Civil, Structural & Geotechnical Engineering
Engelund-Hansen (1967) Total Alluvial Sand-Bed Sediment Transport Formula
Calculate total bed-material sediment load (bedload + suspended sand across dunes and upper-plane beds) via Frank Engelund and Eggert Hansen’s (1967, DTU) energy-balance law f_EH * Phi = 0.1 * theta^2.5 (Phi = 0.05 * (V^2 / (g * R_h * S)) * theta^2.5).
Account automatically for dune form drag via Engelund & Hansen’s energy-work law!
In sand-bed rivers (median grain size d_50 ~= 0.15 to 2.0 mm), ripples and dunes dissipate fluid energy without moving grains, AND sand moves simultaneously as bedload and suspended load! Equating stream-power work against potential energy gained lifting grains over dune crests, Danish hydraulic engineers Frank Engelund and Eggert Hansen (1967, Technical University of Denmark Monograph) derived the remarkably accurate threshold-free Total Bed-Material Load Law:
Defining hydraulic resistance factor f_EH = (2 * g * R_h * S) / (V^2) and total Shields stress theta = (R_h * S) / ((s - 1) * d_50), Engelund & Hansen proved f_EH * Phi = 0.1 * (theta^2.5) -> Dimensionless Total Transport Phi = 0.05 * ((V^2) / (g * R_h * S)) * (theta^2.5), giving Dry Mass Unit Sand Load g_s (kg / (m · s)) = 1000 * s * Phi * SQRT((s - 1) * g * (d_50^3))!
| Alluvial Sand River Variable (V = 1.35 m/s, R_h = 2.20 m, S = 0.0004, d_50 = 0.0004 m) | Formula | Evaluated Value |
|---|---|---|
| Total Shields Mobility Parameter theta | (R_h * S) / ((s - 1) * d_50) | 1.3333 |
| Darcy-Type Channel Friction Factor f_EH | (2 * g * R_h * S) / V^2 | 0.00947 |
| Engelund-Hansen Dimensionless Intensity Phi | 0.1 * theta^2.5 / f_EH | 21.676 |
| Dry Mass Sand Load per Meter Width g_s | rho_s * Phi * SQRT((s-1)*g*d_50^3) | 1.848 kg/(m·s) |
Engelund-Hansen (1967) Dimensionless Total Sand-Load Intensity Phi
Evaluates 0.05 * (V^2 / (9.80665 * R_h * S)) * (((R_h * S) / ((s - 1) * d_50))^2.5) across Mean Velocity V (B2 m/s), Hydraulic Radius R_h (B3 m), Slope S (B4), Median Sand Diameter d_50 (B5 m), and Specific Gravity s (B6 > 1).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 1), "Requires Positive V, R_h, Slope S, Grain d_50, and SG s > 1", ROUND(0.05 * ((B2 ^ 2) / (9.80665 * B3 * B4)) * (((B3 * B4) / ((B6 - 1) * B5)) ^ 2.5), 3))Dry Mass Unit Total Sand Transport Rate g_s (kg / (m · s))
Multiplies Engelund-Hansen Phi by 1000 * s * SQRT((s - 1) * 9.80665 * d_50^3).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 1), "Requires Positive V, R_h, Slope S, Grain d_50, and SG s > 1", ROUND(1000 * B6 * (0.05 * ((B2 ^ 2) / (9.80665 * B3 * B4)) * (((B3 * B4) / ((B6 - 1) * B5)) ^ 2.5)) * SQRT((B6 - 1) * 9.80665 * (B5 ^ 3)), 3))Try the example
Use the original values or change them to test your understanding.