Question

A simple ideal Brayton cycle without regeneration is modified to incorporate multistage compression with intercooling and multistage expansion with reheating, without changing the pressure or temperature limits of the cycle. As a result of these two modifications, (a) Does the net work output increase, decrease, or remain the same? (b) Does the back work ratio increase, decrease, or remain the same? \((c) \quad\) Does the thermal efficiency increase, decrease, or remain the same? (d) Does the heat rejected increase, decrease, or remain the same?

Short answer

Answer: The modified Brayton cycle has increased net work output, decreased back work ratio, increased thermal efficiency, and decreased heat rejected.

Step-by-step solution

(Understanding the Cycle Modifications)

The given initial Brayton cycle is a simple ideal one without regeneration, meaning it consists of one stage of compression, heat addition, expansion, and heat rejection. The modified Brayton cycle incorporates multistage compression with intercooling between each stage and multistage expansion with reheating between each stage.

(Net Work Output)

Multistage compression with intercooling and multistage expansion with reheating reduce the amount of work required for compression and increase the work output in the expansion stage. This is because intercooling decreases the work input for compression and reheating increases the efficiency of the turbine. As a result, the net work output \((W_{net})\) of the modified cycle is greater than the original cycle. Therefore, the net work output increases.

(Back Work Ratio)

The back work ratio is the ratio of the work required for compression \((W_{in})\) to the work extracted in the expansion stage \((W_{out})\). Since multistage compression with intercooling decreases the work input for compression and multistage expansion with reheating increases the turbine work output, the back work ratio will decrease. This is because the denominator (work output) in the ratio increases, while the numerator (work input) decreases, leading to a decreased ratio.

(Thermal Efficiency)

Thermal efficiency \((\eta_{th})\) is the ratio of the net work output \((W_{net})\) to the heat input \((Q_{in})\). Since multistage compression with intercooling and multistage expansion with reheating both increase the net work output and maintain the same heat input (because temperature and pressure limits have not changed), the thermal efficiency of the cycle will increase.

(Heat Rejected)

Heat rejected \((Q_{out})\) is the difference between the heat input \((Q_{in})\) and the net work output \((W_{net})\). Since the net work output has increased but the heat input has remained the same (due to unchanged temperature and pressure limits), the heat rejected in the modified cycle will be less than in the original cycle. Therefore, the heat rejected decreases. In summary, the modified ideal Brayton cycle with multistage compression and intercooling and with multistage expansion and reheating has: a) increased net work output b) decreased back work ratio c) increased thermal efficiency d) decreased heat rejected

Key concepts

Multistage Compression

In a Brayton cycle, introducing multistage compression offers a significant advantage over a single-stage compression system. Instead of compressing the working fluid in one go, it is done in stages, with each stage slightly increasing the pressure. By dividing the compression process, the total work required for compressing the gas can be reduced; this is because compressing a gas in smaller increments requires less energy than doing it all at once due to the gas's increasing resistance to compression as its pressure rises.

Multistage compression also reduces the strain on individual components, leading to less wear and potentially longer equipment life. Students should understand that the implementation of this system within the Brayton cycle tends to increase the overall cycle efficiency, which helps explain why the net work output of the cycle would increase after such a modification.

Intercooling

Intercooling is an important enhancement used in conjunction with multistage compression. It involves cooling the working fluid between compression stages to bring its temperature down before it enters the next compression stage. Cooling the air reduces its volume, making it easier to compress further, and hence less work is required.

Intercooling not only improves efficiency by reducing the compressor work but also helps in controlling the temperature within the system, avoiding potential thermal stresses. A clearer conclusion for students might be observed in practical applications such as gas turbine engines where intercooling contributes to an overall improvement in cycle performance.

Reheating

Reheating in the Brayton cycle refers to the process of adding heat to the working fluid between stages of expansion. Just as intercooling is applied between compression stages, reheating is used between expansion stages. By reheating the gas, its volume increases, and as a result, it can do more work during the subsequent expansion stage.

This approach maximizes the energy extraction from the heat added to the cycle and results in a higher thermal efficiency. It is important for students to recognize that reheating not only enhances power output but also can reduce the thermal load on turbine materials, potentially leading to improved turbine longevity.

Thermal Efficiency

Thermal efficiency is a measure of how well an engine or other thermal system converts heat into work. It is calculated as the ratio of net work output to the heat input, expressed as a percentage. The higher the percentage, the more efficient the cycle is.

Incorporating multistage compression with intercooling and reheating into the Brayton cycle boosts the thermal efficiency by increasing the net work output while maintaining the same level of heat input. This is a critical concept for students to grasp, as it highlights the importance of optimizing cycle components to improve overall efficiency.

Net Work Output

The net work output of a thermal cycle like the Brayton cycle is the difference between the work produced during the expansion phase and the work consumed during the compression phase. Improvements in efficiency through multistage compression and expansion with intercooling and reheating respectively lead to a favorable outcome: an increase in the net work output.

This is because the work required for compression is lower, and the work extracted during expansion is higher. Students must realize that the net work output is an essential indicator of the performance of the cycle.

Back Work Ratio

The term 'back work ratio' is often a new concept for students. It measures the proportion of the energy produced during the turbine's expansion process that must be used to drive the compressor. A high back work ratio indicates a less efficient cycle because more of the energy generated is needed to sustain the cycle's operation.

In the context of the Brayton cycle, modifications such as intercooling reduce the energy required for compression, thus decreasing the back work ratio. It's pivotal for students to understand that lowering the back work ratio is indicative of a higher overall cycle efficiency.

Heat Rejection

Heat rejection in the Brayton cycle occurs when the working fluid is cooled after expansion and before it is reintroduced into the compressor. It is an essential part of the cycle because it ensures that the working fluid is at an appropriate temperature for compression.

When the cycle includes intercooling and reheating, the net work output increases, and, as a consequence, the heat rejected decreases, assuming constant heat input. It is critical for students to comprehend that effective heat rejection is integral to maintaining cycle efficiency and stability.