# The dimensionless Stokes number

by Robert P. Donovan

Last month's column noted the dramatic difference in range and slowing of two differently sized baseballs even when thrown by the Red Sox's Pedro Martinez at 100 mph.

The particle Reynold's number (Re), a dimensionless ratio of inertial forces/viscous forces, proved to be a useful property for predicting which of these forces dominated the behavior of the two different-size baseballs.

The motion of the particle-size baseball is easier to calculate than that of the regulation baseball, because knowing that the particle Re is less than or on the order of Re =1 justifies using Stokes law as the only significant drag force slowing the particle. Inertial forces can be ignored.

A reverse simplification applies to the thrown regulation baseball. Its much larger particle Reynold's number allows viscous drag to be neglected and only the inertial forces to be considered. Here, however, the changing velocity of the slowing baseball changes the relationship between the drag force and velocity, unlike for the slowing particle for which Stokes law holds throughout its deceleration phase. Consequently, the motion of spheres in the Stokes regime, such as 1-µm baseballs and virtually all cleanroom particles, can be described fairly simply by the recognition of some basic properties and relationships that follow from Stokes law.

For example, for particle motion in which particle Re

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It describes the time required for a particle to reach a new steady state when changing from one set of flow conditions to another. For example, t is the time required for a particle inserted at rest into a flowing air stream of velocity, V, to reach a velocity equal to 0.63V. Or, similarly, the time required for the 1-µm baseball to lose 63 percent of its initial velocity after being pitched into still air. The time 3t is the time to reach 95 percent of the final steady state. For all practical purposes 3t can be taken to be 100 percent of the final value.

From equation (1), the relaxation time of a unit density 1-µm sphere in air at standard conditions is 3.1 3 10-6s. Thus, the 1-µm baseball stops about 10 µs after Pedro releases it.

The distance traveled by a Stokes particle injected into still air with a velocity, V, is given by the simple expression:

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where S is called the stopping distance.

The 1-µm baseball moving at 100 mph (4470 cm/s) when it leaves Pedro's hand travels about 0.01 cm before stopping.

Particle relaxation time and stopping distance have importance in understanding when particle inertial impaction is significant in a cleanroom. Inertial impaction refers to particle capture from an aerosol stream by stationary surfaces oriented perpendicular to the flow streamlines.

The streamlines of the airflow separate upstream of the blocking surface and flow around it. Particles entrained in the airflow try to follow these streamlines. Massive particles, like the thrown regulation baseball, are unable to follow the streamlines as they pass around the blocking surface. Their inertia causes them to depart from the streamlines and instead collide with the blocking object. This action is called inertial impaction.

The question then arises, “what mass or size of particle is likely to be captured by what surface at what air velocity because of inertial impaction?” The dimensionless Stokes number (Stk) provides an answer. Stk values less than 1 imply insignificant particle capture by inertial impaction. This quantity helps to understand the behavior of particles entrained in cleanroom airflows and process equipment and provides guidance on optimizing cleanroom and process equipment designs and configurations.

Robert P. Donovan is a process engineer assigned to the Sandia National Laboratories as a contract employee by L & M Technologies Inc., Albuquerque, NM. His Sandia project work is developing technology for recycling spent rinse waters from semiconductor wet benches.

References

• Hinds, W. C., Aerosol Technology, John Wiley & Sons, (1982)