Problem 57 - College Physics

Question

An ideal fluid is moving at \(3.0 \mathrm{~m} / \mathrm{s}\) in a section of a pipe of radius \(0.20 \mathrm{~m}\). If the radius in another section is \(0.35 \mathrm{~m},\) what is the flow speed there?

Step-by-Step Solution

Answer

The flow speed in the second section is approximately 0.98 m/s.

Step 1: Understand the Problem

We have an ideal fluid flowing through two sections of a pipe with different radii. We need to find the flow speed in the second section given the flow speed and radius in the first section.

Step 2: Apply the Continuity Equation

For an ideal fluid, the continuity equation states that the product of cross-sectional area and velocity is constant along a flow. Mathematically, it's expressed as: \( A_1v_1 = A_2v_2 \). Here, \( A \) is the cross-sectional area and \( v \) is the velocity of the fluid.

Step 3: Calculate Areas

The cross-sectional area of a pipe is given by \( A = \pi r^2 \). For the first section: \[ A_1 = \pi (0.20)^2 \] For the second section:\[ A_2 = \pi (0.35)^2 \]

Step 4: Solve for the Unknown Velocity

We need to find \( v_2 \). Use the continuity equation: \[ \pi (0.20)^2 (3.0) = \pi (0.35)^2 v_2 \] Cancel \( \pi \) from both sides and solve for \( v_2 \): \[ v_2 = \frac{0.20^2 \times 3.0}{0.35^2} \]

Step 5: Compute the Result

Calculate \( v_2 \) by substituting the values: \[ v_2 = \frac{0.040 \times 3.0}{0.1225} \approx 0.98 \, \mathrm{m/s} \]

Key Concepts

Continuity Equation

The continuity equation is a fundamental concept in fluid dynamics, particularly when dealing with incompressible, ideal fluids. It asserts that the mass of fluid flowing through a pipe remains constant throughout, despite changes in the pipe's diameter.
This principle is valuable when you want to conserve mass in a fluid flow situation. It is mathematically represented as \( A_1v_1 = A_2v_2 \), where \( A \) indicates the cross-sectional area of the pipe and \( v \) is the flow velocity.

• In simpler terms, the product of the area of the pipe and the velocity of fluid in one section equals the product in another section.
• This equation implies that if the pipe's sectional area increases, the fluid velocity decreases, and vice versa.

It is important to remember, this only applies under ideal conditions.

Ideal Fluid

An ideal fluid is a theoretical fluid that is incompressible and has no viscosity. These assumptions make calculations simpler but do not entirely represent real-world fluids.
A few important traits of an ideal fluid include:

• Incompressibility: Its density is constant, hence the mass continuity can apply without complicating factors.
• Non-viscous: It flows without internal friction. Consequently, there's no energy loss due to resistance.

This idealization can help in understanding the basics of fluid flow, although real fluid scenarios might need adjustments for viscosity and compressibility.

Pipe Flow

Pipe flow refers to the movement of fluids through a closed conduit, typically pipes. Understanding pipe flow is essential in various engineering applications and everyday utilities.
The speed and behavior of fluid within pipes depend on several factors, including:

• The pipe's internal diameter or radius, which directly affects flow rate and velocity due to the continuity equation.
• Fluid characteristics, such as whether it behaves more like an ideal or real fluid.

Engineers use these principles to design efficient piping systems in buildings, water treatment facilities, and more.

Velocity Calculation

Determining the velocity of a fluid in different sections of a pipe is crucial for many engineering tasks. Using the continuity equation, you can compute this velocity by linking area and flow speed.

For example, given two sections of a pipe with different radii, and the speed of flow in one section, you can find the unknown velocity in another section by applying the formula:
\[ v_2 = \frac{A_1v_1}{A_2} \]
Substituting the area formula \( A = \pi r^2 \) and known values allows solving for the desired velocity. The process entails:

• Calculating the area for each pipe section using the radius given.
• Applying the continuity equation to link these areas with their respective velocities.
• Solving for the unknown velocity efficiently.

This calculation helps in confirming efficient fluid transport within various pipe configurations.