Pressure rise: rigid versus elastic pipe

For instantaneous closure the pressure rise is $\Delta p=\rho cV$. The wave celerity $c$ depends on whether the pipe wall is treated as rigid (only the water is compressible) or elastic (the wall also stretches).

Given data

$D=0.3$ m, $t=6$ mm, $Q=0.2$ m³/s, $L=5000$ m, $E=2.25\times10^{11}$ N/m², $K=2.0\times10^9$ N/m².

$$V=\frac{Q}{A}=\frac{0.2}{\frac{\pi}{4}(0.3)^2}=2.829\ \text{m/s}$$

(a) Rigid pipe

$$c_r=\sqrt{\frac{K}{\rho}}=\sqrt{\frac{2\times10^9}{1000}}=1414.2\ \text{m/s}$$ $$\Delta p=\rho c_rV=1000\times1414.2\times2.829=4.001\ \text{MPa}\ (407.9\ \text{m of water})$$

(b) Elastic pipe

$$c_e=\sqrt{\frac{K/\rho}{1+\dfrac{KD}{Et}}}=\sqrt{\frac{2\times10^6}{1+\dfrac{2\times10^9\times0.3}{2.25\times10^{11}\times0.006}}}=1176.7\ \text{m/s}$$ $$\Delta p=\rho c_eV=1000\times1176.7\times2.829=3.329\ \text{MPa}\ (339.4\ \text{m})$$

Comparison

The rigid-pipe assumption over-estimates the pressure by about 20%, because the elastic wall expands and absorbs some energy.

Maximum closing time for the result to be valid

The result holds for rapid closure, i.e. $t_c\le 2L/c$:

• Elastic: $t_{max}=\dfrac{2\times5000}{1176.7}=8.50$ s.
• Rigid: $t_{max}=\dfrac{2\times5000}{1414.2}=7.07$ s.

Answer: elastic pipe: $\Delta p=3.33$ MPa (339 m); rigid pipe: $\Delta p=4.00$ MPa (408 m); closing time must not exceed 8.50 s (elastic) / 7.07 s (rigid).