Gas turbine 5:1: power, efficiency, work ratio

Assumptions: no gas properties are given, so air-standard values $c_p = 1.005$ kJ/kg K and $\gamma = 1.4$ are used throughout ($\frac{\gamma-1}{\gamma} = 0.2857$, $5^{0.2857} = 1.5838$). Fuel mass is neglected. The mechanical efficiency 0.97 applies to the net shaft power delivered to the generator.

Data: $r_p = 5$, $T_3 = 550 + 273 = 823$ K, $T_1 = 20 + 273 = 293$ K, $\eta_c = 0.80$, $\eta_t = 0.83$, $\dot m = 15$ kg/s, $\eta_{mech} = 0.97$.

Compressor

$$\begin{aligned} T_{2s} &= 293 \times 1.5838 = 464.06\ \text{K} \\ T_2 &= 293 + \frac{464.06 - 293}{0.80} = 506.82\ \text{K} \\ W_c &= 1.005(506.82 - 293) = 214.89\ \text{kJ/kg} \end{aligned}$$

Turbine

$$\begin{aligned} T_{4s} &= \frac{823}{1.5838} = 519.63\ \text{K} \\ T_4 &= 823 - 0.83(823 - 519.63) = 571.20\ \text{K} \\ W_t &= 1.005(823 - 571.20) = 253.06\ \text{kJ/kg} \end{aligned}$$

Power output

$$\begin{aligned} W_{net} &= 253.06 - 214.89 = 38.16\ \text{kJ/kg} \\ P &= \eta_{mech}\,\dot m\,W_{net} = 0.97 \times 15 \times 38.16 = 555.3\ \text{kW} \end{aligned}$$

Thermal efficiency

$$\begin{aligned} Q_s &= c_p (T_3 - T_2) = 1.005(823 - 506.82) = 317.76\ \text{kJ/kg} \\ \eta_{th} &= \frac{W_{net}}{Q_s} = \frac{38.16}{317.76} = 0.1201 = 12.01\% \end{aligned}$$

(Overall, including the drive loss: $0.97 \times 12.01 = 11.65\%$.)

Work ratio

$$\text{Work ratio} = \frac{W_{net}}{W_t} = \frac{38.16}{253.06} = 0.151$$

Answer: Power output ≈ 555 kW, thermal efficiency ≈ 12.0 %, work ratio ≈ 0.151.

(If gas properties $c_p = 1.15$ kJ/kg K, $\gamma = 1.333$ are used for the expansion, as in some textbooks, the results become about 657 kW, 12.4 % and 0.174.)