Premed · Premed · Physics 1
Lecture 24: Heat Engines and Refrigerators
Physics I — Mechanics & Thermodynamics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the operation of a heat engine and define thermal efficiency
- Describe the Carnot cycle and calculate Carnot efficiency
- Explain why the Carnot engine sets the upper limit on efficiency
- Analyze refrigerators and heat pumps using the coefficient of performance
- Apply the principles of heat engines to real-world examples
- Connect engine efficiency to the second law of thermodynamics
Lecture Content
I. Heat Engines — Overview
A heat engine is a device that converts thermal energy into mechanical work. It operates cyclically by absorbing heat Q_H from a high-temperature reservoir at T_H, performing work W on the surroundings, exhausting waste heat Q_C to a low-temperature reservoir at T_C, and then returning to its initial state. By the first law applied to a complete cycle (where Delta U = 0), W = Q_H - Q_C.
The thermal efficiency (eta) measures what fraction of the absorbed heat is converted to useful work: eta = W / Q_H = (Q_H - Q_C) / Q_H = 1 - Q_C / Q_H. By the second law, efficiency is always less than 100%. The smaller Q_C is relative to Q_H, the more efficient the engine. Examples of heat engines include internal combustion engines, steam turbines, and jet engines.
<image>An energy flow diagram for a heat engine. A hot reservoir at temperature T_H is at the top, with an arrow labeled Q_H pointing down into the engine (circle labeled "Engine"). An arrow labeled W points to the right from the engine (work output). An arrow labeled Q_C points down from the engine to a cold reservoir at temperature T_C at the bottom. The equation W = Q_H - Q_C is written beside the diagram. The efficiency formula eta = W/Q_H = 1 - Q_C/Q_H is written below.</image>
II. The Carnot Cycle
The Carnot cycle is an idealized thermodynamic cycle that achieves the maximum possible efficiency between two given temperatures. It consists of four reversible steps. First, an isothermal expansion at T_H in which the gas absorbs Q_H from the hot reservoir while maintaining constant temperature. Second, an adiabatic expansion in which the gas continues expanding with no heat exchange, and its temperature drops from T_H to T_C. Third, an isothermal compression at T_C during which the gas releases Q_C to the cold reservoir at constant temperature. Fourth, an adiabatic compression that raises the temperature from T_C back to T_H, completing the cycle.
On a P-V diagram, the Carnot cycle forms a closed loop bounded by two isotherms and two adiabats. All four processes are reversible, meaning there is no friction and no heat transfer across finite temperature differences.
III. Carnot Efficiency
The efficiency of a Carnot engine is eta_Carnot = 1 - T_C / T_H, where temperatures must be in Kelvin. This is the maximum possible efficiency for any engine operating between temperatures T_H and T_C. No real engine can exceed this limit, a result known as Carnot's theorem. Real engines always contain irreversibilities that reduce their efficiency below the Carnot value.
Several important implications follow. Efficiency increases as T_H increases or T_C decreases. An efficiency of 100% is achievable only if T_C = 0 K (absolute zero), which is unattainable. If T_H = T_C, there is no temperature difference and no work can be extracted. As a concrete example, a power plant with T_H = 800 K and T_C = 300 K has a Carnot efficiency of 1 - 300/800 = 62.5%, though real efficiency might be only 35-40% due to irreversibilities.
<image>A P-V diagram of the Carnot cycle. Four processes are drawn and labeled: (1) isothermal expansion along the T_H isotherm from state A to B (Q_H absorbed), (2) adiabatic expansion from B to C (temperature drops to T_C), (3) isothermal compression along the T_C isotherm from C to D (Q_C released), (4) adiabatic compression from D back to A (temperature rises to T_H). The enclosed area is shaded and labeled "W_net." The two isotherms (T_H and T_C) are drawn as hyperbolic curves, and the two adiabats are drawn as steeper curves. The Carnot efficiency equation is written beside the diagram.</image>
IV. Refrigerators and Heat Pumps
A refrigerator (or air conditioner) is a heat engine operating in reverse. Work is done on the system to move heat from a cold reservoir to a hot reservoir. Heat Q_C is absorbed from the cold space (inside the refrigerator), work W is supplied by the compressor, and Q_H = Q_C + W is expelled to the warm surroundings.
The coefficient of performance (COP) for a refrigerator is COP_ref = Q_C / W = Q_C / (Q_H - Q_C), measuring the cooling obtained per unit of work input. A COP greater than 1 is typical and expected. The maximum COP for a Carnot refrigerator is COP_ref,Carnot = T_C / (T_H - T_C).
A heat pump used for heating has COP_HP = Q_H / W = Q_H / (Q_H - Q_C), which always equals COP_ref + 1. The maximum is COP_HP,Carnot = T_H / (T_H - T_C). A heat pump delivers more heat than the energy it consumes because it extracts additional heat from the cold outdoor environment.
V. Real Heat Engines
The Otto cycle, which models the gasoline engine, consists of four strokes (intake, compression, power, exhaust), idealized as two adiabatic and two isochoric processes. Its efficiency is eta_Otto = 1 - 1/r^(gamma-1), where r is the compression ratio. Typical compression ratios of 8-12 yield efficiencies of 25-30%.
The Diesel cycle is similar but has combustion occurring at constant pressure (isobaric) rather than constant volume. Higher compression ratios (15-25) produce higher efficiency (30-40%). The Rankine cycle, used in steam power plants, exploits phase changes (boiling and condensation) and achieves typical efficiencies of 30-40%.
All real engines fall short of Carnot efficiency due to friction in moving parts, heat losses through engine walls, non-quasi-static (rapid) processes, and incomplete combustion.
VI. Entropy and Engine Efficiency
For a Carnot engine, which is reversible, Delta S_universe = 0. The hot reservoir loses entropy Delta S_H = -Q_H / T_H, the cold reservoir gains entropy Delta S_C = +Q_C / T_C, and the total is Delta S_total = -Q_H/T_H + Q_C/T_C = 0. This condition gives Q_C/Q_H = T_C/T_H, from which the Carnot efficiency eta = 1 - T_C/T_H follows directly.
For any real (irreversible) engine, Delta S_universe > 0, which requires Q_C/Q_H > T_C/T_H. This means more heat must be wasted, and therefore eta_real < eta_Carnot. The second law, through entropy, sets the fundamental upper bound on engine performance. No amount of engineering ingenuity can exceed the Carnot limit; it is a law of nature.
VII. Biological and Environmental Applications
The human body is sometimes compared to a heat engine, but this analogy has important limitations. With T_H approximately 310 K (body temperature) and T_C approximately 300 K (environment), the Carnot limit would be eta = 1 - 300/310 = 3.2%, which is extremely low. Yet actual muscle efficiency is approximately 25%, far exceeding this limit. This is possible because muscles are not heat engines; they are chemical engines that directly convert chemical energy into mechanical work without first converting it entirely to heat.
Power plants must dump waste heat Q_C into the environment through rivers, oceans, or cooling towers. A 1000 MW power plant operating at 33% efficiency produces approximately 2000 MW of waste heat. The greenhouse effect has climate implications for atmospheric heat engines: by raising the effective cold reservoir temperature T_C, it reduces the efficiency of the weather patterns that redistribute energy around the globe.
<image>A comparison diagram showing three engine types side by side. Left: A Carnot engine (ideal, reversible) with eta = 1 - T_C/T_H. Center: A real heat engine with eta < eta_Carnot, showing additional arrows for heat loss to friction and irreversibilities. Right: A refrigerator with work input W, heat extracted from cold reservoir Q_C, and heat dumped to hot reservoir Q_H = Q_C + W. COP formulas are written beneath the refrigerator. All three diagrams use the standard hot reservoir (top) / engine (middle) / cold reservoir (bottom) layout with energy flow arrows.</image>
