Particle Suspension in a Rotating Drum Chamber When the Influence of Gravity and Rotation are Both Significant

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1 Aerosol Science and Technology ISSN: (Print) (Online) Journal homepage: Particle Suspension in a Rotating Drum Chamber When the Influence of Gravity and Rotation are Both Significant B. Asgharian & O. R. Moss To cite this article: B. Asgharian & O. R. Moss (1992) Particle Suspension in a Rotating Drum Chamber When the Influence of Gravity and Rotation are Both Significant, Aerosol Science and Technology, 17:4, , DOI: / To link to this article: Published online: 11 Jun Submit your article to this journal Article views: 304 Citing articles: 9 View citing articles Full Terms & Conditions of access and use can be found at

2 Particle Suspension in a Rotating Drum Chamber When the Influence of Gravity and Rotation are Both Significant B. Asgharian* and 0. R. Moss Chemical Industry Institute of Toxicology, P. 0. Box 12137, Research Triangle Park, NC In recent years, rotating chambers have been found to be an effective method of retaining particles suspended in the air for an extended period of time. Rotating drum chambers have the potential of providing a stable atmosphere of well-characterized inhalable particles for periods lasting from hours to days for use in inhalation toxicology studies. To aid in planning for the use of rotating drum chambers in inhalation studies, we created a model that describes (a) the concentration of particles in the chamber under various conditions and (b) the particle sizes for which gravity and rotation influence particle dynamics. Previous publications describe the suspension/deposition of particles when the rotational effect is dominant, but do not describe particle suspension/deposition when gravitational settling is significant as occurs when such drum chambers are operated at optimal conditions for retaining the highest fraction of particles over time. By using the limiting trajectory of particles, the fraction of particles that remain suspended in a 1-m diameter rotating drum chamber was derived for forces of gravity only, rotation only, and gravity plus rotation. For particles between 0.5 and 1 pm in diameter and for suspension times of < 96 h, there was no loss of the suspended particles for drum rotation rates from 0.1 to 10 rpm. For 2- and 5-pm diameter particles, > 98% and 91%, respectively, remain suspended after 96 h under optimal rotation of the drum chamber. Optimal rotation rates were independent of particle size for particles < 10 pm in diameter (agreeing with Gruel et al. [I9871 even though we predicted suspended fractions higher by > 30% for 10-pm particles after 96 h). For 20-pm diameter particles and suspension times < 96 h, the maximum suspended fraction occurred for drum rotation rates between 0.3 and 0.5 rpm. The particles > 2 pm can be selectively removed from an airborne particle size distribution in time periods of < 15 h when the rotational rate is > 5 rpm. INTRODUCTION Stainless steel rotating cylindrical chambers have been found to be an effective method of retaining particles suspended in the air for an extended period of time when operated at 2-3 rmp (Goldberg et al., 1958; Dimmick and Wang, 1969; Frostling, 1973). The axis of such cylinders is horizontal with gravity in the vertical direction. Maintaining a high number concentration of generated particles is desirable in long-term animal exposure studies where the generated particles are *To whom correspondence should be addressed. scarce, expensive, or highly toxic. Rotating drum chambers have potential use in providing a stable atmosphere of wellcharacterized respirable particles for periods lasting from hours to days for use in inhalation toxicology studies. Particle aging by means of rotating chambers was first suggested by Goldberg et al. (1958). In rotating drum chambers, particles remain airborne for a much longer period of time than in simple stirred settling chambers, primarily because of the competition between gravitational and centrifugal forces. Each force attempts to produce a different particle trajectory. When the chamber is not rotat- Acrosol Science and Technology 17: (1992) Elsevier Science Puhl~shing Co., Inc.

3 264 B. Asgharian and 0. R. Moss ing, particles deposit quickly by gravitational settling. When the drum chamber is at a high rotation rate (typically above 10 rpm), the particle trajectory due to gravitational settling is depressed and the acting centrifugal forces move the particles in the radial direction away from the center of the drum chamber until the particles are deposited. For rotation rates < 10 rpm, the two competing forces are of the same order of magnitude and the particle deposition is delayed. For each particle size, there must be a rotation rate between 0 and 10 rpm that optimizes the fraction of suspended particles. A few studies are available on the analysis of rotating drum chambers. Dimmick and Wang (1969), Goldberg (1971), and Frostling (1973) derived an identical expression for the fraction of suspended particles at high drum rotation rates where the centrifugal force is the dominant mechanism of particle deposition. Recently, Gruel et al. (1987) derived a different expression which partly takes into account the effect of gravitational settling. Using this result, they were able to obtain a simple equation predicting the optimum rotation rate of the drum chamber. For chamber rotation rates > 5 rpm, the prediction of the fraction of suspended particles by Gruel et al. (1987) approaches that of the previous findings. However, when the rotation rate is reduced to zero, their solution does not converge to the gravitational settling solution. Their predicted optimum rotation rate can be in error up to 70% for 20-pm diameter particles where gravity is significant. In the current study, rotating drum chambers are reexamined for potential use in inhalation studies. Using the limiting trajectories of the suspended particles (Pich, 1972), expressions are derived for the fraction of suspended particles in the chamber when the driving mechanisms are gravity, rotation, and gravity plus rotation. The optimum drum rotation rate for the chamber is obtained for different particle sizes and different suspension periods. Utilizing these results, particle size dispersion and the overall fraction of the particles that remained suspended, were obtained for the non-uniform initial size distribution. MATHEMATICAL MODELING OF PARTICLE SUSPENSION/DEPOSITION IN A ROTATING DRUM CHAMBER Forces exerted on the suspended particles in a drum chamber may include electrostatic, diffusion, pressure gradient, mutual collision, gravitational, and viscous drag forces. Among these, only the gravitational and viscous forces are relevant since the others are insignificant under normal conditions. Considering the force balance on a single spherical particle, the equation of motion may be cast into the following form: where_ ZD is the drag force on the particle, Fg is the gravitation force, and p,, d, and a' are the particle mass density, diameter, and acceleration, respectively. Omitting the Basset term and the added mass in the expression for drag force (Rudinger, 1980), which are unimportant for aerosols, the equation of motion in x, y Cartesian coordinate system may be expressed as follows: where x and y describe the position of a particle at time t in a coordinate system located at the center of the drum chamber, w is the chamber rotation rate, g is the gravitational constant, and r is the

4 Particle Suspension in a Rotating Drum Chamber 265 particle relaxation time given by where R is the radius of the drum chamber. By substituting x and y from Eqs. 7 and 8 into Eq. 9, we obtain in which E*, is the absolute (dynamic) viscosity of air and C is the slip correction factor given as (Hinds, 1982) Therefore, the initial positions of the limiting trajectories is a circle with its center located at with Kn being the Knudsen number. where h is the mean free path of air. Equations 2 and 3 may be solved for the cases of gravity only, rotation only, and gravity plus rotation. The resulting particle trajectory is consequently used to determine the fraction of suspended particles in the drum chamber. and with radius R as shown in Figure 1. The fraction of initial particles that remain suspended at t = t, will be the overlapping area of the chamber and the limiting trajectory circle divided by the crosssectional area of the drum chamber. Carrying out the algebra we obtain, Gravity Only By setting w = 0 in Eqs. 2 and 3 and solving the resultant equations of motion, the following results for the particle trajectory are obtained: where y, is the y coordinate of the center of the limiting trajectory circle as given by Eq. 12 and has always a positive value. Initially, when y, = 0, N/No = 1, and when y, 2 2R, N/N, = 0. where x, and y, define the initial position of the particle at time t = 0. The fraction of initial particles that remain suspended (N/N,) at time t, is calculated by determining the limiting trajectories of the particles. In this method, we seek the initial positions x = x,, y = y, of those particles that will just deposit on the chamber wall at the end of suspension time, t,. For such particles, Rotation Only The problem of a gas-filled cylinder rotating around its axis with a constant angular velocity was first set up by Lapple and Shepherd (1940) and subsequently solved by Kriebel (1961). The formulation is identical to Eqs. 2 and 3 except for the absence of the gravity term in Eq. 3. For aerosol particles for which the relaxation time (7) is much smaller than unity, the following simplified solution to the equations of motion was obtained by Rudinger

5 B. Asgharian and 0. R. Moss X chamber wall FIGURE 1. Deposition of suspended particles in a stationary drum chamber. x = (x,2 +y:) 1'2erw2t cos cot t, is the area of this circle divided by the cross-sectional area of the chamber. Thus, All particles having such trajectories and just depositing on the drum wall in time t, will initially be on a circle defined by substituting x and y from Eqs. 14 and 15 into Eq. 9 at t = t,. This leads to Thus, the circle has a radius of R~-'""R with its center located at the center of the drum chamber (Figure 2). The fraction of initial particles still suspended after time as previously obtained by Dimmick and Wang (19691, Goldberg (1971), and Frostling (1973). Gravity Plus Rotation When both gravity and rotation are present, Gruel et al. (1987) found the follow-

6 Particle Suspension in a Rotating Drum Chamber Y FIGURE 2. Deposition of suspended particles in a rotating drum chamber by pure rotation. ing solution to the set of Eqs. 2 and 3: x = e'~~'[(x, + ;)COS wt + I y, sin wt where the exponential terms in the above equations are the contribution of centrifugal acceleration (x and y in Eqs. 2 and 3). This approximate solution for the trajectory of each particle is accurate to the first power of term 7. Figure 3 shows the plot of Eqs. 18 and 19 indicated by a solid line, and the numerical solution of the coupled Eqs. 2 and 3 using a fourth-order Runge-Kutta formula (Hornbeck, 1975), indicated by A. The results are calculated for a unit density particle diameter of 20 pm, a 50-cm drum chamber radius, and a rotation rate of 1 rpm. Excellent agreement is observed between the approximate and numerical solutions. Comparison of the numerical values between the two solutions showed an agreement of up to four decimal places. Therefore, Eqs. 18 and 19 may be used to derive the equation describing the initial positions x,, y, of all particles that will just deposit on the wall at time t,. By substituting in Eq. 9 to obtain

7 B. Asgharian and 0. R. Moss pact, expanding spiral trajectory FIGURE 3. Trajectory of suspended particles in a rotating drum chamber when w = 1 rpm, d = 20 pm, for Eqs. 18 and 19 indicated by a solid line (-1 and for the numerical solution of Eqs. 2 and 3, indicated by triangles.(a). where 78 x, = -(I- cos wt,e~rw2t~) 0 rg sin wt, - TW2t, y,= the initial positions lie on the arc of a circle of radius R~-'"~'R and center x, and y,. The center (x,, y,) is located at a different circle of radius (rg)/w eptwzt~ positioned at [-(rg)/w, 01, which is the center of rotation of the suspended particle. Note that when w + 0, Eqs. 21 and 22 reduce to Eqs. 11 and 12 to the first order of r term. Consequently, Eq. 20 converges to Eq. 10, the case of gravitational settling only. On the other hand, when w + co, x, =y, = 0, this corresponds to the case of rotation only. To obtain an expression for the frac- tion of initial particles that remain suspended at t = t,, two situations are encountered: The circle, denoting the initial position of all particles whose trajectories will just intersect the wall of the drum chamber in time t, (Eq. 20), and the chamber wall do not meet (Figure 4). This occurs at higher rotation rates where the gravitational effect is small. In this case, the fraction of initial particles still suspended is simply the ratio of the two cross-sectional areas. If the distance between the centers of the two areas is denoted by H, where

8 Particle Suspension in a Rotating Drum Chamber FIGURE 4. Deposition of suspended particles in a rotating drum when both gravity and rotation are significant. the fraction of initial particles that are suspended is 2. The circle, denoting the initial position of all particles whose trajectories will intersect the wall of the drum chamber in time t, (Eq. 20), and the chamber wall meet. This implies that both gravity and rotation are significant. The fraction of the initial particles remaining suspended in the chamber is the overlapping section (shaded area in Figure 4) of the two areas divided by the drum chamber cross-sectional area: 1 + e-2702tr 2

9 270 B. Asgharian and 0. R. Moss where with H being defined by Eq. 23. In general, N/N, depends on three dimensionless parameters: wr, wtr, and (rg)/(rw). The latter is the dimensionless radius of the center of revolution of the particle at time t = 0. If wt, 2 n-, Eq. 20 will also describe mathematically the initial paths of those particles that have deposited, and thus have escaped the drum chamber but have returned to the chamber wall as result of their spiral paths. To exclude these particles, we let ber distribution of particles, N(x), at time t = t,, we can write AN dn N(x) Lim - - = K(x) Ax+OhNo dno N(,(x) (31) where K(x) is the number fraction of suspended particles at a given diameter as calculated from Eq. 24. By substituting dn,, from Eq. 29 into Eq. 31, the following equation is found dn = qf(,(x) ~ ( x dx. ) (32) Integrating the above equation yields - where N =, - I: d N denotes the total number of particles of all sizes at time t,. The fraction of initial particles that remain suspended at time t, is thus, Equation 24 describes the general situation when both w and g 'are significant. It should be mentioned that this equation will reduce to the limiting solutions of Eqs. 13 and 17 when w becomes small and large. For the case when the initial size distribution of the particles in the chamber is not uniform, Eq. 24 can be used to calculate the fraction and size distribution of particles that remain suspended at time t,. Suppose N,(d) denotes the initial number distribution of particles at t = 0 (i.e., number of particles at each diameter d). For an initial size distribution of particles, fo(x), the following relationship holds: The particle size distribution at time t = t, is: 1 dn f(x) = -- N dx For the above size distribution, the count median aerodynamic diameter (CMAD) of the particles can be calculated from 2 = exp {/-Txf( x ) d x) where in which = 12-, d No is the total number of particles at time t = 0. For a num- RESULTS AND DISCUSSION The fraction of unit density particles that remain suspended after time t, in a 1-m diameter drum chamber is calculated from

10 Particle Suspension in a Rotating Drum Chamber Eq. 24 for different particle diameters and drum rotation rates. For each particle size and suspension time, there is a rate of drum chamber rotation that results in a maximum fraction of suspended particles. This optimum rotation rate may be calculated by differentiating Eq. 24, setting the results to zero, and solving the resultant equation. An alternative approach (followed here) is to calculate the fraction suspended from Eq. 24 for w (rpm) ranging from 0 to R/R = 1 + ep'"''~ to find the drum rotation rate giving the maximum N/N,. In Figure 5, the obtained values of the optimum drum rotation rates are plotted for suspension times of 0-14 days. For suspension time of < 1 day, the optimum rotation rates of all particles sizes are similar but diverge as the suspension time increases. The optimum rotation rate is independent of particle size for particles smaller than 5 pm. The optimum rotation rate for maximal suspension increases with particle size and decreases with suspension time. Figure 6 shows the fraction of initial particles that remain suspended at optimum drum rotation rates for suspension time of 0-14 days. These suspension fractions correspond to the maximum possible values at each suspension time. The maximum fraction of suspended particles decreases with both particle size and suspension time. According to the results, 0.5-pm Suspension Time, days FIGURE 5. Optimum drum rotation rate for suspension times of 0-14 days.

11 B. Asgharian and 0. R. Moss 0.5 pm; / stationary drum Suspension Time, days FIGURE 6. Maximum fraction of initial particles that rcmain suspended in a rotating drum chamber for suspension times of 0-14 days. The dashed line curve (- --) represents the case of a stationary drum. particles have no significant loss in 14 days. However, as the particle size increases, the loss becomes more significant. During this period, there is a 18% loss of 5-pm particles. For 20-pm particles, after 1 day of suspension, > 60% of the particles have deposited and the remainder will deposit gradually in 14 days. Also shown in this figure is the fraction of 0.5-pm particles that remain suspended due to sedimentation only in a stationary drum. As seen, the particles deposit very quickly in - 2 days. This clearly demonstrates the effectiveness of rotating drum chambers as particle aging devices. The optimum rotation rate and fraction of diameters of 0.5, 1, 2, 5, 10, and 20 pm remaining after aging times of 1, 6, 48, and 96 h are listed in Table 1. The values of optimum rotation rate concur with Gruel et al. (1987). The fractions of initial particles that remain suspended are also in good agreement with this previous work for particles smaller than 2 pm in diameter. However, the calculated fraction of

12 Particle Suspension in a Rotating Drum Chamber 273 TABLE 1. Optimum Rotation Rate and Fraction of Suspended Particles Calculated from Equation 24 w(d, t) N/N& w, d, t) Particle (optimum rotation rate, rpm) (fraction of initial particles) size Aging time (h) Aging time (h) d (mm) OO 1.OO suspended particles differs between the two approaches for large particles, differing by as much as a factor of 3 for 20-pm particle diameters after 96 h of storage in a drum chamber operated at the optimal rotation rate of 0.4 rpm. This is expected since the results of Gruel et al. are valid only at moderate drum rotation rates where the effect of gravity is small. When particles become large, the gravity effect becomes more significant. The fraction of initial particles that remain suspended after 1 h are shown in Figure 7 for different values of o and Rotation Rate, rpm FIGURE 7. Fraction of initial particles that remain suspended in a rotating chamber after a 1-h suspension period: (-) present results; (- - -) Gruel et al. (1987).

13 B. Asgharian and 0. R. Moss particle diameters ranging from 0.5 to 20 pm. The results of Gruel et al. (1987) are indicated by dashed lines ( When w = 0, the fraction of particles suspended after time t, is the same as predicted for particle loss due to sedimentation only in the present calculation while the equation of Gruel et al. predicts the fraction to be zero. As w increases, N/N,, the fraction of initial particles suspended after time t,, calculated by both approaches is the same as predicted for particle loss due to rotation only. For 0.5-pm particles, and rotation rates from 0.1 to 10 rpm, the fraction of suspended particles is nearly 100%. The optimum rotation rate can be any value in this range. This explains why previous investigators adopted drum rotation rates of 2-3 rpm (Goldberg et al., 1958, Frostling, 1973), considerably higher than the optimum rates listed in Table 1, and still managed to maintain high concentrations of suspended particles. As the particle size is increased, the particle loss, due to the increasing effect of gravity, is increased and the range of operating w is reduced. For 2-pm particles and 1-h duration, the range is between 0.3 and 3 rpm according to Figure 7. For 1-h duration with 20-pm particles, the operating range of drum rotation rate is narrow and careful adjustment of w at the optimum rate is crucial. The calculated fractions of initial particles that remain suspended after 96 h are shown in Figure 8. The shape of the curves are similar to those shown for 1-h suspension times (Figure 7). However, in contrast to Figure 7, the operating range of the drum chamber rotation rate and the Rotation Rate, rpm FIGURE 8. Fraction of initial particles that remain suspended in a rotating chamber after a 96-h suspension period: (-) present results; (- - -) Guel et al. (1 987).

14 Particle Suspension in a Rotating Drum Chamber fraction of suspended particles at these rotation rates are significantly less. When the initial size of the particles is not monodisperse, it is possible, by using a rotating drum chamber, to eliminate the larger particles and produce a more uniform size dispersion. Figure 9 shows the fraction of suspended particles at a given size in a mixture as a function of particle diameter after a 15-h period for rotation rates of 0.4, 2, 5, and 10 rpm. For small rotation rate (w = 0.4 rpm), there is a small loss of large particles. When w is increased, the fraction of larger diameter particles reduces and at w = 10 rpm, practically all the particles larger than 3 pm are removed. It is of practical interest to generate a mixture of different size particles a few hours before the usage (e.g., 15 h) and obtain a narrow size dispersion of particles using a drum chamber. For a lognormal size distribution of particles, the fraction of particles remaining suspended in the chamber and the size distribution function and CMAD of particles at different drum rotation rates and suspension periods were calculated from Eqs The integrations in these equations were performed using Gauss-Quadrature technique (Lanczos, 1956). For a log-normal size distribution of the initial particles with a CMAD of do = 1.78 pm (corresponding to a 5-pm mass median aerodynamic diameter particle) and geometric standard deviation of a,, = 1.8: Particle Diameter, prn FIGURE 9. Fraction of initial particles that remain suspended in a rotating chamber during a 15-h suspension period.

15 B. Asgharian and 0. R. Moss Since the number of particles < 0.5 pm for the above size distribution is < 2%, the effect of diffusional loss would be small and can be neglected. For the above initial size dispersion, the number size distribution of the particles that remain suspended after 15 h for drum rotation rates of 0.4, 2, 5, and 10 rpm, are calculated and plotted in Figure 10. For small rotation rate of 0.4 rpm, the distribution of airborne particles appears to be undisturbed. When the rotation rate is increased, the distribution becomes narrower which indicates more loss of the larger particles and, consequently, a more uniform size dispersion. Table 2 lists CMAD, the overall frac- tion of initial - - particles that remain suspended (N/No), and the geometric standard deviation (that results from assuming a log-normal distribution of suspended particles at t = t,) of the above results (Figure 10). For a log-normal distribution, cx is calculated from B rg = exp x (x) K(x) dx When w is small, d and a, are not changed and there is a small loss of the suspended particles. As w increases, both 1 10 Particle Diameter, p m FIGUKE: 10. Number size distribution of suspended particles in a rotating chamber after a 15-h suspension period.

16 Particle Suspension in a Rotating Drum Chamber 277 TABLE 2. Calculated values of CMD, N/N,, and u (Based on a Log-normal Size ~istritution) of the Particles Remaining Suspended in a Drum Chamber after a 15-h Period d and a, decrease, which indicates a narrower distribution of the airborne particles. The overall fraction of the initial particles that remain suspended decreases as w increases. For w = 20 rpm, there remains only 8% of the initial particles. In addition, a, has increased. This is due to uneven loss of the particles from the two tails of the particle size spectrum. This in turn implies that particle distribution, though becoming more uniform, is deviating from log-normal. SUMMARY The fraction of suspended particles in a rotating chamber was investigated when the driving mechanisms of depositions were gravity and rotation. It was found that for submicrometer particles, when the suspension time did not exceed 96 h, the optimum drum rotation rate could vary from 0.1 to 10 rpm. As the particle size increased, the range of drum rotation rate decreased. For 20-pm particles, the range of the drum rotation rate producing maximal particle suspension decreased to between 0.3 and 0.5 rpm. Furthermore, the particle loss was significantly enhanced ( > 85% for a 96-h suspension period). Based on the analysis presented here, two applications of rotating drum chambers are envisioned. When operated at low (optimum) rotation rates, the drum chambers are capable of storing particles < 1 pm up to a period of 1 y, and particles between 1 and 2 pm for up to a period of 6 mo with 50% of particles remaining suspended. Drum chambers operated at high rotation rates (> 5 rpm) can be used to remove large particles (> 2 pm) thus producing a relatively narrow particle size dispersion (a, = 1.5). However, since the particle loss increases significantly with increasing the rotation rate, the suspension period for rotations > 5 rpm cannot exceed one day. The present study demonstrates the feasibility of using drum chambers as devices for particle size separation and long-term preservation. Because of their simplicity and easy operation, drum chambers will prove valuable in inhalation studies where the duration of study is normally long thus requiring a stable, wellcharacterized, and cost-effective source of respirable particles. The research.was supported, in part, by a grant from the Thermal Insulation Manufacturers Association, Inc. REFERENCES Dimmick, R. L., and Wang, ~.'(1969). In An Introduction to ExperimentalAerobiology (R. L. Dimmick and A. B. Ekers, eds.). John Wiley & Sons, New York, pp Frostling, H. (1973). J. Aerosol Sci. 4: oldb berg, L. J., Watkins, H. M. S., Boerke, E. E., and Chatigny, M. A. (1958). Am. J. Hyg. 68: Goldberg, L. J. (1971). Appl. Microbiol. 21: Gruel, R. L., Reid, C. R., and Allemann, R. T. (1987). J. Aerosol Sci. 18: Hinds, W. C. (1982). In Aerosol Technology. John Wiley & Sons, New York, pp. 45. Hornbeck, W. H. (1975). In Numerical Methods. Quantum, New York, pp Kriebel, A. R. (1961). J. Basic Eng. Trans. ASME 83D: Lanczos, C. (1956). In Applied Analysis. Prentice Hall, New York. Lapple, C. E., and Shepherd, C. B. (1940). Ind. Eng. Chem. 32: Pich, J. (1972). J. Aerosol Sci. 33: Rudinger, G. (1980). Fundamentals of Gas-Particle Flow (J. C. Williams and T. Allen, eds.). Elsevier, New York, pp Received June 17, 1991; accepted May 4, 1992.