Question

A particle dispersoid has 1510 spherical particles of uniform density. An air purifier is proposed to be used to remove these particles. The diameter-specific number of particles in the dispersoid, along with the number removal efficiency of the proposed purifier is shown in the following table:

The overall mass removal efficiency of the proposed purifier is ________% (rounded off to one decimal place).

Hint:

To calculate mass removal efficiency, consider the number of particles, their sizes, and the removal efficiency for each size group. Multiply the removal efficiency by the mass proportional to the particle size.

Answer 1

Step 1: Calculate the mass of each group of particles.
For each particle group, the mass can be determined by: \[ \text{Mass of a particle} = \frac{4}{3} \pi \left( \frac{d}{2} \right)^3 \cdot \rho \] where:
\( d \) is the diameter of the particle
\( \rho \) is the density of the particle
Since the density is uniform and not given, we can assume a proportional relationship for the mass based on diameter.
The mass of each particle is proportional to \( d^3 \).
So, the mass for each group is proportional to the number of particles times the cube of the particle diameter:
For \( d = 1 \, \mu\text{m} \), mass per particle is proportional to \( 1^3 = 1 \)
For \( d = 10 \, \mu\text{m} \), mass per particle is proportional to \( 10^3 = 1000 \)
For \( d = 100 \, \mu\text{m} \), mass per particle is proportional to \( 100^3 = 10^6 \)
Step 2: Calculate the total mass of particles in each group.
For particles with diameter 1 µm:
Number of particles = 1000, mass per particle proportional to \( 1^3 = 1 \)
Total mass = \( 1000 \times 1 = 1000 \)
For particles with diameter 10 µm:
Number of particles = 500, mass per particle proportional to \( 10^3 = 1000 \)
Total mass = \( 500 \times 1000 = 500000 \)
For particles with diameter 100 µm:
Number of particles = 10, mass per particle proportional to \( 100^3 = 10^6 \)
Total mass = \( 10 \times 10^6 = 10000000 \)
Step 3: Calculate the total number of particles and total mass.
Total number of particles: \[ \text{Total number} = 1000 + 500 + 10 = 1510 \] Total mass: \[ \text{Total mass} = 1000 + 500000 + 10000000 = 10500000 \] Step 4: Calculate the removal efficiency for each group.
The removal efficiency is applied to the number of particles in each group:
For \( d = 1 \, \mu\text{m} \):
Number of particles removed = \( 1000 \times 0.99 = 990 \)
Mass removed is proportional to the particle size:
Mass removed = \( 990 \times 1 = 990 \)
For \( d = 10 \, \mu\text{m} \):
Number of particles removed = \( 500 \times 0.75 = 375 \)
Mass removed = \( 375 \times 1000 = 375000 \)
For \( d = 100 \, \mu\text{m} \):
Number of particles removed = \( 10 \times 0.10 = 1 \)
Mass removed = \( 1 \times 10^6 = 1000000 \)
Step 5: Calculate the total mass removed.
Total mass removed: \[ \text{Total mass removed} = 990 + 375000 + 1000000 = 1374990 \] Step 6: Calculate the overall mass removal efficiency.
The overall mass removal efficiency is given by: \[ \text{Mass removal efficiency} = \frac{\text{Total mass removed}}{\text{Total mass}} \times 100 \] Substitute the values: \[ \text{Mass removal efficiency} = \frac{1374990}{10500000} \times 100 = 13.1 \% \]