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Old Mar 26th 2017, 10:23 AM   #1
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Join Date: Feb 2017
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Finding time for velocity ( water hammer related)

Hello,

The problem is given:


I first found v0(final) from:

$\displaystyle v0=sqrt(\frac {2*g*h}{lambda*\frac {l}{d}+1}) $

$\displaystyle v0=sqrt(\frac {2*9.81*18}{0.03*\frac {400}{2}+1}) $

v0 is 7.1 m/s

90% of v0 is v= 6.39 m/s.

Time to reach v=6.39 m/s:

$\displaystyle t=\frac {l*v0}{2*g*H}*log(\frac {v0+v}{v0-v})$
$\displaystyle t=\frac {400*7.1}{2*9.81*18}*log(\frac {7.1+6.39}{7.1-6.39})
$
$\displaystyle t=10.28 $ s

But the answer key says it is 1 mins 20 secs

Where may I go wrong?

Thanks a lot
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Old Mar 26th 2017, 12:07 PM   #2
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Location: Somerset, England
Posts: 995
Don't know where you got your equations, but the way to solve this is to balance forces (via pressure heads) for the accelerating fluid.

With your setup the main balance equation is

Assuming the tank head is a constant H ( = 18m)

At any time t let the velocity of flow in the pipe be v.
Then if H is the constant head producing flow,

H = Friction head + velocity head at exit + acceleration head + loss at valve + shock loss

If L is the lenght of pipe and a the cross sectional area then

mass of liquid in pipe = waL/g; where w is the specific weight.

If the velocity is (v + dv) at time (t + dt)

the acceleration of liquid in pipe = dv/dt

For for this acceleration = (waL/g) dv/dt

Head required for acceleration = (L/g) dv/dt

Head required toovercome friction = (4fL/d)(v^2/2g)

Loss of head at valve = (4f *25/d ) (v^2/2g)

Loss of head at entry = 0.5v^2/2g

Velocity head at exit = v^2/2g

I'm not sure whether you are meant to include all of these or ignore some.
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