# a)Equation of motion

Gravity separation is conceptually simple. The droplets of any liquid in a vapor flow are acted on by three forces: gravity, buoyancy, and drag. The resultant of these forces causes motion in the direction of the net force. A primary design goal is to size the separator such that the drag and buoyancy forces succumb to the gravity force causing the droplet to disengage, i.e. separate, from the vapor flow. The force balance on a typical liquid droplet can be established by application of Newton’s Law: where the forces, Fi, and acceleration, a, are functions of time, t, and md is the mass of the droplet. The magnitudes of the gravity, buoyancy and drag forces,

respectively, are defined as follows:

FG ρL Vd  g                                  (1)

FB ρv  Vd  g                               (2)

FD ρv  U2  C Ad/2                   (3)

The gravity force is always directed downward, the buoyancy force is opposite the gravity force, and the drag force is opposite the direction of droplet velocity. The droplet Reynolds number is defined as the ratio of inertia and viscous forces and the characteristic length is the droplet diameter. The droplet Reynolds number is defined as follows:

$Re_d=\frac{\rho_vUD_d}{\mu}$

where rhov and μv are the vapor density and absolute viscosity, respectively, and U is the velocity of the vapor past the droplet relative to the droplet’s velocity. The drag coefficient, CD, for a smooth sphere can be numerically estimated using the following (Bird, 1960)

$C_D=\frac{24}{Re_d}$      (Re_d<1)

$C_D=\frac{18.5}{Re_d}$   (1<Re_d<500)

$C_D\sim 0.44$                      ($Re_d<2e10^5$)

While both of the estimates are equally valid, Equation (8) will be used for the present model development because it is defined over the entire Reynolds number range of interest.

The gravity force Fg acts downward and thus incites a velocity in that di- rection while the buoyancy force Fb and the drag force acts FD on the opposite direction of movement and thus upward. This can be expressed as follow :

$\sum F=m_pa=F_g-F_b-F_D$

$\sum F=m_pa=m_pg-\rho_fV_pg-\frac{1}{2}C_DA_p\rho_f(u-v)^2$