Water hammer pressure and stress, 2300 m pipeline

Data. $L = 2300$ m, $D = 0.15$ m, $t = 2.8$ mm, $Q = 2200$ l/min $= 0.03667$ m³/s, $K = 2\times10^9$ Pa, $E = 2.08\times10^{11}$ Pa. Instantaneous closure.

Velocity and wave speed

$$V_0 = \frac{Q}{\pi D^2/4} = \frac{0.03667}{0.017671} = 2.075\ \text{m/s}$$ $$c = \sqrt{\frac{K/\rho}{1 + \dfrac{KD}{Et}}} = \frac{\sqrt{2\times10^{6}}}{\sqrt{1 + \dfrac{2\times10^9(0.15)}{2.08\times10^{11}(0.0028)}}} = \frac{1414.2}{\sqrt{1 + 0.5151}} = 1148.9\ \text{m/s}$$

Check closure type: $2L/c = 4.00$ s, and the valve closes instantaneously, so the closure is rapid (full Joukowsky rise).

Water hammer pressure

$$\Delta p = \rho cV_0 = 1000(1148.9)(2.075) = 2.384\times10^6\ \text{Pa} = 2.38\ \text{MPa}\quad(\Delta H = 243.0\ \text{m of water})$$

Stress developed

Hoop (circumferential) stress due to the pressure rise, thin-walled pipe:

$$\sigma = \frac{\Delta p\,D}{2t} = \frac{(2.384\times10^6)(0.15)}{2(0.0028)} = 63.9\ \text{MPa}$$

(The longitudinal stress is about half of this, $\Delta p D/4t$.)

Pressure-time variation at 1725 m from the reservoir

The point is $d = 2300 - 1725 = 575$ m from the valve ($x = 1725$ m from the reservoir). Wave arrival times ($c = 1148.9$ m/s):

• $t_1 = d/c = 0.500$ s: wave from the valve arrives; pressure rises by $+2.38$ MPa
• $t_2 = (2L - d)/c = 3.503$ s: wave reflected from the reservoir arrives (negative); pressure falls back to the static value
• $t_3 = (2L + d)/c = 4.504$ s: negative wave reflected from the valve arrives; pressure falls to $-2.38$ MPa
• $t_4 = (4L - d)/c = 7.507$ s: positive wave reflected from the reservoir arrives; pressure returns to the static value
• The cycle repeats with period $4L/c = 8.007$ s.

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(top = 1.0, bottom = -1.0; '-' is zero line)

(Pressure change relative to static pressure: +1 = $+\Delta p$, $-1$ = $-\Delta p$; time from 0 to $4L/c = 8.0$ s.)

Answer: $\Delta p = 2.38$ MPa (243 m of water); hoop stress $\sigma = 63.9$ MPa; the pressure at 1725 m from the reservoir is a stepped wave with the changes at $t$ = 0.50, 3.50, 4.50 and 7.51 s.