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Thermodynamic Approaches in Engineering Systems

Stanisław Sieniutycz

Warsaw University of Technology

Faculty of Chemical and Process Engineering

Warsaw, Poland AMSTERDAM

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Preface

The present volume responds to the need for a synthesizing book that throws light upon extensive fields of thermodynamics and entropy theory from the engineering perspective and applies basic ideas and results of these fields in engineering problems. The book is, in brief, a collection of reviews synthesizing important achievements of applied nonequilibrium thermodynamics in some important branches of mechanical, chemical, and biochemical engineering with a few excursions into environmental engineering.

Contemporary applications of thermodynamics, both reversible and irreversible, are broad and diverse. The range of thermodynamic applications is expanding and new areas finding a use for the theory are continually emerging. One of the main goals of this book is to outline the achievements in applied thermodynamics (both equilibrium and nonequilibrium) obtained within the recent 50 years. This goal is very broad since thermodynamics serves currently as a formal and substantive theory in applications involving analyses of physical, chemical, biological, and even economical systems. Dealing with such a broad goal within a single limited volume requires necessary elimination of many attractive mathematical aspects; consequently the book exploits the power of mathematical analysis to a lesser extent than many more specialized while narrower texts. Yet, this disadvantage is compensated by a vast literature analysis, which is a strong asset of the book to serve as a comprehensive reference text.

Let us shortly discuss several basic topics, most of which are analyzed in the present book. Thermodynamics provides theoretical tools to exclude improper kinetic equations or to improve empirical kinetic formulas. It may furnish thermodynamic equilibrium metrics and its disequilibrium extensions to measure a thermodynamic length. It may also help to describe reversible and irreversible dynamics in terms of energy-type Hamiltonians or related Lagrangians. The use of disequilibrium entropies serves to develop a wave theory of heat and mass transfer, applied to overcome the well-known paradox of infinite propagation speeds of thermal and material disturbances, implied by classical parabolic equations. Advanced thermodynamics can provide unifying theoretical framework for both classical and anomalous diffusion. The latter is usually treated by applying Tsallis’s (1988) theory for a generalized definition of entropy, consistent with both thermodynamics and statistical mechanics. Applied

in a chemical context, thermodynamics provides substantiation for the kinetic law of mass action consistent with suitable chemical resistance, a quantity useful in the analysis of chemical networks.

In fact, both classical and nonequilibrium thermodynamics predict the global stability of the thermodynamic equilibrium. However, for disequilibrium singular points and distributed steady states, sufficient stability conditions cannot be formulated in terms of entropy or other thermodynamic potentials, although, occasionally, some excess thermodynamic or quasi-thermodynamic potential can prove their utility as stability criteria (Lapunov functions).

The problem of thermodynamic stability criteria in disequilibrium systems has not yet been completely solved, despite extensive research in this area. Without doubt, thermodynamics may establish exact stability criteria for system trajectories approaching an equilibrium singular point. In this case one may apply the negative excess entropy W = Seq – S as a Lapunov functional for which the entropy density ρs is the same function of the thermodynamic state as prescribed by classical thermodynamics (local equilibrium assumption). Invoking the local form of the second law σ > 0 one obtains in the gradient-less case the conditions dW/dt < 0, which proves the stability of all system trajectories approaching the equilibrium singular point. This means that thermodynamics ensures the global stability of the thermodynamic equilibrium (Tarbell, 1982).

A practical question can be asked: In what sense can it be said that the distribution of the driving forces in one industrial configuration is better than in another? To answer this question realistic engineering examples, involving heat exchangers, plate distillation columns, chromatographic separators, etc., may be analyzed. When “a duty” (useful effect) for a process is defined, the following equipartition principle for the entropy production is advanced: “for a given duty, the best configuration of the process is that in which the entropy production rate is most uniformly distributed” (Tondeur and Kvaalen, 1987). A generalization of the principle may also be proposed for systems with a distributed design variable: “the optimal distribution of an investment is such that the investment in each element is equal to the cost of the energy degradation in this element”. Thus a uniform distribution of the ratio of these two quantities should be preserved. These results show the importance of irreversibility analysis in design.

Thermodynamics may help to predict selling prices and develop paraeconomic analyses and balances which involve unit exergies as measures of unit economic values. Interestingly, thermodynamics and especially exergy theory may be applied to define the so-called proecological tax, the quantity replacing actual personal taxes. A practical field related to thermodynamics is thermoeconomics. It defines the physical–mathematical background for energy–economic–ecological analyses which are made in different fields of knowledge. In this field the concept of exergetic cost is employed (Valero et al., 1994a,b); this is a concept close to that of embodied energy or cumulative exergy consumption (Szargut et al., 1988). The analysis of the exergetic cost focuses rigorously on the process of cost formation.

A relatively new subset of nonequilibrium thermodynamics, called finite-time thermodynamics (FTT), has proved its remarkable effectiveness in modeling the performance and power limits of thermal, chemical, and electrochemical engines, refrigerators and heat pumps. Limits for finite resources are associated with the notion of exergy. Classical thermodynamics is capable of providing energy limits in terms of exergy changes. However, they are often too distant from reality (real energy consumption can be much higher than the lower bound and/or real energy yield can be much lower than the upper bound). Yet, by introducing rate dependent factors, irreversible thermodynamics offers enhanced limits that are closer to reality. Limits evaluated for finite resources refer either to a sequential relaxation of a resource to the environment (engine mode) or to resource’s upgrading in the process undergoing in the inverse direction (heat pump mode). The current research in the field of FTT includes fuel cells which are expected to play a significant role in the next generation of energy systems and road vehicles for transportation. However, substantial progress is required in reducing manufacturing costs and improving performance of FC systems.

Although the field of finite-time thermodynamics is both broad and vital, FTT is only briefly treated in the present book because there is already a number of sources in this area (Berry et al., 2000; Chen and Sun, 2004; Sieniutycz and Jeżowski, 2009, 2013; etc). This treatment refers to chemical reactors analyzed in Section 9.10, where a problem of power yield in thermochemical generators and fuel cells is presented with some detail in the spirit of FTT. A discussion in FTT style is developed for radiation systems (chapter: Radiation and Solar Systems ) in view of recent results of Badescu (2014), who showed that that the upper bound for the reversible work extraction is not always Carnot efficiency and all upper-bound efficiencies depend on the geometric factor of the radiation reservoir.

Evaluation of the entropy lowering in reference to its equilibrium value yields a good indicator of disequilibrium (Ebeling and Engel-Herbert, 1986). Entropy lowering can be linked with the contraction of the occupied part of the phase space due to the formation of attractors. Oscillations in solids and turbulence in liquid flows may serve as examples. Ebeling et al. (1986) investigate these entropy properties in the turbulence context. Self-oscillations and flows in tubes (laminar and turbulent) are considered. The results show that under the condition of a fixed energy the entropy decreases with excitation. Engel-Herbert and Ebeling (1988a) study the Brownian motion of nonlinear oscillators in a heat bath. For the excitation of sustained oscillations, realized by van der Pol oscillators, the entropy, calculated in the limit of weak dissipation and weak noise, decreases monotonically with the feedback strength, at the fixed average energy.

Turbulent flows are common in nature. They are observed in many technological processes, hence they are of interest for thermodynamics. Turbulent flows are a manifestation of spatially extended, nonlinear dissipative systems in which diverse length scales are simultaneously excited and coupled. The significance of turbulence in our physical world cannot be overestimated. Without turbulence, mixing of air and fuel would not occur on suitable

time scales, the transfer and dispersion of chemical reagents, energy, momentum, and pollutants would be far less effective; that is, many observed positive effects would be impossible (Sreenivasan, 1999). Yet, turbulence has also negative consequences, for example, it increases energy consumption in airplanes, automobiles, ships, pipelines, etc., distorts the signal propagation in fluids, decreases safety of aircraft flight, etc. (Sreenivasan, 1999). For an engineer the main goal is prediction and control of turbulence effects, their suppression or enhancement depending on current needs.

In the present book we focus on thermodynamic aspects of turbulence, following mainly the results obtained by Klimontovich and Ebeling, and some coworkers of these researchers who compared with experiments the theory of hydrodynamic turbulence. Thermodynamic aspects of turbulent two-phase flows are also briefly outlined. The structure of a turbulent flow may be expressed by the collective mechanisms for the transfer of momentum, heat, and mass and by entropy lowering determined relative to an equilibrium system of the same energy. The transition from laminar to turbulent motion may be regarded as a nonequilibrium transition to a more ordered state.

One can evaluate the change of the entropy during the transition of a flow from laminar to turbulent motion. As the Reynolds number Re grows, a certain portion of the flow energy is transferred to the collective macroscopic motions of the molecules, which are reflected, for example, by hydrodynamic velocity, by turbulent pulsations, and randomness, which, “apparent from a causal observation, is not without some order” (Sreenivasan, 1999). Fixing the energy as the Reynolds number Re varies, this portion of the transferred energy is taken from the microscopic thermal motion. But the energy contained in the thermal degrees of freedom possesses a higher value of entropy. Thus, the transformation of energy from the microscopic scale of thermal motion to macroscopic scales of collective molecular motion at fixed energy lowers the entropy (Engel-Herbert and Ebeling, 1988b). Inspired by this result Engel-Herbert and Schumann (1987) studied the entropy behavior during a nonequilibrium phase transition. They observed that for the generation of sustained oscillations the entropy decreases monotonously if the average oscillator energy remains fixed. Soon afterward, Klimontovich (1991) showed the link between a turbulent motion and the structure of chaos.

Some problems are discussed here in a necessarily brief way, but the book will serve its purpose if the reader gets encouraged to further studies, to improve his knowledge and understanding of problems which could only be outlined here. In particular, the chapter on thermodynamic aspects of engineering biosystems will require updated studies soon, in view of rapid progress in this area.

This book can be used as a supplementary or reference text in the following courses:

• technical thermodynamics and industrial energetics (undergraduate)

• separation operations and separation systems (undergraduate)

• alternative and unconventional energy sources (graduate)

• stability of chemical reactions and transport processes (graduate)

• thermoeconomics of solar energy conversion (graduate)

The content organization of the book is as follows. Chapters: Contemporary Thermodynamics for Engineering Systems and Variational Approaches to Nonequilibrium Thermodynamics outline main approaches and results obtained in applied disequilibrium thermodynamics, focusing on the methods and examples considered in the whole book. Chapter: Wave Equations of Heat and Mass Transfer shows the role of thermodynamics in setting a simple theory of wave equations for heat and mass transfer. Chapter: Classical and Anomalous Diffusion describes the significance of Tsallis’s (1988) theory for a generalized definition of entropy in the development of the generalized theory of diffusion, which includes diffusive phenomena, both classical and anomalous. Chapter: Thermodynamic Lapunov Functions and Stability displays some stability problems and related Lapunov’s functions derived from thermodynamics. Chapter: Analyzing Drying Operations on Thermodynamic Diagrams presents drying operations with granular solids on thermodynamic diagrams. Chapter: Frictional Fluid Flow through Inhomogeneous Porous Bed describes basic aspects of frictional fluid flow through both inhomogeneous and anisotropic porous media in which fluid streamlines are curved by a location dependent hydraulic conductivity. Chapter: Thermodynamics and Optimization of Practical Processes exposes the role of thermodynamics in optimization of applied processes undergoing in nonreacting systems. Chapter: Thermodynamic Controls in Chemical Reactors extends the related control ideas and results to chemical reactors. Chapter: Power Limits in Thermochemical Units, Fuel Cells, and Separation Systems offers a generalized treatment of power limits in various thermochemical engines, fuel cells, and separation systems. Chapter: Thermodynamic Aspects of Engineering Biosystems considers some basic thermodynamic properties of engineering biosystems. Chapter: Multiphase Flow Systems presents valuable selected results for multiphase flow systems, whereas Chapter: Radiation and Solar Systems—some recent results for applied radiation and solar systems. Chapter entitled Appendix: A Causal Theory of Hydrodynamics and Heat Transfer constitutes a supplementary text which shows the role of entropy four-flux in the dissipative relativistic phenomena and variety of disequilibrium temperatures obtained within a causal approach to hydrodynamics and heat transfer.

Acknowledgments

The author expresses his gratitude to the Polish Committee of National Research (KBN) and the Ministry of National Education of Poland under whose auspices a considerable part of his own research discussed in the book was performed in the framework of two grants: grant 3 T09C 02426 (Nonequilibrium thermodynamics and optimization of chemical reactions in physical and biological systems) and grant N N208 019434 (Thermodynamics and optimization of chemical and electrochemical energy generators and the related applications to fuel cells). A critical part of writing any book is the process of reviewing, thus the authors are very much obliged to the researchers who patiently helped them read through subsequent chapters and who made valuable suggestions. In preparing this volume the authors received help and guidance from Viorel Badescu (Polytechnic University of Bucharest, Romania), Ferenc Markus (Budapest University of Technology and Economics, Hungary), Alina Jeżowska (Rzeszów University of Technology, Poland), Andrzej B. Jarzębski (Institute of Chemical Engineering of Polish Academy of Science, and Faculty of Chemistry at the Silesian University of Technology, Gliwice), Lingen Chen (Naval University of Engineering, Wuhan, P. R. China), Piotr Kuran, Artur Poświata, and Zbigniew Szwast (Faculty of Chemical and Process Engineering at the Warsaw University of Technology), Elżbieta Sieniutycz (University of Warsaw), Anatolij M. Tsirlin (System Analysis Research Center, Pereslavl-Zalessky, Russia), and Anita Koch (Elsevier). We also acknowledge the scientific cooperation of many colleagues of the Institute of Fluid Flow Machinery in Gdańsk, Poland, the former Publisher of Archives of Thermodynamics. Finally, appreciations also go to the whole book’s production team in Elsevier for their cooperation, help, patience, and courtesy.

Stanisław Sieniutycz

Warszawa, August 2015

Contemporary Thermodynamics for Engineering Systems

1.1 Introduction

We begin with some information about the nature of thermodynamics as a macroscopic theory and then describe basic concepts and definitions in thermodynamics. Thermodynamics is a science that includes the study of energy transformations and relationships among the physical properties of substances that are affected by these transformations. The properties which really sets thermodynamics apart from other sciences are energy transformations through heat and work. It is clear that the aforementioned definition is broad and vague, and that various users will apply in their work different aspects of the thermodynamic theory. Chemical engineers typically focus on phase equilibria, stoichiometry, chemical reactions, catalysis, and so forth. Mechanical engineers are more interested in power generators, fuel cells, refrigeration devices, and nuclear reactors. Both users can apply in their research some common methods, such as optimal control theory thus contributing to the field called thermodynamic optimization.

Thermodynamic properties can be determined by studying either macroscopic or microscopic behavior of matter. Classical thermodynamics treats matter as a continuum and studies the macroscopic behavior of matter. Statistical thermodynamics investigates the statistical behavior of large groups of individual particles. It postulates that observed physical property (eg, temperature T, pressure P, energy E, etc.) is equal to the appropriate statistical average of a large number of particles. For a statistical theory of disequilibrium systems, see, for example, Kreuzer (1981). Thermodynamics is based upon experimental observation of macroscopic systems. Its power and beauty follows from a basic property of macrosystems composed of sufficiently large number of particles. Namely, as a consequence of the law of large numbers, these macrosystems can be described well (modulo to fluctuations) in terms of only several variables, for example, temperature, pressure and mole numbers.

Conclusions stemming from observations have been cast as postulates or laws. The study of thermodynamics considers five laws or postulates. Two of them deal with energy transformation and three deal with the physical properties.

1.1.1 Energy Transformation Laws

Energy transformation laws comprise the first and the second law of thermodynamics. Both laws acquired many formulations which are available in numerous books on thermodynamics. Subsequently, we limit ourselves to the following formulations:

First Law of Thermodynamics: Energy is conserved, that is, system’s energy can change only by transfer of heat, work or substances by the system’s boundary, which means that there is no energy generation within the system. Note that this formulation may be invalid for partial energies of subsystems occupying the system’s space.

Second Law of Thermodynamics: Entropy is not conserved and/or there is a non-negative entropy generation within the system. The important consequence of the second law is that all real processes spontaneously proceed in one direction of time. Another basic and useful conclusion stemming from the second law states that various forms of energy have different qualities and the notion of exergy (available energy) can be introduced as a sort of the value measure for the useful energy contained in the substance or attributed to the heat.

1.1.2 Property Relationship Laws

Property relationship laws include zeroth law of thermodynamics, third law of thermodynamics and the state postulate. They may be expressed as follows.

Zeroth Law of Thermodynamics: When each of two systems is in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.

Third Law of Thermodynamics: The entropy of a perfect crystal equals zero at absolute zero temperature.

State Postulate: The state of a simple, single-phase thermodynamic system is completely specified by two independent variables which are the intensive properties.

1.1.3 Energy Conversion and Its Efficiency

Energy conversion and efficiency of this conversion are of primary concern in applied thermodynamics. For energy consuming or producing devices the thermal efficiency is: ηth = energy obtained/(energy paid)

Example efficiencies are:

1. Solar cell: 12%

2. Gas turbine: 12–16%

3. Automobile: 12–25%

4. Steam power plant: 38–41%

5. Fuel cells: 40–60%

6. Electric motor: 90%

Now we shall pass to some definitions and say several words about the thermodynamic vocabulary.

1. Thermodynamic system

Thermodynamic system is a three-dimensional region of the physical space bounded by arbitrary surfaces (which may be real or imaginary and may change size or shape). There are various types of hermodynamic systems, namely the following five.

a. Closed system is a system which is impermeable with respect to the flow of matter, for example, a fixed, closed volume. A property of the closed system is its constant mass.

b. Open system is a system open with respect to the flow of matter such as a flow reactor. The system is defined by a well-defined volume surrounding the region of interest. The surface of this volume is called the control surface. Mass, heat, work and momentum can flow across the control surface into the system and from the system.

c. Isolated system is a system that is not influenced by the part of space which is external to the system boundaries. No heat, work, mass, or momentum can cross the boundary of an isolated system. Variables such as N, V and U are fixed and constant in isolated system.

d. Simple system is a system that does not contain any internal adiabatic, rigid and impermeable boundaries and is not acted upon by external forces.

e. Composite system is a system that is composed of two or more simple systems.

2. Property

Property, by definition, is a characteristic of a system. There are several types of properties:

a. Primitive property is a property that can in principle be specified by describing an operation or a test to which the system is subjected. Examples include mechanical measurements (eg, pressure, volume, and temperature T) and heat capacity.

b. Derived property is a property mathematically defined in terms of primitive properties.

c. Intensive property is a property that is independent of the extent of or mass of the system. Examples are T, P, density, ρ, …, and so forth.

d. Extensive property is a property whose value for the system is dependent upon the mass or extent of the system. Examples are the enthalpy, internal energy, volume, and so forth.

e. Specific property is an extensive property per unit mass. Specific properties are intensive.

f. State property is a property that only depends on the thermodynamic state of the system, not the path taken to get to that state.

3. State of a system

State, by definition, is a characteristic of a system. There are several types of states:

a. Thermodynamic state is the condition of the system as characterized by the values of its properties.

b. Stable equilibrium state is a state in which the system is not capable of finite spontaneous change to another state without a change in the state of the surroundings. Several types of equilibrium must be fulfilled: thermal, mechanical, phase, and chemical.

c. State postulate: The equilibrium state of a simple closed system can be completely characterized by two independent variables and the masses of the species within the system.

4. Thermodynamic process

A thermodynamic process is a transformation from one equilibrium state to another.

a. Quasistatic process is a process where every intermediate state is a stable equilibrium state.

b. Reversible process is one in which a second process could be performed so that the system and surroundings can be restored to their initial states with no change in the system or surroundings.

The following remarks can be made at this point. Reversible processes are quasistatic, but quasi-static processes are not necessarily reversible. A quasistatic process in a simple system is also reversible. Some factors which render processes irreversible are friction, unrestrained expansion of gasses, heat transfer through a finite temperature difference, mixing, chemical reaction, and so forth.

5. Thermodynamic path

Thermodynamic path is the specification of a series of states through which a system passes in a process. Various paths can be specified, such as: isothermal, ∆T = 0; isobaric, ∆P = 0; adiabatic, ∆Q = 0; polytropic, PVk = constant; isochoric, ∆V = 0; isentropic, ∆S = 0 (adiabatic and reversible); isenthalpic, ∆H = 0 (adiabatic and entirely irreversible, as in the throttling process). Path calculations usually involve the knowledge of the properties of gases and liquids (Reid et al., 1987).

6. Heat engines, refrigerators and HPs

Work may be converted to heat directly and completely, but converting heat to work requires the use of a special device called a heat engine. Heat engines may vary considerably from one another, but share the following characteristics: They operate on a cycle, they receive the driving heat from a high temperature source, they convert part of this heat to work, they reject the remaining waste heat to a low-temperature sink.

The work-generating device that fits the definition of a heat engine is the steam power plant, Fig. 1.1. The fraction of the heat input that is converted to network output is a measure of the performance of a heat engine, and is called the thermal efficiency. Most heat engines possess poor thermal efficiencies, with more than one half of the thermal energy supplied to the working fluid ending up in the environment (Naragchi, 2013).

In Nature, heat flows in the direction of decreasing temperature. The reverse process, however, cannot occur spontaneously and requires the use of special devices called refrigerators. Like heat engines, refrigerators are cyclic devices. The working fluid used in the refrigeration cycle is called a refrigerant. The most frequently used refrigeration cycle is called the vapor-compression refrigeration cycle, shown schematically in Fig. 1.2.

The refrigerant enters the compressor as a vapor and is compressed to the condenser pressure. It leaves the compressor at a relatively high temperature and cools down and condenses as it flows through the coils of the condenser by rejecting heat to the surrounding medium. Next, the refrigerant enters an expansion valve, where the JouleThompson effect takes place and the refrigerant’s temperature and pressure decrease due to the throttling. The refrigerant then enters the evaporator, where it evaporates by absorbing heat from the refrigerated space (cooling effect). The cycle is completed as the refrigerant leaves the evaporator and returns to the compressor. The efficiency of a

Figure 1.1: Steam power-plant as a heat engine.

Figure 1.2: A vapor-compression refrigeration cycle.

refrigerator is expressed in terms of the coefficient of performance (COP), denoted by COPR. The value of COPR = QL/Wnet, in can be greater than unity, and usually is for most refrigerators. That is, the amount of heat removed from the refrigerated space can be greater than the amount of work input.

Another device that transfers heat from a low temperature medium to a high temperature one is the heat pump (HP). Refrigerators and heat pumps operate on the same thermodynamic cycle, but have different objectives. The objective of a refrigerator is to maintain the refrigerated space at a low temperature. The objective of an HP is to maintain a heated space at a high temperature. The coefficient of performance of an HP, COPHP, is defined as: COPHP = QH/Wnet, in. Air-conditioning units are refrigerators whose refrigerated space is a room or building. The same air-conditioning unit can operate as an HP in the winter by installing it backwards. In this mode, which we call the HP mode, the air-conditioner absorbs heat from the refrigerated space (outdoors) and rejects it to the heated space (house). The performance indices of refrigerators and air-conditioning units are often expressed in the United States in terms of the Energy Efficiency Rating (EER), which is the amount of heat removed (in BTUs) for one watt-hour of electricity consumed. The relation between EER and COP is: EER = 3.412COPR

The work-energy theorem derived from Newton’s second law is essential for the development of energy concepts in physics (Arons, 1999). The theorem applies to the displacement of a particle or the center of mass of an extended body treated as a particle. Because work, as a quantity of energy transferred in accordance with the first law of thermodynamics, cannot be calculated in general as an applied force times the displacement of center of mass, the work-energy theorem is not a valid statement about energy transformations when work is done against a frictional force or actions on or by deformable bodies. To use work in conservation of energy calculations, work must be calculated as the sum of the products of forces and their corresponding displacements at locations where the forces are applied at the periphery of the system under consideration. Failure to make this distinction results in errors and misleading statements prevalent in textbooks, thus reinforcing confusion about energy transformations associated with the action in everyday experience of zero-work forces such as those present in walking, running, jumping or accelerating a car. Without a thermodynamically valid work definition, it is also impossible to give a correct description of the connection between mechanical and thermal energy changes and of dissipative effects. The situation can be corrected and student understanding of the energy concepts greatly enhanced by introducing the internal energy concept, that is, articulating the first law of thermodynamics in a simple, phenomenological form without unnecessary mathematical encumbrances (Arons, 1999).

Before closing this section it is worth mentioning a beautiful book on energy history, development and current state of research (Badur, 2012). This treatise is along the