Joint distribution functions

What are joint distribution functions? Explain in detail.

Joint distribution functions, as detailed in R.P. King's "Modeling and Simulation of Mineral Processing Systems," describe the combined distribution of two or more properties of particles in a population. These functions are particularly significant when multiple characteristics of particles influence their performance in mineral processing operations.

Definition and Utility

The joint distribution function, P(D,g)P(D, g), represents the fraction of particles that possess a specific size DD and mineralogical composition gg. This function is crucial for modeling processes that depend simultaneously on size and composition, such as comminution or separation.

Discrete Representation

For practical applications, joint distribution functions are often discretized. The particle population is divided into size classes and grade (composition) classes, creating a grid in the size-composition space. The mass fraction in each grid cell is given by:

pij(D,g)=P(Di−1,gj)−P(Di,gj)−P(Di−1,gj−1)+P(Di,gj−1),p_{ij}(D, g) = P(D_{i-1}, g_j) - P(D_i, g_j) - P(D_{i-1}, g_{j-1}) + P(D_i, g_{j-1}),

where pij(D,g)p_{ij}(D, g) denotes the fraction of particles in the ii-th size class (Di−1<D≤DiD_{i-1} < D \leq D_i) and the jj-th grade class (gj−1<g≤gjg_{j-1} < g \leq g_j).

Relation to Single-Property Distributions

The joint distribution function relates directly to individual property distributions:

  • Marginal Distributions: The size distribution P(D)P(D) and grade distribution P(g)P(g) can be obtained by summing over the grade and size dimensions, respectively.
  • Conditional Distributions: These describe distributions conditioned on a specific value of another property, e.g., P(g∣D)P(g | D), the grade distribution for a fixed particle size.

Applications

  1. Modeling Mineral Liberation: Joint distributions link size and mineral composition, enabling the prediction of liberation patterns during comminution.
  2. Simulation of Unit Operations: They provide detailed input for simulating processes like flotation or magnetic separation, where both size and grade influence separation efficiency.
  3. Process Optimization: Understanding joint distributions helps in designing operations to maximize recovery and grade.

Practical Measurement

Joint distribution functions are measured experimentally by combining techniques like:

  1. Screening: To separate particles into size classes.
  2. Dense-Liquid Fractionation: To classify particles by composition within each size class.

These measurements yield the discrete joint distributions, which can be used for process modeling and simulation.

For a detailed discussion, refer to Chapter 2, Section 2.6 and 2.7 (pages 17-23) of the book "Modeling and Simulation of Mineral Processing Systems" by R.P. King

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