Cement, Lime & Mineral Pyroprocessing, USBM Sullivan Rotary Kiln Residence Time
Rotary Kiln Sullivan Residence Time & Fill % Formula
Calculate USBM Sullivan Solids Mean Residence Time (min), Axial Bed Advance Velocity (m/min), In-Kiln Solids Holdup Inventory (tonnes), and Volumetric Fill Degree (%).
Slowing a rotary calciner’s VFD drive from 1.8 to 1.1 RPM stretches residence time 64%—and piles 20 extra tonnes of hot clinker onto the trunnion rollers!
Inclined refractory-lined **Rotary Kilns, Calciners, and Dryers** (Length `L` m, Inside Brick Diameter `D` m, Inclination Slope `S` deg, Shell Speed `N` RPM) transport granular limestone, alumina, or lithium spodumene of Dynamic Angle of Repose `theta` (`deg`) via cascading avalanches down the tumbling chord! Under the classic **U.S. Bureau of Mines (`USBM`) Sullivan Equation** (for un-dammed smooth cylindrical kilns), **Mean Solids Residence Time `tau = (1.77 * L * SQRT(theta)) / (S * D * N)`** (`min`), setting **Axial Bed Velocity `v_{ax} = L / tau`** (`m/min`)! At Steady Mass Feed Rate `dot_m` (`t/h`, Bulk Density `rho_b` `t/m³`), conservation of mass (`M_{hold} = dot_m * (tau / 60)` tonnes) means slowing `N` WITHOUT proportionally cutting feed rate inflates **Volumetric Kiln Fill Degree `f% = ((M_{hold} / rho_b) / ((PI() / 4) * D^2 * L)) * 100%`**—risking burner-hood spillage or radial heat-transfer starvation above `~12%–15%` fill!
Verify on a `55.0` m Lime Calcining Rotary Kiln—Refractory_Length `L` = 55.0 m (B2), Inside_Brick_Diameter `D` = 3.40 m (B3), Shell_Slope `S` = 2.20° (B4), Drive_Speed `N` = 1.80 RPM (B5), Limestone_Repose_Angle `theta` = 38.0° (B6), Kiln_Feed_Rate = 42.0 t/h (B7), and Bed_Bulk_Density `rho_b` = 1.35 t/m³ (B8): Mean Residence Time equals `44.57` min (E2), Axial Velocity runs `1.234` m/min (E3), Hot Holdup Inventory equals `31.20` tonnes (E4), and Volumetric Fill sits in the optimal rolling-cascade regime at `4.63%` (E5)!
| USBM Sullivan Rotary Kiln KPI (L=55 m, D=3.4 m, S=2.2°, theta=38°, 42 t/h, N=1.8 vs 1.1 RPM) | Formula | Verified Output |
|---|---|---|
| Mean Residence Time tau (min) | (1.77 * L_m * SQRT(theta_deg)) / (S_deg * D_m * N) | 44.57 min (72.93 min @ N=1.1 RPM) |
| Axial Bed Velocity (m/min) | Kiln_Length_L_m / Residence_Time_min | 1.234 m/min (0.754 m/min @ N=1.1 RPM) |
| In-Kiln Solids Holdup (tonnes) | Feed_Rate_t_h * (Residence_Time_min / 60) | 31.20 t (51.05 t trunnion load @ 1.1 RPM!) |
| Volumetric Fill Degree (%) | ((Holdup_t / Bulk_rho) / ((PI()/4)*D^2*L))*100 | 4.63% (7.57% deeper bed @ 1.1 RPM) |
USBM Sullivan Tumbling-Cascade Residence Time, Holdup & Fill Degree Formula
Validate `Length L > 3` m, `Diameter D > 0.3` m, `Slope S` in `[0.3°, 8.0°]`, `Speed N` in `(0.1, 15.0]` RPM, `Repose_Angle theta` in `[15°, 55°]`, `Feed > 0` t/h, and `Bulk_Density > 0.2` t/m³!
=LET(lk_m, B2, dk_m, B3, sl_d, B4, nr_p, B5, th_d, B6, fd_t, B7, rb_t, B8, tr_m, (1.77 * lk_m * SQRT(th_d)) / (sl_d * dk_m * nr_p), mh_t, fd_t * (tr_m / 60), vk_m, (PI() / 4) * (dk_m ^ 2) * lk_m, IF(OR(lk_m <= 3, dk_m <= 0.3, sl_d < 0.3, sl_d > 8, nr_p <= 0.1, nr_p > 15, th_d < 15, th_d > 55, fd_t <= 0, rb_t <= 0.2), "Invalid Kiln Input", ROUND(((mh_t / rb_t) / vk_m) * 100, 2)))Slowed 1.1 RPM Shell Speed Holdup Surge Check (Set Kiln Speed B5 = 1.1 RPM)
Turn down kiln rotation from `1.80` to `1.10` RPM (`B5 = 1.1`) while holding `42.0` t/h raw stone feed: Residence Time E2 jumps to `72.93` min, Live Bed Inventory E4 swells by `+19.85` tonnes to `51.05` t, and Volumetric Fill E5 climbs to `7.57%`—explaining why modern DCS kiln controllers ratio-link weigh-belt feed t/h directly to kiln RPM!
=IF(ISTEXT(E5), E5, ROUND((1.77 * B2 * SQRT(B6)) / (B4 * B3 * B5), 2))Try the example
Use the original values or change them to test your understanding.