Thermodynamics

A Dynamical Systems Approach

Wassim M. Haddad, VijaySekhar Chellaboina, Sergey G. Nersesov

• Publisher: Princeton University Press
• Year: 2009
• ISBN:
  9781400826971
• Topic: Ciencias físicas
• Subtopic: Termodinámica

This book places thermodynamics on a system-theoretic foundation so as to harmonize it with classical mechanics. Using the highest standards of exposition and rigor, the authors develop a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic large-scale dynamical system models that bridge the gap between classical and statistical thermodynamics.
The authors' theory is motivated by the fact that a discipline as cardinal as thermodynamics--entrusted with some of the most perplexing secrets of our universe--demands far more than physical mathematics as its underpinning. Even though many great physicists, such as Archimedes, Newton, and Lagrange, have humbled us with their mathematically seamless eurekas over the centuries, this book suggests that a great many physicists and engineers who have developed the theory of thermodynamics seem to have forgotten that mathematics, when used rigorously, is the irrefutable pathway to truth.
This book uses system theoretic ideas to bring coherence, clarity, and precision to an extremely important and poorly understood classical area of science.

Chapter One

Introduction

1.1 An Overview of Thermodynamics

Energy is a concept that underlies our understanding of all physical phenomena and is a measure of the ability of a dynamical system to produce changes (motion) in its own system state as well as changes in the system states of its surroundings. Thermodynamics is a physical branch of science that deals with laws governing energy flow from one body to another and energy transformations from one form to another. These energy flow laws are captured by the fundamental principles known as the first and second laws of thermodynamics.The first law of thermodynamics gives a precise formulation of the equivalence between heat and work and states that among all system transformations, the net system energy is conserved. Hence, energy cannot be created out of nothing and cannot be destroyed; it can merely be transferred from one form to another. The law of conservation of energy is not a mathematical truth, but rather the consequence of an immeasurable culmination of observations over the chronicle of our civilization and is a fundamental axiom of the science of heat.The first law does not tell us whether any particular process can actually occur, that is, it does not restrict the ability to convert work into heat or heat into work, except that energy must be conserved in the process. The second law of thermodynamics asserts that while the system energy is always conserved, it will be degraded to a point where it cannot produce any useful work. Hence, it is impossible to extract work from heat without at the same time discarding some heat, giving rise to an increasing quantity known as entropy.

While energy describes the state of a dynamical system, entropy refers to changes in the status quo of the system and is a measure of molecular disorder and the amount of wasted energy in a dynamical (energy) transformation from one state (form) to another. Since the system entropy increases, the entropy of a dynamical system tends to a maximum, and thus time, as determined by system entropy increase [70, 89, 105], flows on in one direction only. Even though entropy is a physical property of matter that is not directly observable, it permeates the whole of nature, regulating the arrow of time, and is responsible for the enfeeblement and eventual demise of the universe.^1,2 While the laws of thermodynamics form the foundation to basic engineering systems as well as nuclear explosions, cosmology, and our expanding universe, many mathematicians and scientists have expressed concerns about the completeness and clarity of the different expositions of thermodynamics over its long and flexuous history; see [19, 23, 32, 41, 45, 77, 94, 96, 99].

Since the specific motion of every molecule of a thermodynamic system is impossible to predict, a macroscopic model of the system is typically used, with appropriate macroscopic states that include pressure, volume, temperature, internal energy, and entropy, among others. One of the key criticisms of the macroscopic viewpoint of thermodynamics, known as classical thermodynamics, is the inability of the model to provide enough detail of how the system really evolves; that is, it is lacking a kinetic mechanism for describing the behavior of heat. In developing a kinetic model for heat and dynamical energy, a thermodynamically consistent energy flow model should ensure that the system energy can be modeled by a diffusion (conservation) equation in the form of a parabolic partial differential equation. Such systems are infinite-dimensional, and hence, finite-dimensional approximations are of very high order, giving rise to large-scale dynamical systems. Since energy is a fundamental concept in the analysis of large-scale dynamical systems, and heat (energy) is a fundamental concept of thermodynamics involving the capacity of hot bodies (more energetic subsystems) to produce work, thermodynamics is a theory of large-scale dynamical systems.

High-dimensional dynamical systems can arise from both macroscopic and microscopic points of view. Microscopic thermodynamic models can have the form of a distributed-parameter model or a large-scale system model comprised of a large number of interconnected subsystems. In contrast to macroscopic models involving the evolution of global quantities (e.g., energy, temperature, entropy, etc.), microscopic models are based upon the modeling of local quantities that describe the atoms and molecules that make up the system and their speeds, energies, masses, angular momenta, behavior during collisions, etc. The mathematical formulations based on these quantities form the basis of statistical mechanics. Thermodynamics based on statistical mechanics is known as statistical thermodynamics and involves the mechanics of an ensemble of many particles (atoms or molecules) wherein the detailed description of the system state loses importance and only average properties of large numbers of particles are considered. Since microscopic details are obscured on the macroscopic level, it is appropriate to view a microscopic model as an inherent model of uncertainty. However, for a thermodynamic system the macroscopic and microscopic quantities are related since they are simply different ways of describing the same phenomena. Thus, if the global macroscopic quantities can be expressed in terms of the local microscopic quantities, the laws of thermodynamics could be described in the language of statistical mechanics. This interweaving of the microscopic and macroscopic points of view leads to diffusion being a natural consequence of dimensionality and, hence, uncertainty on the microscopic level, despite the fact that there is no uncertainty about the diffusion process per se.

Thermodynamics was spawned from the desire to design and build efficient heat engines, and it quickly spread to speculations about the universe upon the discovery of entropy as a fundamental physical property of matter. The theory of classical thermodynamics was predominantly developed by Carnot, Clausius, Kelvin, Planck, Gibbs, and Carath´eodory,³ and its laws have become one of the most firmly established scientific achievements ever accomplished. The pioneering work of Carnot [24] was the first to establish the impossibility of a per-petuum mobile of the second kind⁴ by constructing a cyclical process (now known as the Carnot cycle) involving two competing cycles, and showing that it is impossible to extract work from heat without at the same time discarding some heat. Carnot’s main assumption (now known as Carnot’s principle) was that it is impossible to perform an arbitrarily often repeatable cycle whose only effect is to produce an unlimited amount of positive work. In particular, Carnot showed that the efficiency of a reversible cycle—that is, the ratio of the total work produced during the cycle and the amount of heat transferred from a boiler (furnace) to a cooler (refrigerator)—is bounded by a universal maximum, and this maximum is only a function of the temperatures of the boiler and the cooler. Both heat reservoirs (i.e., furnace and refrigerator) are assumed to have an infinite source of heat so that their state is unchanged by their heat exchange with the engine (i.e., the device that performs the cycle), and hence, the engine is capable of repeating the cycle arbitrarily often. Carnot’s result was remarkably arrived at using the erroneous concept that heat is an indestructible substance, that is, the caloric theory of heat.⁵

Using a macroscopic approach and building on the work of Carnot, Clausius [26–29] was the first to introduce the notion of entropy as a physical property of matter and to establish the two main laws of thermodynamics involving conservation of energy⁶ and nonconservation of entropy. Specifically, using conservation of energy principles, Clausius showed that Carnot’s principle is valid. Furthermore, Clausius postulated that it is impossible to perform a cyclic system transformation whose only effect is to transfer heat from a body at a given temperature to a body at a higher temperature. From this postulate Clausius established the second law of thermodynamics as a statement about entropy increase for adiabatically isolated systems (i.e., systems with no heat exchange with the environment). From this statement Clausius goes on to state what have become known as the most controversial words in the history of thermodynamics and perhaps all of science; namely, the entropy of the universe is tending to a maximum, and the total state of the universe will inevitably ap proach a limiting state. The fact that the entropy of the universe is a thermodynamically undefined concept led to serious criticism of Clausius’ grand universal generalizations by many of his contemporaries as well as numerous scientists, natural philosophers, and theologians who followed.⁷ In his later work [29], Clausius remitted his famous claim that the entropy of the universe is tending to a maximum.

In parallel research Kelvin [55, 93] developed similar, and in some cases identical, results as Clausius, with the main difference being the absence of the concept of entropy. Kelvin’s main view of thermodynamics was that of a universal irreversibility of physical phenomena occurring in nature. Kelvin further postulated that it is impossible to perform a cyclic system transformation whose only effect is to transform into work heat from a source that is at the same temperature throughout.⁸ Without any supporting mathematical arguments, Kelvin goes on to state that the universe is heading towards a state of eternal rest wherein all life on Earth in the distant future shall perish. This claim by Kelvin involving a universal tendency towards dissipation has come to be known as the heat death of the universe.

Building on the work of Clausius and Kelvin, Planck [82,83] refined the formulation of classical thermodynamics. From 1897 to 1964, Planck’s treatise [82] underwent eleven editions. Nevertheless, these editions have several inconsistencies regarding key notions and definitions of reversible and irreversible processes. Planck’s main theme of thermodynamics is that entropy increase is a necessary and sufficient condition for irreversibility. Without any proof (mathematical or otherwise), he goes on to conclude that every dynamical system in nature evolves in such a way that the total entropy of all of its parts increases. In the case of reversible processes, he concludes that the total entropy remains constant. Unlike Clausius’ entropy increase conclusion, Planck’s increase entropy principle is not restricted to adiabatically isolated dynamical systems. Rather, it applies to all system transformations wherein the initial states of any exogenous system, belonging to the environment and coupled to the transformed dynamical system, return to their initial condition.

Unlike the work of Clausius, Kelvin, and Planck involving cyclical system transformations, the work of Gibbs [39] involves system equilibrium states. Specifically, Gibbs assumes a thermodynamic state of a system involving pressure, volume, temperature, energy, and entropy, among others, and proposes that an isolated system⁹ (i.e., a system with no energy exchange with the environment) is in equilibrium if and only if all possible variations of the state of the system that do not alter its energy, the variation of the system entropy is negative semidefinite. Hence, Gibbs’ principle gives necessary and sufficient conditions for a thermodynamically stable equilibrium and should be viewed as a variational principle defining admissible (i.e., stable) equilibrium states. Thus, it does not provide any information about the dynamical state of the system as a function of time nor any conclusion regarding entropy increase in a dynamical system transformation.

Carathéodory [20, 21] was the first to give a rigorous axiomatic mathematical framework for thermodynamics. In particular, using an equilibrium thermodynamic theory, Carathéodory assumes a state space endowed with a Euclidean topology and defines the equilibrium state of the system using thermal and deformation coordinates. Next, he defines an adiabatic accessibility relation wherein a reachability condition of an adiabatic process¹⁰ is used such that an empirical statement of the second law characterizes a mathematical structure for an abstract state space. Carathéodory’s postulate for the second law states that in every open neighborhood of any state of a system, there exist states such that for some second open neighborhood contained in the first neighborhood, all the states in the second neighborhood cannot be reached by adiabatic processes from states in the first neighborhood. From this postulate Carathéodory goes on to show that for a special class of systems, which he called simple systems, there exists a locally defined entropy and an absolute temperature on the state space for every simple system equilibrium state. One of the key weaknesses of Carath é odory’s work is that his principle is too weak in establishing the existence of a global entropy function.

Adopting a microscopic viewpoint, Boltzmann [15] was the first to give a probabilistic interpretation of entropy involving different configurations of molecular motion of the microscopic dynamics. Specifically, Boltzmann reinterpreted thermodynamics in terms of molecules and atoms by relating the mechanical behavior of individual atoms with their thermodynamic behavior by suitably averaging properties of the individual atoms. In particular, even though individual atoms are assumed to obey the laws of Newtonian mechanics, by suitably averaging over the velocity distribution of these atoms Boltzmann showed how the microscopic (mechanical) behavior of atoms and molecules produced effects visible on a macroscopic (thermodynamic) scale. He goes on to argue that Clausius’ thermodynamic entropy (a macroscopic quantity) is proportional to the logarithm of the probability that a system will exist in the state it is in relative to all possible states it could be in. Thus, the entropy of a thermodynamic system state (macrostate) corresponds to the degree of uncertainty about the actual system mechanical state (microstate) when only the thermodynamic system state (macrostate) is known. Hence, the essence of Boltzmann thermodynamics is that thermodynamic systems with a constant energy level will evolve from a less probable state to a more probable state with the equilibrium system state corresponding to a state of maximum entropy (i.e., highest probability).

In the fir...