Civil, Structural & Geotechnical Engineering
Ackers-White (1973) Sediment Transport Formula: Mobility Number F_gr & Concentration
Calculate Peter Ackers and W. Rodney White’s (1973, HR Wallingford / ASCE) Transition-Exponent Sediment Mobility Number F_gr and Mass Flux Concentration X_ppm across fine transitional sand to coarse gravel.
Unify fine suspended sand (n = 1) and coarse bedload gravel (n = 0) via Ackers & White’s transition exponent n!
Developed at Hydraulics Research Station Wallingford from > 1,000 flume and river datasets by Peter Ackers and W. Rodney White (1973, ASCE Journal of the Hydraulics Division; updated HR Wallingford 1990), the Ackers-White Total Load Method bridges fine suspended load (governed by total bed shear velocity u_* = SQRT(g * d * S)) and coarse bedload (governed by mean channel velocity V and grain-roughness log law) using a single grain-size Transition Exponent n in [0, 1]!
From Dimensionless Grain Size D_gr = d_35 * ((((s - 1) * g) / (nu^2))^(1/3)): for coarse grains D_gr > 60 (d_35 > ~2.4 mm), n = 0, A_gr = 0.17, m = 1.50 (1.78 updated), C_gr = 0.025; for transitional sand 1 <= D_gr <= 60, n = 1 - 0.56 * LOG10(D_gr), A_gr = (0.23 / SQRT(D_gr)) + 0.14! Ackers & White’s General Sediment Mobility Number is F_gr = ((u_*^n) / SQRT(g * d_35 * (s - 1))) * ((V / (SQRT(32) * LOG10(10 * Depth / d_35)))^(1 - n)), generating Dimensionless Transport G_gr = C_gr * ((MAX(0, (F_gr / A_gr) - 1))^m) and Mass Concentration X_ppm = 1,000,000 * (G_gr * s * d_35) / (Depth * ((u_* / V)^n))!
| Ackers-White Coarse-Gravel Case (D_gr > 60 -> n = 0, A_gr = 0.17, m = 1.5, C_gr = 0.025) | Formula at n = 0 | Evaluated Value |
|---|---|---|
| Grain-Log Velocity Factor | V / (SQRT(32) * LOG10(10 * Depth / d_35)) | 0.1397 m/s (at V = 2.65 m/s, Depth = 1.8 m, d_35 = 0.008 m) |
| Coarse Sediment Mobility Number F_gr | Grain_Vel_Factor / SQRT((s - 1) * g * d_35) | 0.3884 (> A_gr = 0.17) |
| Dimensionless Transport G_gr | 0.025 * ((F_gr / 0.17) - 1)^1.5 | 0.03641 |
| Mass Sediment Concentration X (ppm by weight) | 1e6 * G_gr * s * d_35 / Depth | 428.8 ppm (mg/kg) |
Ackers-White Sediment Mobility Number F_gr (General n in [0, 1])
Evaluates ((u_*^n) / SQRT((s - 1) * g * d_35)) * ((V / (SQRT(32) * LOG10(10 * Depth / d_35)))^(1 - n)) across Mean Velocity V (B2 m/s), Flow Depth (B3 m), Shear Velocity u_* (B4 m/s), Grain Size d_35 (B5 m), Specific Gravity s (B6 > 1), and Transition Exponent n (B7 in [0, 1]).
=IF(OR(B2 <= 0, B3 <= B5, B4 <= 0, B5 <= 0, B6 <= 1, B7 < 0, B7 > 1), "Invalid Channel Velocity, Depth, u_*, d_35, s, or Exponent n", ROUND(((B4 ^ B7) / SQRT((B6 - 1) * 9.80665 * B5)) * ((B2 / (SQRT(32) * LOG10((10 * B3) / B5))) ^ (1 - B7)), 4))Ackers-White Dimensionless Sediment Transport Parameter G_gr
Computes C_gr * (MAX(0, (F_gr / A_gr) - 1)^m) given critical threshold A_gr (B8), exponent m (B9), and coefficient C_gr (B10).
=IF(OR(B2 <= 0, B3 <= B5, B4 <= 0, B5 <= 0, B6 <= 1, B7 < 0, B7 > 1, B8 <= 0, B9 <= 0, B10 <= 0), "Invalid Ackers-White Inputs", ROUND(B10 * (MAX(0, ((((B4 ^ B7) / SQRT((B6 - 1) * 9.80665 * B5)) * ((B2 / (SQRT(32) * LOG10((10 * B3) / B5))) ^ (1 - B7))) / B8) - 1) ^ B9), 5))Try the example
Use the original values or change them to test your understanding.