Heat Engines

A heat engine is any device that converts thermal energy (heat) into mechanical work by exploiting a temperature difference between two reservoirs. Every time a car engine burns petrol or a steam turbine spins in a power plant, a heat engine is at work. The core idea is deceptively simple: heat flows naturally from hot regions to cold ones, and we can 'intercept' some of that flow and divert it into useful mechanical energy — turning a shaft, pushing a piston, or propelling a vehicle. The working principle relies on a thermodynamic cycle: a substance (the working fluid — often a gas) absorbs heat Q_H from a hot reservoir, does work W on its surroundings, and then dumps the remaining heat Q_C into a cold reservoir (like the atmosphere or a cooling system). Energy conservation guarantees that Q_H = W + Q_C, so the work extracted is always less than the total heat absorbed. This is not a flaw in our engineering — it is a fundamental law of nature. Heat engines are governed by the First and Second Laws of Thermodynamics. The Second Law, in particular, places a strict upper limit on how efficiently any heat engine can operate, regardless of how cleverly it is built. Understanding heat engines therefore sits at the intersection of practical engineering and deep physical principles, making them one of the most important topics in all of physics.

Key takeaways
  • The Hot Reservoir Supplies Heat
  • The Working Fluid Expands and Does Work
  • The Cold Reservoir Accepts Waste Heat
  • Thermal Efficiency: How Much Work Do We Get?
  • The Carnot Engine: The Ideal Limit

The Hot Reservoir Supplies Heat

The engine absorbs a quantity of heat Q_H from a high-temperature source — this could be burning fuel, a nuclear reactor, or concentrated sunlight. The temperature of this source is T_H. Think of the hot reservoir as a large tank of thermal energy that the engine can draw from without significantly changing the reservoir's temperature.

The Working Fluid Expands and Does Work

The absorbed heat raises the internal energy of the working fluid (e.g., steam or a gas mixture). The fluid expands — pushing a piston or spinning turbine blades — and in doing so, converts some of that internal energy into mechanical work W. This is the 'useful output' of the engine. Analogously, think of squeezing a heated balloon: it pushes outward as it expands, doing work on whatever is containing it.

The Cold Reservoir Accepts Waste Heat

After expansion, the working fluid still contains thermal energy. To complete the cycle and return the fluid to its initial state (ready to absorb heat again), the engine must expel the remaining heat Q_C into a cold reservoir at temperature T_C. This waste heat cannot be avoided — the Second Law forbids a 100% conversion of heat into work. In a car engine, this waste heat is what comes out of the exhaust and radiator.

Thermal Efficiency: How Much Work Do We Get?

Efficiency η (eta) measures the fraction of absorbed heat that becomes useful work. From energy conservation: W = Q_H − Q_C, so η = W/Q_H = 1 − Q_C/Q_H. An efficiency of 1 (100%) would mean no waste heat — forbidden by the Second Law. Real engines have efficiencies typically between 20% and 45%. The absolute theoretical maximum for any heat engine operating between temperatures T_H and T_C is the Carnot efficiency: η_Carnot = 1 − T_C/T_H, where temperatures must be in Kelvin.

The Carnot Engine: The Ideal Limit

Sadi Carnot showed in 1824 that no engine operating between two fixed temperatures can be more efficient than a reversible engine — now called a Carnot engine — running the same cycle. The Carnot cycle consists of two isothermal (constant temperature) and two adiabatic (no heat exchange) processes. It is an idealization: real engines suffer from friction, turbulence, and irreversible heat transfer, all of which reduce efficiency below the Carnot limit.

The Key Equation

η=1−QCQH≤1−TCTH\eta = 1 - \frac{Q_C}{Q_H} \leq 1 - \frac{T_C}{T_H}
  • \eta — Thermal efficiency (dimensionless (0 to 1)): Fraction of the absorbed heat that is converted into useful mechanical work.
  • Q_C — Waste heat expelled (Joules (J)): Heat dumped into the cold reservoir per cycle; always positive.
  • Q_H — Heat absorbed (Joules (J)): Heat taken from the hot reservoir per cycle; always positive.
  • T_C — Cold reservoir temperature (Kelvin (K)): Absolute temperature of the cold sink (e.g., atmosphere at ~300 K).
  • T_H — Hot reservoir temperature (Kelvin (K)): Absolute temperature of the hot source (e.g., steam at ~800 K).

Worked Example

Problem. A steam power plant operates with steam at T_H = 600 K and exhausts heat to a river at T_C = 300 K. In each cycle the engine absorbs Q_H = 8000 J from the hot steam. (a) What is the maximum (Carnot) efficiency? (b) What is the maximum work the engine can do per cycle? (c) If the actual efficiency is 30%, how much work is actually done and how much heat is wasted?

ηCarnot=50%,Wmax⁡=4000 J,Wactual=2400 J,QC=5600 J\eta_{\text{Carnot}} = 50\%,\quad W_{\max} = 4000\text{ J},\quad W_{\text{actual}} = 2400\text{ J},\quad Q_C = 5600\text{ J}

Answer. Carnot efficiency = 50%; max work = 4000 J; actual work = 2400 J; waste heat = 5600 J.

Figure
Diagram
HOTRESERVOIRT_HCOLDRESERVOIRT_CENGINEη = 1 − Q_C/Q_HQ_HQ_CWWORKQ_H = W + Q_C (First Law)
Origins · 1 minSadi Carnot and how this was discovered
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Sadi Carnot
1796–1832 · French

Carnot published 'Reflections on the Motive Power of Fire' in 1824, establishing the theoretical maximum efficiency of any heat engine operating between two temperatures. His reversible Carnot cycle became the cornerstone of thermodynamics, predating the formal statement of the Second Law by decades. Carnot died of cholera at only 36, and much of his unpublished work was burned for fear of contagion. His groundbreaking 1824 book sold fewer than 600 copies in his lifetime, yet it laid the foundation for the entire field of thermodynamics.

  • The Carnot Cycle — The Carnot cycle defines the ideal reversible heat engine and sets the absolute upper bound on thermal efficiency — it is the benchmark against which all real engines are measured.
  • Second Law of Thermodynamics — The Second Law explains why heat engines cannot be 100% efficient and introduces entropy — the fundamental reason waste heat is unavoidable.
  • First Law of Thermodynamics — Energy conservation (Q_H = W + Q_C) is applied directly in every heat engine calculation to relate heat absorbed, work done, and heat expelled.
  • Entropy — Entropy quantifies the disorder in a system; the requirement that total entropy never decreases is precisely what limits engine efficiency to below 100%.