Question

At what pressure ratio does a Brayton cycle using a monatomic gas have an efficiency of$50\%$ ?

Short answer

The pressure ratio using a monatomic gas,$r=11.3.$

Step-by-step solution

Step: 1 Definition of Brayton cycle:

The Brayton cycle is a thermodynamic cycle that works on the basis of constant pressure process . It involves four operation , like isentropic compression , constant pressure heat addition , isentropic expansion and constant pressure heat rejection. It is mostly used in gas turbine power plants.

Step: 2 Equating the Efficiency:

The performance of a Brayton engine operating at a compression ratio of r with a gas of heat capacity ratio of $\gamma$is calculated as follows:

$\begin{array}{r}\eta =1-\frac{1}{r^{\frac{\gamma -1}{\gamma }}}\\ \end{array}$
Simplify as follows.

$\begin{array}{r}\\ \frac{1}{r^{\frac{\gamma -1}{\gamma }}}=1-\eta \end{array}$

Therefore we have

$r^{\frac{\gamma -1}{\gamma }}=\frac{1}{1-\eta }.$

Step: 3 Finding the value of r:

The value of $\gamma$for air is 1.4 . Efficiency is 50% hence$\eta =.50$

The ratio $r$is given by

localid="1649322326172" $\begin{array}{r}r=\sqrt[\frac{\gamma -1}{\gamma }]{\frac{1}{1-\eta }}\\ r=\sqrt[\frac{1.40-1}{1.40}]{\frac{1}{1-0.5}}=11.3.\end{array}$