Dynamic Modeling and Predictive Control in Solid Oxide Fuel Cells E-Book

Biao Huang

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 409

Veröffentlichungsjahr: 2013

Beschreibung

The high temperature solid oxide fuel cell (SOFC) is identified as one of the leading fuel cell technology contenders to capture the energy market in years to come. However, in order to operate as an efficient energy generating system, the SOFC requires an appropriate control system which in turn requires a detailed modelling of process dynamics.

Introducting state-of-the-art dynamic modelling, estimation, and control of SOFC systems, this book presents original modelling methods and brand new results as developed by the authors. With comprehensive coverage and bringing together many aspects of SOFC technology, it considers dynamic modelling through first-principles and data-based approaches, and considers all aspects of control, including modelling, system identification, state estimation, conventional and advanced control.

Key features:

• Discusses both planar and tubular SOFC, and detailed and simplified dynamic modelling for SOFC
• Systematically describes single model and distributed models from cell level to system level
• Provides parameters for all models developed for easy reference and reproducing of the results
• All theories are illustrated through vivid fuel cell application examples, such as state-of-the-art unscented Kalman filter, model predictive control, and system identification techniques to SOFC systems

The tutorial approach makes it perfect for learning the fundamentals of chemical engineering, system identification, state estimation and process control. It is suitable for graduate students in chemical, mechanical, power, and electrical engineering, especially those in process control, process systems engineering, control systems, or fuel cells. It will also aid researchers who need a reminder of the basics as well as an overview of current techniques in the dynamic modelling and control of SOFC.

Leseprobe

Table of Contents

Title Page

Copyright

Preface

Acknowledgments

List of Figures

List of Tables

Chapter 1: Introduction

1.1 Overview of Fuel Cell Technology

1.2 Modelling, State Estimation and Control

1.3 Book Coverage

1.4 Book Outline

Part One: Fundamentals

Chapter 2: First Principle Modelling for Chemical Processes

2.1 Thermodynamics

2.2 Heat Transfer

2.3 Mass Transfer

2.4 Fluid Mechanics

2.5 Equations of Change

2.6 Chemical Reaction

2.7 Notes and References

Chapter 3: System Identification I

3.1 Discrete-time Systems

3.2 Signals

3.3 Models

3.4 Notes and References

Chapter 4: System Identification II

4.1 Regression Analysis

4.2 Prediction Error Method

4.3 Model Validation

4.4 Practical Consideration

4.5 Closed-loop Identification

4.6 Subspace Identification

4.7 Notes and References

Chapter 5: State Estimation

5.1 Recent Developments in Filtering Techniques for Stochastic Dynamic Systems

5.2 Problem Formulation

5.3 Sequential Bayesian Inference for State Estimation

5.4 Examples

5.5 Notes and References

Chapter 6: Model Predictive Control

6.1 Model Predictive Control: State-of-the-Art

6.2 General Principle

6.3 Dynamic Matrix Control

6.4 Nonlinear MPC

6.5 General Tuning Guideline of Nonlinear MPC

6.6 Discretisation of Models: Orthogonal Collocation Method

6.7 Pros and Cons of MPC

6.8 Optimisation

6.9 Example: Chaotic System

6.10 Notes and References

Part Two: Tubular SOFC

Chapter 7: Dynamic Modelling of Tubular SOFC: First-Principle Approach

7.1 SOFC Stack Design

7.2 Conversion Process

7.3 Diffusion Dynamics

7.4 Fuel Feeding Process

7.5 Air Feeding Process

7.6 SOFC Temperature

7.7 Final Dynamic Model

7.8 Investigation of Dynamic Properties through Simulations

7.9 Notes and References

Chapter 8: Dynamic Modelling of Tubular SOFC: Simplified First-Principle Approach

8.1 Preliminary

8.2 Low-order State Space Modelling of SOFC Stack

8.3 Nonlinear State Space Model

8.4 Simulation

8.5 Notes and References

Chapter 9: Dynamic Modelling and Control of Tubular SOFC: System Identification Approach

9.1 Introduction

9.2 System Identification

9.3 PID Control

9.4 Closed-loop Identification

9.5 Notes and References

Part Three: Planar SOFC

Chapter 10: Dynamic Modelling of Planar SOFC: First-Principle Approach

10.1 Introduction

10.2 Geometry

10.3 Stack Voltage

10.4 Mass Balance

10.5 Energy Balance

10.6 Simulation

10.7 Notes and References

Chapter 11: Dynamic Modelling of Planar SOFC System

11.1 Introduction

11.2 Fuel Cell System

11.3 SOFC along with a Capacitor

11.4 Simulation Result

11.5 Notes and References

Chapter 12: Model Predictive Control of Planar SOFC System

12.1 Introduction

12.2 Control Objective

12.3 State Estimation: UKF

12.4 Steady-state Economic Optimisation

12.5 Control and Simulation

12.6 Results and Discussions

12.7 Notes and References

Appendix A: Properties and Parameters

A.1 Parameters

A.2 Gas Properties

References

Index

This edition first published 2013

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Library of Congress Cataloging-in-Publication Data

Huang, Biao, 1962-

Dynamic modeling and predictive control in solid oxide fuel cells / Biao Huang, Yutong Qi, AKM Monjur Murshed. pages cm

Includes bibliographical references and index.

ISBN 978-0-470-97391-2 (hardback)

1. Solid oxide fuel cells–Mathematical models. 2. Solid oxide fuel cells–Simulation methods. 3. Dynamics. I. Qi, Yutong. II. Murshed, Monjur. III. Title.

TK2931H82 2013

621.31′2429–dc23

2012035074

A catalogue record for this book is available from the British Library.

Print ISBN: 9780470973912

Preface

Today's energy-hungry civilization is in search of an alternative source to replace the currently available but continuously depleting energy sources. Stringent environmental regulations restricting emissions of greenhouse gases, SOx and NOx have narrowed down the search for a clean source of energy to few options. It has generated a lot of attention towards the fuel cell as an alternative source of clean energy. Fuel cells are electrochemical devices that directly convert chemical energy to electrical energy. As it does not involve any rotary or thermal components, it does not suffer from any friction and combustion loss. Moreover, the unused fuel from the cell can be used to generate more power, making it attain high overall efficiency.

Among various fuel cells, the low temperature proton exchange membrane fuel cell (PEMFC) and the high temperature solid oxide fuel cell (SOFC) have been identified as the likely fuel cell technologies that will capture the market in the future.

In order to operate and control SOFC systems, it is necessary to investigate dynamic characteristics of SOFC through modelling and simulations. Modelling and controller design are two integral parts of advanced process control strategies that are intricately dependent on each other. From the process control view point, models should be easy to use for designing controller and yet be detailed enough for giving a true account of the system dynamics.

In this book, two types of models, first-principle and data-based, are developed for SOFC. The first-principle models take electrochemical, chemical and thermal aspects into account and provide a set of nonlinear ordinary differential equations (ODE). Zero-dimensional thermal models of fuel cell system component such as heat exchangers, reformer and burner are also provided for fuel cell system simulation and control. In parallel, data-based models are developed through the system identification approach.

Dynamic models can be used to investigate responses of the fuel cells under different operating conditions to account for pitfalls associated with the design and material selections. By means of optimal control, one can steer the operating condition towards a favourable one to improve the durability and efficiency of the fuel cells. Thus, dynamic modelling and control are the essential ingredients in fuel cell developments.

With the advent of cheap computational power, applications of difficult-to-implement complex controllers such as nonlinear model predictive controller, have been seen in the industries. In this book, both conventional controls and advanced model predictive controls are applied in the fuel cell system.

This book attempts to consolidate the results developed or published by authors over the last eight years along with fundamentals in one place and presents them in a systematic way. In this respect, the book is likely to be of use for graduate students and researchers as a textbook or monograph and as a place to look for basics as well as state-of-the-art techniques in dynamic modelling and control, and their applications. As several self-contained fuel cell dynamic models with detailed parameters and explanations are presented in the book, it provides an excellent place for researchers in process systems engineering and control systems engineering to look for challenging problems to test new control theory and algorithms. The readers of this book will be graduate students and researchers in Chemical Engineering, Mechanical Engineering or Electrical Engineering, with the major in process control, fuel cells, process systems engineering or control systems.

The book covers a variety of subjects including Chemical Engineering Fundamentals, System Identification, State Estimation and Process Control but they are not a pre-requisite for understanding the material as the book contains detailed introduction to these subjects. In this respect, this book can also be used as a textbook or as a reference for learning the fundamentals of chemical engineering, system identification, state estimation and process control with vivid illustrations of fuel cell application examples.

Acknowledgments

We would like to specifically thank our colleagues and collaborators, Professors Kumar Nandakumar, Karl Chuang and Jingli Luo, who have inspired many discussions and ideas in fuel cell research over the past years, and also to members of the computer process control group at the University of Alberta, Canada, who have provided a stimulating environment for process control research. The broad range of talent within the Department of Chemical and Materials Engineering at the University of Alberta has allowed cross-fertilisation and nurturing of many different ideas that have made this book possible. We are indebted to many of our industrial collaborators, who have inspired us with practical relevance in broad areas of process control research. We would also like to thank our computing support staff and other supporting staff of the Department of Chemical and Materials Engineering at the University of Alberta. The support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Western Canada Fuel Cell Initiative (WCFCI) for this and related research work are gratefully acknowledged. Last but not least, we would like to acknowledge Shima Khatibisepehr and Seraphina Kwak for their detailed review and comments for many chapters of the book.

List of Figures

List of Tables

4.1

Identification data

6.1

Polynomial roots

x

j

and the weighting functions

W

j

for

m

collocation points

6.2

Matrices for

m

-point orthogonal collocation found from

Equation 6.33

7.1

Input and output variables

8.1

I/O variables of the model

8.2

Variation of species in the SOFC stack from the inlet to the outlet

8.3

SOFC stack design parameters

9.1

Input and output variables

9.2

Time constants for tested processes at three different operation modes

9.3

Gains for tested processes at three different operation modes

9.4

Time delays for tested processes at three different operation modes

9.5

Processes considered for the system identification

9.6

BJ models at three different operation modes

10.1

Model parameters

11.1

Balance of plant (BOP) parameters

12.1

Optimal fuel flows for minimum indirect energy

12.2

Sets of different steady state solution for different initial conditions

A.1

Input conditions for the simulations

A.2

Model parameters

A.3

Inherent property parameters of gas ingredients

A.4

Approximation of specific heat (

Cv

) of gas ingredients between 700 and 1500 K

A.5

Approximation of viscosity of gas ingredients between 700 and 1500 K

A.6

Approximation of thermal conductivity of gases between 700 and 1500 K

A.7

Approximation of enthalpy of formation between 700 and 1500 K

Chapter 1

Introduction

Fuel cells are electrochemical devices that directly convert chemical energy into electrical energy. As the production of energy in fuel cells does not involve any moving parts and their principle of operation varies from that of heat engines, the energy produced by these cells is neither associated with any mechanical friction loss nor limited by Carnot cycle efficiency. Moreover, the unreacted fuel from the cell can be used to generate more power. The overall efficiency of the cell can also be increased by recovering the heat generated during operation from the exhaust gas.

Today's energy-hungry civilisation is in search of an alternative source to replace the currently available but continuously depleting conventional energy sources. Stringent environmental regulations have restricted the emission of greenhouse gases, SOx and NOx, and hence narrowed down the search for a clean source of energy to a few options. These are the main reasons behind the growing interest in the development of fuel cells as an alternative source of clean energy.

However, there are a number of obstacles in the commercialisation of fuel cells as a main source of energy. The main obstacle comes from the high manufacturing cost of the fuel cell. A vast amount of research is being conducted on the design and operation of fuel cells for reducing the cost and hence turning these devices into a viable and competitive source of energy. Selections of materials for electrolyte, catalyst and electrodes also contribute to the cost of a fuel cell. A number of researchers have focussed on this area. It is often required to simulate the fuel cell system under different operating conditions to account for all the pitfalls associated with the design and material selections. Depending on the perspective, the modelling and simulation can range from micro to system levels. This book focuses on solid oxide fuel cell system from the perspective of process control for the safe and efficient operation of the fuel cell system as a power source. It includes control relevant modelling, state estimation and controller design.

1.1 Overview of Fuel Cell Technology

Construction of a unit fuel cell mainly consists of three parts—electrolyte, cathode and anode. Fuel is continuously fed into the anode of the fuel cell, and a suitable oxidant, usually air, is fed into the cathode. The main purpose of the electrolyte is to prevent direct contact of the fuel with the oxidant while connecting the anode and cathode electrically. The electrolyte also allows the passage of the oxidant or reductant ions to the other side to take part in the electrochemical reaction.

1.1.1 Types of Fuel Cells

The classification of fuel cells is based on the choice of electrolyte and fuel. They are as follows:

Solid Oxide Fuel Cell (SOFC):

Solid oxide fuel cell uses a solid ceramic type oxide, and thus receives the name. Y

2

O

3

stabilised ZrO

2

(YSZ) is a common electrolyte used in SOFCs. The operating temperature of the fuel cell is usually high (600–1000 °C). Owing to the solid nature of the electrolyte and electrodes, the SOFC can be designed and fabricated in the most versatile ways, including planar and tubular designs.

Molten Carbonate Fuel Cell (MCFC):

Molten carbonate fuel cells use different combinations of alkali carbonates as an electrolyte. These carbonates are usually contained in a ceramic matrix. The operating temperature of MCFCs is also high, usually between 600 and 700 °C.

Proton Exchange Membrane Fuel Cell (PEMFC):

In this type of fuel cell, a polymeric ion exchange membrane is used as an electrolyte. The operating temperature of these cells is usually low (40–80 °C).

Phosphoric Acid Fuel Cell (PAFC):

The electrolyte in the PAFC is 100% phosphoric acid, which is held in a silicon carbide structure. The operating temperature of the fuel cell is about 150–220 °C, which is one of the attractive features of PAFC. This operating temperature makes it flexible to design the fuel cell and the balance of plant (BOP).

Other types of fuel cells include alkaline fuel cell (AFC), direct methanol fuel cell (DMFC), regenerative fuel cell (RFC) and metal air fuel cell (MAFC). Fue (2004) provides a summary of major differences in different types of fuel cells.

The low-temperature PEMFC and the high-temperature SOFC have been identified as the likely fuel cell technologies that will capture the most significant market in the future.

The basic principle of a typical hydrogen SOFC is shown in Figure 1.1. The chemical reactions inside the cell, which are directly involved in the production of electricity, are as follows:

1.1

At the anode of the SOFC, hydrogen gas reacts with oxygen ions that are migrated through the electrolyte to form water and release electrons. At the cathode, oxygen ionises with electrons and creates O2− ions. O2− ions are transported to anode through the electrolyte. Electrons produced at the anode flow through an external electrical circuit and reach the cathode. These reactions, therefore, both proceed continuously and supply electricity to the external circuit. Usually, SOFCs work at a high temperature, in the range of 600–1000 °C, to meet the electrolyte's ionic conductivity requirement.

Figure 1.1 Principle of solid oxide fuel cell

Hydrogen used as the fuel for SOFCs can be produced by steam reforming of natural gas. For a high-temperature fuel cell such as SOFC, the reforming reaction can be performed internally, within the anode of the cell.

1.1.2 Planar and Tubular Designs

To meet the voltage requirement for most of the applications, fuel cell systems need to be composed of stacks of connected individual cells. An SOFC stack is composed of a number of SOFC cells to produce a high voltage output. In designing SOFC stack and cells, there are many factors that need to be considered, such as gas delivery, thermal stresses, mechanical strength, inherent electrical and ionic resistance and choice of seal materials. SOFCs are manufactured in various geometries, the most common of which are the planar and tubular designs shown in Figures 1.2 and 1.3, respectively.

Figure 1.2 Tubular design of SOFC stack and cell

Figure 1.3 Planar design of SOFC stack and cell

One of the most important advantages of the tubular design is that it does not need the seal to separate fuel and air flow. Another advantage is that the tubular shape can improve the strength of the cell. The tubular shape can also improve the gas delivering property. This kind of design is suitable for stationary and large-scale power generation applications.

On the other hand, the most significant advantage of the planar SOFC design is its lower electrical resistance. Planar SOFCs are more suitable for mobile and low power applications.

1.1.3 Fuel Cell Systems

In an ideal fuel cell, hydrogen is used as a fuel along with air as an oxidant. Such a fuel cell can work as the cleanest possible source of energy—the by-product of the reaction being water. However, as hydrogen is not readily available in nature, in practice, the hydrogen used as fuel for these systems needs to be produced from other sources. Hydrogen-rich fuels are most commonly used to produce hydrogen either internally or externally to the fuel cell. Thus, a fuel cell plant usually involves components for pre- and post-processing of the reactants and products. The components, which are also called BOP, may include compressors, turbines, heat exchangers, reactors for reforming of the fuel and a DC–AC converter or inverter to connect the fuel cell to an existing power grid.

Compressors or blowers are required to build necessary pressure to pass reactants and products through different components. The unreacted fuel from the fuel cell itself can be combusted in a gas turbine for generating more power. The compressor–turbine duo thus provides a net power in addition to the direct power generated by the fuel cell itself. In residential applications, the hot effluent gas can be used to supply hot water and provide heat for the households.

A fuel cell directly converts chemical energy into electrical energy. The output being a DC voltage is appropriate to operate small equipment. For a fuel cell power plant, the DC power needs to be converted to AC in order to be transferable to the power supply grid. Thus, the BOP may also include a power conditioning unit (PCU).

1.1.4 Pros and Cons of Fuel Cells

Fuel cells have various advantages over conventional power generation systems such as batteries and turbines. As with any other technology, a fuel cell comes with some advantages and disadvantages. Some of these are described below.

Advantages:

Unlike turbines, a fuel cell system does not have any moving components, and thus does not have any mechanical friction loss associated with it. It also provides a quiet operation and less maintenance.

Unlike a heat engine, a fuel cell converts chemical energy directly into electrical energy. Thus, it is not limited by Carnot cycle efficiency.

The exhaust (unreacted fuel) gas from the fuel cell can be used to generate excess power by coupling with a heat engine, thereby, increasing the efficiency.

The efficiency of a fuel cell is not limited by size. Thus, a small fuel cell powering a laptop or a personal electronic gadget can generate power at the same efficiency as a 10 MW fuel cell power plant.

A wide range of fuels may be used for fuel cells.

As the reaction inside a fuel cell occurs between specific ions only, it limits the release of NO

x

and SO

x

to the environment.

Disadvantages:

Fuel cells are expensive compared to other energy producing technologies at least at the moment.

Most fuel cells use hydrogen as fuel, and it impedes commercialisation of these devices because of the cost and complexities associated with the production, storage and transportation of hydrogen.

In comparison with batteries, fuel cells have lower power densities and shorter lifetimes.

Impurity of fuel gas may poison catalysts in electrodes.

1.2 Modelling, State Estimation and Control

Process modelling, state estimation and design of the controller are part of advanced process control strategies. They are intricately dependent on each other. For example, building a model (whether it is first-principle or data-based, linear or nonlinear, 0D or 3D model) affects the design of the controller and state estimation techniques. Thus, the modelling of a process should always be based on the objective. A simple model developed for the purpose of control may perform better than a complex 3D model, which, on the other hand, may be suitable for design and performance analysis of the process. In simple words, the modelling objective of this book can be stated as finding a model that is suitable for controller design.

Similarly, controller design and state estimation techniques should be objective-oriented. A process expressed by a complex model may be stable enough to be controlled by a regular proportional integral derivative (PID) controller. On the other hand, a simple process may have a lot of environmental and economic constraints, requiring a multivariate controller to maintain the optimal performance of the system.

An SOFC system, which exhibits highly nonlinear characteristics, needs to be studied by various modelling, estimation and control techniques. This book covers all these three inter-related aspects, that is, modelling, state estimation and control.

1.3 Book Coverage

The book consists of three parts. Part I provides a tutorial of the fundamental principles used in the subsequent chapters. Specifically, chemical engineering principles, system identification, state estimation and model predictive control are applied to fuel cell systems and thus their fundamentals are covered in Part I.

Part II focuses on detailed and simplified dynamic modelling of tubular SOFC cells. The first-principle modelling considers all dynamics of the flow, including mass, energy and moment balances.

The data-based modelling in Part II is based on the system identification approach, which is presented in detail in Part I. Various aspects of system identification are illustrated through applications in the modelling of the fuel cell. As a natural outcome of system identification, the models identified are used for feedback control design including PID and IMC.

Recent advances and growing interest in fuel cells have led to a lot of activities on not only the modelling of fuel cells but also their system components. These models range from zero-dimensional to complex three-dimensional models and also cover the area of performance evaluation and optimal design of the fuel cell. However, little work has been done on developing control relevant models on the system level that sufficiently describe the fuel cell system dynamics, yet are simple enough for control design. This motivated us for developing lumped models of fuel cell and BOP to form a fuel cell system in Part III. To diversify the coverage of the book, Part III is devoted to the planar SOFC.

A wide range of linear and nonlinear control techniques have been developed and implemented in various industries. Especially during the past decade, with the advent of cheap computational power, a trend shifted from traditional PID controller towards previously non-implementable controllers, such as nonlinear model predictive controller (NMPC). This led us to attempting NMPC in SOFC systems along with optimisation to maximise electrical energy generated from SOFCs.

1.4 Book Outline

The book is organised as follows:

Chapter 2 provides an introduction to chemical engineering fundamentals, which are the basis for first-principle modelling in the subsequent chapters.

Chapter 3 provides foundation for system identification, including discrete-time representation of processes, signals, input design and model structures for data-based modelling.

Chapter 4 presents introduction to advanced topics in system identification, including prediction error method, nonlinear identification, model validation, practical issues, close-loop identification and subspace identification.

Chapter 5 introduces state estimation methods along with parameter estimation. The focus of this chapter is on Unscented Kalman Filter, as this will be used in model predictive control of SOFC in the subsequent chapter.

Chapter 6 provides a tutorial overview on both linear and nonlinear model predictive controls (MPC). Following the current industrial practice in the application of MPC, the economic optimisation strategy is also discussed.

Chapter 7 gives a detailed application of the first-principle approach to dynamic modelling of tubular SOFC. This chapter illustrates in detail how the chemical engineering principles, discussed in Chapter 2, can be applied to solve modelling problem for fuel cell processes.

Chapter 8 applies chemical engineering principles to derive a reduced order dynamic model useful for control design. This chapter can also serve as a self-contained introduction to SOFC operation principles and simplified modelling procedure.

Chapter 9 illustrates in detail how a system identification approach can be applied to solve practical data-based modelling problems for SOFCs. On the basis of the identified models, conventional feedback controllers are designed and simulated for tubular SOFC.

Chapter 10 gives a detailed application of the first-principle approach to dynamic modelling of planar SOFC.

Chapter 11 considers first-principle modelling of an entire fuel cell system, including BOPs, and how components can be combined to form a system.

Chapter 12 illustrates the design of model predictive control for the SOFC system, including state estimation, linear MPC and nonlinear MPC.

Depending on readers' background and interest, they can read the entire book according to the sequence of chapters or they can selectively read some of the chapters. In the latter case, the suggested route of readings, based on the reader's background, is as follows:

1. For readers who are interested in first-principle-based modelling, the sequence of reading is

2. For readers who are interested in first-principle-based modelling as well as control, the sequence of reading is

3. For readers who are interested in the data-based system identification approach, the sequence of reading is

For additional supporting material of the book, we refer to the book homepage at http://www.ualberta.ca/~bhuang/SOFCbook/.

Part One

Fundamentals

Chapter 2

First Principle Modelling for Chemical Processes

2.1 Thermodynamics

The name thermodynamics comes from the Greek words ‘therme’ (heat) and ‘dynamics’ (power), which describes the early efforts to convert heat into power. Thermodynamics did not emerge as a science until the construction of the first successful atmospheric steam engine. At present, the name is broadly interpreted to include all aspects of energy and energy transformations.

One of the most fundamental laws of nature is the conservation of energy. Energy can be neither created nor destroyed. It can only be transformed from one form to another and transported from one place to another. The first law of thermodynamics is simply an expression of the conservation of energy, and it asserts that energy is a thermodynamic property. The second law of thermodynamics asserts that energy has quality as well as quantity, and actual processes occur in the direction of decreasing quality of energy.

2.1.1 Forms of Energy

Energy exists in numerous forms such as thermal, mechanical, kinetic, potential, electrical, magnetic, chemical and nuclear. In general, various forms of energy can be classified into two groups: macroscopic and microscopic. A form of energy that a system possesses as a whole with respect to some outside reference is grouped as the macroscopic energy, such as kinetic and potential energies. On the other hand, a form of energy related to the molecular structure and the molecular activity of a system is grouped as the microscopic form of energy, such as thermal, chemical and nuclear energies. The sum of all the microscopic energies of a system is called the internal energy, U, of the system.

The portion of the internal energy that is associated with the kinetic energy of molecules and intermolecular forces between molecules is frequently referred to as heat or thermal energy. The internal energy that is associated with the atomic bonds in a molecule is called chemical (or bond) energy. The internal energy that is associated with the bonds within the nucleus of the atom itself is called nuclear energy.

The forms of energy that are not contained in a system, such as chemical and nuclear energy, can be viewed as dynamic forms of energy. The dynamic forms of energy often appear at the boundary of a system, and they represent the energy gained or lost by a system during a process. Systems may exchange energy via mass transfer because at any time, mass is transferred into or out of a system; the energy contained in the mass is transferred along with it. Even without mass transfer, energy interaction can still occur via heat transfer and work. Heat transfer and work are the only two forms of energy interactions that are associated with a closed system.

2.1.2 First Law

In general, the first law of thermodynamics is simply an expression of the principle of conservation of energy. It provides a sound basis for studying relationships among the various forms of energy and energy interactions. On the basis of the experiments of Joule in the first half of the nineteenth century, the first law states that energy can be neither created nor destroyed; it can only change forms. It implies that in the absence of heat transfer, for a system undergoing a series of adiabatic processes from a specified state to another specified state, the net work done is the same regardless of the nature of the closed system and the details of the process. Similarly, in the absence of any work interactions between a system and its surroundings, the amount of net heat transfer is equal to the change in the total energy of a closed system. So the first law for a closed system can be expressed as

2.1

where Q is the net heat transfer to the system across its boundaries, W is the net work done by the system, ΔU is the change of internal energy and ΔKE and ΔPE are changes of kinetic and potential energies, respectively.

The internal energy expresses itself macroscopically via two intensive properties of the substance, temperature and pressure. Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. It is an intensive property that indicates the activity of random motion of the constituent particles of the matter and is the result of the motion of the particles. Pressure is a measure of the force exerted by the motion of constituent particles of gases and liquids.

The change in the energy of a material is related to its temperature through a material property named specific heat, C. The specific heat is defined as the energy required to raise the temperature of a unit mass of a substance by 1°. Two kinds of specific heat need to be distinguished: specific heat at constant volume Cv, and specific heat at constant pressure Cp. These properties may differ significantly for substances in the gas phase.

The specific heat at constant volume is defined as

2.2

while the specific heat at constant pressure is defined as

2.3

Another important property is called enthalpy, H, defined as follows:

2.4

Enthalpy is a combination of two properties: the internal energy U and the product PV—which is the energy required to make room for the medium by displacing its environment and establishing its volume and pressure. Enthalpy is a measure of the total energy of a thermodynamic system and is widely used in engineering practice.

For an ideal gas, the temperature, pressure and volume are related by the ideal gas equation

2.5

where n is the number of moles and R is called the gas constant.

2.1.3 Second Law

It is a common experience that a cup of hot coffee left in a room eventually cools off. It is impossible for the coffee to spontaneously absorb heat from the cold room and to heat up itself. This example shows that processes occur in a certain direction. The first law places no restriction on the direction of a process, and satisfying the first law does not ensure that a process will actually occur. The second law of thermodynamics is used to identify whether a process can take place or not.

The Kelvin–Planck statement of the second law of thermodynamics is

It is impossible for any device that operates in a cycle to receive heat from a single reservoir and produce a net amount of work.

This means that a heat engine cycle must exchange heat with a low temperature sink, as well as a high temperature source to keep it in continuous operation.

The Clausius statement of the second law of thermodynamics is

It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a lower temperature body to a higher temperature body.

It is a common knowledge that heat does not, of its own volition, flow from a medium with a lower temperature to a medium with a higher temperature. The Clausius statement implies that this process is not impossible, if an external power is introduced to the process, as in the condition of a refrigerator. It simply states that the refrigerator will not work unless its compressor is driven by an external power source.

2.2 Heat Transfer

Heat transfer is a thermal energy in transit due to a spatial temperature difference. When a temperature difference exists in a medium or between media, heat transfer occurs.

Heat transfer has three different modes. When a temperature gradient exists in a stationary medium, which may be a solid or liquid, the heat transfer is called conduction. If the heat transfer occurs between a surface and a moving fluid when they are at different temperatures, the heat transfer is called convection. The third mode of heat transfer is called thermal radiation or simply radiation, which refers to heat transfer between two surfaces at different temperatures in the absence of an intervening medium.

2.2.1 Conduction

The term conduction refers to the transportation of energy in a medium due to a temperature gradient. Conduction is a process facilitated by the activity at atomic and molecular levels. It may be seen as the transfer of energy from the more energetic to the neighbouring less energetic atoms and molecules of a substance.

Consider a substance in which there exists a temperature gradient and assume that there is no bulk or macroscopic motion. Temperature at any point of the substance is associated with the energy of atoms or molecules in proximity to the point. This energy is related to the random translational motion, as well as to the internal rotational and vibrational motions of the atoms and molecules. Higher temperature means higher energy. When neighbouring atoms and molecules collide, a transfer of energy from the more energetic to the less energetic atoms or molecules occurs. Furthermore, because of random motion, a hypothetical plane is constantly crossed by electrons or atoms or molecules from above and below. Particles from above are associated with higher energy than those from below. The net energy is, therefore, transferred from the place where the temperature is higher, to the place where the temperature is lower, as shown in Figure 2.1. Collisions between particles enhance this kind of energy transfer.

Figure 2.1 Mechanism of conduction heat transfer

For heat conduction, the rate of heat transfer is governed by Fourier's law. Fourier's law states that the rate of heat transfer in a given direction is proportional to the gradient of temperature in that direction and to the area normal to the direction of heat flow. For one-dimensional transfer through a wall, the Fourier's law is expressed as

2.6

where qx (Wm−2) represents the heat transfer rate in the x direction per unit area perpendicular to the direction of transfer. It is called heat flux and has a unit of (Wm−2). The heat flux is proportional to the temperature gradient dT/dx in its direction. The parameter k is known as the thermal conductivity (Wm−1 · K), which is a characteristic property of the wall material. The minus sign indicates that heat is transferred in the direction of decreasing temperature.

Knowing the section area A, the Fourier's law can also be written as

2.7

where Qx (W) is called heat flow. It represents the rate of heat flow through the area A in the x direction.

Heat is a kind of energy, with the standard basic unit of joules (J). Therefore, the rate of heat transfer is joule per second (Js−s). It is the more straightforward unit to represent the concept of heat transfer. Some literature defines the unit of heat transfer rate as watt (W) representing the concept of power. Power is the rate at which work is performed or energy is converted. The adoption of this unit is more consistent with the concept of the first law of thermodynamics.

Fourier's law implies that the heat flux or heat flow is a directional quantity. The direction of flow is normal to the cross-sectional area of heat transfer. In other words, the direction of heat flow will always be normal to the surface of constant temperature, called an isothermal surface. Therefore, heat flow is recognised as a vector quantity. A more general statement of Fourier's law is the conduction rate equation:

2.8

Fourier's law is phenomenological or empirical. It is developed from the observed phenomena rather than being derived from first principles. By thermal conductivity, which is an important material property, Fourier's law is defined as the cornerstone of conduction heat transfer. It applies to all materials, regardless of their state—solid, liquid or gas.

2.2.2 Convection

In addition to energy transfer by random motion at the atomic or molecular level, energy is also transferred by the bulk or macroscopic motion