The Swebrec equation is a mathematical model used to describe and predict the particle size distribution (PSD) of fragmented rock, particularly after blasting and crushing in mining operations. It is known for its accuracy across a wide size range, especially in modeling the fine and coarse ends of the PSD curve, where other models often fall short.

The Swebrec Equation

The general form of the Swebrec cumulative distribution function is:

Where:

• P(x)P(x)

  = cumulative percent passing (undersize) for size
  xx

• xx

  = particle size (mm or μm)

• x50x_{50}

  = size at which 50% of the material passes (median)

• xmaxx_{\text{max}}

  = maximum particle size

• bb

  = curve-shaping parameter that controls the steepness and symmetry

How It’s Applied in Rock Fragmentation Analysis

1. Data Collection

• After blasting or crushing, samples of the fragmented rock are taken.

• Sieve analysis or digital image processing is used to determine the PSD (i.e., the proportion of rock that falls under different size categories).

2. Curve Fitting

• The collected PSD data is plotted as cumulative percent passing vs. particle size.

• The Swebrec function is fitted to this data using statistical methods (e.g., least squares).

• The goal is to find the best values of
  x50x_{50}
  ,
  xmaxx_{\text{max}}
  , and
  bb
  that minimize the difference between the actual and modeled data.

3. Model Evaluation

• Goodness of fit is assessed using:

  • Coefficient of determination (
    R2R^2
    )

  • Estimation error (e.g., RMSE or residuals)

  • Statistical criteria like Cochran's or Chi-square tests

4. Interpretation and Use

• The fitted Swebrec curve provides insight into:

  • Blasting efficiency (how well energy was distributed)

  • Fragmentation uniformity

  • Potential grinding energy requirements

• The parameters can guide blast design optimization (e.g., charge size, hole spacing) and comminution circuit adjustments.

Why Use the Swebrec Function?

• Accurate over a wide size range (fine and coarse particles)

• Smooth and continuous: avoids abrupt changes seen in some other models

• Better fit than Rosin-Rammler or Gates-Gauss-Schumann for blast fragmentation

• More physically realistic for rock breakage mechanisms

Example Interpretation:

If a blast produced rock with:

• x50=80x_{50} = 80

  mm,

• xmax=300x_{\text{max}} = 300

  mm,

• b=1.3b = 1.3

  ,

then using the Swebrec function you can calculate what % of the rock is smaller than any given size (say 100 mm), and visualize the entire PSD curve.