Problem 43 - University Physics with Modern Physics

Question

What gauge pressure is required in the city water mains for a stream from a fire hose connected to the mains to reach a vertical height of 15.0 m? (Assume that the mains have a much larger diameter than the fire hose.)

Step-by-Step Solution

Answer

147 kPa gauge pressure is required.

Step 1: Understand the Problem

To find the gauge pressure in the water mains required for a stream of water to reach a height of 15.0 m, we need to apply the concept of fluid dynamics and understand that the pressure must overcome the gravitational force exerted on the column of water to reach that height.

Step 2: Apply the Bernoulli's Equation

We start with Bernoulli's equation, which relates the pressure, velocity, and height in a flowing fluid as: \( P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2 \). Since the fire hose nozzle is open to the atmosphere, \( P_2 \) is atmospheric pressure.

Step 3: Simplify the Equation

Assume \( v_1 = 0 \) since the mains have a much larger diameter than the hose, thus the velocity of water in the mains is negligible, and \( h_1 = 0 \). The equation simplifies to \( P_1 = P_2 + \rho gh \), where \( P_1 \) is the pressure in the mains, \( \rho \) is the density of water (1000 kg/m³), \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h = 15.0 \) m.

Step 4: Calculate the Required Gauge Pressure

Gauge pressure is the difference between \( P_1 \) (pressure in the mains) and \( P_2 \) (atmospheric pressure). Therefore, \( P_{gauge} = \rho gh = (1000\, \text{kg/m}^3)(9.8\, \text{m/s}^2)(15.0\, \text{m}) = 147,000\, \text{Pa} \). So, \( P_{gauge} = 147\, \text{kPa} \).

Key Concepts

Bernoulli's Equation

Bernoulli's Equation is a fundamental principle in fluid dynamics that helps us understand the behavior of fluid flow. It explains the relationship between pressure, velocity, and height within a flowing fluid. This equation is particularly useful in situations where the fluid is incompressible and there are no friction losses.

Here's what the equation looks like:

• \( P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2 \)

Where:

• \( P \) is the pressure at different points in the fluid.
• \( \rho \) is the fluid's density.
• \( v \) is the fluid's velocity.
• \( g \) is the acceleration due to gravity.
• \( h \) is the height above a reference point.

In the context of our fire hose example, Bernoulli’s equation helps us address how much pressure is needed in the water mains to push the water up to a vertical height. By making certain assumptions, such as larger diameter means negligible initial velocity, the equation can simplify our calculations.

Gauge Pressure

Gauge pressure is an important concept used to measure pressure in systems where real-world conditions, like atmospheric pressure, need to be considered. It tells us how much pressure is exerted by a fluid over and above atmospheric pressure.

When you calculate pressure in any system, there are usually two components:

• Absolute Pressure: Total pressure exerted, including atmospheric pressure.
• Atmospheric Pressure: The pressure exerted by the weight of the atmosphere at sea level.

In most practical scenarios like our fire hose problem, we use gauge pressure, which is the pressure above atmospheric pressure. This is simply because it is handy to know how much additional pressure is required to make things work. For example, in the exercise solution, the gauge pressure is the force needed to ensure the water reaches 15 meters up. Calculating gauge pressure gives us a direct, useful value because it subtracts out the baseline atmospheric pressure.

Gravitational Force in Fluids

Gravitational force plays a vital role in fluid dynamics, especially when dealing with vertical movement of fluids. Essentially, gravity's pull affects fluid pressure and movement within a fluid system.

Gravity acts consistently on water, influencing the pressure required to push fluids against its force. The key factors involved are:

• Density of the fluid (\( \rho \)): More dense fluids need more force to move uphill.
• Height (\( h \)): The higher you want the fluid to rise, the more force needed.
• Acceleration due to gravity (\( g \)): On Earth, this constant is approximately 9.8 m/s².

In the exercise, the goal was to calculate how much gauge pressure is needed for water to reach 15 meters high. When you use the formula \( P_{gauge} = \rho gh \), it becomes clear that gravity determines the necessary pressure—a prime example of how gravitational forces work in fluids. This relationship underlines the essential principles that govern how fluids behave when met with gravitational forces.