Finds the terminal settling velocity of a small spherical particle by balancing weight, buoyancy and drag, iterating on the drag coefficient.
Inputs
Particle and fluid properties. Defaults are a 10 micrometer water droplet in air.
Particle diameter in micrometers.
1000 for water droplets, about 2600 for sand.
Air at 15 C is about 1.225.
Air at 15 C is about 1.81e-5.
Results
Terminal velocity
-
Particle Reynolds number
-
Drag coefficient CD
-
Regime
-
How it works
At terminal velocity the drag on the particle exactly balances its net weight (gravity minus buoyancy). The difficulty is that the drag coefficient itself depends on the velocity, so the calculator starts from the Stokes solution and iterates: each pass recomputes the Reynolds number, updates the drag coefficient with the Schiller-Naumann correlation, and corrects the velocity until it stops changing.
Start from the Stokes terminal velocity, which is exact only for Re < 1.
Compute the particle Reynolds number from the current velocity guess.
Evaluate the drag coefficient with the Schiller-Naumann correlation.
Correct the velocity through the drag function f = CD Re / 24, under-relax the update for stability, and repeat until the relative change drops below 1e-6.
Stokes terminal velocity (starting point)
U_stokes = g D^2 (ρp - ρf) / (18 μ)
Particle Reynolds number
Rep = ρf U D / μ
Schiller-Naumann drag coefficient
CD = (24/Rep)(1 + 0.15 Rep^0.687) (Rep < 1000) ; CD = 0.44 (Rep ≥ 1000)
Velocity correction
U_new = U_stokes / f ; f = CD Rep / 24
D
Particle diameter, m
ρp, ρf
Particle and fluid densities, kg/m3
μ
Fluid dynamic viscosity, Pa.s
g
Gravity, 9.81 m/s2
U
Terminal settling velocity, m/s
Rep
Particle Reynolds number
CD
Drag coefficient
f
Drag function, ratio of actual drag to Stokes drag
References
Schiller, L. and Naumann, A., "Uber die grundlegenden Berechnungen bei der Schwerkraftaufbereitung", Zeitschrift des Vereines deutscher Ingenieure, 77, 1933.
Clift, R., Grace, J. R. and Weber, M. E., Bubbles, Drops, and Particles, Academic Press, 1978.
