Introduction & Context

The Gaudin-Schuhmann (G-S) distribution is a power-law model that describes the cumulative mass fraction of particles finer than a given size. It is widely used in comminution and classification circuits to quantify how feed or product material breaks. The two-parameter form \(F(x)=\left(\frac{x}{x'}\right)^n\) is preferred in plant surveys because it can be calibrated from only two sieve data points, making it ideal for rapid field assessments of mills, crushers, and air classifiers.

Methodology & Formulas

Input validation
The cumulative mass fractions \(F_1\) and \(F_2\) must lie strictly between 0 and 1, and the coarse sieve \(x_1\) must be larger than the fine sieve \(x_2\).
Slope exponent \(n\)
The exponent is obtained from the two measured points \((x_1,F_1)\) and \((x_2,F_2)\) by \[ n=\frac{\ln(F_1/F_2)}{\ln(x_1/x_2)} \]
Characteristic size \(x'\)
Once \(n\) is known, the size at which the cumulative mass fraction equals unity is \[ x'=\frac{x_1}{\left(F_1\right)^{1/n}} \] A small positive lower bound (e.g., 1×10⁻⁹) is imposed on \(n\) to avoid division by zero.

Recommended validity regime for food powders
Parameter	Range	Interpretation
\(n\)	0.3 – 1.2	Distribution slope; values outside this band may indicate poor G-S fit

The Gaudin-Schuhmann (G-S) distribution is defined by:

K (size modulus): the theoretical 100 % passing size (µm or mm).
m (distribution modulus): the slope of the line on log-log paper; higher m means a narrower size spread.

To obtain them:

Collect a full particle-size analysis (sieve or laser) and convert cumulative % passing to fractional % retained.
Plot log(cumulative % passing) vs. log(particle size). Fit a straight line through the central portion.
K is the x-intercept at 100 % passing; m is the slope of that line.

The G-S model is linear only over the central size range; fines often deviate due to agglomeration or measurement artefacts. To keep m consistent:

Exclude the top 5 % and bottom 5 % of the data range unless those points lie on the straight-line trend.
Use R² > 0.98 as a cut-off; if adding fines drops R² below this, truncate the fit.
Report the size interval used for fitting so future comparisons are valid.

Re-arrange the G-S equation: P80 = K × (0.8)^(1/m).
Example: K = 2 000 µm, m = 0.85 → P80 = 2 000 × (0.8)^(1/0.85) ≈ 1 540 µm.
Use this P80 directly in Bond/Rowland power-draw models.

Re-fit after grinding.

Impact and attrition shift the slope; m(product) is typically 0.1–0.3 units higher than m(feed) in ball mills.
Using the feed m for product will over-predict the amount of ultrafines and bias classification efficiency calculations.

Worked Example – Determining the Gaudin-Schumann Slope from Plant Survey Data

A metallurgical survey team has collected wet-screen data from the ball-mill discharge stream at the Blue Ridge Concentrator. Two size fractions are available: the 80 %-passing size \(x_{80}\) and the 20 %-passing size \(x_{20}\). These two points are sufficient to back-calculate the Gaudin-Schuhmann exponent \(n\), which is required for the grinding-circuit simulator.

Knowns

Cumulative mass fraction passing 500 µm: \(F_1 = 0.85\)
Cumulative mass fraction passing 200 µm: \(F_2 = 0.46\)
Corresponding sizes: \(x_1 = 500\) µm and \(x_2 = 200\) µm

Step-by-Step Calculation

Compute the ratio of cumulative fractions: \[ \frac{F_1}{F_2} = \frac{0.85}{0.46} = 1.848 \]
Compute the inverse size ratio: \[ \frac{x_2}{x_1} = \frac{200\ \mu m}{500\ \mu m} = 0.400 \]
Take natural logarithms of both sides of the Gaudin-Schuhmann equation: \[ F = \left(\frac{x}{x'}\right)^n \Rightarrow \ln\left(\frac{F_1}{F_2}\right) = n\ln\left(\frac{x_1}{x_2}\right) \]
Solve for the exponent \(n\): \[ n = \frac{\ln(1.848)}{\ln(2.5)} = \frac{0.614}{0.916} = 0.670 \]
Back-substitute to find the reference size \(x'\) (size at 100 % passing): \[ x' = \frac{x_1}{(F_1)^{1/n}} = \frac{500\ \mu m}{(0.85)^{1/0.670}} = 637\ \mu m \]

Final Answer

The Gaudin-Schumann slope for the surveyed stream is n = 0.670 and the reference size is x′ = 637 µm. These parameters can now be entered into the simulator to predict the full size distribution under varying throughput conditions.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

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