Question

Liquid mercury (viscosity \(=1.55 \times 10^{-3} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\) ) flows through a horizontal pipe of internal radius \(1.88 \mathrm{~cm}\) and length \(1.26 \mathrm{~m}\). The volume flux is \(5.35 \times 10^{-2} \mathrm{~L} / \mathrm{min}\). \((a)\) Show that the flow is laminar. (b) Calculate the difference in pressure between the two ends of the pipe.

Short answer

Use the values for the viscosity, radius and volume flux given in the problem to first calculate the Reynolds number. If it is less than 2000, the fluid flow is considered laminar. Then, use the viscosity, length and radius of the pipe, alongside the volume flux to compute the pressure difference between the two ends of the pipe using Poisseuille's law. The result will be in Pascals.

Step-by-step solution

Calculating Reynolds Number

Firstly, the Reynolds number (Re) should be calculated using the following formula: \(Re = \frac{VDρ}{μ}\) where V is the average velocity of the fluid, D is the diameter of the pipe, ρ is the fluid density, and μ is the fluid viscosity. The average velocity can be calculated by dividing the volume flux by the area of the pipe, which in turn is calculated by: \(A=\pi r^{2}\) where r is the radius. The radius of the pipe can be converted to meters, the volume flux to m³/s, then those values can be used to calculate the average velocity. The diameter of the pipe is twice the radius. Using known value of mercury's density (13.6 g/cm⁻³) which should be converted to kg/m⁻³, and given viscosity, now Reynolds number can be calculated. If the Reynolds number is smaller than 2000, this indicates a laminar flow.

Calculating Pressure Difference

Next, the pressure difference between the two ends of the pipe can be calculated using Poiseuille's Law, with the formula: \(ΔP=8μLVQ/πr²\) where μ is the dynamic viscosity, L is the length of the pipe, V is the average velocity calculated before, Q is the volume flow rate, and r is the radius of the pipe. The values given in the problem can be substituted directly into the equation because they are already in proper SI units. The result will be the pressure difference in Pascals (Pa).

Key concepts

Understanding Reynolds Number

The Reynolds number is a dimensionless quantity used to predict the flow pattern of fluid in a pipe. It helps determine whether the flow is laminar or turbulent. To find the Reynolds number, we use the formula: \[ Re = \frac{VDρ}{μ} \] Here,

• \( V \) is the average velocity of the fluid
• \( D \) is the diameter of the pipe
• \( ρ \) is the fluid density
• \( μ \) is the fluid viscosity

For a flow to be considered laminar, the Reynolds number should be less than 2000.

To calculate the average velocity, you must first convert the volume flux into meters cubed per second. Then, calculate the cross-sectional area of the pipe using \( A=\pi r^{2} \), where \( r \) is the radius. Next, find the average velocity by dividing the volume flux by this area.

Finally, apply it to the equation for the Reynolds number, and compare the result to 2000 to determine the flow type.

Exploring Volume Flux

Volume flux, often referred to as the volumetric flow rate, measures how much volume of fluid passes through a section of a pipe over time. It is expressed in units like liters per minute (L/min) or cubic meters per second (m³/s). In this exercise, the volume flux is given as \(5.35 \times 10^{-2} \mathrm{~L/min}\).

To use this value in calculations, we should convert it to cubic meters per second. This involves converting liters to cubic meters and minutes to seconds.

The volume flux can then be used to calculate other properties, such as the average velocity of the fluid by dividing it by the cross-sectional area of the pipe. Knowing the volume flux is essential when predicting the pressure difference and testing for laminar or turbulent flow.

Introduction to Poiseuille's Law

Poiseuille's Law is crucial for understanding fluid dynamics in pipes, especially for laminar flow. It allows us to calculate the pressure difference between the two ends of a pipe. The law is expressed as: \[ ΔP=\frac{8μLQ}{πr^{4}} \] Where:

• \(ΔP\) is the pressure difference
• \(μ\) is the dynamic viscosity of the fluid
• \(L\) is the length of the pipe
• \(Q\) is the volume flux (volume flow rate)
• \(r\) is the radius of the pipe

Poiseuille's Law shows that the pressure difference is directly proportional to the viscosity, length, and flow rate, but inversely proportional to the fourth power of the radius.

This law is applied when calculating the pressure difference in a laminar flow, using the given dimensions and properties of the pipe and fluid as inputs.

Investigating Pressure Difference

The pressure difference across a pipe can tell us a lot about the energy dynamics of fluid flow. It indicates how much energy is required to move the fluid from one end of the pipe to the other. Calculating this difference is essential in designing systems for efficient fluid transport.

In our exercise, after confirming that the flow is laminar using the Reynolds number, we use Poiseuille's Law to determine the pressure difference. You should check that all measurements are in proper SI units before substituting them into the formula for pressure difference. This will help ensure accuracy.

Understanding these relations aids in optimizing conditions like pipe radius or length to control the pressure difference and manage flow more effectively.