Thermodynamics: A Dynamical Systems Approach

VijaySekhar Chellaboina
Sergey G. Nersesov
Pages: 200
https://www.jstor.org/stable/j.ctt7s1k3

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-xii)
4. Chapter One Introduction
(pp. 1-16)

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...

5. Chapter Two Dynamical System Theory
(pp. 17-44)

As discussed in Chapter 1, in this monograph we develop thermodynamic system models using large-scale nonlinear compartmental dynamical systems. The mathematical foundation for compartmental modeling isnonnegative dynamical system theory[47, 90], which involves dynamical systems with nonnegative state variables. Since our thermodynamic state equations govern energy flow between subsystems, it follows from physical arguments that system energy initial conditions give rise to trajectories that remain in the nonnegative orthant of the state space. In this chapter we introduce notation, several definitions, and some key results on nonlinear nonnegative dynamical systems needed for developing the main results of this monograph....

6. Chapter Three A Systems Foundation for Thermodynamics
(pp. 45-102)

The fundamental and unifying concept in the analysis of complex (large-scale) dynamical systems is the concept of energy. As noted in Chapter 1, the energy of a state of a dynamical system is the measure of its ability to produce changes (motion) in its own system state as well as changes in the system states of its surroundings. These changes occur as a direct consequence of the energy flow between different subsystems within the dynamical system. Since heat (energy) is a fundamental concept of thermodynamics involving the capacity of hot bodies (more energetic subsystems) to produce work, thermodynamics is a...

7. Chapter Four Temperature Equipartition and the Kinetic Theory of Gases
(pp. 103-114)

The thermodynamic axioms introduced in Chapter 3 postulate that subsystem energies are synonymous with subsystem temperatures. In this chapter, we generalize the results of Chapter 3 to the case where the subsystem energies are proportional to the subsystem temperatures with the proportionality constants representing the subsystemspecific heatsorthermal capacities.¹ In the case where the specific heats of all the subsystems are equal, the results of this section specialize to those of Chapter 3. To include temperature notions in our large-scale dynamical system model, we replace Axioms i) and ii) of Section 3.3 with the following axioms. Let ßi...

8. Chapter Five Work, Heat, and the Carnot Cycle
(pp. 115-130)

In Chapter 3, we showed that the first law of thermodynamics is essentially a statement of the principle of the conservation of energy. Hence, the variation in energy of a dynamical system G during any transformation is equal to the amount of energy that the system receives from the environment. In Chapter 3, however, the notion of energy that the system receives from the environment and dissipates to the environment was limited to heat and did not include work. When external forces act on the dynamical system G , they can produce work on the system, changing the system’s internal...

9. Chapter Six Thermodynamic Systems with Linear Energy Exchange
(pp. 131-140)

In this chapter we specialize the results of Chapter 3 to the case of large-scale dynamical systems with linear energy exchange between subsystems, that is,w(E) = WEandd(E) = DE, where$W\in{\mathbb{R}^{q\timesq}}$and$D\in{\mathbb{R}^{q\timesq}}$In this case, the vector form of the energy balance equation (3.1), with${t_0}=0$, is given by

$E(T)=E(0)+\int_0^T{WE(t){\rm{d}}t}-\int_0^T{DE(t){\rm{d}}t+\int_0^T{S(t){\rm{d}}t},T\ge0,}$

or, in power balance form,

$\dotE(t)=WE(t)-DE(t)+S(t),E(0)={E_0},t\ge0.$

Next, let the net energy flow from thejth subsystem${{\calG}_j}$to theith subsystem${{\calG}_i}$be parameterized as${\phi_{ij}}(E)=\Phi_{ij}^{\rm{T}}E$, where${\Phi_{ij}}\in{\mathbb{R}^q}$and$E\in\bar\mathbb{R}_+^q$. In this case, since${w_i}(E)=\sum\nolimits_{i=1,j\nei}^q{{\phi_{ij}}(E)}$, it follows that

$W\; = \;{\left[ {\begin{array}{*{20}{c}} {\sum\limits_{j\, = \,2}^q {{\Phi _{1j,\; \ldots \,,}}} } & {\sum\limits_{j\, = \,1,\;j \ne \,i}^q {{\Phi _{ij,\, \ldots ,\;}}} } & {\sum\limits_{j\, = \,1}^{q\, - \,1} {{\Phi _{{q_j}}}} } \\\end{array}} \right]^{\rm{T}}}$.

Since${\phi _{ij}}(E)\; = - {\phi _{ji}}(E),\;i,\;j\; = \;1,\; \ldots ,\;q,\;i\; \ne j$and$E\; \in \;\bar \mathbb{R}_ + ^q$it follows...

10. Chapter Seven Continuum Thermodynamics
(pp. 141-168)

In this chapter we extend the results of Chapter 3 to the case of continuum thermodynamic systems, where the subsystems are uniformly distributed over ann-dimensional space. Since these thermodynamic systems involve distributed subsystems, they are described by partial differential equations and hence are infinite-dimensional systems. Our formulation in this chapter involves a unification of the behavior of heat as described by the equations of thermal transfer and classical thermodynamics. With the notable exception of [11], the amalgamation of these classical disciplines of physics is virtually nonexistent in the literature. Specifically, we consider continuous dynamical systems${\cal G}$defined over a...

11. Chapter Eight Conclusion
(pp. 169-174)

In this monograph, we have outlined a general systems theory framework for thermodynamics in an attempt to harmonize it with classical mechanics. The proposed macroscopic mathematical model is based on a nonlinear (finite-and infinite-dimensional) compartmental dynamical system model that is characterized by energy conservation laws capturing the exchange of energy between coupled macroscopic subsystems. Specifically, using a large-scale dynamical systems perspective, we developed some of the fundamental properties of reversible and irreversible thermodynamic systems involving conservation of energy, nonconservation of entropy and ectropy, and energy equipartition. This model is formulated in the language of dynamical systems and control theory, and...

12. Bibliography
(pp. 175-184)
13. Index
(pp. 185-187)