Mineral Comminution & Wet Classification, Plitt-Arterburn Hydrocyclone d50c & G-Force
Hydrocyclone Cut Size d50c, G-Force & Slurry Flow Formula
Calculate Hydrocyclone Volumetric Slurry Capacity (m³/h), Tangential Inlet Velocity (m/s), Wall Centrifugal G-Force, and Plitt-Arterburn Corrected Cut Size d50c (um).
Let ball-mill cyclone feed solids creep from 18% to 32% by volume and hindered settling coarsens cut size d50c nearly 3x!
Closed-circuit ball-mill grinding loops classify ore slurry inside **Conical Hydrocyclones** of Cylinder Diameter `D_c` (`cm`) fed at Sump Pump Pressure Drop `Delta_P` (`kPa`), Mineral Specific Gravity `rho_s` (`g/cm³`), and Volumetric Feed Solids Concentration `C_v` (`%` by volume). Tangential injection at **`v_{in} = 0.85 * SQRT((2 * Delta_P * 1,000) / 1,280)`** (`m/s`) spins slurry past **`G = v_{in}^2 / (9.80665 * (D_c / 200))`** Gs (`60–300+` Gs!) while delivering **Volumetric Slurry Capacity `Q = 0.016 * D_c^{1.9} * SQRT(Delta_P)`** (`m³/h`). Under the **Plitt / Arterburn Wet-Classification Correlation**, **Corrected `50%` Cut Diameter `d_{50c} = (14.8 * D_c^{0.46} * EXP(0.063 * C_v)) / (Delta_P^{0.28} * SQRT(rho_s - 1))`** (`um`)—exposing the exponential **`EXP(0.063 * C_v)`** hindered-settling term that ruins overflow fineness whenever sump dilution water falters!
Verify on a `25.0` cm (`10`-inch) Copper Concentrator Cyclone—Cylinder_Diameter `D_c` = 25.0 cm (B2), Feed_Header_Pressure `Delta_P` = 120.0 kPa (B3), Sulfide/Silicate_Ore_SG `rho_s` = 2.75 g/cm³ (B4), and Feed_Solids_Volume `C_v` = 18.0% (B5): Volumetric Throughput hits `79.4` m³/h (E2), Inlet Velocity runs `11.64` m/s (E3 = `110.5` Gs at wall in E4), and Corrected Cut Size `d_{50c}` classifies cleanly at `40.01` um (E5)!
| Plitt-Arterburn Hydrocyclone KPI (Dc=25 cm, Ore SG=2.75: 120 kPa/18% vs 65 kPa/32% Cv) | Formula | Verified Output |
|---|---|---|
| Slurry Volumetric Flow Q (m³/h) | 0.016 * (Dc_cm ^ 1.9) * SQRT(dP_kPa) | 79.4 m³/h (58.4 m³/h @ 65 kPa) |
| Tangential Inlet Velocity (m/s) | 0.85 * SQRT((2 * dP_kPa * 1000) / 1280) | 11.64 m/s (8.57 m/s @ 65 kPa) |
| Wall Centrifugal Acceleration (Gs) | (vin_m_s ^ 2) / (9.80665 * (Dc_cm / 200)) | 110.5 Gs (59.9 Gs @ 65 kPa) |
| Corrected Cut Diameter d50c (um) | (14.8*Dc^0.46*EXP(0.063*Cv))/(dP^0.28*SQRT(SG-1)) | 40.01 um (114.74 um coarse grit carryover!) |
Plitt-Arterburn Hydrocyclone Cut Diameter d50c, G-Field & Capacity Formula
Validate `Cyclone_Diam D_c` in `[5, 100]` cm, `Pressure_Drop Delta_P` in `[20, 400]` kPa, `Mineral_SG rho_s` in `(1.20, 8.00]`, and `Volumetric_Solids C_v` in `[1%, 50%]`!
=LET(dc_c, B2, dp_k, B3, sg_s, B4, cv_p, B5, IF(OR(dc_c < 5, dc_c > 100, dp_k < 20, dp_k > 400, sg_s <= 1.2, sg_s > 8, cv_p < 1, cv_p > 50), "Invalid Cyclone Input", ROUND((14.8 * (dc_c ^ 0.46) * EXP(0.063 * cv_p)) / ((dp_k ^ 0.28) * SQRT(sg_s - 1)), 2)))Sump Dilution-Valve Loss & Pump Wear (dP = 65 kPa, Cv = 32%) Roping Coarsening Check
Lose sump dilution water (`C_v` climbs to `B5 = 32%`) on a worn slurry pump impeller (`Delta_P` drops to `B3 = 65` kPa): Wall G-Force E4 nearly halves to `59.9` Gs while `EXP(0.063 * 32)` blows Cut Size `d_{50c}` E5 from `40.01` up to **`114.74` um**—sending unliberated coarse quartz straight into the flotation cells and risking apex "roping"!
=IF(ISTEXT(E5), E5, LET(vi_n, 0.85 * SQRT((2 * B3 * 1000) / 1280), ROUND((vi_n ^ 2) / (9.80665 * (B2 / 200)), 1)))Try the example
Use the original values or change them to test your understanding.