12.1 Thermodynamic cycles for heat engines

Heat Engines and Thermodynamic Cycles

Purpose of Heat Engines

A heat engine converts thermal energy into mechanical work by cycling a working fluid through repeated changes in temperature, pressure, and volume. The goal is to harness energy from a high-temperature source and turn it into useful output, whether that's spinning a generator or pushing a piston.

Two laws of thermodynamics govern every heat engine:

• The first law (energy conservation) tells you that the net work output equals the difference between heat absorbed and heat rejected.
• The second law dictates that you can never convert all the absorbed heat into work. Some heat must always be dumped to a cooler reservoir. This is why no engine is 100% efficient.

Components of Heat Engine Cycles

Every heat engine cycle involves the same basic ingredients:

• Working fluid: the substance (air, steam, a gas mixture) that absorbs and rejects heat as it flows through the cycle
• High-temperature reservoir (heat source): supplies thermal energy to the working fluid (e.g., burning fuel, a nuclear reactor)
• Low-temperature reservoir (heat sink): absorbs the rejected heat (e.g., the atmosphere, a cooling tower)
• Mechanical components: pistons, cylinders, or turbines that convert the fluid's expansion into useful work

The four stages that appear in nearly every cycle are:

1. Heat addition — The working fluid absorbs heat from the hot reservoir, raising its temperature and pressure.
2. Expansion — The hot, high-pressure fluid expands, doing mechanical work (pushing a piston or spinning a turbine blade).
3. Heat rejection — The fluid dumps leftover thermal energy to the cold reservoir, lowering its temperature and pressure.
4. Compression — The fluid is compressed back toward its starting state, completing the loop so the cycle can repeat.

Carnot Cycle and Efficiency Limits

The Carnot cycle is a theoretical, fully reversible cycle that operates between two constant-temperature reservoirs. It consists of four reversible processes:

1. Isothermal expansion — the working fluid absorbs heat from the hot reservoir at constant temperature $T_H$
2. Adiabatic expansion — the fluid continues to expand with no heat transfer, cooling from $T_H$ toward $T_C$
3. Isothermal compression — the fluid rejects heat to the cold reservoir at constant temperature $T_C$
4. Adiabatic compression — the fluid is compressed with no heat transfer, warming back up to $T_H$

The thermal efficiency of any heat engine is defined as the ratio of net work output to heat input:

$\eta = \frac{W_{net}}{Q_{in}}$

For a Carnot engine, this simplifies to a function of the reservoir temperatures alone (in absolute units, i.e., Kelvin):

$\eta_{Carnot} = 1 - \frac{T_C}{T_H}$

For example, an engine operating between a hot reservoir at 600 K and a cold reservoir at 300 K has a maximum possible efficiency of $1 - \frac{300}{600} = 0.50$, or 50%.

Why does the Carnot cycle matter? Because the second law guarantees that no real engine operating between the same two temperatures can exceed Carnot efficiency. It's the absolute ceiling. Every real engine falls short due to friction, heat leaks, and other irreversibilities, but the Carnot cycle gives you the benchmark to measure how close a real design gets.

Comparison of Practical Engine Cycles

Real engines sacrifice some theoretical efficiency in exchange for practicality, durability, and cost. The three most important practical cycles are the Otto, Diesel, and Rankine.

Otto Cycle (Gasoline Engines)

Used in spark-ignition internal combustion engines. Its four idealized processes are:

1. Isentropic compression
2. Constant-volume heat addition (models the rapid combustion after spark ignition)
3. Isentropic expansion (power stroke)
4. Constant-volume heat rejection (exhaust)

Thermal efficiency depends on the compression ratio $r$ (ratio of maximum to minimum cylinder volume) and the specific heat ratio $\gamma$:

$\eta_{Otto} = 1 - \frac{1}{r^{\gamma - 1}}$

Higher compression ratios mean higher efficiency, which is why engine designers push $r$ as high as possible before knocking becomes a problem. Typical gasoline engines have compression ratios around 8–12.

Diesel Cycle (Diesel Engines)

Used in compression-ignition engines, where fuel ignites from the heat of compression rather than a spark. The four processes are:

1. Isentropic compression
2. Constant-pressure heat addition (fuel is injected and burns as the piston moves)
3. Isentropic expansion (power stroke)
4. Constant-volume heat rejection (exhaust)

Efficiency depends on the compression ratio $r$, the cutoff ratio $\rho$ (ratio of volume after to volume before heat addition), and $\gamma$:

$\eta_{Diesel} = 1 - \frac{1}{r^{\gamma - 1}} \left( \frac{\rho^{\gamma} - 1}{\gamma(\rho - 1)} \right)$

Diesel engines run at much higher compression ratios (14–25) than gasoline engines, which is a key reason they tend to be more thermally efficient. The trade-off is heavier construction to handle the higher pressures.

Rankine Cycle (Steam Power Plants)

This is the cycle behind most large-scale electricity generation. Instead of a gas in a cylinder, the working fluid is water/steam flowing through a loop of components:

1. Isentropic compression (pump) — liquid water is pressurized
2. Constant-pressure heat addition (boiler) — water absorbs heat and becomes high-pressure steam
3. Isentropic expansion (turbine) — steam expands through a turbine, producing shaft work
4. Constant-pressure heat rejection (condenser) — exhaust steam is cooled back to liquid

Rankine cycle efficiency depends on the steam's temperature and pressure limits, as well as the real-world performance of the turbine, pump, and boiler. Superheating the steam and using reheat stages are common ways to push efficiency higher.

Why are all these cycles less efficient than Carnot?

Every real cycle includes irreversibilities: friction in moving parts, pressure drops in piping, heat transfer across finite temperature differences, and incomplete combustion. These losses are unavoidable in practice, so engineers design cycles to optimize performance within the constraints of available materials, technology, and cost.