Drag on non-spherical, orthotropic aerosol particles

J. Aerosol Sci., Vol. 18, No. I, pp. 87-97, 1987.

00214502/87 $3.00+0.00 © 1987 Pergamon Journals Ltd.

Printed in Great Britain.

DRAG

ON NON-SPHERICAL, ORTHOTROPIC PARTICLES

AEROSOL

DAVID L. JOHNSON, DAVID LEITH* and PARKER C. REIST* 10th Medical Laboratory, Attn: AEMML-PM-0IH, APe New York 09180, U.S.A., and *Department of Environmental Sciencesand Engineering, Universityof North Carolina at Chapel Hill, Chapel Hill, NC 27514, U.S.A. (Received 26 November 1985;in final form 12 June 1986)

Abstract--Inhaled particleshavebeenimplicatedin manyformsof respiratorydisease.Mathematical lung deposition models have been developedfor spheres and fibers,but not for non-spherical,noncylindricalparticlesbecausetheoryand experimentaldata havebeenunavailable.To satisfythis need, non-spherical,orthotropic particlessettlingin air weremodeledunder dynamicallysimilarconditions using aluminum prisms settling in viscousoil. Twenty-fiverectangular prisms, each with a different length:width:thicknessratio, weresettledin eachof their threeprimaryorientationsand the resulting drag forcesdetermined.Linearregressionusingpredictorsrelatedto prism geometryand orientation resulted in an empiricaldrag force equation. Equation drag predictions are in excellentagreement with published data for rectangular prisms and other orthotropic objects including cylinders, ellipsoids,and double-conicals.The equation was validatedby settlinga quasi-monodisperseaerosol of tungstic acid platelets in a St6ber spiral centrifuge, and comparing predicted with observed aerodynamicbehavior.Usingthese results,lungdepositionmodelsfor flakesand other non-spherical particles are now possible.

BACKGROUND Stokes's solution of the Navier-Stokes equation, describing drag on a spherical body under viscous flow conditions, is well known and has been extensively verified: F = 3FI/~Dv,

(1)

where F is the fluid drag force on the sphere, # is the viscosity of the fluid, D is the sphere diameter, and v is the sphere velocity relative to the fluid. Stokes's equation is exact only when Reynolds number (NRe) is zero, but is in error by < 2 % for NRe < 0.05 (Happel and Brenner, 1964). Viscous drag on non-spherical objects may be predicted using a modification of Stokes's Law: F - 3II#Dsv K ' (2) where K is a shape resistance factor (Happel and Brenner, 1964) and D s is the equivalent volume sphere diameter: the diameter of a sphere with the same volume as the object in question. A theoretical expression for K was developed for ellipsoids by Oberbeck (1876), and is discussed by McNown and Malaika (1950). Thin disks and long cylinders may be approximated by oblate and prelate ellipsoids. Rectangular prisms, like spheres, disks and cylinders, are orthotropic; that is, they possess three mutually perpendicular planes of symmetry. Happel and Brenner (1964) discuss average resistance to translation for orthotropic objects, based on the work of Gans (1911). The expression they derive for average resistance provides an approach to developing shape resistance factors for rectangular prisms or other orthotropic objects. Gans showed that orthotropic objects settling in viscous flow exhibit no tendency to rotate and hence have no preferred settling orientation. These objects will be stable in whatever orientation they possess when settling begins. Squires and Squires (1937), McNown and Malaika (1950), Heiss and Coull (1952), and Gurel et al. (1955) demonstrated this for NRe < 0.1. The average translational velocity of orthotropic objects settling under viscous conditions 87

88

DAVID L. JOHNSONet

aL

is (Happel and Brenner, 1964): v~ -

0 VAp 3/~ (kl + k2 q- k3) k,

(3)

where g is the gravitational constant, Vis the particle volume, Ap the difference in particle and fluid densities, k the unit vector along the vertical axis, z, and k~, k2 and k3 the shape resistance factors along each of the principal translational axes x', y', z'. The average direction of settling is in the direction of gravity. Because g VAp = F and k = (kl + k2 + k3)/3, F = --~ /~v'~,

(4)

or,

F - 3II/~Dsv~

g

(5)

The average resistance/( is then the scalar: /~ = (K1 + K E + K 3 )

3

'

(6)

where K1 = 3H/~Dskl, K2 = 31q/./Dsk2, and K3 = 3H/~Dsk3. Fuchs (1964) points out that Brownian rotation causes aerosol particles to change orientation constantly, so that average resistance for each particle is given by equation (6). Knowing K1, K2 and K3, one can predict the mean settling velocity for randomly oriented, orthotropic particles. Little experimental work has been conducted to characterize the resistance factors K1, K2 and Ka for rectangular prisms, although Kasper et al. (1985) have recently reported careful experiments to determine k values for cylinders and chains of spheres. McNown and Malaika (1950) conducted extensive studies of the viscous settling of orthotropic objects that included cubes, square plates and elongated rectangular prisms. Both "flatwise" and "edgewise" settling were evaluated. Gurel et al. (1955) observed the behavior of isometric and "compact" particles including cubes and one rectangular prism. Heiss and Coull (1952) report resistance shape factors for cubes, plates, rectangular prisms, disks and cylinders. In each study, the prisms had square cross-section, so that only two unique dimensions existed. This limitation forced Heiss and Coull to develop separate models for flatwise and edgewise settling. Their equations employ two descriptors of body orientation and shape: surface sphericity, q~, and the ratio of equivalent volume sphere diameter to projected area diameter, Ds/D . . Surface sphericity describes shape, whereas D J D , addresses both shape and orientation. A comparison of the reported values is in Table 1. Entries for Heiss and Coull were calculated from their equations, which accurately predict drag on objects with two equal dimensions but can not be applied if all three dimensions are different. Development of a model for the general case was the major goal of this research. EXPERIMENTS Aluminum prisms settling in oil

Viscous drag on microscopic objects settling in air can be conveniently studied with macroscale models using dynamic similitude. Fluid forces on an object in one fluid may be studied in another fluid if the Reynolds numbers are the same (Mercer, 1978). The drag forces on small rectangular prisms in air may thus be studied using large prisms with the same shape through appropriate selection of prism material density, fluid density and fluid viscosity. Prisms were dropped in a viscous oil and their settling velocities measured in three orientations. Because slight imperfections in the prisms can cause unstable settling (McNown and Malaika, 1950; Heiss and Coull, 1952), the prism material must be machineable to high

89

Drag on non-spherical, orthotropic aerosol particles Table 1. Shape resistance factors reported in the literature Observed K value

Shape Cube Plate Rectangular prism Rectangular prism

Relative dimensions

Settling orientation*

McNown and Malaika

Gurel et al.

Heiss and Coull

1 1 4 4 1 1

N/A Flatwise Edgewise

0.962 0.714 0.769

0.924 ---

0.962 0.725 0.760

1

Flatwise

--

0.980

0.980

1 1 4 0.475

1

* 'Flatwise'--square projected area; 'edgewise'--rectangular projected area.

tolerance and polishable. Aluminum was used for these reasons and because of its low density, approximately 2.7 g c m - 3. Twenty-five rectangular prisms were machined, each with a unique length: width:thickness ratio. Dimensions of the prism models are in Table 2. The fluid chosen was U C O N IR~ Lubricant 50-HB-5100 (Union Carbide Corporation, Ethylene Oxide Derivatives Division, Danbury, Connecticut) because its viscosity changes little with temperature, it reaches its equilibrium water content in one day, it does not react with air and is not volatile. Figure 1 shows a sketch of the tank. The central 30 cm diameter, 60 cm deep cylinder held the oil, and was surrounded by a large water bath. The tank had clear plastic windows in the front and back, scribed with horizontal lines 20 cm apart, the distance settling models were timed. Each window was identically scribed so that sighting through the oil tank defined planes for starting and stopping the timing. A constant temperature heater/circulator maintained the water bath at 30 + 0.5°C. Rectangular prisms have a maximum of three planes of symmetry. Drag on each model was measured for each of the three orientations where a symmetry plane was perpendicular to the direction of motion. Since 25 models were used, 75 object : orientation combinations were tested. To account for additional drag contributed by tank walls, each of the combinations was evaluated in each of three different-diameter tanks to allow extrapolation to the infinite fluid case (see, for example, Kasper et al., 1985). Clear plastic tubes 12.5 and 9.5 cm in diameter were each placed in the tank so that the tube axis coincided with the tank axis. The cylindrical oil tank itself served as the tank with largest diameter. The oil and water bath temperatures and the oil viscosity were measured daily. The viscosity was corrected for wall effects using Faxen's equation: #o~ =

(1 + 2.4 Ds/D )'

(7)

where #~ is the true fluid viscosity,/~ is the measured viscosity, Ds the sphere diameter, and D the tank diameter. The prisms were placed in the oil using tweezers, positioned just below the surface of the oil, optically aligned in the desired orientation, and released. Settling times were recorded. Observations for which poor orientation or settling behavior was noted were not used. Each model was dropped in each of its three primary orientations five times. The total number of observations taken was 25 prisms × 3 orientations × 3 tanks × 5 replications = 1125 observations. Experimental error may have derived from several sources; however, each was controlled in the work reported here. Variations in viscosity and density of the oil were minimized by controlling bath temperatures as discussed above. Kasper et al. (1985) found that convection currents in their apparatus, which was similar to ours, were on the order of 10 #m s- ~, several

90

DAVID L. JOHNSON et al.

Table 2. Prism shape resistance factors for prisms settling parallel to the A axis Vertical dimension, A (cm) 0.081 0.815 2.443 0.132 0.511 2.537 0.511 3.175 0.638 2.545 0.081 0.815 1.626 0.160 0.648 1.920 0.130 0.511 2.032 0.160 0.635 1.278 0.160 0.320 1.60 O.198 0.409 1.631 0.099 1.021 0.198 0.813 0.259 0.335 1.359 0.257 0.338 1,704 0.231 1.374 0.318 0.427 1.270 1.593 0.320 1.270 0.315 1.110 0.368 0.305 0.508 1.024 0.312 0.422 0.841 0.307 0.635 0.831 0.419 0.439 0.589 0.472

Horizontal dimension, B (cm)

Horizontal dimension, C (cm)

Shape resistance factor, K

2.443 0.081 0.081 2.537 0.132 0.132 0.066 0.066 0.066 0.066 1.626 0.081 0.081 1.920 0.160 0.160 2.032 0.130 0.130 1.278 0.160 0.160 1.600 0.160 O.160 1.631 O.198 O.198 1.021 0.099 0.818 0.198 1.359 0.259 0.259 1.704 0.257 0.257 1.374 0.231 1.270 0.318 0.318 0.312 0.312 0.312 0.312 0.368 1.110 1.024 0.305 0.305 0.841 0.312 0.312 0.635 0.307 0.414 0.414 0.589 0.439 0.470

0.815 2.443 0.815 0.5 l 1 2.537 0.511 3.175 0.511 2.545 0.638 0.815 1.626 0.815 0.648 1.920 0.648 0.511 2.032 0.511 0.635 1.278 0.635 0.320 1.60 0.320 0.409 1.631 0.409 1.019 1.019 0.813 0.818 0.335 1.359 0.335 0.338 1.704 0.338 0.457 0.457 0.427 1.270 0.427 0.320 1.593 0.315 1.270 0.371 0.371 0.508 1.024 0.508 0.422 0.841 0.422 0.635 0.635 0.419 0.831 0.584 0.584 0.470

0.461 0.578 0.679 0.540 0,627 0,783 0,507 0,628 0,549 0.662 0.509 0.648 0.718 0.580 0.701 0.821 0.566 0.658 0.804 0.658 0.777 0.864 0.628 0.679 0.856 0.664 0.727 0.883 0.569 0.750 0.730 0.883 0.739 0.756 0.929 0.690 0.710 0.899 0.717 0.922 0.780 0.809 0.953 0.925 0.737 0.963 0.785 0.957 0.815 0.776 0.855 0.949 0.840 0.878 0.975 0.839 0.930 0.973 0.877 0.898 0.935 0.933

Drag on non-spherical,orthotropic aerosol particles

91

Fig. 1. The prisms settling tank. The cylindrical tank contained synthetic oil with viscosity approximately 20 Poise, and was surrounded by a constant temperature water bath.

orders of magnitude smaller than the sedimentation velocities for the prisms used in the present work and hence negligible. As object orientation affects drag, any observations in which poor orientation or poor settling behavior were noted were not used. The prisms were measured to a tolerance of + 0.0003 cm. The principal translational resistance coefficient K was calculated for each observation from: 18#o~ z

K = D2#Ap---~---~,

(8)

where z is the path length (20 cm) and t is the observed settling time. Brenner (1962) showed that K m = K ( 1 - ctds/D),where Km is the resistance factor measured for an object with diameter d sin a tank with diameter D. K, the resistance factor for drag force in an unbounded fluid, can be found from the intercept for a plot of K,, vs ds/D. Brenner's theory suggests that ct should be constant for all prism: orientation combinations and that the three lines for a particular prism should be parallel (i.e. have a common slope). This allows the data to be collapsed across orientation for a particular prism and perhaps across prisms as well if all prisms have the same slope. Statistical tests of the plots for linearity, parallelism, and differences in slope were performed and the data subsequently pooled over all models. K was then obtained for each prism : orientation combination; these values are also listed in Table 2. An equation for K was then developed. Because the drag on the prism is the drag on an equivalent volume sphere divided by K, developing the equation for K was equivalent to developing the equation to predict drag force. The resistance shape factor K is dimensionless. A linear expression for predicting K must, therefore, contain dimensionless terms. As Fuchs points out (1964), these terms describe deviation from sphericity. For non-isometric objects, at least one term must describe orientation. Candidate predictors were identified through the literature; others were original to this work. Some predictors describe model geometry only, whereas others describe both geometry and orientation. Various combinations of the candidate predictors are evaluated using the SAS procedure STEPWISE, which incorporates a forward-backward elimination scheme to build a regression model (SAS Institute, 1982). The subset o f predictors best describing the data were used to formulate the empirical drag force equation.

92

DAVID L. JOHNSON et al.

Tungstic acid prisms settlin 9 in air

Finally, experiments were carried out to validate the drag equation. This required an aerosol of rectangular prisms. Furusawa and Hachisu (1966) reported work with quasimonodisperse sols of microscopic, tungstic acid platelets grown using modifications to Zocher's method (Zocher and Jacobsohn, 1979). These platelets are rectangular prisms 3-4/~m long, 1.5-2/~m wide, and approximately 0.07/tm thick when grown under controlled conditions. Sols of tungstic acid microcrystals were grown as described by Furusawa and Hachisu, but with a different gel incubation time (Johnson, 1985). A scanning electron micrograph of the crystals is shown in Fig. 2. The St6ber Spiral Duct Aerosol Spectrometer (St6ber and Flachsbart, 1969; 1971; St6ber et al., 1978) was used to characterize the platelets aerodynamically. The centrifuge used in this study is of the Los Alamos Scientific Laboratory type, as described by Moss et al. (1972), which features a deep channel and an Archimedean spiral. Samples of deposited aerosols were collected on 200mesh transmission electron microscope (TEM) grids covered with Formvar film and placed along the centrifuge collection liner at 17.5, 25 and 30cm from the inlet of the instrument, locations corresponding to aerodynamic diameters of 1.7, 1.2 and 1.0/~m. Crystals were aerosolized from water using a Collison nebulizer. Monodisperse polystyrene latex microspheres were aerosolized simultaneously. The test aerosol passed through a diffusion dryer and aerosol charge neutralizer then into the centrifuge where crystals and spheres deposited according to their aerodynamic diameters. The TEM grid samples were shadowed with gold-palladium alloy at an angle of 12 degrees, and photomicrographs taken of individual particles on each grid at 5000 ×. Only single particles lying flat on the Formvar were photographed. Eleven size measurements were taken for each particle, as shown in Fig. 3. The 11 measurements were used to calculate the surface sphericity q', equivalent volume diameter Ds, and projected area diameters D,(1), D, (2), D,(3) for each particle. These were then input to the equation and the K1, K2 and K3 resistance factors obtained. From these, the average translational resistance factor/~ for each particle was predicted. This value and the crystal density Po (5.5 g cm - 3) allowed calculation of aerodynamic diameter, Da, for each particle from: D a = Ds~/pp K .

(9)

Predicted aerodynamic diameters were grouped into size intervals and the fraction of particles in each interval plotted against the interval midpoint size.

RESULTS AND DISCUSSION Three predictors provided a good fit to the data: the surface sphericity, W; the ratio of equivalent volume diameter to projected area diameter, D s / D . ; and the ratio of maximum body dimension (along a principal axis) to projected area diameter, Dmax/D,. A model incorporating these variables provides a multiple correlation coefficient R 2 of 0.992: K = 0.197 + 0.627 (~) + 0.240(Ds/D,) - 0.029 (Dmax/D,).

(10)

Predicted values from this model are in excellent agreement with data reported in the literature for prisms; differences are generally less than 2.5 °/0 (McNown and Malaika, 1950; Pettyjohn and Christiansen, 1948; Heiss and Coull, 1952; Beroes, 1950). For other, nonprismic orthotropic forms such as cylinders and ellipsoids the agreement is also good, with differences generally less than 3-4 ~o (McNown and Malaika, 1950; Heiss and Coull, 1952). This agreement across forms suggests that the model may be applicable to orthotropic forms in general, to approximate shape resistance factors expected over the range of object dimension ratios used here. For spheres, the model predicted a shape resistance factor K = 0.97. To satisfy the boundary condition K = 1.0 when W = D j D , = D m a j D , --- 1.0, the model was constrained

Drag on non-spherical, orthotropic aerosol particles

Fig. 2. Tungstic acid platelet microcrystals were used as a test aerosol to validate the drag force equation (magnification 5000 x ).

93

Drag on non-spherical, orthotropic aerosol particles

95

el<

7

l W

b

L

I__ 0

-f-

Fig. 3. Elevenparticle measurementswere taken from shadowed TEM grid photomicrographsand used to predict each particle's averageshape resistance factor and aerodynamicdiameter.

to satisfy this condition, so that K = 0.246 + 0.531 (~) + 0.258 (Ds/D .) - 0.036 (Dmax/On).

(11)

The percent differences in K values predicted using equation (11) and those experimentally observed are essentially unchanged for prisms, and are still quite small ( < 4 %) for the other forms as well. Plots of equation (11) for K are shown in Fig. 4, using the relative dimensions C = 1 and A and B varied over the range studied (0.1-5.0). Substitution of equation (11) for K into equation (2), Stokes's Law, yields 3H#DsV F = 0.246 + 0.531 (V) + 0.258 (Ds/D,) - O.O36(Dmax/D,, )"

(12)

The predicted aerodynamic diameters obtained from particle dimension measurements of the TEM grid samples were grouped into 0.1 #m size intervals and the fraction of the total present in each interval plotted against the midpoints of that interval, Figs 5-7. These figures show excellent agreement between predicted aerodynamic diameter obtained using equation (12) and true aerodynamic diameter for the samples with true aerodynamic diameter of 1.2

1.0 s ~ s

.9

~ w ~ _

s

f

I

s

.8 I

s S'~

/

-~,.~..,~.

l~

"

.6

"I

.5

, C=l

,

,

4

5

.4 0

I

2

3 A

Fig. 4. Shaperesistancefactor K from equation (11)for various levelsof body dimensionsA and B, with dimension C held constant = 1. A is parallel to the direction of motion, B and C are normal.

96

D A V I D L. JOHNSON e t a l .

35 30

7////

V////, "////~

25

20

~////,,

-

"

"//////

X/////

el/l/l/

f44444•

Y I0

. . . . //,

7////l

V/////,

............

"//////,

::::::::::::::::::

1.4 15 16 17 t8 Predicted Aerodynamic Diameter (/zM)

13

1.9

Fig. 5. Histogram of the 1 7 . 5 c m T E M grid showing the distribution of predicted particle aerodynamic diameters about the true aerodynamic diameter of 1.7 # m (n = 23 particles).

40--

//////L :55

-

50-25

~LL/LLL,

20

'lillll/

15

¢IIII//,

I0

'/////2

5

0

0.9

I Predicted

12 1.3 14 I.I Aerodynamic Diameter (/zM)

1.5

Fig. 6. Histogramof the 25.0 cm TEM grid; trueaerodynamicdiameter = 1.2/am(n = 46 particles). 45 40 ////I

55 30

~ 25

/Z/Z/,

nD /////

I0 5 0 "//]///I 0.6

0.7

0.8

0 9

I

I.I

12

1.5

Predicted Aerodynamic DPameter (,u.M)

Fig. 7. Histogram of the 30.0 c m T E M grid; true aerodynamic diameter = 1.0/~m (n = 52 particles).

and 1.0 #m, Figs 6 and 7. The agreement is very good for the sample with true aerodynamic diameter of 1.7 #m even though that plot is based on only 23 particles, all that could be found lying flat on the grid. The spiral centrifuge calibration was checked for each aerodynamic diameter sampled using latex spheres, which also collected on the grids. Measurement o f the sphere diameters showed each had the physical diameter expected for latex particles o f the

Drag on non-spherical, orthotropic aerosol particles

97

corresponding aerodynamic diameter and provided direct verification of the instrument calibration. CONCLUSIONS

The empirical drag equation provides extremely good estimates of the drag on rectangular prisms and other orthotropic objects settling in air under viscous conditions. The shape resistance factors obtained using the equation are in excellent agreement with those observed by other investigators for both rectangular prisms and other orthotropic forms including cylinders and double-conicals, and also agree well with predictions from theory for ellipsoids of revolution. One other equation has been proposed for predicting drag on triaxial orthotropic bodies (Clift et al., 1978). Its drag predictions for prisms like those used in this study err by as much as 100 %, compared with a maximum error of 10 % for equation (12). The present model provides the best drag force estimates available for nonspherical, non-cylindrical orthotropic objects within the range of relative dimensions examined in this study. REFERENCES Beroes, C. S. (1950) The effect of shape and orientation on the free settling rates of square plates and bars in a viscous medium. Masters thesis, University of Pittsburgh, Pittsburgh, Pennsylvania. Bowen, B. D. and Masliyah, J. H. (1973) Drag force on isolated axisymmetric particles in Stokes flow. Can. J. chem. Engng 51, 8-15. Brenner, H. (1962) Effect of finite boundaries on the Stokes resistance of an arbitrary particle. J. Fluid Mech. 12~ 35-48. Clift, R., Grace, J. R. and Weber, M. E. (1978) Bubbles, Drops, and Particles. Academic Press, New York. Fuchs, N. A. (1964) The Mechanics of Aerosols. Pergamon Press, Oxford. Furusawa, K. and Hachisu, S. (1966) Study of tungstic acid sol (I), method for preparing tungstic acid sol and its properties. Sci. Lt, Tokyo 15, 115-130. Gans, R. (1911) Wie fallen Stabe und Scheiben in einer reibenden Flussigkeit? Sitzungsber, Math-Physik Klasse Akad. Wissen. Muchen 41, 198-203. Gurel, S., Ward, S. G. and Whitmore, R. L. (1955) Studies of the viscosity and sedimentation of suspensions, Part 3: The sedimentation of isometric and compact particles. Br. J. appl. Phys. 6, 83-87. Happel, J. and Brenner, H. (1964) Low Reynolds Number Hydrodynamics. Martinus Nijhoff, Boston, Massachusetts. Harris, R. L. Jr. and Fraser, D. A. (1976) A Model for deposition of fibers in the human respiratory system. Am. ind. Hyg. Ass. J. 37, 73-89. Heiss, J. F. and Coull, J. (1952) The effects of orientation and shape on the settling velocity of non-isometric particles in a viscous medium. Chem. Engng Prog. 48, 133-140. Heyder, J. and Rudolf, G. (1984) Mathematical models of particle deposition in the human respiratory tract. J. Aerosol Sci. 15, 697-707. Johnson, D. L. (1985) Viscous drag on orthotropic aerosol particles. Doctoral dissertation, University of North Carolina at Chapel Hill, North Carolina. Kasper, G., Niida, T. and Yang, M. (1985) Measurement of viscous drag on cylinders and chains of spheres with aspect ratios between 2 and 50. J. Aerosol Sci. 16, 535-556. Kunkel, W. B. (1948) Magnitude and character of errors produced by shape factors in Stokes' law estimates of particle radius. J. appl. Phys. 19, 1056-1058. McNown, J. S. and Malaika, J. (1950) Effects of particle shape on settling velocity at low Reynolds numbers. Trans. Am. geophys. Un. 31, 74-82. Mercer, T. T. (1973) Aerosol Technology in Hazard Evaluation. Academic Press, New York. Moss, O. R., Ettinger, H. J. and Coulter, J. R. (1972) Aerosol density measurements using a modified spiral centrifuge aerosol spectrometer. Envir. Sci. Tech., 6, 614-617. National Institute for Occupational Safety and Health (1977) Occupational Diseases--A Guide to Their Recognition. U.S. Department of Health, Education, and Welfare, Washington, D.C. Oberbeck, A. (1876) J. reine agnew. Math. 81, 62. Pettyjohn, E. S. and Christiansen, E. B. (1948) Effect of particle shape on free-settling rates of isometric particles. Chem. Engng Prog. 44, 157-172. SAS Institute, Inc. (1982) SAS User's Guide: Statistics, 1982 Edition. SAS Institute, Inc., Cary, North Carolina. Squires, L. and Squires, W. (1937) The sedimentation of thin disks. Trans. Am. Inst. chem. Engrs 33, 1-12. St0ber, W. and Flachsbart, H. (1969) Size-separating precipitation of aerosols in a spinning spiral duct. Envir. Sci. Tech. 3, 1280-1296. St0ber, W. and Flachsbart, H. (1971) High resolution aerodynamic size spectrometry of quasi-monodisperse latex spheres with a centrifuge. J. Aerosol Sci. 2, 103-116. St~iber, W., Martonen, T. B. and Osborne, S. Jr. (1978) On the limitations of aerodynamical size separation of dense aerosols with the duct centrifuge, Recent Developments in Aerosol Science. John Wiley, New York. Zocher, H. and Jacobsohn, K. (1979) Concerning tactosols. Kolloidchem. Beih. 28, 167-206.