Water hammer in elastic cast-iron pipe

Water hammer

Water hammer is the sudden rise (and following fall) of pressure in a pipe when the flowing water is suddenly stopped or its velocity is suddenly changed, e.g. by quick closing of a valve or sudden load rejection at a turbine. The kinetic energy of the moving water is turned into pressure energy, and a pressure wave travels back and forth along the pipe at the speed of sound in water.

Problems caused by water hammer

• Very high pressure can burst the pipe or penstock.
• Negative pressure in the return wave can cause collapse of thin pipes and cavitation.
• Damage to valves, joints, anchors and supports; noise and vibration.
• Need for thicker (costlier) pipes and protective devices like surge tanks and relief valves.
• Unstable turbine governing.

Numerical: elastic pipe, sudden closure

Data: $D = 0.15$ m, $t = 0.01$ m, $p = 1.962$ MN/m², $K = 1.962$ GN/m², $E = 117.7$ GN/m², $\rho = 1000$ kg/m³.

For sudden closure in an elastic pipe:

$$p = V\sqrt{\frac{\rho}{\dfrac{1}{K} + \dfrac{D}{E\,t}}}$$ $$\begin{aligned} \frac{1}{K} + \frac{D}{Et} &= \frac{1}{1.962 \times 10^9} + \frac{0.15}{117.7 \times 10^9 \times 0.01} \\ &= 5.097 \times 10^{-10} + 1.274 \times 10^{-10} = 6.371 \times 10^{-10}\ \text{m}^2/\text{N} \\ \sqrt{\frac{1000}{6.371 \times 10^{-10}}} &= 1.2528 \times 10^6 \\ V &= \frac{1.962 \times 10^6}{1.2528 \times 10^6} = 1.566\ \text{m/s} \end{aligned}$$

Answer: Maximum velocity of water ≈ 1.57 m/s.