Problem 48 - University Physics with Modern Physics

Question

A pressure difference of \(6.00 \times 10^{4} \mathrm{Pa}\) is required to maintain a volume flow rate of 0.800 \(\mathrm{m}^{3} / \mathrm{s}\) for a viscous fluid flowing through a section of cylindrical pipe that has radius 0.210 \(\mathrm{m} .\) What pressure difference is required to maintain the same volume flow rate if the radius of the pipe is decreased to 0.0700 \(\mathrm{m} ?\)

Step-by-Step Solution

Answer

The required pressure difference is approximately \(7.42 \times 10^{6} \mathrm{Pa}\).

Step 1: Understand the Problem

We need to find the pressure difference required to maintain the same volume flow rate in a pipe when the radius of the pipe changes. Given initial conditions are a radius of 0.210 m and a pressure difference of \(6.00 \times 10^{4} \mathrm{Pa}\). The final goal is to find out the pressure difference when the radius of the pipe is changed to 0.0700 m.

Step 2: Identify the Formula to Use

For viscous flow in a cylindrical pipe, the Hagen-Poiseuille equation can be used: \[ Q = \frac{\pi \Delta P r^4}{8 \eta L} \] where \(Q\) is the volume flow rate, \(\Delta P\) is the pressure difference, \(r\) is the radius of the pipe, \(\eta\) is the viscosity of the fluid, and \(L\) is the length of the pipe. Since \(Q\), \(\eta\), and \(L\) are constant in both scenarios, we can use the equation: \(\frac{\Delta P_1}{r_1^4} = \frac{\Delta P_2}{r_2^4}\).

Step 3: Use Known Values in the Formula

The initial condition gives us \(\Delta P_1 = 6.00 \times 10^{4} \mathrm{Pa}\), \(r_1 = 0.210 \mathrm{m}\), and \(r_2 = 0.0700 \mathrm{m}\). Substitute these into the relationship \[ \frac{\Delta P_1}{r_1^4} = \frac{\Delta P_2}{r_2^4} \] to find \(\Delta P_2\).

Step 4: Perform the Calculations

Substitute the known values into the proportionality equation: \[ \frac{6.00 \times 10^{4} \mathrm{Pa}}{(0.210)^4} = \frac{\Delta P_2}{(0.0700)^4} \]Calculate \((0.210)^4\) and \((0.0700)^4\), then solve for \(\Delta P_2\).

Step 5: Calculate the Power of the Radius

Calculate \((0.210)^4 = 0.001941\) and \((0.0700)^4 = 0.00002401\).

Step 6: Calculate the New Pressure Difference

Plug the radius powers back into the equation: \[ \frac{6.00 \times 10^{4} \mathrm{Pa}}{0.001941} = \frac{\Delta P_2}{0.00002401} \]Calculate \(\Delta P_2\) by multiplying both sides by \(0.00002401\): \(\Delta P_2 = \frac{6.00 \times 10^{4} \times 0.00002401}{0.001941} \approx 7.42 \times 10^{6} \mathrm{Pa}\).

Key Concepts

Viscous Fluid Flow

Viscous fluid flow refers to the movement of fluid through a medium, such as a pipe, where the fluid's viscosity plays a significant role in determining how the fluid moves. Viscosity is a measure of a fluid's resistance to flow and deformation by shear stress or tensile stress. This concept is crucial when examining how fluids behave in real-world applications.

• Fluids with high viscosity, like honey, flow slowly, while those with low viscosity, like water, flow faster.
• In a pipe, viscous fluid flow results in a parabolic velocity profile, meaning fluid moves fastest at the center and slowest near the pipe's walls.
• Understanding viscous flow is essential for designing systems where fluid transport is involved, such as in pipelines, medical devices, and engineering processes.

By considering fluid viscosity, engineers can predict how changes in system variables, like pressure difference or pipe diameter, affect flow rate.

Pressure Difference

The pressure difference is an essential factor that drives fluid flow through pipes. In the context of the Hagen-Poiseuille equation, it specifically refers to the difference in pressure between two points in a pipe that facilitates the movement of fluid from one end to the other.

• For a fluid to flow, there must be a higher pressure at the inlet than at the outlet; this difference is what propels the fluid.
• The Hagen-Poiseuille equation demonstrates that the flow rate is directly proportional to the pressure difference (\(\Delta P\)).
• When the pressure difference increases, the flow rate also increases, assuming the other variables in the equation remain constant.

This relationship is crucial in applications such as water distribution systems, where maintaining the proper pressure difference ensures efficient fluid transport.

Cylindrical Pipe

A cylindrical pipe is a common structure used for transporting fluids. Its shape influences how the fluid flows within, and this is where the geometry comes into play in fluid dynamics.

• The key geometric factor in a pipe is its radius, which greatly affects flow characteristics. In the Hagen-Poiseuille equation, the flow rate is proportional to the fourth power of the pipe radius.
• This means small changes in the radius result in significant changes to the flow rate and pressure needed to maintain a given flow rate.
• In a practical scenario, decreasing the pipe radius leads to a dramatic increase in pressure needed to maintain a constant flow rate.

Understanding the impact of a pipe's shape is essential for engineers and designers when optimizing systems for fluid conveyance.

Volume Flow Rate

The volume flow rate quantifies how much fluid passes through a section of a pipe in a given time interval. It is a crucial parameter in fluid dynamics, especially when designing systems to transport liquids or gases efficiently.

• It is represented by the symbol \(Q\) and typically measured in cubic meters per second (\(m^3/s\)).
• The Hagen-Poiseuille equation relates the volume flow rate to the pressure difference, fluid viscosity, pipe length, and radius.
• By manipulating any of these variables, one can alter the flow rate as needed for a specific application.

Understanding and controlling the volume flow rate is vital in numerous fields, including engineering, environmental science, and any industry that involves fluid transport.