Francis turbine design (45 m³/s, 150 m)

A Francis turbine is designed by fixing its power, a synchronous speed, the specific speed and then the main runner dimensions and the safe setting (suction head) against cavitation. The empirical relations used are the standard ones of de Siervo and de Leva given in the ESHA small hydro guide.

Data: $Q = 45$ m³/s, $H = 150$ m, $\eta = 0.85$, $H_{atm} = 10.3$ m, $H_v = 0.2$ m of water, frequency $f = 50$ Hz.

1. Power output

$$P = \eta \rho g Q H = 0.85 \times 9.81 \times 45 \times 150 = 56284.9\ \text{kW}$$

2. Trial specific speed and speed

Empirical relation for Francis turbines (metric, $P$ in kW):

$$\begin{aligned} N_s &= \frac{3470}{H^{0.625}} = \frac{3470}{150^{0.625}} = \frac{3470}{22.912} = 151.5 \\ N &= \frac{N_s H^{5/4}}{\sqrt{P}} = \frac{151.5 \times 524.95}{237.24} = 335.1\ \text{rpm} \end{aligned}$$

3. Synchronous speed and actual specific speed

The generator must run at $N = 120f/p$. The nearest synchronous speed is with $p = 18$ poles:

$$\begin{aligned} N &= \frac{120 \times 50}{18} = 333.33\ \text{rpm} \\ N_s &= \frac{333.33\sqrt{56284.9}}{150^{1.25}} = 150.6 \end{aligned}$$

This lies in the Francis range (about 50 to 450).

4. Runner outlet (throat) diameter $D_3$

Peripheral velocity coefficient:

$$\begin{aligned} K_u &= 0.31 + 0.0025 N_s = 0.31 + 0.0025 \times 150.6 = 0.687 \\ D_3 &= \frac{84.5 K_u \sqrt{H}}{N} = \frac{84.5 \times 0.687 \times \sqrt{150}}{333.33} = 2.132\ \text{m} \end{aligned}$$

5. Runner inlet diameter $D_1$ and inlet height $H_1$

$$\begin{aligned} D_1 &= \left(0.4 + \frac{94.5}{N_s}\right) D_3 = 1.027 \times 2.132 = 2.190\ \text{m} \\ H_1 &= (0.094 + 0.00025 N_s) D_3 = 0.1317 \times 2.132 = 0.281\ \text{m} \end{aligned}$$

$H_1$ is the height of the runner inlet, which is also the height of the guide vanes.

6. Cavitation check and setting of the turbine

Critical Thoma cavitation coefficient for Francis turbines:

$$\sigma_c = 7.54 \times 10^{-5} N_s^{1.41} = 7.54 \times 10^{-5} \times 150.6^{1.41} = 0.0888$$

Maximum suction head (height of runner outlet above tailwater):

$$\begin{aligned} H_s &= H_{atm} - H_v - \sigma_c H \\ &= 10.3 - 0.2 - 0.0888 \times 150 \\ &= -3.22\ \text{m} \end{aligned}$$

The negative sign means the runner outlet must be placed 3.22 m below the minimum tailwater level (submerged setting) to avoid cavitation.

Summary of design

A spiral casing, 16 to 24 guide vanes and an elbow draft tube starting at diameter about $D_3$ complete the unit.