Thermodynamics — Engineering Reference

1. At a glance

Thermodynamics is the science of energy, work, heat, and the transformations between them. It is the engineering discipline that bounds what is possible — not what is cheap or efficient, but what is physically allowed by the conservation of energy and the increase of entropy. Every heat engine, every refrigerator, every gas turbine, every chemical reactor, every HVAC system, every internal-combustion engine, every cryogenic plant, every power station — coal, gas, nuclear, geothermal, solar-thermal — operates inside the cage that the two laws of thermodynamics build.

Classical engineering thermodynamics is macroscopic and equilibrium-based. It does not concern itself with the position or momentum of any individual molecule; instead it defines a system (the chunk of matter under study), surroundings (everything else), and a boundary (real or imaginary surface separating them), and tracks bulk properties — pressure P, temperature T, specific volume v, internal energy u, enthalpy h, entropy s — that emerge from molecular averages but obey their own self-contained algebra.

Where statics gives ΣF = 0 and ΣM = 0, and mechanics of materials gives σ = Eε, thermodynamics gives:

• First law: dE = δQ − δW (energy is conserved)
• Second law: dS_universe ≥ 0 (entropy of an isolated system never decreases)

Everything else — Carnot efficiency, Rankine cycles, refrigeration COP, exergy destruction, adiabatic flame temperature, the existence of a maximum-efficiency heat engine — follows from these two statements and the property data of real working fluids.

Where thermodynamics sits in the design stack:

• Prerequisite for heat-transfer (rate equations, conduction/convection/radiation), fluid-mechanics (compressible flow, nozzles), hvac-fundamentals (psychrometrics, refrigeration sizing), propulsion (gas turbines, rockets, ramjets), chemical-process-fundamentals (reactors, distillation, phase equilibria), pumps-turbomachinery (turbine, compressor performance).
• Builds on calculus, statistics (for kinetic-theory derivations), and chemistry (for combustion and reacting flows). The engineering subset rarely needs more than partial derivatives and definite integrals.
• Distinguished from heat transfer by being equilibrium and rate-free. Thermodynamics says how much heat can flow; heat transfer says how fast it actually does.

If you cannot read a P-h diagram, write down the first law for a control volume, or compute the Carnot efficiency between two reservoirs, you cannot design a power plant, a chiller, or an engine.

2. Why it matters

The economic and environmental stakes are enormous. The thermal-to-electric chain runs the world:

• Electric generation: roughly 60 % of global electricity comes from heat engines (coal, natural gas combined cycle, nuclear, oil, biomass, geothermal, concentrated solar). Every Watt of capacity is sized using cycle analysis.
• HVAC and refrigeration: ~15 % of global electricity goes through vapor-compression cycles. ASHRAE Handbook Fundamentals 2025 is the practitioners’ bible.
• Transportation: gasoline (Otto), diesel (Diesel), turbofan (Brayton), rocket (open-cycle Brayton-like) — all heat engines.
• Industrial heat: chemical, petrochemical, steel, cement, glass — all bounded by exergy analysis. A modern refinery’s thermodynamic optimization is worth tens of millions of dollars per year.

The two practical pay-offs of mastering thermodynamics:

1. Efficiency bounds. The Carnot bound η = 1 − T_C/T_H is the maximum possible thermal efficiency of any heat engine between reservoirs at T_H and T_C. No engineering trick — better materials, finer machining, exotic working fluids — can exceed it. Knowing the bound stops the engineer from over-promising and tells the designer how much room there is to grow.

2. Exergy / availability analysis. Where the first law tracks energy quantity, the second law tracks energy quality. Exergy analysis localizes losses to specific components (combustor, turbine, heat exchanger, throttle) and tells the engineer which to attack first. In a modern combined-cycle gas turbine, the combustor alone destroys ~25–30 % of fuel exergy; tightening turbine isentropic efficiency from 0.90 to 0.92 destroys far less and yet wins the same overall efficiency improvement. Exergy points the way.

3. First principles

System types

• Closed system (control mass): fixed quantity of matter, no mass crosses the boundary; energy may. Example: gas in a sealed piston-cylinder.
• Open system (control volume, CV): fixed region in space; mass and energy may both cross the boundary. Example: jet engine, steam turbine, heat exchanger.
• Isolated system: no mass, no energy crosses. Used for entropy-conservation arguments.

Properties

• Intensive properties are independent of system size: P, T, v, ρ, u, h, s.
• Extensive properties scale with system size: V, U, H, S, m.

Specific (per-unit-mass) intensive properties are lowercase (u, h, s, v); extensive are uppercase (U, H, S, V). A common engineering convention.

State and equilibrium

A state is a fully specified condition of a system. The state principle says that for a simple compressible substance (no electromagnetic, surface-tension, or significant gravitational effects), the state is fixed by two independent intensive properties — most commonly (P, T), (P, v), (T, v), (P, x), or (T, x) where x is steam quality. Get any two, and a property table (REFPROP, IAPWS-IF97, CoolProp) gives every other property.

A system is in thermodynamic equilibrium when there are no internal gradients of temperature, pressure, or chemical potential, and no tendency for net energy or mass flow. A quasi-equilibrium (quasi-static) process is one that proceeds so slowly that the system is in equilibrium at every instant — an idealization, but the foundation for reversible-process analysis.

Process types

• Isobaric (constant P), isothermal (constant T), isochoric (constant v), isentropic (constant s, i.e. adiabatic + reversible), adiabatic (no heat transfer, Q = 0), polytropic (Pv^n = const for some exponent n).

The polytropic exponent n covers all the special cases: n = 0 → isobaric, n = 1 → isothermal (ideal gas), n = γ → isentropic (ideal gas), n → ∞ → isochoric.

Sign conventions

The classical (Çengel/Moran) convention: Q positive when added to the system, W positive when done by the system. So a heat engine: Q_in > 0, W_out > 0, Q_out < 0. Some chemistry texts flip W; always state the convention explicitly.

4. Properties and state

State variables

Symbol	Property	SI units	US-customary	Notes
T	Temperature	K (or °C)	°R (or °F)	T[K] = T[°C] + 273.15; T[°R] = T[°F] + 459.67
P	Pressure	Pa, kPa, bar, MPa	psi, psf, atm	1 bar = 100 kPa; 1 atm = 101.325 kPa = 14.696 psi
v	Specific volume	m³/kg	ft³/lbm	v = 1/ρ
ρ	Density	kg/m³	lbm/ft³
u	Internal energy (specific)	kJ/kg	Btu/lbm
h	Enthalpy (specific)	kJ/kg	Btu/lbm	h ≡ u + Pv
s	Entropy (specific)	kJ/(kg·K)	Btu/(lbm·°R)
g	Gibbs free energy (specific)	kJ/kg	Btu/lbm	g ≡ h − Ts
a	Helmholtz free energy (specific)	kJ/kg	Btu/lbm	a ≡ u − Ts
c_p, c_v	Specific heats	kJ/(kg·K)	Btu/(lbm·°R)	At const. P / const. v
γ	Specific heat ratio	—	—	γ ≡ c_p/c_v
x	Quality (vapor mass fraction)	—	—	0 ≤ x ≤ 1 in two-phase

Ideal gas

For temperatures well above the critical point and pressures well below it, real gases approach the ideal-gas equation of state:

$Pv=RT\text{or}PV=n{R}_{u}T$

where R is the specific gas constant (= R_u / M, where R_u = 8.314 J/(mol·K) is the universal constant and M is molar mass), v is specific volume, n is moles. Common R values:

Gas	M (g/mol)	R [kJ/(kg·K)]	c_p [kJ/(kg·K)]	c_v [kJ/(kg·K)]	γ
Air (dry)	28.97	0.2870	1.005	0.718	1.400
N₂	28.01	0.2968	1.040	0.743	1.400
O₂	32.00	0.2598	0.918	0.658	1.395
CO₂	44.01	0.1889	0.846	0.657	1.289
H₂O (vapor)	18.02	0.4615	1.872	1.410	1.327
CH₄ (methane)	16.04	0.5182	2.254	1.735	1.299
H₂	2.016	4.124	14.32	10.20	1.405
He	4.003	2.077	5.193	3.116	1.667
Ar	39.95	0.2081	0.5203	0.3122	1.667

Values at 300 K, 1 atm. Source: NIST Chemistry WebBook + Çengel & Boles 10th ed. Table A-2. c_p, c_v vary with T — these are 300 K reference values; for combustion analysis use polynomial fits (NASA Glenn coefficients, Burcat tables).

For ideal gases the Mayer relation c_p − c_v = R is exact, and Δh = c_p ΔT and Δu = c_v ΔT for any process between two states (only T matters).

Real fluids and property data

Real gases at high pressure or near the critical point deviate from ideal-gas behavior. Engineering practice:

• Compressibility factor Z = Pv/(RT). Z = 1 for ideal; Z deviates near the critical point. Generalized Nelson–Obert charts give Z(P_r, T_r) where P_r = P/P_c, T_r = T/T_c.
• Cubic equations of state (van der Waals, Redlich–Kwong, Soave–Redlich–Kwong, Peng–Robinson) — used in process simulators for hydrocarbons.
• Multi-parameter Helmholtz EOS (modern reference fluids) — NIST REFPROP 10.0 implements 156 pure fluids and mixtures via Helmholtz energy fundamental equations, accurate to within experimental uncertainty.
• Water/steam: IAPWS-IF97 (1997 industrial formulation) is the international standard, embedded in every steam-cycle calculation tool. Reference state h = 0, s = 0 at the saturated liquid triple point (T = 0.01 °C, P = 611.657 Pa).
• Natural gas mixtures: GERG-2008 is the reference EOS for custody-transfer accuracy, 21 components.
• Refrigerants: REFPROP or CoolProp; reference state varies by fluid (IIR uses h = 200 kJ/kg, s = 1.0 kJ/(kg·K) at 0 °C saturated liquid; ASHRAE uses h = 0 at −40 °C saturated liquid).

Phase diagrams

For a pure substance, the P-T diagram shows three lines meeting at the triple point: solid–liquid (fusion), liquid–vapor (vaporization, ending at the critical point), solid–vapor (sublimation). Above the critical point (T_c, P_c) liquid and vapor are indistinguishable — the supercritical region used in ultra-supercritical Rankine plants and supercritical CO₂ cycles.

Substance	T_c	P_c	Notes
Water	373.95 °C	22.064 MPa	IAPWS-IF97
CO₂	31.04 °C	7.377 MPa	R-744, low T_c → easy supercritical
N₂	−146.95 °C	3.396 MPa	Cryogenic
He-4	−267.96 °C	0.2275 MPa	Lowest known T_c
Methane	−82.59 °C	4.599 MPa	LNG benchmark
R-134a	101.06 °C	4.0593 MPa	Common refrigerant
R-1234yf	94.7 °C	3.382 MPa	HFO replacement, GWP < 1

The T-s diagram and h-s (Mollier) diagram are the workhorses of cycle analysis. On T-s, reversible heat transfer appears as area under the process line. On h-s, the vertical drop across a turbine is exactly the specific work output (for adiabatic flow, neglecting KE/PE).

Two-phase quality: in the wet-vapor region, the state is fixed by (T, x) or (P, x). Quality x is the mass fraction of vapor:

$x=\frac{m_{\text{vapor}}}{m_{\text{total}}},u=(1-x)u_f+x{u}_g=u_f+x{u}_{fg}$

(and identically for h, s, v). Subscripts f, g, fg = saturated liquid, saturated vapor, and difference (g − f).

5p. First law — energy conservation

Closed system

For a closed system undergoing any process from state 1 to state 2:

$\Delta U=Q-W\text{or in rate form}\dot{Q}-\dot{W}=\frac{dU}{dt}$

For unit mass: Δu = q − w. Work for a quasi-equilibrium boundary expansion:

$W={\int }_1^{2}PdV$

For specific common closed-system processes (ideal gas):

Process	Work	Heat
Isothermal (T const)	W = mRT ln(V₂/V₁)	Q = W
Isobaric (P const)	W = P(V₂ − V₁) = mR(T₂ − T₁)	Q = mc_p(T₂ − T₁)
Isochoric (V const)	W = 0	Q = mc_v(T₂ − T₁)
Adiabatic (Q = 0)	W = (P₁V₁ − P₂V₂)/(γ−1) = mc_v(T₁ − T₂)	Q = 0
Polytropic (PV^n)	W = (P₁V₁ − P₂V₂)/(n−1), n ≠ 1	Q = mc_v(T₂ − T₁) + W

Open system (control volume), steady state

The steady-flow energy equation (SFEE) is the workhorse:

${\dot{Q}}_{cv}-{\dot{W}}_{cv}={\sum }_{\text{out}}\dot{m}\left(h+\frac{V^2}{2}+gz\right)-{\sum }_{\text{in}}\dot{m}\left(h+\frac{V^2}{2}+gz\right)$

For most engineering devices (single inlet, single outlet, negligible PE):

$\dot{Q}-\dot{W}=\dot{m}\left[(h_2-h_1)+\frac{V_2^2-V_1^2}{2}\right]$

Standard simplifications:

Device	Assumption	Reduced SFEE
Turbine	Adiabatic, ΔKE ≈ 0	w_t = h₁ − h₂
Compressor / pump	Adiabatic, ΔKE ≈ 0	w_c = h₂ − h₁ (input)
Nozzle	Adiabatic, W = 0	V₂² = V₁² + 2(h₁ − h₂)
Diffuser	Adiabatic, W = 0	h₂ = h₁ + (V₁² − V₂²)/2
Throttle (valve, orifice)	Q = W = 0, ΔKE ≈ 0	h₂ = h₁ (isenthalpic)
Heat exchanger	W = 0, ΔKE ≈ 0	ṁ_h(h_h1 − h_h2) = ṁ_c(h_c2 − h_c1)
Mixing chamber	Q = W = 0	Σṁ_i h_i = ṁ_out h_out

The enthalpy h = u + Pv arises naturally in open-system analysis because flow work (Pv) is bundled with internal energy when matter crosses the boundary. This is why turbine work is computed from Δh, not Δu.

6p. Second law — entropy and irreversibility

Statements

• Kelvin–Planck statement: it is impossible to construct a device that, operating in a cycle, produces net work while exchanging heat with only a single thermal reservoir. Equivalent: no heat engine can be 100 % efficient.
• Clausius statement: it is impossible to construct a device that, operating in a cycle, transfers heat from a colder to a hotter body with no other effect. Equivalent: a refrigerator requires work input.

The two are logically equivalent — assume one is violated and you can construct a perpetual-motion machine of the second kind that violates the other.

Carnot cycle and Carnot bound

A Carnot cycle is a reversible heat engine consisting of two isothermals (at T_H and T_C) and two adiabats. Its efficiency:

${\eta }_{\text{Carnot}}=1-\frac{T_C}{T_H}$

where T_H, T_C are absolute temperatures (K or °R). This is the absolute upper bound for any heat engine operating between reservoirs at T_H, T_C — regardless of working fluid, regardless of mechanism.

Corollary: all reversible engines between the same two reservoirs have the same efficiency, equal to Carnot. Any irreversible engine has lower efficiency.

For refrigerators and heat pumps, the analogous bounds:

${\text{COP}}_{\text{Carnot,refrig}}=\frac{T_C}{T_H-T_C},{\text{COP}}_{\text{Carnot,HP}}=\frac{T_H}{T_H-T_C}$

Entropy and entropy generation

The Clausius inequality for any cyclic process:

$\oint \frac{\delta Q}{T}\le 0(=0\text{for reversible})$

defines entropy as a state property dS = (δQ/T)_rev. For any process:

$dS=\frac{\delta Q}{T_b}+\delta S_{gen},\delta S_{gen}\ge 0$

where T_b is the boundary temperature where heat is transferred. Entropy generation never decreases in any real process; it is zero only for reversible processes.

For a control volume at steady state:

$0={\sum }_{\text{in}}\dot{m}s-{\sum }_{\text{out}}\dot{m}s+{\sum }_j\frac{{\dot{Q}}_j}{T_j}+{\dot{S}}_{gen}$

Exergy (availability)

The exergy (or availability) of a stream is the maximum useful work obtainable as the stream is brought to equilibrium with a specified dead state (T₀, P₀):

$\psi =(h-h_0)-T_0(s-s_0)+\frac{V^2}{2}+gz$

For a closed system: φ = (u − u₀) + P₀(v − v₀) − T₀(s − s₀).

The exergy destruction in any component equals T₀ times the entropy generation:

${\dot{X}}_{dest}=T_0{\dot{S}}_{gen}$

This is the Gouy–Stodola theorem, and it is the most useful single equation in second-law analysis. It localizes losses to components and quantifies them in units of lost work, not lost energy.

7p. Power cycles

Rankine (steam) cycle

Components: boiler → turbine → condenser → pump. Working fluid is water/steam.

• Process 1–2 (pump): adiabatic compression of saturated liquid; w_pump = v_f (P₂ − P₁) ≈ small (~1 % of turbine work).
• Process 2–3 (boiler): isobaric heat addition; q_in = h₃ − h₂.
• Process 3–4 (turbine): adiabatic expansion; w_turb = h₃ − h₄.
• Process 4–1 (condenser): isobaric heat rejection; q_out = h₄ − h₁.

Thermal efficiency: η_th = (w_turb − w_pump) / q_in = 1 − q_out/q_in.

Real plant variants: reheat (expand to intermediate P, re-heat in boiler, expand again — raises avg T_H, prevents wet steam in LP turbine), regenerative feedwater heating (bleed steam to preheat boiler feedwater — reduces external heat addition), supercritical (P > 22.064 MPa, no two-phase boiling), ultra-supercritical (P > 25 MPa, T > 600 °C).

Modern Rankine plants (typical η_th):

• Subcritical coal: 36–38 %
• Supercritical coal: 40–42 %
• Ultra-supercritical (USC) coal: 44–46 % (e.g. RDK 8, Karlsruhe, 47.5 %)
• Advanced USC (A-USC, 700 °C class, under development): targeted 48–50 %

The Carnot bound at T_H = 600 °C = 873 K, T_C = 30 °C = 303 K is η_C = 1 − 303/873 = 65 %. Real USC plants thus capture roughly 2/3 of the Carnot maximum.

Brayton (gas turbine) cycle

Components: compressor → combustor → turbine. Working fluid is air (open cycle, dumping exhaust) or sometimes helium / supercritical CO₂ (closed cycle).

• Process 1–2 (compressor): adiabatic compression.
• Process 2–3 (combustor): isobaric heat addition (combustion).
• Process 3–4 (turbine): adiabatic expansion.
• Process 4–1 (atmosphere or HX): isobaric heat rejection.

Cold-air-standard ideal efficiency:

${\eta }_{\text{Brayton}}=1-\frac{1}{r_p^{(\gamma -1)/\gamma }}$

where r_p = P₂/P₁ is the pressure ratio. Modern simple-cycle gas turbines run r_p = 15–25 and reach η = 35–42 %.

Combined-cycle gas turbine (CCGT): Brayton topping cycle with a Rankine bottoming cycle recovering exhaust heat at ~600 °C. Today’s flagships:

• GE 9HA.02 (50 Hz, 571 MW gas turbine, CCGT total ~826 MW): η_CC ≈ 64.0 % (LHV).
• Siemens SGT-9000HL (50 Hz, 593 MW gas turbine, CCGT ~840 MW): η_CC > 63 %.
• Mitsubishi Power M501JAC (60 Hz, 425 MW gas turbine, CCGT ~600 MW): η_CC ≈ 64 %.

These are among the highest-efficiency heat engines ever built; the limit is set by turbine inlet temperature (TIT, ~1700 °C in J-class machines) and combustor NOₓ formation.

Variations: intercooled (cool between compressor stages — reduces compression work), recuperated (recover turbine exhaust heat to preheat combustor inlet — reduces fuel), reheated (re-heat between turbine stages — raises avg T_H, like steam reheat).

Otto cycle (spark-ignition reciprocating engine)

Cold-air-standard idealization of the gasoline 4-stroke. Constant-volume heat addition.

${\eta }_{\text{Otto}}=1-\frac{1}{r^{\gamma -1}}$

where r = V_BDC / V_TDC is the compression ratio. Typical r:

• Naturally aspirated gasoline: 9–12 (knock-limited)
• Atkinson-cycle hybrid (Toyota Prius gen 4): expansion ratio ≈ 13, geometric compression ~10
• High-octane / direct-injection gasoline: up to 14

At r = 10, γ = 1.4: η_Otto = 1 − 10⁻⁰·⁴ = 60.2 %. Real engines deliver 30–38 % BTE (brake thermal efficiency) — losses in heat transfer to coolant, friction, pumping, throttling, incomplete combustion, and accessory drives.

Diesel cycle (compression-ignition)

Constant-pressure heat addition (idealized).

${\eta }_{\text{Diesel}}=1-\frac{1}{r^{\gamma -1}}\cdot \frac{r_c^{\gamma }-1}{\gamma (r_c-1)}$

where r_c = V₃/V₂ is the cut-off ratio (volume at end vs start of heat addition). For r = 18, r_c = 2, γ = 1.4: η_Diesel ≈ 63 %. Real heavy-duty diesel: 42–48 % BTE; the MAN B&W S60ME-C marine 2-stroke achieves > 50 % BTE — the most efficient large-scale heat engine in routine commercial use.

Dual cycle

Real reciprocating engines have heat release that is part constant-volume (rapid premixed burn near TDC) and part constant-pressure (slower diffusion burn). The dual (Sabathé) cycle models both — used for accurate prediction of diesel and high-load gasoline engines.

Stirling and Ericsson cycles

Stirling: two isothermals + two isochorics, with perfect regeneration → reaches Carnot efficiency in the ideal limit. Practical Stirling engines (Philips, Kockums submarine AIP, NASA RPS) achieve 30–40 % at modest size.

Ericsson: two isothermals + two isobarics, with perfect regeneration → also Carnot in the ideal limit. Practical implementation is rare.

8p. Refrigeration and heat-pump cycles

Vapor-compression cycle

The workhorse for AC, refrigeration, heat pumps. Reverse of a Rankine cycle in topology.

• Process 1–2 (compressor): adiabatic compression of saturated vapor; w_in = h₂ − h₁.
• Process 2–3 (condenser): isobaric heat rejection at high P; q_out = h₂ − h₃.
• Process 3–4 (expansion valve / capillary): isenthalpic throttle; h₄ = h₃.
• Process 4–1 (evaporator): isobaric heat absorption at low P; q_in = h₁ − h₄.

${\text{COP}}_{\text{cooling}}=\frac{q_{in}}{w_{in}}=\frac{h_1-h_4}{h_2-h_1},{\text{COP}}_{\text{HP}}=\frac{q_{out}}{w_{in}}$

Absorption cycle

Heat-driven (not work-driven). Uses a fluid pair: NH₃-H₂O (low-T refrigeration, where water is absorbent, ammonia refrigerant) or LiBr-H₂O (chillers, where LiBr is absorbent, water refrigerant). COP ~0.7 in single-effect, ~1.2 in double-effect.

Used where waste heat (e.g. CHP plant) is cheaper than electricity for compressor drive.

Cascade and multi-stage cycles

For very low evaporator T (deep refrigeration, gas liquefaction), a single refrigerant cannot span the T_H–T_C range efficiently. Cascade systems stack two or more vapor-compression cycles, each with a different refrigerant, sharing a counterflow heat exchanger.

Reverse Brayton (air cycle)

Used in aircraft environmental control systems (ECS) and cryogenic liquefaction (Linde, Claude cycles). Less efficient than vapor-compression at moderate T but works with non-condensing working fluids, no oil, no toxicity.

Refrigerant evolution

Class	Examples	Issue	Status
CFC	R-12, R-11	Ozone depletion (Cl)	Banned (Montreal 1987)
HCFC	R-22, R-123	Mild ozone depletion	Phase-out (Kigali 2016)
HFC	R-134a, R-410A, R-404A	High GWP	Phase-down (Kigali)
HFO	R-1234yf, R-1234ze	Low GWP, mildly flammable (A2L)	Current automotive AC since 2017
Natural	R-744 (CO₂), R-717 (NH₃), R-290 (propane)	CO₂ high P; NH₃ toxic; HCs flammable	Growing in commercial/industrial

Refrigerant	GWP (AR5, 100-yr)	Safety class	Typical use
R-12 (CFC-12)	10 900	A1	Legacy MVAC (banned)
R-22 (HCFC-22)	1 810	A1	Legacy split AC
R-134a (HFC)	1 430	A1	MVAC pre-2017, chillers
R-410A (HFC blend)	2 088	A1	Residential AC
R-32 (HFC)	675	A2L	New residential AC
R-1234yf (HFO)	< 1	A2L	Current MVAC
R-744 (CO₂)	1	A1	Trans-critical heat pumps, supermarket
R-717 (NH₃)	0	B2L	Industrial refrigeration
R-290 (propane)	3	A3	Domestic refrigerators, small AC

GWP source: IPCC AR5 (2013). The Kigali Amendment to the Montreal Protocol mandates a phase-down of HFCs starting 2019 (developed countries 85 % cut by 2036).

9p. Worked examples

Example A — Ideal Rankine cycle with superheat

Problem. Steam enters the turbine at P₃ = 8 MPa, T₃ = 500 °C and exits to a condenser at P₄ = 10 kPa. Compute the thermal efficiency assuming isentropic turbine and pump, no pressure drops in boiler or condenser.

Properties (IAPWS-IF97 via steam tables / CoolProp):

Point	P [kPa]	T [°C]	h [kJ/kg]	s [kJ/(kg·K)]	x
1 (sat liq, P₁ = 10 kPa)	10	45.81	191.81	0.6492	0
2 (pump out, isentropic)	8 000	~45.9	199.93	0.6492	—
3 (superheated steam)	8 000	500	3399.3	6.7266	—
4 (turbine out, isentropic)	10	45.81	2117.3	6.7266	0.8059

State 2 computed from h₂ = h₁ + v_f,1 (P₂ − P₁) = 191.81 + 0.001010 × (8000 − 10) = 191.81 + 8.07 = 199.88 kJ/kg. Tables give 199.93 — close.

State 4: At P = 10 kPa, s_f = 0.6492, s_g = 8.1488 kJ/(kg·K). With s₄ = 6.7266: quality x₄ = (6.7266 − 0.6492)/(8.1488 − 0.6492) = 6.0774/7.4996 = 0.8103. Then h₄ = 191.81 + 0.8103 × 2392.1 = 191.81 + 1938.4 = 2130.2 kJ/kg.

(Slight discrepancy from table value 2117 — comes from rounding in the saturation properties; engineers always re-check with the specific table edition.)

Step 1 — Per-kg work and heat:

• Pump work: w_pump = h₂ − h₁ = 199.93 − 191.81 = 8.12 kJ/kg
• Boiler heat: q_in = h₃ − h₂ = 3399.3 − 199.93 = 3199.4 kJ/kg
• Turbine work: w_turb = h₃ − h₄ = 3399.3 − 2130.2 = 1269.1 kJ/kg
• Condenser heat: q_out = h₄ − h₁ = 2130.2 − 191.81 = 1938.4 kJ/kg

Step 2 — Thermal efficiency:

η_th = (w_turb − w_pump) / q_in = (1269.1 − 8.12) / 3199.4 = 1260.98 / 3199.4 = 0.394 or 39.4 %

Check: 1 − q_out/q_in = 1 − 1938.4/3199.4 = 0.394 ✓

Step 3 — Carnot comparison. Reservoir temperatures: T_H = 500 + 273.15 = 773.15 K (turbine inlet); T_C = 45.81 + 273.15 = 318.96 K (condenser). η_C = 1 − 318.96/773.15 = 0.587 or 58.7 %. The ideal Rankine captures 39.4 / 58.7 = 67 % of Carnot — typical.

Step 4 — Why pump work can usually be neglected. w_pump / w_turb = 8.12 / 1269.1 = 0.6 %. In hand calcs without a property database, setting w_pump ≈ 0 introduces < 1 % error in η_th.

Example B — Otto cycle efficiency and state points

Problem. Cold-air-standard Otto cycle: compression ratio r = 10, intake at P₁ = 100 kPa, T₁ = 300 K, peak temperature T₃ = 1800 K (set by combustion). Compute state points and thermal efficiency. Take γ = 1.4, c_v = 0.718 kJ/(kg·K), R = 0.287 kJ/(kg·K).

Step 1 — State 1 (BDC, intake): P₁ = 100 kPa, T₁ = 300 K, v₁ = RT₁/P₁ = 0.287 × 300 / 100 = 0.861 m³/kg.

Step 2 — State 2 (TDC, end of compression, isentropic): v₂ = v₁/r = 0.0861 m³/kg. T₂ = T₁ · r^(γ−1) = 300 · 10^0.4 = 300 × 2.512 = 753.6 K P₂ = P₁ · r^γ = 100 × 10^1.4 = 100 × 25.12 = 2512 kPa

Step 3 — State 3 (TDC, end of combustion, isochoric heat addition): v₃ = v₂ = 0.0861 m³/kg, T₃ = 1800 K. P₃ = P₂ × T₃/T₂ = 2512 × 1800 / 753.6 = 6000 kPa

Heat added: q_in = c_v (T₃ − T₂) = 0.718 × (1800 − 753.6) = 0.718 × 1046.4 = 751.3 kJ/kg

Step 4 — State 4 (BDC, end of expansion, isentropic): v₄ = v₁ = 0.861 m³/kg. T₄ = T₃ / r^(γ−1) = 1800 / 2.512 = 716.6 K P₄ = P₃ / r^γ = 6000 / 25.12 = 238.9 kPa

Heat rejected: q_out = c_v (T₄ − T₁) = 0.718 × (716.6 − 300) = 0.718 × 416.6 = 299.1 kJ/kg

Step 5 — Net work and efficiency: w_net = q_in − q_out = 751.3 − 299.1 = 452.2 kJ/kg η_Otto (computed) = w_net / q_in = 452.2 / 751.3 = 0.6019 = 60.2 %

Check formula: η = 1 − 1/r^(γ−1) = 1 − 1/2.512 = 0.602 ✓

Real-world comparison. A modern direct-injection turbocharged gasoline engine (e.g. Toyota Dynamic Force 2.5L) achieves ~40 % peak BTE — about 2/3 of the cold-air-standard Otto. Losses: heat transfer to coolant (~25 % of fuel energy), exhaust enthalpy (~25 %), friction + pumping (~10 %), incomplete combustion (~2 %).

Example C — Vapor-compression refrigeration with R-134a

Problem. A vapor-compression cycle using R-134a runs between an evaporator at T_L = −10 °C (P_evap ≈ 200.6 kPa) and a condenser at T_H = 40 °C (P_cond ≈ 1017 kPa). Compressor is isentropic. Compute COP and compare to Carnot.

Properties (REFPROP 10.0 / ASHRAE Handbook 2025 Ch. 30):

Point	P [kPa]	T [°C]	h [kJ/kg]	s [kJ/(kg·K)]
1 (sat vap, P_evap)	200.6	−10	244.55	0.9385
2 (compressor out, isentropic)	1017	~47	277.8	0.9385
3 (sat liq, P_cond)	1017	40	108.26	0.3946
4 (post-throttle)	200.6	−10	108.26	0.4124

State 2 found by extending s = 0.9385 kJ/(kg·K) into the superheated region at P = 1017 kPa; the table gives roughly h₂ ≈ 277.8 kJ/kg.

Step 1 — Refrigerating effect: q_L = h₁ − h₄ = 244.55 − 108.26 = 136.3 kJ/kg

Step 2 — Compressor work: w_c = h₂ − h₁ = 277.8 − 244.55 = 33.25 kJ/kg

Step 3 — Heat rejection: q_H = h₂ − h₃ = 277.8 − 108.26 = 169.5 kJ/kg (Energy check: q_L + w_c = 136.3 + 33.25 = 169.6 ≈ q_H ✓)

Step 4 — COP: COP_cooling = q_L / w_c = 136.3 / 33.25 = 4.10 COP_HP = q_H / w_c = 169.5 / 33.25 = 5.10

Step 5 — Carnot comparison: COP_Carnot,cool = T_L / (T_H − T_L) = 263.15 / (313.15 − 263.15) = 263.15 / 50 = 5.26 Second-law efficiency = COP_real / COP_Carnot = 4.10 / 5.26 = 78 %

A real chiller would also include compressor isentropic efficiency (typ. 70–85 %), subcooling of liquid leaving the condenser (+2–5 K), superheating at evaporator exit (+5–10 K), pressure drops, and parasitic losses. Realistic field COP for a residential split AC is 3.0–4.0.

10p. Real-engine and real-cycle corrections

Ideal-cycle predictions are upper bounds. Engineers correct with component isentropic efficiencies:

${\eta }_{turb}=\frac{w_{actual}}{w_{isentropic}}=\frac{h_3-h_4}{h_3-h_{4s}},{\eta }_{comp}=\frac{w_{isentropic}}{w_{actual}}=\frac{h_{2s}-h_1}{h_2-h_1}$

Typical industrial values:

Component	Typical η	Notes
Steam turbine (large utility)	0.88–0.94	HP stage slightly better than LP
Gas turbine expander	0.88–0.92	Cooled blades, modern J-class
Centrifugal gas compressor	0.78–0.88	Multistage, intercooled
Axial gas compressor (jet engine)	0.85–0.90	Multistage
Reciprocating compressor	0.65–0.85	Single-stage
Centrifugal pump (liquid)	0.65–0.85	High for water; lower for viscous
Boiler feedwater pump	0.82–0.92	Multistage
Throttle / expansion valve	(n/a, isenthalpic)	Highly irreversible

Other losses to account for:

• Pressure drops in heat exchangers: 2–5 % of operating P each side.
• Heat losses from insulated components: < 2 % of throughput in modern plants.
• Mechanical losses (bearings, gearboxes): 0.5–2 %.
• Generator electrical efficiency: 98–99 % large machines.

Component-level exergy destruction in a typical CCGT (Bejan 1997 case study + GE/Siemens published data):

Component	% of total exergy destruction
Combustor (chemical reaction + ΔT)	25–30 %
HRSG (heat exchanger ΔT)	10–15 %
Gas turbine expander	8–10 %
Compressor	5–7 %
Steam turbine	4–6 %
Condenser	3–5 %
Stack losses (exhaust to atm)	5–8 %

The combustor dominates — it always does, because combustion of fuel and oxygen at flame temperatures generates the bulk of the cycle’s entropy. Reducing combustor exergy destruction generally means raising turbine inlet temperature, which is bounded by materials and NOₓ.

11p. Combustion fundamentals

Stoichiometry

For complete combustion of a hydrocarbon C_xH_y:

${\mathrm{C}}_{\mathrm{x}}{\mathrm{H}}_{\mathrm{y}}+(\mathrm{x}+y/4){\mathrm{O}}_2\to \mathrm{x}\mathrm{C}{\mathrm{O}}_2+(y/2){\mathrm{H}}_2\mathrm{O}$

With air (21 % O₂, 79 % N₂ by mole), each mole of O₂ comes with 3.76 mol N₂ (which passes through as inert at modest T). The air-fuel ratio AF_stoich [kg air / kg fuel] for common fuels:

Fuel	AF_stoich	LHV [MJ/kg]	HHV [MJ/kg]	T_ad [°C, stoich, air]
Methane (CH₄)	17.2	50.0	55.5	~1950
Propane (C₃H₈)	15.7	46.4	50.4	~1980
Octane (C₈H₁₈, gasoline)	15.1	44.4	47.3	~2270
Diesel (~C₁₂H₂₃)	14.5	42.6	45.6	~2300
Ethanol (C₂H₅OH)	9.0	26.8	29.7	~1920
Hydrogen (H₂)	34.3	120	142	~2210

LHV (lower heating value) excludes water-vapor latent heat (typical for engines that exhaust water as vapor). HHV (higher heating value) includes it (for condensing boilers).

Equivalence ratio

$\varphi =\frac{(F/A{)}_{actual}}{(F/A{)}_{stoich}}$

φ = 1 stoichiometric, φ < 1 lean, φ > 1 rich. Gasoline engines run φ ≈ 1.0 (three-way catalyst window). Diesel runs lean overall (φ_avg ≈ 0.4–0.7) but with locally-rich diffusion flames. Gas turbines run very lean (φ ≈ 0.4) to limit NOₓ and turbine inlet T.

Adiabatic flame temperature

The temperature attained if all heat of reaction goes into raising the product gas temperature, with no heat loss. Computed by energy-balance with c_p(T) integrals (NASA Glenn polynomials):

${\sum }_{\text{reactants}}{n}_i{h}_i(T_{in})={\sum }_{\text{products}}{n}_j{h}_j(T_{ad})$

Real flame temperatures are lower (radiation losses, dissociation of CO₂ and H₂O above ~1800 K, NOₓ formation).

NOₓ formation

Thermal (Zeldovich) NOₓ dominates above 1800 K:

${\mathrm{N}}_2+\mathrm{O}\rightleftharpoons \mathrm{NO}+\mathrm{N},\mathrm{N}+{\mathrm{O}}_2\rightleftharpoons \mathrm{NO}+\mathrm{O}$

Rate goes as exp(−38000/T) — extremely T-sensitive. Every 100 K reduction in peak flame T roughly halves NO formation. This is the driver behind lean-premixed combustors, exhaust gas recirculation (EGR), and staged combustion.

12p. Edge cases / gotchas

• Compressibility at low pressure. Ideal gas is fine for air at atmospheric P up to high T, but steam below 1 atm or near saturation is not ideal — always use IAPWS-IF97 tables.
• Throttle ≠ valve sometimes. The isenthalpic assumption assumes negligible inlet/outlet KE. For high-velocity valves (Mach > 0.3) the kinetic-energy term matters and the process is no longer purely isenthalpic in the usual hand-calc sense.
• Stagnation vs static states. In nozzles, diffusers, and any compressible flow, the difference between stagnation (total) and static enthalpy is the KE term V²/2. Confusing them is a classic compressible-flow blunder. Stagnation T₀ = T + V²/(2c_p).
• Specific heat temperature dependence. c_p(T) varies substantially: air c_p rises from 1.005 at 300 K to 1.146 at 1500 K. Using a single mid-T average is OK for ΔT < 200 K; for combustion analyses, use temperature-dependent polynomials or table lookups.
• Wet-steam fog and blade erosion. Steam turbine exit quality must stay above ~0.88 to avoid liquid droplets eroding LP blades. This is why reheat is mandatory in modern Rankine plants — without it, the expansion would end deep in the wet region.
• Pinch-point analysis. In any counterflow heat exchanger, the minimum ΔT (pinch) governs both UA size and exergy destruction. Steam plant evaporators: pinch ≈ 5–10 K. Cryogenic LNG plate-fin HX: pinch < 2 K. Approach-temperature optimization is a routine first-law/second-law trade-off.
• Reference state matters. Enthalpy and entropy are defined up to an additive constant. IAPWS uses h = 0, s = 0 at the triple-point saturated liquid (0.01 °C, 611.657 Pa). Refrigerant tables use different conventions (IIR: h = 200 kJ/kg, s = 1.0 at 0 °C sat liquid; ASHRAE pre-2017: h = 0 at −40 °C). When mixing data sources, always verify the reference state.
• Heat-rate vs efficiency. Power-plant engineers quote heat rate in kJ/kWh or Btu/kWh. Conversion: 3600 kJ/kWh ÷ heat rate = thermal efficiency. So a USC plant at 8000 kJ/kWh runs η = 3600/8000 = 45 %. A “10000 Btu/kWh” plant runs at 3412.14/10000 = 34.1 %.
• Negative work, sign errors. The single most common engineering-thermo mistake is mishandling the W sign. Always write down ΔU = Q − W or ΔU = Q + W explicitly at the top of a problem and stick to it.
• Open vs closed system on the same problem. A piston-cylinder with mass flow is a control volume, not a closed system, despite looking like one. Choose the boundary deliberately.

13p. Tools & software

Property data:

• NIST REFPROP 10.0 — gold standard, 156 pure fluids and mixtures via Helmholtz EOS. Commercial; Fortran core with Excel/MATLAB/Python wrappers. Used in every serious research and design org.
• CoolProp — open-source REFPROP-like, ~122 fluids, Python/MATLAB/C++ bindings. Adequate for most engineering work; not always REFPROP-accurate for new mixtures.
• IAPWS-IF97 implementations — for steam: NIST/NBS Steam Tables 1984, X Steam (Magnus Holmgren), IAPWS-IF97 in CoolProp, libSeawater for seawater.
• NIST Chemistry WebBook — free property data (h, s, c_p as f(T)) for thousands of compounds; the citation for one-off lookups.

Cycle and process simulation:

• EES (Engineering Equation Solver) — F-Chart Software, textbook-style equation solver with built-in property tables (steam, refrigerants, air). The default tool for undergraduate and early-career thermo work. Light footprint, excellent for parametric studies.
• Aspen HYSYS and Aspen Plus — heavyweight process simulators, used for refineries, petrochem, LNG, gas processing. HYSYS for dynamic and fluid-mechanics-heavy, Plus for steady-state chemical processes.
• Aspen HTFS+ — specialized for heat-exchanger design (TEMA shell-and-tube, plate-fin, air-cooled).
• gPROMS (PSE) — dynamic process simulation and optimization, used heavily in pharma and chemicals.
• DWSIM — open-source process simulator (.NET, cross-platform). Cubic EOS, NRTL, UNIFAC. Adequate for teaching and small-scale design.
• ChemCAD, ProSim, PRO/II (AVEVA) — commercial competitors to Aspen Plus.

Combustion and reacting flows:

• Cantera — open-source (Python/C++/MATLAB), the reference for 0-D and 1-D combustion modeling, reaction-kinetics integration. GRI-Mech 3.0 mechanism for natural gas.
• CHEMKIN (Ansys) — long-standing commercial combustion-kinetics solver.

Steam tables and Mollier diagrams:

• NBS/NIST Steam Tables (Haar, Gallagher, Kell 1984) — pre-IF97 tables, still widely cited.
• Mollier h-s diagram — paper Mollier charts are still on every Rankine designer’s wall; the diagonal-isobars layout makes turbine work visualizable at a glance.
• R&S (Rogers & Mayhew) Thermodynamic and Transport Properties of Fluids — the British “blue book”, standard in UK/Commonwealth engineering education.

HVAC and refrigeration:

• ASHRAE Handbook Fundamentals (2025) — the practitioners’ reference; psychrometric chart, refrigerant property tables, load calculation methods.
• EnergyPlus (DOE/LBNL) — open-source building energy simulation; embeds psychrometric and HVAC equipment models.
• TRNSYS — transient systems simulation for solar-thermal and HVAC.

14. Cross-references

• heat-transfer — rate equations (conduction, convection, radiation) that put time scales on thermodynamic equilibria. Sister discipline; together they form thermal engineering.
• fluid-mechanics — compressible flow, nozzles, diffusers — apply SFEE with KE terms.
• pumps-turbomachinery — Euler turbine equation, performance maps, specific speed; thermodynamic states map directly to component design.
• hvac-fundamentals — psychrometrics, refrigeration system sizing, building loads. ASHRAE-grounded.
• propulsion — jet engines (turbojet, turbofan, turboprop, ramjet), rockets (chemical, electric); Brayton-cycle extensions.
• chemical-process-fundamentals — chemical-engineering thermodynamics, multicomponent VLE, distillation, reactor energy balance.
• electric-motors — efficiency framing parallels but mechanism differs (no Carnot bound on electric→mechanical).
• circuit-analysis — passive-component thermal analog (Joule heating, RC time constant ↔ thermal mass × thermal resistance).
• statics-fundamentals — control-volume concept generalizes from force balance to energy balance.
• mechanics-of-materials — thermal stress at the boundary between thermo (ΔT) and solid mechanics (E·α·ΔT).
• scientific — Aspen, HYSYS, gPROMS, DWSIM input formats; CAPE-OPEN; Modelica/Modelica Standard Library thermal models.

15. Citations

1. Çengel, Y. A.; Boles, M. A.; Kanoğlu, M. Thermodynamics: An Engineering Approach, 10th ed. McGraw-Hill, 2023. ISBN 978-1264512676. The dominant undergraduate text in the US; clear, problem-heavy, integrated property tables.
2. Moran, M. J.; Shapiro, H. N.; Boettner, D. D.; Bailey, M. B. Fundamentals of Engineering Thermodynamics, 10th ed. Wiley, 2024. ISBN 978-1119820079. The most rigorous mainstream text; preferred in graduate-prep curricula.
3. Sonntag, R. E.; Borgnakke, C. Fundamentals of Thermodynamics, 10th ed. Wiley, 2019. ISBN 978-1119494966. Lineage from Van Wylen, the classical statement of engineering thermo.
4. Bejan, A. Advanced Engineering Thermodynamics, 4th ed. Wiley, 2016. ISBN 978-1119052098. The reference for exergy analysis, entropy-generation minimization, and constructal theory.
5. Annamalai, K.; Puri, I. K.; Jog, M. A. Advanced Thermodynamics Engineering, 2nd ed. CRC Press, 2011. ISBN 978-1439805718. Graduate-level treatment of combustion and multicomponent systems.
6. Smith, J. M.; Van Ness, H. C.; Abbott, M. M.; Swihart, M. T. Introduction to Chemical Engineering Thermodynamics, 9th ed. McGraw-Hill, 2022. ISBN 978-1260597684. The standard ChemE thermo text; deeper on phase equilibria and mixtures.
7. Glassman, I.; Yetter, R. A.; Glumac, N. G. Combustion, 5th ed. Academic Press, 2014. ISBN 978-0124079137. The reference for flame structure, kinetics, and reacting flow.
8. IAPWS — International Association for the Properties of Water and Steam. Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam, IAPWS R7-97 (2012). The legally-binding steam-table standard for utility plant.
9. IAPWS — Guideline on the Henry’s Constant and Vapor-Liquid Distribution Constant for Gases in H₂O and D₂O at High Temperatures, IAPWS G7-04 (2004).
10. NIST — REFPROP Version 10.0 Reference Manual. Lemmon, E. W.; Bell, I. H.; Huber, M. L.; McLinden, M. O. National Institute of Standards and Technology, Standard Reference Database 23, 2018.
11. ASHRAE — 2025 ASHRAE Handbook — Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 2025. ISBN 978-1947192-99-2. The HVAC/refrigeration practitioner’s primary reference.
12. ASME PTC 6-2020 — Steam Turbines: Performance Test Code. American Society of Mechanical Engineers, 2020. Acceptance-test method for utility steam turbines.
13. ASME PTC 22-2014 (R2019) — Gas Turbines: Performance Test Code. American Society of Mechanical Engineers, 2014. Acceptance-test method for stationary gas turbines.
14. ASME PTC 46-2015 — Overall Plant Performance. ASME, 2015. CCGT and overall power-plant performance testing.
15. GERG-2008 — Kunz, O.; Wagner, W. The GERG-2008 Wide-Range Equation of State for Natural Gases and Other Mixtures. J. Chem. Eng. Data 57 (2012) 3032–3091. ISO 20765-2/-3 reference EOS for custody-transfer natural gas.
16. Burcat, A.; Ruscic, B. Third Millennium Ideal Gas and Condensed Phase Thermochemical Database for Combustion (with Update from Active Thermochemical Tables). ANL-05/20, Argonne, 2005. The combustion-thermochemistry reference; ongoing online updates.
17. CoolProp — Bell, I. H.; Wronski, J.; Quoilin, S.; Lemort, V. Pure and Pseudo-pure Fluid Thermophysical Property Evaluation and the Open-Source Thermophysical Property Library CoolProp. Ind. Eng. Chem. Res. 53 (2014) 2498–2508. https://coolprop.org
18. Kigali Amendment to the Montreal Protocol (UNEP, 2016). HFC phase-down schedule. https://ozone.unep.org/treaties/montreal-protocol/amendments/kigali-amendment-2016