Polluted air with particulate matters of diameter 50 µm enter with a horizontal velocity of 1.0 m/s at a height of 0.5 m from the bottom of a dry settling chamber. The density of t... Polluted air with particulate matters of diameter 50 µm enter with a horizontal velocity of 1.0 m/s at a height of 0.5 m from the bottom of a dry settling chamber. The density of the particle is 2000 kg/m^3 and dynamic viscosity of the air is 1.8 × 10^−5 kg/m-s. Assume streamline flow and the density of air is negligible as compared to particles and uniform horizontal velocity of 1.0 m/s of gas and particles within the chamber. Considering particle settling follows Stokes' law, the minimum length in m, of the chamber required for settling of the particle at its bottom is ______ (round off up to 2 decimals).
Understand the Problem
The question is asking for the calculation of the minimum length of a settling chamber required for particles to settle at the bottom, following Stokes' law. It provides details about the particle's size, velocity, density, and viscosity of air, which are necessary for the calculation.
Answer
The minimum length of the settling chamber required for the particles to settle at its bottom is \(0.50 \, m\).
Answer for screen readers
The minimum length of the settling chamber required for the particles to settle at its bottom is (0.50 , m).
Steps to Solve
- Identify the parameters from the problem
From the problem statement, we have the following values:
- Diameter of the particle, (d = 50 , \mu m = 50 \times 10^{-6} , m)
- Velocity of the gas, (v = 1.0 , \text{m/s})
- Density of the particle, (\rho_p = 2000 , \text{kg/m}^3)
- Density of the air, (\rho = 1.8 \times 10^{-5} , \text{kg/m}^3)
- Dynamic viscosity of the air, (\mu = 1.8 \times 10^{-5} , \text{kg/m-s})
- Use Stokes' law to find the settling velocity
According to Stokes' law, the settling velocity (v_s) of a spherical particle in a fluid is given by:
$$ v_s = \frac{2}{9} \frac{(d^2)(\rho_p - \rho)g}{\mu} $$
Where:
- (g \approx 9.81 , \text{m/s}^2) (acceleration due to gravity)
- Calculate the settling velocity
Substituting the values into the equation:
$$ v_s = \frac{2}{9} \frac{(50 \times 10^{-6})^2 \cdot (2000 - 1.8 \times 10^{-5}) \cdot 9.81}{1.8 \times 10^{-5}} $$
- Determine the required length of the settling chamber
The length (L) of the settling chamber can be determined based on the relationship:
$$ L = \frac{v_s}{v} \cdot t $$
Where (t) is the time taken to settle from a height (h = 0.5 , m).
- Calculate the time taken to settle
The time (t) for the particle to settle can be determined from:
$$ t = \frac{h}{v_s} $$
- Final Equation for Length
Now substituting this back into the length equation to find (L):
$$ L = \frac{v_s}{v} \cdot \frac{h}{v_s} = \frac{h}{v} $$
Substituting (h) and (v):
$$ L = \frac{0.5 , m}{1.0 , m/s} = 0.5 , m $$
The minimum length of the settling chamber required for the particles to settle at its bottom is (0.50 , m).
More Information
This solution demonstrates the application of Stokes' law in determining the settling velocity of particles in a fluid. The calculations show how parameters like particle size, density, and fluid viscosity influence the settling process.
Tips
- Misunderstanding Stokes' law: It's essential to apply the correct formula depending on the flow conditions.
- Neglecting unit conversions: Ensure all measurements are in standard units when performing calculations.
- Not considering the effects of the velocity of both gas and particles correctly can lead to errors in the length calculation.
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