Question

A settling chamber is used for the removal of discrete particulate matter from air with the following conditions. Horizontal velocity of air = 0.2 m/s; Temperature of air stream = 77°C; Specific gravity of particle to be removed = 2.65; Chamber length = 12 m; Chamber height = 2 m; Viscosity of air at 77°C = 2.1 × 10\(^{-5}\) kg/m·s; Acceleration due to gravity (g) = 9.81 m/s²; Density of air at 77°C = 1.0 kg/m³; Assume the density of water as 1000 kg/m³ and Laminar condition exists in the chamber. 
The minimum size of particle that will be removed with 100% efficiency in the settling chamber (in $\mu$m is .......... (round off to one decimal place). 
 

Hint:

When calculating the minimum size of a particle removed with 100\% efficiency in a settling chamber, use the formula involving the particle's velocity, chamber dimensions, and the physical properties of air and water. Ensure that all units are consistent and in SI units.

Answer 1

Given: \[ \begin{aligned} \mu &= 2.1 \times 10^{-5} \, \text{kg/m} \cdot \text{s} \\ \rho_{\text{air}} &= 1 \, \text{kg/m}^3 \\ \rho_{\text{water}} &= 1000 \, \text{kg/m}^3 \\ V &= 0.2 \, \text{m/s} \\ G &= 2.65 \\ L &= 12 \, \text{m} \\ H &= 2 \, \text{m} \\ g &= 9.81 \, \text{m/s}^2 \end{aligned} \]
Formula for minimum particle size:
The minimum particle size that will be removed with 100% efficiency in a settling chamber is given by: \[ d = C \sqrt{\frac{18 \mu V H}{g L \rho_p}} \] where:

• \(C = 1\) (constant for laminar flow)
• \(\mu\) is the dynamic viscosity of air
• \(V\) is the velocity of air
• \(H\) is the height of the chamber
• \(g\) is the acceleration due to gravity
• \(L\) is the length of the chamber
• \(\rho_p\) is the density of the particle (\(\rho_p = G \times \rho_{\text{water}}\))

Substituting the known values:
\[ d = 1 \times \sqrt{\frac{18 \times 2.1 \times 10^{-5} \times 0.2 \times 2}{9.81 \times 12 \times (2.65 \times 10^3)}} \] \[ d = \sqrt{\frac{2.52 \times 10^{-4}}{3.12 \times 10^3}} \] \[ d = 2.201 \times 10^{-5} \, \text{m} = 22 \times 10^{-6} \, \text{m} = 22 \, \mu\text{m} \] Final Answer:
The minimum particle size that will be removed with 100% efficiency is: \[ \boxed{22 \, \mu\text{m}} \]