Engineering & Fluid Mechanics
Gates-Gaudin-Schuhmann (GGS) Particle Size Formula: Power-Law Passing & P80 Interpolation
Characterize jaw/cone crusher products and fine mineral tails using the Gates-Gaudin-Schuhmann power law P(d) = 100 * MIN(1, (d / k)^m) and solve directly for 80%-passing size P80 = k * 0.8^(1/m).
Why log-log straight lines govern fine mineral breakage fragments below top size k!
Developed independently by Arthur O. Gates (1915), Antoine M. Gaudin (1926), and Reinhardt Schuhmann Jr. (1940, AIME Technical Publication 1189), the Gates-Gaudin-Schuhmann (GGS) distribution models cumulative mass fraction passing sieve aperture d <= k as a pure power law: P(d) = 100 * (d / k)^m (capped at 100% for d >= k), where k (k_100) is the theoretical 100%-passing top size and m (typically 0.4–0.9 in crushing/grinding) is the log-log distribution modulus!
Inverting the GGS power law gives an exact closed-form P80 input for Bond comminution power calculations: d_80 = k * (0.80)^(1 / m)! For k = 500 um and m = 0.65, cumulative passing at d = 125 um is 40.61% (rising to 63.73% at d = 250 um), and exact P80 = 500 * 0.8^(1/0.65) = 354.71 um!
| Sieve Aperture d (Top Size k = 500 um, m = 0.65) | GGS Cumulative Passing P(d) % | Derived P80 = k * 0.8^(1/m) |
|---|---|---|
| Sieve d = 125 um (Quarter Top Size) | 40.61% Passing | 354.71 um |
| Sieve d = 250 um (Half Top Size) | 63.73% Passing | 354.71 um |
| Sieve d = 600 um (> Top Size k = 500 um) | 100.00% (Capped at MIN(1, ...)) | 354.71 um |
Gates-Gaudin-Schuhmann Cumulative Passing P(d) = 100 * MIN(1, (d / k)^m)
Evaluates GGS cumulative percent passing at sieve size d (B2) with top-size cap at 100% for d >= k (B3).
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0), "Invalid GGS Size Parameters", ROUND(100 * MIN(1, (B2 / B3) ^ B4), 2))Inverted GGS 80%-Passing Size P80 = k * 0.8^(1 / m)
Solves P(d) = 80% directly for Bond comminution size P80 in the same length units as k.
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0), "Invalid GGS Size Parameters", ROUND(B3 * (0.8 ^ (1 / B4)), 2))Try the example
Use the original values or change them to test your understanding.