Settling velocity of discrete particle 0.14 mm at 25°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

$d=0.14$ mm $=0.014$ cm, $S_s=2.61$, $\nu=0.00893$ cm²/s (at 25 °C, about 0.893 centistokes), $g=981$ cm/s².

Step 1: Stokes' law (trial)

$$v_s=\frac{g(S_s-1)d^2}{18\nu}=\frac{981\times1.61\times0.014^2}{18\times0.00893}=1.926\ \text{cm/s}$$ $$Re=\frac{v_s d}{\nu}=\frac{1.926\times0.014}{0.00893}=3.02$$

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 $v_s=1.57$ cm/s, $Re=2.5$, $C_d=12.04$.

Answer: settling velocity $v_s\approx1.57$ cm/s ($15.7$ mm/s), with $Re=2.5$. (Stokes' law would overestimate it as 1.93 cm/s.)