Cyclonic Separation - Alternative Steady State Analysis

Alternative Steady State Analysis

Assume we have a particle of radius and density moving with a parcel of fluid of viscosity and density . The particle and the fluid are moving along a curved trajectory with tangential velocity with a radius of curvature of .

If we view the particle in a frame of reference moving with the fluid, we can describe the behavior of the particle by invoking the imaginary, inertial centrifugal force acting as a form of gravity directed outward, away from the axis of rotation. The magnitude of the centrifugal force will be given by

.

where is the mass of the particle.

If we ignore the universal downward force of gravity and viscous drag between the particle and the fluid parallel to the velocity, there are two other forces acting on the particle - radial viscous drag and buoyancy.

The viscous drag ( ) between the particle and the fluid resulting from radial movement of the particle through the fluid is given by

where is the radial drift velocity of the particle through the fluid and the sign reflects the opposition of the force to the motion.

The buoyancy force exerted on the particle by the fluid is given by

where is the volume of the particle

If we assign upward (toward the center of rotation) as the positive radial direction (+) in our frame of reference, then will be pointed in the negative direction, will be pointed in the positive direction and the direction of will depend on the direction of .

If we assume the system has reached dynamic equilibrium then the sum of the forces is zero

.

After applying the appropriate signs and expanding and explicitly we have

Solving this equation for we have

.

Notice that if the density of the fluid is greater than the density of the particle, the motion is (+), toward the center of rotation and if the particle is denser than the fluid, the motion is (-), away from the center.

Expressing the motion in terms of angular velocity we have

Substituting into the equation above yields

.

In this analysis, is the drift velocity at which dynamic equilibrium is attained - the drag friction generated by the movement of the particle through the fluid balances the centrifugal force of the rotation and the particle has no radial acceleration, traveling at a constant velocity. In the extreme case where (a fluid with no viscosity) the equilibrium drift velocity is undefined – the particle can accelerate without ever reaching equilibrium. In the opposite extreme, ∞, the equilibrium drift velocity is 0, there is no outward radial movement and the particle is frozen in the fluid

In non-equilibrium conditions, the general case equation F=ma must be solved

The presence of both and makes this a differential equation and complicates the solution. Note that if the densities of the particle and fluid are equal, the solution is and cyclonic separation is not possible.

In a cyclone particle separator, the design objective is to control the system geometry and the operating parameters so that the drift velocity will move the particle out of the carrier fluid, before the fluid exits via the outlet tube. In most cases, the steady state solution is used as guidance in designing a separator, but the actual performance must be evaluated and modified empirically.