To calculate the breakage function

 Define and calculate the breakage function for a given dataset. (p. 136–150)

Definition of the Breakage Function

The breakage function (denoted as ${𝐵}_{𝑖𝑗}$) describes how particles of a given size class $𝑗$ break into smaller size classes $𝑖$ during the comminution process. It represents the fraction of material from size class $𝑗$ that ends up in size class $𝑖$ after breakage.

The breakage function satisfies the following condition:

$$\sum _{𝑖=1}^{𝑗}{𝐵}_{𝑖𝑗}=1$$

This ensures mass conservation, as the total mass of fragments produced from a size class equals the original mass of the size class.

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Steps to Calculate the Breakage Function

1. Experimental Setup

   • Collect a representative sample of material and subject it to controlled comminution (e.g., in a laboratory crusher or ball mill).
   • Measure the size distribution of the feed and the product using standard sieves or particle size analyzers.

2. Determine Size Fractions

   • Divide the feed and product into discrete size classes, typically using sieve analysis.
   • Denote:
     • ${𝐹}_{𝑗}$: Mass fraction of feed material in size class $𝑗$.
     • ${𝑃}_{𝑖}$: Mass fraction of product material in size class $𝑖$.

3. Calculate Breakage Function (${𝐵}_{𝑖𝑗}$)

   • For each size class $𝑗$, calculate the fraction of material that breaks into size class $𝑖$:$${𝐵}_{𝑖𝑗}=\frac{\text{Mass of particles in size class }𝑖\text{ produced from }𝑗}{\text{Total mass of particles in size class }𝑗\text{ in feed}}$$

4. Verify Mass Conservation

   • Ensure that the sum of breakage fractions for each parent size class equals 1:$$\sum _{𝑖=1}^{𝑗}{𝐵}_{𝑖𝑗}=1$$

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Example Calculation

Given Dataset:

• Feed Size Class $𝑗$: 2 mm to 4 mm (${𝐹}_{𝑗}=0.4$)
• Product Size Classes:
  • ${𝑖}_1$: 0.5 mm to 1 mm (${𝑃}_{{𝑖}_1}=0.1$)
  • ${𝑖}_2$: 1 mm to 2 mm (${𝑃}_{{𝑖}_2}=0.2$)
  • ${𝑖}_3$: 2 mm to 4 mm (${𝑃}_{{𝑖}_3}=0.1$)

Calculation:

$${𝐵}_{{𝑖}_1,𝑗}=\frac{0.1}{0.4}=0.25,{𝐵}_{{𝑖}_2,𝑗}=\frac{0.2}{0.4}=0.5,{𝐵}_{{𝑖}_3,𝑗}=\frac{0.1}{0.4}=0.25$$

Verification:

$${𝐵}_{{𝑖}_1,𝑗}+{𝐵}_{{𝑖}_2,𝑗}+{𝐵}_{{𝑖}_3,𝑗}=0.25+0.5+0.25=1$$

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Applications of the Breakage Function

• Grinding Circuits: Predicting the size distribution of products from crushers and mills.
• Optimization: Tuning mill parameters to achieve desired product specifications.
• Simulation: Used in population balance models to simulate comminution processes.

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Reference: R.P. King, Modeling and Simulation of Mineral Processing Systems, p. 136–150.