Francis runner design (40 m head, 10 m³/s)

Data

$H = 40$ m, $Q = 10\ \text{m}^3/\text{s}$, $N = 300$ rpm, $K_u = 0.85$, $K_f = 0.3$, $\eta_o = 0.80$, $\eta_h = 0.90$. Radial discharge at outlet: $V_{w2} = 0$, and $V_{f2} = V_{f1}$.

$\sqrt{2gH} = \sqrt{2 \times 9.81 \times 40} = 28.01$ m/s.

a) Power developed

$$P = \eta_o\gamma Q H = 0.8 \times 9.81 \times 10 \times 40 = 3139\ \text{kW}$$

b) Inlet diameter and width

$u_1 = K_u\sqrt{2gH} = 0.85 \times 28.01 = 23.81$ m/s

$$D_1 = \frac{60u_1}{\pi N} = \frac{60 \times 23.81}{\pi \times 300} = 1.516\ \text{m}$$

$V_{f1} = K_f\sqrt{2gH} = 0.3 \times 28.01 = 8.40$ m/s

$$B_1 = \frac{Q}{\pi D_1 V_{f1}} = \frac{10}{\pi \times 1.516 \times 8.40} = 0.25\ \text{m}$$

(vane thickness neglected)

c) Guide vane angle at inlet

From $\eta_h = \dfrac{V_{w1}u_1}{gH}$:

$$V_{w1} = \frac{0.9 \times 9.81 \times 40}{23.81} = 14.83\ \text{m/s}$$ $$\tan\alpha = \frac{V_{f1}}{V_{w1}} = \frac{8.40}{14.83} \Rightarrow \alpha = 29.5^\circ$$

Runner vane angle at inlet: since $V_{w1} < u_1$, $\tan\phi = \dfrac{V_{f1}}{u_1 - V_{w1}} = \dfrac{8.40}{8.98}$, so $\phi = 43.1^\circ$ with the backward tangent. Measured from the direction of motion of the wheel, the vane angle is $\theta = 180^\circ - 43.1^\circ = 136.9^\circ$ (an obtuse, backward-curved vane).

d) Specific speed

$$N_s = \frac{N\sqrt{P}}{H^{5/4}} = \frac{300\sqrt{3139}}{40^{1.25}} = 167.1$$

e) Outlet diameter

The flow velocity is constant, $V_{f2} = 8.40$ m/s, and the water leaves without whirl in the axial direction, so the outlet flow area is a full circle:

$$\frac{\pi}{4}D_2^2 = \frac{Q}{V_{f2}} \Rightarrow D_2 = \sqrt{\frac{4 \times 10}{\pi \times 8.40}} = 1.23\ \text{m}$$

Powerhouse dimensions for three units (assumed thumb rules)

Each unit: 3.14 MW, $D_1 = 1.52$ m. Assume a vertical shaft and a spiral casing outer diameter of about $2.7D_1 \approx 4.1$ m.

  PLAN (3 units + erection bay)        Section
 +--------+--------+--------+-----+      roof
 | unit 1 | unit 2 | unit 3 | EB  |    ==========  crane rail
 |  (G)   |  (G)   |  (G)   |     |     |  G  |  generator floor
 +--------+--------+--------+-----+     | TURB|  turbine floor
  25.5 m x 13 m                          \DT /  draft tube

Answer: $P = 3139$ kW; $D_1 = 1.52$ m, $B_1 = 0.25$ m; $\alpha = 29.5^\circ$; $N_s = 167.1$; $D_2 = 1.23$ m; powerhouse about 25.5 m x 13 m x 19 m.