Problem 68 - College Physics

Question

A hypodermic needle is \(3.0 \mathrm{~cm}\) in length and \(0.30 \mathrm{~mm}\) in diameter. What pressure difference between the input and output of the needle is required so that the flow rate of water through it will be \(1 \mathrm{~g} / \mathrm{s}\) ? (Use \(1.0 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}\) as the viscosity of water.)

Step-by-Step Solution

Answer

The calculated pressure difference between the input and output of the needle required to maintain a flow rate of 1 g/s is approximately \(2.0 \times 10^{7} \, \mathrm{Pa}\).

Step 1: Identify key parameters

First, identify the known values from the problem. The diameter \(d=0.30 \, \mathrm{mm}\), length \(l=3.0 \, \mathrm{cm}\), volumetric flow rate \(Q=1 \, \mathrm{g/s} = 1 \, \mathrm{cm}^3/\mathrm{s}\) (since the density of water is \(1 \, \mathrm{g/cm^3}\)) and viscosity \(\eta=1.0 \times 10^{-3} \, \mathrm{Pa.s}\). Remember to use SI units in your calculations.

Step 2: Use the Poiseuille’s Law

Use Poiseuille’s Law, which describes the flow rate for a liquid passing through a long cylindrical pipe. The formula is: \(Q=\frac{π \times (d/2)^4 \times \Delta P}{8 \times \eta \times l}\). We need to solve this equation for \(\Delta P\) (pressure difference), which gives us \(\Delta P=\frac{8 \times \eta \times l \times Q}{\pi \times (d/2)^4}\).

Step 3: Substitute and evaluate

Substitute the known values into above derived equation to get the pressure difference. Therefore, \(\Delta P = \frac{8 \times 1.0 \times 10^{-3} \, \mathrm{Pa.s} \times 0.03 \, \mathrm{m} \times 0.001 \, \mathrm{m^3/s}}{\pi \times (0.15 \times 10^{-3})^4 \, \mathrm{m^4}}\). Calculate the above expression to get the value of pressure difference.

Key Concepts

Poiseuille's Law

Poiseuille's Law describes how fluids flow through a pipe or cylindrical structure. It is critical in understanding fluid dynamics, especially for liquids like water flowing through narrow channels or pipes. This law provides an equation to relate the flow rate of a liquid to different parameters of the pipe:

• The radius of the pipe: A crucial factor, where an increase in radius results in a significant increase in flow rate.
• The length of the pipe: Longer pipes result in a reduced flow rate due to increased friction and resistance.
• The viscosity of the fluid: Higher viscosity means thicker fluid that flows more slowly.
• The pressure difference between the two ends of the pipe: An increased pressure difference can drive more fluid through the pipe.

Poiseuille's equation is expressed as:\[ Q = \frac{\pi \times (d/2)^4 \times \Delta P}{8 \times \eta \times l} \]where \(Q\) is the volumetric flow rate, \(d\) is the diameter, \(\Delta P\) is the pressure difference, \(\eta\) is the viscosity, and \(l\) is the length of the pipe. This formula helps determine the necessary pressure difference to achieve a desired flow rate.

Pressure Difference

The pressure difference, denoted as \(\Delta P\), plays a crucial role in fluid dynamics, particularly when using Poiseuille's Law.
Imagine a pipe where we need water to flow from one end to the other. Without a pressure difference, there is no driving force to push the liquid along the pipe. Therefore, this difference is essential to maintain and regulate the flow rate through a cylindrical conduit.Calculating the pressure difference requires rearranging Poiseuille's formula to produce:\[ \Delta P = \frac{8 \times \eta \times l \times Q}{\pi \times (d/2)^4} \]This equation shows that to achieve a target flow rate, adjusting the pressure difference can compensate for variation in other factors like pipe length or fluid viscosity. Consequently, a greater pressure difference is necessary if: - The pipe is longer.- The fluid has higher viscosity.- You require a higher flow rate.

Viscosity of Water

Viscosity is a measure of a fluid's resistance to deformation and flow. For liquids like water, it represents how thick or thin the liquid is, impacting how quickly it moves through pipes or porous media. The viscosity of water is typically quite low, making it a fluid that flows easily. In standard conditions, water's viscosity is around \(1.0 \times 10^{-3} \mathrm{Pa.s}\). This value is used in calculations involving Poiseuille's Law to determine the necessary pressure difference for a specified flow rate.Using Poiseuille’s Law, a lower viscosity (as is the case with water) allows for a smoother, faster flow under the same pressure difference. In contrast, higher viscosity fluids, like honey or oil, require more force (higher pressure difference) to achieve similar flow rates. Understanding viscosity is crucial for designing systems where fluid flow efficiency is a priority.