Brayton cycle efficiency and optimum pressure ratio

Ideal Brayton cycle

Processes: 1-2 isentropic compression; 2-3 constant-pressure heat addition; 3-4 isentropic expansion; 4-1 constant-pressure heat rejection.

 T-s diagram                      p-v diagram
 T                                p
 |             3                  |  2 ___________ 3
 |            /|                  |   \             \
 |           / |                  |    \             \
 |   p2     /  |                  |     \             \
 |         2   |                  |      1 ___________ 4
 |         |   4                  +------------------------ v
 |         |  /   p1
 |         | /
 |         1
 +-------------------- s

On the T-s diagram 1-2 and 3-4 are vertical (isentropic); on the p-v diagram 2-3 and 4-1 are horizontal (constant pressure).

For a perfect gas with constant $c_p$, and pressure ratio $r_p = p_2/p_1 = p_3/p_4$:

$$q_{in} = c_p (T_3 - T_2), \qquad q_{out} = c_p (T_4 - T_1)$$ $$\eta = 1 - \frac{q_{out}}{q_{in}} = 1 - \frac{T_4 - T_1}{T_3 - T_2}$$

From the isentropic relations, $\dfrac{T_2}{T_1} = \dfrac{T_3}{T_4} = r_p^{(\gamma-1)/\gamma}$, hence $\dfrac{T_4}{T_1} = \dfrac{T_3}{T_2}$ and $\dfrac{T_4 - T_1}{T_3 - T_2} = \dfrac{T_1}{T_2}$:

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

The efficiency depends only on the pressure ratio and rises with it.

Pressure ratio for maximum net work

Let $x = r_p^{(\gamma-1)/\gamma}$, $T_2 = T_1 x$, $T_4 = T_3/x$:

$$w_{net} = c_p (T_3 - T_4) - c_p (T_2 - T_1) = c_p \left[T_3\left(1 - \frac1x\right) - T_1(x - 1)\right]$$ $$\frac{dw_{net}}{dx} = c_p\left[\frac{T_3}{x^2} - T_1\right] = 0 \;\Rightarrow\; x^2 = \frac{T_3}{T_1}$$ $$\boxed{r_{p,opt} = \left(\frac{T_3}{T_1}\right)^{\gamma/[2(\gamma-1)]}}$$

At this ratio $T_2 = T_4 = \sqrt{T_1 T_3}$ and the maximum work is

$$w_{net,max} = c_p\left(\sqrt{T_3} - \sqrt{T_1}\right)^2$$

The efficiency keeps rising above this pressure ratio, but the net work falls and becomes zero at $x = T_3/T_1$. Practical gas turbines are therefore designed at a pressure ratio between the maximum-work value and this limit, usually closer to the maximum-work value.