Rosin Ramler Equation

What is the Rosin Ramler Equation for particle size distribution. Explain in detail. 

The Rosin-Rammler Equation for particle size distribution is expressed as:

$$P(D) = 1 - \exp\left[-\left(\frac{D}{D_{63.2}}\right)^n\right]$$

Where:

• $P(D)$ = Cumulative fraction of the total mass of particles with sizes less than or equal to $D$.
• $D$ = Particle size.
• $D_{63.2}$ = Particle size at which 63.2% of the material passes (a characteristic parameter of the distribution).
• $n$ = Distribution modulus, indicating the spread of the size distribution. A larger $n$ corresponds to a narrower distribution.

Explanation

1. Mathematical Representation:

   • $P(D)$ ranges from 0 to 1 as $D$ increases.
   • At $D = D_{63.2}$, $P(D) = 0.632$, which is why $D_{63.2}$ is used as the reference size.

2. Log-Log Transformation:

   • The Rosin-Rammler distribution is often analyzed in its linearized form: $$\ln\left[\ln\left(\frac{1}{1-P(D)}\right)\right] = n\ln(D) - n\ln(D_{63.2})$$ This linear relationship allows for easy determination of $n$ and $D_{63.2}$ using a graph of $\ln\left[\ln\left(\frac{1}{1-P(D)}\right)\right]$ against $\ln(D)$.

3. Application:

   • The Rosin-Rammler equation is widely used in mineral processing to describe size distributions of comminuted materials.
   • It helps predict performance of size reduction units and evaluate process efficiency.

Reference

"Modeling and Simulation of Mineral Processing Systems" by R.P. King on pages 7–9