Page 136 - Petroleum and Gas Field Processing

balances F g ; thereafter, the oil droplet continues to rise at a constant
velocity known as the settling or terminal velocity. Similarly, a water
droplet, being higher in density than the oil, tends to move vertically
downward under the gravitational or buoyant force, F g . The continuous
phase (oil), on the other hand, exerts a drag force, F d , on the water droplet
in the opposite direction. The water droplet will accelerate until the
frictional resistance of the fluid drag force, F d , approaches and balances
F g ; thereafter, the water droplet continues to rise at a constant velocity
known as the settling or terminal velocity. Upward settling of oil droplets
in water and downward settling of water droplets in oil follow Stokes’ law
and the terminal settling velocity can be obtained as follows. The drag
force, F d , is proportional to the droplet surface area perpendicular to the
direction of flow, and its kinetic energy per unit volume; Hence,
2 c u 2
F d ¼ C d d ð2Þ
4 2g
whereas F g is given by
3
F g ¼ d ð Þ ð3Þ
6
where d is the diameter of the droplet (ft), u is the settling velocity of the
3
droplet (ft/s), c is the density of the continuous phase (lb/ft ), g is
gravitational acceleration (ft/s), and C d is the drag coefficient. For a low
Reynolds number, Re, flow, the drag coefficient is given by
24 24 g
0
C d ¼ ¼ ð4Þ
Re du
2
0
where m is the viscosity of the continuous phase (lb-s/ft ).
Substituting for C d from Eq. (4) into Eq. (2) yields
0
F d ¼ 3 du ð5Þ
The settling terminal velocity, u, is reached when F d ¼ F g . Therefore,
equating Eqs. (3) and (5) and solving for u, the droplet settling velocity, we
obtain

2
ð Þd
u ¼
18 0
The typical units used for droplet diameter and viscosity are the
micrometers and centipoise, respectively. Letting m be the viscosity in

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