Question

A heat engine cycle often used in refrigeration, is the Brayton cycle, which involves an adiabatic compression. followed by an isobaric expansion, an adiabatic expansion and finally an isobaric compression. The system begins at temperature \(T_{1}\) and transitions to temperatures \(T_{2}, T_{3},\) and \(T_{4}\) after respective parts of the cycle. a) Sketch this cycle on a \(p V\) -diagram. b) Show that the efficiency of the overall cycle is given by \(\epsilon=1-\left(T_{A}-T_{1}\right) /\left(T_{3}-T_{2}\right)\)

Short answer

Question: Sketch the Brayton cycle on a pV-diagram and discuss the factors affecting its efficiency. Answer: The Brayton cycle consists of four processes: adiabatic compression, isobaric expansion, adiabatic expansion, and isobaric compression. To sketch the cycle on a pV-diagram, mark the initial point T₁ and follow through each process as described below: 1. Adiabatic compression: Draw an upward curve from T₁ to T₂, representing the increase in temperature and pressure. 2. Isobaric expansion: Connect T₂ to T₃ with a horizontal line, as pressure remains constant during this process. 3. Adiabatic expansion: Draw a downward curve from T₃ to T₄, marking the decrease in temperature and pressure. 4. Isobaric compression: Connect T₄ to T₁ with a horizontal line, as pressure remains constant. The efficiency of the Brayton cycle depends on the temperature differences between each stage, which in turn are influenced by the properties of the working substance and the pressure and volume variations during each process. However, without specific information or numerical values related to the working substance or the initial and final conditions, we are unable to calculate the actual efficiency.

Step-by-step solution

Sketch the Brayton cycle on a \(pV\)-diagram

To sketch the Brayton cycle, start by drawing a \(pV\)-diagram, with pressure (\(p\)) on the vertical axis and volume (\(V\)) on the horizontal axis. The four processes of the cycle are represented by four points: 1. Adiabatic compression: Mark the initial point (\(T_1\), with the lowest temperature) on the curve, then draw an upward curve to represent the adiabatic compression, which increases both temperature and pressure, leading to point \(T_2\). 2. Isobaric expansion: Connect point \(T_2\) to point \(T_3\) with a horizontal line since the pressure remains constant during this process. 3. Adiabatic expansion: Draw a downward curve from point \(T_3\) to point \(T_4\), as the process involves a decrease in pressure and temperature. 4. Isobaric compression: Lastly, connect point \(T_4\) to the initial point \(T_1\) with a horizontal line, as the pressure remains constant during this process. We now have our \(pV\)-diagram depicting the Brayton cycle.

Re-write the given expression for efficiency

The efficiency expression given in the problem is: \(\epsilon = 1-\frac{(T_{A}-T_{1})}{(T_{3}-T_{2})}\) However, the exercise mentions that the system begins at temperature \(T_{1}\) and transitions to temperatures \(T_{2}, T_{3},\) and \(T_{4}\) after respective parts of the cycle. There is no mention of \(T_A\) in the question. We assume that \(T_A\) should actually be \(T_4\). So, we re-write the efficiency expression as: \(\epsilon = 1-\frac{(T_{4}-T_{1})}{(T_{3}-T_{2})}\)

Calculate the efficiency of the cycle

According to the exercise, the efficiency of the Brayton cycle is given by the expression: \(\epsilon = 1-\frac{(T_{4}-T_{1})}{(T_{3}-T_{2})}\) To determine the efficiency, we need to find the values of \(T_{1}, T_{2}, T_{3}\) and \(T_{4}\). However, we are not given any information about the working substance or any specific numerical values in the exercise. In this situation, we are limited to analyze the qualitative behavior of the efficiency of the Brayton cycle. The efficiency depends on the temperature differences of each stage of the cycle, which in turn depend on the properties of the working substance and the pressure and volume variations during each process. Based on the given information in the exercise, we are unable to calculate the actual efficiency or the specific values for \(T_{1}, T_{2}, T_{3}\), and \(T_{4}\).

Key concepts

Adiabatic Compression

The Brayton cycle starts with adiabatic compression, a process where the working fluid is compressed, causing the temperature and pressure to increase without any heat exchange with the surroundings. Imagine squeezing a gas into a smaller container with perfect insulation—this compresses the gas without losing heat to the outside, thus conserving energy internally.

In technical terms, during adiabatic compression (\(T_1\rightarrow T_2\) in the Brayton cycle), work is done on the gas, which increases its internal energy and consequently its temperature. It's a cornerstone of the cycle, as it prepares the working fluid for the next phase of energy transfer.

For a better understanding, remember that 'adiabatic' refers to the absence of heat transfer and 'compression' refers to reducing the volume of the gas. There's a direct relationship between volume and pressure in this phase—when the volume decreases, the pressure increases significantly.

Isobaric Expansion

Following adiabatic compression, the Brayton cycle features isobaric expansion. During this phase, the gas expands at a constant pressure and does work on the surroundings. Think of it like letting a hot, high-pressure gas push a piston outwards—it does work by moving the piston, and since the pressure stays the same, the process evolves along a straight horizontal line on a pressure-volume diagram.

It involves heat transfer to the working fluid while maintaining a steady pressure (\(T_2\rightarrow T_3\) in the cycle). The energy added during this phase can be visualized as fuel being burned in a jet engine to propel an airplane forward. Here, 'isobaric' means constant pressure, and 'expansion' is the increase in volume as the gas pushes against the piston or turbine.

Adiabatic Expansion

The third phase in the Brayton cycle is adiabatic expansion. Now, the heated and pressurized gas expands, doing work but without heat exchange—similar to the gas freely expanding into a vacuum, except here it's doing work such as turning a turbine.

This expansion causes a drop in both temperature and pressure (\(T_3\rightarrow T_4\) in the cycle). As the gas expands, it cools down, which is useful in various real-world applications like jet engines, where the expanded, cooler gas is expelled to generate thrust. The concept 'adiabatic' again implies no heat transfer to the surroundings, and 'expansion' here results in a reduced pressure.

Isobaric Compression

The cycle closes with isobaric compression. This is where the volume of the gas decreases at a constant pressure, typically rejecting heat to the surroundings. Picture a piston compressing a gas slowly enough that heat can escape, keeping the pressure steady.

This process brings the system back to its initial state (\(T_4\rightarrow T_1\) in the cycle). In most practical realizations of the Brayton cycle, like in gas turbine engines, this phase can coincide with the exhaust of gases in an environment where the expelled gases are at constant atmospheric pressure. The term 'isobaric' means constant pressure during compression, and 'compression' in this context involves decreasing the volume while rejecting heat.