An air standard Brayton cycle receives air at 101.325 kPa and 21°C. The peer pressure and temperature limits of the cycle are 414 kPa and 816°C respectively. The compression effici... An air standard Brayton cycle receives air at 101.325 kPa and 21°C. The peer pressure and temperature limits of the cycle are 414 kPa and 816°C respectively. The compression efficiency is 85% and the turbine efficiency is 96%. Determine the thermal efficiency of the cycle.
Understand the Problem
The question is asking us to determine the thermal efficiency of an air standard Brayton cycle given specific parameters such as pressure, temperature limits, and efficiencies of the compressor and turbine. We will use the given information to apply the formulas related to Brayton cycle efficiency.
Answer
$$ \eta_{th,actual} = \eta_{th,ideal} \cdot \eta_c \cdot \eta_t $$
Answer for screen readers
The thermal efficiency of the air standard Brayton cycle can be calculated as follows, assuming we have the necessary parameters.
Final answer for thermal efficiency is given by:
$$ \eta_{th,actual} = \eta_{th,ideal} \cdot \eta_c \cdot \eta_t $$
We will need values for $T_1$, $T_3$, $\eta_c$, $\eta_t$, and $r$ to compute this.
Steps to Solve
- Identify the parameters
Write down the given parameters:
- Compressor efficiency ($\eta_c$)
- Turbine efficiency ($\eta_t$)
- Pressure ratio ($r$)
- Maximum temperature ($T_3$)
- Minimum temperature ($T_1$)
- Calculate the ideal thermal efficiency ($\eta_{th,ideal}$)
For an ideal Brayton cycle, the thermal efficiency can be calculated by the formula:
$$ \eta_{th,ideal} = 1 - \frac{1}{r^{(\gamma - 1)/\gamma}} $$
where $\gamma$ is the specific heat ratio (usually taken as $1.4$ for air).
- Account for efficiencies in the actual cycle
The actual thermal efficiency ($\eta_{th,actual}$) can be calculated using:
$$ \eta_{th,actual} = \eta_{th,ideal} \cdot \eta_c \cdot \eta_t $$
This means we multiply the ideal thermal efficiency by the efficiencies of the compressor and turbine.
- Substitute the known values into the equations
Insert the known values into the formulas derived in the previous steps to find the thermal efficiency.
Calculate the ideal thermal efficiency first and then use it to find the actual thermal efficiency.
- Final Calculation
Perform the calculations step-by-step to obtain the final value for thermal efficiency, $ \eta_{th,actual}$.
The thermal efficiency of the air standard Brayton cycle can be calculated as follows, assuming we have the necessary parameters.
Final answer for thermal efficiency is given by:
$$ \eta_{th,actual} = \eta_{th,ideal} \cdot \eta_c \cdot \eta_t $$
We will need values for $T_1$, $T_3$, $\eta_c$, $\eta_t$, and $r$ to compute this.
More Information
The Brayton cycle is an important thermodynamic cycle used in gas turbines and jet engines. Understanding the thermal efficiency allows for improvements in design and performance.
Tips
- Forgetting to use the correct formula for thermal efficiency.
- Using incorrect values for the specific heat ratio ($\gamma$), which is crucial for accurate calculations.
- Not accounting for the efficiencies of the compressor and turbine when calculating actual thermal efficiency.
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