Rosin–Rammler Distribution: Step-by-Step Numerical Example

Rosin–Rammler Distribution: Step-by-Step Numerical Example

The Scenario

You have performed a sieve analysis on a ball mill product. After fitting the data, you obtained:

• Size Modulus (D_63.2): 150 µm
• Distribution Modulus (λ): 1.2

The Rosin–Rammler Formula:

P(D) = 1 − exp[ − ( D / D₆₃.₂ ) ^ λ ]

Question 1: How much "fines" do we have?

Goal: Find what percentage passes through a 75 µm screen.

Step 1: Plug in values

P(75) = 1 − exp[ − (75 / 150) ^ 1.2 ]

Step 2: Ratio
75 / 150 = 0.5

Step 3: Apply λ
0.5^1.2 ≈ 0.435

Step 4: Exponential
e^−0.435 ≈ 0.647

Step 5: Final calculation

P(75) = 1 − 0.647 = 0.353 (→ 35.3% fines)

Question 2: What is the P₈₀? (Reverse Calculation)

Goal: Find D such that P(D) = 0.80.

0.80 = 1 − exp[ − (D / 150) ^ 1.2 ]

Step 1: Rearrange

0.20 = exp[ − (D / 150) ^ 1.2 ]

Step 2: Take natural log

ln(0.20) = − (D / 150) ^ 1.2
1.609 = (D / 150) ^ 1.2

Step 3: Remove exponent

(D / 150) = 1.609 ^ (1 / 1.2) ≈ 1.486

Step 4: Solve for D

D = 1.486 × 150 = 223 µm
→ P₈₀ = 223 µm

Meaning: 80% of the particles are finer than 223 microns.