b)Terminal Velocity

Terminal Velocity Most separators will have a given dimension stated by the terminal or maximum velocity achieved by particles. So in that manner it is the first expression to be obtained. Furthermore, for the case of maximum velocity , there will be no acceleration( a=0). Hence the equation becomes :

$m_pg-\rho_fV_pg-\frac{1}{2}C_DA_P\rho_f(u_{max})^2$

Given the volume and area of a sphere and the density expressed via Vp= corresponding volume of the particles, rp=volumetric radius 

$d=2r_p=\left(\frac{6V_p}{\pi})^\frac{1}{3}\right)$

$A_p=A_{sphere}=4\pi r_{p}^{2}$

$A_p>A_{sphere}(non-sphereical)$

$\rho_p=\frac{m_p}{V_p}$

To consider the shape of the particle, surface area is used. An increase in it may cause an increase in drag. 

$\frac{1}{2}C_D\pi r^2\rho_fu_{max}^2=V_fg(\rho_p-\rho_f)$

$\frac{1}{2}C_D\pi r^2\rho_fu_{max}^2=\frac{4}{3}\pi r^3g(\rho_p-\rho_f)$

$u_{max}^{2}=\frac{8rg(\rho_p-\rho_f)}{3\rho_fC_D}$

Leading to the expression for maximum velocity expressed as 

$u_{max}=\sqrt{\frac{8rg(\rho_p-\rho_f)}{3\mu_f}}$

Which can also be expressed as : But due to the drag coeddiecent used ( based on stokes ), only applicable at low Re 

$u_{max}=\frac{2r^2g(\rho_p-\rho_f)}{9\mu_f}$

Terminal velocity using the Stokes correction factor : Another way to express the terminal velocity is by the use of a correction factor . This cor- rection factor is a coefficient by which the flow in question would be assimilated to the one assumed by stokes following 

$F_D=-3\pi d\mu_fu_f=-\frac{1}{8}\pi d^2\rho_f\vert u \vert C_D$

Thus yielding 

$u_{max}\frac{2}{9}\frac{r^2(\rho_p-\rho_f)}{\mu_ff}$

And would be applicable for every interval in which a set of correction f exist being calculated experimentally for every case of flow and factors involved.