Energy 26 (2001) 307–319www.elsevier.com/locate/energyPrinciples of control thermodynamicsP. Salamona,*, J.D. Nultonb, G. Siragusac, T.R. Andersend, A. LimonaaDepartment of Mathematical and Computer Sciences, San Diego State University, 5300 Campanile Drive, SanDiego, CA 92182, USAbDepartment of Mathematics, San Diego City College, San Diego, CA 92101, USAcDepartment of Chemistry, San Diego State University, 5300 Campanile Drive, San Diego, CA 92182, USAdCAPEC, Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, DenmarkReceived 28 September 1999AbstractThe article presents a partial synthesis of progress in control thermodynamics by laying out the mainresults as a sequence of principles. We state and discuss nine general principles (0–8) for finding boundson the effectiveness of energy conversion in finite-time.  2001 Elsevier Science Ltd. All rights reserved.1. IntroductionThis article presents a synthesis of progress using a particular approach to the meaning oftime for thermodynamic processes. The approach captures one aspect of the flavor of traditionalthermodynamics: that of providing bounds. Our aim is to understand the limiting role of time ina thermodynamic process. Specifically, our quest is to understand the limits to energy conversionprocesses in which the time evolution is only partially specified, i.e. the sequence of states tra-versed by some part of the system is given. We then ask the question: Of what total processmight this given time evolution of our subsystem be a part? In general there are many possibleanswers to this question. One of the goals of the endeavors described below is to examine themathematical structure of this set of possible co-evolutions of our subsystem and its sequence ofenvironments. In particular, we look for extreme points in this set; notably ones that maximizework or minimize entropy production. As a simple example, consider the operation of a heatengine in which a gaseous working fluid traverses a given cycle as specified by a quasistatic locusin its (p,V) plane, i.e. by an indicator diagram. This example in various guises has resurfaced in* Corresponding author.E-mail address: [email protected] (P. Salamon).0360-5442/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved.PII: S0360-5442(00)00059-1308 P. Salamon et al. / Energy 26 (2001) 307–319all approaches to what we classify here as control thermodynamics. The set of co-evolutions ofthis working fluid with an environment has been modeled and studied in various ways and withvarious possible thermal contacts with the co-evolving environment. Interesting general statementscan be made about minimizing entropy production and maximizing power given the path traversedby the working fluid.Our framework for attacking this problem of course builds on prior developments in thermodyn-amics and allied fields such as heat transfer and fluid mechanics.1These fields specify the dynami-cal equations for systems of interest. Generally, the framework we consider has some incompletelyspecified set of dynamical equations which leave some parameters available for control. Suchparameters could represent, for example, the state of the environment. In the standard controlproblem formulation, this means that some of our variables appear in the dynamical equationsbut do not themselves have specified time derivatives. In this formulation the variables are dividedinto two classes: x=(x1,x2,…,xn) and u=(u1,u2,…,un) where the xs are those variables for which wehave a dynamical equation and the us are the rest. Thus the dynamical equations take the formx˙⫽f(x,u). (1)In the control theory literature, the xs are called the state variables and the us are called thecontrol variables.2Specifying the values of the us results in a complete system of equations forthe xs. One can then ask for the optimal controls u*(t).3A weaker, but much more generallytractable, question concerns finding bounds on the resultant optimal values of the objectives. Thesequestions constitute the essence of control thermodynamics.This finite-time control approach to thermodynamics has been pursued by engineers for a longtime [3]. The present paper is concerned with the extraction of general principles which apply tostudies using this approach.42. A little historyIn the early 1970s, four “groups”5working independently developed some general principlesgoverning 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, Andr-esen, Salamon, and Nitzan in Chicago, Rozonoer and Tsirlin in the Soviet Union, and Curzon1This dependence of our approach on its allied fields has been emphasized in Bejan’s textbooks where he depicts this new subjectusing a triangle with edges labeled by thermodynamics, heat transfer, and fluid mechanics [2].2Note that this usage conflicts with the thermodynamic use of these words since at least some of the control variables in a problemmay well coincide with a thermodynamic state variable of some system.3Or, better still, the optimal feedback controls u*(x) where the values of the control variables are given in terms of the statevariables rather than in terms of the time.4We caution the reader that the authors whose works are discussed here do not necessarily see their work as “control thermodyn-amics”.5Strictly speaking, the first of these “groups” (Adrian Bejan) was a single individual. Furthermore, his association with MIT wasonly as an undergraduate and a graduate student. Despite this, we refer to four groups for convenience.309P. Salamon et al. / Energy 26 (2001) 307–319and Ahlborn in Canada. While many of their results coincided, each group has made importantcontributions; 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] treatedreciprocating operation of heat engines, the approach of Bejan [2,9] was based on steady-stateoperation of a distributed cycle. The difference in focus between these two approaches implies areal physical difference in the way the processes are conceptualized. In the case of a steady-stateoperation, we envision the working fluid as flowing continuously around the apparatus, with someportion of the fluid in each of the states along the quasistatic locus at each instant of time. Eachpoint on this