Engineering & Science
Richardson-Zaki Hindered Settling Velocity & Kynch Gravity Thickener Solids Flux
Calculate concentrated slurry hindered settling velocity v_h = v_t * ((1 - C_v) ^ n) and batch gravity-thickener solids flux G_s = C_v * v_h for mineral tailings, wastewater clarifiers, and fluidized beds.
Why concentrated slurries settle 7x slower than a single isolated Stokes particle!
Drop a single sand grain into water and it settles at isolated terminal velocity v_t (say 12.5 mm/s). Now pour a 35% solids-by-volume mining slurry into a gravity thickener: every falling particle displaces upward return water, elevates effective suspension viscosity, and crowds neighboring wakes! John Richardson and W. N. Zaki (1954) proved that hindered zone-settling velocity obeys the remarkably clean power law: v_h = v_t * ((1 - C_v) ^ n), where C_v is volumetric solids fraction (0 <= C_v < 1) and exponent n = 4.65 in the creeping laminar Stokes regime (decreasing toward n = 2.39 in turbulent Newton settling)!
In wastewater clarifier and mineral thickener sizing (Kynch theory), engineers multiply hindered velocity by solids concentration to obtain volumetric batch solids flux G_s = C_v * v_h = C_v * v_t * ((1 - C_v) ^ n): at C_v = 0.20 (n = 4.65, v_t = 12.5 mm/s), settling slows to v_h = 4.429 mm/s (flux 0.886 mm/s); at C_v = 0.35, v_h plunges to 1.686 mm/s (flux 0.590 mm/s)—identifying the bottleneck thickener layer!
| Slurry Concentration C_v (vt = 12.5 mm/s, n = 4.65) | Voidage Fraction (1 - Cv) | Hindered Speed vh (mm/s) | Batch Solids Flux Gs = Cv * vh |
|---|---|---|---|
| Infinite Dilution Limit (Cv = 0.00) | 1.00 | 12.500 mm/s | 0.000 mm/s |
| Moderately Concentrated (Cv = 0.20) | 0.80 (0.80^4.65 = 0.354) | 4.429 mm/s | 0.886 mm/s (Near peak flux) |
| Dense Thickener Zone (Cv = 0.35) | 0.65 (0.65^4.65 = 0.135) | 1.686 mm/s | 0.590 mm/s (Flux bottleneck!) |
Richardson-Zaki Hindered Zone-Settling Velocity v_h = v_t * (1 - C_v)^n
Validates 0 <= C_v < 1 and positive v_t, n, then evaluates v_t * ((1 - C_v) ^ n).
=IF(OR(B2 <= 0, B3 < 0, B3 >= 1, B4 <= 0), "Invalid Slurry Inputs", ROUND(B2 * ((1 - B3) ^ B4), 3))Volumetric Batch Gravity-Thickener Solids Flux G_s = C_v * v_h (mm/s)
Multiplies solids volume fraction B3 by hindered settling speed v_h for Kynch flux-curve thickener design.
=IF(OR(B2 <= 0, B3 < 0, B3 >= 1, B4 <= 0), "Invalid Slurry Inputs", ROUND(B3 * B2 * ((1 - B3) ^ B4), 3))Try the example
Use the original values or change them to test your understanding.