Subtopic: Mechanics and energy requirement in size reduction of agriculture produce
Question Statement:
A hammer mill grinds grain from an initial average particle diameter of $4.0\text{ mm}$ to a final product diameter of $1.0\text{ mm}$ using $8.0\text{ kW}$ of power. Assuming Rittinger's law applies, calculate the power in $\text{kW}$ required to further reduce the particle diameter from $1.0\text{ mm}$ to $0.5\text{ mm}$ at the same feed throughput.
Answer Key:
10.67
Step-by-Step Solution & Explanation
Rittinger's law states that power is proportional to the increase in specific surface area:
$P = K_R \left( \frac{1}{d_2} - \frac{1}{d_1} \right)$
For initial reduction ($4.0\text{ mm} \to 1.0\text{ mm}$):
$8.0 = K_R \left( \frac{1}{1.0} - \frac{1}{4.0} \right) = K_R (1.0 - 0.25) = 0.75 K_R$
$K_R = \frac{8.0}{0.75} = 10.667\text{ kW}\cdot\text{mm}$
For second reduction ($1.0\text{ mm} \to 0.5\text{ mm}$):
$P_2 = 10.667 \left( \frac{1}{0.5} - \frac{1}{1.0} \right) = 10.667 (2.0 - 1.0) = 10.667\text{ kW}$