Question

The average particle size $D_p$ (in mm) represented in terms of fineness modulus (FM) can be estimated by the equation:

• A. $D_p = 0.135 \, (\text{FM})^{1.366}$
• B. $D_p = 0.135 \, (1.366)^{\text{FM}}$
• C. $D_p = 1.366 \, (\text{FM})^{0.135}$
• D. $D_p = 1.366 \, (0.135)^{\text{FM}}$

Hint:

Remember: Particle size relation is power law form → $D_p = k(\text{FM})^n$.

Answer 1

The Correct Option is A

Concept: Fineness modulus (FM) is an empirical index number that gives an idea about the average size of particles in a sample. It is commonly used in food engineering and material science to estimate particle size distribution. An empirical relationship exists between fineness modulus and average particle size.

Step 1: Understanding fineness modulus.
Fineness modulus is calculated as:
• Sum of cumulative percentages retained on standard sieves divided by 100 It gives a single numerical value representing overall coarseness or fineness.

Step 2: Relationship between FM and particle size.
Based on experimental correlations, the average particle size $D_p$ is related to FM using an empirical power law relation: \[ D_p = k \, (\text{FM})^n \] where:
• $k$ and $n$ are empirical constants

Step 3: Known empirical constants.
From standard food engineering correlations: \[ k = 0.135 \quad \text{and} \quad n = 1.366 \]

Step 4: Substituting values.
\[ D_p = 0.135 \, (\text{FM})^{1.366} \]

Step 5: Evaluating options.
• Option (A) matches the correct empirical relation
• Option (B) incorrectly places exponent
• Option (C) swaps constants incorrectly
• Option (D) incorrect exponential form Final Conclusion:
The correct relation is $D_p = 0.135 (\text{FM})^{1.366}$. Hence, option (1) is correct.