Francis design (150 m, 25 m³/s)

Power

$$P = \eta \gamma Q H = 0.82 \times 9.81 \times 25 \times 150 = 30{,}166\ \text{kW}$$

(With $\eta = 0.81$: $P = 29{,}798$ kW.) One unit is adopted.

Specific speed

For Francis turbines, an empirical (Schweiger and Gregory) relation in metric units (kW, m, rpm) is

$$N_s = \frac{3763}{H^{0.854}} + 49.4 = \frac{3763}{150^{0.854}} + 49.4 = 101.5$$

Speed from $N_s = \dfrac{N\sqrt{P}}{H^{5/4}}$:

$$N = \frac{101.5 \times 150^{1.25}}{\sqrt{30166}} = 306.9\ \text{rpm}$$

Synchronous speed for 50 Hz is $N = 6000/p$ rpm. Nearest value: $p = 20$ poles, $N = 300$ rpm. Actual specific speed $N_s = \dfrac{300\sqrt{30166}}{150^{1.25}} = 99.3$ (98.7 for $\eta=0.81$).

Runner diameter

Peripheral velocity $u = K_u\sqrt{2gH}$ with $K_u = 0.7$ (typical 0.6 to 0.9 for this $N_s$):

$$u = 0.7\sqrt{2 \times 9.81 \times 150} = 37.97\ \text{m/s}$$ $$D = \frac{60u}{\pi N} = \frac{60 \times 37.97}{\pi \times 300} = 2.42\ \text{m}$$

Setting (Thoma cavitation coefficient)

$$\sigma_c = 0.0432\left(\frac{N_s}{100}\right)^2 = 0.0432(0.993)^2 = 0.0426$$

Assume sea-level site: $H_{atm} = 10.3$ m, $H_v = 0.24$ m.

$$H_s = H_{atm} - H_v - \sigma_c H = 10.3 - 0.24 - 0.0426 \times 150 = 3.68\ \text{m}$$

This is the highest permissible position of the runner outlet above tailwater level. A lower setting (a safety margin of 1 to 2 m, or lower for a higher site elevation) is adopted in practice. At higher elevation reduce $H_{atm}$ by about 1 m per 900 m.

Answer: $N_s \approx 99$ (target 101.5); $N = 300$ rpm; $D \approx 2.42$ m; $H_s \le 3.7$ m above tailwater level (practically set lower).