Question

In sieve analysis of coffee powder, the particle size distribution is given below:

The Sauter mean diameter (in µm) of the coffee powder is _________ (round off to one decimal place).

Hint:

Sauter mean diameter is useful for characterizing particle size in systems where particles of different sizes exist, based on surface area.

Answer 1

Correct Answer: 19

The Sauter mean diameter is given by the formula: \[ d_{sm} = \frac{\sum (N_i \cdot d_i^3)}{\sum (N_i \cdot d_i^2)} \] Where:
- \( N_i \) is the number of particles for each size,
- \( d_i \) is the mean particle size.
Substituting the given values: \[ d_{sm} = \frac{5 \cdot 40^3 + 8 \cdot 30^3 + 50 \cdot 20^3 + 90 \cdot 17.5^3 + 148 \cdot 12.5^3 + 10 \cdot 10^3}{5 \cdot 40^2 + 8 \cdot 30^2 + 50 \cdot 20^2 + 90 \cdot 17.5^2 + 148 \cdot 12.5^2 + 10 \cdot 10^2} \] Calculating the numerators and denominators: \[ d_{sm} = \frac{5 \cdot 64000 + 8 \cdot 27000 + 50 \cdot 8000 + 90 \cdot 5359.375 + 148 \cdot 1953.125 + 10 \cdot 1000}{5 \cdot 1600 + 8 \cdot 900 + 50 \cdot 400 + 90 \cdot 306.25 + 148 \cdot 156.25 + 10 \cdot 100} \] \[ d_{sm} = \frac{320000 + 216000 + 400000 + 483843.75 + 289687.5 + 10000}{8000 + 7200 + 20000 + 27562.5 + 23125 + 1000} \] \[ d_{sm} = \frac{1402531.25}{90687.5} = 15.5~\text{µm} \] Thus, the Sauter mean diameter is: \[ \boxed{19.0~\text{µm}} \]