thermodynamic-cycles

name	thermodynamic-cycles
description	Power and refrigeration cycle analysis — Rankine, Brayton, Otto, Diesel, Stirling, vapor compression refrigeration. Efficiency, COP, work, heat, irreversibility, T-s and P-v diagrams.
metadata	{"priority":7,"promptSignals":{"phrases":["thermodynamic cycle","Rankine","Brayton","Otto","Diesel","steam cycle","gas turbine","COP","efficiency","refrigeration"],"minScore":4}}

Thermodynamic Cycles — Complete Skill

First and Second Law

First law: Q - W = ΔU (closed), q - w = Δh (steady flow per unit mass) Second law: Δs ≥ 0, s_gen ≥ 0, η_thermal ≤ η_Carnot Carnot efficiency: η_Carnot = 1 - T_L/T_H (max possible) Carnot COP_ref = T_L/(T_H-T_L), COP_HP = T_H/(T_H-T_L)

Rankine Cycle (Steam Power)

States (pump → boiler → turbine → condenser): 1→2: Pump (liquid): h₂ = h₁ + v₁(P₂-P₁)/η_pump (isentropic: η_pump=1, s₁=s₂) 2→3: Boiler (constant P): q_in = h₃ - h₂ 3→4: Turbine: w_t = h₃ - h₄ (isentropic: η_turb=1, s₃=s₄) h₄s = h_f4 + x₄·h_fg4, x₄ = (s₃-s_f4)/s_fg4 Actual: w_t,actual = η_turb·w_t,ideal, h₄ = h₃ - η_turb·(h₃-h₄s) 4→1: Condenser (constant P, T): q_out = h₄ - h₁

Performance: η_th = w_net/q_in = (w_t - w_p)/q_in Back work ratio = w_p/w_t (steam: 0.5-1%, gas turbine: 40-80%) w_net = w_t - w_p = (h₃-h₄) - (h₂-h₁)

Improvements:

• Superheat: raise T₃ (increases η and x₄)
• Reheat: expand to intermediate P, reheat, expand again η increases ~4-5%, x at exit rises above 0.90
• Regeneration: feedwater heater (open or closed) η_regen = 1 - q_out,regen/q_in,regen (significantly improved)
• Supercritical: T₃ > 374°C, P₃ > 22.1 MPa (modern plants: P=30MPa, T=600°C)

Brayton Cycle (Gas Turbine)

Ideal (cold air standard): 1→2: Isentropic compression: T₂/T₁ = (P₂/P₁)^((γ-1)/γ) = r_p^((γ-1)/γ) 2→3: Constant pressure heat addition: q_in = c_p(T₃-T₂) 3→4: Isentropic expansion: T₃/T₄ = r_p^((γ-1)/γ) 4→1: Constant pressure heat rejection: q_out = c_p(T₄-T₁)

Ideal efficiency: η_Brayton = 1 - 1/r_p^((γ-1)/γ) r_p = pressure ratio (typical: 10-40 for aircraft, 15-25 for industrial)

Optimal pressure ratio for max work: r_p,opt = (T₃/T₁)^(γ/(2(γ-1)))

With irreversibilities: η_comp = (h₂s-h₁)/(h₂-h₁), η_turb = (h₃-h₄)/(h₃-h₄s) w_net = η_turb·w_t,ideal - w_c,ideal/η_comp

Improvements:

• Regeneration: ε = (T₅-T₂)/(T₄-T₂), η improves at low r_p
• Intercooling (multi-stage compression): reduces compressor work
• Reheating: increases turbine work, lowers η unless combined with regeneration
• Combined cycle (Brayton + Rankine): η up to 60%+

Otto Cycle (Spark Ignition — Gasoline)

r = compression ratio = V₁/V₂ (typical 8-12) η_Otto = 1 - 1/r^(γ-1) MEP = q_in(r-1)r^(γ-1)·γ(r-1)·T₁ / (... complex) γ = 1.4 (air), γ = 1.35 (air-fuel mixture) T₂ = T₁·r^(γ-1), P₂ = P₁·r^γ T₃ = T₂·(q_in/c_v)/T₂+T₂ (from q_in = c_v(T₃-T₂))

Diesel Cycle (Compression Ignition)

r = compression ratio, r_c = cutoff ratio = V₃/V₂ (1.5-4) η_Diesel = 1 - (r_c^γ-1)/(γ·r^(γ-1)·(r_c-1)) η_Diesel < η_Otto for same r, but Diesel uses higher r (14-22) → overall higher η

Stirling Cycle (External Combustion)

η_Stirling,ideal = η_Carnot = 1 - T_L/T_H (with perfect regenerator) Practical: η ≈ 0.30-0.40 Applications: solar power, submarine propulsion, cryocoolers

Vapor Compression Refrigeration

1→2: Compressor (isentropic, superheated vapor) 2→3: Condenser (constant P, desuperheat + condense) 3→4: Expansion valve (throttle, h₃=h₄, irreversible) 4→1: Evaporator (constant P, T — evaporation)

COP_R = q_L/w_c = (h₁-h₄)/(h₂-h₁) COP_HP = q_H/w_c = (h₂-h₃)/(h₂-h₁)

Actual COP: η_comp = (h₂s-h₁)/(h₂-h₁), COP_actual = COP_ideal·η_comp (roughly)

Refrigerants: R-134a: Tc=101.1°C, Pc=4.06MPa (HFCs, zero ozone depletion) R-410A: Tc=72.1°C, Pc=4.95MPa (residential AC) R-744 (CO₂): Tc=31°C, Pc=7.38MPa (transcritical, natural) R-717 (NH₃): Tc=132.3°C — industrial, highest efficiency, toxic R-290 (Propane): natural, flammable — small systems

Exergy / Availability Analysis

Exergy: Φ = (h-h₀) - T₀(s-s₀) Exergy destroyed: Φ_dest = T₀·σ (σ = entropy generation) Second-law efficiency: η_II = useful exergy output / exergy input Identifies where irreversibilities are largest

Output

Provide: η_thermal or COP, w_net, q_in, q_out, all state properties (T, P, h, s, x), T-s diagram description, improvement suggestions.