Thermodynamics

Thermodynamics Mr. V. Prabaharan Assistant Professor Department of Mechanical Engineering Shree Venkateshwara Hi-Tech Engineering College Gobichettipalayam - 638455 Dr. J. M. Prabhudass Associate Professor Department of Mechanical Engineering Sri Sairam Institute of Technology Sai Leo Nagar, West Tambaram, Chennai - 600044 Dr. N. Poyyamozhi Assistant Professor Grade-I Department of Mechanical Engineering Panimalar Engineering College Bangalore Trunk Road, Varadharajapuram, Poonamallee, Chennai - 600123 Dr. V. T. Vimalananth Assistant Professor Department of Mechanical Engineering Academy of Maritime Education and Training Kanathur, Chennai - 603112 Edition Details (I,II,III): I ISBN: 978-93-6786-795-2 Month & Year: September, 2025 Copyright @ Mr. V. Prabaharan Dr. J. M. Prabhudass Dr. N. Poyyamozhi Dr. V. T. Vimalananth Pages: 249 Price: 850/- About the Authors’ Mr. V. Prabaharan , ME (THERMAL ENGINEERING) is an Assistant Professor in Mechanical Engineering at Shree Venkateshwara Hi-Tech Engineering College, Erode. He holds a B.E. and M.E. from Anna University . He has published in reputed journals .With academic and industry experience, he teaches subjects like Heat and mass Transfer, Thermodynamics, Thermal Engineering, Strength of Materials and Engineering Graphics. He is a Life Member of ISTE . Dr. J. M. Prabhudass is an Associate Professor in the Department of Mechanical Engineering at Sri Sairam Institute of Technology with over 20 years of teaching experience. He holds a B.E in Mechanical Engineering, M.Tech in Thermal Engineering and Ph.D. with research interests in composite materials. He has authored international publications in reputed Journals. Dr. Prabhudass has successfully guided and mentored numerous student teams in national competitions. His academic excellence is reflected in achieving 100% results in multiple core mechanical subjects. A member of SAE, ISTE, and IEEE, he has organized and participated in several workshops, conferences, and FDPs. He also serves as Department NAAC Coordinator and Department Strategist, fostering innovation and technical excellence among students Dr. N. Poyyamozhi earned his undergraduate degree in Mechanical Engineering in 2009 and completed his postgraduate studies in 2011. Currently serving as an Assistant Professor Grade-I at Panimalar Engineering College, he specializes in Thermal Energy Storage Systems. With over 34 research papers published, he has presented his work at both national and international conferences. Dr. Poyyamozhi is also actively involved in organizing workshops and delivering guest lectures. In addition to his academic contributions, he holds more than four patents and has authored a book. Dr. V. T. Vimalananth is an Assistant Professor in Mechanical Engineering at AMET University, Chennai. He holds a B.E. and M.E. from Anna University and a Ph.D. in Mechanical Engineering from University College of Engineering, Villupuram. His research focuses on Alternate Fuels, IC Engines, and Pollution Control. He has published in reputed journals and presented at international forums. With academic and industry experience, he teaches subjects like Thermodynamics, TQM, and Engineering Graphics. He is a Life Member of ISTE, SAE, and IAENG. Dr.Vimalananth has also trained in EV Technology and Additive Manufacturing. Research Profile:researchgate.net/profile/Vimalananth-V-T Preface Thermodynamics is one of the fundamental pillars of science and engineering, providing the principles that govern energy, heat, and work, and their transformations. Its applications span across mechanical engineering, chemical processes, power generation, refrigeration, aerospace, renewable energy, and countless other domains. Understanding these principles is essential not only for engineers and scientists but also for anyone seeking to comprehend the physical processes that drive our world. This book has been carefully structured to provide both clarity and depth. It introduces the core concepts of thermodynamics in a systematic manner, starting from the basic laws and extending to advanced applications in real-world systems. Emphasis has been placed on blending theoretical knowledge with practical illustrations, problem-solving approaches, and examples drawn from engineering practice. The text is designed for students, educators, and professionals alike. For learners, it serves as a comprehensive guide to mastering fundamental principles and solving numerical problems. For teachers, it offers a structured resource to support instruction and classroom discussions. For researchers and practitioners, it provides a reliable reference to explore thermodynamic concepts in the context of modern technologies. We have also made conscious efforts to present the subject in an accessible way by using clear explanations, diagrams, solved examples, and exercises that encourage self-learning and application. The integration of modern tools and computational methods has been highlighted wherever relevant, reflecting the evolving nature of engineering education. While preparing this book, we have drawn inspiration from the pioneering contributions of scientists, engineers, and educators whose work continues to shape the field of thermodynamics. We humbly hope that this book will serve as a valuable companion for all readers who wish to gain both foundational knowledge and practical insight into the subject. Finally, we welcome constructive feedback and suggestions from readers, as this book is a result of continuous learning and refinement. Mr. V. Prabaharan Dr. J. M. Prabhudass Dr. N. Poyyamozhi Dr. V. T. Vimalananth Acknowledgement The completion of this book on Thermodynamics has been a deeply rewarding and enlightening journey, made possible through the unwavering support, guidance, and contributions of many individuals and institutions. We extend our heartfelt gratitude to all who have played a vital role in bringing this work to fruition. First and foremost, we express our sincere appreciation to our teachers, mentors, and colleagues, whose valuable insights, constructive feedback, and constant encouragement have significantly enriched the depth and clarity of this book. Their expertise has guided us in presenting both the foundational principles and the practical applications of thermodynamics in a comprehensive manner. We are especially grateful to our families for their unconditional love, patience, and continuous support throughout this endeavor. Their belief in our work and their sacrifices have been a source of strength and inspiration, enabling us to complete this book successfully. Our deepest thanks also go to the academic and research communities in the fields of thermodynamics, energy systems, and applied sciences. Their pioneering research, innovations, and scholarly contributions have laid the groundwork upon which this book is built, and their work continues to inspire us to delve deeper into this essential subject. We also wish to acknowledge the assistance of modern computational tools and resources, which have facilitated problem-solving, data analysis, and the presentation of complex concepts in a clear and accessible way. These tools have highlighted the relevance of technology in advancing the understanding and teaching of thermodynamics. Above all, we express our profound gratitude to Almighty God for His guidance, wisdom, and blessings throughout this journey. His grace has been our constant source of motivation in overcoming challenges and ensuring the successful completion of this book. We hope this book serves as a valuable resource for students, educators, and professionals, fostering a deeper understanding of thermodynamics and its vital role in engineering, science, and technology. Mr. V. Prabaharan Dr. J. M. Prabhudass Dr. N. Poyyamozhi Dr. V. T. Vimalananth Thermodynamics 3 Applications of Continuum Concept Valid for atmospheric air, water, steam, and common engineering fluids under normal pressure and temperature conditions. Not valid in rarefied gases (high altitudes, vacuum chambers, space environments). Used extensively in thermodynamics, fluid mechanics, and heat transfer where bulk properties are more useful than molecular data. 1.2 Comparison of microscopic and macroscopic approach Microscopic approach: Deals with the behavior of individual molecules, requiring detailed knowledge of molecular motion and interactions.  Studies the behavior of individual molecules that make up the system.  Based on the principles of statistical mechanics.  Requires knowledge of the position, velocity, and energy of each molecule.  System properties (pressure, temperature, etc.) are obtained by statistical averaging of molecular data.  More accurate at the molecular level but complex and computationally heavy for large systems. Macroscopic approach (continuum assumption): Considers matter as continuously distributed, ignoring discrete molecular nature. Properties such as pressure, temperature, and velocity are described as point functions.  Studies the system as a whole, without considering individual molecules.  Based on classical thermodynamics and the continuum hypothesis.  Properties like pressure, temperature, volume, and density are treated as continuous functions of space and time.  Easier and more practical for engineering analysis.  Works well when continuum assumption (Kn < 0.01) is valid. Aspect Microscopic Approach Macroscopic Approach Basis Individual molecules and their interactions Overall system treated as a continuum Foundation Statistical mechanics, kinetic theory Classical thermodynamics Properties Derived from molecular motion (e.g., pressure from molecular collisions) Measured directly as bulk properties (P, V, T) Level of Detail Provides molecular-level details Provides average/system-level information Thermodynamics 4 Complexity Very high (needs data of millions of molecules) Relatively simple and practical Applicability Rarefied gases, nano/micro scale flows, plasma physics Engineering systems, fluids, gases, heat engines Example Determining velocity distribution of molecules Measuring temperature of steam in a boiler Table. 1.1 Microscopic vs. Macroscopic Approach. 1.3 Path and Point Functions Path Functions A path function is a quantity whose value depends on the specific path followed during a thermodynamic process, rather than just the initial and final states of the system. Unlike properties (such as temperature, pressure, or volume) which are state functions, path functions are not properties of the system. Instead, they represent boundary interactions like heat and work. Consider a system that changes from an initial state (point 1) to a final state (point 2) on a P– V diagram. Different process paths (A, B, and C) can connect these two states. Fig. 1.2 Graph of P-V showing Example of the path function. Thermodynamics 5 Although the initial and final states are the same, the area under each curve (which represents work done) is different for each path. Hence, the work done and heat transferred depend on the path, not just the end states. Examples of Path Functions 1. Work (W): Work done by or on a system depends on how pressure and volume change during the process. For the same initial and final states, the work differs depending on whether the process is isothermal, adiabatic, or polytropic. Example: Gas expanding slowly vs. rapidly - work values differ. 2. Heat Transfer (Q): Heat absorbed or rejected depends on how energy is supplied or removed. The amount of heat differs if heating is constant, rapid, or involves a phase change, even if the system reaches the same final state. Characteristics of Path Functions 1. Process Dependent: Their values depend on the manner in which the process occurs. 2. Different Paths, Different Values: Same initial and final states can yield different amounts of heat or work. 3. Inexact Differentials: Path functions are expressed using δQ, δW instead of dQ, dW, indicating they are not exact differentials and not properties. Point Functions (State Functions) A point function, also known as a state function, is a property whose value depends only on the state of the system and not on the path taken to reach that state. In other words, point functions are determined solely by the initial and final states, regardless of the process in between. Examples of point functions include internal energy (U), entropy (S), enthalpy (H), and temperature (T). Examples of Point Functions 1. Internal Energy (U): If a gas undergoes expansion or compression, its internal energy change depends only on the starting and ending states. The manner of energy transfer (slow, fast, heat, or work) does not affect the total change in internal energy. Thermodynamics 6 2. Temperature (T): If water is heated from 20°C to 80°C, the temperature rise is always 60°C, regardless of how slowly or rapidly the heating takes place. Fig. 1.3 Graph of P-V showing Example of the point function. 3. Enthalpy (H): When water is converted into steam at a given pressure, the enthalpy change is fixed and depends only on the initial liquid state and the final vapor state. Characteristics of Point Functions 1. State-Dependent: Their values are determined only by the condition (state) of the system, not the path taken. 2. Exact Differentials: Point functions can be expressed in the form of exact differentials (dU, dH, dT, etc.), meaning they can be integrated directly. 3. Unique at a State: For a given thermodynamic state, point functions have definite values that remain the same no matter how the state was achieved. Relationship between Work and Internal Energy In thermodynamics, point functions (state functions) and path functions are interconnected.  Point functions describe the condition of a system at a given state (e.g., internal energy, enthalpy, entropy). Thermodynamics 7  Path functions describe how the system reaches that state through energy interactions (e.g., heat, work). Path functions like heat (Q) and work (W) cause changes in point functions such as internal energy (U), enthalpy (H), and entropy (S). Key Relationships 1. Path functions drive changes in point functions: Heat and work transfer are responsible for altering internal energy, enthalpy, or entropy. 2. Different paths - different Q and W, but same ΔU: For a given change of state, the work and heat values depend on the process path, but the change in internal energy (ΔU) remains the same because it is a state property. 3. Point functions set limits for path functions: The change in internal energy defines how heat and work interact in any thermodynamic process. Aspect Point Function (State Function) Path Function Definition Depends only on the state of the system Depends on the path taken between states Dependence Independent of the path followed Dependent on the path followed Examples Pressure, Temperature, Volume, Internal Energy, Enthalpy, Entropy Work (W), Heat (Q) Value for Same States Same value, no matter how the state is reached Different values for different processes Process Information Only initial and final states are needed Exact path of the process must be known Differentials Exact (perfect) differentials, written as dU, dH, dS Inexact (imperfect) differentials, written as δQ, δW Cyclic Process Net change over a cycle = 0 Net value over a cycle ≠ 0 (can be positive or negative) Table. 1.2 Difference between point and path function. 1.4 Intensive and Extensive Properties In thermodynamics, properties of a system are classified into intensive and extensive depending on whether they vary with the amount of matter present. Thermodynamics 8 1. Intensive Properties Intensive properties are those that do not depend on the size or quantity of matter in a system. They remain the same regardless of how much substance is present. For example, whether you have a drop of water or a bucket of water, the temperature and boiling point remain identical. Characteristics  Independent of mass or volume.  Describe the intrinsic nature of a substance.  Useful in identifying and comparing materials.  Often expressed as ratios or derived properties (e.g., density = mass/volume). Examples 1. Temperature: A sample of water at 25°C stays at that temperature irrespective of quantity. 2. Pressure: A gas divided between two containers still exerts the same pressure. 3. Density: Pure gold always has a density of ~19.3 g/cm³, no matter the sample size. 4. Melting/Boiling Points: Ice always melts at 0°C and water boils at 100°C (at 1 atm). 5. Other examples: Color, refractive index, conductivity, hardness, surface tension. 2. Extensive Properties Extensive properties are those that depend on the amount of matter present in the system. Their values scale with the size or mass of the system. For instance, doubling the mass of a substance also doubles its volume and total energy. Characteristics  Directly proportional to the amount of matter.  Additive in nature: The total value is the sum of the values of all parts.  Provide quantitative rather than qualitative information.  Not suitable for substance identification, but essential for measuring total system behavior. Thermodynamics 9 Examples 1. Mass: Two 500 g samples together make 1000 g. 2. Volume: Adding more matter increases the occupied space. 3. Energy: Total internal, kinetic, or potential energy increases with system size. 4. Other Examples: Enthalpy, entropy, heat capacity, number of moles, length. Relationship between Intensive and Extensive Properties Intensive and extensive properties are closely related in thermodynamics and material science. In many cases, ratios or combinations of extensive properties give rise to intensive properties. This relationship helps explain why some properties remain constant for a substance, while others vary with the amount of matter. Example: Density Density is defined as: ρ= m V  Mass (m) and Volume (V) are both extensive properties.  Their ratio, density (ρ), is an intensive property. No matter if you have 1 mL, 1 L, or 100 L of pure water, the density remains approximately 1 g/cm³ at room temperature. Examples of Intensive Properties Derived from Extensive Ones 1. Molar Volume (V/n): Ratio of volume to number of moles. 2. Specific Heat (C/m): Heat capacity per unit mass. 3. Molar Mass (M/n): Ratio of mass to number of moles. These derived intensive properties are crucial in chemistry, thermodynamics, and material science because they describe the intrinsic behavior of substances independent of quantity. Thermodynamics 10 Aspect Intensive Properties Extensive Properties Dependence Independent of the amount of matter Dependent on the amount of matter Effect of Subdivision Remain unchanged when the system is divided Change proportionally with subdivision Usage Identify and characterize substances Measure the total amount of substance Examples Temperature, Pressure, Density, Boiling Point Mass, Volume, Energy, Entropy Additivity Not additive Additive in nature Relation Linked to the internal structure and interactions of matter Linked to the overall size or extent of the system Table. 1.3 Difference between Intensive and Extensive Properties. 1.5 Total and Specific Quantities In thermodynamics, many properties are expressed in terms of total or specific quantities. These terms are particularly important when dealing with extensive properties, since they depend on the overall mass or size of the system. By distinguishing between total and specific values, it becomes easier to analyze thermodynamic systems of varying scales, from a small sample of fluid in a test tube to large industrial systems such as boilers and turbines. Total Quantities A total quantity refers to the complete or overall value of an extensive property for the entire system. These values depend directly on the size or mass of the system, which means that if the mass of the system is doubled, the total property value also doubles. Total quantities are generally denoted using uppercase symbols in thermodynamics. For example, Volume (V) represents the total space occupied by the system, Enthalpy (H) represents the total heat content, Internal Energy (U) represents the total stored energy, and Entropy (S) measures the total disorder of the system. Each of these values gives a complete picture of the energy or property content of the system as a whole. Thus, when we say a gas has a total enthalpy of 500 kJ or a volume of 10 m³, we are referring to the total quantities of the system, which scale with the amount of matter present. Thermodynamics 11 Specific Quantities To make thermodynamic analysis more convenient, especially when comparing systems of different sizes, it is common to express properties in terms of specific quantities. A specific property is obtained by dividing an extensive property by the mass of the system. Since mass is also an extensive property, this division produces a value that is independent of the system size, making it an intensive property. Specific properties are generally denoted using lowercase symbols. For instance, specific volume (v) is defined as the volume per unit mass (V/m), expressed in m³/kg. Similarly, specific enthalpy (h) is the enthalpy per unit mass (H/m), expressed in J/kg, and specific internal energy (u) is internal energy per unit mass (U/m). Likewise, specific entropy (s) is the entropy per unit mass (S/m). These specific quantities are particularly useful when working with thermodynamic tables, as they provide standardized values that can be easily scaled up or down depending on the actual mass of the system. As an example, if 2 kg of gas occupies a total volume of 10 m³, its specific volume can be calculated as: 𝑣=𝑉 𝑚=10 2=5 m3/kg This value is independent of the system’s total size and would remain the same even if we considered a smaller or larger portion of the same gas. Importance of Total and Specific Quantities The distinction between total and specific quantities is vital in thermodynamic analysis. Total properties provide essential information when calculating overall energy balances in largescale engineering systems. For example, the total enthalpy of steam entering a turbine is necessary for determining the total work output. On the other hand, specific properties simplify the analysis by giving values per unit mass. This makes it easier to compare substances, irrespective of the amount available. Engineers and scientists often use specific values from steam or refrigerant property tables, which can then be multiplied by the actual mass of the system to obtain the corresponding total property. This dual use of total and specific quantities provides flexibility in thermodynamics, allowing both large-scale system analysis and small-scale property comparison. Thermodynamics 12 1.6 System and their types In thermodynamics, the concept of a system forms the foundation of all analysis. A thermodynamic system is defined as a specified quantity of matter or a particular region in space selected for study. The behavior of this system is analyzed in terms of its properties, interactions, and energy exchanges. Everything that lies outside the chosen system is referred to as the surroundings. In practice, the surroundings usually denote the region in the immediate neighborhood that can have a measurable influence on the system. The separation between the system and its surroundings is defined by a boundary. This boundary may be real or imaginary, and it can either be fixed (such as the walls of a rigid container) or movable (such as a piston head in a cylinder). The boundary plays an essential role in determining whether mass or energy crosses into or out of the system. Fig. 1.4 Thermodynamic System, boundary, surroundings. Types of Thermodynamic Systems Thermodynamic systems are broadly classified into three categories depending on whether energy and mass can cross the boundary. 1. Closed System A closed system (also called a control mass system) consists of a fixed amount of matter.  Mass: Remains constant; no mass crosses the system boundary.  Energy: Heat and work interactions are possible across the boundary.  Volume: May change, as the system’s boundary can expand or contract. Example:  Gas confined in a piston–cylinder arrangement.  A sealed container of water being heated (no mass transfer, but energy transfer occurs). Thermodynamics 19 Characteristics  Process can be retraced in the opposite direction.  Produces maximum possible work output (for a given change of state).  Idealization - no real process is perfectly reversible. Examples  Isothermal expansion or compression of an ideal gas carried out infinitely slowly.  Frictionless piston–cylinder movement.  Heat transfer between two bodies at the same temperature (theoretically). Types of Reversible Processes a) Internally Reversible Process A process is said to be internally reversible if no irreversibility occurs within the system during the process.  The system passes through a continuous series of equilibrium states.  If the process is reversed, the system retraces the same path and returns to its original state.  However, irreversibilities may still occur outside the system (in the surroundings). Example: Slow, frictionless compression or expansion of a gas in a piston–cylinder device. b) Externally Reversible Process A process is externally reversible if no irreversibility occurs outside the system boundaries during the process.  The interaction between the system and surroundings is perfectly balanced.  For instance, when heat transfer takes place between two bodies at the same temperature, there is no external irreversibility. Example: Heat transfer between a system and a thermal reservoir at exactly the same temperature. 2. Irreversible Process An irreversible process is a real process that cannot be exactly reversed, because it leaves permanent changes in the system or surroundings. When such a process is reversed, the system and surroundings do not return to their initial states. Thermodynamics 20 All natural processes are irreversible to some extent, because they involve finite differences in driving forces (temperature, pressure, chemical potential) and include dissipative effects such as friction, turbulence, electrical resistance, or inelastic deformation. Causes of Irreversibility 1. Frictional effects: Mechanical friction, viscous drag in fluids. 2. Unrestrained expansion of gases: Sudden expansion into a vacuum. 3. Heat transfer across a finite temperature difference: Flow of heat from a hot body at 500 K to a cold body at 300 K. 4. Mixing of different substances: e.g., diffusion of two gases. 5. Inelastic deformations: Plastic deformation of solids. 6. Natural spontaneous processes: Flow of heat from hot to cold, spontaneous chemical reactions. Characteristics  Process cannot be retraced to restore both system and surroundings.  Work output is always less than that of a reversible process between the same states.  All real processes are irreversible in practice. Examples  Free expansion of a gas into a vacuum.  Sudden compression or expansion of gases.  Heat transfer through a finite temperature difference.  Natural mixing of gases or liquids. Types of Irreversible Processes a) Internal Irreversibility An internal irreversibility arises due to effects within the system itself that disturb equilibrium.  These include factors like friction within the working fluid, turbulence, viscous dissipation, or non-equilibrium chemical reactions.  They prevent the system from passing through a series of equilibrium states. Example:  Viscous effects in fluid flow.  Combustion reactions inside a cylinder. Thermodynamics 21 b) External Irreversibility An external irreversibility occurs due to disturbances or dissipative effects in the surroundings of the system.  These are caused by interactions between the system and its environment that are not perfectly balanced. Example:  Mechanical friction between a moving piston and its cylinder walls.  Heat transfer through a finite temperature difference between a hot source and a cold system. Consider a thermodynamic system undergoing a change from state A to state B as shown in the diagram. The manner in which this change occurs determines whether the process is reversible or irreversible. Reversible Process A reversible process is an ideal process that can be completely reversed, such that both the system and surroundings are restored to their original states without leaving any trace of the process.  In the figure, the curved path (A → B) represents the reversible process.  During this path, the system passes through a continuous series of equilibrium states, where pressure, temperature, and other properties are well-defined at every stage.  For a process to be reversible, it must occur infinitely slowly, ensuring that the system remains in thermodynamic equilibrium throughout. Thermodynamics 22 Key Conditions for Reversibility 1. The process must be quasi-static (infinitely slow). 2. There should be no dissipative effects such as friction, turbulence, viscosity, or heat transfer through a finite temperature difference. 3. Each intermediate state must be an equilibrium state. Irreversible Process An irreversible process is a real process that cannot be completely reversed, because it leaves permanent effects on the system and surroundings.  In the figure, the straight dashed line (A → B) shows an irreversible path.  The system does not pass through equilibrium states during the process, and intermediate states cannot be defined precisely.  Irreversibility arises due to factors like friction, unrestrained expansion, turbulence, heat transfer across finite temperature differences, or mixing of substances. Special Case: Reversible Adiabatic (Isentropic) Process  If the process is adiabatic (no heat transfer, Q=0) and reversible, it becomes an isentropic process.  In such a case, the entropy remains constant (ΔS=0).  These processes are important in analyzing ideal cycles like the Carnot cycle, Otto cycle, and Rankine cycle. 1.10 Heat and Work Transfer and It’s Sign convention Work In thermodynamics, work is defined in terms of its external effect. A system is said to perform positive work during a process if the sole external effect produced by the system can be reduced to the lifting of a weight against gravity. Consider a gas enclosed in a piston–cylinder arrangement. When the gas expands, the piston is pushed upward. In this arrangement, no mass is directly lifted against gravity. However, if the piston is connected to an external mechanism (such as pulleys and weights), the upward movement of the piston can result in the lifting of a mass. Thus, although the piston itself only moves, the external effect equivalent to lifting a weight allows us to define this expansion as work done by the system. Thermodynamics 23 Fig. 1.9 Expansion of actual lifting of mass. External Effects Only When evaluating thermodynamic work, only the effects external to the system boundaries are considered. Events or energy transfers occurring within the system itself are not counted as work. Fig. 1.10 Expansion without actual lifting of mass. Example: Suppose a lift (elevator) with a person and suitcase is considered as a thermodynamic system. If the person lifts the suitcase inside the lift, this event occurs within the system boundaries and therefore does not qualify as thermodynamic work. Thermodynamics 24 On the other hand, if the entire lift moves upward due to an external force, then that motion can be considered as work done on or by the system. Units of Work and Power In thermodynamics, work is the product of force and displacement in the direction of the force. In the SI system, the unit of force is the Newton (N), and the unit of distance is the metre (m). Hence, the unit of work becomes: Work = Newton-metre (N.m )  One Newton-metre is also known as one Joule (J). Since thermodynamic processes often involve large quantities of energy, work is frequently expressed in kilojoules (kJ), where: 1 kJ=1000 J The rate of doing work is called power.  Power is expressed as work per unit time: Power = Work Time  Its unit in SI is Joule per second (J/s), which is given the special name Watt (W). Thus, 1Watt=1 Joule second =1𝑁⋅𝑚 𝑠 For large-scale applications, power is also expressed in kilowatts (kW) or megawatts (MW). Sign Convention of Work In thermodynamics, a clear sign convention is followed to avoid confusion: 1. Work done by the system on the surroundings - Positive Example: Expansion of gas in a piston-cylinder does work on the piston. Thermodynamics 25 2. Work done on the system by the surroundings - Negative Example: Compression of gas by applying external pressure. This convention ensures consistency when applying the First Law of Thermodynamics. Fig. 1.11 Sign convention of work. Heat In thermodynamics, heat is defined as a mode of energy transfer between systems (or between a system and its surroundings) that occurs solely due to a temperature difference. When two bodies at different temperatures come in contact, energy in the form of heat flows from the body at a higher temperature to the body at a lower temperature until thermal equilibrium is reached. An important point to note is that:  Heat is not a property contained in a body.  Heat can only be recognized as it crosses the system boundary during an interaction.  Just like work, heat is a path function, meaning its value depends on the process path taken, not just the end states. Modes of Heat Transfer Heat can be transferred from one system to another, or between a system and its surroundings, by three distinct modes: conduction, convection, and radiation. 1. Conduction Conduction occurs without any bulk movement of matter. The transfer of energy takes place through molecular vibrations in solids and through the motion of free electrons in Thermodynamics 26 metals. It is the dominant mode of heat transfer in solids. Example: Heat transfer through the wall of a furnace or along a metal rod heated at one end. 2. Convection Convection occurs due to bulk movement of fluid particles and is therefore significant in liquids and gases. It combines the effects of conduction within the fluid and advection due to fluid motion. Types of Convection: 1. Forced Convection: When fluid motion is induced by an external device such as a fan, pump, or blower. 2. Natural Convection: When fluid motion arises naturally due to density differences caused by temperature variations. Example: Cooling of hot water in a vessel (natural convection), cooling of a car radiator using a fan (forced convection). 3. Radiation Radiation is the transfer of heat in the form of electromagnetic waves (primarily infrared radiation). Unlike conduction and convection, radiation does not require a medium and can occur even through a vacuum. Generally, it is a surface phenomenon, as surfaces emit and absorb radiant energy. However, in certain cases like gases containing water vapor or carbon dioxide, radiation can be a volumetric phenomenon. Example: Solar energy reaching the Earth, heat transfer between two plates separated by a vacuum. Sign Convention of Heat To maintain consistency in thermodynamic analysis, the following sign convention is used:  Heat transferred into the system → Positive  Heat transferred out of the system → Negative For example, if heat is supplied to a gas in a piston–cylinder device, it is considered positive. Conversely, if the gas rejects heat to the surroundings, it is negative. Thermodynamics 27 Fig. 1.12 Sign convention of heat. 1.11 Displacement work Consider a piston-cylinder arrangement containing a fluid as shown in Figure 1.13. If the pressure of the fluid inside the cylinder is greater than the pressure of the surroundings, there will be an unbalanced force acting on the piston. As a result, the piston will move outward. Fig. 1.13 Piston-cylinder arrangement containing a fluid. The force acting on the piston is given by: 𝐹=𝑃×𝐴 where,  𝑃= pressure of the fluid  𝐴= area of piston Thermodynamics 28 The work done by this force when the piston moves a distance 𝑑𝑥 is: 𝛿𝑊=𝐹×𝑑𝑥=(𝑃⋅𝐴)⋅𝑑𝑥 Since the change in volume of the system is 𝑑𝑉=𝐴⋅𝑑𝑥, 𝛿𝑊=𝑃𝑑𝑉 This expression is called displacement work or 𝑝𝑑𝑉 work. To obtain the total work done in a process, this incremental work is integrated from the initial to the final volume: 𝑊=∫ 𝑉2 𝑉1𝑃𝑑𝑉 Thus, the area under the process curve in a P-V diagram represents the displacement work. Evaluation of Displacement Work Constant Pressure Process Consider a piston-cylinder arrangement (Figure 1.14) in which the fluid expands while the pressure remains constant throughout the process. Fig. 1.14 Piston-cylinder arrangement. Thermodynamics 35 Fig. 1.19 Temperature Scales. a) Celsius Scale ( ∘𝐂 ) This is also called the centigrade scale. Defines the freezing point of water as 0∘C and the boiling point of water as 100∘C at 1 atm pressure. It is a relative scale, widely used in daily life, laboratories, and engineering practice. Relationship with Kelvin scale: 𝑇(𝐾)=𝑇( ∘𝐶)+273.15 b) Fahrenheit Scale ( ∘𝐅 ) Commonly used in the United States and some other countries. Defines the freezing point of water as 32∘F and the boiling point as 212∘F at 1 atm pressure. Thermodynamics 36 It divides the temperature difference between ice and steam points into 180 equal divisions. Conversion formulas: 𝑇( ∘𝐹) =9 5𝑇( ∘𝐶)+32 𝑇( ∘𝐶) =5 9(𝑇( ∘𝐹)−32) c) Kelvin Scale (K) The absolute thermodynamic scale of temperature. Absolute zero (0 K) is the lowest possible temperature, corresponding to −273.15∘C, where molecular motion theoretically ceases. The freezing point of water =𝟐𝟕𝟑.𝟏𝟓 𝐊, and the boiling point =𝟑𝟕𝟑.𝟏𝟓 𝐊. The Kelvin scale is the SI unit of temperature and is essential for all scientific and thermodynamic calculations. Relation with Celsius: 𝑇(𝐾)=𝑇( ∘𝐶)+273.15 d) Rankine Scale ( ∘𝐑 ) The absolute temperature scale used in countries where the Fahrenheit system is preferred. Zero Rankine (0∘R) corresponds to absolute zero. Freezing point of water =491.67∘R; Boiling point =671.67∘R. Conversion formulas: 𝑇( ∘𝑅)=𝑇( ∘𝐹)+459.67 𝑇( ∘𝑅)=1.8𝑇(𝐾) 1.14 First Law of Thermodynamics and Application to closed and open systems The First Law of Thermodynamics is a statement of the principle of conservation of energy applied to thermodynamic systems. It asserts that energy can neither be created nor destroyed; it can only be transferred from one system to another or transformed from one form to another. In its most general form, the First Law is expressed as: “When energy is transferred or transformed, the total energy of an isolated system remains constant. The final total energy in all forms must be equal to the original total energy.” Thermodynamics 37 First Law of Thermodynamics for a Closed System Undergoing a Process The First Law of Thermodynamics expresses the principle of conservation of energy. For a closed system of constant mass, energy can cross the system boundary only in two forms: heat (Q) and work (W). General Formulation The First Law can be expressed as: Energy Entered into the System - Energy Left the System = Change in the Energy Content of the System If a closed system initially has an energy content of 𝐸1, receives a net heat input 𝑄, and performs work 𝑊, the final energy content of the system becomes 𝐸2. Thus, 𝑄−𝑊=(𝐸2−𝐸1) Fig. 1.20 First law for closed system. Total Energy Content of a System The energy content of a system (𝐸) can be expressed as the sum of three major components: 𝐸=𝑈+ Kinetic Energy + Potential Energy 𝐸=𝑈+1 2𝑚𝐶2+𝑚𝑔𝑧 Where,  𝑈 : Internal energy of the system  1 2𝑚𝐶2 : Kinetic energy due to system velocity 𝐶 Thermodynamics 38  mgz: Potential energy due to elevation 𝑧 in a gravitational field Hence, 𝑄−𝑊=Δ𝑈+Δ𝐾𝐸+Δ𝑃𝐸 Internal Energy (U) The internal energy represents the microscopic energy stored within the system, arising from:  Molecular motion (translational, rotational, vibrational)  Intermolecular forces  Electron spin and vibrations  Chemical bonds Internal energy is denoted by 𝐔, and for many thermodynamic analyses, it plays the most significant role. Kinetic Energy Kinetic energy accounts for the motion of the system as a whole with velocity 𝐶. 𝐾𝐸=1 2𝑚𝐶2 For stationary systems, this term is zero. In thermodynamics, unless specifically considered (e.g., in nozzles or turbines), the kinetic energy term is often neglected. Potential Energy Potential energy arises due to the position of the system in a gravitational field. 𝑃𝐸=𝑚𝑔𝑧 For systems that remain at the same elevation, this term remains constant. In practical thermodynamic systems, unless there is significant elevation change (e.g., hydro turbines, water pumps), potential energy is often neglected. Units The SI unit of energy is the Joule (J). For large-scale systems, energy is often expressed in kilojoules (kJ). Thermodynamics 39 The Thermodynamic Property: Enthalpy Consider a stationary system of fixed mass undergoing a quasi-equilibrium process at constant pressure. Applying the First Law of Thermodynamics: 𝑄1−2−𝑊1−2=𝐸2−𝐸1 For a general system: 𝐸2−𝐸1=(𝑈2−𝑈1)+𝑚(𝐶22−𝐶12 2)+𝑚𝑔(𝑍2−𝑍1) Since the system is stationary, changes in kinetic energy and potential energy are negligible: 𝐸2−𝐸1=𝑈2−𝑈1 Work Term at Constant Pressure The displacement work done is: 𝑊1−2=𝑃(𝑉2−𝑉1)=𝑃2𝑉2−𝑃1𝑉1 Substituting in the First Law: 𝑄1−2=(𝑃2𝑉2−𝑃1𝑉1)+(𝑈2−𝑈1) Rearranging, 𝑄1−2=(𝑈2+𝑃2𝑉2)−(𝑈1+𝑃1𝑉1) Definition of Enthalpy The terms within the brackets depend only on the end states of the system. This combination of properties can be grouped into a single thermodynamic property, known as enthalpy. 𝐻=𝑈+𝑃𝑉 For specific properties (per unit mass): ℎ=𝑢+𝑝𝑣 where,  𝐻= enthalpy (kJ) Thermodynamics 40  ℎ= specific enthalpy ( kJ/kg )  𝑈= internal energy (kJ)  𝑢= specific internal energy ( kJ/kg )  𝑉= volume (m3)  𝑣= specific volume (m3/kg) Significance of Enthalpy  Enthalpy is a convenient property when analyzing processes at constant pressure.  It combines internal energy ( U ) and the flow energy ( PV ) required to push fluid across a system boundary.  Many practical devices such as boilers, turbines, condensers, and pumps are analyzed in terms of enthalpy changes. Flow Energy Flow energy is defined as the energy required to move a mass into or out of a control volume against a pressure. It represents the work needed to push fluid across the boundary of a control volume. Consider a mass of volume 𝑉 entering a control volume against a pressure 𝑝. To push this volume across the boundary, work must be done. Fig. 1.21 Flow energy. Thermodynamics 41 The flow energy can be derived as follows: Flow Energy = Work done in moving the mass = Force × Distance =(𝑝⋅𝐴)⋅𝑑𝑥 =𝑝⋅(𝐴⋅𝑑𝑥) =𝑝𝑉 where,  𝑝= pressure  𝐴= cross-sectional area  𝑑𝑥= displacement  𝑉=𝐴⋅𝑑𝑥= volume of the mass Thus, the flow energy per unit mass is: 𝑝𝑉 𝑚=𝑝𝑣 where 𝑣= specific volume. Relation to Enthalpy Since enthalpy is defined as the sum of internal energy and flow energy, we can write: 𝐻=𝑈+𝑝𝑉 or in terms of specific properties, ℎ=𝑢+𝑝𝑣 where,  𝐻= Enthalpy (kJ)  𝑈= Internal energy (kJ)  𝑝𝑉= Flow energy (kJ)  ℎ= Specific enthalpy (kJ/kg)  𝑢= Specific internal energy (kJ/kg)  𝑝𝑣= Flow energy per unit mass (kJ/kg) Thermodynamics 42 Significance of Flow Energy  Flow energy arises only in open systems (control volumes) where mass crosses the system boundary.  It represents the work required to push fluid into or out of a system.  In closed systems, no flow energy exists because there is no mass transfer.  Flow energy, combined with internal energy, gives enthalpy, which is extensively used in analyzing steady-flow devices such as turbines, compressors, nozzles, pumps, boilers, and condensers. First Law of Thermodynamics for a Control Volume (Open Systems) In many engineering applications, mass continuously enters and leaves a system. Such systems are best analyzed using the concept of a control volume bounded by a control surface. The First Law of Thermodynamics for a control volume combines the principles of conservation of mass and conservation of energy to account for all forms of energy transfer across the control surface. Fig. 1.22 First law for open system. Conservation of Energy Energy may cross the control surface in two forms: 1. Heat transfer (Q) Thermodynamics 43 2. Work transfer (W) Additionally, each unit of mass that enters or leaves the control volume carries energy in the form of:  Internal energy (𝑢)  Flow energy (pv)  Kinetic energy (𝐶2 2)  Potential energy (gz) Thus, the general energy balance for a control volume is: Net energy crossing boundary as heat and work + Energy carried in by mass flow - Energy carried out by mass flow = Net change in energy content of the control volume Considering a control volume with mass flow across its boundary (see Figure), the energy balance becomes: 𝑄˙−𝑊 ˙+∑ in 𝑚˙in (ℎ+𝐶2 2+𝑔𝑧)−∑ out 𝑚˙out (ℎ+𝐶2 2+𝑔𝑧)=𝑑𝐸𝐶𝑉 𝑑𝑡 where,  𝑄˙= Rate of heat transfer to the control volume  𝑊 ˙= Rate of work transfer from the control volume  𝑚˙in ,𝑚˙out = Mass flow rates at inlet and outlet  ℎ=𝑢+𝑝𝑣= Enthalpy (includes internal + flow energy)  𝐶2 2= Kinetic energy per unit mass  𝑔𝑧= Potential energy per unit mass  𝑑𝐸𝐶𝑉 𝑑𝑡 = Rate of change of total energy within the control volume 1.15 The Steady-State Flow Process In engineering applications such as turbines, compressors, nozzles, boilers, and condensers, mass and energy continuously flow into and out of the system. When certain conditions are satisfied, the flow process is referred to as a steady-state flow process. Thermodynamics 44 Conditions for Steady-State Flow A process is called a steady flow process if: 1. The mass and energy content of the control volume remain constant with time. 2. The state and energy of the fluid at the inlet, exit, and every point inside the control volume are time-independent. 3. The rate of energy transfer in the form of heat and work across the control surface remains constant with time. These conditions ensure that although mass and energy are crossing the system boundary, the properties inside the control volume do not change with time. Applications of the Steady Flow Energy Equation (SFEE) The Steady Flow Energy Equation (SFEE) governs the working of many thermodynamic devices used in engineering applications. These devices operate under steady-flow conditions, where mass and energy continuously cross the system boundary but conditions inside remain constant with time. Below is a summary of the working principles and governing equations for key components: 1. Turbines Turbines are devices used in steam, gas, and hydraulic power plants to produce work. As the fluid expands through the turbine, it transfers energy to the blades, which rotate a shaft and generate mechanical work. Fig. 1.23 Turbines. Thermodynamics 51 For a constant pressure process: 𝑑𝑝=0 ⇒ 𝛿𝑄=𝑑𝐻 Thus, 𝑑ℎ=𝐶𝑝𝑑𝑇 Hence, 𝐶𝑝 relates the change in enthalpy to the change in temperature. Important Relations 1. Difference between specific heats: 𝐶𝑝−𝐶𝑣=𝑅 where 𝑅= gas constant. 2. Ratio of specific heats (adiabatic index): 𝛾=𝐶𝑝 𝐶𝑣 3. Since both enthalpy (h) and internal energy (u) are properties: 𝑑ℎ=𝐶𝑝𝑑𝑇,𝑑𝑢=𝐶𝑣𝑑𝑇 (valid for all processes). Work Interaction in a Reversible Steady Flow Process In a steady flow process, the work interaction per unit mass between an open system (control volume) and its surroundings can be expressed using the first law of thermodynamics in differential form. From the energy balance for a steady flow process: 𝛿𝑞−𝛿𝑤=𝑑ℎ+𝐶𝑑𝐶+𝑔𝑑𝑧 Rearranging for work: 𝛿𝑤=𝛿𝑞−(𝑑ℎ+𝐶𝑑𝐶+𝑔𝑑𝑧) Also, we know: 𝛿𝑞=𝑑𝑢+𝑝𝑑𝑣 (or) 𝛿𝑞=𝑑ℎ−𝑣𝑑𝑝 Thermodynamics 52 Substituting, 𝛿𝑤=𝑑ℎ−𝑣𝑑𝑝−(𝑑ℎ+𝐶𝑑𝐶+𝑔𝑑𝑧) 𝛿𝑤=−𝑣𝑑𝑝−(𝐶𝑑𝐶+𝑔𝑑𝑧) Work Done in a Process Integrating between states 1 and 2: 𝑊=−∫ 2 1𝑣𝑑𝑝−(𝐶22−𝐶12) 2−𝑔(𝑧2−𝑧1) Special Case: Stationary System For a stationary system (no change in velocity or elevation): 𝑊=−∫ 2 1𝑣𝑑𝑝 1. The work interaction in a steady flow process depends on pressure-volume changes as well as changes in kinetic and potential energy. 2. For a stationary control volume with negligible kinetic and potential energy variations, the work reduces simply to: 𝑊=−∫ 𝑣𝑑𝑝 which represents the flow work contribution. First Law for an Open System under Unsteady Flow Conditions In many practical engineering applications, the flow of mass and energy into and out of a system is unsteady, meaning the amount of energy and mass within the control volume varies with time. Unlike steady flow systems, in unsteady flow systems, the properties of the system are not constant over time. Examples of Unsteady Flow Processes 1. Filling of closed tanks with gas or liquid. 2. Discharge of fluid from closed vessels. 3. Fluid flow in reciprocating machines (e.g., compressors, pumps) during each cycle. Thermodynamics 53 Assumptions for Analysis To develop a mathematical model for such processes, the following assumptions are made: 1. The control volume remains fixed relative to the coordinate system. 2. The state of the mass within the control volume may change with time, but at any instant, the state is uniform throughout the control volume. 3. The state of the mass crossing each inlet and outlet section of the control surface is uniform with respect to time, even though the mass flow rates may vary with time. Mass Balance for Unsteady Flow Let the mass of fluid inside the control volume at the beginning and end of a time interval Δ𝑡 be 𝑚1 and 𝑚2, respectively. Applying conservation of mass: (𝑚2−𝑚1)𝐶𝑉=Σ𝑚in −Σ𝑚out Where:  Σ𝑚in = mass entering the control volume during Δ𝑡.  Σ𝑚out = mass leaving the control volume during Δ𝑡. Energy Balance for Unsteady Flow Applying the first law of thermodynamics to the control volume: [𝑄𝐶𝑉−𝑊𝐶𝑉]+∑ 𝑖𝑛 𝑚𝑖𝑛[ℎ+𝐶2 2+𝑍𝑔]−∑ 𝑜𝑢𝑡 𝑚𝑜𝑢𝑡[ℎ+𝐶2 2+𝑍𝑔]=Δ𝐸𝐶𝑉 Where:  Δ𝐸𝐶𝑉= Change in energy of the control volume during Δ𝑡.  𝑄𝐶𝑉= Heat energy transferred into the control volume.  𝑊𝐶𝑉= Work done by the control volume.  ℎ= Specific enthalpy of inlet/outlet streams.  𝐶2 2= Specific kinetic energy of inlet/outlet streams.  𝑍𝑔= Specific potential energy of inlet/outlet streams. Thermodynamics 54 Unlike steady flow, in unsteady flow systems the energy and mass within the control volume vary with time. The governing equation considers both transient accumulation (or depletion) of energy and the net transfer of energy associated with mass crossing the boundaries. Such analysis is vital for studying charging and discharging of tanks, start-up and shutdown of turbines/compressors, and reciprocating machines. Thermodynamics 55 CHAPTER -2 SECOND LAW AND AVAILABILITY ANALYSIS 2.1 Heat Reservoir, source and sink A thermal energy reservoir is a hypothetical concept used in thermodynamics to represent a body with a very large heat capacity. Because of this, its temperature remains essentially constant even when a finite amount of heat is added or removed. The heat capacity (𝐶) of any material is defined as: 𝐶=𝑚⋅𝑐 Where,  𝑚= mass of the body  𝑐= specific heat of the material From the first law of thermodynamics: 𝑄=𝑚𝑐Δ𝑇=𝐶Δ𝑇 For a reservoir, since 𝐶→∞, any finite heat transfer 𝑄 results in Δ𝑇≈0. Hence, the temperature of the reservoir is considered constant. Key Note: A thermal energy reservoir can interact with other systems by supplying or absorbing heat indefinitely, without experiencing a measurable change in temperature. Examples of Thermal Energy Reservoirs 1. Natural reservoirs: Atmosphere, oceans, large rivers, lakes, geothermal bodies, and the Sun. 2. Engineered reservoirs: Large furnaces, industrial cooling ponds, power plant condensers. 3. Phase-change systems: Melting ice, boiling water, or any material undergoing a phase change at constant temperature. These act as reservoirs because they absorb or reject large amounts of heat (latent heat) without temperature variation. Thermodynamics 56 5. Everyday cases: Even smaller systems like the air in a room absorbing CPU heat can be treated as a reservoir when its temperature change is negligible compared to the process under study. Why Does the Reservoir’s Temperature Not Vary? In reality, any body will experience some temperature change when heat transfer occurs. However, in thermodynamics, a reservoir is treated as an idealized concept. This assumption simplifies analysis by allowing systems to exchange heat with a constant-temperature body. 1. For real systems, slight variations occur. 2. For ideal analysis, we assume ΔT=0, ensuring accurate modeling of reversible cycles like the Carnot cycle. Heat Source and Heat Sink  A heat source is a high-temperature reservoir that supplies energy in the form of heat.  A heat sink is a low-temperature reservoir that absorbs the rejected heat. Note: Both source and sink are essential for the functioning of thermodynamic devices. Heat transfer always occurs from a source at higher temperature to a sink at lower temperature. Applications in Thermodynamic Systems 1. Heat Engines (e.g., steam engines, gas turbines, internal combustion engines):  Source: Boiler or combustion chamber  Sink: Atmosphere or condenser 2. Refrigerators & Heat Pumps:  Source: Cold space or refrigerated chamber (low-temperature reservoir)  Sink: Surroundings or atmosphere (high-temperature reservoir) 3. Power Plants:  Source: Steam produced by burning coal, oil, or nuclear fuel  Sink: Cooling water from rivers, lakes, or cooling towers Without both a source and a sink, no cyclic thermodynamic device can operate. If only one reservoir existed, it would violate the Second Law of Thermodynamics, which requires heat to flow from a high-temperature body to a low-temperature body to produce work. Thermodynamics 57 2.2 Heat Engine A heat engine is a device that operates on a thermodynamic cycle and produces useful work by transferring heat energy from a high-temperature reservoir to a low-temperature reservoir. In this process, part of the heat supplied from the hot reservoir is converted into work, while the remaining portion is rejected to the cold reservoir. Examples of practical heat engines include internal combustion (I.C.) engines, steam engines, gas turbines, and boilers. Despite differences in construction and operation, all heat engines work on the same fundamental principle: heat flows from a source (hot reservoir) to a sink (cold reservoir), and a fraction of this energy is converted into mechanical work. General Representation of a Heat Engine A heat engine typically interacts with three elements: 1. Hot reservoir (source) at temperature 𝑇1, which supplies heat 𝑄1. 2. Working fluid or system, which converts part of this heat into work output 𝑊. 3. Cold reservoir (sink) at temperature 𝑇2, which absorbs the rejected heat 𝑄2. The schematic diagram of a general heat engine is shown in Figure 2.1. Fig. 2.1 Heat Engine. Thermodynamics 58 Work and Efficiency of a Heat Engine The performance of any heat engine is measured in terms of its thermal efficiency (𝜂), defined as the ratio of net work output to the heat input supplied to the engine: 𝜂= Net work output Total heat input =𝑊 𝑄1(2.1) where,  𝑊= Net work produced by the engine  𝑄1= Heat supplied from the hot reservoir at temperature 𝑇1 From the energy balance of the cycle: 𝑊=𝑄1−𝑄2(2.2) where,  𝑄2= Heat rejected to the cold reservoir at temperature 𝑇2 Substituting equation (2.2) into (2.1), the efficiency becomes: 𝜂=𝑄1−𝑄2 𝑄1=1−𝑄2 𝑄1(2.3) Efficiency in Terms of Reservoir Temperatures For a reversible heat engine (such as the Carnot engine), the heat transfer ratios are directly related to the absolute temperatures of the reservoirs: 𝑄2 𝑄1=𝑇2 𝑇1 Hence, the efficiency can be expressed as: 𝜂=1−𝑇2 𝑇1(2.4) where,  𝑇1= Temperature of hot reservoir  𝑇2= Temperature of cold reservoir This shows that the efficiency of a heat engine depends only on the temperatures of the two reservoirs. Thermodynamics 59 Observations: 1. Efficiency is always less than unity (η<1): This means no engine can convert the entire heat input into work. A portion of the energy must always be rejected to a sink, in accordance with the Second Law of Thermodynamics. 2. Higher T1 and lower T2 improve efficiency: Increasing the source temperature or decreasing the sink temperature increases efficiency. This is why power plants aim to achieve high boiler temperatures and use efficient cooling systems for condensers. 3. Absolute temperatures must be used: The temperatures T1 and T2 in the efficiency expression are always measured in Kelvin. Significance of Heat Engine Analysis Practical insight: The analysis of heat engines highlights the inevitable losses in real systems and the impossibility of achieving 100% efficiency. Design objective: Engineers aim to design systems that operate as close as possible to the Carnot efficiency, which serves as the theoretical upper limit. Applications: Heat engine principles are applied in automotive engines, aircraft propulsion systems, steam and gas power plants, and other energy conversion devices. 2.3 Refrigerator and Its Performance A refrigerator is a thermodynamic device that operates on a cyclic process to remove heat from a low-temperature region and reject it to a high-temperature region. Its main purpose is to maintain the temperature of a space or body lower than that of the surroundings. In simple terms, a refrigerator extracts heat from a cold reservoir (the refrigerated space) and rejects it to the warmer surroundings, with the aid of external work input. Examples: Air conditioners, coolers, domestic refrigerators, and freezers. Working Principle As illustrated in Figure 2.2, the refrigerator operates between two thermal reservoirs: Thermodynamics 60 1. Low-temperature reservoir (the refrigerated space), from which heat 𝑄4 is absorbed. 2. High-temperature reservoir (surroundings/atmosphere), to which heat 𝑄3 is rejected. An external work input 𝑊 is supplied to maintain the continuous cycle of operation. Fig. 2.2 Refrigerator. Coefficient of Performance (COP) of a Refrigerator Unlike a heat engine, where the measure of performance is thermal efficiency, the performance of a refrigerator is expressed in terms of the Coefficient of Performance (COP). The COP of a refrigerator is defined as the ratio of the heat extracted from the lowtemperature reservoir to the net work input required: COPref = Heat extracted from cold reservoir Work input COPref =𝑄4 𝑊=𝑄4 𝑄3−𝑄4=𝑇4 𝑇3−𝑇4 Thermodynamics 67 From the above Figure 2.4: 1. The impossible engine: A device operating with a single reservoir, delivering all the absorbed heat as work ( 𝑊=𝑄1 ), violates the Kelvin-Planck statement. 2. The possible engine: A real engine absorbs heat 𝑄1 from a hot reservoir, converts part into work ( 𝑊=𝑄1−𝑄2), and rejects the remainder 𝑄2 to a cold reservoir, thereby complying with the second law. 2. Clausius Statement The Clausius statement may be expressed as: “It is impossible to construct a device operating in a cycle that produces no other effect than the transfer of heat from a colder body to a hotter body.” In simple terms, heat cannot flow spontaneously from a cold reservoir to a hot reservoir without external assistance. However, heat can naturally flow in the opposite direction from hot to cold without any aid. Implications:  Refrigerators and heat pumps, which transfer heat from low to high temperatures, require external work input to function.  A spontaneous transfer of heat from cold to hot would violate the Clausius statement. Fig. 2.5 Clausius Statement. Thermodynamics 68 From the above Figure 2.5: 1. Possible system: Heat transfer from hot to cold reservoir without external work is natural and allowed. 2. Impossible system: Direct heat transfer from cold to hot reservoir without external aid is not possible. 3. Possible system with work input: Heat can be made to flow from cold to hot reservoir when assisted by external work, as in refrigerators or heat pumps. Equivalence of the Two Statements Although the Kelvin–Planck and Clausius statements appear different, they are equivalent expressions of the second law:  A violation of the Kelvin–Planck statement would imply the construction of a 100% efficient heat engine, which could then be used to drive a refrigerator without work input, violating the Clausius statement.  Similarly, if the Clausius statement were violated, a heat pump could operate without work input, leading to a heat engine with complete conversion of heat into work, violating the Kelvin–Planck statement. Thus, both statements reinforce the impossibility of perpetual motion machines of the second kind (PMM-II). Significance of the Second Law  It defines the direction of energy transfer processes (from hot to cold).  It sets a limit on efficiency for cyclic devices such as engines, refrigerators, and heat pumps.  It emphasizes that work is a higher-grade energy form than heat, as heat cannot be completely converted into work. Corollaries of the Second Law of Thermodynamics The Second Law of Thermodynamics establishes the direction and limitations of energy interactions. Several important consequences, known as corollaries, can be derived from the second law with the help of reversible cycles. These corollaries provide deeper insights into the performance of heat engines, refrigerators, and heat pumps. Thermodynamics 69 Corollary 1: Impossibility of Heat Transfer from Cold to Hot without Work Input It is impossible to construct a system that operates in a cycle and transfers heat from a cold body to a hot body without any external work input. Proof: Fig. 2.6 Heat pump operating with work input 𝑾=𝟎. If this were possible, the system would act like a heat pump operating with work input 𝑊= 0, as shown in Figure 2.6. If it absorbs 𝑄4 units of heat from the cold reservoir, it must deliver the same amount, 𝑄3=𝑄4, to the hot reservoir in order to satisfy the first law. Now consider a heat engine operating between the same two reservoirs. The engine receives 𝑄1 units of heat from the hot reservoir, produces work 𝑊, and rejects 𝑄2 to the cold reservoir. If the rejected heat 𝑄2 is supplied to the heat pump as input, the cold reservoir becomes unnecessary. The combined system would then extract (𝑄1) heat from the hot reservoir and convert it entirely into work 𝑊, which is a violation of the Kelvin-Planck statement. Thermodynamics 70 Hence, this proves that heat transfer from a cold to hot reservoir is impossible without work input, and Corollary 1 is true. Corollary 2: Efficiency of an Irreversible Engine No heat engine operating between two given heat reservoirs can be more efficient than a reversible engine working between the same temperature limits. This corollary emphasizes that reversible cycles, such as the Carnot cycle, represent the upper limit of efficiency for any engine. Corollary 3: Efficiency of Reversible Engines All reversible engines operating between the same two heat reservoirs have the same efficiency, regardless of the working substance or cycle details. This means the efficiency depends only on the temperatures of the reservoirs and not on the type of system or medium. Corollary 4: Engines Operating Between Multiple Reservoirs The efficiency of a reversible engine operating between more than two reservoirs is always less than the efficiency of a reversible engine operating between just two reservoirs at the highest and lowest temperatures of the working fluid. This stresses the importance of considering extreme temperature limits when evaluating performance. Corollary 5: Cyclic Integral of Heat-to-Temperature Ratio Whenever a system undergoes a thermodynamic cycle: ∮ 𝛿𝑄 𝑇≤0  For a reversible cycle, ∮ 𝛿𝑄 𝑇=0  For an irreversible cycle, ∮ 𝛿𝑄 𝑇<0 This forms the basis of the Clausius inequality, which is fundamental to entropy analysis. Thermodynamics 71 Corollary 6: Entropy of a Thermally Isolated System The entropy of any closed system that is thermally isolated from its surroundings remains constant. If the process is reversible, the entropy remains unchanged; however, if the process is irreversible, the entropy increases. This is the foundation of the principle of entropy increase. Perpetual Motion Machine of the Second Kind (PMM-II) A PMM-II is a hypothetical machine that receives heat energy from a single hot reservoir and converts it entirely into an equivalent amount of work, i.e., it would operate with 100% efficiency.  Such a machine is impossible to construct because it directly violates the second law of thermodynamics.  In other words, no cyclic device can convert all the heat supplied into work without rejecting some heat to a sink. The concept of PMM-II clearly illustrates the fundamental limitations of real energy conversion systems. Significance of Corollaries  They help establish the limits of performance for real thermodynamic devices.  They form the basis for understanding entropy, irreversibility, and efficiency.  They confirm the impossibility of perpetual motion machines.  They provide practical guidelines in designing engines, refrigerators, and heat pumps. 2.6 Carnot Cycle and Its Performance The Carnot cycle, introduced by Sadi Carnot, is a theoretical cycle that serves as the ideal standard of performance for all heat engines. It is also known as the constant temperature cycle, as two of its processes occur at constant temperature. The Carnot cycle consists of four reversible processes: 1. Two isothermal (constant temperature) processes 2. Two isentropic (reversible adiabatic) processes Since all four processes are reversible, the Carnot cycle is itself a reversible cycle. The p–V and T–s diagrams for the Carnot cycle are shown in Figure 2.7. Thermodynamics 72 Fig. 2.7 Carnot cycle. Processes of the Carnot Cycle Process 1-2: Isentropic Compression The working fluid (air or gas) is compressed isentropically from state 1 to state 2. During this process, pressure and temperature increase from 𝑝1 to 𝑝2 and 𝑇1 to 𝑇2, while the volume decreases from 𝑉1 to 𝑉2. Since the process is isentropic, no heat is transferred, and the entropy remains constant ( 𝑠1= 𝑠2 ). Process 2-3: Isothermal Heat Addition The working fluid absorbs heat 𝑄1 at a constant high temperature 𝑇2=𝑇3. During this process, both volume and entropy increase (from 𝑉2 to 𝑉3, and 𝑠2 to 𝑠3 ), while pressure decreases from 𝑝2 to 𝑝3. Heat supplied: 𝑄1=𝑇3(𝑠3−𝑠2) Process 3-4: Isentropic Expansion The fluid expands isentropically from state 3 to state 4. Both pressure and temperature decrease from 𝑝3 to 𝑝4, and 𝑇3 to 𝑇4. Since the process is adiabatic and reversible, entropy remains constant ( 𝑠3=𝑠4 ). During this step, the system performs useful work on the surroundings. Thermodynamics 73 Process 4-1: Isothermal Heat Rejection Heat is rejected at the constant low temperature 𝑇1=𝑇4. Both volume and entropy decrease from 𝑉4 to 𝑉1, and 𝑠4 to 𝑠1, while pressure also decreases. Heat rejected: 𝑄2=𝑇1(𝑠4−𝑠1) The system then returns to its initial state, completing one full thermodynamic cycle. Work and Efficiency of Carnot Cycle The net work done during the cycle is the difference between the heat absorbed and the heat rejected: 𝑊=𝑄1−𝑄2 The thermal efficiency of the Carnot engine is: 𝜂=𝑊 𝑄1=1−𝑄2 𝑄1 Since heat transfers in reversible isothermal processes are directly proportional to absolute temperatures: 𝑄2 𝑄1=𝑇1 𝑇2 Therefore, 𝜂Carnot =1−𝑇1 𝑇2 where:  𝑇1= Temperature of cold reservoir (minimum temperature)  𝑇2= Temperature of hot reservoir (maximum temperature) Key Features of the Carnot Cycle 1. Reversibility: Since all processes are reversible, the Carnot cycle represents the upper limit of performance for any heat engine. 2. Maximum Efficiency: The efficiency depends only on the temperature limits of the reservoirs, not on the working fluid or system design. Thermodynamics 74 3. Ideal Standard: No real engine can exceed the Carnot efficiency; at best, real engines can only approach it. 4. Benchmarking: The Carnot cycle provides a reference to evaluate actual cycles like Otto, Diesel, Rankine, and Brayton cycles. Significance of Carnot Cycle Establishes the theoretical upper bound on efficiency for all heat engines. Demonstrates the importance of temperature difference between reservoirs: higher T2 or lower T1 improves efficiency. Forms the basis of the Carnot Theorem, which states: No engine operating between two reservoirs can be more efficient than a Carnot engine operating between the same reservoirs. 2.7 Reversed Carnot Cycle and Its Performance Since all the processes in the Carnot cycle are reversible, the cycle can be operated in reverse. When reversed, it is called the Reversed Carnot Cycle, which forms the theoretical basis for refrigerators and heat pumps.  In refrigeration mode, the cycle extracts heat from a low-temperature body (cold reservoir) and rejects it to a high-temperature body (surroundings).  In heat pump mode, the cycle extracts heat from the surroundings (cold reservoir) and delivers it to a warmer space (hot reservoir) to maintain comfortable indoor conditions. The cycle consists of two isothermal processes and two isentropic processes, and its p–V and T–s diagrams are shown in Figure 2.8. Processes of the Reversed Carnot Cycle Process 1-2: Isentropic Compression (Compressor) The working fluid is compressed isentropically from state 1 to state 2. Pressure and temperature both increase, while entropy remains constant ( 𝑠1=𝑠2 ). Work input is required for this step. Process 2-3: Isothermal Heat Rejection (Condenser) At constant high temperature 𝑇3, heat is rejected to the hot reservoir (atmosphere). Pressure remains constant while entropy decreases from 𝑠2 to 𝑠3. Thermodynamics 75 Heat rejected: 𝑄3=𝑇3(𝑠2−𝑠3) Fig. 2.8 Reversed carnot cycle. Process 3-4: Isentropic Expansion (Expansion Valve/Device) The working fluid expands isentropically from state 3 to state 4. Pressure and temperature decrease, while entropy remains constant ( 𝑠3=𝑠4 ). Process 4-1: Isothermal Heat Absorption (Evaporator) At constant low temperature 𝑇4, heat is absorbed from the cold reservoir. Entropy increases from 𝑠4 to 𝑠1. Heat absorbed (useful effect for refrigeration): 𝑄4=𝑇4(𝑠1−𝑠4) The fluid then returns to its initial state, completing one cycle. Work and COP of the Reversed Carnot Cycle The net work input is the difference between heat rejected and heat absorbed: 𝑊=𝑄3−𝑄4=(𝑇3−𝑇4)(𝑠1−𝑠4) Thermodynamics 76 Coefficient of Performance (COP) for Refrigerator The performance of a refrigerator is expressed by COP, defined as: COPRef = Heat absorbed from cold body Work input COPRef =𝑄4 𝑊=𝑇4 𝑇3−𝑇4 where:  𝑇3= High temperature (hot reservoir)  𝑇4= Low temperature (cold reservoir) Thus, COP of a refrigerator depends only on the temperature limits. Coefficient of Performance (COP) for Heat Pump When operated as a heat pump, the useful effect is the heat rejected to the high-temperature reservoir: COP𝐻𝑃= Heat delivered to hot body Work input COP𝐻𝑃 =𝑄3 𝑊=𝑇3 𝑇3−𝑇4 Relation between COP of Refrigerator and Heat Pump For the same temperature limits, COP𝐻𝑃=COP𝑅𝑒𝑓+1 This shows that a heat pump always has a COP greater than that of a refrigerator operating between the same reservoirs. Significance of Reversed Carnot Cycle 1. Provides the ideal standard of performance for all refrigerators and heat pumps. 2. Demonstrates that COP depends only on the temperature limits (𝑇3,𝑇4) and not on the working fluid. 3. Shows that higher COP is obtained when the temperature difference between reservoirs is smaller. Thermodynamics 83  In Figure 2.10 (b), a process between states 1 and 2 is shown. The shaded area under the curve 1→2 corresponds to the heat transferred during the process: 𝑄1−2, rev =Area(1−2−𝑠2−𝑠1−1) Thus, the T-s diagram provides a direct and convenient means of measuring and visualizing heat transfer in reversible processes. Special Features 1. Isentropic Processes: Shown as vertical lines (s=constants). 2. Isothermal Processes: Shown as horizontal lines (T=constant). 3. Reversible Heat Transfer: Area under the curve directly gives the heat transferred. 4. Irreversible Processes: Deviations from ideal vertical/horizontal lines highlight losses and entropy generation. Applications of T–s Diagram 1. Heat Transfer Evaluation:  Since heat transfer equals the area under the curve, T–s diagrams allow engineers to evaluate heat interactions without solving complex equations. 2. Cycle Representation:  Thermodynamic cycles (such as Carnot, Rankine, Otto, Diesel, and Brayton cycles) are often represented on T–s diagrams.  The enclosed area in a cycle on a T–s diagram corresponds to the net work output of the cycle. 3. Entropy Change Visualization:  Entropy changes are represented as horizontal displacements along the sss-axis.  Isentropic processes appear as vertical lines (constant entropy).  Isothermal processes appear as horizontal lines (constant temperature). 4. Performance Analysis:  In power plants and refrigeration systems, T–s diagrams help in identifying irreversibilities, energy losses, and efficiency improvements. Thermodynamics 84 2.11 Entropy Relations: Tds Equations Entropy (s) is a thermodynamic property that can be expressed as a function of different sets of independent variables such as temperature (T), pressure (p), and specific volume (v). The relationships between entropy and these variables lead to the well-known Tds equations, which are fundamental in thermodynamics. These equations are derived using the first and second laws of thermodynamics, along with Maxwell relations. Entropy as a Function of 𝑇 and 𝑝 Consider entropy expressed as: 𝑠=𝑓(𝑇,𝑝) The total differential is: 𝑑𝑠=(𝜕𝑠 𝜕𝑇)𝑝𝑑𝑇+(𝜕𝑠 𝜕𝑝)𝑇𝑑𝑝 From the definition of specific heat at constant pressure: 𝐶𝑝=𝑇(𝜕𝑠 𝜕𝑇)𝑝⇒(𝜕𝑠 𝜕𝑇)𝑝=𝐶𝑝 𝑇 From Maxwell's relation: (𝜕𝑠 𝜕𝑝)𝑇=−(𝜕𝑣 𝜕𝑇)𝑝 Substituting these into the differential entropy expression: 𝑑𝑠=𝐶𝑝 𝑇𝑑𝑇−(𝜕𝑣 𝜕𝑇)𝑝𝑑𝑝 Multiplying through by : 𝑇𝑑𝑠=𝐶𝑝𝑑𝑇−𝑇(𝜕𝑣 𝜕𝑇)𝑝𝑑𝑝 This is known as the First Tds Equation or First Form of the Entropy Equation. Thermodynamics 85 Entropy as a Function of 𝑇 and 𝑣 Now consider entropy as a function of temperature and specific volume: 𝑠=𝑓(𝑇,𝑣) The total differential is: 𝑑𝑠=(𝜕𝑠 𝜕𝑇)𝑣𝑑𝑇+(𝜕𝑠 𝜕𝑣)𝑇𝑑𝑣 From the definition of specific heat at constant volume: 𝐶𝑣=𝑇(𝜕𝑠 𝜕𝑇)𝑣⇒(𝜕𝑠 𝜕𝑇)𝑣=𝐶𝑣 𝑇 From Maxwell's relation: (𝜕𝑠 𝜕𝑣)𝑇=(𝜕𝑝 𝜕𝑇)𝑣 Substituting into the entropy expression: 𝑑𝑠=𝐶𝑣 𝑇𝑑𝑇+(𝜕𝑝 𝜕𝑇)𝑣𝑑𝑣 Multiplying through by : 𝑇𝑑𝑠=𝐶𝑣𝑑𝑇+𝑇(𝜕𝑝 𝜕𝑇)𝑣𝑑𝑣 This is known as the Second Tds Equation or Second Form of the Entropy Equation. Entropy as a Function of 𝑝 and 𝑣 Entropy can also be expressed as a function of pressure and specific volume: 𝑠=𝑓(𝑝,𝑣) The total differential is: 𝑑𝑠=(𝜕𝑠 𝜕𝑝)𝑣𝑑𝑝+(𝜕𝑠 𝜕𝑣)𝑝𝑑𝑣 Thermodynamics 86 Multiplying through by : 𝑇𝑑𝑠=𝑇(𝜕𝑠 𝜕𝑝)𝑣𝑑𝑝+𝑇(𝜕𝑠 𝜕𝑣)𝑝𝑑𝑣 From Maxwell relations, (𝜕𝑠 𝜕𝑣)𝑝=(𝜕𝑝 𝜕𝑇)𝑣,(𝜕𝑠 𝜕𝑝)𝑣=−(𝜕𝑣 𝜕𝑇)𝑝 By combining these relationships and applying thermodynamic identities, one can simplify and derive alternative useful forms. Applications of Tds Equations 1. Entropy Change Calculations: Useful for evaluating entropy changes in processes involving ideal gases, real gases, and vapors. 2. Thermodynamic Property Relations: Provide links between measurable properties (T,p,v) and entropy. 3. Cycle Analysis: Widely applied in analyzing power cycles (Rankine, Brayton) and refrigeration cycles. 4. Irreversibility Measurement: Help in entropy generation studies and efficiency evaluations. 2.12 Entropy Change of Ideal Gas Entropy is a key thermodynamic property that helps measure the irreversibility of processes and the availability of energy for doing useful work. For an ideal gas, entropy change can be derived from the first law of thermodynamics and expressed in terms of measurable properties like temperature, pressure, and volume. Consider an ideal gas being heated from state 1 to state 2, with its temperature rising from 𝑇1 to 𝑇2, as shown in Figure 2.11. For a reversible process: 𝑑𝑠=𝛿𝑄𝑟𝑒𝑣 𝑇 Thermodynamics 87 From the First Law of Thermodynamics: 𝛿𝑄=𝑑𝑈+𝛿𝑊=𝑚𝐶𝑣𝑑𝑇+𝑝𝑑𝑉 Fig. 2.11 An ideal gas being heated from state 1 to state 2, with its temperature rising from 𝑻𝟏 to 𝑻𝟐. Since for an ideal gas, 𝑝𝑉=𝑚𝑅𝑇, we can write: 𝛿𝑄 𝑇 =𝑚𝐶𝑣𝑑𝑇 𝑇+𝑝𝑑𝑉 𝑇 𝑑𝑆 =𝑚𝐶𝑣𝑑𝑇 𝑇+𝑚𝑅𝑑𝑉 𝑉 Integrating between states 1 and 2: Δ𝑆=𝑆2−𝑆1=𝑚𝐶𝑣ln (𝑇2 𝑇1)+𝑚𝑅ln (𝑉2 𝑉1) This is the general entropy change expression in terms of temperature and volume. Entropy Change in Terms of Temperature and Pressure From the ideal gas equation: 𝑝1𝑉1 𝑇1=𝑝2𝑉2 𝑇2 Thermodynamics 88 Rearranging, 𝑉2 𝑉1=𝑝1 𝑝2⋅𝑇2 𝑇1 Substituting this into the general entropy expression: Δ𝑆=𝑚𝐶𝑣ln (𝑇2 𝑇1)+𝑚𝑅ln (𝑝1 𝑝2⋅𝑇2 𝑇1) Simplifying: Δ𝑆=𝑚𝐶𝑝ln (𝑇2 𝑇1)−𝑚𝑅ln (𝑝2 𝑝1) Thus, entropy change in terms of temperature and pressure is: Δ𝑆=𝑚𝐶𝑝ln (𝑇2 𝑇1)+𝑚𝑅ln (𝑝1 𝑝2) Entropy Change in Terms of Pressure and Volume From the ideal gas law: 𝑝1𝑉1 𝑇1=𝑝2𝑉2 𝑇2 Rearranging: 𝑇2 𝑇1=𝑝2 𝑝1⋅𝑉2 𝑉1 Substituting this into the general entropy expression: Δ𝑆=𝑚𝑅ln (𝑉2 𝑉1)+𝑚𝐶𝑣ln (𝑝2 𝑝1⋅𝑉2 𝑉1) Simplifying: Δ𝑆=𝑚𝐶𝑝ln(𝑉2 𝑉1)+𝑚𝐶𝑣ln(𝑝2 𝑝1) Thermodynamics 89 Key Observations 1. The entropy change of an ideal gas depends only on its end states (since entropy is a property) and not on the path taken. 2. Entropy increases when the gas is heated or expanded. 3. For reversible adiabatic processes, entropy remains constant (ΔS=0). 4. These expressions are extremely useful in cycle analysis (Otto, Diesel, Rankine, and refrigeration cycles). 2.13 Entropy Change for Different Processes 1. Entropy Change in Constant Volume Process A thermodynamic process in which the volume remains constant throughout is known as a constant volume process or isochoric process. Since there is no change in volume, the work done by the system is zero ( 𝑊=𝑝Δ𝑉=0 ). The heat supplied to the system is therefore entirely used to increase the internal energy. A system undergoing a constant volume process from state 1 to state 2 is shown in the 𝐩−𝐕 and 𝐓−𝐬 diagrams in Figure 2.12. Fig. 2.12 Constant volume process. Thermodynamics 90 Entropy Change Derivation From the general entropy change expression for an ideal gas: Δ𝑆=𝑚𝑅ln (𝑉2 𝑉1)+𝑚𝐶𝑣ln (𝑇2 𝑇1) For a constant volume process: 𝑉1=𝑉2⇒ln (𝑉2 𝑉1)=0 Thus, entropy change reduces to: Δ𝑆=𝑚𝐶𝑣ln (𝑇2 𝑇1) Since for an ideal gas, 𝑇2 𝑇1=𝑝2 𝑝1 the entropy change can also be expressed as: Δ𝑆=𝑚𝐶𝑣ln (𝑝2 𝑝1) Graphical Representation  p-V Diagram (Figure 2.12a): The process is represented by a vertical line since volume is constant while pressure decreases from 𝑝1 to 𝑝2.  T-s Diagram (Figure 2.12b): The process appears as a curve rising from 𝑇1 to 𝑇2. The area under the curve between entropies 𝑠1 and 𝑠2 represents the heat transfer during the process. Key Features of Constant Volume Process 1. Work Done: 𝑊=∫ 𝑝𝑑𝑉=0 No work is done since volume remains fixed. Thermodynamics 91 2. Heat Transfer: 𝑄=Δ𝑈=𝑚𝐶𝑣(𝑇2−𝑇1) The heat supplied is completely utilized in raising the internal energy. 3. Entropy Change: Δ𝑆=𝑚𝐶𝑣ln (𝑇2 𝑇1)=𝑚𝐶𝑣ln (𝑝2 𝑝1) 4. Practical Application:  This process occurs during the compression and expansion in closed combustion chambers (e.g., spark ignition engines during combustion).  Useful for analyzing constant-volume heat addition in the Otto cycle. 2. Entropy Change in Constant Pressure Process A thermodynamic process in which the pressure remains constant throughout is known as a constant pressure process or isobaric process. In this case, the system undergoes a change in volume and temperature while maintaining constant pressure. Such processes are very common in practical thermodynamic cycles-for example, in boilers where heat is supplied at constant pressure. A system undergoing a constant pressure process from state 1 to state 2 is represented in the 𝐩−𝐕 and 𝐓−𝐬 diagrams shown in Figure 2.13. Fig. 2.13 Constant pressure process. Thermodynamics 92 Entropy Change Derivation From the general entropy change expression for an ideal gas: Δ𝑆=𝑚𝑅ln (𝑉2 𝑉1)+𝑚𝐶𝑣ln (𝑇2 𝑇1) For a constant pressure process, the relation between temperature and volume is: 𝑇2 𝑇1=𝑉2 𝑉1 Substituting into the above equation: Δ𝑆=𝑚(𝐶𝑝−𝐶𝑣)ln (𝑉2 𝑉1)+𝑚𝐶𝑣ln (𝑉2 𝑉1) Δ𝑆=𝑚𝐶𝑝ln (𝑉2 𝑉1) Since at constant pressure 𝑉2 𝑉1=𝑇2 𝑇1, Δ𝑆=𝑚𝐶𝑝ln (𝑇2 𝑇1) Thus, entropy change in a constant pressure process depends only on the ratio of final to initial temperatures. Graphical Representation  p-V Diagram (Figure 2.13a): The process appears as a horizontal line since pressure remains constant while volume increases from 𝑉1 to 𝑉2. The area under this line represents the work done during the process.  T-s Diagram (Figure 2.13b): The process is represented by a rising curve from 𝑇1 to 𝑇2, with entropy increasing from 𝑠1 to 𝑠2. The shaded area under the curve gives the heat transfer to the system. Key Features of Constant Pressure Process 1. Work Done: 𝑊=𝑝(𝑉2−𝑉1) The work is equal to the product of constant pressure and change in volume. Thermodynamics 99 Dividing through by temperature : 𝑑𝑆=(𝛾−𝑛 𝛾−1)𝑚𝑅𝑑𝑉 𝑉 Integrating from state 1 to state 2 gives: 𝑆2−𝑆1=(𝛾−𝑛 𝛾−1)𝑚𝑅ln (𝑉2 𝑉1) This is one of the standard entropy relations for a polytropic process. Alternative Derivations Using gas laws and thermodynamic identities, the change in entropy can also be written as: 𝑑𝑆=𝑚[𝛾−𝑛 𝛾−1]𝐶𝑣ln (𝑝1 𝑝2) Or, 𝑑𝑆=𝑚𝐶𝑣(𝛾−𝑛) 𝑛ln (𝑝1 𝑝2) These alternative expressions are useful in cases where pressure and volume are known instead of temperature. Graphical Representation Fig. 2.16 Polytropic process. Thermodynamics 100  p-V diagram: Shows a curved path depending on the value of 𝑛. For 𝑛=1, it reduces to an isothermal process; for 𝑛=𝛾, it becomes an adiabatic process.  T-s diagram: Illustrates that entropy increases when heat is absorbed and decreases when heat is rejected. Special Cases of Polytropic Index  𝑛=0 : Constant pressure process  𝑛=1 : Isothermal process  𝑛=𝛾 : Reversible adiabatic (isentropic) process  𝑛=∞ : Constant volume process Thus, the polytropic process serves as a generalized form of thermodynamic processes, making it an essential tool in the analysis of practical cycles like compressors, turbines, and engines. 2.14 Principle of increase in entropy The concept of entropy is central to the second law of thermodynamics. For a reversible process, the change in entropy is given by: 𝑑𝑠=𝑑𝑄 𝑇 This relation indicates that entropy change can be directly evaluated from the heat transfer at a given absolute temperature. However, in real systems, processes are not always reversible. Therefore, it is essential to extend the principle to irreversible processes. Entropy in Irreversible Processes Consider a thermodynamic system undergoing a change of state from point 1 to point 2 by a reversible process ( 1−A−2 ), and then returning to state 1 either through an internally reversible process ( 2−B−1 ) or through an irreversible process (2-C-1), as illustrated in Figure 2.17. For the reversible cycle 1-A-2-B-1, the Clausius equality applies: ∫ 2𝐴 1𝐴 𝑑𝑄 𝑇+∫ 1𝐵 2𝐵 𝑑𝑄 𝑇=0 (2.5) Thermodynamics 101 Fig. 2.17 Entropy in irreversible process. For the irreversible cycle 1-A-2-C-1, the Clausius inequality holds: ∫ 2𝐴 1𝐴 𝑑𝑄 𝑇+∫ 1𝐶 2𝐶 𝑑𝑄 𝑇≤0 (2.6) Derivation of Entropy Inequality Subtracting equation (2.5) from (2.6), we get: ∫ 1𝐶 2𝐶 𝑑𝑄 𝑇−∫ 1𝐵 2𝐵 𝑑𝑄 𝑇≤0 (2.7) Reversing limits and rearranging: ∫ 2𝐶 1𝐶 𝑑𝑄 𝑇≥∫ 2𝐵 1𝐵 𝑑𝑄 𝑇(2.8) Since the path 2−𝐵−1 is reversible, the entropy change along it can be written as: 𝑑𝑠=𝑑𝑄 𝑇 Substituting into equation (2.8): ∫ 2 1𝑑𝑠≥∫ 2𝐶 1𝐶 𝑑𝑄 𝑇 Thermodynamics 102 or 𝑑𝑠≥𝑑𝑄 𝑇(2.9) Interpretation of the Principle Equation (2.9) establishes the Principle of Increase of Entropy, which states:  For a reversible process, 𝑑𝑠=𝑑𝑄 𝑇  For an irreversible process, 𝑑𝑠>𝑑𝑄 𝑇  For an impossible process, 𝑑𝑠<𝑑𝑄 𝑇 This principle implies that irreversibility always increases entropy in a system. In other words, entropy is a measure of energy degradation or the tendency of systems to move toward disorder. Entropy of an Isolated System For an isolated system, no heat or mass interaction occurs with the surroundings. Therefore: 𝑑𝑆≥0  If the process is reversible, entropy remains constant.  If the process is irreversible, entropy increases.  Entropy can never decrease in an isolated system. 2.15 Applications of II Law: Available Energy and Unavailable Energy Available Energy (A.E.) Available energy is defined as the portion of the energy supplied as heat that can be effectively converted into useful work by a reversible engine. In thermodynamics, this concept plays a crucial role in determining the practical utilization of heat energy. Thermodynamics 103 Consider a Carnot engine operating between the temperature limits 𝑇1 and 𝑇2, as illustrated in Figure 2.18. According to the second law of thermodynamics, the rectangular area 1−2− 3−4 on the T−s diagram represents the useful energy. However, the area 2−3−5−6 corresponds to the energy lost to the surroundings due to irreversibilities, such as friction, turbulence, and other uncontrollable factors. Fig. 2.18 T-s diagram. The primary objective of engineering design is to minimize these energy losses. One effective method is to reduce the temperature of heat rejection, 𝑇2, closer to the ambient or surrounding temperature 𝑇0. As the rejection temperature decreases, the useful energy that can be converted into work increases, thereby reducing the unproductive energy loss. It is important to note that, as per the Carnot theorem, no real or ideal engine can ever surpass the efficiency of a Carnot engine operating between the same temperature limits. This fact establishes the theoretical boundary for the conversion of available energy into work. The efficiency of the Carnot engine is given by: 𝜂max=1−𝑇2 𝑇1 Thus, the useful energy output becomes: 𝑊=(1−𝑇2 𝑇1)𝑄 Thermodynamics 104 For available energy: 𝐴.𝐸=(1−𝑇0 𝑇1)𝑄 where 𝑇0 is the temperature of the surroundings. By the entropy principle: Δ𝑆=𝑄 𝑇0 Hence, 𝐴.𝐸=𝑄−𝑇0Δ𝑆 Unavailable Energy (U.A.E.) Unavailable energy is defined as the portion of the energy supplied as heat that cannot be converted into useful work. This loss is unavoidable because of irreversibilities such as friction, heat transfer across finite temperature differences, and entropy generation. Mathematically, unavailable energy is expressed as: 𝑈⋅𝐴⋅𝐸=𝑄−𝐴⋅𝐸 Substituting the relation for available energy: 𝑈⋅𝐴⋅𝐸=𝑄−[𝑄−𝑇0Δ𝑆] 𝑈⋅𝐴⋅𝐸=𝑇0Δ𝑆 Thus, the unavailable energy is directly related to entropy generation and the surrounding temperature. Note: Unavailable energy can also be viewed as:  The loss in available energy,  A measure of irreversibility, or  The consequence of the principle of entropy generation. Thermodynamics 105 Decrease in Available Energy Through a Finite Temperature Difference In practical heat transfer processes, energy is seldom transferred at an infinitesimal temperature difference. Instead, heat flows through a finite temperature difference, which inevitably results in a loss of available energy. This concept can be illustrated using the T-s diagram shown in Figure 2.19. Fig. 2.19 T-s diagram. Ideal Case: Reversible Heat Transfer Consider a reversible heat engine operating between a high-temperature reservoir at 𝑇1 and a lowtemperature reservoir at 𝑇0.  Heat supplied from the source: 𝑄1=𝑇1Δ𝑆  Heat rejected to the sink: 𝑄2=𝑇0Δ𝑆  Work output, which represents the available energy (A.E.): 𝑊=𝑄1−𝑄2=(𝑇1−𝑇0)Δ𝑆 Thermodynamics 106 In this ideal case, the work output is maximized because the process is fully reversible, and there are no losses due to finite temperature gradients. Practical Case: Heat Transfer Through a Finite Temperature Difference In reality, when heat 𝑄1 is transferred to the heat engine, it does not reach the engine at the reservoir temperature 𝑇1. Instead, due to finite temperature resistance, the effective temperature at which the heat enters the engine is slightly lower, denoted as 𝑇1′. The hightemperature reservoir remains at 𝑇1, but the working fluid of the engine receives heat at 𝑇1′.  Heat supplied to the engine: 𝑄1′=𝑇1Δ𝑆=𝑇1′Δ𝑆′  Heat rejected to the sink at 𝑇0 : 𝑄2′=𝑇0Δ𝑆′  Actual work output: 𝑊′=𝑄1′−𝑄2′=(𝑇1′−𝑇0)Δ𝑆′ Clearly, the actual work 𝑊′ is less than the ideal work 𝑊, since 𝑇1′<𝑇1 and Δ𝑆′>Δ𝑆. Decrease in Available Energy The decrease in available energy is caused by the excess heat rejected to the sink: Δ𝑊=𝑊−𝑊′=𝑄2′−𝑄2 Thus, the loss in available energy directly depends on the difference between 𝑇1 and 𝑇1′.  If the temperature difference ( 𝑇1−𝑇1′ ) is small, the decrease in available energy is minimal.  If the temperature difference is large, the decrease in available energy becomes significant. 2.16 Concept of Irreversibility (I) In thermodynamics, irreversibility is a measure of the loss of work potential in a process due to the presence of non-idealities such as friction, unrestrained expansion, mixing of different substances, heat transfer through a finite temperature difference, and other dissipative effects. Thermodynamics 107 It can be formally defined as the difference between the maximum possible work (reversible work) and the actual work obtained in a real process. 𝐼=𝑊max−𝑊act Where:  𝑊max = Maximum work obtainable if the process were completely reversible.  𝑊act = Actual work obtained in the real, irreversible process. Expression for Irreversibility By applying the second law of thermodynamics, irreversibility can also be expressed in terms of entropy generation. For a process that exchanges heat with surroundings at ambient temperature 𝑇0 : 𝐼=𝑇0Δ𝑆 Here,  𝑇0 is the temperature of the surroundings (dead state),  Δ𝑆 is the entropy generated during the process. Thus, irreversibility quantifies the energy lost to the surroundings that cannot be converted into useful work. Physical Significance  In an ideal reversible process, irreversibility is zero, since no entropy is generated and all available energy can be converted into useful work.  In real processes, irreversibility is always positive, because entropy generation is unavoidable.  A higher degree of irreversibility means greater energy degradation, reducing the efficiency of the system. Examples of Sources of Irreversibility 1. Mechanical irreversibility: Friction in moving parts, inelastic deformation. 2. Thermal irreversibility: Heat transfer across a finite temperature difference. 3. Chemical irreversibility: Combustion, mixing of gases. 4. Electrical irreversibility: Resistance heating, eddy currents. Thermodynamics 108 2.17 Expressions for Energy of a Closed Systems in Terms of Availability and Second Law Efficiency In thermodynamics, the analysis of closed systems is of significant importance, particularly when studying the useful portion of energy and the associated losses. The concepts of availability (exergy), unavailable energy (anergy), and second law efficiency are vital for quantifying the true effectiveness of energy utilization. The total energy supplied to a system is never fully converted into useful work due to irreversibilities and entropy generation. Hence, the second law of thermodynamics guides us to distinguish between the useful (available) and the wasted (unavailable) parts of energy. For any closed system undergoing a process: Heat Transfer: 𝑄=(𝛾−𝑛 𝛾−1)⋅𝑝1𝑉1−𝑝2𝑉2 𝑛−1 or 𝑄=(𝛾−𝑛 𝛾−1)⋅𝑚𝑅(𝑇1−𝑇2) 𝑛−1 Change in Entropy: Δ𝑆=𝐶𝑝ln (𝑇2 𝑇1)−𝑅ln (𝑝2 𝑝1) or Δ𝑆=𝐶𝑣ln (𝑇2 𝑇1)+𝑅ln (𝑉2 𝑉1) Availability (Maximum Work): 𝑊max=𝑄−𝑇0Δ𝑆 Irreversibility: 𝐼=𝑊max−𝑊=𝑇0Δ𝑆 Second Law Efficiency: 𝜂𝐼𝐼=𝑊act 𝑊max Case Studies (a) Constant Volume Process When air is heated at constant volume from 𝑇1 to 𝑇2 with surroundings at temperature 𝑇0 : Heat supplied: 𝑄=𝑚𝐶𝑣(𝑇2−𝑇1) Thermodynamics 115 Entropy Balance The concept of entropy balance plays a fundamental role in thermodynamics, as it accounts for the transfer and generation of entropy within a system and its surroundings. According to the second law of thermodynamics, the entropy change of any system is always greater than or equal to the entropy transferred to it, the difference being the entropy generated within the system due to irreversibilities. This generation of entropy is the reason for the imbalance between entropy transfer and entropy change. Mathematically, the entropy balance equation can be expressed as: (Entropy entering the system) + (Entropy generation within the system) = (Entropy leaving the system) + (Entropy change of the surrounding This relation can be rewritten as: 𝑆in +𝑆gen =𝑆out +Δ𝑆nurt 𝑆gen =(𝑆out −𝑆in )+Δ𝑆nurr Since (𝑆out −𝑆in )=Δ𝑆system , the expression becomes: 𝑆gen =Δ𝑆syptem +Δ𝑆surr (2.10) For unit mass, this entropy balance reduces to: 𝑠gen −Δ𝑠ovgreen +Δ𝑠gurr (2.11) Interpretation of the Entropy Balance The entropy balance equation (2.10) is universal in nature and applies to any system undergoing any thermodynamic process, whether reversible or irreversible. It highlights that the entropy generation in a process is equal to the combined entropy changes of the system and its surroundings. For a reversible process: Entropy generation is zero, i.e., 𝑆gen =0. In this case, the entropy change of the system is exactly balanced by the entropy transfer to or from the surroundings. For an irreversible process: Entropy generation is always positive, i.e., 𝑆𝑔𝑒𝑛>0. This indicates that entropy increases because of irreversibilities such as friction, unrestrained expansion, mixing of fluids, and heat transfer across a finite temperature difference. Thermodynamics 116 For an impossible process: If entropy generation is negative, i.e., 𝑆𝑔𝑒𝑛<0, the process violates the second law of thermodynamics and is therefore physically impossible. Practical Importance The entropy balance provides a powerful tool for evaluating the performance and feasibility of thermodynamic systems. It allows engineers to:  Quantify irreversibilities in real processes.  Assess the efficiency of systems like turbines, compressors, and heat exchangers.  Evaluate the degradation of energy quality through entropy generation.  Apply the principle of entropy balance for designing energy-efficient systems by minimizing entropy production. 2.20 First Law and Second Law Efficiencies In thermodynamics, the performance of a system can be evaluated using two distinct measures of efficiency: the First Law Efficiency and the Second Law Efficiency. While both are related to energy utilization, they differ significantly in terms of what they measure and how they reflect the quality of energy conversion. First Law Efficiency ( 𝜼𝟏 ) The First Law Efficiency is based on the principle of energy conservation, i.e., the First Law of Thermodynamics. It is defined as the ratio of the net work output of the system to the total heat supplied to it. Mathematically, it is expressed as: 𝜂1= Net work output Heat supplied This efficiency indicates how effectively the supplied energy is converted into useful work. However, it does not account for the quality of energy or the irreversibilities present in the process. For instance, two processes may have the same first law efficiency, but one may involve greater losses due to irreversibility, which is not reflected by this measure. Second Law Efficiency ( II) The Second Law Efficiency is based on the Second Law of Thermodynamics and considers not only the amount of energy but also the quality and availability of energy. It is defined as Thermodynamics 117 the ratio of the change in available energy of the system to the change in available energy of the source. Alternatively, it can be expressed as the ratio of the availability of the output to the availability of the input. Mathematically: 𝜂𝐼𝐼= Change in available energy of the system Change in available energy of the source or, 𝜂𝐼𝐼=𝐴out 𝐴in Comparison and Practical Insight  First Law Efficiency (𝜼𝟏) focuses only on energy conservation and does not distinguish between useful and wasted energy. It provides a basic measure of performance but overlooks irreversibilities and degradation of energy.  Second Law Efficiency ( 𝜂II ) goes further by evaluating how effectively the system utilizes the available energy (exergy). It accounts for irreversibilities, such as friction, mixing, and heat transfer through finite temperature differences, which reduce the useful energy that can be extracted. Engineering Significance  𝜂1 is often used in simple performance analysis, such as in engines, boilers, and power cycles, where energy conservation is the main concern.  𝜂 II is more meaningful in advanced system evaluations, as it highlights the gap between actual performance and the ideal reversible process. It allows engineers to pinpoint losses, optimize system design, and move toward energy-efficient and sustainable solutions. Thermodynamics 118 CHAPTER -3 PROPERTIES OF PURE SUBSTANCE AND STEAM POWER CYCLE 3.1 Formation of Steam and Its Thermodynamic Properties When a pure substance such as water is heated under a constant pressure, it undergoes distinct phase changes before reaching the state of superheated steam. To illustrate, consider 1 kg of water in a closed vessel maintained at a constant pressure of 1.01325 bar (atmospheric pressure) and an initial temperature of –20°C. The various stages of heating are represented in Figure 3.1, which shows the temperature–heat added relationship. Fig. 3.1 Formation of steam. Thermodynamics 119 (a) Heating of Ice (Line 1-2) Initially, as heat is supplied, the temperature of ice rises from −20∘C to the freezing point of water (0∘C). During this stage, the ice remains in solid form, and the rise in temperature is proportional to the heat input. The enthalpy change in this stage is given by: ℎ=𝑚𝐶𝑝𝑖𝑐𝑒(𝑇2−𝑇1) where 𝐶𝑝ice is the specific heat of ice. (b) Melting of Ice (Line 2-3) Once the temperature reaches 0∘C, any further heat supplied does not increase the temperature but is instead used to convert solid ice into liquid water. This process is known as fusion or melting. The heat required to convert ice at 0∘C into water at 0∘C is called the latent heat of fusion or latent heat of ice. During this stage, both ice and water coexist in equilibrium until the entire ice has melted. (c) Heating of Water (Line 3-4) After melting, the liquid water is heated from 0∘C to its boiling or saturation temperature, which is 100∘C at atmospheric pressure. The heat supplied in this process is called the sensible heat of water and is denoted by ℎ𝑓. At this stage:  The temperature at which boiling starts (for a given pressure) is the saturation temperature 𝑇sat .  The pressure corresponding to this temperature is called the saturation pressure 𝑝sat . Both 𝑇sat and 𝑝sat are dependent on each other. The enthalpy change can be expressed as: ℎ𝑓=𝑚𝐶𝑝uuter (𝑇sat −𝑇initial ) (d) Vaporization (Line 4-5) Once the water reaches the saturation temperature, further heat addition does not increase the temperature but converts the liquid water into vapor. Initially, steam contains water particles in suspension, forming a mixture known as wet steam. Thermodynamics 120 The heat supplied during this constant-temperature phase change is called the latent heat of vaporization (or enthalpy of vaporization), denoted by ℎ𝑓𝑔. The total enthalpy of dry saturated steam is expressed as: ℎ𝑔=ℎ𝑓+ℎ𝑓𝑔 For wet steam, the enthalpy is calculated using the dryness fraction ( x ): ℎ𝑤𝑒𝑡=ℎ𝑓+𝑥⋅ℎ𝑓𝑔 where 0≤𝑥≤1. Here,  𝑥=0→ saturated liquid (all water),  𝑥=1→ dry saturated steam (no water droplets). At atmospheric pressure, ℎ𝑓≈419 kJ/kg and ℎ𝑓𝑔≈2257 kJ/kg. (e) Dry Saturated Steam (Point 5) When all the water has completely evaporated, the steam is said to be dry saturated steam. At this stage, the steam exists at saturation temperature and pressure, and no liquid phase remains in suspension. The enthalpy of dry saturated steam is: ℎ𝑔=ℎ𝑓+ℎ𝑓𝑔 (f) Superheating of Steam (Line 5-6) If further heat is supplied after dry saturation, the temperature of the steam rises above the saturation temperature. This steam is called superheated steam, and the process is referred to as superheating. The additional heat supplied is known as the heat of superheat or superheat enthalpy, denoted by ℎsup . It is expressed as: ℎsup =ℎ𝑔+𝐶𝑝(𝑇sup −𝑇sat ) where 𝐶𝑝 is the specific heat of superheated steam, and 𝑇𝑠𝑢𝑝 is the superheated temperature. Superheated steam is particularly useful in power generation cycles because it reduces moisture content in turbines, minimizes blade erosion, and increases thermal efficiency. Thermodynamics 121 3.2 Thermodynamic Properties 1. Dryness Fraction The dryness fraction of steam is a fundamental property that describes the quality of wet steam. It is defined as the ratio of the mass of the dry steam actually present to the total mass of the steam mixture (which consists of both dry steam and water droplets in suspension). It is denoted by the symbol 𝐱. 𝑥= 𝑚𝑔 𝑚𝑓+𝑚𝑔 where:  𝑚𝑔= Mass of dry steam (kg)  𝑚𝑓= Mass of water droplets in suspension (kg) This term is applicable only for wet steam because dry steam contains no suspended water particles.  For dry saturated steam, 𝑚𝑓=0, hence: 𝑥=1 Thus, the dryness fraction varies between 0 and 1:  𝑥=0→ saturated liquid (all water)  𝑥=1→ dry saturated steam (no moisture present)  0<𝑥<1→ wet steam When expressed in percentage, the dryness fraction is called the quality of steam: Quality of steam =100×𝑥% This parameter is very important in power plants, as the presence of moisture in steam reduces turbine efficiency and causes blade erosion. Hence, higher values of dryness fraction are desirable for efficient operation. 2. Wetness Fraction The wetness fraction is a complementary property to the dryness fraction. It is defined as the ratio of the mass of suspended water droplets to the total mass of the steam mixture. Thermodynamics 122 Mathematically, Wetness fraction =𝑚𝑓 𝑚𝑓+𝑚𝑔 Since 𝑥= 𝑚𝑔 𝑚𝑓+𝑚𝑔, the wetness fraction can be written as: Wetness fraction =1−𝑥 Thus, the wetness fraction is simply the moisture content of steam. For example:  If 𝑥=0.9 (dryness fraction), the wetness fraction =0.1, i.e. 10% moisture content. When expressed as a percentage, the wetness fraction is sometimes referred to as priming. High wetness fraction in steam is undesirable as it leads to turbine blade wear, corrosion, and efficiency losses. Hence, steam is often superheated to eliminate moisture before being expanded in turbines. 3. Phase Rule The behavior of multi-component, multi-phase systems is governed by the Gibbs Phase Rule, which establishes the number of independent intensive variables required to completely define the state of a system at equilibrium. The rule is expressed as: 𝑛=𝐶−𝜙+2 where:  𝑛= Number of independent variables (degrees of freedom)  𝐶= Number of components  𝜙= Number of phases in equilibrium Application Examples: 1. Two-Phase, Single-Component System For water (𝐶=1) existing in two phases (say, liquid + vapor, so =2 ): 𝑛=1−2+2=1 This means that only one independent property (either temperature or pressure) is sufficient to define the state. The other property is automatically fixed, as pressure and temperature are related through the saturation condition. Thermodynamics 123 2. Triple Point of Water At the triple point, water exists in three phases simultaneously ( 𝜙=3 ) and =1 : 𝑛=1−3+2=0 Thus, the triple point has no degree of freedom. Neither temperature nor pressure can be changed; both are fixed uniquely at the triple point. 3. Single-Phase, Single-Component System For water existing entirely in one phase (say, liquid only or vapor only): 𝑛=1−1+2=2 This implies that two independent intensive properties (e.g., pressure and temperature, or temperature and specific volume) are required to fix the state of the system. Significance of the Phase Rule  The phase rule is a powerful tool in thermodynamics and physical chemistry to understand equilibrium states.  For pure substances, it simplifies property calculations since fewer variables are needed.  For engineering applications like boilers, condensers, and turbines, the phase rule provides clarity on how many properties must be specified to determine the system's condition. 3.3 p–v Diagram of Pure Substances A p–v diagram represents the relationship between the specific volume (v) of a substance, plotted along the X-axis, and the pressure (p), plotted along the Y-axis. It is a fundamental diagram in thermodynamics used to describe the phase-change processes of pure substances. When a pure substance is gradually heated or cooled at constant pressure, it undergoes a series of state changes that can be represented on the p–v diagram. Figures 3.2 and 3.3 show two typical p–v diagrams:  Figure 3.2 – For substances that contract during freezing (most substances).  Figure 3.3 – For substances that expand during freezing (water is the most notable example). The difference between these two diagrams lies primarily in the slope of the solid–liquid equilibrium line. Thermodynamics 124 Fig. 3.2 p-v diagram of a substance during contraction on freezing. Fig. 3.3 p-v diagram of a substance during expansion on freezing. Regions of the p–v Diagram 1. Solid Region: To the left of the saturated solid line, the substance exists entirely in the solid state. Thermodynamics 131  Constant quality (x = 0.2, 0.4, 0.6, etc.) lines are drawn to show the condition of steam within this region. 3. Superheated Region (Right of Dry Steam Line) The region to the right of the saturated vapour line corresponds to superheated steam. Here, steam exists at temperatures higher than the saturation temperature for a given pressure. Critical Point As temperature and pressure increase, the saturated liquid line and saturated vapour line approach each other and meet at the critical point. At this point, the liquid is converted directly into vapour without forming a two-phase mixture. For water, the critical temperature is 374.15°C and the corresponding critical pressure is 221.2 bar. Beyond this state, there is no distinction between liquid and vapour, and the substance exists as a supercritical fluid. Representation of Thermodynamic Processes The T–s diagram is particularly valuable because different thermodynamic processes appear as simple curves or straight lines: 1. Isothermal Process: A process at constant temperature appears as a horizontal line. 2. Isentropic Process: A reversible adiabatic process (constant entropy) appears as a vertical line. This property is especially useful for analysing expansion and compression processes in turbines and compressors. 3. Constant Pressure Lines: In the superheated region, constant pressure lines are drawn. These lines slope upward to the right, reflecting the increase in entropy with temperature at constant pressure. 4. Constant Volume Lines: In the wet region, constant specific volume lines can also be represented. These are useful for calculating changes in specific volume during phase change processes. Applications The T–s diagram is widely used in steam power cycle analysis (e.g., Rankine cycle, reheat cycle, regenerative cycle). Thermodynamics 132 It provides a clear visual representation of heat addition, heat rejection, expansion, and compression processes. By reading values directly from the chart, engineers can quickly determine properties like entropy, dryness fraction, and enthalpy differences. 3.6 h-s diagram or Mollier chart The h–s diagram, more commonly known as the Mollier chart, is a graphical representation of the thermodynamic properties of steam. In this diagram:  The vertical axis (ordinate) represents the enthalpy (h), expressed in kJ/kg.  The horizontal axis (abscissa) represents the entropy (s), expressed in kJ/kg·K. This chart is an invaluable tool for solving problems in steam power engineering, as it allows quick and direct determination of thermodynamic properties without lengthy calculations. Regions of the h–s Diagram 1. Wet Region (Below Dry Steam Line) The region below the saturated vapour line (dry steam line) represents wet steam, where steam exists as a mixture of water and vapour.  Dryness fraction lines are drawn parallel to the saturated vapour line, helping to determine the quality of steam.  For example, a dryness fraction of 0.8 indicates that the mixture consists of 80% dry steam and 20% suspended water. 2. Superheated Region (Above Dry Steam Line) The area above the dry steam line represents superheated steam.  In this region, steam exists at a temperature higher than the saturation temperature for a given pressure.  Constant temperature curves are plotted, which slope upward to the right, indicating entropy increase with temperature. Thermodynamics 133 3. Critical Point At the critical pressure (221.2 bar), the saturated liquid line and saturated vapour line meet. Beyond this point, the distinction between liquid and vapour phases disappears, and steam behaves as a supercritical fluid. Fig. 3.7 Mollier chart. Representation of Thermodynamic Processes The Mollier chart is particularly useful because various thermodynamic processes can be easily represented: 1. Isentropic (Reversible Adiabatic) Process: Since entropy remains constant, an isentropic process appears as a vertical line. Example: Expansion of steam in a turbine. 2. Throttling Process: In throttling, enthalpy remains constant. Therefore, the process is represented as a horizontal line. Thermodynamics 134 Example: Expansion through a throttle valve in refrigeration or calorimeter experiments. 3. Constant Pressure Process: In the wet region, constant pressure lines appear straight, while in the superheated region, they are curved. 4. Constant Temperature Process: These are shown as sloping lines in the superheated region. 5. Constant Volume Process: Such processes can also be traced using the pre-drawn lines in the chart. Engineering Significance  The Mollier chart is extensively used for steam turbine analysis, since it allows the enthalpy drop (heat drop) during adiabatic expansion to be directly measured as the vertical distance between two points.  It reduces the effort of consulting detailed steam tables, making it convenient for practical calculations in power plants.  Processes like compression, expansion, heating, cooling, and throttling can all be quickly visualized. 3.7 p-v-T surface In thermodynamics, the behaviour of a pure substance can be described using three important thermodynamic properties: pressure (p), specific volume (v), and temperature (T). When these three variables are plotted in a three-dimensional coordinate system, the resulting plot is known as the p–v–T surface. In this representation:  Specific volume (v) and temperature (T) are taken as the independent variables (forming the horizontal plane).  Pressure (p) is taken as the dependent variable (represented on the vertical axis). Thus, every point on the surface corresponds to a unique equilibrium state of the substance. Thermodynamics 135 Characteristics of the p–v–T Surface 1. Single-Phase Regions: The single-phase regions (solid, liquid, or vapour) appear as continuous curved surfaces on the p–v–T plot. 2. Two-Phase Regions: The regions where two phases coexist in equilibrium (solid–liquid, liquid–vapour, solid– vapour) appear as surfaces perpendicular to the p–T plane. These represent saturation conditions. 3. Triple Point Line: The line where the three phases (solid, liquid, and vapour) coexist is represented as the triple point line on the surface.  For water, this corresponds to a temperature of 273.16 K and a pressure of 0.6113 kPa.  At this line, specific volume changes, but temperature and pressure remain fixed. 4. Critical Point: At the end of the saturated liquid and vapour surfaces lies the critical point, beyond which the distinction between liquid and vapour phases disappears. Substances Expanding and Contracting on Freezing 1. Figure 3.8 (a) – Water (Expands on Freezing): For water and similar substances, freezing causes expansion due to the open crystal lattice of ice. This is reflected in the slope of the solid–liquid equilibrium surface. 2. Figure 3.8 (b) – CO₂ (Contracts on Freezing): For substances like carbon dioxide, freezing causes contraction, and the solid–liquid surface has an opposite slope compared to water. This distinction between contraction and expansion on freezing is a critical property for material behaviour under varying pressures and temperatures. Thermodynamics 136 Fig. 3.8 p-v-T surface for water and CO2. Relation with Two-Dimensional Diagrams The commonly used thermodynamic diagrams such as p–v, p–T, and T–v diagrams are simply two-dimensional projections of the complete p–v–T surface onto their respective coordinate planes:  The p–v diagram is obtained by projecting onto the p–v plane.  The p–T diagram is a projection onto the p–T plane.  The T–v diagram is a projection onto the T–v plane. Although the p–v–T surface provides the most comprehensive description of substance behaviour, in practice, two-dimensional diagrams are used more frequently since they are simpler and convenient for engineering applications. Engineering Importance  The p–v–T surface provides a complete thermodynamic map of a pure substance. Thermodynamics 137  It helps explain the interrelation between different property diagrams used in practice.  Understanding this surface is essential for analysing phase changes, locating the triple point, and interpreting the critical point behaviour.  While the surface offers a wealth of information, engineers typically rely on simplified 2D diagrams or steam tables for practical calculations. 3.8 Steam Tables In thermodynamics, it is often tedious and impractical to calculate steam properties such as pressure, temperature, specific volume, enthalpy, and entropy using theoretical equations alone. To simplify this, the properties of water and steam have been experimentally determined and systematically arranged in tabular form. These tabulations are collectively known as Steam Tables. Steam tables provide the thermodynamic properties of 1 kg of steam, usually under different conditions of saturation or superheating. For wet steam, the required properties can be obtained with the help of the dryness fraction in combination with the steam tables. Steam tables are broadly divided into three main parts: 1. Saturated Water Table (Pressure Scale) This table presents the properties of steam at different saturation pressures. The key properties listed include:  Saturation temperature (T_sat)  Specific volume of liquid water (𝑣𝑓) and dry saturated steam (𝑣𝑔)  Enthalpy of saturated liquid (ℎ𝑓), latent heat of vaporisation (ℎ𝑓𝑔), and dry saturated steam (ℎ𝑔)  Entropy of saturated liquid (𝑠𝑓), evaporation (𝑠𝑓𝑔), and dry saturated steam (𝑠𝑔) If the required pressure is not exactly listed, the values at intermediate pressures can be obtained by interpolation. Thermodynamics 138 Press ure in kPa Saturat ion temper ature in ∘C Spec ific volu me 𝑣𝑓 ( m3/ kg ) Spec ific volu me 𝑣𝑔 ( m3/ kg ) Enth alpy ℎ𝑓 (kJ/k g) Enth alpy ℎ𝑓𝑔 (kJ/k g) Enth alpy ℎ𝑔 (kJ/k g) Entro py 𝑠𝑓 (kJ/k g•K) Entr opy 𝑠𝑓𝑔 ( kJ/ kg⋅ K ) Entr opy 𝑠𝑔 ( kJ/ kg⋅ K ) 10.0 45.81 1.01 0 14.6 7 191.8 3 2392. 8 2584. 7 0.649 3 7.50 09 8.15 02 15.0 53.97 1.04 0 10.0 2 225.9 4 2373. 1 2599. 1 0.754 9 7.25 36 8.00 85 20.0 60.06 1.01 7 7.64 9 251.4 0 2358. 3 2609. 7 0.832 0 7.07 65 7.90 85 Table. 3.1 Saturated Water Table (Pressure Scale). 2. Saturated Water Table (Temperature Scale) This table provides the same set of properties but arranged according to saturation temperature instead of pressure. Since for every pressure there is only one unique saturation temperature, this form of the table is particularly useful when the temperature of steam is known. Like the pressure-based table, properties include specific volume, enthalpy, and entropy values for liquid, vapour, and mixtures. Saturati on tempera ture in °C Press ure in kPa Speci fic volu me (vf) (m³/k g) Speci fic volu me (vg) (m³/k g) Entha lpy (hf) (kJ/kg ) Entha lpy (hfg) (kJ/kg ) Entha lpy (hg) (kJ/kg ) Entro py (sf) (kJ/kg ·K) Entro py (sfg) (kJ/kg ·K) Entro py (sg) (kJ/kg ·K) 70 31.19 1.023 5.042 292.98 2333.8 2626.8 0.9549 6.8761 7.8310 Thermodynamics 139 75 38.58 1.026 4.131 313.93 2321.4 2635.3 1.0155 6.6669 7.6824 80 47.39 1.029 3.407 334.91 2308.8 2643.7 1.0753 6.5369 7.6122 Table. 3.2 Saturated Water Table (Temperature Scale). 3. Superheated Steam Table The superheated steam table lists the properties of steam at temperatures above the saturation temperature for a given pressure. The tabulated properties include:  Specific volume (v)  Enthalpy (h)  Entropy (s) The data are given for different superheated temperatures (e.g., 100°C, 200°C, 300°C, etc.) and corresponding pressures. This table is especially useful in steam turbine analysis, where superheated steam is widely employed to improve efficiency and reduce moisture content at turbine exhaust. Temp. (°C) Specific Volume (m³/kg) Enthalpy (kJ/kg) Entropy (kJ/kg·K) 100 17.20 2687.5 8.449 150 19.51 2783.1 8.689 200 21.83 2879.6 8.905 250 24.14 2977.4 9.101 300 26.45 3076.6 9.282 350 28.75 3177.3 9.450 400 31.06 3279.6 9.608 500 35.68 3489.1 9.898 600 40.30 3705.5 10.162 Table. 3.3 Superheated Steam Table. Variation of Properties From the study of steam tables, the following trends are observed:  As pressure increases, the sensible heat (hf) gradually increases.  The latent heat of vaporisation (hfg) decreases with pressure.  The enthalpy of dry saturated steam (hg) initially increases with pressure, reaching a maximum at approximately 33.5 bar, and then begins to decrease. These variations are of great importance in the design and analysis of boilers, condensers, and steam turbines. Thermodynamics 140 Engineering Applications of Steam Tables  Used in Rankine cycle analysis to evaluate heat addition, heat rejection, turbine work, and pump work.  Essential for determining steam quality (dryness fraction) in power plants.  Provides quick reference for property evaluation without solving complex thermodynamic equations.  Used in both laboratory experiments and industrial applications for accurate and convenient property determination. 3.9 Determination of dryness fraction The dryness fraction (x) of steam indicates the proportion of vapour present in a wet steam mixture. It is a critical parameter in steam engineering, as the efficiency and performance of steam power plants largely depend on the steam quality. The dryness fraction can be experimentally measured by using different types of steam calorimeters. The commonly used calorimeters are: 1. Bucket or Barrel Calorimeter 2. Throttling Calorimeter 3. Separating Calorimeter 4. Combined Separating and Throttling Calorimeter 1. Bucket or Barrel Calorimeter In this method, the calorimeter is placed inside an insulated vessel filled with a known mass of water. Steam from the main line is passed into the calorimeter through a steam pipe and bubbled into the water. The steam condenses, transferring its latent heat (and part of its sensible heat) to the water, thereby raising its temperature. By measuring the temperature rise of water and knowing the mass of water and condensed steam, the dryness fraction can be estimated. Limitation: This method is simple but not very accurate, as some heat is lost to the surroundings and the measurement does not account for exact energy transfer. 2. Throttling Calorimeter This calorimeter is best suited for steam with a high dryness fraction (low moisture content). It works on the principle of the throttling process, where steam expands through a throttling valve or orifice. Thermodynamics 243 From the chart (at 100 kPa ):  Enthalpy at Point 1: ℎ1=16.5 kJ/kg dry air  Enthalpy at Point 2: ℎ2=28 kJ/kg dry air  Humidity ratio at Point 2: 𝜔2=0.003 kg/kg dry air  Humidity ratio at Point 3: 𝜔3=0.012 kg/kg dry air Step 2: Mass flow rate of dry air Use ideal-gas relation for the dry-air stream: 𝑚𝑎=𝑝𝑉 ˙ 𝑅𝑎𝑇1=100×(45/60) 0.287×283 =0.923 kg/s Step 3: Heat added in heating section (𝟏→𝟐) 𝑄 ˙=𝑚𝑎(ℎ2−ℎ1)=0.923(28−16.5)=10.615 kJ/s Step 4: Moisture added in humidifying section ( 𝟐→𝟑 ) Humidity increase: Δ𝜔 =𝜔3−𝜔2=0.012−0.003=0.009 kg vapour /kg dry air Thermodynamics 244 Steam mass flow: Δ𝜔 = 𝑚˙𝑣 𝑚𝑎⇒𝑚˙𝑣=Δ𝜔𝑚𝑎=0.009×0.923=0.0083 kg/s 5.8 Psychrometric Processes In air-conditioning and environmental control systems, it is often necessary to alter the thermodynamic state of moist air to meet specific requirements of comfort or process needs. These transformations of moist air are known as psychrometric processes. Each process involves a change in one or more psychrometric properties, such as temperature, humidity ratio, enthalpy, or relative humidity. The main psychrometric processes are: 1. Sensible heating 2. Sensible cooling 3. Humidification and dehumidification 4. Cooling and dehumidification 5. Cooling with adiabatic humidification 6. Cooling and humidification by water injection (evaporative cooling) 7. Heating and humidification 8. Humidification by steam injection 9. Adiabatic chemical dehumidification 10. Adiabatic mixing of two air streams Each process can be represented on the psychrometric chart and analyzed mathematically to determine heat and moisture transfers. 1. Sensible Heating Sensible heating refers to heating air without changing its specific humidity (i.e., no addition or removal of moisture). This is achieved by passing air over a hot surface such as a heating coil. On the psychrometric chart, sensible heating is represented by a horizontal line moving to the right, since the dry-bulb temperature increases while the humidity ratio remains constant. The final air temperature is always less than the coil surface temperature, since full thermal equilibrium is not reached. Thermodynamics 245 This process is commonly used in winter air-conditioning, preheating of ventilation air, and drying systems. Fig. 5.4 Psychrometric process. 2. Sensible Cooling Sensible cooling is the reverse of sensible heating, where air is cooled without changing its specific humidity. This is usually achieved by passing air over a cooling coil with a surface temperature above the dew-point of the entering air, ensuring that condensation does not occur. On the psychrometric chart, sensible cooling is represented by a horizontal line moving to the left, as the dry-bulb temperature decreases but the moisture content remains unchanged. This process is used in summer air-conditioning where cooling is required but dehumidification is not necessary. 3. Humidification and Dehumidification Humidification is the process of adding moisture to air without changing its dry-bulb temperature. This increases both the specific humidity and relative humidity. On the psychrometric chart, the process is shown as a vertical upward line, since moisture increases but temperature remains constant. Thermodynamics 246 Dehumidification, on the other hand, removes moisture from air at constant dry-bulb temperature, decreasing both the specific humidity and relative humidity. On the chart, it is represented as a vertical downward line. These processes are essential for controlling comfort conditions and for industrial operations such as textile and paper manufacturing. 4. Cooling and Dehumidification This process is widely used in summer air-conditioning to reduce both the temperature and humidity of moist air. It occurs when air is passed over a cooling coil with a surface temperature below the dew-point of the incoming air. As air cools, its dry-bulb temperature decreases, and once the dew point is reached, condensation begins, reducing the humidity ratio. On the psychrometric chart, this is represented by a sloping line moving downward to the left. The concept of Apparatus Dew Point (ADP) is important here. The ADP is the effective surface temperature of the cooling coil, and dehumidification occurs only when the coil surface temperature is below the dew point of the incoming air. 5. Cooling with Adiabatic Humidification This process involves passing air through a chamber containing sprays of recirculated water. The spray water temperature is lower than the dry-bulb temperature but higher than the dewpoint temperature of the entering air. Since no external heat is added or removed (adiabatic process), the air approaches the condition of adiabatic saturation. On the psychrometric chart, the process follows a line of constant wet-bulb temperature or nearly constant enthalpy. In practice, the final state of air is not fully saturated (point 3 on the chart) but lies somewhere along the line (point 2), depending on the effectiveness of the system. This process is commonly used in cooling towers and evaporative coolers. 6. Cooling and Humidification by Water Injection (Evaporative Cooling) In this process, water is directly injected into the airstream. As the water evaporates, the air is both cooled and humidified. The final condition depends on the extent of evaporation. When the water is injected at a temperature equal to the wet-bulb temperature of the entering air, the process follows a constant wet-bulb line on the chart. This is the principle of evaporative cooling, widely used in hot and dry climates where cooling is needed but refrigeration is impractical. Thermodynamics 247 7. Heating and Humidification This process is typical of winter air-conditioning, where cold, dry outdoor air must be warmed and humidified to provide comfort indoors. The air is passed through a humidifier with spray water at a temperature higher than the dry-bulb temperature of the entering air. On the psychrometric chart, the process is shown as a diagonal upward line to the right, indicating that both temperature and humidity ratio increase simultaneously. The final relative humidity may be higher or lower than the initial, depending on the extent of heating and moisture addition. Thermodynamics 248 Fig. 5.5 Psychrometric process-II. 8. Humidification by Steam Injection In this process, steam is injected directly into the air to increase its humidity ratio. The drybulb temperature changes very little, since the latent heat of steam is absorbed while its sensible heat contribution is minimal. On the psychrometric chart, the process is represented by a vertical upward movement, similar to humidification, but with a slight temperature rise. This is commonly used in textile industries, where high humidity levels must be maintained for fabric processing. 9. Adiabatic Chemical Dehumidification This process is mainly used in industrial air-conditioning systems that require low humidity levels or low dew-point temperatures. In this method, air is passed over chemicals (such as silica gel or lithium chloride) that have a high affinity for moisture. Thermodynamics 249 As air passes over the chemicals, water vapor is absorbed, reducing the specific humidity. At the same time, the latent heat released during condensation is converted into sensible heat, thereby increasing the dry-bulb temperature. On the psychrometric chart, the process follows a line of nearly constant enthalpy or wetbulb temperature, but moving downward in humidity and upward in temperature. 10. Adiabatic Mixing of Two Air Streams When two air streams of different conditions mix adiabatically, the final state of the mixture depends on the enthalpy, humidity ratio, and mass flow rate of each stream. Let:  𝑚1,ℎ1,𝑊1= mass flow, enthalpy, and humidity ratio of stream 1  𝑚2,ℎ2,𝑊2= mass flow, enthalpy, and humidity ratio of stream 2  𝑚3,ℎ3,𝑊3= corresponding properties of the mixture The energy and mass balance gives: 𝑚1 𝑚2=ℎ3−ℎ2 ℎ1−ℎ3=𝑊3−𝑊2 𝑊1−𝑊3 On the psychrometric chart, the final state (point 3 ) lies on the straight line connecting the two states (points 1 and 2). Its exact location depends on the mass ratio of the mixing streams. This process is important in ventilation systems, where outdoor and recirculated air streams are mixed to achieve desired indoor conditions.