6 Thermodynamic Cycles

What makes a cycle

A thermodynamic cycle is a sequence of processes that returns a working substance to its starting state. Since internal energy is a property of state, its net change over the complete cycle is zero:

Cycles can produce useful work, as in a power plant, or use work to move heat, as in a refrigerator. To analyze a cycle, track the working fluid’s states, energy transfers, and direction of operation.

Reading cycle diagrams

Cycle diagrams show the working fluid’s path through a sequence of states. Each point represents a state, and each curve represents a process.

On a pressure–volume (P–V) diagram, the signed area enclosed by the cycle gives net boundary work:

A clockwise loop generally indicates positive net work done by the working fluid, while a counterclockwise loop indicates net work input.

A temperature–entropy (T–s) diagram is also useful. For an internally reversible process, the area under a path represents heat transfer:

Diagrams help visualize a cycle, but energy balances and property data are needed to calculate its performance.

Energy balances

Using the convention that heat entering the system and work done by the system are positive, the first law for a closed system is

Over a complete cycle, the system returns to its starting state and $\Delta U=0$. Therefore, net heat transfer equals net work:

Here, $Q_{\text{net}}$ and $W_{\text{net}}$ refer to the net heat and work over all processes in the cycle. For a power cycle, net work output equals heat supplied minus heat rejected. For a refrigeration cycle, net work input equals heat rejected to the warm surroundings minus heat absorbed from the cooled space.

For steady-flow devices such as turbines, compressors, pumps, and heat exchangers, energy balances are usually applied to each component. In an introductory analysis, changes in kinetic and potential energy are often neglected. A turbine typically produces work as the fluid’s enthalpy decreases; a compressor or pump requires work to raise the fluid’s pressure.

Power cycles and efficiency

A power cycle receives heat, turns part of that energy into net work, and rejects the rest to a cooler environment. Its simplified energy balance is

Thermal efficiency compares net work output with heat input:

For example, a cycle that receives $1000\ \text{kJ}$ of heat and rejects $600\ \text{kJ}$ produces $400\ \text{kJ}$ of net work. Its thermal efficiency is $0.40$, or $40\%$. By the second law, a cycle operating between hot and cold surroundings cannot convert all supplied heat into work; some heat must be rejected.

The Rankine cycle is a common model for steam power plants. Its pump raises liquid pressure, its boiler adds heat, its turbine produces work as steam expands, and its condenser rejects heat and returns the fluid to liquid. Real devices are not perfectly reversible, so their actual work output and efficiency are below ideal predictions.

Refrigeration and heat pumps

A refrigerator or heat pump uses work input to transfer heat from a colder region to a warmer one. The energy balance is

where $Q_{\text{in}}$ is heat absorbed from the cold space and $Q_{\text{out}}$ is heat delivered to the warm surroundings.

A refrigerator is judged by how much heat it removes, while a heat pump is judged by how much heat it supplies. Their coefficients of performance are

For example, if a refrigerator removes $300\ \text{kJ}$ of heat while using $100\ \text{kJ}$ of work, its coefficient of performance is $3$. A coefficient of performance can exceed $1$ because the device moves heat rather than converting work entirely into heat.

The common vapor-compression cycle circulates refrigerant through a compressor, condenser, expansion device, and evaporator. The refrigerant absorbs heat at the evaporator and releases it at the condenser.