# Principles of Control Thermodynamics (13 pages)

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Energy 26 2001 307 319 www elsevier com locate energy Principles of control thermodynamics P Salamon a J D Nulton b G Siragusa c T R Andersen d A Limon a a Department of Mathematical and Computer Sciences San Diego State University 5300 Campanile Drive San Diego CA 92182 USA b Department of Mathematics San Diego City College San Diego CA 92101 USA c Department of Chemistry San Diego State University 5300 Campanile Drive San Diego CA 92182 USA d CAPEC Department of Chemical Engineering Technical University of Denmark DK 2800 Lyngby Denmark Received 28 September 1999 Abstract The article presents a partial synthesis of progress in control thermodynamics by laying out the main results as a sequence of principles We state and discuss nine general principles 0 8 for finding bounds on the effectiveness of energy conversion in finite time 2001 Elsevier Science Ltd All rights reserved 1 Introduction This article presents a synthesis of progress using a particular approach to the meaning of time for thermodynamic processes The approach captures one aspect of the flavor of traditional thermodynamics that of providing bounds Our aim is to understand the limiting role of time in a thermodynamic process Specifically our quest is to understand the limits to energy conversion processes in which the time evolution is only partially specified i e the sequence of states traversed by some part of the system is given We then ask the question Of what total process might this given time evolution of our subsystem be a part In general there are many possible answers to this question One of the goals of the endeavors described below is to examine the mathematical structure of this set of possible co evolutions of our subsystem and its sequence of environments In particular we look for extreme points in this set notably ones that maximize work or minimize entropy production As a simple example consider the operation of a heat engine in which a gaseous working fluid traverses a given cycle as specified by a quasistatic locus in its p V plane i e by an indicator diagram This example in various guises has resurfaced in Corresponding author E mail address salamon math sdsu edu P Salamon 0360 5442 01 see front matter 2001 Elsevier Science Ltd All rights reserved PII S 0 3 6 0 5 4 4 2 0 0 0 0 0 5 9 1 308 P Salamon et al Energy 26 2001 307 319 all approaches to what we classify here as control thermodynamics The set of co evolutions of this working fluid with an environment has been modeled and studied in various ways and with various possible thermal contacts with the co evolving environment Interesting general statements can be made about minimizing entropy production and maximizing power given the path traversed by the working fluid Our framework for attacking this problem of course builds on prior developments in thermodynamics and allied fields such as heat transfer and fluid mechanics 1 These fields specify the dynamical equations for systems of interest Generally the framework we consider has some incompletely specified set of dynamical equations which leave some parameters available for control Such parameters could represent for example the state of the environment In the standard control problem formulation this means that some of our variables appear in the dynamical equations but do not themselves have specified time derivatives In this formulation the variables are divided into two classes x x1 x2 xn and u u1 u2 un where the xs are those variables for which we have a dynamical equation and the us are the rest Thus the dynamical equations take the form x f x u 1 In the control theory literature the xs are called the state variables and the us are called the control variables 2 Specifying the values of the us results in a complete system of equations for the xs One can then ask for the optimal controls u t 3 A weaker but much more generally tractable question concerns finding bounds on the resultant optimal values of the objectives These questions constitute the essence of control thermodynamics This finite time control approach to thermodynamics has been pursued by engineers for a long time 3 The present paper is concerned with the extraction of general principles which apply to studies using this approach 4 2 A little history In the early 1970s four groups 5 working independently developed some general principles governing optimal control of thermodynamic processes in finite time These groups were Bejan working alone as an undergraduate student and later as a graduate student at MIT Berry Andresen Salamon and Nitzan in Chicago Rozonoer and Tsirlin in the Soviet Union and Curzon 1 This dependence of our approach on its allied fields has been emphasized in Bejan s textbooks where he depicts this new subject using a triangle with edges labeled by thermodynamics heat transfer and fluid mechanics 2 2 Note that this usage conflicts with the thermodynamic use of these words since at least some of the control variables in a problem may well coincide with a thermodynamic state variable of some system 3 Or better still the optimal feedback controls u x where the values of the control variables are given in terms of the state variables rather than in terms of the time 4 We caution the reader that the authors whose works are discussed here do not necessarily see their work as control thermodynamics 5 Strictly speaking the first of these groups Adrian Bejan was a single individual Furthermore his association with MIT was only as an undergraduate and a graduate student Despite this we refer to four groups for convenience P Salamon et al Energy 26 2001 307 319 309 and Ahlborn in Canada While many of their results coincided each group has made important contributions a sample result from each group is presented later in this paper While the approach of Berry and co workers 4 7 and of Curzon and Ahlborn 8 treated reciprocating operation of heat engines the approach of Bejan 2 9 was based on steady state operation of a distributed cycle The difference in focus between these two approaches implies a real physical difference in the way the processes are conceptualized In the case of a steady state operation we envision the working fluid as flowing continuously around the apparatus with some portion of the fluid in each of the states along the quasistatic locus at each instant of time Each point on this locus then corresponds to a particular physical location in the apparatus In the case of reciprocating operation we