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

$$v_0 = \sqrt{2 g h}$$

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

$$P_G + \frac{1}{2} \rho v^2 + 0 = 0 + 0 + \rho g h$$

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

$$P_G + P_{atmos}+ \frac{1}{2}\rho v^2_g = P_{atmos} + \frac{1}{2} \rho v^2_{opening}$$

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

 

[HalfLostAndSearching]

I would like to clarify a few points about this problem and provide a more complete solution.

Firstly, the problem statement does not provide enough information to accurately calculate the gauge pressure in the water mains. We would need to know the diameter of the hose and the flow rate of the water to accurately calculate the pressure. However, we can make some assumptions and use Bernoulli's equation to provide an estimate.

Assuming the hose diameter is small enough that we can neglect any changes in pressure due to friction, we can use Bernoulli's equation in its simplified form:

P + 1/2ρv^2 + ρgh = constant

Where:
P = pressure
ρ = density of water
v = velocity of water
g = acceleration due to gravity
h = height

Since the water is spraying up to a height of 25 meters, we can set h = 25 m and solve for the velocity v:

v = √(2gh)

Substituting this into Bernoulli's equation, we get:

P + ρgh = constant

Since we are interested in the gauge pressure, we can set the constant to be the atmospheric pressure (P_atm = 0), giving us:

P_gauge = -ρgh = -(1000 kg/m^3)(9.8 m/s^2)(25 m) = -245,000 Pa

Note that this is a negative value, indicating that the gauge pressure must be below atmospheric pressure. This is because the water is being sprayed upwards, creating a decrease in pressure at the bottom of the hose.

However, this value is an underestimate since we have neglected friction and other losses in the system. In reality, the gauge pressure would need to be higher to overcome these losses and provide enough pressure for the water to spray to a height of 25 meters.

In conclusion, while Bernoulli's equation can provide an estimate for the gauge pressure in this scenario, it is not enough information to accurately calculate the pressure. More information about the system, such as the hose diameter and flow rate, would be needed for a more precise calculation.