Regenerative gas turbine: 1 to 6.9 bar

Assumptions: the inlet air temperature is not given; it is taken as 32 °C (305 K), the value used in the standard textbook version of this problem. Air-standard values $c_p = 1.005$ kJ/kg K and $\gamma = 1.4$ are used throughout; fuel mass neglected.

Data: $p_1 = 1$ bar, $p_2 = 6.9$ bar ($r_p = 6.9$), $T_3 = 816 + 273 = 1089$ K, $\eta_c = 0.84$, $\eta_t = 0.85$, $\varepsilon = 0.60$; $6.9^{0.2857} = 1.7365$.

       +------ Regenerator ------+
       |  x                 y    |
 1     | 2                  4    |
 --> [C] --> [CC] --3--> [T] ----+

Compressor

$$\begin{aligned} T_{2s} &= 305 \times 1.7365 = 529.63\ \text{K} \\ T_2 &= 305 + \frac{529.63 - 305}{0.84} = 572.41\ \text{K} \\ W_c &= 1.005(572.41 - 305) = 268.75\ \text{kJ/kg} \end{aligned}$$

Turbine

$$\begin{aligned} T_{4s} &= \frac{1089}{1.7365} = 627.13\ \text{K} \\ T_4 &= 1089 - 0.85(1089 - 627.13) = 696.41\ \text{K} \\ W_t &= 1.005(1089 - 696.41) = 394.55\ \text{kJ/kg} \end{aligned}$$

Regenerator

$$T_x = T_2 + \varepsilon(T_4 - T_2) = 572.41 + 0.6(696.41 - 572.41) = 646.81\ \text{K}$$

(a) Thermal efficiency

$$\begin{aligned} W_{net} &= 394.55 - 268.75 = 125.80\ \text{kJ/kg} \\ Q_s &= 1.005(1089 - 646.81) = 444.40\ \text{kJ/kg} \\ \eta_{th} &= \frac{125.80}{444.40} = 0.2831 = 28.31\% \end{aligned}$$

(b) Air rate

Air needed per kWh of net output:

$$\text{Air rate} = \frac{3600}{W_{net}} = \frac{3600}{125.80} = 28.62\ \text{kg/kWh}$$

(c) Work ratio

$$\text{Work ratio} = \frac{W_{net}}{W_t} = \frac{125.80}{394.55} = 0.319$$

Answer: $\eta$ = 28.31 %, air rate = 28.62 kg/kWh, work ratio = 0.319.

(If the inlet air is taken as 15 °C = 288 K instead, the same method gives $\eta$ = 30.79 %, air rate = 25.57 kg/kWh and work ratio = 0.357.)