Penstock thickness with gate closure time

Data

$L = 600$ m, $D = 0.8$ m $= 80$ cm, $Q = 5\ \text{m}^3/\text{s}$, static head $H = 80$ m, $a = 1200$ m/s, $\sigma = 1326\ \text{kg/cm}^2$, $\eta_j = 0.85$, closure time $T_c = 30$ s.

Type of closure

Velocity: $v = Q/A = 5/(\frac{\pi}{4}0.8^2) = 9.947$ m/s.

Critical time $2L/a = 2(600)/1200 = 1$ s. Since $T_c = 30\ \text{s} \gg 1$ s, the closure is slow, so the Joukowsky value $av/g$ does not apply; use the Michaud (Allievi) formula.

Water hammer rise

$$\Delta H = \frac{2Lv}{gT_c} = \frac{2 \times 600 \times 9.947}{9.81 \times 30} = 40.56\ \text{m}$$

Design pressure and thickness

$$H_{total} = 80 + 40.56 = 120.56\ \text{m} \;\Rightarrow\; p = 12.06\ \text{kg/cm}^2$$ $$t = \frac{pD}{2\sigma\eta_j} = \frac{12.06 \times 80}{2 \times 1326 \times 0.85} = 0.428\ \text{cm} = 4.28\ \text{mm}$$

Practical checks: minimum thickness for handling, $t_{min} = \dfrac{D(\text{mm}) + 800}{400} = \dfrac{800+800}{400} = 4$ mm. Allow 1.5 to 2 mm corrosion allowance.

Answer: calculated $t = 4.3$ mm; recommend a 6 mm thick steel penstock (4.3 mm plus corrosion allowance, rounded to a standard plate).