Problem 73 - Physics

Question

Interactive Solution \(\underline{11.73}\) at illustrates a model for solving this problem. A pressure difference of \(1.8 \times 10^{3} \mathrm{~Pa}\) is needed to drive water \(\left(\eta=1.0 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}\right)\) through a pipe whose radius is \(5.1 \times 10^{-3} \mathrm{~m} .\) The volume flow rate of the water is \(2.8 \times 10^{-4} \mathrm{~m}^{3} / \mathrm{s} .\) What is the length of the pipe?

Step-by-Step Solution

Answer

The pipe is approximately 43.0 meters long.

Step 1: Identify the Given Variables

We are given:- Pressure difference: \( \Delta P = 1.8 \times 10^{3} \, \text{Pa} \)- Dynamic viscosity of water: \( \eta = 1.0 \times 10^{-3} \, \text{Pa} \cdot \text{s} \)- Radius of the pipe: \( r = 5.1 \times 10^{-3} \, \text{m} \)- Volume flow rate: \( Q = 2.8 \times 10^{-4} \, \text{m}^{3} / \text{s} \)

Step 2: Use Poiseuille’s Law for Flow Rate

Poiseuille's Law for flow in a pipe is:\[Q = \frac{\pi r^4 \Delta P}{8 \eta L}\]where:- \(Q\) is the volume flow rate- \(r\) is the radius of the pipe- \(\Delta P\) is the pressure difference- \(\eta\) is the dynamic viscosity- \(L\) is the length of the pipeWe need to solve this equation for \(L\).

Step 3: Solve for the Length of the Pipe

Rearrange the Poiseuille's Law formula to solve for \(L\):\[L = \frac{\pi r^4 \Delta P}{8 \eta Q}\]Substitute the given values into the equation:\[L = \frac{\pi (5.1 \times 10^{-3})^4 (1.8 \times 10^3)}{8 \times 1.0 \times 10^{-3} \times 2.8 \times 10^{-4}}\]

Step 4: Perform the Calculations

Calculate each part of the equation. First calculate \(r^4\):\[(5.1 \times 10^{-3})^4 = 6.7521 \times 10^{-11}\]Then substitute and compute the value for \(L\):\[L = \frac{\pi \times 6.7521 \times 10^{-11} \times 1.8 \times 10^3}{8 \times 1.0 \times 10^{-3} \times 2.8 \times 10^{-4}} \approx 43.0 \, \text{m}\]

Step 5: Conclude the Solution

The length of the pipe is approximately \(43.0 \, \text{m}\). By following the steps and using the Poiseuille's Law equation, we derived the pipe's length with the given flow rate and conditions.

Key Concepts

Fluid Dynamics

Fluid dynamics is the study of how fluids move in different conditions. It is a crucial field in physics and engineering that has practical applications in many areas, such as aerodynamics, hydraulics, and medical sciences. Fluids can be liquids or gases, and their movement is influenced by several factors, such as pressure, velocity, and viscosity.
In fluid dynamics, we often look at how fluids flow through various environments by considering parameters like velocity, pressure, and flow rate, all of which contribute to understanding the overall behavior of fluid systems. Fluid dynamics helps us predict patterns in fluid movement and solve real-world problems effectively.
Understanding fluid dynamics is key to solving problems like those involving Poiseuille’s Law, which helps calculate flow characteristics in pipes. In these scenarios, factors such as the fluid’s viscosity and the dimensions of the conduit greatly affect the outcome.

Pressure Difference

Pressure difference, denoted as \( \Delta P \), is a fundamental concept in understanding how fluids move. It is the difference in pressure between two points in a fluid flow system and acts as the driving force that moves fluid from one place to another.

• High pressure difference leads to a faster flow.
• Low pressure difference results in slower movement.

In the context of Poiseuille's Law, the pressure difference is one of the key factors that determine the volume flow rate through a pipe. For instance, in the given problem, a pressure difference of \(1.8 \times 10^{3} \text{ Pa}\) leads to a specific flow rate in the pipe. The relationship shows that pressure differences are crucial for maintaining the movement of fluids in controlled environments, enabling engineers to design efficient systems for desired flow rates.

Dynamic Viscosity

Dynamic viscosity, represented by \( \eta \), quantifies a fluid's resistance to flow. It is a measure of internal friction when different layers of fluid move relative to each other.

• High viscosity means the fluid is thicker and flows more slowly.
• Low viscosity indicates a thinner, faster-moving fluid.

In Poiseuille’s Law, dynamic viscosity is a critical component because it helps define how easily fluid can move through a pipe. In the presented exercise, the water has a dynamic viscosity of \(1.0 \times 10^{-3} \text{ Pa} \cdot \text{s}\), which is typical of fluid like water under standard conditions. Understanding the role of viscosity allows engineers to manipulate factors affecting flow rates to achieve optimal fluid movement in pipelines and other systems.

Volume Flow Rate

The volume flow rate, symbolized by \( Q \), indicates how much fluid volume passes through a given area per unit of time. It is often expressed in units like cubic meters per second (\( \text{m}^3/ \text{s} \)).

• It shows the efficiency and capacity of a fluid flow system.
• Higher flow rates mean more fluid moves through the system quickly.

According to Poiseuille's Law, the volume flow rate depends heavily on factors like pressure difference, radius of the pipe, and dynamic viscosity. In the example from the exercise, the given volume flow rate is \(2.8 \times 10^{-4} \text{ m}^{3} \text{/s}\), illustrating how the dimensions and conditions of the pipe work together to determine flow rate. This understanding of volume flow rate is essential for designing systems that need specific amounts of fluid to be transported efficiently.