Page contents
How do I calculate the settling velocity of a particle?
The settling velocity (terminal velocity) is the steady speed reached by a particle falling through a fluid at rest, when its apparent weight is balanced by drag. For a small sphere in laminar flow (particle Reynolds number below 1), Stokes' law gives w = (ρp − ρf) g d² / (18 μ): the velocity grows with the square of the diameter and decreases with viscosity. Beyond that, drag is no longer purely viscous and the drag coefficient CD of the sphere must be used. For a pipe to convey particles without deposits, the fluid velocity must be 1.5 to 2.5 times the settling velocity.
Stokes' law and general formula
- w: settling velocity, in m/s
- ρp, ρf: densities of the particle and of the fluid, in kg/m³ (if ρp < ρf the particle rises: oil droplets, bubbles)
- d, r: diameter and radius of the particle, in m; g = 9.81 m/s² (ω² r in a centrifuge)
- μ (mu): dynamic viscosity of the fluid, in Pa·s (water at 20 °C: 0.001 Pa·s; air: 1.8 × 10⁻⁵ Pa·s)
- Rep: Reynolds number of the particle; CD: drag coefficient of the sphere (24 / Rep in the Stokes regime, 0.44 beyond Rep ≈ 1,000)
Calculated settling velocities
| Particle | Diameter | Fluid | Settling velocity w | Rep | Regime |
|---|---|---|---|---|---|
| Fine silt | 10 µm | Water | 0.09 mm/s (32 cm/h) | 0.001 | Stokes |
| Coarse silt | 50 µm | Water | 2.2 mm/s | 0.11 | Stokes |
| Fine sand | 100 µm | Water | 8 to 9 mm/s | 0.8 | Stokes limit |
| Medium sand | 200 µm | Water | 25 mm/s (Stokes would give 36 mm/s) | 5 | intermediate, CD needed |
| Coarse sand | 1 mm | Water | 0.15 m/s | 155 | intermediate, CD needed |
| Dust | 10 µm | Air | 6 mm/s (22 m/h) | 0.004 | Stokes |
Detailed explanations and calculation in Mecaflux
Sedimentation velocity of a particle is the minimum speed that a flood must have to carry, rather than drop, particles or sediment.. Sedimentation velocity of a particle is given by Stokes' law:where w is the sedimentation rate, ρ is the density ( p and f respectively indicate particle and fluid), g is the acceleration due to gravity, r is the radius of the particle and μ is the dynamic viscosity of the fluid.
Low Sedimentation Rate particle means that the particle has a tendency to "float" in the fluid. A high sedimentation rate indicates that the particle "quickly sank to the bottom of fluid". To increase the sedimentation velocity of a particle, to make it sinking in a fluid, in order to separate particles , we can increase the gravity factor "g" of equation above, using a centrifuge.
It is thus possible to adjust the speed of the centrifuge so that the sedimentation velocity causes separation of different particles, of a heterogeneous fluid (comprising particles of various types). (see gravity / centrifuge)
If the fluid velocity is greater than the deposition rate (sedimentation) the aggregate continues downstream.. But even at the speed of sedimentation in the pipes, deposits can accumulate, agglomerate and create caps ... There are therefore the sedimentation rate, the transport speed. The conveying speed of avoiding aglomeration the particles is 1.5 to 2.5 times higher than the speed of sedimentation.
The interface of calculating Sedimentation Rate in the Tools MECAFLUX
Alert "minimum Sedimentation Rate" can be adjusted in the interface sizing of pipes to detect the maximum diameters, not to exceed, in a project network with fluid countaining particles.
The pipelines transporting particules as vacuuming chips or the rainwater, must be dimensioned according to the conveying speed, 1.5 to 2.5 times greater than the rate of sedimentation in order to prevent particle deposition.
detail of the interface sizing, hydraulic or air ducts.
Frequently asked questions
Up to which particle size is Stokes' law valid?
As long as the particle Reynolds number Rep = ρf w d / μ stays below 1, i.e. about 100 µm for sand in water and 50 to 80 µm for dust in air. Beyond that, Stokes' law overestimates the velocity (36 mm/s instead of 25 mm/s for a 200 µm grain) and the velocity must be computed iteratively with the drag coefficient of the sphere, which itself depends on Rep.
What conveying velocity for a pipe carrying particles?
The fluid velocity must exceed the settling velocity by a factor of 1.5 to 2.5 so that particles do not settle and form plugs, the higher factor applying to horizontal pipes. In practice, wood chip extraction ducts are sized for 20 m/s and sewers carrying sand for at least 0.6 to 1 m/s. Mecaflux Standard lets you set a minimum velocity alert in the duct sizing interface.
How does a centrifuge speed up settling?
By replacing g with the centrifugal acceleration ω² r in Stokes' law. At 3,000 rpm (ω = 314 rad/s) and a 10 cm radius, ω² r = 9,870 m/s², i.e. 1,000 times gravity: a particle that would take an hour to settle under gravity settles in a few seconds. This is how the particles of a heterogeneous fluid are separated according to their size or density.
How do I size my particle conveying pipe?
Mecaflux Standard computes the settling velocity of your particles in your fluid and lets you set a minimum velocity alert in the duct sizing, to choose the diameter that avoids deposits. For a chip, dust or sand-laden water extraction network, Mecaflux carries out the study of your project (network and fan or pump) and delivers the calculation report, or offers training on the software applied to your own project.