As mentioned in the previous question, a force balance is made for the rising sphere. The buoyancy and weight forces are relatively simple to determine. The drag force, however, varies with the velocity. Under conditions of very slow motion of the sphere, the drag force is inversely proportional to the velocity. Under other conditions, the drag force is a constant.It is customary to express drag force in terms of a drag coefficient for the sphere, and to express velocity in terms of Reynolds number Re = VD/µOne familiar theory used to determine the drag force on a submerged oil particle in water is Stokes' Law. Stokes Law can be derived from purely theoretical considerations and applies when the Reynolds number of the moving sphere is less than or equal to 1. (Actually, Stokes' Law relates the drag on a sphere to its velocity as the fluid flows past a stationary sphere.)When a sphere begins moving through a fluid, it accelerates and eventually reaches a constant velocity, called the terminal velocity. Combining Stokes' Law with the preceding force balance equation, the terminal velocity of a sphere rising in a quiescent liquid can be determined using the following equation(4):whereV = The rise velocity of a sphereg = The acceleration of gravityµ = The dynamic viscosity of the carrying liquid (water)w = The density of the carrying liquid (water)o = The density of the oil sphereD = The diameter of the oil particleBecause this equation for terminal velocity was derived for a sphere rising in a quiescent (not in motion) liquid, it is more suitable for decanters. In fact, it is only valid when the Reynolds number is less than 1.(5)Note that there are two Reynolds numbers associated with flow through a separator in which there are oil bubbles suspended in the water (the carrying fluid). One Reynolds number applies to the flow of the mixture as it moves through the separator. The other applies to motion of the oil bubbles as they rise upward through the water. In fact, each oil bubble will, in turn, have a separate Reynolds number. In a well-designed separator, all the oil bubbles will move upward, and droplet coalescence occurs. Applying Stokes' Law to this moving mass of oil, in an effort to model separation, is not possible. Thus, due to the water and oil velocities and motion within the separator, Stokes Law does not apply to the oil bubbles. While Stokes' Law cannot be used to calculate terminal velocity in this case, the overall effect of the parameters (weight, buoyancy and drag) on bubble rise still applies. The only difference is in how the drag force is determined.Stokes' Law can be applied successfully for determining the terminal velocity of a single sphere moving in another fluid. Stokes' Law can also be used to predict trends such as how terminal velocity is affected by other parameters.