Prediction of aerodynamic diameter of particles with rough surfaces

Powder Technology 147 (2004) 64 – 78 www.elsevier.com/locate/powtec

Prediction of aerodynamic diameter of particles with rough surfaces P. Tanga,b, H.-K. Chana,*, J.A. Raperb,1 a

b

Faculty of Pharmacy, University of Sydney, NSW 2006, Australia Department of Chemical Engineering, University of Sydney, NSW 2006, Australia Received 20 September 2004; accepted 22 September 2004 Available online 5 November 2004

Abstract This work demonstrates a simple means of predicting aerodynamic diameter of particles having fractal surfaces. A number of drag coefficient expressions were employed for the calculation of aerodynamic diameter. A set of model objects having surface fractal dimensions varying from 2.00 to 2.55 was constructed, from which all the parameters required to compute shape and surface roughness factors can be obtained. Drag coefficients (C D) formulated by Ro and Neethling, Haider and Levenspiel, Thompson and Clark, Ganser, and Tran-Cong et al. were shown to perform similarly within the specified criteria. The aerodynamic diameters calculated follow the expected trend of decreasing diameter with increasing surface roughness. Comparisons with literature data show that the C D expressions by Ro and Neethling, Haider and Levenspiel, and Ganser, on average, agree to within 17–23% of the measured values. D 2004 Elsevier B.V. All rights reserved. Keywords: Fractal; Aerodynamic diameter; Drag coefficients

1. Introduction Reports have shown that surfaces of most materials, including natural and synthetic, porous and non-porous, amorphous and crystalline are fractal on a molecular scale [1– 4]. Since two-thirds of drag on particles is friction drag (related to the total surface area) and one-third is the form drag (related to projected area in the settling direction), the effect of surface roughness is crucial [5]. Surface roughness increases the drag force of a particle as it settles and therefore reduces the settling velocity. Accurate prediction of settling velocity is crucial in many processes in various engineering fields, e.g., flotation, thickening, and purification. In pharmaceutical applications, a precise determination of aerodynamic diameter is required for assessing the performance of aerosol particles widely used for inhalation therapy [6,7]. The difference in aerosol performance between smooth and rough surfaced objects (Fig. 1) was reported [8], where * Corresponding author. Tel.: +61 29351 6072; fax: +61 29351 4391. E-mail address: [email protected] (H.K. Chan). 1 Current affiliation: Department of Chemical Engineering, University of Missouri-Rolla, MO 65409-1230, USA. 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.09.036

rough particles showed better dispersion coupled with more fine particles of less than 5 Am generated in the aerosol cloud. The surface roughness of these particles was successfully characterized by gas adsorption and light scattering and found to be fractal [9]. Smooth particles (Fig. 1a) were found to have a surface fractal dimension (D S) of 2.06 while the rough particles (Fig. 1b) possess a higher D S of 2.45. The improved dispersion and increased fine particles in the aerosol could be due to two factors, namely the contact area between particles and their aerodynamic behaviour. Rough particles are less cohesive than smooth particles because of less surface contact area. Once they are dispersed into individual particles, it is their aerodynamic behaviour that determines the extent of fine particles generation. However, determination of aerodynamic diameter of individual particles is not simple. It is recognized that complete dispersion of dry particulate solids, especially in the size range below 20 Am, is difficult due to strong cohesive forces between the particles. Particle sizing equipment that measures aerodynamic diameter of individual particle is widely available, but there are a number of biases associated with the use of these

P. Tang et al. / Powder Technology 147 (2004) 64–78

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Fig. 1. Scanning electron micrographs of (a) smooth and (b) rough bovine serum albumin particles.

instruments. Mitchell and Nagel [10] described in detail the working principle and limitations of measuring the aerodynamic diameters of aerosols from pressurized metered dose inhalers (pMDI) using equipment based on time-of-flight (TOF). The problems associated with the use of these equipment include: (i) coincidence effect when the aerosol concentration is too high; (ii) possibility of missing out particles if the intensity of scattered light drops below a threshold value; (iii) error associated with transforming the number to a volume (mass) weighted bases for particle size distribution; (iv) distortion of liquid droplets larger than 5 Am, having low surface tension and viscosity, due to high acceleration (ultra-Stokesian motion) in the measurement zone. Work involving non-spherical solid crystalline particles has shown an undersizing error of 20– 50% [11,12]. Careful verification of the results by an independent technique was strongly recommended [10]. However, it may be difficult to find a reliable independent technique. It is the aim of this work to develop a computational method to obtain aerodynamic diameters of corrugated objects with fractal surfaces. The computational method was developed using C D correlations reported in literature [5,13–17]. We did not make separate attempts to compute or measure the coefficients used in these correlations. Instead, our objective was to test which of these correlations is the most suitable for particles with fractal surface. To the authors’ best knowledge, no work has been reported in correlating surface fractal dimension and aeorodynamic diameter. Section 1.1 briefly describes the many attempts to formulate drag coefficient for non-spherical particles since the calculation of aerodynamic diameter requires the use of drag coefficient. Section 1.2 introduces the concept of a fractal object, which is followed by the method to construct the objects in Section 2.3. Sections 2.1 and 2.2 describe in detail the computational method used to obtain aerodynamic diameter of fractal objects. 1.1. Drag coefficient It is widely known that properties of a particle (e.g., size, shape, density, and orientation), together with properties of

the surrounding fluid (e.g., density and viscosity), significantly affect the terminal settling velocity. There is a vast number of literature reporting derived expressions of drag coefficient for non-spherical particles in Newtonian and non-Newtonian fluids [18–22]. The works on formulating the drag force for irregular particles are mainly dealt with for a specific shape, such as cylindrical particles [17,23,24], right triangles and rectangles [17], chains of spheres and cylinders [25,26], deformable bubbles and drops [23]. A method using self-similarity to numerically compute drag force exerted in Stokes’ regime (Reb0.1) has been reported Y [27]. In this method, the velocity field V is represented by superposition of n spheres of radius R i each moving with Y Y the virtual velocity U i (pU , the cluster velocity) and the drag coefficient was expressed three-dimensionally as [27]:

Y

CD ¼

6lL 1 8

n Y P Ui Ri i Y

ð1Þ

qL V 2 DA2 Y

Y

Y

The virtual velocity, U i , was chosen so that U ¼ V at the centre of each of the n spheres. The velocity of a primary particle at position j due to particle at position i was expressed as a matrix to represent the three-dimensional position of the particles. The calculation becomes cumbersome for large aggregates because it involves a matrix of n n size and requires a computer calculation (a computer code called Cluster Drag was used in their work). Chhabra et al. [28] wrote an excellent report comparing the available expressions to calculate drag on nonspherical particles by using 1900 data points covering 104bReb5105 and 0.09bwb1. Expressions derived by Haider and Levenspiel [13], Ganser [5], Hartman et al. [29], Swamee and Ojha [30], and Chien [31] 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 accurate with an average error of 16.3%. Based on the discussion by Chhabra et al., the drag coefficients derived by Haider and Levenspiel and Ganser will be used in this work. Furthermore, expres-

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P. Tang et al. / Powder Technology 147 (2004) 64–78

sions derived by Ro and Neethling [14], Tran-Cong et al. [16], Xie and Zhang [17], and Thompson and Clark [15] will also be employed because of the simplicity and ease to use which make them more appealing and attractive. Ro and Neethling derived a drag force relationship for biocoated particles, which are microorganisms coated with organic compounds removed from wastewater. As a result of the biofilm growth, there will be increase of both the skin drag caused by the rough surface, and form drag caused by non-sphericity of the particles. The most recent work on drag coefficient or irregularly shaped particles is that by Tran-Cong et al. covering shapes including plane and elongated conglomerates of spheres, orthotropic, axisymmetric, and isometric objects. Experimental data was shown to agree with the calculated drag coefficient within 5% for 0.15bReb3, 12% for 3bReb200 and 20% for 200bReb1500. Comparison with previous literature data [13,32] at low and intermediate Re numbers agreed to within 10%. At higher Re numbers, the discrepancy increased due to the formation of a strong wake behind the object causing them to wobble or spin during settling. To avoid the difficulty of obtaining surface area of irregular particles in order to get the sphericity values (defined in Eq. (9)), Thompson and Clark used a parameter called scruple, J, to take into account non-sphericity and surface roughness [15]. Scruple is defined as the ratio of drag coefficient of the particle and of a sphere of equivalent volume at Re=10,000. The value of 10,000 was chosen because most of the irregular shaped particles are fully in turbulent regime at this Re number and therefore the drag coefficient is constant. Then they constructed a unified diagram of C D as a function of Re for different scruple. Ganser later extended the works by Haider and Levenspiel [13] and Thompson and Clark [15] by including scruple in their shape factor, K 2 [5]. Due to the large number of shape descriptors required to classify irregular shapes, many works have been attempted to formulate a simple relationship to predict settling velocity of irregularly shaped particles without the necessity of determining sphericity [33,34]. Tsakalakis and Stamboltzis [34] presented an empirical relationship to predict settling velocities of crushed particles (galena, quartz) in water, involving only four parameters including size and density of particles together with density and viscosity of water. The difference between calculated and measured settling velocities was reported to be within 30%. Since the empirical formula does not include any shape and surface roughness factor, its applicability to determine settling velocity of other shaped particles should be further tested to confirm that the coefficients in the formula are not system specific.

Fig. 2. Self similarity of fractal surface [39].

roughness [35–37]. Mandelbrot [38] defines a fractal object as one having a dimension D which is greater than the topological dimension, but less than or equal to the dimension of the embedding Euclidean space. The value of D S varies from 2 for a perfectly smooth surface to 3 for a very rough surface. The most important property that distinguishes a fractal surface from a non-fractal one is self-similarity. This means that irrespective of length scales or magnification, the repetition of disorder within the structure takes place. This concept is illustrated in Fig. 2 [39]. As a first technique to determine the fractal dimension of a rugged boundary, a series of polygons of side length b is constructed. The perimeter of the polygon, P, then becomes the approximation of the perimeter at resolution b. The boundary fractal dimension, D L, can then be determined as follows [40,41]: P~b1DL

ð2Þ

Eq. (2) shows that D L can be obtained from the slope of a plot of perimeter, P, against the yard stick, b, on a log–log scale. This method is employed in this work to obtain D L and shown in detail in Section 3.1. D L, which varies from 1 to 2, describes the boundary line of a fractal object while D S describes a surface morphology. The relationship between D L and D S is [42]: DS ¼ ð1 þ DL Þ

ð3Þ

D S can also be obtained using other available methods include light scattering and gas adsorption. Light scattering [43–45] is usually favoured because of its simplicity and rapidness. However, the refractive indices of the particles and the suspending fluid has to be similar and the size of particle (R) is limited such that 2kRjm1jV1, where k=2p/ k. Gas adsorption method [35,37,46], on the other hand, is a much longer process. However, this method also yields surface area of the particles which is required to calculate sphericity, as shown in Eq. (9).

2. Computation method

1.2. Surface fractals

2.1. Drag coefficient of rough surfaced particles

A non-integer dimension called surface fractal dimension, D S, has been widely used to characterize particle surface

Ro and Neethling [14] extended the drag coefficient proposed earlier by Schiller and Naumann [47] to take into

P. Tang et al. / Powder Technology 147 (2004) 64–78

account the whole range of non-sphericity (01), S F, and surface roughness v: CD ¼

24v þ aSF2 Reb Re

ð4Þ

where daT and dbT are correlation coefficients. S F is a shape factor in this case and is defined as: SF ¼

dA dV

ð5Þ

and v is a skin factor to take into account the surface roughness, defined as: v¼

dS dV

ð6Þ

The equivalent surface area diameter was chosen to represent the surface roughness effect because the viscousdrag force (first term in Eq. (4)) is proportional to the wetted surface area of the settling particle. Reynolds number is calculated as [48]: Re ¼

U qL d V lL

ð7Þ

Equivalent volume diameter is the relevant dimension mostly used to calculate Re number of non-spherical particles [5,13,16,32]. Schiller and Naumann reported that for 0.001bReb1000, daT and dbT were found to be 3.6 and 0.313 [47], respectively. Ro and Neethling found that the rough and deformable biocoated particles settling with Reynolds number (15bReb87), daT and dbT were 21.55 and 0.518, respectively. Another expression of drag coefficient employed here is the one derived by Haider and Levenspiel [13] for Reb25,000 and 0.026bwb1.0. It was derived from the expressions obtained by Clift and Gauvin [49] and by Turton and Levenspiel [50], which were based on experimental data and by fitting the experimental drag data for non-spherical particles obtained by Pettyjohn and Christiansen [32]. The following expression was obtained: CD ¼

24 ½1 þ expð2:3288  6:4581w Re þ 2:4486w2 ÞReð0:0964þ0:5565wÞ þ

Re:expð4:905  13:8944w þ 18:4222w2  10:2599w3 Þ Re þ expð1:4681 þ 12:2584w  20:7322w2 þ15:8855w3 Þ ð8Þ

where sphericity,w, is defined as [32]:

W¼

As Ans

ð9Þ

67

As mentioned in Section 1.1., Thompson and Clark define scruple as [15]: J¼

CD CDS

ð10Þ

where C D and C DS are drag coefficients of the particle and a sphere (having equivalent volume) at Re=10,000. The drag coefficient, C D, at the actual settling velocity can then be determined either from the constructed unified diagram or from expression shown below: CD ¼

0:208 24 ð1 þ f0:178  100:056J gRe0:677J Þ Re

þ

Re:ð0:101 þ 0:366  JÞ Re þ 5732:1  J1:96

ð11Þ

Eq. (11) is more accurate for Reb10,000 and a wide range of scruple than the unified diagram. As with sphericity, scuple for sphere is 1. As explained in Section 1.1, Chhabra et al. found that the expressions derived by Haider and Levenspiel and Ganser were found to be accurate and therefore will be used in this work. The expression derived by Ganser [5] for ReK 1K 2b105 and 0.026bwb1 is:   CD 24 0:4305 0:6567 ¼ 1þ0:1118ðReK1 K2 Þ þ 3305 K2 ReK1 K2 1þ ReK1 K2 ð12Þ where K 1 and K 2 for isometric particles are:     1 1 2 0:5 K1 ¼ þ w 3 3 K2 ¼ 101:8148ðlogwÞ

0:5743

ð13Þ

ð14Þ

Two shape factors, K 1 and K 2, are needed in Eq. (12) because it was derived assuming that every isolated particle experiences a Stokes’ regime where drag is proportional to the velocity, and a Newton’s regime where drag is proportional to the square of velocity. These assumptions result in advantages that the way a particle behaves in these two regimes can be utilised to calculate the drag for a wide range of Reynolds number and enables the identification of distinctions in shape and orientation [5]. Xie and Zhang [17] derived an empirical shape correction factor and drag coefficients of non-spherical particles in the laminar regime (Reb1) for specially prepared particles with sphericity varying from 0.2 to 1: CD ¼ w0:83

24 Re

ð15Þ

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P. Tang et al. / Powder Technology 147 (2004) 64–78

In our work, the distance between the container wall and the settling particles is assumed to be sufficient, so that no effect is exerted by the wall during particle settling. Furthermore, the slip correction factor is not taken into account in the calculation of aerodynamic diameter because the particles of interest are larger than 1 Am [48]. Tran-Cong et al. formulated C D for 0.05bReb1500, 0.80bd A/d Vb1.50, and 0.4bcb1.0 [16]: 0:687    24 dA 0:15 dA Re CD ¼ 1 þ pffiffiffi Re dV c dV  2 0:42 ddAV þ 1:16    pffiffiffi c 1 þ 4:25  104 ddAV Re

ð16Þ

where c is the particle circularity defined as: c¼

pdA Pp

ð18Þ

From dimensional analysis, F D can be expressed in terms of drag coefficient, C D, as [21]: 2FD AA qL U 2

2 dae g 18lL

In our calculation, we compared the aerodynamic diameters obtained from Eqs. (20) and (21) and found agreement only for smooth spherical particles. This is because Eq. (21) was derived by assuming that the particle is a sphere having C D=24/Re and did not take into account the surface roughness. It is therefore not suitable for calculation of the aerodynamic diameter of rough particles.

The drag coefficients employed in this work require the use of shape factor involving the equivalent diameters. To be able to obtain surface fractal dimension of various objects with their equivalent diameters, a set of model objects having self-similar surfaces was built. These objects were built from cubes and spheres to allow simple determination of d A, d V, d S required in drag coefficient computation. The perimeter fractal dimension, D L, was then obtained from the plot of perimeter, P, against yardstick, b, as shown in Eq. (2). Once D L is obtained then the surface fractal dimension, D S, can easily be calculated using Eq. (3).

3. Results and discussion ð19Þ 3.1. Surface fractal dimension of model objects

According to the definition of aerodynamic diameter (d ae), which is the diameter of a unit density sphere (q O=1 g/ cm3) that has the same settling velocity as the particle, d ae can be computed from the Stokes’ equation once the settling velocity is obtained:

U¼

ð21Þ

2.3. Construction of objects with fractal surface

Based on a balance of the three forces, namely, gravity, bouyancy, and drag force, acting on a single particle settling in a fluid, the drag force, F D, for each corresponding particle can be computed using the formula below:

CD ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6V qP dae ¼ pdO qO

ð17Þ

2.2. Aerodynamic diameter of rough surfaced particles

FD ¼ VP ðqP  qL Þg

difference between settling velocities obtained from Eqs. (4), (8), (11), (12), (15), or (16) and Eq. (19). Gonda and Khalik [51] derived a simple expression to compute aerodynamic diameter as:

ð20Þ

In summary, the drag coefficient is first calculated using Eqs. (4), (8), (11), (12), (15), or (16), then the settling velocity is calculated using Eq. (19) followed by determination of aerodynamic diameter from Eq. (20). As can be seen from Eqs. (4), (8), (11), (12), (15), or (16), the computation of drag coefficient involves Reynolds number which is a function of settling velocity. Therefore, an iteration process is required and stopped when there is no

Objects having surface fractal dimension varying from 2.17 to 2.55 were successfully built. They are shown in Fig. 3. From these objects, projected area can be easily calculated. For example, for the model object shown in Fig. 3(c), projected area equals 25s 2 where s is the length of each primary cube. These objects are built identically in three dimensions having equal axes at right angles to each other, i.e., isometrically, so that there is no preferred orientation in the settling direction. In other words, the side view is identical to the front view, as illustrated in Fig. 4. Once the three-dimensional structure of the object is completed then the volume and total external surface area can be computed. From this point, all the necessary parameters needed to calculate shape and roughness correction factors can be obtained.

P. Tang et al. / Powder Technology 147 (2004) 64–78

69

3.2. Aerodynamic diameter of the fractal objects Aerodynamic diameter will be affected by the volume, projected area, and the nature of the external surface of a particle. Since it is our objective to compare the aerodynamic

Fig. 3 (continued).

diameter of smooth and rough particles, it is necessary that the objects have the same volume and projected area. Therefore, the aerodynamic behaviour will depend only on the nature of the external surface, which is reflected in the surface fractal dimension. Any values of projected area and volume of the model fractal objects can be chosen as long as the w, Re, ReK 1K 2, d A/d V, and c criteria for drag coefficients expressed in Eqs. (4), (8), (11), (12), (15), and (16) are satisfied. In this work, projected area and volume of the fractal objects were set to be 300 and 4500 Am 3 , respectively. Then the length of each cube in Fig. 3(b)–(e) and diameter of each sphere in Fig. 3(a) were calculated based on the set values of projected area, A A, and volume, V. These values are listed in Table 1 together with the average value of size calculated from the projected area and volume. Objects 2 to 6 have rough surfaces while a model of an object with a smooth surface having surface fractal dimension of 2.00 is represented by a single sphere (object 1). For the calculation of Reynolds number, the values of density and viscosity of air at 20 8C and atmospheric pressure used are 1.2 kg/m3 and 1.8105 Pa s, respectively [48]. The density of the model particle used for calculation is 8000 kg/m3 (this value was chosen to satisfy the Re

Fig. 3. Determination of surface fractal dimension for object (a) D S=2.17; (b) D S=2.21 (c) D S=2.24; (d) D S=2.42; (e) D S=2.55. Encircled unit shows a repeating unit.

Fig. 4. (a) Front view and (b) side view of an object with surface fractal dimension of 2.24. The white cubes in (a) are the cubes from the front view that will be seen from the viewing direction. These cubes are again shown in white in (b). To make the side view exactly the same as the front view, 16 cubes have to be added to the units. These cubes are shown in grey in (b).

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P. Tang et al. / Powder Technology 147 (2004) 64–78

Table 1 Summary of primary unit size based on projected area and volume Objects

Object 1, DS=2.00,

Object 2, DS=2.17,

Size of primary unit (Am) based on projected area of 300 Am2

Size of primary unit (Am) based on volume of 4500 Am3

Average size of primary unit (Am)

19.54

20.98

20.01

6.18 (diameter of the primary spheres)

7.97

Table 2 Summary of settling velocity, U, Reynolds number, Re, drag coefficient, C D, and aerodynamic diameter, d ae, of fractal objects

7.07

Object 3, DS=2.21,

4.33 (length of the primary cube)

5.44

4.88

Object 4, DS=2.24,

3.46 (as above)

4.79

4.13

Object 5, DS=2.42,

Object 6, DS=2.55,

1.92 (as above)

0.91 (as above)

3.09

1.85

2.51

1.38

criterion in the equations used). The model objects were first assumed to settle in the region 0.001bReb1000 and therefore the value of a and b used in Eq. (4) were 3.6 and 0.313, respectively. This can be confirmed later once the settling velocity is obtained from iteration. To confirm the validity of the calculation, the aerodynamic diameter of object 1 is also calculated using Eq. (22) which is derived simply from Stokes’ equation: rffiffiffiffiffiffi qP dae ¼ dV qO

and A A used in Eq. (19) are 4195 Am3 and 314 Am2, respectively (based on d V=20.01 Am). The apparent difference between the value calculated here (44.74 Am) and the presented value in Table 2 (58.49Am b d aeb 60.02Am) is

ð22Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ¼ The aerodynamic diameter (dae ¼ 20:01 lm 8000kg=m 1000kg=m3 56:60 lm) indeed agrees between the two methods when V

(a) Calculated using Eq. (4) by Ro and Neethling Object DS U (m/s) Re CD

d ae (Am)

1 2 3 4 5 6

60.02 44.66 44.63 42.38 41.16 34.78

2.00 2.17 2.21 2.24 2.42 2.55

1.09e01 6.04e02 6.03e02 5.44e02 5.13e02 3.66e02

1.46e01 7.32e02 7.40e02 6.40e02 5.69e02 3.73e02

1.65e+02 5.38e+02 5.40e+02 6.64e+02 7.46e+02 1.46e+03

(b) Calculated using Eq. (8) by Haider and Levenspiel Object D S w U (m/s) Re CD

d ae (Am)

1 2 3 4 5 6

60.02 46.06 45.78 42.95 42.18 33.17

2.00 2.17 2.21 2.24 2.42 2.55

1.00 0.39 0.38 0.33 0.30 0.25

1.09e01 6.42e02 6.35e02 5.59e02 5.39e02 3.33e02

1.46e01 7.79e02 7.78e02 6.57e02 5.97e02 3.40e02

1.65e+02 4.75e+02 4.87e+02 6.29e+02 6.76e+02 1.77e+03

(c) Calculated using Eq. (11) by Thompson and Clark Object D S J U (m/s) Re CD

d ae (Am)

1 2 3 4 5 6

1.83e+02 3.54e+02 3.62e+02 4.83e+02 5.14e+02 1.60e+03

58.49 49.58 49.32 45.89 45.17 34.01

(d) Calculated using Eq. (12) by Ganser Object D S ReK 1K 2 U (m/s) Re

CD

d ae (Am)

1 2 3 4 5 6

1.77e+02 4.67e+02 4.68e+02 5.59e+02 6.16e+02 1.10e+03

58.94 46.27 46.24 44.23 43.17 37.32

2.00 2.17 2.21 2.24 2.42 2.55

1.00 12.31 12.90 15.80 28.44 43.44

2.00 2.17 2.21 2.24 2.42 2.55

1.04e01 7.44e02 7.37e02 6.37e02 6.18e02 3.50e02

1.40e01 6.86e01 7.16e01 7.17e01 6.34e01 6.98e01

1.38e01 9.03e02 9.03e02 7.50e02 6.85e02 3.57e02

1.05e01 6.48e02 6.47e02 5.92e02 5.64e02 4.22e02

1.40e01 7.86e02 7.94e02 6.97e02 6.25e02 4.30e02

(e) Calculated using Eq. (15) by Xie and Zhang Object DS U (m/s) Re

CD

d ae (Am)

1 2 3 4 5 6

1.65e+02 4.16e+02 3.88e+01 3.44e+01 3.97e+01 2.01e+01

60.02 84.66 86.18 88.81 85.69 101.57

2.00 2.17 2.21 2.24 2.42 2.55

1.09e01 2.17e01 2.25e01 2.39e01 2.22e01 3.12e01

1.46e01 2.63e01 2.76e01 2.81e01 2.46e01 3.18e01

(f) Calculated using Eq. (16) by Tran-Cong et al. Object D S d A/d V c U (m/s) Re

CD

d ae (Am)

1 2 3 4 5 6

1.78e+02 3.23e+02 3.09e+02 4.07e+02 5.98e+02 1.14e+03

58.90 50.74 51.30 47.88 43.49 37.04

2.00 2.17 2.21 2.24 2.42 2.55

1.00 1.23 1.20 1.32 1.53 1.57

1.02 1.03 0.47 0.41 0.80 0.52

1.05e01 7.80e02 7.97e02 6.94e02 5.73e02 4.15e02

1.40e01 9.45e02 9.77e02 8.17e02 6.35e02 5.24e02

P. Tang et al. / Powder Technology 147 (2004) 64–78

simply because of the different set values of volume (4500 Am3) and projected area (300 Am2) used for the sphere (object 1). However, since our intention is to compare the aerodynamic diameter of smooth and rough particles, it is necessary that the objects have the same volume and projected area. Therefore, the set value of volume (4500 Am3) and projected area (300 Am2) are used for calculation. Table 2 shows the values of calculated aerodynamic diameters for objects with different surface fractal dimensions using the drag coefficient C D calculated from Eqs. (4), (8), (11), (12), (15), and (16). In Table 2, object 1 having a smooth surface has the largest aerodynamic diameter. As the object surface becomes rough, the aerodynamic diameter gets smaller due to the higher drag forces exerted by the surrounding fluid. It is also interesting to note that the difference of aerodynamic diameter between the smooth surface object and the roughest object 6 having D S of 2.55, shown in Table 2(a)–(d) and (f), is less than 44%. From Table 2, it is also shown that the criteria of w, Re, ReK 1K 2, d A/d V, c are satisfied and therefore insignificant error can be expected. Furthermore, Table 2(a) shows that the Reynolds number being higher than 0.001 and less than 1000 confirm the use of a and b of 3.6 and 0.313, respectively, in Eq. (4). Fig. 5 shows that the aerodynamic diameters, calculated using Eqs. (4), (8), (11), (12), and (16), decrease with surface fractal dimension. When Eq. (15) was used, the dependence of the aerodynamic diameter opposes the expected trend. In order to check whether the coefficient used in Eq. (15) is system specific or not, the density and viscosity of suspending liquid and density of particle were changed to the values used by Xie and Zhang [17]. However, the dependence of aerodynamic diameter on the sphericity still shows opposite trend which demonstrates that the coef-

71

ficient in Eq. (15) is not system specific. Xie and Zhang mainly used three different particle shapes, namely, cylinder, right triangle, and rectangle, to formulate the expression of drag coefficient and, furthermore, no effect of settling orientation was mentioned during the formulation. Therefore, Eq. (15) might be shape and orientation specific. In the work by Xie and Zhang, errors carried in the calculation were not reported. The similar performance between the equation derived by Haider and Levenspiel and Ganser is expected, as reported by Chhabra et al. [28]. The two methods were reported to show overall mean errors ranging from 16.6% to 24.1% when compared with data from the literature. Although Eqs. (8) and (12) only include sphericity, w, of the particle in the calculation of drag coefficient, the aerodynamic diameters calculated follow the expected trend. This is because in calculating the sphericity, w, as expressed in Eq. (9), the effect of surface roughness might have been already taken into account by including the surface area of the particle. The uses of scruple( J), d A/d V, and c in Eqs. (11) and (16) are obviously sufficient to take into account roughness and non-sphericity. It can be observed also from Table 2 that differences in aerodynamic diameter calculated using drag coefficients shown in Eqs. (4), (8), (11), (12), and (16) are small. These values are again listed together with the average, standard deviation and percentage variation in Table 3. Differences in aerodynamic diameter when C D was calculated by Eqs. (4), (8), (11), (12), and (16) are within 6% (Table 3), indicating that the drag coefficients of the various equations perform similarly. Since Eqs. (4), (8), (11), (12), and (16) have been well developed and shown to agree with extensive collections of literature data, their similar performance lends further support to the

Fig. 5. Plot of aerodynamic diameter, calculated using drag coefficient obtained from Eqs. (4), (8), (11), (12), (15), and (16), as a function of surface fractal dimension, D S.

72

P. Tang et al. / Powder Technology 147 (2004) 64–78

Table 3 (a) The aerodynamic diameters calculated using drag coefficients shown in Eq. (4) by Ro and Neethling, Eq. (8) by Haider and Levenspiel, Eq. (11) by Thompson and Clark, Eq. (12) by Ganser, and Eq. (16) by Tran-Cong et al. Object D S Aerodynamic diameter (Am)

1 2 3 4 5 6

2.00 2.17 2.21 2.24 2.42 2.55

Ro and Neething

Haider and Levenspiel

Thompson and Clark

Ganser

TransCong et al.

60.02 44.66 44.63 42.38 41.16 34.78

60.02 46.06 45.78 42.95 42.18 33.17

58.49 49.58 49.32 45.89 45.17 34.01

58.94 46.27 46.24 44.23 43.17 37.32

58.90 50.74 51.30 47.88 43.49 37.04

(b) Percentage variation of all the aerodynamic diameters listed in (a) Object DS Average Standard Variation deviation (%) 1 2 3 4 5 6

2.00 2.17 2.21 2.24 2.42 2.55

59.27 47.46 47.45 44.66 43.03 35.27

0.70 2.57 2.77 2.25 1.50 1.84

1.19 5.42 5.83 5.03 3.50 5.22

3.3. Comparison between calculated settling velocities and literature data By referring to the original works of Haider and Levenspiel, Thompson and Clark, Ganser, and Tran-Cong et. al, one can see how the settling velocities calculated from the C D expressions have been compared with literature data and shown to agree well. Since Ro and Neethling have not done so, to be extensive, we also conducted a comparison to confirm the accuracy of the C D correlation. Since Table 3 shows similar results from Eqs. (4), (8), (11), (12), and (16), we did random test using Eqs. (8) and (12) only. Chhabra et al. [28] had used 1900 individual data collected from 19 works investigating the effect of nonsphericity on settling velocities. However, since the fractal objects employed in this work are isometric with flow in the Table 4 Percentage difference between calculated settling velocity using Eq (4), U C, and those measured by Pettyjohn and Christiansen, U M, for isometric particles of different sizes and materials in glucose–water solution (see Appendix A for more data) Glucose–water solution

validity of the present work as a quick estimation of aerodynamic diameter of rough surfaced particles. The particle density used in this work (8000 kg/m3) is higher than those usually encountered in pharmaceutical applications (densityb2000 kg/m3). However, as long as the criteria for the chosen C D expression are satisfied, the aerodynamic diameter calculated should be accurate regardless of the density. Furthermore, since Eqs. (4), (8), (11), (12), and (16) generate similar results, one can use any of these equations to calculate the aerodynamic diameter of particles. There was an interesting finding as we tried to calculate aerodynamic diameters using Eq. (21). When the fractal objects are set to have the same volume and projected area, the aerodynamic diameters of the constructed objects with different surface fractal dimensions is the same. This can be explained by referring to Eq. (19). By combination with Eq. (18) and re-arranging it, the following relationship is obtained: 2VP ðqP  qL Þg CD U ¼ AA qL 2

Density Viscosity

1.3456 g/ml 7.201 P

Cubes

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Lead Lead tin

1.77e02 1.13e02

2.04e02 1.31e02

6.12e02 3.88e02

13.20 14.05

Octahedron

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Lead Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium

2.01e02 1.20e02 2.85e03 6.10e03 1.02e02 1.02e02 1.13e02 3.82e02

1.67e02 1.01e02 2.39e03 5.04e03 8.48e03 8.42e03 9.34e03 3.16e02

5.33e02 3.12e02 7.33e03 2.27e02 5.05e02 4.91e02 5.77e02 3.78e01

20.71 18.90 19.19 21.03 20.79 21.38 21.00 20.82

Glucose–water solution Density Viscosity

1.3796 g/ml 36.1 P

Cubes

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

3.52e03 4.80e03 1.37e02 2.20e03 3.37e03 8.83e02 1.95e03 3.02e03 6.67e03 7.71e03 1.17e02

4.12e03 5.67e03 1.61e02 2.62e03 4.08e03 1.05e02 2.34e03 3.63e03 8.04e03 9.27e03 1.42e02

2.52e03 4.07e03 1.95e02 1.59e03 3.09e03 1.27e02 2.84e03 5.38e03 1.81e02 2.25e02 4.26e02

14.55 15.41 15.08 16.06 17.39 15.62 16.73 16.81 17.00 16.83 17.33

ð23Þ

From Eq. (23), when volume and projected area of the objects are fixed, the drag coefficient and settling velocity will be constant, resulting in non-dependence of aerodynamic diameter on the surface fractal dimension. This is because Eq. (21) was derived without taking into account the surface roughness and non-sphericity of the particles. It is therefore not suitable for calculation of aerodynamic diameter of rough particles.

P. Tang et al. / Powder Technology 147 (2004) 64–78

73

Table 5 Percentage difference between calculated settling velocity using Eq (8), U C, and those measured by Pettyjohn and Christiansen, U M, for isometric particles of different sizes and materials in glucose–water solution (see Appendix B for more data)

Table 6 Percentage difference between calculated settling velocity using Eq (12), U C, and those measured by Pettyjohn and Christiansen, U M, for isometric particles of different sizes and materials in glucose–water solution (see Appendix C for more data)

Glucose–water solution

Glucose–water solution

Density Viscosity

1.3456 g/ml 7.201 P

Density Viscosity

1.3456 g/ml 7.201 P

Cubes

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Cubes

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Lead Lead tin

1.77e02 1.13e02

2.18e02 1.41e02

6.52e02 4.17e02

18.54 20.02

Lead Lead tin

1.77e02 1.13e02

2.06e02 1.33e02

6.19e02 3.95e02

14.15 15.38

Octahedron

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Octahedron

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Lead Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium

2.01e02 1.20e02 2.85e03 6.10e03 1.02e02 1.02e02 1.13e02 3.82e02

1.76e02 1.08e02 2.57e03 5.39e03 8.98e03 8.93e03 9.88e03 3.25e02

5.64e02 3.32e02 7.89e03 2.42e02 5.35e02 5.21e02 6.11e02 3.88e02

13.98 11.64 10.75 13.31 13.99 14.51 14.36 17.60

Lead Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

2.01e02 1.20e02 2.85e03 6.10e03 1.02e02 1.02e02 1.13e02

1.69e02 1.03e02 2.44e03 5.13e03 8.60e03 8.54e03 9.47e03

5.40e02 3.17e02 7.49e03 2.31e02 5.12e02 4.99e02 5.86e02

19.00 16.95 16.67 18.89 19.05 19.61 19.32

Glucose–water solution Glucose–water solution Density Viscosity

1.3796 g/ml 36.1 P

Cubes

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

3.52e03 4.80e03 1.37e02 2.20e03 3.37e03 8.83e03 1.95e03 3.02e03 6.67e03 7.71e03 1.17e02

4.54e03 6.24e03 1.74e02 2.90e03 4.50e03 1.14e02 2.58e03 3.99e03 8.73e03 1.00e02 1.52e02

2.78e03 4.48e03 2.11e02 1.76e03 3.40e03 1.38e02 3.13e03 5.91e03 1.97e02 2.44e02 4.57e02

22.57 23.15 21.77 24.09 25.06 22.64 24.50 24.29 23.60 23.23 22.94

Density Viscosity

1.3796 g/ml 36.1 P

Cubes

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

3.52e03 4.80e03 1.37e02 2.20e03 3.37e03 8.83e03 1.95e03 3.02e03 6.67e03 7.71e03 1.17e02

24.24e03 5.84e03 1.64e02 2.71e03 4.20e03 1.07e02 2.41e03 3.73e03 8.21e03 9.46e03 1.44e03

2.60e03 4.19e03 1.99e02 1.64e03 3.18e03 1.30e02 2.92e03 5.53e03 1.85e02 2.29e02 4.32e02

17.05 17.78 16.84 18.60 19.77 17.59 19.15 19.06 18.75 18.47 18.53

Glucose–water solution Glucose–water solution Density Viscosity

1.425 g/ml 915.7 P

Cubes

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

5.18e04 1.98e03 7.30e03 3.36e04 1.29e03 4.78e03 2.88e04 4.33e04 1.23e03 1.05e03 1.57e03

7.07e04 2.78e03 1.11e02 4.58e04 1.83e03 7.31e03 3.96e04 6.08e04 1.86e03 1.59e03 2.47e03

3.49e05 2.74e04 2.18e03 2.26e05 1.81e04 1.44e03 3.91e05 7.44e05 3.99e04 3.13e04 6.07e04

26.78 28.77 34.28 26.76 29.64 34.66 27.25 28.81 34.05 33.63 36.23

Density Viscosity

1.425 g/ml 915.7 P

Cubes

U M (m/s)

U C (m/s)

Re

D(U MU C) (%)

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

5.18e04 1.98e03 7.30e03 3.36e04 1.29e03 4.78e03 2.88e04 4.33e04 1.23e03 1.05e03 1.57e03

6.58e04 2.59e03 1.04e02 4.27e04 1.70e03 6.82e03 3.69e04 5.66e04 1.73e03 1.48e03 2.30e03

3.24e05 2.55e04 2.03e03 2.11e05 1.68e04 1.35e03 3.65e05 6.93e05 3.72e04 2.92e04 5.66e04

21.29 23.48 29.57 21.33 24.41 29.93 21.84 23.49 29.17 28.73 31.53

74

P. Tang et al. / Powder Technology 147 (2004) 64–78

Newtonian liquid and laminar region, the simplest and most readily available data for the calculation of d V, d S, d A, V, and A A are those from Pettyjohn and Christiansen. Table 4 shows the comparison between calculated and measured settling velocities. Table 4 shows that Eq. (4) predicts quite accurately the settling velocity, with maximum difference less than 22%. The disparity could be due to dependency of daT and dbT on the properties of particle and suspending medium since it was derived for sand particles covered by microorganisms [14]. Table 5 also shows that Eq. (8) predicts settling velocity accurately especially for cube and octahedron particles (within 20%) settling in glucose–water solution with viscosity 7.201 P. However, for solutions with higher viscosity of 36.1 and 915.7 P, the percentage differences are higher. Even for the same shape particles, particles settled in solution with viscosity 915.7 P have, on average, higher percentage difference. This might be because during the development of their empirical correlation, Haider and Levesnpiel did not use the whole set of data presented by Pettyjohn and Christiansen. Ganser extended the drag coefficients formulated by Haider and Levenspiel and Thompson and Clark used the same data utilised by Haider and Levenspiel. In general, Table 6 shows lower percentage differences between calculated and measured settling velocities when Eq. (12) was used instead of Eq. (8), as also found by Chhabra et al. [28]. Table 5 shows lower percentage differences when the liquid viscosity was lower, as low as 10% for viscosity of 7.201 P. The settling velocity comparisons shown in Tables 4–6 support the choice of the drag coefficients used in our work being suitable because of the smaller percentage difference between measured and calculated values. This comparison shows, in the first instance, that it is possible to predict the aerodynamic diameters of particles with fractal surfaces with acceptable accuracy using the proposed method.

4. Conclusions A computational method to obtain aerodynamic diameter of an object with fractal surfaces has been developed. Construction of model objects with fractal surfaces enabled the determination of surface fractal dimension and equivalent diameters associated with volume, projected area, and external surface area. A number of drag coefficient expressions was employed to assess their suitability to predict the aerodynamic diameter of these fractal objects. The drag coefficient expressions derived by Ro and Neethling, Haider and Levenspiel, Thompson and Clark, Ganser, and TranCong et al. generate similar aerodynamic diameters with percentage variation less than 6%. The results calculated from those drag coefficients follow the expected trend of decreasing aerodynamic diameter with increasing surface roughness. Comparison with literature data from Pettyjohn and Christiansen shows that the C D expressions by Ro and Neethling, Haider and Levenspiel, and Ganser were able to predict the

aerodynamic diameter with average difference varies from 17% to 23%. This calculation technique serves as a quick estimation of aerodynamic diameter that might be applied to situations when direct measurement of aerodynamic diameter using available equipment fails due to equipment limitations or difficulty in dispersing powders into individual particles in the aerosol. Nomenclature Surface fractal dimension DS Re Particle Reynolds number Y V Velocity field (see Eq. (1)) Y Ui Virtual velocity of each primary sphere comprising an aggregate (see Eq. (1)) Y U Cluster velocity (see Eq. (1)) Ri Radius of each primary sphere comprising an aggregate (see Eq. (1)) D A Diameter of an aggregate w Sphericity lL viscosity of suspending fluid qL density of suspending fluid J Hydrodynamic particle shape descriptor called scruple P Perimeter of the polygon b Yardstick used to measure perimeter of object with fractal surface D L Boundary fractal dimension k Wavelength of the incident light m Complex refractive index of the particle relative to that of the surrounding medium SF Shape factor (see Eq. (5)) v Skin factor (see Eq. (6)) dA Equivalent projected area diameter (diameter of a sphere having the same projected area as the particle) dV Equivalent volume diameter (diameter of a sphere having the same volume as the particle) dS Equivalent surface area diameter (diameter of a sphere having the same surface area as the particle) U Particle terminal settling velocity A s, A ns Surface area of spherical and non-spherical particles of equal volume and density, respectively C D, C DS Drag coefficient of the particle and a sphere (having equivalent volume) at Re=10,000 (see Eq. (11)) K 1, K 2 The Stokes’ and Newton’s shape factors, respectively (see Eq. (12)) c Particle circularity (see Eq. (17)) Pp Projected perimeter of the particles in its direction of motion FD Drag force V P, q P Volume and density of the particle, respectively g Gravitational acceleration AA Projected area of the particle normal to the settling direction d ae Aerodynamic diameter dO Diameter of a sphere having the same viscous drag as the particle in interest.

P. Tang et al. / Powder Technology 147 (2004) 64–78

75

Appendix A

Glucose–water solution Density Viscosity

1.3456 g/ml 7.201 P

Cubes

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead tin

1.12e+04 7.74e+03

1.60e03 1.59e03

1.46e03 1.44e03

1.79e03 1.77e03

4.44e+02 6.97e+02

Octahedron

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium

1.12e+04 7.76e+03 2.87e+03 2.85e+03 2.79e+03 2.84e+03 2.83e+03 2.77e+03

1.71e03 1.65e03 1.64e03 2.40e03 3.19e03 3.12e03 3.31e03 6.39e03

1.86e03 1.79e03 1.78e03 2.61e03 3.47e03 3.40e03 3.60e03 6.95e03

1.86e03 1.79e03 1.78e03 2.61e03 3.47e03 3.40e03 3.60e03 6.95e03

5.01e+02 8.50e+02 3.58e+03 1.17e+03 5.28e+02 5.42e+02 4.62e+02 7.49e+01

Glucose–water solution Density Viscosity

1.3796 g/ml 36.1 P

Cubes

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

1.12e+04 1.12e+04 1.12e+04 7.74e+03 7.74e+03 7.77e+03 2.80e+03 2.85e+03 2.79e+03 2.80e+03 2.79e+03

1.60e03 1.88e03 3.17e03 1.59e03 1.98e03 3.17e03 3.18e03 3.88e03 5.91e03 6.35e03 7.87e03

1.46e03 1.71e03 2.88e03 1.44e03 1.80e03 2.88e03 2.89e03 3.53e03 5.37e03 5.77e03 7.16e03

1.79e03 2.09e03 3.53e03 1.77e03 2.21e03 3.53e03 3.54e03 4.33e03 6.58e03 7.07e03 8.76e03

1.06e+04 6.58e+03 1.38e+03 1.68e+04 8.68e+03 2.12e+03 9.44e+03 4.98e+03 1.48e+03 1.20e+03 6.36e+02

Appendix B

Glucose–water solution Density Viscosity

1.3456 g/ml 7.201 P

Cubes

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead tin

1.12e+04 7.74e+03

1.60e03 1.59e03

1.46e03 1.44e03

1.79e03 1.77e03

3.91e+02 6.03e+02

Octahedron

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead tin Aluminium Octahedron

1.12e+04 7.76e+03 2.87e+03 Density (kg/m3)

1.71e03 1.65e03 1.64e03 d V (m)

1.86e03 1.79e03 1.78e03 d A (m)

1.86e03 1.79e03 1.78e03 d S (m)

4.46e+02 7.49e+02 3.09e+03 CD

Aluminium Aluminium Aluminium Aluminium Aluminium

2.85e+03 2.79e+03 2.84e+03 2.83e+03 2.77e+03

2.40e03 3.19e03 3.12e03 3.31e03 6.39e03

2.61e03 3.47e03 3.40e03 3.60e03 6.95e03

2.61e03 3.47e03 3.40e03 3.60e03 6.95e03

1.02e+03 4.70e+02 4.82e+02 4.13e+02 7.09e+01

76

P. Tang et al. / Powder Technology 147 (2004) 64–78

Appendix B (continued) Glucose–water solution Density Viscosity

1.3796 g/ml 36.1 P

Cubes

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

1.12e+04 1.12e+04 1.12e+04 7.74e+03 7.74e+03 7.77e+03 2.80e+03 2.85e+03 2.79e+03 2.80e+03 2.79e+03

1.60e03 1.88e03 3.17e03 1.59e03 1.98e03 3.17e03 3.18e03 3.88e03 5.91e03 6.35e03 7.87e03

1.46e03 1.71e03 2.88e03 1.44e03 1.80e03 2.88e03 2.89e03 3.53e03 5.37e03 5.77e03 7.16e03

1.79e03 2.09e03 3.53e03 1.77e03 2.21e03 3.53e03 3.54e03 4.33e03 6.58e03 7.07e03 8.76e03

8.71e+03 5.43e+03 1.17e+03 1.37e+04 7.14e+03 1.78e+03 7.76e+03 4.13e+03 1.26e+03 1.02e+03 5.53e+02

Glucose–water solution Density Viscosity

1.425 g/ml 915.7 P

Cubes

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

1.12e+04 1.12e+04 1.12e+04 7.77e+03 7.75e+03 7.77e+03 2.80e+03 2.79e+03 2.79e+03 2.80e+03 2.80e+03

3.17e03 6.32e03 1.26e02 3.71e03 6.53e03 1.27e02 6.35e03 7.87e03 1.38e02 1.27e02 1.58e02

2.88e03 5.75e03 1.15e02 2.88e03 5.77e03 1.16e02 5.77e03 7.16e03 1.25e02 1.15e02 1.44e02

3.53e03 7.04e03 1.40e02 3.53e03 7.07e03 1.41e02 7.07e03 8.76e03 1.53e02 1.41e02 1.76e02

6.89e+05 8.80e+04 1.11e+04 1.06e+06 1.33e+05 1.67e+04 6.14e+05 3.23e+05 6.04e+04 7.68e+04 3.97e+04

Appendix C

Glucose–water solution Density Viscosity

1.3456 g/ml 7.201 P

Cubes

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead tin

1.12e+04 7.74e+03

1.60e03 1.59e03

1.46e03 1.44e03

1.79e03 1.77e03

4.34e+02 6.75e+02

Octahedron

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

1.12e+04 7.76e+03 2.87e+03 2.85e+03 2.79e+03 2.84e+03 2.83e+03

1.71e03 1.65e03 1.64e03 2.40e03 3.19e03 3.12e03 3.31e03

1.86e03 1.79e03 1.78e03 2.61e03 3.47e03 3.40e03 3.60e03

1.86e03 1.79e03 1.78e03 2.61e03 3.47e03 3.40e03 3.60e03

4.87e+02 8.22e+02 3.43e+03 1.13e+03 5.13e+02 5.26e+02 4.50e+02

Glucose–water solution Density Viscosity

1.3796 g/ml 36.1 P

P. Tang et al. / Powder Technology 147 (2004) 64–78

77

Appendix C (continued) Cubes

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

1.12e+04 1.12e+04 1.12e+04 7.74e+03 7.74e+03 7.77e+03 2.80e+03 2.85e+03 2.79e+03 2.80e+03 2.79e+03

1.60e03 1.88e03 3.17e03 1.59e03 1.98e03 3.17e03 3.18e03 3.88e03 5.91e03 6.35e03 7.87e03

1.46e03 1.71e03 2.88e03 1.44e03 1.80e03 2.88e03 2.89e03 3.53e03 5.37e03 5.77e03 7.16e03

1.79e03 2.09e03 3.53e03 1.77e03 2.21e03 3.53e03 3.54e03 4.33e03 6.58e03 7.07e03 8.76e03

1.00e+04 6.22e+03 1.33e+03 1.58e+04 8.18e+03 2.02e+03 8.90e+03 4.72e+03 1.42e+03 1.15e+03 6.18e+02

Glucose–water solution Density Viscosity

1.425 g/ml 915.7 P

Cubes

Density (kg/m3)

d V (m)

d A (m)

d S (m)

CD

Lead Lead Lead Lead tin Lead tin Lead tin Aluminium Aluminium Aluminium Aluminium Aluminium

1.12e+04 1.12e+04 1.12e+04 7.77e+03 7.75e+03 7.77e+03 2.80e+03 2.79e+03 2.79e+03 2.80e+03 2.80e+03

3.17e03 6.32e03 1.26e02 3.17e03 6.35e03 1.27e02 6.35e03 7.87e03 1.38e02 1.27e02 1.58e02

2.88e03 5.75e03 1.15e02 2.88e03 5.77e03 1.16e02 5.77e03 7.16e03 1.25e02 1.15e02 1.44e02

3.53e03 7.04e03 1.40e02 3.53e03 7.07e03 1.41e02 7.07e03 8.76e03 1.53e02 1.41e02 1.76e02

7.96e+05 1.02e+05 1.28e+04 1.23e+06 1.54e+05 1.93e+04 7.09e+05 3.73e+05 6.96e+04 8.86e+04 4.58e+04

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