Heat Engines

Power cycles, conversion of heat into mechanical work, and thermodynamic thermal efficiencies.

A heat engine is a device that operates cyclically to convert thermal energy extracted from a high-temperature heat reservoir into net useful mechanical work, inevitably rejecting an unavoidable fraction of residual heat to a lower-temperature sink. The theoretical analysis of heat engines established the historical and physical cornerstone of classical thermodynamics, formalizing the Second Law and determining the absolute upper limits of energy conversion efficiency.

Operating Principles and Energy Balance

Over each completed thermodynamic cycle, the working fluid returns to its exact initial state, ensuring zero net change in internal energy ($\Delta U_{cycle} = 0$). By the First Law of Thermodynamics, net mechanical work generated equals the difference between heat absorbed and heat rejected:

$W_{net} = Q_h - |Q_c|$

The thermal efficiency $\eta$ characterizes the ratio of useful work extracted to the total heat input supplied by the hot reservoir:

$\eta = \frac{W_{net}}{Q_h} = 1 - \frac{|Q_c|}{Q_h}$

The Kelvin-Planck statement of the Second Law of Thermodynamics dictates that it is impossible to construct a cyclic heat engine whose sole effect is to absorb heat from a single reservoir and convert it completely into work ($\eta < 1$). Every continuous cyclic engine strictly requires at least two heat reservoirs operating at different temperatures.

The Carnot Cycle and the Universal Efficiency Bound

In 1824, Nicolas Léonard Sadi Carnot demonstrated that no heat engine operating between two thermal reservoirs at absolute temperatures $T_h$ and $T_c$ can exceed the efficiency of an ideal reversible engine executing the Carnot cycle, consisting of:

1. Reversible isothermal expansion at temperature $T_h$.
2. Reversible adiabatic expansion dropping from $T_h$ to $T_c$.
3. Reversible isothermal compression at temperature $T_c$.
4. Reversible adiabatic compression restoring temperature $T_h$.

The Carnot efficiency depends strictly on the reservoir temperatures:

$\eta_{Carnot} = 1 - \frac{T_c}{T_h}$

This fundamental relation proves that maximizing thermal conversion efficiency requires operating at the highest feasible source temperature $T_h$ and discharging to the lowest available ambient temperature $T_c$.

Reversed Heat Engines: Refrigerators and Heat Pumps

When an external work input $W_{in}$ drives the cycle in the counter-clockwise direction on a pressure-volume indicator diagram, the system functions as a reversed heat engine:

• Refrigerator: Designed to remove heat $Q_c$ from a refrigerated low-temperature space and exhaust it into warmer surroundings. Its effectiveness is quantified by the refrigeration coefficient of performance: $COP_R = \frac{Q_c}{W_{in}}$.
• Heat Pump: Designed to deliver thermal energy $Q_h$ into a heated space by absorbing heat from a cold exterior environment. Its performance metric is: $COP_{HP} = \frac{Q_h}{W_{in}} = COP_R + 1$.

Reversed cycles are governed by the Clausius statement of the Second Law: heat cannot spontaneously flow from a colder body to a hotter body without an accompanying expenditure of external mechanical work.