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

• bigplanet401
In summary: The question is asking for what gauge pressure is necessary in order to spray water up to a height of 25 meters. The answer is that the pressure should be zero, and so the flow rate should be low enough that the water velocity inside the hose is negligible.
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.

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.

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?

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 PG.

Chet

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.

## 1. How do you use Bernoulli's Law to calculate gauge pressure for a fire hose height of 25m?

To calculate gauge pressure for a fire hose height of 25m using Bernoulli's Law, you need to first determine the velocity of the water exiting the hose. This can be done by using the equation: v = √(2gh), where v is the velocity, g is the acceleration due to gravity (9.8 m/s²), and h is the height of the water column (25m). Once you have the velocity, you can use the Bernoulli's Law equation: P + (ρgh) + (1/2)ρv² = constant, where P is the gauge pressure, ρ is the density of water (1000 kg/m³), and v is the velocity. Rearrange the equation to solve for P and you will have the gauge pressure for a fire hose height of 25m.

## 2. What is gauge pressure and how is it different from absolute pressure?

Gauge pressure is the pressure measured relative to atmospheric pressure. It does not take into account the atmospheric pressure, which is typically around 14.7 psi. Absolute pressure, on the other hand, includes atmospheric pressure in its measurement. This means that absolute pressure will always be higher than gauge pressure by about 14.7 psi. Gauge pressure is often used in situations where the effects of atmospheric pressure are negligible, such as in the case of a fire hose.

## 3. Why is Bernoulli's Law used to calculate gauge pressure for a fire hose height of 25m?

Bernoulli's Law is used because it relates the pressure of a fluid to its velocity and height. In the case of a fire hose, the water is under constant pressure as it exits the hose, which means that the only factors affecting the pressure are the velocity and height of the water. Bernoulli's Law allows us to take these factors into account and calculate the gauge pressure at the exit point of the hose.

## 4. Are there any other factors that may affect the gauge pressure and need to be considered?

Yes, there are a few other factors that may affect the gauge pressure for a fire hose. These include the diameter of the hose, the type of nozzle being used, and any obstructions or bends in the hose. These factors can cause changes in the velocity of the water and may need to be taken into account in the calculation of gauge pressure.

## 5. Can Bernoulli's Law be used for other types of fluid flow calculations?

Yes, Bernoulli's Law can be used for a variety of fluid flow calculations, including calculating the pressure in a pipe, the lift force on an airplane wing, and the flow rate through a nozzle. It is a fundamental principle in fluid mechanics and is used in many different applications.

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