Consider an air standard cycle in which the air enters the compressor at 1.0 bar and 20°C. The pressure of air leaving the compressor is 3.5 bar and the temperature at the turbine inlet is 600°C. For 1 kg of air, determine the efficiency of the cycle: 

Concept:

The efficiency of an ideal Brayton cycle can be determined using the given pressure ratios and specific heat ratio.

The thermal efficiency formula is given by:

\(\eta = 1 - \frac{T_4 - T_1}{T_3 - T_2}\)

Calculation:

Given:

Initial pressure at compressor inlet, \( P_1 = 1.0 \, \text{bar} \)

Initial temperature at compressor inlet, \( T_1 = 20^\circ \text{C} = 293 \, \text{K} \)

Pressure at compressor outlet, \( P_2 = 3.5 \, \text{bar} \)

Temperature at turbine inlet, \( T_3 = 600^\circ \text{C} = 873 \, \text{K} \)

Specific heat ratio, \(\gamma = 1.4\)

Calculate the temperature after compression ( T₂ ):

\( T_2 = T_1 \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} = 293 (3.5)^{0.2857} \approx 420 \, \text{K} \)

Calculate the temperature after expansion ( T₄ ):

\( T_4 = T_3 \left( \frac{P_1}{P_2} \right)^{0.2857} = 873 \left( \frac{1}{3.5} \right)^{0.2857} \approx 608 \, \text{K} \)

Efficiency Calculation:

Using the temperatures, the efficiency can also be given by:

\(\eta = 1 - \frac{T_4 - T_1}{T_3 - T_2} = 1 - \frac{608 - 293}{873 - 420} = 1 - \frac{315}{453} = 1 - 0.695 \approx 0.305 \approx 30.5\% \)

The efficiency of the Brayton cycle is approximately 30.5%.