Question

Suppose a Brayton engine (see Problem 20.26 ) is run as a refrigerator. In this case, the cycle begins at temperature \(T_{1}\), and the gas is isobarically expanded until it reaches temperature \(T_{4}\). Then the gas is adiabatically compressed, until its temperature is \(T_{3}\). It is then isobarically compressed, and the temperature changes to \(T_{2}\). Finally, it is adiabatically expanded until it returns to temperature \(T_{1}\) - a) Sketch this cycle on a \(p V\) -diagram. b) Show that the coefficient of performance of the engine is given by \(K=\left(T_{4}-T_{1}\right) /\left(T_{3}-T_{2}-T_{4}+T_{1}\right) .\)

Short answer

Question: Sketch the cycle of a Brayton engine running as a refrigerator on a pV-diagram and find the formula for the coefficient of performance using the given temperatures. Answer: To sketch the cycle of a Brayton engine running as a refrigerator on a pV-diagram, follow these steps: (1) Plot isobaric expansion process as a horizontal line from the initial point (1) to a point (2) with a higher volume and same pressure. (2) Plot adiabatic compression process as an upward curve from point (2) to a point (3) with higher pressure and lower volume. (3) Plot isobaric compression process as a horizontal line from point (3) to a point (4) with a lower volume and same pressure. (4) Plot adiabatic expansion process as a downward curve from point (4) to the initial point (1) with a lower pressure and higher volume. The formula for the coefficient of performance (K) for the refrigerator is: \( K = \frac{T_{4}-T_{1}}{T_{3}-T_{2}-T_{4}+T_{1}} \).

Step-by-step solution

Refreshing basics of a Brayton engine cycle

A Brayton cycle consists of four processes in an engine: 1) Isobaric expansion, 2) Adiabatic compression, 3) Isobaric compression, and 4) Adiabatic expansion. These processes are executed in a closed system with an ideal gas.

Sketch the Brayton cycle on a pV-diagram for a refrigerator

To sketch the cycle on a pV-diagram, we will label the four stages we mentioned earlier as (1), (2), (3), and (4). The temperatures given are \(T_1, T_2, T_3, and T_4\) for these four stages, respectively. Plot the isobaric expansion process (1) to (2) as a horizontal line from (1) to a point (2) with a higher volume and same pressure. Plot the adiabatic compression process (2) to (3) as an upward curve from point (2) to a point (3) with higher pressure and lower volume. Plot the isobaric compression process (3) to (4) as a horizontal line from (3) to a point (4) with a lower volume and same pressure. Plot the adiabatic expansion process (4) to (1) as a downward curve from point (4) to the initial point (1) with a lower pressure and higher volume. We have now sketched the pV-diagram of the refrigerator Brayton engine cycle.

Calculate the coefficient of performance

To find the coefficient of performance \(K\) for the refrigerator, we use the given formula: $$ K=\frac{T_{4}-T_{1}}{T_{3}-T_{2}-T_{4}+T_{1}} $$ It is important to note that, for this problem, we are not given the actual numerical values for these temperatures. What we have instead is a general formula for the coefficient of performance. If the problem required a numerical solution, we could just plug in the given temperatures and calculate the value of \(K\). In summary, we have sketched the Brayton engine cycle as a refrigerator on a pV-diagram and found the formula for the coefficient of performance using the given temperatures.

Key concepts

Isobaric Expansion

Imagine a gas within an engine, expanding while maintaining the same pressure; this process is known as isobaric expansion. During this stage in a Brayton engine running as a refrigerator, the gas starts at a temperature of \(T_1\) and expands until it reaches \(T_4\). During isobaric expansion, the work done by the system is represented by the area under the horizontal process path on a pV-diagram. It's essential for the heat absorption process that enables a refrigerator to cool its contents.

In terms of the Brayton cycle, the isobaric expansion corresponds to the cooling phase wherein the refrigerant absorbs heat from the environment at a constant pressure, causing a temperature rise. The work done on the gas is converted into internal energy, leading to an increase in temperature without a change in pressure.

Adiabatic Compression

When a gas is compressed with no heat exchange with its surroundings, the process is called adiabatic compression. In the Brayton cycle, this step follows the isobaric expansion phase. The temperature of the gas increases from \(T_4\) to \(T_3\) due to the work done on the gas, leading to an increase in both pressure and temperature without a change in entropy.

In a pV-diagram, adiabatic processes are represented by steeper curves than those for isobaric processes. This compression is crucial because it increases the refrigerant's temperature, preparing it to release the absorbed heat as it moves into the next phase of the cycle. Understanding adiabatic processes is fundamental to thermodynamics, where they represent idealized situations with no thermal energy entering or leaving the system during this stage.

Coefficient of Performance

In thermodynamics, the Coefficient of Performance (COP) is a measure of a refrigerator's efficiency. It is defined as the ratio of heat removed from the cold reservoir to the work input required to remove that heat. For the Brayton engine running as a refrigerator, the COP is denoted by \(K\) and given by the formula:

\[K = \frac{T_4 - T_1}{T_3 - T_2 - T_4 + T_1}\]
Understanding the COP helps us compare the efficacy of different refrigeration systems. A higher COP indicates a more efficient refrigerator, as it requires less work to transfer the same amount of heat. This concept is foundational not just for theoretical thermodynamics but also for practical engineering and environmental sustainability.

Thermodynamics

Thermodynamics is the study of the relations between heat, work, temperature, and energy. The Brayton cycle is one of the many applications of these principles, showcasing how energy is transformed and conserved within a thermodynamic system. It governs the behavior of refrigeration cycles, such as the modified Brayton cycle used in the problem.

Thermodynamics lays out several fundamental laws which predict how systems behave under different conditions. For the Brayton engine as a refrigerator, it is imperative to understand how heat transfer and work relate to each other and how they affect the temperatures and pressures at different points in the cycle. The efficiency, signified by the COP, reflects how well the cycle follows the laws of thermodynamics.

pV-diagram

A pV-diagram is a graphical representation that plots pressure (p) against volume (V) for a given thermodynamic process. For any thermodynamic cycle, including the Brayton cycle, the pV-diagram visually depicts the stages as curved or straight lines, which represent different thermodynamic processes (isobaric, adiabatic, isothermal, etc.).

In the case of a Brayton engine operating as a refrigerator, the diagram helps to illustrate how the gas moves through each of the four stages: isobaric expansion, adiabatic compression, isobaric compression, and adiabatic expansion. From the diagram, we can gather information about the work done and heat transferred during each stage of the cycle, which is fundamental for understanding and calculating the system's efficiency or COP.