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/m³ and dynamic viscosity of the air is 1.8 × 10⁻⁵ kg/m-s. 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) Read more

Steps to Solve

1. Identify the Parameters

   From the problem, we have the following parameters:

   • Diameter of the particle, $d = 50 , \mu m = 50 \times 10^{-6} , m$
   • Velocity of the particle (and air), $v = 1.0 , m/s$
   • Density of the particle, $\rho_p = 2000 , kg/m^3$
   • Dynamic viscosity of air, $\mu = 1.8 \times 10^{-5} , kg/(m \cdot s)$

2. Calculate the Settling Velocity Using Stokes' Law

   Stokes' law gives the settling velocity $v_s$ as:
   $$ v_s = \frac{d^2 (\rho_p - \rho) g}{18 \mu} $$
   Here, $\rho$ (density of air) is negligible compared to $\rho_p$, so we can approximate:
   $$ v_s = \frac{d^2 \rho_p g}{18 \mu} $$
   With $g \approx 9.81 , m/s^2$, we proceed to calculate $v_s$.

3. Substituting the Known Values

   Substitute the values into the equation:
   $$ v_s = \frac{(50 \times 10^{-6})^2 (2000)(9.81)}{18(1.8 \times 10^{-5})} $$

4. Calculate the Settling Velocity

   Perform the calculations step by step:

   • Calculate $(50 \times 10^{-6})^2 = 2.5 \times 10^{-9}$
   • Then calculate the numerator:
     $$ 2.5 \times 10^{-9} \times 2000 \times 9.81 = 4.9025 \times 10^{-6} $$
   • Calculate the denominator:
     $$ 18 \times (1.8 \times 10^{-5}) = 3.24 \times 10^{-4} $$
   • Finally, the settling velocity becomes:
     $$ v_s = \frac{4.9025 \times 10^{-6}}{3.24 \times 10^{-4}} \approx 0.0151 , m/s $$

5. Calculate the Minimum Length of the Chamber

   To find the minimum length ( L ), we can use the formula that relates length, velocity, and settling velocity:
   $$ L = \frac{v \cdot h}{v_s} $$
   Here, ( h = 0.5 , m ):
   $$ L = \frac{1.0 \cdot 0.5}{0.0151} $$

6. Final Calculation of Length

   Calculate ( L ):
   $$ L = \frac{0.5}{0.0151} \approx 33.11 , m $$