Question Bank • QB_APE_030 NAT • 2 Marks

Section 6: Agricultural Process Engineering

Size Reduction and Material Handling

Subtopic: Effectiveness of separation

Question Statement:

A vibrating screen separates fine grain from feed. The mass fraction of desired undersize material in the feed ($x_F$) is $0.40$. The oversize product stream (overflow) contains a mass fraction $x_D = 0.05$ of undersize material, while the undersize product stream (underflow) contains $x_B = 0.90$ of undersize material. The overall effectiveness of the screen ($E$) calculated as $E = \frac{(x_F - x_D)(x_B - x_F) x_B (1 - x_D)}{(x_B - x_D)^2 x_F (1 - x_F)}$ is ________ (round off to 3 decimal places).

Answer Key:

0.863

Step-by-Step Solution & Explanation

Screen effectiveness formula: $E = \frac{(x_F - x_D)(x_B - x_F) x_B (1 - x_D)}{(x_B - x_D)^2 x_F (1 - x_F)}$ 1. Numerator terms: • $x_F - x_D = 0.40 - 0.05 = 0.35$ • $x_B - x_F = 0.90 - 0.40 = 0.50$ • $x_B = 0.90$ • $1 - x_D = 1 - 0.05 = 0.95$ $\text{Numerator} = 0.35 \times 0.50 \times 0.90 \times 0.95 = 0.149625$ 2. Denominator terms: • $(x_B - x_D)^2 = (0.90 - 0.05)^2 = (0.85)^2 = 0.7225$ • $x_F = 0.40$ • $1 - x_F = 0.60$ $\text{Denominator} = 0.7225 \times 0.40 \times 0.60 = 0.1734$ 3. Overall effectiveness: $E = \frac{0.149625}{0.1734} = 0.862889 \approx 0.863$
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