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Power cycles convert heat input to net work output with thermal efficiency η = W_net/Q_in. The Carnot cycle sets an upper bound: η_Carnot = 1 - T_cold/T_hot. Real cycles (Rankine, Brayton, Otto) operate below Carnot due to irreversibilities and practical constraints. Cycle efficiency improves through higher pressure ratios, superheat, reheat, regeneration, and reduced losses.
You've already studied the second law and know that no heat engine can be 100% efficient — some heat must be rejected to a cold reservoir. Thermal efficiency is the quantitative expression of this constraint: η = W_net / Q_in, the fraction of heat input that becomes net work. For a cycle operating between a hot source at T_hot and a cold sink at T_cold (measured in Kelvin), the Carnot efficiency η_Carnot = 1 − T_cold/T_hot sets the absolute upper bound. No cycle, no matter how cleverly designed, can exceed Carnot efficiency between those two temperature limits. A power plant drawing heat from steam at 600°C (873 K) and rejecting to cooling water at 30°C (303 K) has a Carnot limit of about 65% — real plants achieve 40-45%, the gap representing irreversibilities.
The Carnot cycle itself is a theoretical benchmark, not a practical design: it requires processes that are infinitely slow (to remain reversible) and involves heat exchange at exactly T_hot and T_cold. Real cycles accept irreversibilities in exchange for finite power output. The Rankine cycle (steam power plants) replaces Carnot's isothermal compression of a wet vapor with easy pump compression of liquid water — far more practical, though less efficient. The Brayton cycle (gas turbines) operates entirely in the gas phase with continuous compression and expansion. The Otto cycle (gasoline engines) approximates the rapid combustion and expansion of a piston engine. Each is analyzed by tracking W_net = Q_in − Q_out across all components and computing η = W_net / Q_in.
The key to improving efficiency is to raise the average temperature at which heat is added and lower the average temperature at which it is rejected — getting as close to operating between T_hot and T_cold as possible. Superheat (heating steam above saturation) raises the average temperature of heat addition. Higher pressure ratios in Brayton or Rankine cycles allow expansion to extract more work before heat rejection. Reheat (expanding partially, reheating, then expanding again) keeps the working fluid hotter longer. Regeneration (using exhaust heat to preheat the incoming fluid) reduces Q_in for the same W_net by internal heat exchange — it does not break the Carnot limit, but it reduces the required fuel by recycling energy that would otherwise be wasted.
When analyzing a cycle, the systematic approach is: label each state point (1, 2, 3, 4, ...) around the cycle, write the first law for each device (w = h_in − h_out for turbines and compressors, q = h_out − h_in for boilers and condensers), sum to find W_net and Q_in, then compute η. Each device's first law is just the steady-flow energy equation applied to one component. The cycle analysis knits those device-level balances into a system-level efficiency. This framework carries directly into Rankine, Brayton, and Otto analysis, where you'll apply these same steps to specific working fluids and real operating conditions.