If a particle is allowed to settle in a fluid under gravity, its velocity will increase until the accelerating force, gravity force, is exactly balanced by the resistance forces, buoyancy and drag. The video below shows the derivation for this force balance on a spherical particle to derive the drag coefficient.
The drag coefficient variation with particle Reynolds number has been measured experimentally and a large number of equations have been proposed to allow calculation of the drag coefficient from this Reynolds number. However, a useful set of equations for ease of calculation can be given by, $$C_D=\begin{cases} \frac{24}{\text{Re}_p} & \text{Re}_p < 0.3 \\ \frac{24}{\text{Re}_p}(1+0.15\text{Re}_p^{0.687}) & 0.3 \le\text{Re}_p < 500 \\ 0.44 & 500 \le\text{Re}_p < 4\times 10^5 \\ 0.19 & \text{Re}_p\ge 4\times 10^5 \end{cases}$$ As particles fall in different flow regimes the key parameters have differing effects. The graph below is always linear due to the constant terminal velocity, but the $x$-axis timescale changes based on the particle and fluid properties.