Problem 60 - Physics

Question

Water is circulating through a closed system of pipes in a two-floor apartment. On the first floor, the water has a gauge pressure of \(3.4 \times 10^{5} \mathrm{~Pa}\) and a speed of \(2.1 \mathrm{~m} / \mathrm{s}\). However, on the second floor, which is \(4.0 \mathrm{~m}\) higher, the speed of the water is \(3.7 \mathrm{~m} / \mathrm{s}\). The speeds are different because the pipe diameters are different. What is the gauge pressure of the water on the second floor?

Step-by-Step Solution

Answer

The gauge pressure on the second floor is \(2.9576 \times 10^5 \mathrm{~Pa}\).

Step 1: Understand the Bernoulli's Equation

We’ll use Bernoulli's equation which is \( P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \) where \( P \) is pressure, \( \rho \) is the water density, \( v \) is the speed, \( h \) is the height, and \( g \) is the gravitational acceleration.

Step 2: Substitute the Known Values

Set \( P_1 = 3.4 \times 10^5 \mathrm{~Pa} \), \( v_1 = 2.1 \mathrm{~m/s} \), \( v_2 = 3.7 \mathrm{~m/s} \), \( h_1 = 0 \), \( h_2 = 4 \mathrm{~m} \), and \( g = 9.81 \mathrm{~m/s^2} \). Assume the density of water \( \rho = 1000 \mathrm{~kg/m^3} \).

Step 3: Calculate the Pressure Difference Due to Speed Change

Apply the terms related to velocity: \( \frac{1}{2} \rho v_1^2 = \frac{1}{2} \times 1000 \times (2.1)^2 = 2205 \mathrm{~Pa} \) and similarly \( \frac{1}{2} \rho v_2^2 = 6845 \mathrm{~Pa} \). The difference due to speed change is \( 6845 - 2205 = 4640 \mathrm{~Pa} \).

Step 4: Calculate the Pressure Difference Due to Height Change

Include the gravitational potential term: \( \rho g h = 1000 \times 9.81 \times 4 = 39240 \mathrm{~Pa} \).

Step 5: Calculate the Gauge Pressure on the Second Floor

Using the Bernoulli equation, calculate \( P_2 = P_1 + (\frac{1}{2} \rho v_1^2 - \frac{1}{2} \rho v_2^2) - \rho g h \). Substitute values: \( P_2 = 3.4 \times 10^5 - 4640 - 39240 = 2.9576 \times 10^5 \mathrm{~Pa} \).

Key Concepts

Fluid Dynamics

Fluid dynamics is a field of physics concerned with the movement of liquids and gases. In this context, we focus on how water moves through pipes within a building.
Bernoulli’s Equation is a fundamental principle used in fluid dynamics to describe the conservation of energy in a fluid flow.

• It states that the sum of pressure energy, kinetic energy, and potential energy in a flowing fluid remains constant, provided the fluid flow is steady and frictionless.
• This principle is especially useful for understanding behaviors like how fluid speeds increase as the diameter of a pipe decreases.

With Bernoulli’s Equation, it becomes easier to predict how changes in fluid speed and height affect pressure, which is crucial when dealing with multi-level systems like the apartment scenario outlined above.

Gauge Pressure

Gauge pressure is the pressure within a system above atmospheric pressure. It is essential in assessing the operational efficiency of a fluid system.

• Unlike absolute pressure, which includes atmospheric pressure in its measurement, gauge pressure zeroes in solely on the pressure exerted by a fluid.
• In the given exercise, understanding gauge pressure helps us determine how pressure changes from one floor to another.

This understanding is critical as it highlights how pressure can drop due to increased height or speed, affecting the flow dynamics within the system.

Pipe Flow

Pipe flow refers to the movement of fluids through a closed conduit, typically round in cross-section. In our exercise, water moves through the apartment's pipe system, subject to changes in diameter.

• Variation in pipe diameters causes fluid velocities to change, due to the principle of continuity, which states that the mass flow rate must be constant along the pipe.
• As a result, smaller diameter sections lead to higher velocities, and larger diameter sections lead to lower velocities.

Understanding pipe flow is vital in designing piping systems that maintain optimal fluid delivery across various floors without unnecessary pressure losses.

Water Pressure Calculation

Water pressure calculation involves applying Bernoulli’s Equation to determine pressure changes in a fluid network. To solve the exercise provided, apply these steps:

• Identify the initial and final points in the pipe system, considering changes in height and velocity.
• Substitute given values such as initial gauge pressure, fluid density, gravity, and height difference.

By carefully following these calculations, one can determine the new pressure at any point in a piping system. Correctly calculating water pressure ensures the system operates within its designed parameters, avoiding potential disruptions and maintaining efficient water flow.