# The Stokes number (Stk) and inertial impaction

by Robert P. Donovan

Particle properties known as the “relaxation time” and the “stopping distance” were defined in last month's column. These definitions will be used here in defining a dimensionless number called the Stokes number (Stk).

Stk is the ratio of the particle stopping distance (S) in a given flow divided by a characteristic dimension (C) of the object perpendicular to the flow:1

Stk = S/C

Consider the example of vertical laminar flow (VLF) in a cleanroom. Assume that the air flow velocity from the ceiling to the floor is 100 ft/min (~ 50 cm/s) and that the density of a 1-µm particle is 2 g/cm.3 The particle Reynold's number Re = pair Vdp/n—(see the October column) is 0.03, well within the Stokes regime so that the equations presented in the November column apply, as does equation (1) above. (Remember none of these equations are valid for Re much larger than 1.) The relaxation time (eq. 1, November column) and stopping distance (eq. 2, November column) of these 1-µm particles in a VLF cleanroom are 6.2 x 10-6 s and 0.0003 cm, respectively. Using the rule of thumb that says that Stk less than 1 means insignificant particle capture by inertia, an obstruction would have to be about 3-µm in width in order for inertial impaction to be significant. Impact with a bench top of typical width (C ~ 50 cm) is insignificant. Even a perforated bench top with many circular holes separated by 1mm walls will not collect 1 µm particles because of inertial impaction. And indeed inertial impaction of particles entrained in the ambient air at the flow velocities typical of VLF cleanrooms is seldom of consequence.

At the low operating pressures of some semiconductor processing equipment, however, this conclusion does not necessarily hold.2

Back to baseball at Fenway Park. Let's assume there is a 40 mph wind blowing in from centerfield, exactly aligned with the centerline between the pitcher and the catcher. Pedro Martinez's fastball now gets a convective assist from the wind and reaches the batter at a higher speed than on a windless day. So does the 1 µm baseball, which now reaches homeplate at 40 mph by being entrained in the convective airflow approximately 10-20 µs after Pedro releases it. Both the swinging bat and the stationary catcher's mitt represent obstructions that the air streamlines easily flow around but not the massive regulation baseballs, which collide with one or the other. Not so the 1 µm baseball — its Reynold's number with respect to the stationary catcher's mitt, calculated at

V = 40 mph (~1790 cm/s), is now about 1.2, which makes the Stokes assumptions of the above equations marginal but still usable. As noted earlier, the relaxation time of a 1 µm baseball is about 3 µs. (This number assumes that the density of a baseball is about 1 g/cm3, the nominal density of water. It's probably a little denser since it seems to me that baseballs don't float. None of the conclusions that follow, however, depend on a more precise estimate of a baseball's density.) The stopping distance (eq. 2, November column) under the postulated windy conditions of Fenway Park becomes ~ 0.006 cm, which means that any object larger than about 60 µm (such as the catcher's mitt or the swinging bat) has a low probability of catching or being impacted by this baseball. Assuming the batter could see this ball, he couldn't hit it no matter how well he aligns his swing with the trajectory of a ball this size. But even if the winds enabled Pedro to get the 1 µm baseball over the plate and the helpless batter couldn't touch it, neither could the catcher—like trying to catch air.

None of the above analyses apply to the regulation baseball—its particle Re is far removed from the Stokes regime; using the relaxation time and stopping distance equations to predict their motion is totally inappropriate, so that no meaningful Stokes number can be calculated for the regulation baseball from equation (1).

References

1. Hinds, W. C., Aerosol Technology, John Wiley & Sons, (1982)
2. Rader, D. J. and A. S. Geller, “Transport and Deposition of Aerosol Particles”, Chap. 6 in Contamination Free Manufacturing for Semiconductors and Other Precision Products, Robert P. Donovan, Editor, Marcel Dekker, Inc (to be published).

Robert P. Donovan is a process engineer assigned to the Sandia National Laboratories as a contract employee by L & M Technologies Inc., Albuquerque, NM.