removed by desorption at fixed moments of time.
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 =
Ai fi , n ∑i=1 Ai fi
(7.91)
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 nth 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 Figure 7.68.
Figure 7.68: 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
242  7 Transducers and measuring systems operating conditions specific to the analyzing unit 4. For the case of the singleflux chromatograph, the conditioned sample enters directly in the analyzing unit 4. For the multiflux 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 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 samplecarrying 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, xray, ultraviolet, visible, nearinfrared, midinfrared, farinfrared, 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 spec
7.9 Problems  243
trometers, quadrupole mass spectrometer, ion cyclotrone resonance mass analyzer, ultraviolet and visible spectrometers, Raman scattering emission spectrophotometers, nuclear magnetic resonance spectrometers, Xray fluorescence spectrometers, microwave spectrometers, neutron activation analyzers and atomic emission spectrometers. Fault detection and diagnosis of sensor operation has continuously grown during the last decades, leading to the development of the family of intelligent sensors [12]. The multivariate statistical analysis based on data driven approaches offers sound theoretical methods to serve them [13]. Some of the most frequently used techniques are: Principal Components Analysis and Regression, Partial Least Squares, Support Vector Machine, Artificial Neural Networks, Kalman Filtering and Adaptive Learning [13, 14]. They are also the core of the soft sensors used to build inference instrumentation which is able to compete with the online measuring equipment in terms of cost and promptitude of the response [15, 16].
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?
244  7 Transducers and measuring systems (5) Consider a volumetric flow rate measuring system based on the orifice plate, differential pressure transducer and square root extractor, presented in Figure 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 3 200 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. (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 conductivityelectrical 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) 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 mm H2 O to 2 000 mm H2 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 7.5. Table 7.5: Pressure drop coefficients. (D0 /Dp )2 ξ
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
7 000
1 670
730
400
245
165
117
96
65.6
51.5
(8) Consider a level transducer based on float with the following characteristics: crosssectional area of the float A = 18 cm2 , fluid density ρ = 1 000 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.
References  245
(9) 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 Rintglass = 100 MΩ and of the calomel electrode is Rintcal = 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] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16]
Millea, A., Cartea metrologului Metrologie generală, Editura Tehnică, Bucureşti, 1985. http://www.frenchmetrology.com/en/more/glossary.asp. Agachi, S., Automatizarea Proceselor Chimice, Editura Casa Cărţii de Ştiinţă, ClujNapoca, 1994. McMillan, G. K. (editorinchief), Considine, D. M. (late editorinchief), Process/Industrial Instruments and Controls Handbook, 5th Edition, McGrawHill, 1999. Bsata, A., Instrumentation et Automatisation dans le Controle des Procedes, Deuxieme edition, Les Editions Le Griffon d’argile, Quebec, 1994. http://www.systechillinois.com/en/zirconiaoxygenanalysis_51.html. http://www.torayeng.com/measuring/tec/zirconia.html. Agachi, S., Cristea, M. V., Lucrări Practice de Automatizarea Proceselor Chimice, Tipografia Univ., ClujNapoca, 1996. Cristea, M. V., Agachi, S. P. Elemente de Teoria Sistemelor, Editura Risoprint, ClujNapoca, 2002. Stephanopoulos, G., Chemical Process Control. An Introduction to Theory and Practice, Englewood Cliffs, NewJersey 07632, Prentice Hall, 1984. Marinoiu, V., Paraschiv, N., Automatizarea Proceselor Chimice, Vol. I, Vol. II, Editura Tehnică, Bucureşti, 1992. Qin, S. J., Li, W., Detection and identification of faulty sensors in dynamic processes. AIChE J., 47 (7), (2001), 1581–1593. Kadlec, P., Gabrys, B., Strandt, S., Datadriven soft sensors in the process industry, Comput. Chem. Eng., 33 (4), (2009), 795–814. Kadlec, P., Grbic, R., Gabrys, B., 2011. Review of adaptation mechanisms for data driven soft sensors. Comput. Chem. Eng., 35 (1), (2011), 1–24. He, Y. L., Geng, Z. Q., Zhu, Q. X., Soft sensor development for the key variables of complex chemical processes using a novel robust bagging nonlinear model integrating improved extreme learning machine with partial least square, Chemom. Intell. Lab. Syst., 151, (2016), 78–88. Mendes, J., Souza, F., Araújo, R., Gonçalves, N., Genetic fuzzy system for datadriven soft sensors, Appl. Soft Comput., 12 (10), (2012), 3237–3245.
8 Controllers 8.1 Classification of controllers The controller is the element of the control system (Figure 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 (ProportionalIntegral), PID (ProportionalIntegralDerivative), 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 continuousaction controller (Figure 8.1).
Figure 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 discreteaction one (Figure 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), twoposition or threeposition controllers can be used. This is the case with domestic applications where the thermostats of the central heating, of the flat iron or of the refrigerator are usually onoff controllers (Figures 8.2a and 8.3). https://doi.org/10.1515/9783110647938008
248  8 Controllers
Figure 8.2: Two types of discrete actions of the controller: (a) onoff, twoposition controller; (b) digital controller.
Figure 8.3: Bimetallic controller of the flat iron (see SAMSON http://www.docentes.unal. edu.co/sorregoo/docs/sistemas%20de% 20control.pdf) [2].
8.1 Classification of controllers  249
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 (Figure 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
c = c0 + Kc (e +
1 ∫ e (τ) dτ), Ti
(8.2)
0
1 Ti
t ∫0
where Ti is the integral time constant and e(τ)dτ is the integral component of the control signal. The proportional integral derivative (PID) controller has the control action expressed by equation (8.3): t
de (t) 1 ), c = c0 + Kc (e + ∫ 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 (Figure 8.4). The specialized controllers have the actuator embedded in the controller (e. g. the gas pressure controller on the methane gas pipelines, or the waterlevel controller in the reservoir of the flush toilet, see Figure 8.5). 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 Figure 8.5 is an example of a controller using the energy of the controlled process: accumulation of mass in a reservoir.
250  8 Controllers
Figure 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).
Figure 8.5: The level controller in the flush toilet reservoir.
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 Figure 8.6.
8.2 Classical control algorithms  251
Figure 8.6: P controller response at a step input.
The result of such a behavior on the heater (Figure 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 . Δr
(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 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 Figure 8.7.
Figure 8.7: Steady state characteristic of the proportional controller.
252  8 Controllers The relationship between the Proportional Band and the controller gain is equation (8.5) PB =
100 . Kc
(8.5)
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 acidstrong base and the other one weak acidweak base (Figure 8.8), the titration curve is different, presenting different Kpr , much larger in the first case (the slope is about 1 000 : 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.
Figure 8.8: Two neutralization processes at which the pH is controlled: (a) strong acidstrong base; (b) weak acidweak base.
In order to properly adjust the pH to the value 7, the system in Figure 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 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 Figure 4.2.
8.2 Classical control algorithms  253
The P control with steadystate 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 (Figure 8.9), which controls the pressure produced by an airblower in a pipeline.
Figure 8.9: Pressure control system with a P controller.
In the normal steady state condition, the values of the pressure are p0 = 200 mm H2 O;
Δp1 = 50 mm H2 O;
Δpv = 50 mm H2 O;
pi = 150 mm H2 O;
p = 100 mm H2 O.
The steady state characteristics of the elements in the control system are Transducer: Measurement range of the pressure transducer: 0 ⋅ ⋅ ⋅ 200 mm H2 O Output range of the transducer: 2 ⋅ ⋅ ⋅ 10 mA Measuring 100 mm H2 O, the output signal xr = 6 mA The steady state characteristic is given in Figure 8.10.
Figure 8.10: Steady state characteristic of the pressure transducer in the example.
254  8 Controllers 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 mm H2 O to be maintained) At p = 100 mm H2 O, xr = xr0 = 6 mA, e = 0, and c = c0 = 6 mA The steady state characteristic of the controller is given in Figure 8.11.
Figure 8.11: Steady state characteristic of the controller in the example.
Actuator (control valve) + electropneumatic convertor: Input range of signal: 2 ⋅ ⋅ ⋅ 10 mA (0 ⋅ ⋅ ⋅ 100 %) Stem variation range: 0 ⋅ ⋅ ⋅ 100 % Pressure drop range: 0 ⋅ ⋅ ⋅ 100 mm H2 O Stem position at steady normal state: h = 50 % Pressure drop value at normal steady state: Δpv = 50 mm H2 O When p = 100 mm H2 O, xr = 6 mA, e = 0 and c = c0 = 6 mA, Δpv = 50 mm H2 O The steady state characteristic of the control valve is given in Figure 8.12.
Figure 8.12: Steady state characteristic of the control valve in the example (h = h(xc )) and (Δpv = Δpv (h)).
8.2 Classical control algorithms  255
The control system aims to keep 100 mm H2 O in the pipeline in the presence of disturbances. Let us analyze the behavior of the system. Suppose a disturbance of +20 mm H2 O at the tap after the blower, meaning that Δp1 = 70 mm H2 O. This can be caused by the manipulation of the tap during an ordinary maintenance operation. The consequences are: pi = 130 mm H2 O, the valve does not react yet and Δpv = 50 mm H2 O, the pressure in the pipeline is p = 80 mm H2 O, giving a deviation of −20 mm H2 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 mm H2 O. The result after the first intervention of the controller is thus p = 130 mm H2 O– 40 mm H2 O = 90 mm H2 O, an increase of the pressure with 10 mm H2 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 Table 8.1 and at different controller gain factors. Table 8.1: Functioning of the pressure control system in time at different controller gain factors. t p [s] [mm H2 O] Kp = 0.5 0 100 2 80 4 85 6 83.75 8 84.37 10 84 Kp = 1 0 100 2 80 4 90 6 85 8 87.5 10 86.25 Kp = 2 0 100 2 80 4 100 6 80 8 100 10 80
Δp [%]
ΔXr [%]
Xr [mA]
ΔXc [%]
0 −10 2.5 −1.25 0.625
0 −10 2.5 −1.25 0.625
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 50 −10 40 5 45 −2.5 42.5 1.25 43.75
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
6 0 5.2 −5 5.4 1.25 5.3 −0.125 5.35 0.625
Xc [mA]
Δh [%]
h
6 0 50 5.6 −5 45 5.7 1.25 46.25 5.65 −0.625 46.62 5.67 0.31 45.93
50 30 50 30 50
Δpr
[mm H2 O]
50 45 46.25 46.62 45.93
est1 = 100 − 84 = 16 mm H2 O = 8%
50 40 45 42.5 43.75
est2 = 100 − 86.7 = 16 mm H2 O = 6%
50 30 Oscillating 50 operation 30 mode 50
256  8 Controllers
Figure 8.13: Dynamic behavior of the pressure control system in Figure 8.9.
The dynamic behavior of the control system is illustrated in Figure 8.13. 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 (Table 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).
s→0
(8.6)
t→∞
For the pressure control system formerly discussed, the block diagram is given in Figure 8.14.
Figure 8.14: Block diagram of the pressure control system.
For this example mA mA Kv mm H2 O 100 Hv (s) = ; Kv = = 12.5 ; Tv s + 1 8 mA mm H2 O Hpr d (s) = Kpr d = 1 mm H2 O Hc (s) = Kc = 1
Tv = 3 s
8.2 Classical control algorithms  257
(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 ; KT = = 0.04 ; TT s + 1 200 mm H2 O mm H2 O Hpr m (s) = Kpr m = 1 mm H2 O
TT = 1 s
(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) =
1+
Kpr d Kv K KT K Tv s+1 c TT s+1 pr m
=
1+
1
12.5 3s+1
1 0.04 1 s+1
.
Then lim sH(s)D(s) = lim s
s→0
s→0
−20 (3s + 1)(s + 1) ⋅ = lim Δp(t) = −13.5 mm H2 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 from 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 (Figure 3.1). The heated fluid flow is of F = 5 m3 /h, having the density of ρ = 1 000 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 ⋅ 1 000 ⋅ 1 ⋅ 2 = 10−2 Gcal/h. In one year, a plant is working 7 200 hours and with a cost of a Gcal of around 70 Euros, the total money loss is Economic loss = ΔQ ⋅ 7 200 ⋅ 70 = 5 040 Euros/year.
258  8 Controllers 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 ProportionalIntegral 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 Figure 8.15. Ti , the integral time, representing the time until the initial proportional step doubles. The demonstration of this statement is in the case of a forced constant error e = A produced by the step input Δr = A: t
K A 1 Δc = Kc (A + ∫ Adτ) = Kc A + c t = 2Kc A Ti Ti
(8.7)
0
when t = Ti .
Figure 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
8.2 Classical control algorithms  259
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 Figure 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.
Figure 8.16: The response of a slow controlled process to a too fast control action imposed by a small integral time.
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 (Figure 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 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)
260  8 Controllers
Figure 8.17: Pressure control in recipients with different volumes.
and considering the expression of density ρ = (8.12) and (8.13) Rp1 Rp2
Rp1 + Rp2
pM , RT ∘
the equations (8.8)–(8.11) become
Rp1 Rp2 V1 M dp1 pi + p + p1 = ∘ RT dt Rp1 + Rp2 Rp1 + Rp2 0
Rp1 Rp2 V2 M dp2 Rp1 Rp2 p + p . + p2 = Rp1 + Rp2 RT ∘ dt Rp1 + Rp2 i Rp1 + Rp2 0
(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 ), Ti s
(8.14)
1 1 ) = Kc (1 − j ), Ti jω Ti ω
(8.15)
HPI (s) = Kc (1 + the frequency function HPI (jω) = Kc (1 + and the module and phase angle M(ω) = Kc √1 +
1 Ti2 ω2
and φ(ω) = − tan−1
The Bode plots are given in Figure 8.18.
1 . Ti ω
(8.16)
8.2 Classical control algorithms  261
Figure 8.18: Bode plots for the PI controller.
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 antiwindup solutions to be applied, but the most reliable is to disconnect the I com
262  8 Controllers ponent of the controller until the controlled variable enters in a controllable range (say, ± 10 % of the setpoint). Modern controllers have the antiwindup setup [5].
8.2.3 ProportionalIntegralDerivative controller (PID) [6] The equation describing the PID algorithm is (8.3). The step response of the controller is given in Figure 8.19.
Figure 8.19: Step response of the PID controller.
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 like 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 (Figure 8.20). One can observe the volumes are different, V1 > V2 . According to the heat balances (3.11) where the term of reaction is 0, ∘ Fvi ρcpA Ti∘ − Fvo ρcp T ∘ − KT AT (T ∘ − Tag )=
d (Vρcp T ∘ ), dt
(8.17)
8.2 Classical control algorithms  263
Figure 8.20: Choosing comparatively the values of Td for the temperature controllers in two different drying chambers.
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 ρcp
Fρcp + KT AT
and Tpr2 =
V2 ρcp
Fρcp + 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 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 + Td s). HPID (s) = Kc (1 + 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 −
The Bode diagram is presented in Figure 8.21.
1 ). Ti ω
(8.22)
264  8 Controllers
Figure 8.21: Bode plots for the PID controller.
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
8.2 Classical control algorithms  265
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 (iPID) 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.
8.2.4 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. 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 (Figure 8.22).
Figure 8.22: Steady state characteristic of the reactor producing aqueous chloramine solution in the hydrazine hydrate process [12].
266  8 Controllers 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 Figure 8.23 [13].
Figure 8.23: The structure of an adaptive controller.
There are several surveys and comprehensive scientific contributions to the theory and practice of selfadaptive controllers [14–16]. The authors preferred to mention a practical selfadaptive controller produced by the company ASEA Novatune and used later on to control an ethylene oxide reactor in the presence of a catalyst [17]. The control scheme is given in Figure 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 Figure 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 Figure 8.23, and is presented in Figure 8.25.
8.2 Classical control algorithms  267
Figure 8.24: Adaptive control of an ethylene oxide reactor [17].
Figure 8.25: The structure of a selfadaptive controller taking into consideration the setpoint value.
Adaptive control can be used in many processes such as PVC batch polymerization or εcaprolactam, 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 (εcaprolactam) influencing the initial tuning of the controller’s parameters. Optimal controllers The optimal control implies minimizing or maximizing a global performance index: N
J(k, x(k)) = ∑ f (x(k), u(k)). k=1
(8.23)
268  8 Controllers 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 Figure 8.26, x(k) is the distillate average concentration, and the control action is the optimal reflux ratio time function R(k) = u(k) (Figure 8.26). The general structure of such an optimal control scheme is given in Figure 8.27.
Figure 8.26: Optimal operation of a batch distillation.
Figure 8.27: The structure of an optimal control system using the change of the setpoint value.
8.2 Classical control algorithms  269
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 Figure 8.28.
Figure 8.28: Principle of model predictive control.
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)
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 (Figure 8.29).
270  8 Controllers
Figure 8.29: PID and NMPC control of the PVC reactor for the optimal temperature profile.
8.2.5 Distributed Control Systems and Supervisory Control and Data Acquisition Systems Distributed Control Systems (DCS) The development of Distributed Control Systems (DCS) is directly related to the huge potential offered by the microprocessor and data communication technologies that emerged in the 1970s, with the most representative product at that time the Honeywell TDC2000. The DCS [24] principally consists of integrated hardware and software systems accomplishing two basic functions for the realtime management of the enterprise: monitoring and control (Figure 8.30). Its architecture is designed so that the computedbased 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
8.2 Classical control algorithms  271
Figure 8.30: Block diagram of a distributed control system.
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.
272  8 Controllers The distributed architecture of the DCS demonstrates very great flexibility due to the straight reconfiguration of its software. Builtin 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 modelbased 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 conceptbased design, implemented in a harmonious architecture that integrates different subsystems communicating which each other, having distributed functions and responsibilities in such a way as to make the DCS work as a whole and offer the means for finding the globally optimal control solutions. Example 8.5. The control system in Figure 8.31 has the following characteristics: Hc (s) = kc ;
HAD (s) =
Figure 8.31: Feedback control system.
10 ; 2s + 1
Hpr (s) =
0.5 ; 30s + 1
HT (s) =
1 10s + 1
8.2 Classical control algorithms  273
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 %? For PB = 50 % kc =
100 =2 PB
The Bode plots and Nyquist plot for the system in Example 8.5 with PB = 50 % are presented in Figures 8.32 and 8.33.
Figure 8.32: Bode plot for PB = 50 %.
274  8 Controllers
Figure 8.33: Nyquist plot for PB = 50 %.
For PB = 10 % For PB = 10 %, the Bode and Nyquist plots are presented in Figure 8.34 and Figure 8.35. Looking at the Bode plots, it is obvious that the overall gain increases with the gain of the controller (or decrease of the PB). At the same time, since the controller has P configuration, the crossover frequency does not change.
8.2 Classical control algorithms  275
Figure 8.34: Bode plot for PB = 10 %.
276  8 Controllers
Figure 8.35: Nyquist plot for PB = 10 %.
Supervisory Control and Data Acquisition (SCADA) [25] SCADA stands for Supervisory Control and Data Acquisition Systems and are designed to collect field information, transfer it to a central computer facility, and display the information to the operator, rather than to properly control the process. They are not fully fledged control systems being focused more on supervision and monitoring. They collect the information from the process monitored, transfer it to the central station, process the data collected and then display the information or the results of the analysis (in graphic or text form) on a central or on several distributed displays. The four parts of a SCADA system are: – Field Instruments – PLC/Remote Terminal Unit – Communications Link – Central Computer Station A possible scheme of a SCADA system is given in Figure 8.36.
8.2 Classical control algorithms  277
Figure 8.36: SCADA system; PLC Programmable Logic Controller; IED – Intelligent Electronic Device; RTU – Remote Terminal Unit; MTU – Master Terminal Unit.
While a SCADA system and a DCS system are similar, there is a basic difference between them: a SCADA system is event driven and operator centric. It is a data gathering orientated system and eventually coupled to a remote control station; a DCS is process state driven. It is directly connected with field devices and control is done locally and automatically. The operator is just informed of what has happened having no essential role in the process of monitoring and control. A SCADA master station generally considers changes of state having as the main criterion driving the system of data gathering and presentation. A change of state will cause the system to generate database updates, alarms, events, and any special processing required relating to that (analysis). Event lists and alarm lists are of major importance to the operator. Conversely, DCS systems are process control systems that are state based, and the process variable’s present and past states are the main criteria driving them. PLC (see paragraph 8.2.5) protocols are generally register scanning based, with no specific change of state processing provided. DCS software tasks are generally run sequentially, rather than event driven. If a process starts to move from a set parameter, the DCS responds to maintain that parameter value. Notifying the operator is a secondary consideration. Events and alarm lists are secondary in importance to the process displays, importance being given to keeping the parameters at the set points or following a certain trend.
8.2.6 Programmable Logic Controllers (PLC) [26–28] A PLC is a controller based on a microprocessor which is using a programmable memory to store instructions and implement functions, usually logic functions (such as AND, OR, IF. . . ) used in controlling, timing, sequencing, switching. The term logic is used because programming is primarily concerned with implementing logic and switching operations; for example, if A (temperature in the reactor is higher than
278  8 Controllers 250 °C) and B (pressure in the reactor is higher than 12 bar) occurs, switch on C (open the evacuation valve of the reactor); if A or B occurs, switch on D (cut the supply flow of the reactant). Input devices (that is, sensors such as switches) and output devices (motors, valves, etc.) in the system being controlled are connected to the PLC. The operator then enters a sequence of instructions, a program, into the memory of the PLC. The controller then monitors the inputs and outputs according to this program and carries out the control rules for which it has been programmed. The majority of PLCs today are modular, allowing the user to add an assortment of functionality including: discrete control, analog control, PID control, position control, motor control, serial communication and highspeed networking [27]. Emergence of PLCs induced a huge progress in the operation and maintenance of the control and switching in process industries. Before PLCs, main tasks were performed by Electromechanical Relays to switch power, to control power relays, there were used Timers and Counters. The maintenance was a nightmare, troubleshooting was a nightmare, everything was hardwired, coils burnt, contacts wear out. It was time consuming to wire, debug, and change and more than that, it consumed lot of power (Figure 8.37).
Figure 8.37: Electromechanical panel for switching and interlocking [27] (Electromechanical Relays Panel of the Monomer Plant Rîșnov, Romania, 1973).
Nowadays, PLCs use the solid state technology instead of mechanical one, and are much more reliable; they are easily programmed and maintained, easy to modify the input and output devices, able to function in an industrial environment. At the same time, they are more costeffective (if a system has more than 6 relays it will be cheaper to install a PLC), very easy to replicate, far more versatile, easily expandable (modular); they are easy to network and communicate with other systems and are reusable. PLC Structure The general scheme of the PLC is given in Figure 8.38 and its architecture is given in Figure 8.39.
8.2 Classical control algorithms  279
Figure 8.38: PLC structure [29].
Figure 8.39: PLC architecture [29].
It can be observed that it is very similar to a computer, including CPU, Program, Memory, Inputs, Outputs. A typical physical operating PLC is shown in Figure 8.40 together with its components. Concerning the CPUs, one has to consider following features: Program Memory (Complexity, # loops), Data Memory (how many devices), Scan time (controller speed), if PID control is required, Integrated I/O etc. For the Input Modules (Figure 8.41) [27], there are two classes of inputs: Digital (either 1 or 0 signals) – Switch, Push Button, Proximity Sensors etc. and Analog (continuous range) – Temperature, Pressure, Flowrate, Weight, Interval of the Current Range (4–20 mA e. g.), or Voltage (±10 V e. g.), etc. For the Output Module (Figure 8.42), output signals from PLC to field devices there are of two classes: Digital (either 1 or 0 signals) – Relays, Solenoid valves, LED indicators; and Analog (continuous variable for continuous actuators) – Voltage (±10 V e. g.), Current (4–20 mA e. g.).
280  8 Controllers
Figure 8.40: PLC structure with the component parts. The PLC is part of the pyrolysis plant in the Chemical, Materials and Metallurgical Department, Botswana International University of Science and Technology.
Figure 8.41: Input PLC module.
8.2 Classical control algorithms  281
Figure 8.42: PLC Output module.
PLC programming [27, 28] Antonsen [28] gives a detailed picture of the possibilities of programming the PLCs. Normally, the programming is done externally and downloaded to the PLC memory. The main category of programming languages – IEC611313 languages – comprises of function block diagram (FBD), ladder diagram (LD), sequential flow chart (SFC), structured text (ST; similar to the Pascal programming language), instruction list (IL; similar to assembly language). All these are presented in Figure 8.43. Additionally, Wiring Diagrams (Figure 8.44), showing the actual connection points for the wires to/from the components and terminals of a controller, and Line Diagrams (Figure 8.45), also called schematic or elementary diagrams, show the circuits which form the basic operation of the controller. They do not indicate the physical relationships of the various components in the controller. They are an ideal means for troubleshooting a circuit. Example 8.6. In the following sequence, an example of a mixing process controlled by a PLC (Figure 8.46) is described. In a process of mixing wax, the motor must automatically start when predetermined pressure and temperature are reached; pressure switch closes when pressure
282  8 Controllers
Figure 8.43: The components of PLC programming [27].
Figure 8.44: Wiring diagrams.
is reached and temperature switch closes when temperature is reached. Manual push button switch is required to start the motor to override the pressure and temperature switches.
8.2 Classical control algorithms  283
Figure 8.45: Ladder Diagrams.
Figure 8.46: PLC of a mixing process [27].
The Control System design needs to assure the functions: Motor starter coil is energized when manual switch button is pressed OR when both the Pressure AND the Temperature set points have been reached (Figure 8.47a,b).
284  8 Controllers
Figure 8.47: (a) Wiring diagram. (b) Ladder diagram.
Figure 8.48: Input/Output connections.
The I/O connections for this case are presented in Figure 8.48. Example 8.7. Industrial application to measurement of the components of the flue gas in a steel plant. Plant In a Steel Plant, one has to monitor the content of CO, CO2 , H2 and O2 in a steel furnace. The scope of the CO2 measurement is to eliminate the infrared crossinterference with the CO measurement. Usually this is done by IR analyzers, installing appropriate filters.
8.2 Classical control algorithms  285
Analyzer The process analyzer was equipped with two different O2 measurement channels due to the fact that the process values could vary between low values (approx. 1 000 ppmv) and high values (up to 21 % vol.). The low concentration O2 channel uses an electrochemicaltype sensor while the high concentration O2 channel uses a paramagnetictype sensor (see Chapter 7). PLC Application If high levels of O2 enter for a sufficient amount of time in the electrochemical sensor, it will quickly consume its electrolyte solution rendering the sensor inactive. To solve this problem, a PLC program was written for the internal PLC of the analyzer. The PLC inputs were the O2 concentration (from both channels) and the analyzer calibration status. Based on this information, the PLC switched between O2 benches to avoid electrolyte depletion of the electrochemical cell. Whenever the O2 concentration measured by “low concentration” electrochemical sensor went above 8 000 ppmv, the PLC sends a digital signal to a 3/2way solenoid valve changing the gas path and sending the gas sample to the paramagnetic sensor. Whenever the O2 concentration measured by “high concentration” paramagnetic sensor dropped below 1 % vol. (10 000 ppmv), the PLC sends a digital signal to a 3/2way solenoid valve changing the gas path and sending the gas sample to the electrochemical sensor (Figure 8.49).
Figure 8.49: The setup for the selection of the analyzers.
Programming PLC In the following section, we present a part of the industrial PLC program operating sequentially for the selection of the analyzers. The solenoid valves are those specified in the specification above. Excerpts from such a program are presented in the following lines.
286  8 Controllers
# PLC program # Purpose: Solenoid Valve Control # Date: DD.MM.YYYY ################################################################### # PLC Timer Setup # # Syntax: # .. .. .. # Name  Id  Value ; comment # `' `' `' # ## TMR_MODE 1..8 OFFDELAY,ONDELAY, REPPULSE, # SINGLEPULSE, RETRIGSPULSE, INHIBSPULSE, CLKTRGPULSE , # COUNTER # TMR_DURATION 1..8 1..3600 # TMR_PERIOD_CNT 1..8 1..3600 (REPPULSE: sec, CLKTRGPULSE: min, COUNTER: counts) # TMR_TRIG_TIME 1..8 YYYY,MM,DD,hh,mm ################################################################### # # PLC Sequencer Setup # # Syntax: # .. .. .. # Name  Id  Value ; comment # `' `' `' # ## SEQ_DURATION 1..4 1..3600 # SEQ_STAB_TIME 1..4 0..3600 (max SEQ_DURATION1) # SEQ_NUM_OUTS 1..4 2..16 # SEQ_SNGL_CYCLE 1..4 FALSE,TRUE ################################################################### #Example for Timer 1TMR_MODE 1 OFFDELAY; TMR_DURATION 1 5; delay 5 sec TMR_PERIOD_CNT 1 10; do not care TMR_TRIG_TIME 1 2009,11,28,23,40; do not care
8.3 Problems  287
#Example for Timer 2TMR_MODE 2 ONDELAY; TMR_DURATION 2 5; delay 5 sec TMR_PERIOD_CNT 2 10; do not care TMR_TRIG_TIME 2 2009,11,28,23,40; do not care #Example for Timer 3 …………………………… #Example for Timer 8 …………………………… ################################################################### # # Programming Quick Reference # ################################################################### # .. #  OPERATORS # `' # CLR Set register to FALSE # AND Logical AND of register and # OR Logical OR of register and # NEG Negate register # STO Store register to # END End of program This example is intended just to give an idea, for those interested, of the programming issues linked to PLCs. It is to be mentioned that this is not the whole program to operate the selection of the solenoid valves. As it may be seen from the examples above, PLC is usually used for the sequential operations, but the newer versions, such as PLC RIO (Reconfigurable Inputs Outputs), can undertake more complex algorithms.
8.3 Problems (1) The automatic control system in Figure 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.
288  8 Controllers (2) The phase margin of the PD controller of the ACS in Figure 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 %? (3) Consider the heat exchanger temperature ACS from Figure 8.50.
Figure 8.50: 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 consumption 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 kg kcal kcal ρ = 1 000 m 3 ; specific heat cp = 1 kg K ; heat transfer coefficient KT = 1 000 m2 hK ; heat
transfer area AT = 1 m2 .
What is the offset and the extra heat consumption if the setpoint is changed with +10 °C? (4) Two CSTRs have the following characteristics: kg kcal 2 Reactor 1: V1 = 1 m3 ; KT1 = 1 000 mkcal 2 hK ; AT1 = 3 m ; ρ1 = 1 000 m3 ; cp1 = 1 kg K ; F1 = 1 2 Reactor 2: V2 = 2 m3 ; KT2 = 800 mkcal 2 hK ; AT2 = 5 m ; ρ2 = 1 000
kg ; cp2 m3
=1
kcal ; F2 kg K
=1
m3 . h 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. (5) Design the behavior of the autoadaptive controller for the process described in Figure 8.22.
References [1]
Agachi P. S., Automatizarea proceselor chimice (Automation of chemical processes), Casa Cartii de Stiinta, ClujNapoca, 1994, p. 108.
References  289
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
[22] [23] [24] [25] [26]
*** 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, ClujNapoca, 1994, pp. 110–116. Agachi P. S., Automatizarea proceselor chimice (Automation of chemical processes), Casa Cartii de Stiinta, ClujNapoca, 1994, 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, ClujNapoca, 1994, pp. 120–124. Ang, K. H., Chong, G., Li, Y., PID control system analysis, design, and technology, IEEE Trans. Control Syst. Technol., 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 Syst. Mag., 26, (2006), 32–41. Fliess, M., Join C., Intelligent PID Controllers, Proc. of 16th Mediterranean Conf. on Control and Automation, Ajaccio, France, 2008. 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. Silberg A. I., Topan V. A., Agachi P. S., et al. Procedure and reactor for manufacturing chloramine, Romanian Patent no. 120893/25.11.1985. Calin S., Dumitrache I., Regulatoare automate (Automatic controllers), Ed. Didactica si Pedagogica, Bucuresti, 1985, Chap. 8,11. Åström, K. J., Theory and applications of adaptive control — a survey, Automatica, 19, (1983), 471–486. Åström, K. J., Wittenmark, B., On selftuning regulators, Automatica, 9, (1973), 185–199. Åström, K. J., Adaptive Control, Dover, 2008, pp. 25–26. Bengtsson G., Egart B., Experience with Selftuning Control in the Process Industry, Proc. IX IFAC Congress, Budapest, 1984, Section Case Studies. Pontryagin, L. S., The Mathematical Theory of Optimal Processes, Mir Publishers, Moscow, 1962. Doya, K., How can we learn efficiently to act optimally and flexibly, Proc. Natl. Acad. Sci. USA, 106 (28), (2009), 11429–11430. Betts, J. T., Practical Methods for Optimal Control Using Nonlinear Programming, SIAM Press, Philadelphia, Pennsylvania, 2001. 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. Agachi, P. Ş., Nagy, Z. K., Cristea, M. V., ImreLucaci, A., Model Based Control, Case Studies in Process Engineering, WileyVCH, Weinheim, 2006, pp. 17–20, ISBN 3527315454. Nagy Z., Agachi S., Model predictive control of a PVC batch reactor, Comput. Chem. Eng., 6, (1997), 571–591. http://www.automation.com/library/articleswhitepapers/fieldbusserialbusionetworks/ ieee1394andindustrialautomationaperfectblend. Adams, Tracy, SCADA Systems Fundamentals, Course No: E01007, CED Engineering.com, 2014, https://www.cedengineering.com/. Bolton, W., Programmable Logic controllers, 6th Edition, Elsevier, Amsterdam, Boston, Chapter 1, 2015, pp. 1–22.
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[27] Botha, D., Introduction in PLCs, CHEE 320, Botswana International University of Science and Technology, May 2019, pp presentation. [28] Antonsen, T. M., PLC Controls with Structured Text (ST): IEC 611313 and best practice ST programming, BooksOnDemand, ISBN: 9788743002413, 2018, pp. 232. [29] Maged,Sh., Programmable Logic Controllers (PLC’s), Lecture 10, Mechatronic 2 course, Ain Shams University, May 16, 2017. https://www.slideshare.net/MohamedAtef80/lecture10cont11plc.
9 Final control elements (actuating devices) The Final Control Element 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 a 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 with Variable Speed Devices (VSD), the frequency of switching on/off a switch device (relay) – Figure 9.13. But by far the most popular way of manipulating the flow is using the control valves, due to their simplicity and reliability [1], but there are also other types of Final Control Elements described in the text.
9.1.1 Control valves The most frequently used final control elements are the control valves (Figure 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. Actuators and accessories Pneumatic actuator The most frequently used final control elements are the control valves (Figure 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 Figure 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 https://doi.org/10.1515/9783110647938009
292  9 Final control elements (actuating devices)
Figure 9.1: Pneumatic control valve: P – plug; pc – control pressure; Ac – actuator; Sp – spring; M – membrane; SR – seat ring.
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 electropneumatic convertor (EPC) (Figure 9.5b 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 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 (Figure 9.5c) 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. Electric actuators Most actuators in the chemical industries, historically, were pneumatic due to the inherent antiexplosion protection. In the meantime, the electronics evolved tremendously and intrinsically safe barriers (Zener, isolation etc.) were invented, limiting the electrical signals in the field to harmless values.
9.1 Types of final control elements  293
Electric motors permit manual, semiautomatic, and automatic operation of the valve. Motors are used mostly for openclose functions, although they are adaptable to positioning the valve to any point opening as illustrated in Figure 9.2.
Figure 9.2: Electric motor actuator.
The motor is usually reversible, connected through a hight speed type gear train to reduce the motor speed and thereby increase the torque at the stem. Direction of motor rotation determines direction of stem motion. Solenoid actuators (Figure 9.3) The position of an iron core in the solenoid is depending on the electrical signal sent by the controller determining the opening of the valve.
Figure 9.3: Solenoid actuators.
294  9 Final control elements (actuating devices) Hydraulic actuators (Figure 9.4) Hydraulic actuators are used when an actuator has to develop large forces to overcome some large pressure drops. Sometimes, for the sake of the space occupied by actuators, one prefers hydraulic ones.
Figure 9.4: Hydraulic actuator.
Control valves We may find different types of control valves on free sources (Figure 9.5): 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 [1, 2]: – globe valve with direct flow single or double ported (Figure 9.6), threeway (Figure 9.7); – angle valves in which one port is colinear with the valve stem, and the other port is at a right angle to the valve stem (Figure 9.8); – butterfly valve (Figure 9.9) used for large pipelines and low pressure fluids (gas usually); – ball or Vnotch control valve (Figure 9.10) 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 Vnotch of Vshape in the ball allows a greater range ability of the manipulation of the flow.
9.1 Types of final control elements  295
Figure 9.5: Different types of industrial control valves with electropneumatic convertors aside: (a) regular industrial valve; (b) control valve with electropneumatic convertor; (c) actuator with electropneumatic convertor and positioner.
–
slide/gate control valves (Figure 9.11) 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.
296  9 Final control elements (actuating devices)
Figure 9.6: Globe valve twoway singleported.
Figure 9.7: Globe valve threeway singleported.
There are other types of control valves specifically for nonusual fluids manipulation: – membrane control valves (Figure 9.12) 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.
9.1 Types of final control elements  297
Figure 9.8: Angle valve.
Figure 9.9: Butterfly valve.
Figure 9.10: Ball control valve.
298  9 Final control elements (actuating devices)
Figure 9.11: Slide/gate control valve.
Figure 9.12: Membrane valve.
9.1.2 Other types of final control elements –
Conveyor belts with variable speed (Figure 9.13);
Figure 9.13: Variable speed conveyor belt.
9.2 Sizing the control valve
–
 299
screw pumps or screw conveyors (Figure 9.14);
Figure 9.14: 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) or Variable Speed Drive (VSD) [2] (Figure 9.15);
Figure 9.15: Left: gear pump and right: screw drive pump.
–
reflux distributors (Figure 9.16) are used for very small reflux ratio flows. These distributors operate by varying the ratio between the reflux time and evacuation of 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].
300  9 Final control elements (actuating devices)
Figure 9.16: Reflux distributor at the top of a distillation column.
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 (Figure 9.17). It is important because it synthesizes, as seen in Figure 9.17, 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.
Figure 9.17: Hydraulic circuit including a control valve.
If one expresses the volumetric flow Fv function of √p0 –p1 (Figure 9.18), 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)
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.
9.2 Sizing the control valve
 301
Figure 9.18: The dependence of the volumetric flow on the pressure difference (p0 –p1 )1/2 .
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, Δpv = ξv
ρv2 , 2
(9.2)
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 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 crosssection 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)
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.
302  9 Final control elements (actuating devices) The relationship (9.4) becomes Kv = Fv [
kg ρ [m 3] m3 √ 735 = 6.8√ ] = 6.5. h 1 000Δpv [bar] 1 000 ⋅ 0.8
For laminar flows in the pipeline, Bernoulli’s law is applied and Reynold’s number can be calculated with (9.5): Re = ρ
vD , ν
(9.5)
where D is the pipeline diameter [m] and ν the viscosity [Ns/m2 ]. The profile of the flow tube is presented in Figure 9.19.
Figure 9.19: 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]: ReR = Re
D 1 . 2.59 √K v
(9.6)
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 √ReR relationship (9.4) berecalculated using the corrected ReR with the factor C = 20 coming Kv =
Fv ρ . √ C 1 000Δpv
(9.7)
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 (Figure 9.20) 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.
9.2 Sizing the control valve
 303
Figure 9.20: 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, pF pN FvN = 2 ∘v , TN∘ T
(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. Expressing the volumetric flow as the ratio between the mass flow and its density, F
pN ρ m TN∘
N
=
Fm ρ ∘ T
p2
.
(9.11)
304  9 Final control elements (actuating devices) From both equations (9.10) and (9.11), Fv = FvN
pN T ∘ p2 TN∘
and
p2 TN∘ , pN T ∘
ρ = ρN
(9.12) (9.13)
and with (9.4), the expression of Kv becomes Kv =
FvN pN T ∘ √ , 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 Kv = Fv √
1 , 1 000Δpv 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 wetsaturated steam.
9.3 Inherent characteristics of control valves  305
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 Kv = √
1 A (h) = Kv (h), ξv (h) v
(9.18)
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 (Figure 9.21):
Figure 9.21: Typical inherent characteristics of the control valves [9].
306  9 Final control elements (actuating devices) (a) The linear characteristic, where expression
ΔKv Δh
= ct for all values of Kv and with the analytical
K K Kv h = v0 + (1 − v0 ) Kv100 Kv100 Kv100 h100
(9.19)
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 K K Kv h = v0 exp [ln ( v100 ) ]. Kv100 Kv100 Kv0 h100
(9.20)
Using the differentiated expression of (9.20), K dKv 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 inbetween the linear and the equal percentage characteristic, providing a fine control at low flow values and approximately linear characteristic at higher flow capacity: h
K Kv h100 = v0 + . Kv100 Kv100 3 − 2 ( h )2 h
(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 inbetween 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 (Figure 9.22), or via a special characteristic of the electropneumatic converter. 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
(9.22)
9.4 Installed characteristics of the control valves  307
Figure 9.22: Special form of the plugs and seats for corresponding valve characteristics.
x Kv = f2 ( c ) , Kv100 xc100
and
Kv = f1 ∘ f2 . Kv100
(9.23) (9.24)
9.4 Installed characteristics of the control valves Once included in the hydraulic circuit (Figure 9.23), the control valve changes its inherent characteristics depending on the structure of the system and the total pressure drop on the system.
Figure 9.23: The control valve does not work independently but is a part of a hydraulic 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 it
308  9 Final control elements (actuating devices) self (Δpv ). n
Δploc = ∑ ξi
ρv2 2
(9.25)
Δplin = ∑ λj
lj ρv2 Dj 2
(9.26)
i=1 m j=1 l
Δpz = ∑ ρghk
(9.27)
k=1
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 ρv2 m lj ρv2 + ∑ λj + ∑ ρghk + 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 = ρF 2 ( K12 + v
1 ), KL2
Δp ΔpL = Fv = Kv √ v = F. ρ ρ
(9.29)
and from this results F= F=
Kv √1 +
2
K ( Kv ) L
√
Δps ρ
or
Kv 2
√1 + ( Kv ) ( Kv100 ) K K v100
2
√
(9.30) Δps . ρ
(9.31)
L
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)
9.4 Installed characteristics of the control valves  309
Supposing that the flow does not change its regime and the fluid is incompressible, KL stays constant and replacing (9.32) in (9.31), F=
Kv 2 Δp −Δp s v100 Δpv100
√1 + ( Kv ) K v100
√
Δps ρ
and using (9.32) F
F100
Kv Kv100
= √1 +
K (K v ) v100
2
Δp ( Δp s v100
√ − 1)
Δps . Δpv100
(9.33)
Δp
If we denote the parameter m = Δpv100 where Δps = p1 max − p2 (p1 max is sometimes s called maximum pumping height), the relation (9.32) becomes (9.34), representing the installed characteristic of the valve: Kv F = F100 Kv100
1 √m + (
Kv 2 ) (1 Kv100
. − m)
(9.34)
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
√m + ( Kv (h) ) (1 − m) K
.
(9.35)
v100
The installed characteristics of the control valves are plotted in Figure 9.24 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, Figures 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 equalpercentage; for flow, if the setpoint varies – linear; if load varies – equalpercentage; for temperature – equalpercentage.
310  9 Final control elements (actuating devices)
Figure 9.24: The installed characteristics of the control valve.
9.5 The dynamic characteristics of a control valve
 311
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 electropneumatic 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 KEPC =
ΔpAC Δic
and for the first case, KEPC =
bar 0.8 bar = 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 =a dh
(9.37)
and for the equal percentage (logarithmic) dF = bF. dh
(9.38)
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 Figure 9.25 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
312  9 Final control elements (actuating devices)
Figure 9.25: The change of the control valve gain, function of travel and m coefficient in the case of: (a) linear valves; (b) logarithmic valves.
transport line delay, the behavior is in the most cases described by a transfer function of second order Hv (s) =
a3 , a2 s2 + a1 s + 1
(9.39)
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].
9.6 Sizing and choice of the control valves [12, 13] 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, Figure 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),
9.6 Sizing and choice of the control valves  313
(a.5) (a.6) (a.7)
(a.8)
the compensating characteristic of the valve is chosen (Section 6.1, Figure 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 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; the superior value Kvs > Kv100 is chosen from the manufacturer’s tables; 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; 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, Figure 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 (Figure 9.24) should be chosen, but also a logarithmic characteristic with m = 0.2 fits. Choosing a logarithmic valve, Δpv100 = 0.2 Δps
(9.40)
314  9 Final control elements (actuating devices) Δpv100 = 0.2Δps = 0.2(Δpv100 + Δpp100 ) 0.2Δpp100
Δpv100 =
1 − 0.2
=
(9.41)
0.2 ⋅ 12.6 = 3.15 bar. 0.8
(9.42)
According to (9.4) and from Example 9.1, Kvmax = F100 √
ρ 875 = 147√ = 74. 1 000Δpv100 1 000 ⋅ 3.15
(9.43)
We choose from the catalogue Kvs = 90 We recalculate the pressure drop Δpv100 =
2 F100 ρ 1472 875 = = 2.2 bar 2 Kvs 1 000 902 1 000
and m=
(9.44)
Δpv100 2.2 = = 0.15, Δps 12.6 + 2.2
(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 (Figure 5.3). The characteristics 3 of the heat exchanger are: nominal flow through the heat exchanger – FN = 4 mh ;
kg density of the fluid and heating agent ρ = ρag = 1 000 m 3 ; specific heat cp = cp ag =
kcal 2 ; heat transfer coefficient KT = 1v000 mkcal 1 kg 2 hK ; heat transfer area AT = 4 m ; Δpp100 = K ∘ 1 bar; Ti∘ = 20°; Tiag = 50°. The temperature which has to be kept constant is Tn∘ = 30°. The temperature fluctuations around the normal Ti∘ = 20° are of ± 5°. The pressure is given by a centrifugal pump with the operation characteristic given in Figure 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): ∘ Fvi ρcp Ti∘ − Fvo ρcp T ∘ − KT AT (T ∘ − Tag ) − VkCA ΔHr = 0
(6.1)
∘ Fvag ρag cpag Tiag
(6.2)
−
∘ Fvag ρag cpag Tag
∘
+ KT AT (T −
∘ Tag )
= 0,
where VkCA ΔHr = 0, Fvi = Fvo = FN . ∘ Eliminating Tag , and replacing the values in the problem, one obtains the steady state characteristic of the process: T∘ =
Ti ag 4Ti∘ + . 4 + 2Fag 2 + 4 F ∘
ag
(9.47)
9.6 Sizing and choice of the control valves  315
Data calculated for the disturbances of ± 5° in the input temperature and for different values of the agent flow are given in Table 9.1.
Fag [m3 /h]
2 4 6 8 10 12
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
Table 9.1: Variation of the temperature of the heated fluid in the heat exchanger function of the disturbances.
Figure 6.9 is the graphic representation of the temperature variations in this case. (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 3
3
3
are 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 Figures 6.9 and 9.24, 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 √ 1 000Δp
v100
000 = 12√ 11000⋅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. Example 9.5. 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 is constant. The dilution is done at the atmospheric pressure and the pump pumps the concentrate at 2 bar. The pipeline pressure drop is Δpp100 = 1 bar and the density of the concentrate is ρ = 800 kg/m3 . The control valve has to be sized and chosen for manipulating the concentrate (Figure 12.15).
316  9 Final control elements (actuating devices) The component and total mass conservation for the active component are described below: F = Fc + D
D.xD + Fc xc = F.x
x=
Fc Fc 1 .x = x = F c D + Fc c 1 + D F
c
The MATLAB program and the steady state characteristic for the dilution process is presented in Figure 9.26.
Figure 9.26: The plot of the steady state of the dilution process.
The following steps for sizing the valve are given: – Establish the steady state process characteristic on the transfer path manipulated variable → controlled variable. – Based on the maximum and minimum disturbance, the minimum and maximum flow through the valve are calculated, respectively: Fcmax ↔ 0.5 =
0.9
1+
1 Fcmax
Fcmax = 1.25m3 /h 1 Fcmin ↔ 0.5 = 1+ F1
cmin
Fcmax = 1.00 m3 /h
– –
Δp100 = 1 bar from the problem data from the steady state characteristic of the process established in the first bullet above, the compensating characteristic is chosen in order to linearize the process.
9.7 Problems  317
From Figure 9.24, a logarithmic characteristic of the valve with m = 0.5 is chosen. Δp – m = 0.5 = Δpv100 s
Δpv100 = Δps = Δpv100 + Δpp100
∴ 0.5Δpv100 + 0.5pp100 = Δpv100 0.5Δpv100 = 0.5 Δpv100 = 1 bar
Thus KVmax = Fmax √ = 1.25√ = 1.12
ρ 1 000.Δpv100
1 000 1 000 ∗ 1
9.7 Problems (1) 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 4 bar. The density of the product which passes through the control valve of the level kg ACS is ρ = 700 m 3 . 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? (2) 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 through the kg valve ρ = 700 m 3 . The control valve has to be sized and the linearizing characteristic to be chosen. (3) 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 electropneumatic convertors have the input signal 4–20 mA and the output 0.2–1 bar; the drive has the input 0.2–1 bar and the output stem travel 0–100 %. (4) 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;
318  9 Final control elements (actuating devices) 4. 5. 6. 7. 8.
sulfuric acid 60 % produced through the contact chamber process; water emulsion of polyvinyl acetate flow; air conditioning air flow at 300 mm H2 O pressure; slurry flow at the bottom of a distillation column; sand flow for the glass production process.
References [1]
[2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13]
Emerson Process Management Handbook, Fisher, Control Valve Handbook, Fourth Edition, Cernay, France, 2005, pp. 1425. http://www.documentation.emersonprocess.com/groups/ public/documents/book/cvh99.pdf. Campbell, S. J., SolidState AC Motor Controls. New York: Marcel Dekker, Inc., pp. 79–189. ISBN 082477728X. 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. Handbook for control valve sizing, Parcol Bulletin 1I, 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/controlvalvesflowcharacteristicsd_485.html. Headley, M., Guidelines for selecting proper valve characteristics, Valve Mag., 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 customtailored 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 lifecycle [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 the design methodology and special attention should be given to considering both https://doi.org/10.1515/9783110647938010
320  10 Safety interlock systems normal operation conditions and startup 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 (pretest) 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 prestartup tested. A solid state construction is desired and usually thorough verification is made to ensure longlife 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 onsite 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 lifecycle 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 Figure 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 involving raw materials and products that exhibit reduced hazard for explosion or for harming human health [3, 4].
10.2 Safety layers  321
Figure 10.1: Typical configuration of the safety multilayer system.
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 startup 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 implementation of this safety layer benefits from different options. The most common is based on the traditional relays. Nonetheless, hardwired electronic implementation using com
322  10 Safety interlock systems bined digital and analog electronic devices and softwarebased 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 softwarebased interlock systems have gained increasing acceptance under the implementation of the Programmable Logic Controllers, although they are not yet considered as reliable and lowcost 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 selftesting 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 firefighter 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. 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].
10.4 Safety instrumented systems  323
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: prealarms, 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 Figure 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]. 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
324  10 Safety interlock systems
Figure 10.2: Typical configuration of an electrical Safety Instrumented System.
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 onoff signal. This signal is changed when the process parameter exceeds the predefined upperlower 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 tradeoff 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 selfdiagnosis. 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 fieldwiring. 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 separated 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.
10.4 Safety instrumented systems  325
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 (see paragraph 8.2.6). An example of the logic for the activation of the final control element is presented in Figure 10.3, using the logical AND and OR gates and Boolean logic.
Figure 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
326  10 Safety interlock systems 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 nofault working regime. Figure 10.4 presents the 1oo1 (one out of one) architecture.
Figure 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 deenergized (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 Figure 10.5.
Figure 10.5: 1oo2 architecture.
10.4 Safety instrumented systems  327
This architecture has two channels for executing the safety function on the final element. The channels work in parallel and their deenergized 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 Figure 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.
Figure 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 deenergized 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 actuatorfinal 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 shutoff 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
328  10 Safety interlock systems 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 nonredundant 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 softwarebased safety system that allows the simple change of configuration and multiplication of the inputoutput 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 softwarebased safety systems are recommended. Another trend is the advent of smart instruments that may have implemented safety functions, such as selfstate diagnosis, comprehensive measurement analysis and communication with the safety instrumented system [9].
10.5 Problems (1) Please draw the 2oo2 architecture with four channel configuration and describe the operation of the safety system.
References  329
(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., Berger, S., Process safety: How are you doing?, Chem. Eng. Prog., 104 (5), (2008), 40–43. [2] Overton, T. A., and King, G., Inherently safer technology: An evolutionary approach, Process Saf. Prog., 25 (2), (2006), 116–119. [3] Gentile, M., Summers, A., Cookbook versus performance SIS practices, Process Saf. Prog., 27 (3), (2008), 260–264. [4] Summers, A. E., Hearn, W. S., Quality assurance in safe automation, Process Saf. Prog., 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. (editorinchief), Considine, D. M. (late editorinchief), Process/Industrial Instruments and Controls Handbook, 5th ed., McGrawHill, 1999. [7] http://www.aiche.org/ccps/topics/elementsprocesssafety. [8] http://www.processoperations.com/SafeInstrSy/SS_Main.htm. [9] Luyben, W. L., Use of dynamic simulation for reactor safety analysis, Comput. Chem. Eng., 40 (11), (2012), 97–109.
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 (Figure 11.1).
Figure 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 “applies” the same reverse force, F, at midperiod. Similar examples can be given for a pendulum, for springmass systems, for playground swings etc. https://doi.org/10.1515/9783110647938011
334  11 Design and tuning of the controllers
Figure 11.2: The closed control loop oscillates when a disturbance D occurs.
A similar phenomenon takes place in a control loop (Figure 11.2). 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 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
11.2 Control quality criteria
 335
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° i=1
and
5
∏ Mi = MS MC MAD MPr MT ωosc = 1. i=1
(11.1)
(11.2)
Since φS = −180° (Figure 5.4) and MS = 1, the equations (11.1) and (11.2) become φC + φAD + φPr + φT = −180°
MC MAD MPr MT ωosc = 1.
(11.3) (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: (1) Criterion of the integral of the error (IE) (Figure 11.3) expressed through (11.5): ∞
IIE = ∫ e(t)dt = ∑ Si = min 0
(11.5)
336  11 Design and tuning of the controllers
Figure 11.3: The IE criterion.
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. 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) (Figure 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 (+/−).
Figure 11.4: The IAE criterion.
(3) Criterion of the integral squared error (ISE) (Figure 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 in this case: ∞
IISE = ∫ e2 (t)dt = ∑ Si2 = min. 0
(11.7)
11.2 Control quality criteria
 337
Figure 11.5: The ISE criterion.
(4) Criterion of integral timeweighted absolute error (ITAE): ∞
IITAE = ∫ te(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 (Figure 11.6): SSR = min .
(11.9)
Figure 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 (Figure 11.7): σ3 =
1 σ 4 1
(11.10)
Figure 11.7: Quarter decay ratio response criterion.
338  11 Design and tuning of the controllers 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.
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 Tables 11.1 and 11.2. 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
Practically, the influence of the disturbance is not attenuated (large overshoot and large offset)
good
Small overshoot, quick dampening and small offset
too small
Undamped oscillations
11.3 Parameter influence on the quality of the control loop
 339
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
There is no influence of the I component and a very weak one of the P component
good
large
The behavior is of a P controller with “good” PB
small
large
The behavior is of a P controller with “small” PB
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
(continued on next page)
340  11 Design and tuning of the controllers Table 11.2: (continued). PB
Ti
Step response to disturbance, of the classic (feedback) ACS
Observations
good
good
Small overshoot, quick dampening and 0 offset
small
good
Because of the dominant P action, oscillations appear
large
small
Large overshoot, slow attenuation and noise with small oscillations
good
small
Small overshoot, quick dampening and small repeated oscillations around the setpoint
small
small
Undamped, amplified oscillations
11.3 Parameter influence on the quality of the control loop

341
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 = 1 000 % 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 Figure 11.8.
Figure 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].
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
342  11 Design and tuning of the controllers 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]. 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. 5. 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. 6. 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 . 7. PB is fixed at PB = 2PBu value. 8. Ti is reduced at half value from the maximum; the operation mode is M. 9. 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 ); 10. Ti is fixed at = 2Tiu value. 11. Td is increased step by step until the same oscillatory regime is obtained (Tdu ); 12. The derivative component is fixed at Td = 0.5 Tdu . 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 “badlucky” number of steps; there are systems of “conditioned” stability that become unstable both at too small or too large PBs.
11.4 Controller tuning methods 
343
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 Figure 11.9 and the ultimate period Pu is measured on the diagram.
Figure 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 Table 11.3. Table 11.3: Optimal controller parameters proposed by Ziegler and Nichols. Optimal controller parameter
Controller structure P PI
PID
PB Ti Td
2PBu – –
1.7PBu 0.5Pu 0.12Pu
2.2PBu 0.83Pu –
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 ).
344  11 Design and tuning of the controllers –
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 (Figure 3.13) [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 Table 11.4. Table 11.4: The optimal controller settings in the case of the three CSTR control loop. Optimal controller parameter
Controller structure P PI
PB [%] Ti [min] Td [min]
3.12 – –
3.6 3.03 –
PID 2.8 1.82 0.453
The behavior of the control loop in the three cases of optimal setting is given in Figure 11.10.
Figure 11.10: Behavior of the concentration control system in the three CSTR cascade.
11.4 Controller tuning methods  345
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. 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 Figure 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 identi
Figure 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.
346  11 Design and tuning of the controllers fied 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 r change in the measured variable (ME) by the change of the setpoint r ( Δx ). Δr
r . The “process” gain is given by the ratio Kpr = Δx Δc The values recommended by Ziegler and Nichols for optimal settings of the controller are given in Table 11.5.
Table 11.5: Optimal settings of the controller based on the process response curve. Optimal controller parameter
Controller structure P PI
PID
PB [%] Ti [min] Td [min]
Kpr ⋅ τ/T ⋅ 100 − −
0.7Kpr ⋅ τ/T ⋅ 100 2.5τ 0.4τ
1.1Kpr ⋅ τ/T ⋅ 100 2τ −
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 (Figure 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 Table 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 controllability 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.
11.4 Controller tuning methods  347
Figure 11.12: The process step response gives details about the steady state and dynamics of the process.
Table 11.6: Optimal stings calculated through the Cohen–Coon method. www.opticontrols.com
Controller Gain
PB Controller
Kc = ( 1.03 ) ( Tτ + 0.34) g
Integral Time
Derivative Time
p
) ( Tτ ( 0.9 gp
+ 0.092)
PI Controller
Kc =
PD Controller
Kc = ( 1.24 ) ( Tτ + 0.129) g
PID Controller (Noninteracting)
) ( Tτ + 0.185) Kc = ( 1.35 g
TI = 3.33τ ( TT+0.092τ ) +2.22τ −0.324τ TD = 0.27τ ( TT −0.129τ )
p
p
+0.185τ ) TI = 2.5τ ( TT +0.611τ
T ) TD = 0.37τ ( T +0.185τ
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.
348  11 Design and tuning of the controllers Figure 11.13 gives the way to determine the process parameters, which is very similar to the Cohen–Coon method.
Figure 11.13: Step response experiment for determining the controller parameters using Lambda method; c is the controller output and y is the process variable.
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: Controller gain (Kc ): Kc = T/(Kpr ⋅ (λ + τ));
Kpr =
Δr[%] . 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.
11.4 Controller tuning methods 
349
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 “setup” 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
350  11 Design and tuning of the controllers controller parameters should be calculated otherwise. There are theoretical methods close at hand. Method based on Routh stability criterion Considering the control system from Figure 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
HC (s)HAD (s)Hpr m (s) Y(s) = . D(s) 1 + HT (s)HC (s)HAD (s)Hpr m (s)
(11.15) (11.16)
Figure 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 halfplane s, the closed system is unstable.
11.4 Controller tuning methods  351
Routh stability criterion Consider a closedloop 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 halfplane 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 coefficients 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 = b2 = b3 =
a1 a2 − a0 a3 a1
a1 a4 − a0 a5 a1
a1 a6 − a0 a7 a1
...... generated until subsequent coefficients are 0. Similarly, c1 =
b1 a3 − a1 b2 b1
352  11 Design and tuning of the controllers b1 a5 − a1 b3 b1 b1 a7 − a1 b4 c3 = b1 c2 =
⋅⋅⋅⋅⋅⋅ c1 b2 − b1 c2 c1 c1 b3 − b1 c3 d2 = c1 d1 =
until the nth 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 halfplane 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 (Figure 3.13) [6], we want to determine the condition of stability of a concentration ACS with a P controller. The transfer function of the process is ( 21 )
Hpr m (s) =
3 3
(s + 1)
=
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
=1
3 1 ] ] ] ] 0 ] ] ]
(11.24)
]
and does not present any sign change, meaning the system is stable in open loop.
11.4 Controller tuning methods  353
Considering (11.18), the stability of the closed loop is given by the condition 1 + Kc
( 21 )
3
(s + 1)3
= 0,
(11.25)
Kc ) = 0. 8
(11.26)
which becomes s3 + 3s2 + 3s + (1 + Routh’s matrix is [ [ [ [ [ [ [ [
1
3
3
1+ K
9−(1+ 8c ) 3 K 1 + 8c
] ] ] ]. ] ] ]
Kc 8
0
(11.27)
]
The only term capable of becoming negative and thus bringing instability to the closed loop is 9 − (1 + 3
Kc ) 8
< 0,
(11.28)
and the condition means (8 − 3
Kc ) 8
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 − jω3 − 3ω2 + 3jω + 1 + (1 +
Kc =0 8
Kc − 3ω2 ) + j(3ω − ω3 ) = 0. 8
(11.31) (11.32)
354  11 Design and tuning of the controllers This means 3ω − ω3 = 0
from which ω = ±√3
Kc − 3ω2 = 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) (Figure 11.15) is equal with the number of poles of the transfer function H(s) placed in the right halfside of the plane s. H(s) is the transfer function of the open loop.
Figure 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 halfplane 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 (Figure 11.16) is controlled with a P controller. The maximum Kc to assure the stability of the loop is to be determined.
11.4 Controller tuning methods  355
Figure 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. The transfer functions are Kcv Hcv (s) = = Tcv s + 1
0.022 Nm3 /min 0.01 bar
2s + 1
=
2.2 2s + 1
(11.34)
356  11 Design and tuning of the controllers 0.6 bar 0.1⋅2 Nm3 /min
Hv1 (s) =
Kv1 = Tv1 s + 1
Hv2 (s) =
Kv2 0.8 = 1 bar = Tv2 s + 1 20s + 1 20s + 1
HD (s) = HT (s) =
15s + 1
=
3 15s + 1
0.8 bar
0.4bar 0.1⋅2 Nm3 /min
20s + 1
=
2 20s + 1
(11.35) (11.36) (11.37)
1–0.2 bar = 0.16. 5–0 bar
(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 halfplane and thus P = 0. The hodograph of the function is presented in Figure 11.17.
Figure 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
and
Im[H(jω)]ωosc = 0 Re[H(jω)]ωosc > −1 Re[H(jω)] =
(11.40) 2
0.844(1 − 370ω )Kc (1 − 370ω2 )2 + (37 − 600ω2 )2 ω2
11.4 Controller tuning methods  357
and Im [H (jω)] =
0.844(37–600ω2 )ωKc 2
(11.41)
2
(1 − 370ω2 ) + (37 − 600ω2 ) ω2
37 ; from these values the one From Im[H(jω)]ωosc = 0 results ω1 = 0 and ω2 = ±√ 600 verifying Re[H(jω)]ωosc > −1 is chosen:
Re[H(jω)] =
0.844(1–370ω2 )Kc (1 −
2 370ω2 )
The result is Kc < 25.8;
=
0.844Kc
37 1–370 600
the period of oscillation is Posc = Pu =
> −1 (ω2 = √
1 37 2π √ 600
37 ). 600 (11.42)
= 0.64 min.
(11.43)
Method based on the criterion of Quarter Amplitude Damping/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 (Figure 11.2), its steady state open loop gain is 21 . 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); 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).
358  11 Design and tuning of the controllers Example 11.5. Consider a flow ACS, in a pipeline (Figure 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
Figure 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 ω2
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 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 ωc =
2π , Tmax
where Tmax = TAD = 3 s.
11.4 Controller tuning methods  359
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) 1.3 1 2.5 Kc = 0.5, √1 + 0.52 3.52 √1 + 0.22 3.52 √1 + 32 3.52 Kc opt = 4.0
or
PBopt = 25 %.
(b) Consider a PI controller at which MPI (ω) = Kc √1 +
1
Ti2 ω2
and φPI (ω) = − tan−1
1 . Ti ω
Choosing Ti = 5s, we can find the crossover frequency ωosc ω [rad s−1 ]
φpr [grd]
φT [grd]
φc [grd]
φAD [grd]
∑ φi
0.1 1.0 10.0 3.3
−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 . From (11.21) 1
2
2
2.5
2
√1 + 0.5 2.1 √1 + 0.2 2.1
2
Kc √ 1 +
1
52 2.12
√1 + 32 2.12
results Kc opt = 3.52
or
1.3
PBopt ≈ 28.3 %
= 0.5
(11.47)
360  11 Design and tuning of the controllers 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 ). Ti ω
(11.48)
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. Td =
tan 30° + 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)ωosc = 0.5 i
is equivalent with
∏ Mi (ω)ωosc = 1.0. i
(11.49) (11.50)
Thus, 1
2.5
√1 + 0.52 3.52 √1 + 0.22 3.52
Kc √1 + (0.148 ⋅ 3.5 −
2 1.3 1 = 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 Table 11.3, we observe these values are confirmed orientatively by the Ziegler–Nichols method as well. 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. There many other authors browsing through all methods as Ogunnaike and Ray [8], O’Dwyer [16]. Their works deserve to be cited.
11.5 Tuning controllers for some “difficult to be controlled” processes  361
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 (Figure 11.19).
Figure 11.19: Step responses for “easy to be controlled” processes.
There are three classes of processes which are difficult to be controlled: 1. processes with dead time (see Sections 4.7 and 4.8) for which the transfer function has a e−τs term; 2. 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); 3. processes presenting openloop instability where the transfer function has an RHP pole. These types of behavior are depicted in Figure 11.20. 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 inverseresponse 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].
362  11 Design and tuning of the controllers
Figure 11.20: Step response for “not easy to be controlled” processes.
2.
In the case of openloop 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 K Ts − 1
H(s) =
(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
(11.53)
s=
(11.54)
1+
1 − KKc T
is the root.
(11.52)
If we want the root to be negative (and the closed loop stable), the condition is Kc >
1 . K
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. Example 11.7. A nonisothermal CSTR has a temperature control system. Determine the structure and optimal values of the temperature controller. Knowing the 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 m°C 3 /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
11.5 Tuning controllers for some “difficult to be controlled” processes  363
– –
3
of the actuator: KAD = 3.5 mmA/h ; , where the delays of the actuator and transducer are of the transducer: KT = 2 mA °C considered 0.
Determine the structure and optimal values of a temperature controller. The physical transfer path for propagation of the signal from the temperature of the jacket to the inner temperature of the reactor is presented in Figure 11.21. The change of temperature in the jacket changes the temperature of the wall and from this, the heat is transferred to the mass of reaction. All processes might be considered capacitive, characterized by time constant and gain.
Figure 11.21: Feedback control system for Example 11.7. The “process” to be controlled is a heat transfer one, made of three elements in series (jacket – wall – mass of reaction), each with its steady state and dynamic properties.
The transfer functions for the final control element and the transducer are considered proportional, since their delays are negligible compared with those of the “process”; the transfer function of the process is shown below: product of the jacket, { { { { wall and 0.63e−0.5s 0.45 0.8 ( )( )( ) { 4s + 1 4s + 1 26s + 1 { { mass of reaction { { transfer functions
} } } } } } } } }
364  11 Design and tuning of the controllers The expressions defining the first order capacity element are: Mj (ω) =
Kj √1 +
Tj2 ω2
φj (ω) = − tan−1 (Tj ω)
;
For dead time: Mj (ω) = 1;
φj (ω) = −360τω
The calculation of the implicit equations is done using a MATLAB program presented in Figure 11.22.
Figure 11.22: The MATLAB program calculating the crossover frequency of the control loop with a P controller.
a.
Considering a P controller with Mc (ω) = Kc and φj (ω) = 0 The phase angle plot is given in Figure 11.23. The crossover frequency, ωc = 0.163 rad/min, for the system described: rad The crossover frequency being 0.163 min , the disturbed control loop oscillates, until stabilization, with a period Tosc = 2π radrad = 38.5 min. 0.163
min
From equation (11.44), Kc is deduced using the value of ω = 0.163: (Mc ) (MAD ) (Mpr ) (MT ) = 0.5 Mj (ω) = Mw (ω) =
Kj
√1 + Tj2 ω2
(1) = 0.528
Kw
√1 + Tw2 ω2
= 0.377
11.5 Tuning controllers for some “difficult to be controlled” processes  365
Mw (ω) =
Kr √1 + Tr2 ω2
= 0.184
∴ Kc (3.5)(0.528)(0.377)(0.184)(2) = 0.5
Kc = 1.95
or PB = 51.3 %
b. Considering a PI controller at which MPI (ω) = Kc √1 +
1 Ti2 ω2
and φPI (ω) = − tan−1
1 Ti ω
Choosing Ti = 26 min (Tmax ), we can find the crossover frequency ωosc = 0.153 rad/min from the table below 0.63 √1 +
42 (0.1532 )
(1)
0.45 √1 + 42 (0.1532 )
0.8 √1 + 262 (0.1532 ) or
Kcopt = 1.72
1
(Kc ) √1 +
262 0.1532
(2) (3.5) = 0.5.
PBopt = 58.1 %
Tiopt = 26 min
ω (rad/min)
φc
φj
φw
φr
φFCE
φT
∑ φi
0.01 0.1 1.0 10 0.153
−75.4 −21.0 −2.2 −0.22 −14.1
−4.1 −39.8 −256 −1 888.6 −59.0
−2.3 −21.8 −76 −88.6 −31.5
−14.6 −69 −87.8 −89.8 −75.9
0 0 0 0 0
0 0 0 0 0
−21 −130.6 −419.8 −2 067.2 −180.5
c.
Consider a PID at which MPID (ω) = Kc √1 + (Td ω − φPID = − tan−1 (Td ω −
1 2 ) Ti ω
1 ) Ti ω
Choosing Ti = 26 min and calculating Td from the phase margin condition φc = tan−1 (Td ωosc −
1 ) = +30° Ti ωosc
And in the first approximation, considering ωosc = 0.176 rad/min without influence of I and D components of the controller. Td =
tan 30° +
1 26(0.176)
0.176
= 4.52 min
366  11 Design and tuning of the controllers 0.63 √1 + 42 (0.1762 )
(1)
0.45 √1 + 42 (0.1762 )
0.8 √1 + 262 (0.1762 ) Kcopt = 3.8
or
(Kc ) √1 + (4.52 (0.176) −
1
262 0.1762
2
) (2) (3.5) = 1.0
PBopt = 26 %
Tiopt = 26 min Tiopt = 4.5 min
11.6 Problems (1) A distillation column (Figure 11.23) 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.
Figure 11.23: Distillation column with a concentration ACS.
11.6 Problems  367
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 . The weight transducer (2) The speed of the conveyor belt (Figure 11.24) is v = min 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.
Figure 11.24: Conveyor belt with a flow ACS.
(3) Having a two CSTR cascade with a concentration ACS (Figure 11.25), using Nyquist criterion, determine the maximum controller gain for which the loop is stable.
Figure 11.25: Two CSTR cascade with concentration ACS.
368  11 Design and tuning of the controllers The data are the following: V1 = V2 = 1 m3 ,
F1 = F2 = 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 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 . (4) The components of a heater control system are shown in Figure 11.26. 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.
Figure 11.26: Temperature ACS of a heater.
Draw the diagram of the ACS response to a reference temperature change of 10 % in both cases. (5) 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. (6) 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.
References  369
(7) 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? (8) The ACS of a heat exchanger has the following transfer functions: 0.047 m/bar m3 /min ⋅ 112 0.083s + 1 m 2 °C/(m3 /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. (9) Determine the same parameters as in Problem (8), using the Nyquist method.
References [1] [2]
[3] [4] [5] [6] [7] [8]
Agachi, S., Automatizarea proceselor chimice (Chemical Processes Control), Ed. Casa Cartii de Stiinta, ClujNapoca, 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 (Chemical Processes Control), Ed. Casa Cartii de Stiinta, ClujNapoca, 1994, p. 163. Agachi, S., Automatizarea proceselor chimice (Chemical Processes Control), Ed. Casa Cartii de Stiinta, ClujNapoca, 1994, p. 166. Ziegler, J. G., Nichols, N. B., Optimum settings for automatic controllers, Trans. Am. Soc. Mech. Eng., (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. 524.
370  11 Design and tuning of the controllers
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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.onlinecourses.vissim.us/Strathclyde/tuning_pid_controllers.htm. Agachi, S., Automatizarea proceselor chimice (Chemical Processes Control), Ed. Casa Cartii de Stiinta, ClujNapoca, 1994, p. 170. Ahmad, A., Somanathi, M., PID controller tuning for integrating processes, ISA Trans., 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, Ind. Eng. Chem. Fundam., 14, (1975), 221. Agachi, P., Automatizarea proceselor chimice (Chemical Processes Control), Ed. Casa Cartii de Stiinta, ClujNapoca, 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 interelectrode voltage (all in electrochemical processes), rotation speed, torque, weight or turbidity etc. The title of the chapter indicates “process industries”, since all other than chemical process industries (mineral ore processing, beverage processing, food processing, energy production, water treatment etc.) monitor and control parameters of the same type. 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 Figure 12.1. The flow can be controlled either by choking (strangling) the material vein (Figure 12.1a), modifying the centrifugal or dosage pump motor speed by a Variable Frequency Drive (VFD) (Figure 12.1b, c, d), modifying the number of strokes of the metering (piston) pump (Figure 12.1e), or modifying the rotation speed of the motor of a conveyor belt or of a screw pump (Figure 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 Figure 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 https://doi.org/10.1515/9783110647938012
372  12 Basic control loops in process industries
Figure 12.1: Automatic control systems for solid or liquid flow.
Figure 12.2: Steady state characteristics of the centrifugal pump and of attached pipeline.
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 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.
12.1 Flow automatic control systems  373
When the flow rate control is done by modifying the rotational speed n, (Figure 12.3), the characteristics of the pump change with ni .
Figure 12.3: Steady state characteristics of the centrifugal pump and of attached pipeline when the flow is controlled by the rotational speed.
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 (Figure 12.1a) but through the pressure drop on the pipeline (Figure 12.2). The dynamic characteristic is given by the law of momentum conservation (3.16): ρghAp − (∝
lp ρv2 ρv2 d +λ ) Ap = (Ap lp ρv) 2 dp 2 dt
and if on the lefthand 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 ρF 2 . A = Ap dt 2A2p p
(12.2)
Deriving from (12.2), one can deduce the steady state characteristic ρF 2 A = 0, 2A2p p
(12.3)
2 Δp. Aρ
(12.4)
Ap Δp − C F = Ap √
The steady state characteristic F = f (Δp) is nonlinear and is shown in Figure 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
374  12 Basic control loops in process industries
Figure 12.4: Steady state characteristic of the flow.
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 %. 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 d2 f 1 df 2 Δx + Δx + ⋅ ⋅ ⋅ . 1! dx x0 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 ) 2 2 (F − Fn ) = Fn + 2Fn (F − Fn ) = 2Fn F − Fn , dF Fn
(12.7)
where Fn is the nominal operating flow desired as controlled variable value. Thus, (12.2) becomes Ap Δp − C
ρF 2 Ap lp ρ dF ρ 2Fn F + C n = . 2Ap 2Ap Ap dt
(12.8)
Arranged in another form, lp ρFn dF F + F = n Δp, 2Ap Δpn dt 2Δpn
(12.9)
12.1 Flow automatic control systems  375
it depicts a capacitive behavior with a time constant and gain factor of the process Tpr =
lp ρFn
2Ap Δpn
and Kpr =
Fn . 2Δpn
(12.10)
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), 3
Tpr
kg 2.4 m 50 ⋅ 1000 m 3 ⋅ 3600 s LρFn = = 1.08 s, = 2Ap Δpn 2 ⋅ 3.14 ⋅ 0.0004 m2 ⋅ 105 N2 m
Kpr
F = n = 3600 5 sN = 0.33 ⋅ 10−7 2Δpn 2 ⋅ 10 2 m
2.4 m3
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
376  12 Basic control loops in process industries process is noisy, very unstable around the average values of the flow rate, inducing instability in the control loop. 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
mA , l/min bar . = 0.05 mA
l/min m3 /s , = 20 2 bar N/m
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.
A new trend in controlling the flow nowadays is not to choke the flow using a control valve, with high energy costs, but to change the rotation speed of motor of the pump creating the flow. In the traditional industries (chemical, ore processing, beverage), in spite of the demonstrated advantage of the flow control using Variable Frequency Drives (VFD), still this method is not used on a large scale.
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 (Figure 12.5)
Figure 12.5: Controlling the pressure in a distillation column.
12.2 Pressure automatic control systems  377
can be modified either through the outflow of the noncondensable gases when these exist, or through the cooling agent flow rate of the condenser at the top of the 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 (Figure 12.6).
Figure 12.6: The pressure can be controlled through the recipient inflow (a) or outflow (b).
Process characteristics Considering the first situation in Figure 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 Table 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 electricpneumatic equivalence (valid for laminar flow): u ≡ Δp and i ≡ Fm , by p −p p − p1 and Fm2 = , Fmi = 0 R1 R2 for pneumatic circuits, equivalent with Ohm’s law i=
u V1 − V2 = . R R
378  12 Basic control loops in process industries 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), R1 R2 R1 R2 VM dp +p= p + p R1 + R2 RT ∘ dt R1 + R2 1 R1 + R2 0
(12.14)
where Tpr =
R1 R2 VM R1 + R2 RT ∘
and
Kpr =
R2 R1 + R2
(12.15)
(if the control is done as in case (a)). 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 (Figure 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?
12.3 Level automatic control systems  379
Using equation (12.15), 2
Tpr
2
−3 3 1.2 ⋅ 108 N/m ⋅ 1.6 ⋅ 108 N/m kg/s kg/s 200 ⋅ 10 m ⋅ 29 kg/kmol = = 159 s, 2 2 8314 J/kmolK ⋅ 300 K 1.2 ⋅ 108 N/m + 1.6 ⋅ 108 N/m kg/s
Kpr =
1.6 ⋅ 1.2 ⋅
kg/s
2 108 N/m kg/s
2 108 N/m kg/s
2
+ 1.6 ⋅ 108 N/m kg/s
= 0.57.
The process has a capacitive dynamic behavior. 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 +
12
⋅
12
√1 +
5
12
⋅
12
0.57
2 2
√1 + 159 ⋅1
Kc √1 +
1
2
120 ⋅ 12
= 0.5,
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 (Table 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.
380  12 Basic control loops in process industries 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 ensure the presence of the liquid phase in the two pieces of equipment [4]. The level control (Figure 12.7) can be assured through the manipulation either of the inflow or the outflow in a recipient.
Figure 12.7: Level control in a recipient. (a) the outflow is constant; (b) the outflow depends on the level.
Process characteristics In the first situation (Figure 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 Applying the mass conservation law Ar ρ
dh = Fi 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 ρghAp − (∝
lp ρv2 lp ρv2 ρv2 +λ ) Ap = ρghAp − (∝ +λ ) A = 0, 2 dp 2 dp 2 p
(12.17)
from which v=√
2gh α+
l λ dp p
and
Fo = Ap √
2gh
l
α + λ dp
.
p
Equation (12.16) becomes Ar
dh 2gh . = Fi − Ap √ l dt α+λ p
(12.18)
dp
If we denote = Ap √
2g
l
α+λ dp
, equation (12.18) becomes
p
Ar
dh = Fi − C √h, dt
a differential nonlinear first order equation.
(12.19)
12.3 Level automatic control systems  381
The steady state characteristic, h = f (Fi ), h=
A2p
1
2g
l α+λ dp p
Fi2
(12.20)
is represented in Figure 12.8.
Figure 12.8: Nonlinear behavior of the level in the tank.
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 − hn ) = √hn + 1 (h − hn ), dh hn 2√hn
(12.21)
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 +h= Fi − hn , dt Ap √ 2g lp Ap √ 2g lp α+λ α+λ dp
(12.22)
dp
where Tpr =
2√hn Ar Ap √
2g
[s] and Kpr =
l
α+λ dp
p
2√hn Ap √
2g
[ l
α+λ dp
m
m3 s
].
(12.23)
p
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
382  12 Basic control loops in process industries
Figure 12.9: Placing the transducer in a lateral chamber induces oscillations.
boiling process in the reboiler. These processes (stirring, boiling) permanently disturb the surface measured. If the transducer is one with a side measuring chamber (Figure 7.35), the equivalent scheme of functioning is given in Figure 12.9. 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): ρgh1 A1 − ρgh2 A2 − R1 A1 v1 − R2 A2 v2 = M1
dv dv1 + 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 v1 A1 ρ = v2 A2 ρ
(12.26)
and the equality between the quantities displaced in both recipient and chamber, (h − h1 ) A1 = (h2 − h)A2 ,
(12.27)
12.3 Level automatic control systems  383
equation (12.24) becomes ρgh (A1 + A2 ) − 2ρgh2 − (R1 + R2 ) A2 or
dh2 d2 h = (L1 + L2 )A2 ρ 22 dt dt
(L1 + L2 ) d2 h2 R1 + R2 dh2 A + A2 + h2 = 1 h, + 2g ρg dt A2 dt 2
(12.28)
(12.29)
which describes a damped oscillations behavior (equations (4.27) and (4.28) and Figure 4.12) with ωn = √
2g L1 + L2
and ζ =
R1 + R2 . ρ√2g(L1 + L2 )
(12.30)
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. 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 Figure 6.13 and schematically represented in Figure 12.8. 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 (Figure 12.10) with the following characteristics? The tank has Ar = 1 m2 ,
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 .
384  12 Basic control loops in process industries
Figure 12.10: Tank with level control loop.
The transducer has measuring range: 0–2 m; output signal range: 2–10 mA; capacitive behavior: TT = 5 s.
The control valve has electropneumatic converter input signal range: 2–10 mA; electropneumatic 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 =
Kpr =
2√0.5 ⋅ 1 3⋅
10−4 √ 2⋅9.8 1 2√0.5
3⋅
10−4 √ 2⋅9.8 1
= 0.104 ⋅ 104 s = 0.28 h = 17.4 min
= 1064
m
m3 s
= 0.295
The time constant and gain of the transducer: TT = 10 s = 0.17 min; KT =
mA 10 − 2 =4 . 2−0 m
m
m3 h
= 17.73
m
m3 min
.
12.4 Temperature automatic control systems  385
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. The module condition: 17.73
2
4
2
0.14
√1 + 17.4 ⋅ 3.2 √1 + 0.08 ⋅ 3.2 √1 + 0.082 ⋅ 3.22 2
Kc = 2.8;
2
Kc = 0.5
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 (Figure 12.11). 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 ∘ ) (Figure 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 “thermally thin”; CSTR is perfectly mixed – lumped parameter system; the reaction is
386  12 Basic control loops in process industries
Figure 12.11: Temperature control of a CSTR or at the bottom of a distillation column.
Figure 12.12: The capacitive elements in series describing the process behavior in the temperature control loop.
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 Figure 12.12. The equations are those are from the Example 3.1: ∘ ∘ ∘ Fvag ρag cpag Tiag − Fvag ρag cpag Tag + KTag AT (Tw∘ − Tag )= ∘ KTi AT (T ∘ − Tw∘ ) − KTag AT (Tw∘ − Tag )=
d (M c T ∘ ) dt w pw w
d (V ρ c T ∘ ) dt j ag pag ag
Fvi ρA cpA Ti∘ − Fρcp T ∘ − KTi AT (T ∘ − Tw∘ )−K TT ATT (T ∘ − TT∘ ) − VkCA ΔHr =
(12.31) (12.32)
d (Vρcp T ∘ ), dt (12.33)
where ΔHr = 0. Besides these equations, we should write the equation describing the functioning of the thermocouple: KTT ATT (T ∘ − TT∘ ) − Qel =
d (M c T ∘ ), dt T pT T
(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 negligible.
12.4 Temperature automatic control systems  387
For the steady state characteristic we notice that it is nonlinear, the nonlinearity being given by ∘ ∘ ∘ Fvag ρag cpag Tiag − Fvag ρag cpag Tag + KTag AT (Tw∘ − Tag ) = 0.
(12.35)
The other characteristics, give a linear behavior Fvi ρA cpA Ti∘ − Fρcp T ∘ − KTi AT (T ∘ − Tw∘ )−K TT ATT (T ∘ − TT∘ ) = 0
(12.36)
(when the heat of reaction is neglected) ∘ KTi AT (T ∘ − Tw∘ ) − KTag AT (Tw∘ − Tag ) = 0.
(12.37)
The combined three equations can be assembled in T ∘ = f (Fv ag ), 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 ∘ nonlinearity Fvag Tag . The nonlinearity is linearized around the steady state point, ∘ (Fvag n , Tag n ) ∘ ∘ ∘ ∘ ∘ Fvag Tag = Fvag n Tag n + Fvag (Tag − Tag n ) + Tag n (Fvag − Fvag n ) ∘ ∘ ∘ Q = Fvag Tag n + Tag Fvag n − Fvag n Tag n .
(12.38)
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 ∘ ∘ ∘ ∘ = Fvag ρag cpag (Tiag − Tag n ) + Fvag n ρag cpag Tag n + KTag AT Tw .
(12.39)
Trying to obtain the canonic form of the capacity, we obtain Mj cpag
∘ dTag
Fvag n ρag cpag + KTag AT dt +
∘ + Tag =
∘ ∘ ρag cpag (Tiag − Tag n)
Fvag n ρag cpag + KTag AT
∘ Fvag n ρag cpag Tag n
Fvag n ρag cpag + KTag AT
+
Fvag
KTag AT
Fvag n ρag cpag + KTag AT
Tw∘ .
(12.40)
388  12 Basic control loops in process industries The jacket time constant and jacket gain are Tj =
Mj cpag
Fvag n ρag cpag + KTag AT
and Kj =
∘ ∘ ρag cpag (Tiag − Tag 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) Tw = Tr =
Mw cpw
and Kw =
KTag AT + KTi AT Mr cpr
KTi AT + Fρcp + KTT ATT
and Kr =
KTag AT
KTag AT + KTi AT KTi AT
KTi AT + Fρcp + 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)
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 Figure 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,
12.4 Temperature automatic control systems  389
cpw = 0.15 kcal/kg.K; partial heat transfer coefficient heating agentwall: KTag = 500 kcal/m2 K; partial heat transfer coefficient wallreaction mass: KTr = 1500 kcal/m2 K; ∘ ∘ characteristics of the fluids: ρ = ρag = 1000 kg/m3 ; Tiag = 90 °C; Tagn = 78 °C; 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 massthermowell: 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: electropneumatic converter input signal range: 2–10 mA; electropneumatic 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): 0.3 ⋅ 1000 ⋅ 1 = 0.05 h = 3 min 3 ⋅ 1000 ⋅ 1 + 500 ⋅ 6
Tj =
and Kj = Tw =
°C 1000 ⋅ 1 ⋅ (90 − 78) =2 3 m 3 ⋅ 1000 ⋅ 1 + 500 ⋅ 6 h
350 ⋅ 0.15 = 0.0044 h = 0.27 min 500 ⋅ 6 + 1500 ⋅ 6 500 ⋅ 6 °C = 0.25 500 ⋅ 6 + 1500 ⋅ 6 °C
and Kw = Tr =
1.5 ⋅ 1000 ⋅ 1 = 0.13 h = 8.2 min 1500 ⋅ 6 + 2 ⋅ 1000 ⋅ 1 + 1500 ⋅ 0.02 and Kr =
1500 ⋅ 6 °C = 0.82 . 1500 ⋅ 6 + 2 ⋅ 1000 ⋅ 1 + 1500 ⋅ 0.02 °C
The temperature transducer: TTw =
0.5 ⋅ 0.15 = 0.025 h = 1.5 min 1500 ⋅ 0.02
and KTw = 1
°C °C
390  12 Basic control loops in process industries KT = KTw Kad = 1
8 mA mA °C ⋅ = 0.2 . °C 80 − 40 °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° 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 0.2 2 0.82 √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
Ti = 8 min
or PB = 9 %
Td = 2 min. The simulator below (Figure 12.13), created in the Laboratory of Process Control from the University BabesBolyai, Romania [5] shows the behavior of temperature control systems at different combinations of controller’s parameters (KP , Ti , Td ). The upper diagram in the caption shows the controlled temperature in the reactor, and the
12.5 Composition automatic control systems  391
Figure 12.13: LabView simulator panel for the Temperature Control System of a CSTR.
lower diagram shows the temperature in the jacket. In the first part of the time diagram, the behavior of the loop is quite stable at the set point at a certain combination of effects (KP = 10, integral and derivative effect being 0). Though, the operation of the ACS is not so good because of the quick and repeated changes of the temperature of the jacket (yellow plot), meaning a quick and repeated opening/closing of the control valve which has a result the wearing of it. Introducing the integral component, the system becomes totally unstable both for jacket and reaction mass temperature.
12.5 Composition automatic control systems Since chemical products are the main goal of the chemical/pharma industry, not forgetting other industries as well, composition control seems to be one of the most important of all control systems. More recently, environmental engineering is also focused on the quality of the environment, measuring or controlling the composition of water, soil or air [6–8]; waste or drinkable water treatment also focuses on controlling quality parameters of the water as pH or different ions as carbon, nitrogen or phosphorous content [9–11]. 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).
392  12 Basic control loops in process industries 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 [12]. According to our knowledge, but it is also evidenced in [13], 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. 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 Figure 12.14. The considered process is one of diluting a concentrate C with a solvent S.
Figure 12.14: 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)
12.5 Composition automatic control systems  393
Figure 12.15: Nonlinear characteristic of a diluting process.
The steady state characteristic is presented in Figure 12.15. The gain of the process in the nominal operating point (nominal concentrate flow rate Cn ) is Kpr n =
Sxcn Sx Cn dx 1 . − Cn = xcn [ ]= = cn 2 2 dC S + Cn (S + Cn ) F2 (S + Cn )
(12.47)
The dynamic characteristic, as mentioned before, depends on the agitator’s efficiency The efficiency of the mixing agitator is defined by [7] ηs =
Fa , Fa + F
(12.48)
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 = = ; Tpr = 0. (12.50) 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 F V ⋅ = (1 − ηa ) . F Fa + F F
(12.51)
394  12 Basic control loops in process industries 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. Control valve Due to the nonlinear characteristic of the process (Figure 12.15), 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 (Figure 12.16). The very steep slope of the characteristic induces a cycle of oscillations of the control system.
Figure 12.16: pH steady state characteristics for different combinations: acid neutralized with NaOH.
12.5 Composition automatic control systems  395
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. A usual pH control system in a CSTR use the cascade control, the inner loop being that of a ratio flow control (Figure 12.17). The pH control in a wastewater treatment plant is more complicated and is approached in the second volume of this book.
Figure 12.17: pH control system in a CSTR reactor.
Example 12.5. Consider a dilution process control system in a CSTR (Figure 12.14), 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 %.
396  12 Basic control loops in process industries 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: electropneumatic converter input signal range: 4–20 mA; electropneumatic 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 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 F V 350 ⋅ = (1 − ηa ) = 0.05 = 0.25 min F Fa + F F 70
Kpr n =
Sxcn
2
(S + Cn )
=
(F − Cn )xcn 35 ⋅ 1 − . = 0.0071 = 2 l/min F2 70
The transducer is a gas chromatograph with τT = 0.25 min
TT = 0.05 min KT =
mA 20 − 4 = 160 . 0.55–0.45 −
The final control element is a valve with electropneumatic convertor and actuator with the characteristics: KAD = KEPC ⋅ KAC ⋅ Kcv = TAD = 3 s = 0.05 min.
1 − 0.2 bar 100 − 0 % l/min l/min ⋅ ⋅ 3 ⋅ 0.7 = 13.12 and 20 − 4 mA 1 − 0.2 bar % mA
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°.
12.6 Problems  397
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.
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
Ti = 5 min
PB = 476 %
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 (Figure 12.18). 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 correction 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
398  12 Basic control loops in process industries
Figure 12.18: Composition ACS for a cascade of 3 CSTR.
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).
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[10] VasileMircea, C., Zuza, A., Agachi P.S., Nair, A., Ngoc, T. N., HorjuDeac C., Case study on energy efficiency of biogas production in industrial anaerobic digesters at municipal wastewater treatment plants, Environ. Eng. Manag. J., 14 (2), (2015), 357–360. [11] Brehar, M., Varhelyi, M., Cristea, V. M., Crîstiu, D., Agachi, P. S., Influent temperature effects on the activated sludge process at municipal wastewater treatment plant, Stud. Univ. Babeş–Bolyai Chem., 64 (1), (2019), 113–123, DOI:10.24193/subbchem.2019.1.09. [12] Shinskey, G., Process Control Systems, 4th Edition, McGraw Hill, NY, 1996, pp. 275. [13] Agachi, S., Automatizarea Proceselor Chimice (Chemical Process Control), Casa Cartii de Stiinta, ClujNapoca, 1994, pp. 234.
Index ABI Research 6 abstract oriented system 20, 21 accuracy 83, 165, 167, 171, 177, 178, 191–193, 204, 206, 210, 211, 213, 214, 218, 221, 222, 228, 231, 234, 253, 394 accuracy class 168, 171, 180, 184–186, 197, 199–202, 204, 206, 207, 209, 210, 235, 236, 239, 240 activation energy 88 actuator 323, 327 adaptive controller 265, 266 advanced process control (APC) 6 Advanced Process Control (book) 265, 269 alarm and monitoring system 321 analytical model 68, 78 Artificial Intelligence 6 Artificial Neural Network (ANN) 68, 85 automatic control 4, 135, 137, 139 – system 135, 138 automatic control system (ACS) 331, 371, 372, 376, 379, 385, 391 ball or Vnotch control valve 294 Barton cell 182 biocybernetic control systems (BCS) 6 Bode diagram 50–52, 57, 108, 122 bottom molar flow 95 Bourdon tube 177, 178, 180, 181 butterfly valve 294 capacitive level transducer 209 capacitive pressure transducer 184 capacitive system 105, 107, 112, 116 cavitation 301, 302 Central Processing Unit (CPU) 279 centrifugal pump 139, 299, 371 centrifugal pump head pressure 371 chemical engineering 16 chromatograph 240 coal and gas industry VI Cohen–Coon 348 Cohen–Coon method 345, 346, 348, 357 cold junction 175–177 composition control 391, 395 Compound Annual Growth Rate (CAGR) 7 condenser 95, 100, 184, 229, 231, 377
conductive level transducers 211 continuous stirred tank reactor 113 continuous stirred tank reactor (CSTR) 70 control quality criteria 335 control valve 266, 272, 291, 294, 295, 394 controllability 153, 162, 249, 335, 346, 361 controller tuning 151, 333, 342 ConvergingInput ConvergingOutput stability (CICO) 31 conveyor belt 291, 371 Coriolis mass flow meter 200 criterion of integral timeweighted absolute error (ITAE) 337 crossover frequency 261, 264, 385 crystallization 207 CSTR cascade 367 damping ratio 141 data reconciliation 272 decay ratio 141, 337, 349, 357 delimiting surface 10 density transducer 213 derivative system 105, 109–111 derivative time 112, 249, 262 deterministic system 18, 20 differential equation 22, 37, 93 digital controller 247, 249 dissolved oxygen transducer 226 distillate drum 95, 96 distillate molar flow 95 distillation column 93, 138, 153, 318, 380 distributed control system 321 distributed control system (DCS) 6 Distributed Control System (DCS) 270 distributed parameter system 69, 70, 74 Doppler effect 198, 199 dynamic system 14, 15, 19 electrical conductivity transducer 218 electrical motor actuator 293 electropneumatic convertor 254, 292, 311, 396 elementary signal 24, 27 energy 88 energy conservation law 73 error 85, 137, 138, 141, 151, 155, 166–168, 171, 210, 247, 249, 250, 255, 257, 258, 261, 269, 336, 341, 348
402  Index
feed flow 95, 138 feed forward control 153 feedforward control 6 feedback control system 137, 138, 143, 145, 155 final control element 291, 298, 323, 325 First Industrial Revolution 3 first order system 112–114, 123 flow control 371, 395 flow controller 249 flow factor 300–302 flow factor (Kv ) – for gases 303 – for incompressible fluids 300 – for steam 304 flow sheet 68 food and beverage VI Fourth Industrial Revolution 6 frequency response 19, 32–35, 46, 59 gas analyzer 228 gas analyzers based on paramagnetism 234 gear pump 299 globe valve 294 harmonic signal 32 heat capacity 88, 263 heat of reaction 74, 88, 125 heat transfer area 74, 174, 288, 377, 386 higher order capacitive system 118 higher order system 117 holdup 95 homogeneous differential equation 28, 29, 31 hot junction 175, 176 humidity gas analyzer 236 hydraulic actuator 294 impulse response matrix 40, 50 inductive pressure transducer 185 inferential mass flow meter 200, 202 infrared thermometer 179 infrared gas analyzer 228 input module 279 input signal 19, 27, 32–34, 46, 64, 121, 250, 272, 317, 369, 384, 389, 396 input variable 13, 17, 35 inputoutput path with logic solver – configuration 328 – multiplication 328 inputoutput relationship 46, 64, 105, 109
inputoutput representation 21, 36, 37 inputstateoutput relationship 21, 22, 40, 48, 49 installed characteristic 307, 309 integral of the absolute error (IAE) 336 integral of the error (IE) 335 integral of the squared error 268 integral system 107, 108, 110 integral windup 261 interlocking system 319, 328 internal molar reflux 95 inverse response 361 ionization gauge 186, 187 kinetic equation 68 Ladder diagrams 283 Lambda method 347 laminar flow 302 large overshoot 338 latent heat of vaporization 79, 95 level control 89, 249, 380 level controller 383 level transducers based on floats 205 linear characteristic 306, 311 linear timeinvariant system 28, 30, 36, 40, 45 linearity 18, 32, 48, 84 logarithmic (equal percentage) characteristic 306 logic functions (AND, OR, IF) 277 logic solver 323, 325, 326, 328 looms 3 lumped parameters system 69 magnetic flow meter 196, 198 manual control 5, 135 mass conservation law 69, 70, 380 Mathematica 73 mathematical model 68, 78, 87 MATLAB VI, 62, 63, 73, 126–128, 158, 195, 196, 316, 364, 398 measurand 165–168 measuring device 169, 191 membrane control valve 296 Metallurgical Engineering VI method of limit of stability 343 MIMO system 145 mining VI model predictive control 269
Index  403
model predictive control (MPC) 6, 153 modified parabolic characteristic 306, 375 molar mass 88 momentum conservation law 68, 76, 77 multiple effects evaporator 153 nonhomogeneous differential equation 30, 39 nonlinearity 151, 189 Nyquist diagram 59 Nyquist stability criterion 354 object 9 onoff controller 247 optimal parametrical control system (OPCS) 6 Ore Processing VI orifice plate 188–190, 312, 375 oscillation 114, 141, 155, 182, 206, 261, 333, 335, 336, 342, 343, 357, 383, 394 output module 279, 281 output variable 13, 17, 18, 22, 35, 38, 45, 67, 71, 87, 92, 109, 113, 119, 122, 146, 162 oval shaped gear meter 203 overshoot 140, 340, 342 Peltier effect 175, 176 pH meter 224 PI controller 258, 261 piezoelectric effect 185 Pirani gauge 186, 187 Pitot tube 191 plug 291 Plug Flow Reactor (PFR) 71 polarographic cell 226, 227 positioner 292, 328 predictive controller 269 pressure control 253, 256, 309, 377 pressure controller 249 pressure drop 188–190 PRISM technology 6 process control 3, 5, 135, 140, 169, 242, 269, 333 process engineering 4, 172, 180, 187, 203 Programmable Logic Controller (PLC) 277, 322, 325 Proportional Band (PB) 251 proportional controller 250 – gain 250 proportional system 105, 107, 221 psychrometric transducer 239
pure delay system 121 PVC 171, 193, 267 quarter decay response criterion 337 radar level transducer 210 radiometric level transducers 210 reaction rate 23 – constant 125 reaction rate constant 379 reboiler 291, 382 reflux flow 366 regression analysis 81 relative gain array (RGA) 145 relative volatility 79, 93 resistance temperature detector 172, 202 Resistance Temperature Detector (RTD) 172 response curve 345 Reynolds number 190, 200 Reynold’s number 302 rotameter 192 Routh stability criterion 350, 351 safety instrumented system 321, 328 safety integrity 328 safety layer 321 seat ring 291 Second Industrial Revolution 3 second order system 105, 114–116, 119 Seebeck effect 175, 176 SemiContinuous STR 99 sensor 172, 180, 182, 200, 323, 325 sensors 322 sinusoidal signal 27 SISO system 146 smart transducer or control valve 272 solenoid actuator 293 solid state technology 278 solvate 79 solvent 79, 392, 393 stability 30, 31 state variable 21, 22, 36, 37 static system 14 statistical model 78 steady state 27, 34 – error (SSE) 140, 141, 151, 253, 256, 257, 339 – gain 112–114, 147, 149, 373 stem 5, 254, 291, 292, 327, 328, 384, 389 stem travel 292, 300, 306, 309, 311
404  Index
stochastic system 18 subject 9 successive trial in the process 342 sum of square residuals SSR 82 system concept 9 temperature control 3, 142, 145, 262, 265, 269, 385 thermal conductivity gas analyzer 232 thermistor 172, 178, 200 thermocouple 174, 176, 177, 179 thermodynamic and kinetic equations 68 thermodynamic equation 68 Third Industrial Revolution 5 Thomson effect 176 transducer 14, 137, 145, 154, 169–172, 177, 182, 186, 187, 189, 191, 193, 202, 206, 208, 211, 213, 215, 219, 221, 225, 227, 228, 247, 251, 253, 255, 312, 333, 345, 355, 375, 378, 382, 383, 386, 388, 389, 394, 396 transfer function 45, 46 transfer matrix 49, 50 transfer path 116, 124, 137, 312, 313, 385 transient response 34 transient time 140 transition matrix 39 transmitter 169, 171, 173, 174, 178, 179, 182, 184, 191, 193, 197, 205, 209–211, 214, 226, 227, 231, 235, 239, 240, 324, 375
tray 94, 95 turbulent flow 198, 301 ultrasonic level transducer 209 unified signal 170, 388 unit impulse function 25, 27 unit ramp function 24 unit step function 24 vapor flow 95, 96 variable frequency drive 299 Variable Frequency Drive 371 Venturi tube 187 viscosity transducer 217 Vortex flow meter 196 Vortex shedding flow 199 weir 94, 95, 213, 296 Wheatstone bridge 174, 183, 186, 202, 212, 225, 226, 232, 233, 235 wiring diagrams 281, 282 wise machinery (WM) 6 Woltman turbine meter 203 Ziegler–Nichols 348 zirconia oxygen gas analyzer 235