Number and runner diameter of turbines (20 m head)

Given

$Q = 23\ \text{m}^3/\text{s}$, $H = 20$ m, $\eta = 0.85$, $N = 170$ rpm, $N_s = 230$, $K_u = 0.76$.

Total power available

$$P_{total} = \eta\gamma Q H = 0.85 \times 9.81 \times 23 \times 20 = 3836\ \text{kW}$$

Power per turbine from specific speed

$$N_s = \frac{N\sqrt{P}}{H^{5/4}} \;\Rightarrow\; P = \left(\frac{N_s H^{5/4}}{N}\right)^2 = \left(\frac{230 \times 20^{1.25}}{170}\right)^2 = 3274\ \text{kW}$$

Number of turbines

$$n = \frac{3836}{3274} = 1.17 \Rightarrow 2\ \text{turbines}$$

(One turbine of 3274 kW cannot carry the whole 3836 kW, so the next whole number is used. Each then develops $3836/2 = 1918$ kW and takes $Q = 11.5\ \text{m}^3/\text{s}$.)

Runner diameter

$$u = K_u\sqrt{2gH} = 0.76\sqrt{2 \times 9.81 \times 20} = 15.05\ \text{m/s}$$ $$D = \frac{60u}{\pi N} = \frac{60 \times 15.05}{\pi \times 170} = 1.69\ \text{m}$$

Answer: 2 turbines; runner diameter $\approx 1.69$ m.