Thermodynamics – All These Worlds

Foundations & Thermodynamic Language

Concepts Systems State/Process

Thermodynamics studies energy, entropy, and equilibrium constraints. A course begins by defining systems, states, and processes, distinguishing properties (state functions) from path functions, and establishing consistent sign conventions and units.

Core definitions

System vs surroundings; boundary
Closed system: fixed mass; control volume: mass flow allowed
State: described by properties (P, T, v, u, h, s, …)
Process: change of state; cycle: returns to initial state
Equilibrium: mechanical/thermal/phase/chemical equilibrium
Quasi-equilibrium: process proceeds through near-equilibrium states
Intensive vs extensive properties

Sign conventions (typical engineering)

Q > 0 : heat added to system W > 0 : work done by system ΔE = E2 − E1

Many texts define work/heat with these signs; always verify in your course.

Properties of Pure Substances

Properties Phase behavior Tables

You’ll learn to compute thermodynamic properties from equations of state and property tables (water/steam, refrigerants). Central ideas: phase diagrams, saturation, quality, and identifying states from given property pairs.

Phase equilibrium & quality

x = (m_vapor) / (m_total) (in saturated mixture) Specific volume: v = (1−x) v_f + x v_g Internal energy: u = (1−x) u_f + x u_g Enthalpy: h = (1−x) h_f + x h_g Entropy: s = (1−x) s_f + x s_g

Subscripts: f saturated liquid, g saturated vapor
Use if you are in the two-phase dome

Equations of state (EOS)

Ideal gas: PV = nRT or Pv = RT Compressibility: Z = Pv/(RT) ⇒ Pv = ZRT Common real-gas EOS (course-dependent): Van der Waals: (P + a/v^2)(v − b) = RT

Critical point, triple point, saturation lines
P–v and T–s diagrams for process visualization

Work, Heat, and Process Paths

Energy transfer Path functions Sign-sensitive

Heat and work are modes of energy transfer across a boundary, not stored in a system. Boundary work depends on the path (e.g., quasi-static compression/expansion).

Boundary work

δW_b = P dV (quasi-equilibrium) W_b = ∫(1→2) P dV Polytropic: PV^n = constant W_b = (P2V2 − P1V1)/(1 − n) (n ≠ 1) Isothermal ideal gas (T constant): W_b = nRT ln(V2/V1) = mRT ln(v2/v1)

On a P–V diagram, work is the area under the curve.

Other work modes (typical)

Shaft work (turbines/compressors): Ẇ_shaft Electrical work: Ẇ_elec = V I (if applicable) Spring work (simple model): W = ∫ F dx

Many problems separate boundary work from shaft work.
Heat transfer rate is Q̇; work rate is Ẇ.

First Law (Closed Systems)

Law Energy accounting ΔU, ΔKE, ΔPE

The First Law is energy conservation. For a closed system, changes in total energy equal heat in minus work out (with the chosen sign convention).

Energy balance

E = U + KE + PE ΔE = Q − W ΔU + ΔKE + ΔPE = Q − W KE = (1/2) m V^2 PE = m g z

Many textbook problems neglect ΔKE, ΔPE unless stated.

Special processes (common)

Constant volume: W_b = 0 ⇒ Q = ΔU (if only boundary work) Adiabatic: Q = 0 ⇒ ΔE = −W Cyclic: ΔE_cycle = 0 ⇒ Q_net = W_net

Be explicit about which work terms exist in the model.

Control Volumes & Steady-Flow Energy Equation (SFEE)

Open systems Enthalpy Devices

For flowing systems, energy crosses the boundary via mass. This motivates enthalpy and the standard energy equation used for turbines, compressors, nozzles, throttles, heat exchangers, and mixing chambers.

Enthalpy & flow work

h = u + Pv (specific enthalpy) Flow work per unit mass ~ Pv Total specific energy for flow: e = u + Pv + (V^2/2) + gz = h + (V^2/2) + gz

SFEE (single-inlet/single-outlet, steady)

Q̇ − Ẇ = ṁ[(h2 − h1) + (V2^2 − V1^2)/2 + g(z2 − z1)] Per unit mass: q − w = (h2 − h1) + (V2^2 − V1^2)/2 + g(z2 − z1)

Apply device assumptions (adiabatic, negligible ΔKE, etc.) to simplify.

Common device models (typical simplifications)

Nozzle/Diffuser: usually q≈0, w≈0 → Δh ≈ −Δ(V²/2)
Turbine: usually q≈0, ΔKE small → w_out ≈ h1 − h2
Compressor/Pump: usually q≈0 → w_in ≈ h2 − h1 (pump often uses vΔP)
Throttling valve: q≈0, w≈0 → h1 ≈ h2 (Joule–Thomson context)
Heat exchanger: w≈0, ΔKE small → heat transfer changes enthalpy streams
Mixing chamber: combine mass + energy balances (often adiabatic)

Continuity (mass conservation)

Steady flow: ṁ_in = ṁ_out For a stream: ṁ = ρ A V = A V / v

Second Law, Entropy, and Irreversibility

Law Entropy Irreversibility

The Second Law introduces entropy as a state function and imposes directionality: real processes generate entropy. Reversible processes are idealizations that define upper bounds on performance.

Clausius inequality & entropy

Cyclic form: ∮ (δQ/T) ≤ 0 (equality for reversible) Between states: ΔS ≥ ∫(1→2) (δQ/T) (equality for reversible path) Definition (reversible): dS = δQ_rev / T

Entropy balance (control volume / system)

Closed system: ΔS = ∫(δQ/T_boundary) + S_gen Control volume (rate form): dS_cv/dt = Σ( Q̇_k / T_k ) + Σ(ṁ_in s_in) − Σ(ṁ_out s_out) + Ṡ_gen Second law constraint: S_gen ≥ 0 and Ṡ_gen ≥ 0

Typical entropy-generation sources: friction, unrestrained expansion, mixing, finite-ΔT heat transfer, shocks, viscous dissipation, chemical reactions.

Heat Engines, Refrigerators, and Carnot Limits

Performance limits Carnot Uses absolute temperature

The Second Law implies that converting heat fully into work is impossible in a cycle; maximum efficiency is bounded by reversible (Carnot) performance between two reservoirs.

Heat engine

Thermal efficiency: η_th = W_net,out / Q_in = 1 − (Q_out / Q_in) Carnot (reversible, between T_H and T_C): η_Carnot = 1 − T_C/T_H

T must be in absolute units (K or R).

Refrigerator / heat pump

COP_R = Q_C / W_net,in COP_HP = Q_H / W_net,in Carnot refrigerator: COP_R,Carnot = T_C/(T_H − T_C) Carnot heat pump: COP_HP,Carnot = T_H/(T_H − T_C)

T–ds Relations, Maxwell Relations, and Property Relations

Derivations Property math Partial derivatives

A core mid-course goal is to connect measurable variables (P, T, v) to energy/entropy properties (u, h, s) via exact differentials and potentials. These relations let you compute Δu, Δh, Δs for real substances using tables, ideal-gas models, or departure functions (advanced).

Fundamental relations

From definitions: h = u + Pv Differentials (simple compressible system): du = T ds − P dv dh = T ds + v dP

Thermodynamic potentials

Helmholtz: a = u − Ts Gibbs: g = h − Ts Differentials: da = −s dT − P dv dg = −s dT + v dP

Maxwell relations (common set)

From da(T,v): (∂s/∂v)_T = (∂P/∂T)_v From dg(T,P): (∂s/∂P)_T = −(∂v/∂T)_P From du(s,v): (∂T/∂v)_s = −(∂P/∂s)_v From dh(s,P): (∂T/∂P)_s = (∂v/∂s)_P

Useful identities

For any property φ: dφ = (∂φ/∂T)_P dT + (∂φ/∂P)_T dP (or in other variable pairs) Specific heats: c_v = (∂u/∂T)_v c_p = (∂h/∂T)_P

These set up ideal-gas relations and real-fluid corrections.

Ideal Gas Thermodynamics (c_p, c_v, Δu, Δh, Δs)

Model Common calculations Check assumptions

For many gases at moderate pressures and temperatures, the ideal-gas model gives accurate property changes: u(T), h(T), and entropy changes from (T, P) or (T, v). Specific heats may be treated constant or variable with temperature.

Core relations

Pv = RT h = h(T) only, u = u(T) only (ideal gas) Δu = ∫(T1→T2) c_v(T) dT Δh = ∫(T1→T2) c_p(T) dT If c_p, c_v constant: Δu = c_v (T2 − T1) Δh = c_p (T2 − T1) Relation: c_p − c_v = R

Entropy change (ideal gas)

General: Δs = ∫(T1→T2) [c_p(T)/T] dT − R ln(P2/P1) = ∫(T1→T2) [c_v(T)/T] dT + R ln(v2/v1) Constant specific heats: Δs = c_p ln(T2/T1) − R ln(P2/P1) = c_v ln(T2/T1) + R ln(v2/v1)

Ideal-gas isentropic relations (below) require additional assumptions: internally reversible + adiabatic + (often) constant specific heats.

Isentropic Processes and Ideal-Gas Relations

Reversible adiabatic Upper bounds Often uses γ

“Isentropic” usually means adiabatic and internally reversible (so Δs=0). These relations are core to turbine/compressor analysis and to basic compressible-flow device models.

Ideal gas, constant γ = cₚ/cᵥ

PV^γ = constant (or Pv^γ = constant) T2/T1 = (P2/P1)^((γ−1)/γ) T2/T1 = (v1/v2)^(γ−1) P2/P1 = (v1/v2)^γ

Isentropic efficiencies (devices)

Turbine isentropic efficiency: η_t = (W_actual,out)/(W_isentropic,out) = (h1 − h2)/(h1 − h2s) Compressor isentropic efficiency: η_c = (W_isentropic,in)/(W_actual,in) = (h2s − h1)/(h2 − h1)

“2s” denotes the outlet state if the process were isentropic to the same outlet pressure.

Real Substances, Compressibility, and Generalized Charts

Real-fluid behavior Z Chart/table dependence

When ideal-gas assumptions break, courses introduce the compressibility factor, corresponding states, and generalized charts (or modern EOS software). This bridges toward real-gas property estimation.

Compressibility and reduced variables

Z = Pv/(RT) Reduced temperature: T_r = T / T_c Reduced pressure: P_r = P / P_c Principle of corresponding states: Z ≈ Z(T_r, P_r) (plus acentric factor ω for better accuracy)

Departure functions (advanced in some courses)

Real property = ideal-gas property at same T + departure (correction) Example concept: h(T,P) = h_ig(T) + h_dep(T,P) s(T,P) = s_ig(T,P) + s_dep(T,P)

Often implemented via charts or EOS; detailed derivation may be optional.

Vapor Power Cycles (Rankine) and Steam Tables

Cycles Water/steam Power plants

The Rankine cycle models steam power plants: pumping liquid water to high pressure, boiling/superheating, expanding through a turbine, then condensing. Analysis uses steady-flow devices + tables for h and s.

Basic Rankine metrics

Thermal efficiency: η_th = W_net / Q_in Net work per unit mass: w_net = w_turbine − w_pump Heat input: q_in = h3 − h2 (boiler) [state numbering varies]

Typical device models: turbine/pump adiabatic; boiler/condenser no shaft work.
Pump work (incompressible approx): w_p ≈ v (P2 − P1)

Cycle improvements (conceptual)

Superheating: increases average T of heat addition, reduces moisture at turbine exit
Reheat: reduces turbine exhaust quality issues, can improve efficiency
Regeneration: feedwater heaters raise feedwater temperature, improving efficiency
Reheat + regen: common in modern plants

Gas Power Cycles (Otto, Diesel, Brayton)

Ideal cycles Engines/turbines Air-standard assumptions

Idealized air-standard cycles approximate internal combustion engines (Otto/Diesel) and gas turbines (Brayton). They use ideal-gas relations and isentropic compression/expansion to derive efficiency trends.

Otto cycle (spark ignition, ideal)

Compression ratio: r = v1/v2 η_Otto = 1 − 1/(r^(γ−1))

Heat addition at constant volume (idealization).

Diesel cycle (compression ignition, ideal)

Cutoff ratio: ρ = v3/v2 η_Diesel = 1 − (1/r^(γ−1)) * [(ρ^γ − 1)/(γ(ρ − 1))]

Heat addition at constant pressure (idealization).

Brayton cycle (gas turbine, ideal)

Pressure ratio: r_p = P2/P1 For ideal Brayton (constant γ): η_Brayton = 1 − 1/(r_p^((γ−1)/γ))

Efficiency increases with pressure ratio (within limits), but real components have losses.

Brayton enhancements (conceptual)

Regeneration: recover exhaust heat to preheat compressed air
Intercooling: reduce compressor work across stages
Reheat: increase turbine work across stages
Tradeoffs: pressure drops, added complexity, diminishing returns

Refrigeration Cycles (Vapor Compression) and Psychrometrics

Refrigeration COP Refrigerants

Refrigeration reverses the “heat engine” purpose: it uses work to move heat from cold to hot. The ideal vapor-compression cycle uses an evaporator, compressor, condenser, and expansion device. Many courses also include basic moist-air (psychrometric) relationships for HVAC context.

Vapor-compression cycle (key relations)

COP_R = Q_L / W_in Per unit mass: q_L = h1 − h4 (evaporator) w_in = h2 − h1 (compressor) q_H = h2 − h3 (condenser) Throttling valve: h3 ≈ h4 (States depend on diagram/numbering convention)

Psychrometrics (if included)

Humidity ratio: ω = m_vapor / m_dry-air Relative humidity: φ = p_v / p_sat(T) Moist air enthalpy (approx): h ≈ h_da + ω h_v

Often uses psychrometric charts; detailed formulas vary by text.

Mixtures, Phase Equilibrium, and Property Estimation

Mixtures Equilibrium Model choices

Depending on the course (engineering thermo vs physical chemistry emphasis), mixtures may include ideal-gas mixtures, partial pressures, and simple phase equilibrium. Advanced treatments use chemical potential and fugacity.

Ideal-gas mixture basics

Mole fraction: y_i = n_i / n_total Dalton’s law (ideal gases): P = Σ p_i , with p_i = y_i P Mixture gas constant: R_mix = Σ (y_i R_i) (or based on molecular weight) Mixture properties (ideal mixing, for some properties): h_mix ≈ Σ (y_i h_i(T)) u_mix ≈ Σ (y_i u_i(T))

Phase equilibrium (typical exposure)

Gibbs phase rule concept (degrees of freedom)
Simple VLE ideas (Raoult’s law in ideal solutions) if covered
Recognize when “ideal” assumptions are invalid (non-ideal solutions)

Exergy (Availability) and Second-Law Efficiency

Exergy Destruction via irreversibility Design insight

Exergy quantifies the maximum useful work obtainable as a system comes to equilibrium with a reference environment (the “dead state”). It links irreversibility to lost work and provides a sharper design diagnostic than energy alone.

Key ideas & equations (common form)

Exergy destruction (lost work): X_destroyed = T0 S_gen Rate form: Ẋ_destroyed = T0 Ṡ_gen Second-law (exergetic) efficiency (concept): η_II = (useful exergy out) / (exergy in)

T0 is the environment (dead-state) temperature.
Exergy analysis highlights where “quality” of energy is degraded.

Physical meaning

Energy is conserved; exergy is consumed by irreversibility.
Minimize entropy generation to preserve work potential.
Especially useful for heat exchangers, throttles, combustion, and power cycles.

Transient (Unsteady) Systems and Lumped Analysis

Unsteady Tanks Control-mass vs control-volume choice

Many practical problems are unsteady: filling/emptying tanks, start-up/shut-down, charging vessels. You’ll apply integral mass and energy balances over time, often with simplifying assumptions (uniform properties in a tank).

General unsteady control volume (conceptual forms)

Mass: d(m_cv)/dt = Σṁ_in − Σṁ_out Energy: d(E_cv)/dt = Q̇ − Ẇ + Σṁ_in e_in − Σṁ_out e_out Entropy: d(S_cv)/dt = Σ(Q̇_k/T_k) + Σṁ_in s_in − Σṁ_out s_out + Ṡ_gen

Typical modeling moves

Lumped property assumption in tanks: uniform T, P (approx).
Neglect KE/PE except where flow speed/elevation matters.
Choose control mass vs control volume for algebraic simplicity.

Heat Transfer Interfaces (as Used in Thermo)

Interface tools Often a separate course Heat exchangers

Many thermo courses include “just enough” heat-transfer modeling to analyze reservoirs and heat exchangers. Full derivations usually belong to a dedicated heat transfer course, but the following show up often.

Basic modes (common engineering forms)

Conduction (1-D, steady): Q̇ = k A (T_hot − T_cold)/L Convection: Q̇ = h A (T_s − T_∞) Radiation (to large surroundings): Q̇ = ε σ A (T_s^4 − T_sur^4)

Use consistent absolute temperatures for radiation.

Heat exchanger analysis (if included)

LMTD method: Q̇ = U A ΔT_lm Effectiveness–NTU method: ε = Q̇ / Q̇_max NTU = (U A) / C_min

Thermo typically uses results; heat transfer derives them.

Cycle Analysis Pattern (A Repeatable Method)

Workflow Problem-solving

Across Rankine, Brayton, Otto/Diesel, and refrigeration, analysis is mostly the same: define states, apply device models, compute properties, then compute work/heat and performance metrics. The difference is the working fluid and which tables/models you use.

Standard workflow

Sketch the cycle on P–v and/or T–s
Label states; identify known property pairs (e.g., P & T)
State assumptions per device (adiabatic, isentropic efficiency, negligible KE/PE)
Use tables/EOS/ideal-gas relations to compute unknown properties
Apply mass & energy balances (SFEE) to each device
Compute performance (η, COP, specific work, heat rates)
Use Second Law / exergy to diagnose losses (optional but powerful)

Common pitfalls

Mixing units (kPa vs Pa; kJ/kg vs J/kg)
Confusing “adiabatic” with “isentropic” (needs reversibility)
Using quality relations outside the two-phase region
Forgetting pump/compressor efficiency definitions are inverted

Thermo Math Tools & Notation

Notation Differentials Partial derivatives

Thermodynamics relies on careful calculus notation: exact vs inexact differentials, and the meaning of partial derivatives with held variables. Getting notation right prevents conceptual errors (especially around Q and W).

Exact vs inexact

State functions (properties): dU, dH, dS, dP, dT, dv … (exact differentials) Path functions: δQ, δW (inexact differentials; depend on path)

Common derivative shorthand

(∂u/∂T)_v means: change u with T, holding v constant (∂h/∂P)_T means: change h with P, holding T constant

Maxwell relations connect mixed partial derivatives via potentials.

Common Symbols and What They Mean

Reference Units

A consistent symbol map avoids confusion when switching between closed-system and control-volume forms.

State variables / properties

P : pressure (Pa, kPa, bar) T : temperature (K) v : specific volume (m^3/kg) ρ : density (kg/m^3), ρ = 1/v u : specific internal energy (kJ/kg) h : specific enthalpy (kJ/kg) s : specific entropy (kJ/(kg·K)) g : Gibbs free energy (kJ/kg) a : Helmholtz free energy (kJ/kg) x : quality (two-phase), 0→1

Transfers and rates

Q : heat transfer (kJ) Q̇ : heat transfer rate (kW) W : work (kJ) Ẇ : power (kW) ṁ : mass flow rate (kg/s) V : velocity magnitude (m/s) z : elevation (m) c_p, c_v : specific heats (kJ/(kg·K)) R : gas constant (kJ/(kg·K)) γ : ratio of specific heats, γ=c_p/c_v