Calculate Gauge Pressure for Fire Hose Height of 25m w/ Bernoulli's Law

[bigplanet401]

Homework Statement

What gauge pressure in the water mains is necessary if a fire hose is to spray water to a height of 25 m?

Homework Equations

Bernoulli's equation(?)

The Attempt at a Solution

I tried figuring out what water velocity was needed to make the water spray up 25 meters. The expression I got (using conservation of energy) was

<br /> v_0 = \sqrt{2 g h}<br />

But then I get confused and try to argue that, by Bernoulli's law,

<br /> P_G + \frac{1}{2} \rho v^2 + 0 = 0 + 0 + \rho g h<br />

and so the gauge pressure should be zero (using the expression above for the velocity). Thanks for any help.

 

[vela]

When the water exits the hose, the gauge pressure is, in fact, 0. The question is asking what's the pressure inside the hose. You're likely expected to assume that the flow rate is low enough that you can approximate the speed of the water inside the hose to be negligible.

 

[bigplanet401]

How does water move through the hose if its velocity is negligible? Wouldn't this break conservation of mass (as the flow rate near the gauge would be different from that at the end of the hose)?

So if the gauge pressure is zero at the end of the hose, there is only atmospheric pressure: P(absolute) = P(gauge)+P(atmospheric)

Then

<br /> P_G + P_{atmos}+ \frac{1}{2}\rho v^2_g = P_{atmos} + \frac{1}{2} \rho v^2_{opening}<br />

But again, how can you claim the velocity at the gauge is zero (especially when the hose has been turned on for some time)? In order for water to flow, it has to move, right?

 

[Chestermiller]

In this problem, please consider choosing the two points for applying the Bernoulli equation as (1) inside the hose (before the fluid enters the nozzle) and (2) at the top of the spray height z. Also please recognize, as Vela pointed out, that, inside the hose, the ρv²/2 term is implicitly assumed (in the problem statement) to be small compared to the gauge pressure P_G.

Chet