Calculate the viscosity, μ, given that D = 1.0 mm, L = 1.00 m, mass flow rate of oil ρ QL = 0.260 kg/h, pressure drop, Δp = 640 kPa, and ρ = 920 kg/m3.
Understand the Problem
The question is asking to calculate the viscosity of a fluid (oil) given several parameters: diameter (D), length (L), mass flow rate (ρ QL), pressure drop (Δp), and density (ρ). We will use the fluid dynamics principles to derive viscosity using the given values.
Answer
The viscosity \( \mu \) is calculated using the formula: $$ \mu = \frac{\Delta p \cdot \pi D^4}{8 L Q} $$ Where \( Q \) is the volumetric flow rate calculated from the mass flow rate and density.
Answer for screen readers
The viscosity of the fluid ( \mu ) can be calculated using the formula:
$$ \mu = \frac{\Delta p \cdot \pi D^4}{8 L Q} $$
where you would substitute in the known values for ( \Delta p ), ( D ), ( L ), and ( Q ) to find the numerical answer.
Steps to Solve
- Identify the Viscosity Formula
To calculate viscosity, we can use the Hagen-Poiseuille equation for laminar flow in a circular pipe, which is given by:
$$ \Delta p = \frac{8 \mu L Q}{\pi D^4} $$
where ( \Delta p ) is the pressure drop, ( \mu ) is the dynamic viscosity, ( L ) is the length of the pipe, ( Q ) is the volumetric flow rate, and ( D ) is the diameter.
- Rearranging the Equation for Viscosity
To find the dynamic viscosity ( \mu ), we need to rearrange the equation:
$$ \mu = \frac{\Delta p \cdot \pi D^4}{8 L Q} $$
This equation will allow us to isolate viscosity.
- Substituting Parameters
Now, substitute the known values into the equation. Ensure you have your units consistent, especially for pressure (in Pascals), length (in meters), diameter (in meters), and flow rate in cubic meters per second.
- Calculating the Volumetric Flow Rate
If the mass flow rate ( \rho QL ) is provided and the density ( \rho ) is known, you can calculate the volumetric flow rate ( Q ) using:
$$ Q = \frac{\text{mass flow rate}}{\rho} $$
- Final Calculation
Once you have ( Q ) and all other parameters, substitute them back into the viscosity formula to calculate ( \mu ):
$$ \mu = \frac{\Delta p \cdot \pi D^4}{8 L Q} $$
Compute the final value to find the viscosity of the fluid.
The viscosity of the fluid ( \mu ) can be calculated using the formula:
$$ \mu = \frac{\Delta p \cdot \pi D^4}{8 L Q} $$
where you would substitute in the known values for ( \Delta p ), ( D ), ( L ), and ( Q ) to find the numerical answer.
More Information
Viscosity is a measure of a fluid's resistance to flow. It is an important property in fluid dynamics and is essential for engineering applications, such as designing piping systems and predicting flow rates in different materials.
Tips
- Forgetting to convert units consistently can lead to incorrect results.
- Not accounting for the correct fluid density can affect the calculation of volumetric flow rate ( Q ).
- Misapplying the formula or using incorrect values for pressure drop or dimensions.
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