Sources and References

The content above is synthesized from the standard canon of undergraduate engineering thermodynamics and classical-thermodynamics textbooks. Major sources include:

• Cengel, Y. A. and Boles, M. A. Thermodynamics: An Engineering Approach. McGraw-Hill.
• Moran, M. J., Shapiro, H. N., Boettner, D. D., and Bailey, M. B. Fundamentals of Engineering Thermodynamics. Wiley.
• Borgnakke, C. and Sonntag, R. E. Fundamentals of Thermodynamics. Wiley.
• Van Wylen, G. J. and Sonntag, R. E. Fundamentals of Classical Thermodynamics. Wiley (earlier editions of the above).
• MIT OpenCourseWare, 2.005 / 2.006 Thermal-Fluids Engineering, Massachusetts Institute of Technology.
• Stanford University, ME 170 Thermodynamics, course outline and standard treatment.
• Kroos, K. A. and Potter, M. C. Thermodynamics for Engineers. Cengage.
• Rogers, G. F. C. and Mayhew, Y. R. Engineering Thermodynamics: Work and Heat Transfer. Pearson.

These notes consolidate the material traditionally taught across the two-course sequence in Thermodynamics 1 and Thermodynamics 2 at the second-year undergraduate level. Topic weights reflect the union of the two courses: the first half (Chapters 1-10) corresponds to the first-course material on First Law, Second Law, and entropy; the second half (Chapters 11-18) corresponds to the second-course material on power and refrigeration cycles, real fluids, mixtures, psychrometrics, combustion, and chemical equilibrium. All formulations are paraphrased from the cited textbook treatments; no course-specific or offering-specific materials were used.

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Thermodynamics is the engineering science of energy — what forms it takes, how it transforms between those forms, and what fundamental limits govern those transformations. These notes consolidate the material traditionally split across the University of Waterloo’s ME 250 (Thermodynamics 1) and ME 354 (Thermodynamics 2) sequence, providing a single textbook-style treatment of the subject from first principles through reacting mixtures and chemical equilibrium.

The macroscopic, or classical, viewpoint adopted throughout treats matter as a continuum characterized by bulk properties such as pressure, temperature, and specific volume. This deliberately ignores the underlying molecular motions that a statistical-mechanical treatment would emphasize. The payoff is a framework in which the engineer can reason about steam turbines, refrigerators, jet engines, and combustion chambers using a small set of balance equations and carefully tabulated property data.

Chapter 1: Fundamental Concepts

1.1 System, Surroundings, and Boundary

A thermodynamic system is the collection of matter or region of space selected for analysis. Everything outside the system is the surroundings, and the real or imagined surface separating them is the boundary. Boundaries may be fixed or movable, real (a cylinder wall) or conceptual (an imagined envelope around a turbine).

Two categories of system dominate engineering work. A closed system, or control mass, contains a fixed quantity of matter; mass cannot cross its boundary, though energy may. A piston–cylinder device with the valves shut is the canonical example. An open system, or control volume, is a region in space through which mass flows; a nozzle, a turbine, or a heat exchanger is analyzed this way. An isolated system exchanges neither mass nor energy with its surroundings.

1.2 Property, State, Process, and Cycle

A property is any characteristic of a system — pressure \( P \), temperature \( T \), specific volume \( v \), internal energy \( u \), entropy \( s \), and so on. Properties are classified as intensive (independent of the amount of mass: \( P \), \( T \), \( v \)) or extensive (proportional to mass: volume \( V \), internal energy \( U \), entropy \( S \)). Every extensive property has an intensive counterpart on a per-unit-mass basis, denoted by the lowercase symbol.

The state of a system is the condition described by its properties at a given instant. A key empirical result, the state postulate, asserts that the state of a simple compressible system is fixed by two independent intensive properties. “Simple compressible” means the only relevant reversible work mode is compression or expansion; electrical, magnetic, and surface-tension effects are absent or negligible.

A process is any change from one state to another; the series of states traversed is the path. A cycle is a process sequence that returns the system to its initial state, so all properties return to their original values. Distinguishing processes with special constraints simplifies analysis: isothermal (constant \( T \)), isobaric (constant \( P \)), isochoric or isometric (constant \( V \)), adiabatic (no heat transfer), and isentropic (constant \( s \)).

Quasi-equilibrium (or quasi-static) processes proceed slowly enough that the system remains infinitesimally close to equilibrium throughout. Real processes are never truly quasi-equilibrium, but the idealization is useful because work and heat along such a path can be computed from properties alone.

1.3 Temperature and the Zeroth Law

The zeroth law of thermodynamics states that if two bodies are each in thermal equilibrium with a third body, they are in thermal equilibrium with one another. This seemingly obvious statement is the logical basis for temperature measurement: a thermometer (the “third body”) provides a consistent reading of an empirical quantity that all bodies in mutual thermal equilibrium share.

The absolute, or thermodynamic, temperature scale is defined such that \( T \) is always positive and independent of any particular substance. In SI, the kelvin (K) is the unit, with the Celsius scale related by \( T_{\mathrm{C}} = T_{\mathrm{K}} - 273.15 \). Absolute scales are mandatory in thermodynamic relations involving ratios of temperatures (Carnot efficiency, ideal gas law).

1.4 Pressure

Pressure is the normal force exerted by a fluid per unit area on a real or imagined surface. In SI, \( 1 \text{ Pa} = 1 \text{ N/m}^2 \). Engineering units include the bar (\( 10^5 \) Pa), atmosphere (\( 101{,}325 \) Pa), and psi. Absolute pressure is measured relative to a perfect vacuum; gauge pressure is measured relative to local atmospheric pressure:

\[ P_{\mathrm{gauge}} = P_{\mathrm{abs}} - P_{\mathrm{atm}}. \]

Thermodynamic relations always use absolute pressure. In a static fluid column, hydrostatic variation gives \( \Delta P = \rho g \Delta z \), the basis for manometers.

Chapter 2: Properties of Pure Substances

2.1 Phases and Phase-Change Processes

A pure substance has a fixed chemical composition throughout; water is pure whether liquid, vapor, or a mixture, and air (though a mixture of gases) is treated as pure in the absence of condensation. Substances exist in solid, liquid, and vapor phases, distinguished by molecular spacing and order.

Consider heating liquid water in a piston–cylinder at constant pressure. Initially it is a compressed liquid (or subcooled liquid). As heat is added, temperature rises until the liquid reaches saturation — the point where vapor begins to form. Continued heat addition produces a saturated liquid–vapor mixture in which temperature stays fixed at the saturation temperature \( T_{\mathrm{sat}} \) while the fraction of vapor grows. Once the last drop of liquid evaporates, the substance is a saturated vapor, and further heating produces superheated vapor with \( T > T_{\mathrm{sat}} \) at the given \( P \).

At each pressure there is exactly one saturation temperature, and vice versa; the relationship \( P_{\mathrm{sat}}(T_{\mathrm{sat}}) \) is the vapor-pressure curve. On a \( P \)–\( v \) or \( T \)–\( v \) diagram, the locus of saturated liquid states and saturated vapor states meet at the critical point, above which no distinct liquid–vapor phase change occurs. The critical point for water is near \( T_c = 374\,^\circ\mathrm{C} \), \( P_c = 22.06 \) MPa.

2.2 Property Tables and the Quality

For substances with complex equations of state, properties are tabulated. Standard steam tables give \( v \), \( u \), \( h \), and \( s \) for saturated and superheated states. Within the two-phase dome, the quality \( x \) is defined as the mass fraction of vapor:

\[ x = \frac{m_g}{m_g + m_f}, \]

where subscripts \( f \) and \( g \) denote saturated liquid and saturated vapor. Any specific property \( y \) (where \( y \) is \( v \), \( u \), \( h \), or \( s \)) in the two-phase region is the mass-weighted average:

\[ y = y_f + x\left(y_g - y_f\right) = y_f + x\, y_{fg}. \]

For compressed-liquid states where no table is available, the approximation \( y \approx y_f(T) \) — take the saturated-liquid value at the same temperature — is usually acceptable, with a small correction to \( h \) for pressure when precision demands it.

2.3 Enthalpy

The combination \( u + Pv \) appears so often, especially in flow problems and constant-pressure processes, that it receives its own name and symbol:

\[ h = u + Pv. \]

Enthalpy \( h \) is a property because \( u \), \( P \), and \( v \) are properties. Its importance is tied to the flow work \( Pv \) that accompanies mass crossing a control volume boundary.

Chapter 3: Ideal Gas and Compressibility

3.1 The Ideal Gas Equation of State

At pressures far below critical and temperatures well above critical, many gases obey

\[ Pv = RT, \qquad PV = mRT = nR_u T, \]

where \( R \) is the specific gas constant (\( R = R_u / M \), with \( R_u = 8.314 \) kJ/(kmol·K) and \( M \) the molar mass). This is the ideal gas law. Air at atmospheric conditions is modelled as ideal to high accuracy; water vapor at low partial pressure (as in humid air) likewise.

For an ideal gas, internal energy is a function of temperature alone: \( u = u(T) \). Consequently \( h = u + RT \) is also a function of temperature only. This yields the specific heats

\[ c_v = \left(\frac{\partial u}{\partial T}\right)_v = \frac{\mathrm{d}u}{\mathrm{d}T}, \qquad c_p = \left(\frac{\partial h}{\partial T}\right)_p = \frac{\mathrm{d}h}{\mathrm{d}T}, \]

related by \( c_p - c_v = R \). The ratio \( k = c_p/c_v \) is the specific-heat ratio, roughly 1.4 for diatomic gases near room temperature and 1.67 for monatomic gases.

3.2 Compressibility Factor

Real gases deviate from ideal behaviour near the critical point and at high pressure. The compressibility factor

\[ Z = \frac{Pv}{RT} \]

measures this deviation; \( Z = 1 \) exactly for an ideal gas. Using reduced properties \( P_R = P/P_c \) and \( T_R = T/T_c \), the principle of corresponding states asserts that \( Z \) is approximately the same function of \( (P_R, T_R) \) for all gases. Generalized compressibility charts leverage this to give engineering estimates when tabulated data are unavailable.

Chapter 4: Work and Heat

4.1 Sign Conventions and Boundary Work

Thermodynamics distinguishes two modes of energy transfer across a system boundary: work \( W \) and heat \( Q \). Work is organized energy transfer — the action of a force through a distance, torque through an angle, voltage driving charge — whereas heat is energy transfer driven by a temperature difference.

A common sign convention treats heat added to the system and work done by the system as positive. Under this convention, the First Law for a closed system reads \( Q - W = \Delta U \).

For a quasi-equilibrium expansion or compression of a simple compressible substance, boundary work is

\[ W_b = \int_1^2 P\, \mathrm{d}V. \]

Evaluating this integral requires knowledge of \( P(V) \) along the path. Some useful closed-form results:

Process	Path	Boundary work
Isobaric	\( P = \text{const} \)	\( W_b = P(V_2 - V_1) \)
Isochoric	\( V = \text{const} \)	\( W_b = 0 \)
Isothermal ideal gas	\( PV = \text{const} \)	\( W_b = mRT \ln(V_2/V_1) \)
Polytropic	\( PV^n = \text{const} \), \( n \ne 1 \)	\( W_b = (P_2 V_2 - P_1 V_1)/(1 - n) \)

Other work modes relevant in engineering — shaft, spring, electrical, magnetic — do not involve \( P\, \mathrm{d}V \) and must be added separately to any energy balance.

4.2 Heat Transfer

Heat transfer may be modelled through conduction, convection, and radiation, but for thermodynamic bookkeeping the total \( Q \) at a boundary is what matters. An adiabatic process has \( Q = 0 \). Heat and work are both path functions: their values depend on how the change of state was accomplished, not merely on the endpoints. This is why we write \( \delta Q \) and \( \delta W \) rather than \( \mathrm{d}Q \) and \( \mathrm{d}W \).

Chapter 5: First Law for Closed Systems

5.1 Statement

The First Law of Thermodynamics is the principle of energy conservation extended to include heat. For a closed system undergoing any process between states 1 and 2,

\[ Q_{12} - W_{12} = \Delta E = \Delta U + \Delta \mathrm{KE} + \Delta \mathrm{PE}. \]

When kinetic and potential energy changes are negligible, as is typical for fluids in a stationary piston–cylinder,

\[ Q_{12} - W_{12} = U_2 - U_1. \]

In differential form, \( \delta Q - \delta W = \mathrm{d}U \). For a cycle, \( \oint \mathrm{d}U = 0 \), so \( \oint \delta Q = \oint \delta W \): the net heat transferred to a closed system over a cycle equals the net work output. This corollary underpins all heat-engine analysis.

5.2 Specific Heats in Calorimetric Calculations

For a closed system of incompressible substance (liquid or solid at modest pressure), \( \mathrm{d}u \approx c\, \mathrm{d}T \) with a single specific heat \( c \). For ideal gases, \( \mathrm{d}u = c_v\, \mathrm{d}T \) regardless of process, and \( \mathrm{d}h = c_p\, \mathrm{d}T \). These simple integrals, with constant or polynomial \( c(T) \), close the First Law in the absence of phase change.

Chapter 6: First Law for Control Volumes

6.1 Conservation of Mass

A control volume admits mass flow at inlets and outlets. Conservation of mass reads

\[ \frac{\mathrm{d}m_{\mathrm{cv}}}{\mathrm{d}t} = \sum_{\mathrm{in}} \dot{m}_i - \sum_{\mathrm{out}} \dot{m}_e, \]

where \( \dot{m} = \rho V_n A \), with \( V_n \) the normal velocity component and \( A \) the cross-sectional area.

6.2 Energy Balance for a Control Volume

Mass crossing a boundary carries internal energy, kinetic energy, potential energy, and does flow work \( Pv \) on the adjacent fluid. Grouping \( u + Pv = h \), the rate form of the First Law for a control volume is

\[ \frac{\mathrm{d}E_{\mathrm{cv}}}{\mathrm{d}t} = \dot{Q}_{\mathrm{cv}} - \dot{W}_{\mathrm{cv}} + \sum_{\mathrm{in}} \dot{m}_i\left(h_i + \tfrac{1}{2}V_i^2 + g z_i\right) - \sum_{\mathrm{out}} \dot{m}_e\left(h_e + \tfrac{1}{2}V_e^2 + g z_e\right). \]

Here \( \dot{W}_{\mathrm{cv}} \) excludes flow work, which is already absorbed into the enthalpies; it is typically shaft work plus any other non-flow work mode.

6.3 Steady-State, Steady-Flow Analysis

A control volume operating at steady state, steady flow (SSSF) has no time dependence in properties or mass within the control volume, so \( \mathrm{d}m_{\mathrm{cv}}/\mathrm{d}t = 0 \) and \( \mathrm{d}E_{\mathrm{cv}}/\mathrm{d}t = 0 \). For a single inlet and outlet,

\[ \dot{Q} - \dot{W} = \dot{m}\left[(h_e - h_i) + \tfrac{1}{2}(V_e^2 - V_i^2) + g(z_e - z_i)\right]. \]

Standard engineering devices reduce this balance further:

Device	Simplifying assumptions	Reduced balance
Nozzle / diffuser	\( \dot{W} = 0 \), \( \dot{Q} \approx 0 \)	\( h_i + \tfrac{1}{2}V_i^2 = h_e + \tfrac{1}{2}V_e^2 \)
Turbine / compressor	\( \dot{Q} \approx 0 \), \( \Delta \mathrm{KE}, \Delta \mathrm{PE} \approx 0 \)	\( \dot{W} = \dot{m}(h_i - h_e) \)
Throttling valve	\( \dot{W} = 0 \), \( \dot{Q} \approx 0 \), \( \Delta \mathrm{KE} \approx 0 \)	\( h_i = h_e \)
Heat exchanger	\( \dot{W} = 0 \), negligible KE/PE	\( \sum \dot{m}_i h_i = \sum \dot{m}_e h_e \)
Mixing chamber	As heat exchanger, single stream out	\( \sum_{\mathrm{in}} \dot{m}_i h_i = \dot{m}_e h_e \)

6.4 Unsteady (Transient) Flow

When the mass or energy of the control volume is changing in time — filling a tank from a supply line, discharging a pressure vessel, starting up a turbine — the unsteady form must be integrated from an initial to a final state. A common idealization is the uniform-state, uniform-flow model, in which inlet and outlet properties are constant while the control-volume state evolves uniformly. Integrating the rate equations over the process interval gives algebraic balances analogous to those of a closed system but with boundary mass-flow terms.

Chapter 7: Second Law of Thermodynamics

7.1 Motivation

The First Law says energy is conserved, but it does not forbid absurdities. A cup of hot coffee cooling on a desk and spontaneously reheating while the room cools further satisfies energy conservation — yet it is never observed. The Second Law enforces the direction of spontaneous processes and bounds the efficiency of energy conversion.

7.2 Thermal Reservoirs, Heat Engines, and Refrigerators

A thermal reservoir is a body with a thermal capacity so large that heat transfer to or from it does not change its temperature. A heat engine is a device that, operating in a cycle, receives heat \( Q_H \) from a high-temperature reservoir, rejects \( Q_L \) to a low-temperature reservoir, and produces net work \( W_{\mathrm{net}} = Q_H - Q_L \). Its thermal efficiency is

\[ \eta_{\mathrm{th}} = \frac{W_{\mathrm{net}}}{Q_H} = 1 - \frac{Q_L}{Q_H}. \]

A refrigerator or heat pump reverses this flow using work input to transfer heat from cold to hot. Performance is expressed as a coefficient of performance:

\[ \mathrm{COP}_R = \frac{Q_L}{W_{\mathrm{net,in}}}, \qquad \mathrm{COP}_{HP} = \frac{Q_H}{W_{\mathrm{net,in}}} = 1 + \mathrm{COP}_R. \]

7.3 Kelvin–Planck and Clausius Statements

Two classical statements capture the Second Law:

• Kelvin–Planck. No heat engine operating in a cycle can receive heat from a single reservoir and produce an equivalent amount of work. Some heat must be rejected to a colder reservoir.
• Clausius. No device operating in a cycle can transfer heat from a colder body to a hotter one without external work input.

The two statements are logically equivalent: a violation of one implies a violation of the other. Both rule out the perpetual-motion machines of the second kind that would convert heat to work with 100% efficiency.

7.4 Reversibility

A reversible process is one that can be reversed without leaving any trace in either the system or the surroundings. All real processes are irreversible because of friction, heat transfer across finite temperature differences, unrestrained expansion, mixing of different substances, chemical reaction, and other dissipative phenomena. Reversibility is an idealization that bounds what any real device can achieve.

Chapter 8: The Carnot Cycle

8.1 Carnot’s Theorems

Sadi Carnot, in 1824, asked a deceptively simple question: what is the best possible heat engine operating between two reservoirs? His answers, now stated as Carnot’s theorems:

1. The efficiency of an irreversible heat engine operating between two reservoirs is always less than that of a reversible engine operating between the same two reservoirs.
2. All reversible heat engines operating between the same two reservoirs have the same efficiency.

The second theorem is crucial because it means the maximum efficiency depends only on the reservoir temperatures, not on the working substance or the mechanical details.

8.2 The Carnot Cycle and Carnot Efficiency

A Carnot cycle consists of four reversible processes:

1. Reversible isothermal heat addition at \( T_H \).
2. Reversible adiabatic (isentropic) expansion from \( T_H \) to \( T_L \).
3. Reversible isothermal heat rejection at \( T_L \).
4. Reversible adiabatic compression from \( T_L \) back to \( T_H \).

Using these reversible processes to define the thermodynamic temperature scale, Carnot’s theorems yield \( Q_H/Q_L = T_H/T_L \) for any reversible engine, so

\[ \eta_{\mathrm{th,Carnot}} = 1 - \frac{T_L}{T_H}. \]

No engine operating between reservoirs at \( T_H \) and \( T_L \) can exceed this efficiency. Likewise \( \mathrm{COP}_{R,\mathrm{Carnot}} = T_L/(T_H - T_L) \) and \( \mathrm{COP}_{HP,\mathrm{Carnot}} = T_H/(T_H - T_L) \).

Chapter 9: Entropy

9.1 The Clausius Inequality

Consider any cycle, reversible or not. The Clausius inequality asserts

\[ \oint \frac{\delta Q}{T} \le 0, \]

where \( T \) is the temperature at the boundary at which \( \delta Q \) crosses. Equality holds for a reversible cycle. This inequality is a mathematical restatement of the Second Law.

9.2 Definition of Entropy

For a reversible cycle, \( \oint (\delta Q/T)_{\mathrm{rev}} = 0 \). Because the integral vanishes around any reversible closed path, the quantity \( \delta Q/T \) integrated along a reversible path between two states depends only on the endpoints. It therefore defines a property. Clausius named it entropy:

\[ \mathrm{d}S = \left(\frac{\delta Q}{T}\right)_{\mathrm{rev}}, \qquad \Delta S = S_2 - S_1 = \int_1^2 \left(\frac{\delta Q}{T}\right)_{\mathrm{rev}}. \]

Entropy is an extensive property with units of kJ/K; the specific entropy \( s \) has units kJ/(kg·K).

9.3 The Increase-of-Entropy Principle

Applying the Clausius inequality to any process between states 1 and 2 and comparing with a reversible reference yields the entropy balance for a closed system:

\[ \Delta S_{\mathrm{sys}} \ge \int_1^2 \frac{\delta Q}{T}. \]

Equality holds for internally reversible processes. For an adiabatic process, \( \Delta S_{\mathrm{sys}} \ge 0 \), with equality if and only if reversible; an adiabatic reversible process is therefore isentropic. More generally, the total entropy generation \( S_{\mathrm{gen}} \) of system plus surroundings is non-negative, vanishing only for reversible processes:

\[ S_{\mathrm{gen}} = \Delta S_{\mathrm{sys}} + \Delta S_{\mathrm{surr}} \ge 0. \]

This is the increase-of-entropy principle: the entropy of an isolated system can never decrease.

9.4 Property Relations (Tds Equations)

Combining the First Law for a reversible process with the definition of entropy yields two Gibbs relations, often called the Tds equations:

\[ T\, \mathrm{d}s = \mathrm{d}u + P\, \mathrm{d}v, \]\[ T\, \mathrm{d}s = \mathrm{d}h - v\, \mathrm{d}P. \]

Though derived from reversible-process arguments, they relate properties only and therefore hold for any process. They are the workhorse equations for computing entropy changes from tabulated property data.

9.5 Entropy Change of Ideal Gases

For an ideal gas with constant specific heats, the Tds equations integrate to

\[ s_2 - s_1 = c_v \ln\frac{T_2}{T_1} + R \ln\frac{v_2}{v_1}, \]\[ s_2 - s_1 = c_p \ln\frac{T_2}{T_1} - R \ln\frac{P_2}{P_1}. \]

When specific heats vary with temperature, tabulated \( s^\circ(T) \) values replace the \( c_p \ln(T_2/T_1) \) term:

\[ s_2 - s_1 = s^\circ(T_2) - s^\circ(T_1) - R \ln\frac{P_2}{P_1}. \]

9.6 Isentropic Processes of Ideal Gases

Setting \( s_2 = s_1 \) for an ideal gas with constant \( k \) yields the familiar polytropic relations

\[ \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(k-1)/k} = \left(\frac{v_1}{v_2}\right)^{k-1}, \]\[ P v^k = \text{const}. \]

These underpin compressor and turbine analysis as reversible adiabatic benchmarks. Real turbomachines deviate through isentropic efficiencies:

\[ \eta_{T} = \frac{h_1 - h_{2,\mathrm{actual}}}{h_1 - h_{2s}}, \qquad \eta_{C} = \frac{h_{2s} - h_1}{h_{2,\mathrm{actual}} - h_1}, \]

where state \( 2s \) is the isentropic exit at the actual outlet pressure.

9.7 Entropy Balance for Control Volumes

Extending the Clausius inequality to open systems:

\[ \frac{\mathrm{d}S_{\mathrm{cv}}}{\mathrm{d}t} = \sum \frac{\dot{Q}_k}{T_k} + \sum_{\mathrm{in}} \dot{m}_i s_i - \sum_{\mathrm{out}} \dot{m}_e s_e + \dot{S}_{\mathrm{gen}}, \]

with \( \dot{S}_{\mathrm{gen}} \ge 0 \). For SSSF with one inlet and one outlet and adiabatic operation, this reduces to \( s_e \ge s_i \). Entropy generation is the central quantitative measure of irreversibility in any device.

Chapter 10: Exergy (Availability) Analysis

10.1 Dead State and Exergy

While energy is conserved, not all forms are equally useful. Exergy, also called availability, measures the maximum useful work obtainable as a system comes into equilibrium with a reference environment at \( T_0, P_0 \) (the dead state). Exergy vanishes at the dead state and is consumed by irreversibilities.

For a closed system, neglecting kinetic and potential changes,

\[ \Phi = (U - U_0) + P_0(V - V_0) - T_0(S - S_0). \]

For a flowing stream, the specific flow exergy is

\[ \psi = (h - h_0) - T_0(s - s_0) + \tfrac{1}{2}V^2 + gz. \]

10.2 Exergy Destruction and the Gouy–Stodola Theorem

The rate of exergy destruction equals the environment temperature times the rate of entropy generation:

\[ \dot{X}_{\mathrm{dest}} = T_0\, \dot{S}_{\mathrm{gen}} \ge 0. \]

This is the Gouy–Stodola theorem. Every irreversibility in a plant — pressure drop, heat transfer across a temperature difference, mixing, combustion — destroys exergy and is visible on a plant-wide exergy accounting as a direct reduction in useful work potential. Second-law efficiency of a device is the ratio of actual exergy output to exergy input, typically much lower than first-law efficiency and a far more honest measure of thermodynamic quality.

Chapter 11: Vapor Power Cycles — The Rankine Cycle

11.1 Ideal Rankine Cycle

The Rankine cycle is the thermodynamic basis for virtually all central-station steam power plants. In its simplest form it comprises four steady-flow devices connected in a loop:

1. Pump (1 → 2): Compressed liquid is pumped isentropically from condenser pressure to boiler pressure.
2. Boiler (2 → 3): Heat \( q_{\mathrm{in}} \) is added at constant pressure, producing superheated steam.
3. Turbine (3 → 4): Steam expands isentropically to condenser pressure, producing shaft work.
4. Condenser (4 → 1): Heat \( q_{\mathrm{out}} \) is rejected at constant pressure to form saturated liquid.

Per unit mass of working fluid, applying steady-flow energy balances to each device:

\[ w_{\mathrm{pump}} \approx v_1(P_2 - P_1), \qquad q_{\mathrm{in}} = h_3 - h_2, \]\[ w_{\mathrm{turb}} = h_3 - h_4, \qquad q_{\mathrm{out}} = h_4 - h_1, \]\[ \eta_{\mathrm{th}} = \frac{w_{\mathrm{turb}} - w_{\mathrm{pump}}}{q_{\mathrm{in}}}. \]

11.2 Improvements

The ideal Rankine cycle’s thermal efficiency is improved by raising boiler pressure, lowering condenser pressure (limited by cooling-water temperature), and superheating to raise average heat-addition temperature. Reheat cycles expand steam in a high-pressure turbine, return it to the boiler for reheating, then expand it further in a low-pressure turbine; this allows higher boiler pressure without excessive turbine exit moisture. Regeneration extracts partially expanded steam to preheat feedwater, raising the average temperature at which heat is received and substantially improving efficiency.

Chapter 12: Gas Power Cycles — The Brayton Cycle

12.1 Ideal Brayton Cycle

The Brayton cycle models gas-turbine engines and modern jet propulsion. In air-standard form:

1. Compressor (1 → 2): Isentropic compression.
2. Combustor (2 → 3): Constant-pressure heat addition.
3. Turbine (3 → 4): Isentropic expansion.
4. Heat rejection (4 → 1): Constant-pressure cooling (in open cycle, exhaust to atmosphere).

With constant specific heats,

\[ \eta_{\mathrm{th,Brayton}} = 1 - \frac{1}{r_p^{(k-1)/k}}, \]

where \( r_p = P_2/P_1 \) is the pressure ratio. Efficiency rises with pressure ratio, but at high \( r_p \) the net work per unit mass falls, setting a practical optimum for each turbine-inlet temperature.

12.2 Enhancements

Regeneration, feasible when turbine exit temperature exceeds compressor exit temperature, uses exhaust heat to preheat compressed air. Intercooling between compressor stages reduces compression work; reheat between turbine stages increases turbine work. Combined intercooled–reheated–regenerated cycles approach an Ericsson-like limit, approaching Carnot efficiency in principle.

12.3 Other Gas Cycles

The Otto cycle (spark ignition engines) uses constant-volume heat addition, with \( \eta_{\mathrm{th,Otto}} = 1 - r^{-(k-1)} \) in terms of compression ratio \( r \). The Diesel cycle uses constant-pressure heat addition; its efficiency depends on both compression and cutoff ratios and is lower than the Otto at the same compression ratio but higher in practice because diesels tolerate larger compression ratios.

Chapter 13: Refrigeration Cycles

13.1 Vapor-Compression Refrigeration

The dominant household and commercial refrigeration technology reverses a Rankine-like path with a two-phase working fluid. The four processes:

1. Evaporator (4 → 1): Low-pressure refrigerant absorbs \( q_L \) from the cold space, evaporating.
2. Compressor (1 → 2): Isentropic (idealized) compression to condenser pressure.
3. Condenser (2 → 3): Rejection of \( q_H \) to ambient, condensing the refrigerant.
4. Expansion valve (3 → 4): Throttling (isenthalpic) to evaporator pressure.

\[ \mathrm{COP}_R = \frac{h_1 - h_4}{h_2 - h_1}, \qquad \mathrm{COP}_{HP} = \frac{h_2 - h_3}{h_2 - h_1}. \]

The expansion valve, chosen for simplicity and reliability over a turbine, introduces unavoidable irreversibility — an explicit trade-off of exergy destruction for cost.

13.2 Gas Refrigeration and Absorption

Gas refrigeration runs a reversed Brayton cycle with air as working fluid; coefficients of performance are low but the system is lightweight, dominant in aircraft cabin cooling. Absorption refrigeration replaces the compressor with an absorber–generator loop driven by heat (e.g., from combustion or solar) instead of high-grade work, useful where waste heat is abundant.

Chapter 14: Real Fluids and State Functions

14.1 Real-Gas Equations of State

Beyond the ideal gas and corresponding-states approach, cubic equations such as van der Waals, Redlich–Kwong, Soave–Redlich–Kwong, and Peng–Robinson are widely used. They take the form

\[ P = \frac{RT}{v - b} - \frac{a(T)}{v^2 + \alpha v b + \beta b^2}, \]

with \( a \) and \( b \) substance-specific constants fit to critical-point data and vapor-pressure curves. These equations capture both vapor and liquid branches and enable engineering calculations from first principles rather than from tables.

14.2 Maxwell Relations and Departure Functions

The four Maxwell relations follow from the exactness of \( \mathrm{d}u \), \( \mathrm{d}h \), \( \mathrm{d}a \) (Helmholtz), and \( \mathrm{d}g \) (Gibbs). They permit calculation of entropy and enthalpy changes from measurable data — chiefly \( P\)–\( v\)–\( T \) data and specific heats at low pressure. A common engineering construction computes departure functions \( h - h^* \) and \( s - s^* \) from a real-gas equation of state, where starred quantities are the ideal-gas reference values; these departures are then tabulated or charted as functions of reduced properties.

14.3 Generalized Enthalpy and Entropy Charts

Parallel to the compressibility chart, generalized charts of \( (h^* - h)/(RT_c) \) and \( (s^* - s)/R \) versus \( P_R \) and \( T_R \) give engineering accuracy for any substance once critical constants are known, closing the gap between ideal-gas and substance-specific tables.

Chapter 15: Non-Reacting Gas Mixtures

15.1 Composition Measures

A mixture of \( N \) components is described by the mass fractions \( mf_i = m_i/m \) and mole fractions \( y_i = n_i/n \). The apparent (average) molar mass is \( M = \sum y_i M_i \), and the apparent gas constant is \( R = R_u/M \).

15.2 Dalton and Amagat Models

Two equivalent idealizations describe ideal-gas mixtures:

• Dalton’s model treats each component as if it alone occupied the mixture volume at the mixture temperature, defining its partial pressure \( P_i = y_i P \). Mixture pressure is \( P = \sum P_i \).
• Amagat’s model treats each component as if it alone existed at the mixture temperature and pressure, defining its partial volume \( V_i = y_i V \). Mixture volume is \( V = \sum V_i \).

Both models give \( P_i V = n_i R_u T \) for ideal-gas mixtures and coincide in that limit.

15.3 Mixture Properties

For ideal-gas mixtures, extensive properties are additive:

\[ U = \sum m_i u_i(T), \qquad H = \sum m_i h_i(T), \qquad S = \sum m_i s_i(T, P_i). \]

The entropy term uses component partial pressures: mixing different gases at the same \( T \) and \( P \) generates entropy because each species expands from \( P \) to \( P_i = y_i P \). This entropy of mixing is purely a statistical phenomenon — it vanishes when the gases are identical.

Chapter 16: Psychrometrics

16.1 Humid Air

Psychrometrics is the thermodynamics of mixtures of dry air and water vapor, foundational for HVAC and process engineering. Both components are treated as ideal gases, a good approximation at atmospheric conditions. Key definitions, with subscripts \( a \) for dry air and \( v \) for water vapor:

• Specific humidity (humidity ratio): \( \omega = m_v/m_a \).
• Relative humidity: \( \phi = P_v / P_{g}(T) \), where \( P_g \) is the saturation pressure of water at dry-bulb temperature \( T \).

The two are related by \( \omega = 0.622\, P_v / (P - P_v) \).

• Dew-point temperature \( T_{dp} \): the temperature at which water vapor begins to condense when the mixture is cooled at constant pressure.
• Wet-bulb temperature \( T_{wb} \): the temperature indicated by a thermometer whose bulb is covered with a wet wick in an airstream; closely related to the adiabatic-saturation temperature.

16.2 The Psychrometric Chart and Processes

A psychrometric chart plots \( \omega \) against \( T \) for a given total pressure (usually 101.325 kPa), with curves of constant \( \phi \), constant \( T_{wb} \), and constant enthalpy. Common air-conditioning processes map onto simple moves on the chart:

Process	Chart path	Comment
Sensible heating	Horizontal right	\( \omega \) constant
Sensible cooling	Horizontal left	\( \omega \) constant
Cooling with dehumidification	Down and left	Passes below dew point
Heating with humidification	Right and up	Add moisture after heating
Evaporative cooling	Along \( h \approx \) const	Useful in dry climates
Adiabatic mixing of two streams	Straight line between states	Lever rule by mass-flow ratio

16.3 Energy and Mass Balances

For a steady air-conditioning device, independent mass balances on dry air and water vapor together with an energy balance close the problem:

\[ \dot{m}_{a,1} = \dot{m}_{a,2}, \qquad \dot{m}_{a}\omega_1 + \dot{m}_{w,\mathrm{in}} = \dot{m}_{a}\omega_2 + \dot{m}_{w,\mathrm{out}}, \]\[ \dot{Q} - \dot{W} = \dot{m}_a(h_2 - h_1) - \dot{m}_{w,\mathrm{in}} h_{w,\mathrm{in}} + \dot{m}_{w,\mathrm{out}} h_{w,\mathrm{out}}, \]

with \( h \) the enthalpy of moist air per unit mass of dry air: \( h = c_{p,a} T + \omega h_{g}(T) \).

Chapter 17: Reacting Mixtures and Combustion

17.1 Stoichiometry

A combustion reaction converts fuel and oxidizer into products, releasing chemical energy as thermal energy. For a hydrocarbon \( \mathrm{C}_x\mathrm{H}_y \) with stoichiometric (theoretical) air,

\[ \mathrm{C}_x\mathrm{H}_y + \left(x + \tfrac{y}{4}\right)\left(\mathrm{O}_2 + 3.76\,\mathrm{N}_2\right) \to x\,\mathrm{CO}_2 + \tfrac{y}{2}\,\mathrm{H}_2\mathrm{O} + 3.76\left(x + \tfrac{y}{4}\right)\mathrm{N}_2, \]

using the dry-air ratio of 3.76 mol N\(_2\) per mol O\(_2\). Practical combustion uses excess air, reported as percent theoretical air (e.g., 150% theoretical air = 50% excess air). The air–fuel ratio AF is the mass ratio of air to fuel consumed; its reciprocal is the fuel–air ratio. Equivalence ratio \( \phi = \mathrm{AF}_{\mathrm{stoich}}/\mathrm{AF}_{\mathrm{actual}} \) characterizes stoichiometry: \( \phi = 1 \) is stoichiometric, \( \phi < 1 \) is lean, \( \phi > 1 \) is rich.

17.2 Enthalpy of Formation and Energy Balance

Because the chemical bonds of products differ from those of reactants, the familiar \( \Delta h = c_p \Delta T \) is insufficient. Each species is assigned a standard enthalpy of formation \( \bar{h}_f^\circ \) at the reference state \( (25\,^\circ\mathrm{C}, 1\text{ atm}) \), defined as zero for stable elements in their natural form. The enthalpy of any species at state \( (T, P) \) is

\[ \bar{h} = \bar{h}_f^\circ + \left[\bar{h}(T) - \bar{h}(T_{\mathrm{ref}})\right], \]

with the bracketed sensible-enthalpy change taken from ideal-gas tables.

The steady-flow energy balance for a combustion chamber is

\[ \dot{Q} - \dot{W} = \sum_{\mathrm{prod}} \dot{n}_i \bar{h}_i - \sum_{\mathrm{react}} \dot{n}_i \bar{h}_i. \]

For complete combustion of a fuel with reactants and products at the reference state, \( -\dot{Q}/\dot{n}_{\mathrm{fuel}} \) equals the enthalpy of combustion \( \bar{h}_C \); its absolute value for the water-as-vapor case is the lower heating value (LHV), and for the water-as-liquid case the higher heating value (HHV).

17.3 Adiabatic Flame Temperature

If a combustion chamber is adiabatic and produces no shaft work, the energy balance collapses to \( H_{\mathrm{prod}} = H_{\mathrm{react}} \). Given reactant temperature and composition, this implicit equation determines the adiabatic flame temperature \( T_{\mathrm{af}} \), the highest temperature the products can attain. Real flames fall short because of heat loss and incomplete combustion, but \( T_{\mathrm{af}} \) bounds material-selection decisions and informs emissions predictions (in particular thermal NO\(_x\)).

17.4 Second-Law Analysis of Combustion

Combustion is highly irreversible: reacting a hot fuel with hot air generates entropy even when adiabatic, because of the uncontrolled chemical transformation. Absolute entropy values \( \bar{s}^\circ \) tabulated per the third law (entropy vanishes at \( T = 0 \) for a perfect crystal) are required; entropy of an ideal-gas component at partial pressure \( y_i P \) is

\[ \bar{s}_i(T, y_i P) = \bar{s}_i^\circ(T) - R_u \ln\frac{y_i P}{P_{\mathrm{ref}}}. \]

An exergy balance on a combustor typically finds 20–30% of the fuel’s chemical exergy destroyed — the largest single loss in a power plant.

Chapter 18: Chemical Equilibrium

18.1 Criterion for Equilibrium

A closed system at fixed \( T \) and \( P \) reaches equilibrium at the composition that minimizes the Gibbs function \( G = H - TS \). For a reacting mixture undergoing \( \nu_A A + \nu_B B \rightleftharpoons \nu_C C + \nu_D D \), the condition \( \mathrm{d}G = 0 \) at fixed \( T, P \) yields

\[ \sum_i \nu_i \bar{\mu}_i = 0, \]

where \( \bar{\mu}_i \) is the chemical potential of species \( i \), and \( \nu_i \) is positive for products, negative for reactants.

18.2 Equilibrium Constant \( K_p \)

For ideal-gas reactions, substituting \( \bar{\mu}_i = \bar{\mu}_i^\circ(T) + R_u T \ln(P_i/P_{\mathrm{ref}}) \) yields

\[ K_p(T) = \prod_i \left(\frac{P_i}{P_{\mathrm{ref}}}\right)^{\nu_i} = \exp\!\left(-\frac{\Delta G^\circ(T)}{R_u T}\right), \]

the equilibrium constant in terms of partial pressures. Because \( \Delta G^\circ \) depends only on \( T \), so does \( K_p \); \( K_p \) is tabulated (often as \( \ln K_p \)) for many reactions. Expressed in mole fractions with Dalton’s model,

\[ K_p = \left(\prod_i y_i^{\nu_i}\right) \left(\frac{P}{P_{\mathrm{ref}}}\right)^{\Delta \nu}, \]

where \( \Delta \nu = \sum \nu_i \) (products minus reactants). Two direct consequences:

• Temperature dependence (van ’t Hoff): \( \mathrm{d}\ln K_p / \mathrm{d}T = \Delta \bar{h}^\circ/(R_u T^2) \). Exothermic reactions have \( K_p \) that decreases with \( T \) — higher temperatures drive dissociation.
• Pressure dependence: If \( \Delta\nu \ne 0 \), increasing \( P \) shifts equilibrium toward the side with fewer moles of gas (Le Chatelier).

18.3 Simultaneous Reactions

When multiple reactions proceed simultaneously — characteristic of high-temperature combustion products involving CO, CO\(_2\), H\(_2\), H\(_2\)O, O\(_2\), OH, NO, and others — one equilibrium constant is written per independent reaction, and the resulting nonlinear algebraic system is solved numerically. The phase rule \( F = C - P + 2 \) (with \( C \) independent components and \( P \) phases) indicates how many intensive variables may be independently specified.

18.4 Phase Equilibrium

The Gibbs-minimization criterion also governs phase equilibrium: liquid in contact with vapor at fixed \( T \) and \( P \) sits at the composition for which \( \bar{\mu}_i^{\mathrm{liq}} = \bar{\mu}_i^{\mathrm{vap}} \) for every component. For a pure substance this recovers the Clausius–Clapeyron equation

\[ \frac{\mathrm{d}P_{\mathrm{sat}}}{\mathrm{d}T} = \frac{h_{fg}}{T v_{fg}}, \]

the differential equation of the vapor-pressure curve, used to extrapolate or interpolate saturation data across temperature.

Chapter 19: Synthesis — Thermodynamics as an Engineering Tool

The Second Law provides two distinct but complementary ways to evaluate engineering systems. First-law analysis tracks where energy goes; second-law (exergy) analysis tracks where useful work capacity is lost. A typical power plant’s first-law accounting shows most energy loss at the condenser, a cold low-grade stream, while exergy accounting places the largest loss at the combustor, where fuel at ideal exergetic quality burns irreversibly with air. The two perspectives produce different — and both valid — maps of where improvements can plausibly be made.

Across the subject, a small toolkit recurs. Identify the system or control volume; apply mass balance, energy balance, and (when irreversibilities matter) entropy balance; close with an equation of state or tabulated property data. The chapters above mostly specialize this toolkit — to the piston-cylinder, to the turbine, to the Rankine loop, to the combustor — with the right simplifications for each case.