Erosion & Sediment formula solvers

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API RP 14E Erosional Velocity

V_e = \frac{C}{\sqrt{\rho_m}}

Erosion & SedimentThe velocity limit every offshore and onshore piping specification quotes, from API Recommended Practice 14E: the mixture density under a square root, divided into an empirical constant C. It is the first number written on a flowline sizing sheet — and it is a rule of thumb with no published experimental basis, which is why C is yours to choose here rather than ours to assume.

Bed Shear Stress in an Open Channel

\tau_0 = \rho \, g \, R \, S

Erosion & SedimentThe force the flow drags along the bed, per unit of bed area, from nothing but a weight balance: the downslope component of the water's own weight in a reach has nowhere to go except into the boundary. Hydraulic radius times energy slope times unit weight, and it is the number every sediment threshold is compared against.

Critical Shear Stress for Sediment Entrainment

\tau_c = \theta_c \, (s - 1) \, \rho \, g \, d

Erosion & SedimentThe Shields relation turned around and used the way a designer uses it: pick a critical Shields number, multiply by the submerged weight of a grain, and you have the shear stress at which the bed begins to move. Below it a channel is stable; above it, it is not. This is the tractive force method behind stable channel design.

Finnie Erosion of a Ductile Metal

W = \frac{c \, m \, V^{2}}{8 \, p} \, f(\alpha)

Erosion & SedimentIain Finnie's 1960 model in Wear treats an impacting hard particle as a tiny cutting tool: it ploughs a groove out of a ductile surface, and the volume it removes goes as the square of the impact velocity, inversely with the target's flow stress, and with a strongly angle-dependent shape factor that peaks at a SHALLOW angle. That shallow-angle peak is the signature of ductile erosion, and it is the opposite of how brittle materials behave.

Local Scour Depth at a Bridge Pier (HEC-18)

y_s = 2.0 \, K_1 K_2 K_3 K_4 \, y_1 \left( \frac{a}{y_1} \right)^{0.65} Fr_1^{\,0.43}

Erosion & SedimentThe equation the US Federal Highway Administration recommends in HEC-18, "Evaluating Scour at Bridges": the depth of the hole a pier digs for itself in an erodible bed. Scour is the leading cause of bridge failure in North America, and this is the number that decides how deep the foundation goes.

Minimum Pipe Bore at the Erosional Limit

d = \sqrt{\frac{4Q}{\pi V_e}}, \qquad V_e = \frac{C}{\sqrt{\rho_m}}

Erosion & SedimentThe line-sizing form of API RP 14E: for a known flow and mixture density, the smallest inside diameter that still keeps the velocity at or under the erosional limit. Round up to the next available schedule and the pipe is compliant; round down and it is not.

Rouse Suspension Number

P = \frac{w_s}{\kappa \, u_*}

Erosion & SedimentHunter Rouse's 1937 ratio: how fast a grain falls, against how strongly the turbulence lifts it. Below about 0.8 the grain travels in full suspension; around 1.2 to 2.5 it is suspended near the bed; above about 2.5 it stays on the bed and rolls, hops and slides. One number decides which of two entirely different transport regimes you are in.

Shear Velocity (Friction Velocity u*)

u_* = \sqrt{\frac{\tau_0}{\rho}}

Erosion & SedimentA shear stress divided by a density has the units of a velocity squared, and the square root of it is the shear velocity — not a velocity anything actually travels at, but the natural velocity scale of a turbulent boundary layer. The log law, the Rouse number and every roughness correlation are written in terms of it.

Shields Parameter (Dimensionless Shear Stress)

\theta = \frac{\tau}{(\rho_s - \rho) \, g \, d}

Erosion & SedimentAlbert Shields' 1936 dissertation reduced the whole question of whether a grain moves to one ratio: the shear stress the flow applies to the bed, divided by the submerged weight of a grain spread over its own area. Above about 0.03 to 0.06 the bed is in motion; below it, mostly not. It is the single most used number in sediment transport.

Universal Soil Loss Equation (USLE)

A = R \, K \, L \, S \, C \, P

Erosion & SedimentSix factors multiplied together give the long-term average annual soil loss from sheet and rill erosion on a defined slope: rainfall erosivity, soil erodibility, slope length, slope steepness, cover and management, and support practice. Wischmeier and Smith assembled it from more than ten thousand plot-years of measured runoff, and it remains the backbone of conservation planning on every continent.

Wall Thinning Rate and Remaining Life from Metal Loss

L = \frac{w}{P}, \qquad P = \frac{\Delta m}{\rho_M \, A \, \Delta t}

Erosion & SedimentA weighed loss of metal, spread over the area it came off and the time it took, is a thinning rate; the wall you can afford to lose divided by that rate is the remaining life. This is the arithmetic that turns an inspection result — a weight on a balance, or two ultrasonic readings — into a date on a maintenance plan.