Hammer pressure for 4 s and 60 s closure

First, the velocity before closure is found from the head loss in the pipe. Then the closure time is compared with $2L/c$: if $t_c<2L/c$ the closure is rapid ($\Delta p=\rho cV$); otherwise it is gradual ($\Delta p=\rho LV/t_c$).

Assumptions

The pipe discharges freely at the outlet under a head $H=61$ m. Take $\nu=10^{-6}$ m²/s and $g=9.81$ m/s². Losses = friction plus the exit velocity head.

Step 1: Initial velocity

$\varepsilon/D=0.046\times10^{-3}/0.61=7.5\times10^{-5}$. Solve

$$H=\left(1+f\frac{L}{D}\right)\frac{V^2}{2g}$$

with the Colebrook equation for $f$ (iterated): $f=0.0121$ at $Re\approx2.62\times10^6$.

$$V=\sqrt{\frac{2\times9.81\times61}{1+0.0121\times3200/0.61}}=4.30\ \text{m/s}$$

Step 2: Wave celerity

$$c=\sqrt{\frac{K/\rho}{1+KD/(Et)}}=\sqrt{\frac{2.17\times10^6}{1+\dfrac{2.17\times10^9\times0.61}{1.9\times10^{11}\times0.005}}}=952.2\ \text{m/s}$$ $$\frac{2L}{c}=\frac{6400}{952.2}=6.72\ \text{s}$$

Step 3: Hammer pressure

Static pressure: $p_s=\rho g H=1000\times9.81\times61=0.598$ MPa.

Closure in 4 s ($<6.72$ s, rapid):

$$\Delta p=\rho cV=1000\times952.2\times4.30=4.09\ \text{MPa}\ (417\ \text{m})$$

Closure in 60 s ($>6.72$ s, gradual):

$$\Delta p=\frac{\rho LV}{t_c}=\frac{1000\times3200\times4.30}{60}=0.229\ \text{MPa}\ (23.4\ \text{m})$$

Step 4: Stress in the pipe wall, $\sigma=pD/2t$

Step 5: Minimum safe closure time

Working stress $=16000$ psi $=16000\times6894.76=110.3$ MPa. Allowable pressure:

$$p_{all}=\frac{2t\sigma}{D}=\frac{2\times0.005\times110.3}{0.61}=1.808\ \text{MPa}$$

Allowable hammer rise $=p_{all}-p_s=1.808-0.598=1.210$ MPa. Using the gradual-closure formula:

$$t_{min}=\frac{\rho LV}{\Delta p_{all}}=\frac{1000\times3200\times4.30}{1210051}=11.4\ \text{s}$$

This is greater than $2L/c=6.72$ s, so the gradual-closure formula is valid.

Answer: hammer pressure = 4.09 MPa for 4 s closure and 0.229 MPa for 60 s closure; wall stress = 286 MPa and 50.5 MPa respectively; minimum safe closure time ≈ 11.4 s.

The 4 s closure gives a stress far above the working stress, so it is unsafe.