Settling velocity of silica particle 0.05 cm at 10°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

$d=0.5$ mm $=0.05$ cm, $S_s=2.65$, $\nu=0.0091$ cm²/s (given: 0.91 centistokes), $g=981$ cm/s².

Step 1: Stokes' law (trial)

$$v_s=\frac{g(S_s-1)d^2}{18\nu}=\frac{981\times1.65\times0.05^2}{18\times0.0091}=24.705\ \text{cm/s}$$ $$Re=\frac{v_s d}{\nu}=\frac{24.705\times0.05}{0.0091}=135.74$$

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 $v_s=9.40$ cm/s, $Re=51.6$, $C_d=1.22$.

Answer: settling velocity $v_s\approx9.40$ cm/s ($94.0$ mm/s), with $Re=51.6$. (Stokes' law would overestimate it as 24.70 cm/s.)