Problem — Efficiency of the Brayton cycle

Problem

A Brayton gas turbine has pressure ratio $r_p = 8$ with a diatomic gas ($\gamma = 7/5$). Calculate the ideal efficiency $\eta = 1 - r_p^{-(\gamma-1)/\gamma}$.

Solution

Brayton cycle formula. The ideal efficiency depends on the pressure ratio $r_p$ and on $\gamma$:

$\eta = 1 - r_p^{-(\gamma - 1)/\gamma}$

Exponent. With $\gamma = 7/5 = 1{,}4$:

$\frac{\gamma - 1}{\gamma} = \frac{0{,}4}{1{,}4} \approx 0{,}286$

Numerical substitution. With $r_p = 8$:

$\eta = 1 - 8^{-0{,}286}$

Compute $8^{-0{,}286}$: since $8^{0{,}286} \approx 1{,}811$, we get $8^{-0{,}286} \approx 0{,}552$. So:

$\eta = 1 - 0{,}552 = 0{,}448$

$\ev{\eta \approx 44{,}8\%}$

As with the Otto cycle, increasing the pressure ratio improves the efficiency; in real turbines the limit is the maximum temperature the blades can withstand.

Collegamenti

Argomenti: Heat engines Concetti: Efficiency · Thermodynamic transformations Competenze: Solving a thermodynamic cycle Oggetti: Heat engine