Number of propeller turbine units

The number of units is found by comparing the total power available with the power one turbine of the given specific speed can deliver at the given speed.

Data: $Q = 150$ m³/s, $H = 46$ m, $N = 165$ rpm, $\eta_o = 0.865$, $N_s = 234$ (metric, $P$ in kW).

Step 1: Total power available

$$\begin{aligned} P &= \eta_o \, \rho g Q H \\ &= 0.865 \times 9.81 \times 150 \times 46 \ \text{kW} \\ &= 58551.0\ \text{kW} \approx 58.55\ \text{MW} \end{aligned}$$

Step 2: Power of one unit from the specific speed

Specific speed of a turbine is

$$N_s = \frac{N\sqrt{P}}{H^{5/4}}$$

so the power of one unit is

$$\begin{aligned} \sqrt{P_1} &= \frac{N_s H^{5/4}}{N} = \frac{234 \times 46^{1.25}}{165} = \frac{234 \times 119.80}{165} = 169.89 \\ P_1 &= 169.89^2 = 28864\ \text{kW} \end{aligned}$$

Step 3: Number of units

$$n = \frac{P}{P_1} = \frac{58551}{28864} = 2.03 \approx 2$$

Take 2 units (a whole number; 2.03 is practically 2).

Step 4: Output of each unit

$$P_{unit} = \frac{58551}{2} = 29275\ \text{kW} \approx 29.28\ \text{MW}$$

Check: actual specific speed $= \dfrac{165\sqrt{29275}}{46^{1.25}} = 235.7$, very close to the selected 234, so the choice is correct. Discharge per unit $= 150/2 = 75$ m³/s.

Answer: 2 propeller units, each giving about 29275 kW (29.28 MW).