Settling velocity of discrete particle 0.17 mm at 21°C

Settling velocity of a discrete spherical particle is first tried with Stokes' law, and the Reynolds number is checked. If $Re>1$ the transition formula must be used.

Data

$d=0.17$ mm $=0.017$ cm, $S_s=2.65$, $\nu=0.00978$ cm²/s (at 21 °C, about 0.978 centistokes), $g=981$ cm/s².

Step 1: Stokes' law (trial)

$$v_s=\frac{g(S_s-1)d^2}{18\nu}=\frac{981\times1.65\times0.017^2}{18\times0.00978}=2.657\ \text{cm/s}$$ $$Re=\frac{v_s d}{\nu}=\frac{2.657\times0.017}{0.00978}=4.62$$

Since $Re>1$, Stokes' law is not valid and the flow is in the transition zone ($1<Re<1000$).

Step 2: Transition zone (Fair and Geyer)

$$C_d=\frac{24}{Re}+\frac{3}{\sqrt{Re}}+0.34,\qquad v_s=\sqrt{\frac{4g(S_s-1)d}{3C_d}}$$

Trial and error:

The values converge to $v_s=2.06$ cm/s, $Re=3.6$, $C_d=8.61$.

Answer: settling velocity $v_s\approx2.06$ cm/s ($20.6$ mm/s), with $Re=3.6$. (Stokes' law would overestimate it as 2.66 cm/s.)