(233b) A Method to Calculate Aerodynamic Diameter of Particles with Fractal Surface

Surfaces of most materials, including natural and synthetic, porous and non-porous, amorphous and crystalline are fractal on a molecular scale [1, 2]. Surface roughness plays an important role by increasing the drag force of a particle as it settles and therefore reduces the settling velocity. The degree of surface corrugation is commonly quantified by a surface fractal dimension (D_S) with values varying from 2 (smooth) to 3 (very rough). The difference in aerosol performance between smooth (D_S=2.06) and rough (D_S>2.18) surfaced particles has been reported, where rough particles showed better dispersion along with more fine particles of less than 5 mm generated in the aerosol cloud [3]. In pharmaceutical applications, a precise determination of aerodynamic diameter (D_A) is required for assessing the performance of aerosol particles widely used for inhalation therapy. Particle sizing equipment that measures aerodynamic diameter is widely available but there is a number of biases associated with the use of these instruments. In particular, complete dispersion of micron sized dry particulate solids may be difficult to achieve due to strong cohesive forces.

This work describes a computational method to predict D_A of particles with rough surfaces. A set of model objects having DS varying from 2.00-2.55 was constructed [4]. The DS was determined from Richardson's plot (log perimeter versus log yardstick). From these objects, all the parameters required to compute shape and surface roughness factors required to calculate the drag coefficient can be obtained.

Chhabra et.al.[5] compared five available expressions to calculate drag on non-spherical particles by using 1900 data points covering 10^-4 Re<5x10⁵ and 0.09<ψ>1 (see below). Expressions derived by Haider and Levenspiel [6], Ganser [7], Hartman et al.[8], Swamee and Ojha [9], and Chien [10] were assessed by comparing the overall mean and maximum percent errors using the same set of data. It was concluded that the expressions derived by Haider and Levenspiel and Ganser are the most accurate with an average error of 16.3%. For this reason, the drag coefficients (C_D) formulated by Haider and Levenspiel, and Ganser were employed in this work. Additionally, C_D formulated by Ro and Neethling was included because of its simplicity and because their correlation covers the whole range of sphericity, ψ, 0-1. Our objective is to test which of these correlations is most suitable for particles with fractal surface.

The method to compute D_A is illustrated schematically below:

where Re is particle Reynolds number, m_L and ρ_L viscosity and density of suspending fluid, respectively, d_V equivalent volume diameter; U Particle terminal settling velocity; V_P and ρ_P volume and density of the particle, respectively, g gravitational acceleration, A_A projected area of the particle normal to the settling direction.

The C_D equations used in this work formulated by Ro and Neethling, Haider and Levenspiel; and Ganser are shown in Eqs. (1)-(5).

(1) , ,

where d_A and d_S are equivalent projected area and surface area diameter, respectively.

(2)

(3)

where A_S and A_ns are surface area of spherical and non-spherical particles of equal volume and density, respectively.

(4)

(5)

where ψ is as defined in Eq. (3).

3. Results

3.1 Computation Model

Figure 1 shows the relationship between D_A and D_s as determined by our predictive model. For all the correlations used, aerodynamic diameter decreases with increasing fractal dimension due to the increased drag forces exerted by the surrounding fluid. For the calculation, the particle projected area (A_A) and volume (V_P) were set to be constant. Therefore, the aerodynamic behavior will depend only on the nature of the external surface, which is reflected in the DS. Using A_A and V_P values, the dimensions of the primary units in the constructed fractal objects were calculated. The values of AA and VP used were chosen so that the criteria of all the C_D correlations were satisfied.

Figure 1 Plot of aerodynamic diameter, calculated using drag coefficient obtained from Eqs. (1), (2), and (4), as a function of surface fractal dimension, D_S . A_A and V_P were set to be 300mm² and 4500mm³, respectively.

3.2 Validation of the Computation Model

To validate the results obtained from the computational method, we produced particles having different surface corrugation by spray drying bovine serum albumin at different conditions (Figure 2).

(a)                                                                        (b)

Figure 2 Scanning electron micrographs at 8000 magnification of (a) smooth, D_S=2.06 and (b) rough, D_S=2.41 bovine serum albumin (BSA) particles

The D_S and the equivalent volume diameter of BSA particles were determined using laser scattering technique (Malvern Mastersizer S, Worcs, UK). Using these two values, D_A was computed with our model. The calculated DA was then compared with values measured using Particle Size Distribution Analyser (PSD) 3603 (TSI, Shoreview, USA). The PSD measures the time taken to accelerate a particle between two laser beams. The time is then converted to aerodynamic diameter using a simple reference table. The performance of PSD3603 was tested using standard polystyrene, glass and soda lime beads. The results agree within 10% of the certified diameter. The D_As measured by PSD3603 were recalculated after eliminating the big aggregates shown in the particle size distribution. The absence of particles >10?m in both the electron microscope photos (Figure 2) and laser diffraction results showed that complete dispersion could not be achieved using the PSD3603's dry powder disperser. Hence these big aggregates have to be eliminated to give accurate results in the instrument calculation.

The comparison between measured and computed D_As (Figure 3) shows that:

a). D_As computed by the Ro and Neethling C_D correlation agree for particles with D_S 2.35 and 2.41. The maximum differences for particles with DS 2.06 and 2.18 are 18 and 26%, respectively.

b). Using the Haider and Levenspiel correlation, D_A computed matches the measured value only for the particles with D_S 2.18. Particles with other D_S have maximum difference approximately of 30%.

c). D_As calculated using the Ganser C_D correlation agree with the measured values, except for particles with D_S 2.06 with the maximum difference of 18%.

The values of ?a' and ?b' in Eq. (1) were found to be 3.6 and -0.313 for 0.001<Re<1000 [11]. Using the Ro and Neethling correlation, the difference between the computed and measured D_As may be due to the Reynolds number of the spray dried BSA particles, which varies from 8x10^-5 to 1.2x10^-4. CDs computed by the by Haider and Levenspiel model (applicable when Re<25,000) and the Ganser (applicable for ReK₁K₂<10⁵) are more suitable for the BSA particles since the required conditions are satisfied. Tang et al [4] computed settling velocity using the size and density of the particles employed by Pettyjohn and Christiansen. In general, it was also found that the percentage differences between calculated and measured settling velocities were lower when Eq. (4) (maximum difference 36%) was used instead of Eq.(2) (maximum difference 31%), as also found by Chhabra et. al. [5].

(a)                                                                        (b)

(c)

Figure 3 Aerodynamic diameters measured by Particle Size Distribution Analyser (PSD)3603 and computed from the model utilizing the drag coefficient from (a) Ro and Neethling; (b) Haider and Levenspiel; (c) Ganser

The large differences in measured and calculated D_AS for smooth particles is due to inadequate dispersion as these particles are more cohesive than the wrinkled particles. This calculation technique, therefore, can serve to readily estimate D_A in situations when direct experimental measurement is insufficient or inaccurate due to equipment limitations or difficulty in dispersing powders into individual particles in the aerosol.

4. Conclusion

A computational method to obtain D_A of an object with fractal surface has been developed. A number of C_D expressions was employed to assess their suitability to predict the aerodynamic diameter of these fractal objects. The results calculated from the C_Ds derived by Ro and Neethling, Haider and Levenspiel, and Ganser, follow the expected trend of decreasing aerodynamic diameter with increasing surface roughness. Comparison with the measured values from PSD3603 shows that Ganser's correlation predicts D_A most accurately followed by Ro and Neethling's and then Haider and Levenspiel's. When effective dispersion is difficult to achieve, this calculation technique is useful to predict D_A.

References

1. Farin, D., and Avnir, D., in Characterisation of Porous Solids, K.K. Unger, Rouquerol, J., Sing, K.S.W., Kral, H., Editor. 1988, Elsevier, Science: Amsterdam. p. 421-432.
2. Kaye, B.H., Powder Technol., 1987. 50: p. 177-191.
3. Chew, N.Y.K., Tang, P., Chan, H-K., Raper, J.A., Pharm. Res., 2005. 22(1): p. 148-152.
4. Tang, P., Chan, H-K., Raper, J., Powder Tech., 2004. 147: p. 64-78.
5. Chhabra, R.P., Agarwal,L., and Sinha, N.K., Powder Tech., 1999. 101: p. 288-295.
6. Haider, A., and Levenspiel, O, Powder Technology, 1989. 58: p. 63-70.
7. Ganser, G.H., A Powder Tech., 1993. 77: p. 143.
8. Hartman, M., Trnka, O., Svoboda, K., Ind. Eng. Chem. Res., 1994. 33: p. 1979.
9. Swamee, P.K., and Ojha, C.P., J. Hyd. Engrg., 1991. 117: p. 660-669.
10. Chien, S.F., SPE Drilling and Completion, 1994. 9: p. 281.
11. Schiller, L., and Naumann, A., Z. Ver. Deutsch. Ing., 1933. 77: p. 318-320.
12. Clift, R., and Gauvin, W.H., Can. J. Chem. Eng., 1970. 49(4): p. 439-48.
13. Turton, R., and Levenspiel, O., Powder Tech., 1986. 47(1): p. 83-6.
14. Pettyjohn, E.S., and Christiansen, E.B., Chem. Eng. Prog, 1948. 44: p. 157.