Question

Using EES (or other) software, determine the effect of the number of compression and expansion stages on the thermal efficiency of an ideal regenerative Brayton cycle with multistage compression and expansion. Assume that the overall pressure ratio of the cycle is \(18,\) and the air enters each stage of the compressor at \(300 \mathrm{K}\) and each stage of the turbine at \(1200 \mathrm{K}\). Using constant specific heats for air at room temperature, determine the thermal efficiency of the cycle by varying the number of stages from 1 to 22 in increments of 3. Plot the thermal efficiency versus the number of stages. Compare your results to the efficiency of an Ericsson cycle operating between the same temperature limits.

Short answer

Answer: The thermal efficiency of an ideal regenerative Brayton cycle increases with the number of compression and expansion stages. However, the efficiency starts to diminish after a certain number of stages. When comparing the efficiency of the Brayton cycle to that of an Ericsson cycle, the Brayton cycle's efficiency is generally lower. The Ericsson cycle achieves higher efficiency due to its constant-temperature heat addition and rejection processes.

Step-by-step solution

Calculate efficiency of the Brayton cycle for different number of stages

We will use the following equation to determine the efficiency of the regenerative Brayton cycle: \(\eta_b = \frac{1 - \frac{1}{\text{(Pressure ratio)}^\frac{\gamma -1}{\gamma}}}{1 - \frac{1}{\text{(Pressure ratio)}^\frac{n(\gamma -1)}{\gamma} }} \times 100 \%\) Where \(n\) is the number of stages and \(\gamma\) is the specific heat ratio for air (assuming constant specific heat at room temperature). With the given pressure ratio of 18, air inlet temperature for compression and expansion stages 300K and 1200K respectively, we can calculate the efficiency for different values of stages (1 to 22 in increments of 3).

Plot the efficiency against the number of stages

Using the calculated efficiency values for different numbers of stages, we can create a plot (using software such as Excel or Python) with the number of stages on the x-axis and the efficiency on the y-axis. This will visually demonstrate the effect of the number of compression and expansion stages on the thermal efficiency of the Brayton cycle.

Compare to Ericsson cycle efficiency

To compare the Brayton cycle's efficiency to that of an Ericsson cycle, we will use the following equation for the efficiency of an Ericsson cycle operating between the same temperature limits: \(\eta_e = 1 - \frac{T_1}{T_2} \times 100 \%\) Where \(T_1\) is the lower temperature limit (300K in this case) and \(T_2\) is the higher temperature limit (1200K in this case). Calculate the efficiency of the Ericsson cycle and compare it to the efficiency values obtained for the Brayton cycle in step 1. By following these steps, you'll be able to analyze the effect of the number of compression and expansion stages on the thermal efficiency of an ideal regenerative Brayton cycle and compare it to the efficiency of an Ericsson cycle operating between the same temperature limits.

Key concepts

Regenerative Brayton Cycle

The regenerative Brayton cycle is an enhanced version of the standard Brayton cycle which aims to improve thermal efficiency by recovering some of the heat contained in the exhaust gas. This is achieved by passing the exhaust through a heat exchanger, known as a regenerator, before it is rejected from the cycle. The regenerator preheats the air entering the combustion chamber, resulting in a decrease in the fuel requirement.

By doing so, the temperature of the working fluid (often air) is increased before additional heat is added in the combustion process, resulting in higher thermal efficiency. The efficiency gain is especially significant when the cycle includes multiple compression and expansion stages because the regenerative process takes advantage of the incremental temperature differences afforded by these stages.

The exercise you mentioned focuses on analyzing how adding more stages to a regenerative Brayton cycle impacts its overall thermal efficiency. Intuitively, as the number of stages increases, the temperature differential across each heat exchanger decreases, allowing for more effective heat transfer and better utilization of thermal energy.

Thermal Efficiency Calculation

Thermal efficiency is a measure of how well a cycle converts heat into work. It's a crucial parameter for understanding the performance of any heat engine, including the Brayton cycle. When calculating the thermal efficiency \( \eta \), it's important to consider both the energy being added to the system and the work being extracted from it.

For the regenerative Brayton cycle, the formula you noted in the exercise demonstrates how efficiency is dependent on the pressure ratio and the number of stages, both of which are significant factors in overall performance. The pressure ratio affects the maximum temperature that can be achieved in the cycle, influencing the thermodynamic efficiency.

Moreover, the equation also incorporates the specific heat ratio \( \gamma \), highlighting the role of material properties in the efficiency calculation. It's clear that as you increase the number of stages, up to a certain point, the efficiency of the cycle increases because the compressor and the turbine can operate closer to ideal isentropic conditions. However, there comes a point where adding more stages will no longer result in significant efficiency gains due to practical limitations such as increased complexity and parasitic losses.

Ericsson Cycle Comparison

Comparing the Brayton and Ericsson cycles provides insight into how different thermodynamic cycles perform under similar conditions. The Ericsson cycle is similar to the Brayton cycle but includes an isothermal compression and expansion process. Contrast that with the Brayton cycle, which ideally involves adiabatic (isentropic) processes.

In your comparison exercise, an Ericsson cycle equation is employed to calculate thermal efficiency using temperature limits \(T_1\) and \(T_2\). Despite the similarity in operating temperature ranges, the Ericsson cycle can potentially achieve higher efficiency due to its ideal constant-temperature processes.

However, in practical applications, the Brayton cycle, especially with regenerative components, often strikes a better balance between efficiency and feasibility. The regenerative Brayton cycle attempts to emulate the continuous heat exchange of the Ericsson cycle by heating the compressed air using exhaust gases. Your calculations and the plotted efficiency against the number of stages reveal the influence of staging on enhancing the Brayton cycle's performance, allowing for a comparison on more equal footing with the theoretically more efficient Ericsson cycle.