Problem 10 - Applied Physics

Question

A pipe system with a radius of \(0.060 \mathrm{~m}\) has a liquid flowing through it at a speed of \(3.96 \mathrm{~m} / \mathrm{s}\). What is the rate of flow in \(\mathrm{L} / \mathrm{min}\) ?

Step-by-Step Solution

Answer

The flow rate is approximately 8454 L/min.

Step 1: Understand the Problem

We need to find the rate of flow in liters per minute for a liquid flowing through a pipe system. We are given the radius of the pipe and the speed of the liquid flow.

Step 2: Convert Units if Necessary

Currently, the speed is given in meters per second, and we need to find the rate in liters per minute. We will first calculate the rate in terms of cubic meters per second, and then convert it to liters per minute.

Step 3: Calculate the Cross-sectional Area of the Pipe

The area of the circular cross-section for the pipe can be found using the formula: \[A = \pi r^2,\]where \(r = 0.060 \text{ m}\). So,\[A = \pi (0.060)^2 = \pi \, \times \, 0.0036 \, \text{m}^2.\]

Step 4: Calculate the Volume Flow Rate in Cubic Meters Per Second

The volume flow rate, \(Q\), can be calculated by multiplying the cross-sectional area \(A\) by the flow speed \(v\): \[Q = A \times v = \pi \, \times \, 0.0036 \, \text{m}^2 \, \times \, 3.96 \, \text{m/s}. \]

Step 5: Perform the Calculation - Cubic Meters Per Second

Calculate \(Q\): \[Q = 0.0036 \pi \, \times \, 3.96 \, \approx \, 0.04485 \pi \, \text{m}^3/\text{s}\]Using \(\pi \approx 3.1416\),\[Q \approx 0.1409 \, \text{m}^3/\text{s}. \]

Step 6: Convert Cubic Meters Per Second to Liters Per Minute

Since 1 cubic meter is equivalent to 1000 liters, first convert \(Q\) to liters per second:\[Q \approx 0.1409 \times 1000 = 140.9 \, \text{L/s}. \]Convert liters per second to liters per minute by multiplying by 60:\[140.9 \, \text{L/s} \, \times \, 60 \, = \, 8454 \, \text{L/min}.\]

Step 7: Final Result

The final rate of flow of the liquid through the pipe is approximately 8454 liters per minute.

Key Concepts

Pipe Flow

Pipe flow refers to the movement of a fluid within a closed conduit, typically a pipe. It is an essential topic in fluid dynamics and a critical component in many engineering applications. In a pipe flow scenario, fluids flow due to a difference in pressure from one end to the other. This pressure difference causes the fluid to move, overcoming frictional forces that might be present.
A few factors affect how fluid flows through a pipe, including:

• Pipe diameter and length: Larger diameters and shorter pipe lengths can reduce resistance and increase flow rate.
• Fluid velocity: The speed at which the fluid travels inside the pipe.
• Fluid type and properties: Viscosity and density dictate how easily fluid flows.

Understanding these factors helps to analyze and optimize systems like water supply networks, oil pipelines, or cooling systems in engines.

Volume Flow Rate

The volume flow rate, often denoted by the symbol \(Q\), is a measure of the volume of fluid that passes through a surface or a pipe over a set duration of time. It is a pivotal concept in fluid dynamics, especially in assessing pipe performance.
The formula to calculate the volume flow rate is:

• \(Q = A \times v\),

where \(A\) is the cross-sectional area, and \(v\) is the fluid's velocity.
This formula tells us how different factors, like the area of the pipe and the speed of the fluid, contribute to the flow rate. A larger cross-sectional area or higher velocity results in a higher volume flow rate.
Considering the practical aspect, it helps in systems where consistent fluid delivery to different parts is critical, such as irrigation systems or firefighting hoses.

Unit Conversion

Unit conversion is a crucial skill in engineering and science, allowing us to translate measurements from one set of units to another. In fluid dynamics, it helps standardize the data we use since the SI units might differ from those traditionally used in various industries.
For instance, converting between cubic meters per second (\(m^3/s\)) and liters per minute (\(L/min\)) involves knowing specific conversion factors.
Some basic conversions include:

• 1 cubic meter (\(m^3\)) equals 1000 liters.
• To convert from liters per second (\(L/s\)) to liters per minute (\(L/min\)), multiply by 60.

Converting units ensures that measurements are compatible, facilitating communication and collaboration in projects that involve fluid flow, like designing a water distribution system or calculating chemical mix rates in industrial processes.

Cross-Sectional Area

The cross-sectional area plays a pivotal role in determining how a fluid flows through a pipe. It is defined as the area of the pipe's cross-section perpendicular to the direction of flow.
For pipes with a circular cross-section, the area \(A\) can be calculated using the formula:

• \(A = \pi r^2\)

where \(r\) is the radius of the pipe. The area is a crucial factor in the volume flow rate equation, as a larger area allows more fluid to pass through per unit time.
This concept is not only limited to fluids in pipes but extends to design and analysis of systems like air ducts, blood vessels, and even wind tunnels. Understanding how to calculate and optimize the cross-sectional area can efficiently enhance the performance of fluid systems.