Zeroth Law

If A is in thermal equilibrium with C, and B is in thermal equilibrium with C, then A is in thermal equilibrium with B. (Defines temperature.)

First Law — Closed System

Q = ΔU + W

Q = heat added (J), ΔU = change in internal energy (J), W = work done by system (J)

For a cycle: Q_net = W_net (since ΔU = 0)

Second Law

ΔS_universe = ΔS_system + ΔS_surroundings ≥ 0

Clausius Inequality: ∮ δQ/T ≤ 0

Equation of State

PV = nRT (molar form)

PV = mR_sT (mass form, R_s = R/M)

R = 8.314 J/(mol·K), R_s,air = 287 J/(kg·K)

Specific Heat Relations

C_p − C_v = R (Mayer’s relation)

γ = C_p / C_v (ratio of specific heats)

For air: C_p = 1.005 kJ/kg·K, C_v = 0.718 kJ/kg·K, γ = 1.4

Internal Energy & Enthalpy (Ideal Gas)

ΔU = nC_vΔT (always, for ideal gas)

ΔH = nC_pΔT (always, for ideal gas)

Isentropic (Adiabatic Reversible) Relations

TV^γ−1 = constant

TP^(1−γ)/γ = constant

PV^γ = constant

General Definition

dS = δQ_rev / T

Ideal Gas Entropy Change

ΔS = nC_v ln(T&sub2;/T&sub1;) + nR ln(V&sub2;/V&sub1;)

ΔS = nC_p ln(T&sub2;/T&sub1;) − nR ln(P&sub2;/P&sub1;)

Phase Change

ΔS = mL / T

L = latent heat (kJ/kg), T = phase change temperature (K)

Entropy Generation (Heat Transfer Across ΔT)

S_gen = Q(1/T_L − 1/T_H)

Definition

H = U + PV, specific: h = u + Pv

Constant Pressure

Q_P = ΔH

Wet Steam

h = h_f + x·h_fg

x = dryness fraction (quality), h_f = saturated liquid, h_fg = latent heat

General Heat Engine Efficiency

η = W_net/Q_H = 1 − Q_L/Q_H

Carnot Cycle

η_Carnot = 1 − T_L/T_H

Maximum possible efficiency between temperatures T_H and T_L (in Kelvin)

Otto Cycle (Petrol Engine)

η_Otto = 1 − 1/r^(γ−1)

r = compression ratio, γ = C_p/C_v

Diesel Cycle

η_Diesel = 1 − [1/r^(γ−1)] × [(ρ^γ − 1) / γ(ρ − 1)]

r = compression ratio, ρ = cut-off ratio = V&sub3;/V&sub2;

Dual Cycle (Mixed)

η_Dual = 1 − [1/r^(γ−1)] × [(r_p·ρ^γ − 1) / (r_p − 1 + γ·r_p(ρ − 1))]

r_p = pressure ratio during constant-V heat addition

Rankine Cycle (Steam Power Plant)

η_Rankine = [(h&sub1; − h&sub2;) − (h&sub4; − h&sub3;)] / (h&sub1; − h&sub4;)

h values from steam tables. Pump work: W_pump = h&sub4; − h&sub3; ≈ v_f(P&sub4; − P&sub3;)

Brayton Cycle (Gas Turbine)

η_Brayton = 1 − 1/r_p^(γ−1)/γ

r_p = pressure ratio = P&sub2;/P&sub1;

Coefficient of Performance — Refrigerator

COP_R = Q_L/W = Q_L/(Q_H − Q_L)

Carnot: COP_R,Carnot = T_L/(T_H − T_L)

Coefficient of Performance — Heat Pump

COP_HP = Q_H/W = Q_H/(Q_H − Q_L)

Carnot: COP_HP,Carnot = T_H/(T_H − T_L)

COP_HP = COP_R + 1

General Form

Q − W = ˙m[(h&sub2; − h&sub1;) + (V&sub2;² − V&sub1;²)/2 + g(z&sub2; − z&sub1;)]

Conduction (Fourier’s Law)

Q = kA(T&sub1; − T&sub2;)/L

k = thermal conductivity (W/m·K), A = area (m²), L = thickness (m)

Thermal resistance: R = L/(kA)

Convection (Newton’s Law of Cooling)

Q = hA(T_s − T_∞)

h = convection coefficient (W/m²·K)

Radiation (Stefan-Boltzmann Law)

Q = εσA(T_s&sup4; − T_surr&sup4;)

ε = emissivity (0–1), σ = 5.67 × 10⁻⁸ W/(m²·K&sup4;), T in Kelvin

K = °C + 273.15

°F = (9/5) × °C + 32

°C = (5/9) × (°F − 32)

ΔT in K = ΔT in °C (temperature differences are the same in both scales)