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Mineral Processing, Composting & Recycling Separation, Rotary Trommel Screen Kinetics

Rotary Trommel Screen Critical RPM & Bed Fill Formula

Calculate Drum Critical Centrifuging Speed Ncrit (RPM), Operating % Critical Speed, Cascade Helix Residence Time (s), and Cross-Sectional Bed Holdup Fill (%) for Undersize Screening.

Flatten a trommel screen slope to 1.8° at 28% critical speed and a 22.6% deep material bed blinds screen apertures so fines ride straight out the oversize chute!

**Inclined Cylindrical Rotary Trommel Screens** (`Internal Diameter D` m, `Screen Length L` m, `Inclination Slope theta` deg, `Spin Speed N` RPM) classify alluvial gold gravel, municipal compost, and C&D waste by lifting material up the perforated/woven drum wall and cascading it forward along a sawtooth helix! Relative to **Centrifuging Critical Speed `N_{crit} = 42.3 / SQRT(D)`** (`RPM`, where `g = omega^2 * R` pins rock to the shell at `100%` `N_{crit}`!), clean screening demands **`38% <= N / N_{crit} <= 50%`** (`active cascading cataract` without blinding!). Each revolution advances cascading particles axially by `~PI() * D * TAN(theta)` (`Axial Velocity v_a = N * PI() * D * TAN(theta * PI() / 180)` m/min), setting **Drum Screening Residence Time `t_{res} = (L / v_a) * 60`** (`s`) and **Cross-Sectional Bed Fill Fraction `Fill% = (((Q_{t/h} / 60) / rho_b) / v_a) / ((PI() / 4) * D^2) * 100%`**—which MUST stay **`<= 12%–15%`** so fines reach open screen holes!

Verify on a `2.20` m ID x `7.50` m Compost / Aggregate Trommel—Drum_ID `D` = 2.20 m (B2), Screen_Length `L` = 7.50 m (B3), Drum_Speed `N` = 13.5 RPM (B4), Downhill_Slope `theta` = 4.50° (B5), Feed_Rate `Q` = 85.0 t/h (B6), and Loose_Bulk_Density `rho_b` = 0.95 t/m³ (B7): Critical Speed `N_{crit}` equals `28.52` RPM (E2), Operating Speed hits an ideal `47.3%` `N_{crit}` (E3), Residence Time equals `61.3` s (E4), and Cross-Sectional Bed Fill stays thin and un-blinded at `5.34%` (E5)!

Rotary Trommel Screening KPI (D=2.2 m, L=7.5 m, Q=85 t/h, rho=0.95 t/m³)FormulaVerified Output
Critical Centrifuging Speed (RPM)42.3 / SQRT(Drum_ID_D_m)28.52 RPM
Operating % Critical Speed (%)(Operating_RPM / Critical_RPM) * 10047.3% active cascade (28.1% sluggish)
Drum Residence Time tres (s)(Length_m / (RPM * PI() * D * TAN(rad))) * 6061.3 s (259.0 s over-retained!)
Cross-Sectional Bed Fill (%)((Feed_m3_min / Axial_m_min) / Drum_Area) * 1005.34% clean (<15%) vs 22.58% FLOODED!

Trommel Critical Speed Ratio, Helical Cascade Transport & Bed Holdup Formula

Validate `Drum_ID D` in `[0.5, 5.0]` m, `Length L` in `[1.5, 25]` m, sub-centrifuging `0.15 * N_{crit} <= RPM <= 0.85 * N_{crit}`, `Slope theta` in `[1.0°, 12.0°]`, `Feed Q > 0.5` t/h, and `Bulk_Density` in `[0.2, 3.0]` t/m³!

=LET(dd_m, B2, ls_m, B3, nr_p, B4, sl_d, B5, qf_t, B6, rb_t, B7, nc_r, 42.3 / SQRT(dd_m), va_m, nr_p * PI() * dd_m * TAN((sl_d * PI()) / 180), ad_m, (PI() / 4) * (dd_m ^ 2), IF(OR(dd_m < 0.5, dd_m > 5, ls_m < 1.5, ls_m > 25, nr_p < 0.15 * nc_r, nr_p > 0.85 * nc_r, sl_d < 1, sl_d > 12, qf_t <= 0.5, rb_t < 0.2, rb_t > 3), "Invalid Trommel Input", ROUND(((((qf_t / 60) / rb_t) / va_m) / ad_m) * 100, 2)))

Shallow 1.8° Slope + VFD Turn-Down to 8.0 RPM Bed-Flooding Check

Lower support jacks to `B5 = 1.8°` and dial the VFD down to `B4 = 8.0` RPM (`28.1%` critical speed in E3) thinking "longer residence time (`259.0` s in E4) will screen cleaner": instead, Bed Holdup Fill E5 quadruples to **`22.58%`** (`> 15%` flood ceiling!)—burying fine undersize particles atop a thick rolling mat so they discharge unscreened into the oversize pile!

=IF(ISTEXT(E5), E5, LET(va_m, B4 * PI() * B2 * TAN((B5 * PI()) / 180), ROUND((B3 / va_m) * 60, 1)))

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