Formulae for terminal settling velocity
• for a perfect sphere, the standard drag curve is given by Clift et al. (1978):
24
3
+ ,
Re 16
24
[1 + 0.1315Re0.82−0.05Ω ],
=
Re
24
=
[1 + 0.1935Re0.6305 ],
Re
log10 Cd = 1.6435 − 1.1242Ω + 0.1558Ω2 ,
Cd =
Re < 0.01
0.01 ≤ Re ≤ 20
20 ≤ Re ≤ 260
260 ≤ Re ≤ 1500
2
3
1500 ≤ Re ≤ 1.2 × 104
= −2.4571 + 2.5558Ω − 0.9295Ω + 0.1049Ω ,
where Ω ≡ log10 Re, Cd ≡ FD /(ρws2 Ap /2), Re ≡ ws d/ν, and Ap = πd2 /4.
• for a perfect sphere, a standard settling velocity curve is given by Dietrich (1982):
log10 (w∗ )sph = −1.2557 + 1.9294∆ − 0.2944∆2 − 0.0518∆3 + 0.0151∆4 ,
∆ < 2.6
where (w∗ )sph = ws /[g(s − 1)ν]1/3 , ∆ ≡ log10 d∗ , and d∗ = d/[ν 2 /{g(s − 1)}]1/3 .
• for a non-spherical particle, Dietrich (1982) also proposed a correction factor, Ksf , based on the Corey
shape factor, Csf such that w∗ = (w∗ )sph Ksf (Csf )
log10 Ksf
where again ∆ ≡ log10 d∗ , and the Corey shape factor, Csf = c/ (ab), is restricted to values larger
than 0.2.
– Dietrich assumes d is the nominal diameter (how is this defined? and measured?)
– Dietrich also provides a correction factor for roundness effects.
• Various regression equations have been developed to relate density (ρ, kg/m3 ), dynamic viscosity (µ,
Ns/m2 ) and kinematic viscosity (ν, m2 /s) of water and to its temperature (T , ◦ C). An example is
ρ=
999.84 + 18.225T − 7.922 × 10−3 T 2 − 55.448 × 10−6 T 3 + 0.1498 × 10−6 T 4 − 3.933 × 10−10 T 5
1 + 18.16 × 10−3 T
247.8
References
1. Clift, R., Grace, J. R., and Weber, M. E. (1978). Bubbles, drops, and particles, Academy Press.
2. Dietrich, W. E. (1982). Settling velocity of natural particles, Water Resources Research, 18(6), 16151626.