5 Thermodynamic Cycles and Devices

Cycles and energy accounting

A Thermodynamic cycle returns its working substance to its initial state. Since internal energy depends on state, its net change over a complete cycle is zero:

Using the convention that heat entering the working substance and work done by it are positive, the first law gives:

Thus, net heat supplied over a full cycle becomes net work produced, or net work supplied enables net heat to be transferred. On a pressure–volume diagram, the enclosed area represents net work for the cycle. A clockwise cycle generally produces net work; running a cycle in reverse requires work input.

Heat engines and efficiency

A Heat engine operates between a hot reservoir and a cooler reservoir. During each cycle, it absorbs heat $Q_H$, produces net work $W_{\text{out}}$, and rejects heat $Q_C$ to the cooler surroundings. Energy accounting gives:

Thermal efficiency is the fraction of input heat converted into net work:

For example, an engine that absorbs $800\,\text{J}$ and rejects $500\,\text{J}$ per cycle produces $300\,\text{J}$ of work. Its efficiency is $\frac{300}{800}=0.375$, or $37.5\%$.

Rejecting some heat does not violate energy conservation; it is required by the second law. No cyclic engine operating between two reservoirs can convert all its absorbed heat into work.

The Carnot limit for engines

The Carnot cycle is an ideal, reversible cycle that sets the maximum possible engine efficiency between fixed hot and cold reservoir temperatures. For absolute temperatures $T_H$ and $T_C$, measured in kelvins, the maximum efficiency is:

Real engines are less efficient because of friction, heat transfer across finite temperature differences, and other irreversibilities. The Carnot result is a limit, not a description of how most practical engines operate.

Refrigerators and heat pumps

A refrigerator uses work input to remove heat from a cooler space. A heat pump uses work input to deliver heat to a warmer space. They operate on the same basic reversed-cycle principle; the difference is which heat transfer is treated as useful.

For either device, energy conservation over a complete cycle gives:

where $Q_C$ is heat removed from the cold region, $Q_H$ is heat delivered to the warm region, and $W_{\text{in}}$ is supplied work. Their coefficients of performance are:

Since $Q_H=Q_C+W_{\text{in}}$, the two measures are related by:

COP is not thermal efficiency: it compares heat moved with work supplied, rather than work output with heat input, so it can exceed $1$. For example, if a refrigerator removes $600\,\text{J}$ from its interior using $200\,\text{J}$ of work, its COP is $3$. It delivers $800\,\text{J}$ to the surroundings, so the same device operating as a heat pump has COP $4$.

Reversible performance limits

For ideal reversible devices operating between reservoirs at temperatures $T_C$ and $T_H$, the maximum coefficients of performance are:

Temperatures must be measured in kelvins. These limits show that moving heat across a larger temperature difference requires more work for the same heating or cooling effect. Real refrigerators and heat pumps have lower COPs because their cycles are irreversible.

The vapor-compression cycle

Many household refrigerators and air conditioners use a vapor-compression cycle. A refrigerant circulates through four main components:

1. Compressor: Work is supplied to compress the vapor, raising its pressure and temperature.

2. Condenser: The hot refrigerant releases heat to warmer surroundings and condenses into a liquid.

3. Expansion device: The refrigerant’s pressure drops, cooling it.

4. Evaporator: The cold refrigerant absorbs heat from the refrigerated space and evaporates, then returns to the compressor.

The cycle continuously transports heat from the cooler interior to the warmer surroundings, using compressor work to make the transfer possible. In heating mode, a heat pump delivers the condenser’s heat to the space being warmed.

• Coefficient of performance