12.4 Applications of Thermodynamics: Heat Engines, Heat Pumps, and Refrigerators

Thermodynamic Principles and Applications

Heat engines, heat pumps, and refrigerators all move energy between thermal and mechanical forms. They operate between two temperature reservoirs, and the laws of thermodynamics place hard limits on how well they can perform. Understanding these devices means understanding both the energy flows and the efficiency ceilings that govern them.

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Operation of Heat Transfer Devices

Heat engines convert thermal energy into mechanical work. They absorb heat $Q_H$ from a high-temperature reservoir (the heat source), convert part of it into work $W$, and reject the remaining heat $Q_L$ to a low-temperature reservoir (the heat sink).

• Internal combustion engines (gasoline and diesel engines in vehicles)
• Steam turbines in power plants
• Stirling engines used in solar thermal power systems

Heat pumps do the reverse: they move thermal energy from a low-temperature reservoir to a high-temperature reservoir, but this transfer requires work input. Think of them as engines running backward.

• Air-source heat pumps for space heating in buildings
• Ground-source (geothermal) heat pumps that extract heat from the ground, which stays at a relatively stable temperature year-round

Refrigerators operate on the same principle as heat pumps but with a different goal. Instead of delivering heat to the warm side, the purpose is to remove heat from the cold side.

• Household refrigerators and freezers
• Air conditioners, which are really just refrigerators for a room

All three devices are governed by the same two laws:

• First Law of Thermodynamics (energy conservation): The change in internal energy $\Delta U$ equals the heat added $Q$ minus the work done $W$. For a complete cycle, $\Delta U = 0$, so the energy balance becomes $Q_H = W + Q_L$.
• Second Law of Thermodynamics (entropy and directionality): Heat flows spontaneously only from hot to cold. The Kelvin-Planck statement says no heat engine can convert 100% of absorbed heat into work. The Clausius statement says no device can transfer heat from cold to hot without work input.

Thermal Efficiency in Thermodynamic Systems

Thermal efficiency $\eta$ tells you what fraction of the input heat a heat engine actually converts to useful work:

$\eta = \frac{W_{out}}{Q_H}$

Since $W_{out} = Q_H - Q_L$, you can also write this as:

$\eta = 1 - \frac{Q_L}{Q_H}$

This is always less than 1 (less than 100%) because some heat must be rejected to the cold reservoir. Typical gasoline engines achieve about 20–30% efficiency.

Carnot efficiency is the theoretical maximum for any heat engine operating between two specific temperatures:

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

where $T_L$ and $T_H$ are the absolute temperatures (in Kelvin) of the cold and hot reservoirs. No real engine can reach this limit because it assumes perfectly reversible processes with zero friction, zero heat leaks, and infinitely slow operation. But it's the benchmark you compare real engines against.

Coefficient of Performance (COP) measures how effectively heat pumps and refrigerators use work input. Unlike efficiency, COP can be greater than 1.

• For a heat pump (goal is to deliver heat to the warm side): $COP_{heating} = \frac{Q_H}{W_{in}}$

• For a refrigerator (goal is to remove heat from the cold side): $COP_{cooling} = \frac{Q_L}{W_{in}}$

Notice that for the same device, $COP_{heating} = COP_{cooling} + 1$. This follows directly from the energy balance $Q_H = W_{in} + Q_L$.

A higher COP means less work is needed to move a given amount of heat. COP improves when the temperature difference between the two reservoirs is smaller, which is why ground-source heat pumps (with a relatively warm ground temperature in winter) outperform air-source heat pumps on very cold days.

Thermodynamic Processes and Cycles

A thermodynamic cycle is a series of processes that returns a system to its initial state. Because the system returns to the same state, $\Delta U = 0$ over one complete cycle, and the net work equals the net heat transfer.

Several types of processes show up repeatedly in these cycles:

• Reversible process: An idealized process that can be undone without any net change to the system or surroundings. These are used in theoretical analysis (like the Carnot cycle) to determine maximum possible efficiencies. No real process is truly reversible.
• Irreversible process: Every real process. Friction, turbulence, rapid compression, and heat transfer across finite temperature differences all create irreversibilities that increase entropy and reduce efficiency below the Carnot limit.
• Adiabatic process: No heat is exchanged between the system and surroundings ($Q = 0$). Rapid compression and expansion strokes in engines approximate adiabatic processes because they happen too fast for significant heat transfer.
• Isothermal process: Temperature stays constant. Heat is absorbed or rejected slowly enough to maintain thermal equilibrium with a reservoir. The Carnot cycle uses two isothermal and two adiabatic steps.

Applications of Thermodynamic Principles

Analyzing power plants involves connecting efficiency to real energy flows.

• Thermal efficiency: $\eta = \frac{W_{out}}{Q_H}$
• Heat rejected to the environment: $Q_L = Q_H - W_{out}$
• Fossil fuel power plants typically achieve 30–40% efficiency, meaning 60–70% of the fuel's energy is rejected as waste heat.

Designing heating and cooling systems requires choosing the right device and sizing it based on COP.

• Work input needed: $W_{in} = \frac{Q_{transferred}}{COP}$
• A heat pump with a COP of 4 delivers 4 units of heat for every 1 unit of electrical work, making it far more efficient than a simple electric heater (which has an effective COP of 1).

Optimizing energy consumption comes down to a few key strategies:

1. Minimize the temperature difference between reservoirs (this directly improves both Carnot efficiency and COP)
2. Reduce heat losses through better insulation and system design
3. Use regenerative processes to recover waste heat (preheating intake air with exhaust gases, for example)
4. Consider renewable energy sources (solar, geothermal) to reduce reliance on fossil fuels and lower greenhouse gas emissions

Environmental considerations tie directly back to efficiency. Lower efficiency means more fuel burned per unit of useful energy, which means more emissions. Waste heat recovery in industrial processes can capture some of that rejected $Q_L$ and put it to use, improving the overall energy utilization even if the thermodynamic efficiency of the engine itself doesn't change.