Empirical distribution functions

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Empirical Distribution Functions

Empirical distribution functions are mathematical tools used to describe how the values of a particular property (such as particle size) are distributed across a population. In mineral processing, empirical distribution functions are vital for modeling and understanding particle populations in comminution and classification operations.

Key points about empirical distribution functions:

1. Definition:

   • An empirical distribution function quantifies the cumulative fraction of particles with a property value (e.g., size) less than or equal to a specific threshold.
   • For particle size $d_p$, the distribution function $P(d_p)$ is defined as the mass fraction of particles with size $\leq d_p$.

2. Common Forms:

   • Rosin-Rammler Distribution: $$P(d) = 1 - \exp\left(-\left(\frac{d}{d_{63.2}}\right)^n\right)$$ Where $d_{63.2}$ is the size corresponding to $P(d) = 0.632$, and $n$ is a measure of spread.
   • Log-Normal Distribution: $$P(d) = G\left(\frac{\ln(d) - \ln(d_{50})}{\sigma}\right)$$ Here, $d_{50}$ is the median size, $\sigma$ is the standard deviation in logarithmic terms, and $G(x)$ is the Gaussian cumulative distribution function.
   • Logistic Distribution: $$P(d) = \frac{1}{1 + \exp\left(-\frac{\ln(d/d_{50})}{s}\right)}$$ Where $s$ is a spread parameter.

3. Properties:

   • These distributions help fit experimental data and analyze particle size distributions.
   • Specialized coordinate systems are used to linearize these functions for graphical representation (e.g., Rosin-Rammler plot).

4. Applications:

   • Modeling comminution processes.
   • Analyzing particle size distributions after sieving.
   • Aiding in the design and simulation of mineral processing circuits.

5. Practical Representation:

   • The empirical functions can be modified for truncated distributions to account for real-world scenarios where particle size is limited by processing equipment or feedstock properties.

In R.P. King’s book Modeling and Simulation of Mineral Processing Systems, empirical distribution functions are discussed in detail in Chapter 2, Section 2.2.1 (pages 8–12). Various forms of these functions are described, along with their derivations, properties, and practical significance.