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Modeling of chord length distributions
Chemical Engineering Science 61 (2006) 3962 – 3973 www.elsevier.com/locate/ces
Modeling of chord length distributions Marie-Noëlle Pons a,∗ , Kim Milferstedt b , Eberhard Morgenroth b,c a Laboratoire des Sciences du Génie Chimique, CNRS–ENSIC–INPL, 1, rue Grandville, BP 20451, F-54001 Nancy cedex, France b Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 N. Mathews Avenue, Urbana, IL 61801, USA c Department of Animal Sciences, University of Illinois at Urbana-Champaign, 205 N. Mathews Avenue, Urbana, IL 61801, USA
Received 24 May 2005; received in revised form 13 January 2006; accepted 27 January 2006 Available online 20 March 2006
Abstract A procedure for the calculation of chord length distributions (CLD) of populations of rigid and opaque particles of any size and shape distribution is given. It combines the capabilities of a virtual reality renderer to create 2D projections of particles and of an image analysis software which determine their chord lengths. The procedure has been validated on simple shapes (spheres, ellipsoids, parallelepipeds, cuboids, uniform polyhedra) that can be combined to simulate agglomerates or twinned crystals. The procedure has been used to discuss the experimental results obtained on gibbsite particles in different size ranges and to compare the mean chord length to the average particle size. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Crystallization; FBRM; Imaging; Particle; Polyhedra; Powder technology
1. Introduction In the focused beam reflectance method (FBRM), a laser beam, which is rotating at a high speed, is projected through a sapphire window and can be focused either inside or outside the probe into a suspension of particles. Due to the rotation speed (ranging from 2 to 6 m/s) the measurement is not affected significantly by the displacement of the particles. When a particle is hit by the beam, it reflects the light back toward the probe, where it is detected. This retro-propagation stops when the beam does not hit the particle anymore. The time of retrodiffusion of the light is measured by the probe and corresponds to a chord length, as the speed of the beam is known. The main advantage of the FBRM probe is that it can be placed in situ in a reactor or a pipe to monitor in real-time processes such as crystallization or flocculation (either induced (Alfano et al., 2000; Owen et al., 2002; Richmond et al., 1988; Swift et al., 2004) or natural as reported in Law et al., 1997). Applications in bioreactors (Jeffers et al., 2003; Pearson et al., 2003) have been reported as well as in the field of pulp and paper (Dunham et al., 2000, 2002; Fuente et al., 2003). ∗ Corresponding author. Tel.: +33 3 83 17 52 77; fax: +33 3 83 17 53 26.
E-mail address: [email protected] (M.-N. Pons). 0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.01.036
Bubble size distributions have been obtained with this device, especially by Clark and coworkers (Clark and Turton, 1988; Liu and Clark, 1995; Liu et al., 1998). Recently applications in the field of water treatment have been reported to investigate settling processes in clarifiers (De Clercq et al., 2004). The FBRM device provides a chord length distribution (CLD), which is not a particle size distribution (PSD). The back-calculation of PSD from CLD has been the topic of many papers and discussions (Heath et al., 2002; Hukkanen and Braatz, 2003; Ruf et al., 2000; Wynn, 2003). It is necessary to understand correctly the effects of particle properties (shape, size, transparency, etc.) and of the instrument settings on measured CLD to make full use of the information given by the probe. The research objective for many researchers has been to obtain the equivalent sphere size distribution from an experimental chord length distribution. Some have considered 2D-ellipses. Ruf et al. (2000) have investigated more complex shapes such as ellipsoids and rounded-corners parallelepipeds, that can be represented by x k y k z k (1) + + =1 a b c with a, b and c being the half-lengths of the three axes and k a constant 1.
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In a recent contribution, Li and Wilkinson (2005) has performed extensive theoretical work on the development of analytical solutions for PSD-CLD models for spherical and ellipsoidal particles. But particles presenting concavities such as twinned crystals and agglomerates cannot be modeled by such a function. These objects are frequently found in crystallization processes. Another key issue is the effect of the position of the focal point. Beyond the focal point the laser beam broadens and looses power. Our purpose has been to use a scene-rendering software to represent 3D rigid and opaque particles of any convex or concave shape. 2D-projections of these objects on a plane are images of how the particle is “seen” by the probe. For each image the chord length distribution is calculated with the help of an image analysis software. This general scheme is applied to a set of random positions to finally obtain the average CLD of a particle of given shape and size. Finally, the CLD of a particle population is determined by combining individual particle CLDs weighted using the PSD. The procedure has first been applied to simple shapes (ellipsoids, cuboids) for comparison with literature data and then assemblies of parallelepipeds and uniform polyhedra, simulating crystalline aggregates such as gibbsite particles. 2. Materials Gibbsite particles (HS 90, Alcan-Pechiney, Gardanne, France) have been classified by sieving in seven size classes: < 32, 32–50, 50–71, 71–90, 90–112, 112–125, 125–140, 140–160, > 160 m. A subsample from each class was examined by optical microscopy (OM) (Dialux 20, Leitz, Wetzlar, Germany) fitted with a Sony 3CCD camera. Particles were deposited on a glass slide, immersed in a drop of linseed oil and covered by a cover-slip. Images were grabbed using Visilog 5 (Noesis, Les Ulis, France) and analyzed according to procedures previously described (Belaroui et al., 2002). A second sub-sample of each class was embedded in resin and polished: the polished section was covered with a thin layer of graphite prior to examination on a TSM 330 (Jeol, Croissy-surSeine, France) scanning electron microscope fitted with a back-scattered electron detector (BS-SEM). Some images of gibbsite powder deposited on carbon tape and metal-plated have been obtained on same device with a secondary electron detector. SEM images were grabbed using a Spirit system (PGT, Rocky Hill, NJ, USA) and analyzed according to procedures previously described using Visilog 5 (Pons et al., 2005). For both types of visualization the particle size was defined by equivalent diameters: Deq of the projected area for OM and Ds of the section for BS-SEM. The shape was described by the elongation (E), the circularity (C), the index of concavity (I C) and the robustness (R) (Belaroui et al., 2002; Pons et al., 2005). E is calculated as the ratio of the largest to the smallest Feret diameter and expresses the slenderness of the convex bounding polygon to the projected particle or to its section. The circularity is equal to P 2 /(4 · A) where P and A are respectively the Crofton perimeter and the area (projected for OM or of the section for BS-SEM). I C is the ratio of A to the
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√ surface of the convex bounding polygon. R = (2 · ne )/ A, where ne is the number of morphological erosions to observe a complete disappearance of the object of interest, describes the slenderness of the object. More details on its calculations can be found in Pons et al. (1997). CLD measurements were made using a Lasentec D600L F/C FBRM (Lasentec, Richmond, WA, USA). Measurements were performed using the coarse electronics setting except when specified otherwise. The focal point was set at −20 m. Suspensions of each particle size class were analyzed with the FBRM probe in a 500 mL beaker (tall form, diameter 75 mm) using the stand provided by Lasentec. The speed of the four angled bladed impeller was set to 400 rpm. Data were collected using 90 log-channels over the range 2 to 2000 m (scanning speed set to 4 m/s). A measurement duration of 30 s was used, with each distribution being the average of 15 consecutive measurements. The count axis on the experimental chord length distributions represents the number of counts per second. 3. Methods 3.1. Single particle CLD The basic particle, centered at the origin of the coordinates system, is modeled using the software Persistence of Vision (POV-ray) 3.5 (POV-Team, Williamstown, Victoria, Australia). The software offers a range of primary particles such as sphere, ellipsoids, parallelepipeds, cylinders, cuboids. They can be combined to simulate complex shapes. Basic polyhedra are included in POV-ray but more complex uniform polyhedra (icosahedron with 20 facets, pentakisdodecahedron with 60 facets and disdyakistriacontahedron with 120 facets), can be built as triangular mesh objects: their coordinates have been obtained with kaleido (Dr. Z. Har’el, Department of Mathematics, Technion University, Haifa, Israel) available on the [email protected] site. Polyhedron volumes (V ) have been calculated by dividing the polyhedron into n tetrahedra, each of them defined by one of the polyhedron facets and the coordinate system origin (Sheynin and Tuzilov, 2001). Polyhedron surfaces (S) have been calculated as the sum of the facet surfaces. Table 1 summarizes their volume and surface shape factors ka and kv , respectively: ka = S/L2 ,
(2)
kv = V /L3 ,
(3)
Table 1 Geometrical characteristics of the sphere and some uniform polyhedra. L is the diameter of the sphere having the same volume as the solid Solid
L
ka
kv
Lmax
Sphere Cube Icosahedron Pentakisdodecahedron Disdyakistriacontahedron
1 1.238 0.848 0.934 0.918
3.14 (=) 3.89 3.34 3.21 3.48
0.52 (=/6) 0.52 0.52 0.53 0.53
1 1.732 1.000 1.000 1.000
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where L is the characteristic size of the particle, i.e., the diameter of the sphere having the same volume as the solid. Also their largest diagonal the largest distance between two vertices (Lmax, ) is given. Twinned crystals were simulated by associations of parallelepipeds (Fig. 1a). The opaque particle is rotated in space with respect of the three axes Ox, Oy and Oz to obtain nr random positions. For each of them an image (512 × 384 pixels) is created, which contains the projection of the particle in a plane parallel to xOy, at zcam = −3, where the virtual camera is located (Fig. 1b). The following procedure is applied for image analysis using Visilog 5 (Noesis, Les Ulis, France):
Fig. 1. 3D view of a (a) twinned crystal and (b) of its projection (grey-level image) on a plane.
• thresholding of the 8-bits (i.e., 256 gray levels) intensity image, at a fixed threshold of 128, to get the particle binary silhouette (Fig. 2a); • determination of the bounding box of the silhouette (Fig. 2b); • creation of a chord image, which contains a single line with a width of 1 pixel. The number of lines to be tested corresponds to the height of the silhouette, i.e., the number of lines of the bounding box (Fig. 2c); • logical operation (AND) between the particle binary silhouette and the tested line in order to calculate the overlap between the two images: the result is an image which contains
(a)
(b)
(c)
(d) Fig. 2. Steps of image treatment: (a) binary image, (b) bounding box, (c) line image, (d) chord.
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Fig. 3. Effect of the broadening of the laser beam on the apparent size of particles depending upon their distance to the focal point.
the chord(s) (Fig. 2d). Several chords may exist if the particle is concave. The chord length(s) is (are) calculated as the area of the line fragment(s). The normalized CLD of a single particle is q(s, L) where q(s, L) is the fraction of chords between s and ∞ s + ds for a particle with a characteristic size L, assuming 0 q(s, L) ds = 1 . It is the average of the nr CLDs corresponding to the nr random positions. 3.2. Population CLD A population of particles of identical shape but different characteristic size L is considered. Its size distribution (PSD) is n(L) where n(L) dL represents the number of particles of characteristic size between L and (L + dL). The global CLD is obtained as a weighted sum of the single particle CLDs (Ruf et al., 2000). ds ∞ qp (s) ds = q(s, L) · n(L) · L · dL, (4) m1 0 ∞ where m1 = 0 L · n(L) · dL.
(equivalent to 7.7 × 10−3 unit length) independent of the distance to the particle. Experimentally a broadening is expected as shown in Fig. 3. On the one hand the particle will appear larger when it is far from the focal point. On the other hand, as simultaneously the power of the beam decreases, a “blurring” of the particle contour occurs: it will be noisier and more difficult to detect. The blurring effect has been simulated by modifying the position of the focal point of the virtual camera, without changing the distance between the camera and the particle (Fig. 4a). If that distance had been changed, the perspective would have been modified, with a change in the apparent size of the particle. The intensity of blurring can be changed by adjusting the virtual camera aperture: with a low aperture almost everything is in focus. The threshold to get the binary image is kept at 128. The broadening of the laser beam has been simulated by applying to the silhouette a dilation step, whose size is function of the distance between the particle and the focal point (Fig. 4b). 4. Results 4.1. Single particle
3.3. Focal point The CLDs that have been calculated so far (Eq. (4)) considering that the laser beam is ideal: it has a thickness of 1 pixel
Fig. 5 compares the simulated CLD, obtained for a ideal laser beam (i.e., without taking into consideration the distance between the object and the focal point), with the theoretical
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Fig. 4. Simulation of the (a) blurring effect and (b) the beam broadening.
Fig. 5. Theoretical (- - - -) and simulated (—) CLDs of a sphere.
CLD given by q(s, dp ) =
s
dp dp2 − s 2
for 0 < s < dp ,
(5)
where dp is the diameter of the sphere. The simulated CLD is noisier than the theoretical curve, due to the discrete nature of images. Ellipsoids are described by x 2 y 2 z 2 (6) + + = 1, a b c where a, b and c are the half-lengths of the three axes. In Fig. 6 the average CLDs of two types of ellipsoids, the “elongated” one with c = 2a = 2b and the “flat” one with b = c = 2a, obtained over nr = 500 random positions in space are given. They are within 5% of those obtained for 1000 random positions in space. These curves are similar to the ones presented by Ruf et al. (2000). The reduced chord length, cl r , is here defined as cl r = s/c. In Fig. 7 the CLDs of a sharp-corner cube and of two roundcorner cubes with different degrees of roundness have been plotted. The cube CLD exhibits a local maximum at cl r = 1, with√cl r = s/a. The largest chord would be obtained for cl r = 3, which corresponds to the internal diagonal. It has not been detected due to the very low probability to observe it. This largest chord decreases when the roundness increases. Simultaneously the secondary maxima tend to disappear. This reflects the change that would be observed for cubic crystals,
such as those of sodium chloride, when they are subject to abrasion (Marrot et al., 2000). Fig. 8 compares the CLDs for a cube (side lengths a = b = c) and two square-based parallelepipeds, a flat one (a = b, c = a/2) and an elongated one (a = b, c = 2a). As in the case of the elongated and flattened cuboids described by Ruf et al. (2000), peaks arise, which correspond to diagonals. Any large elongation of the shape in one direction will result in a relatively broad CLD, with a larger contribution of the shorter chords (related to the particle “thickness”) with respect to the larger ones (related to the particle “length”). The effect of faceting has been evaluated using uniform polyhedra (Fig. 9). When the number of facets increases, the overall shape tends towards a sphere. The differences start to be clearly visible for the icosahedron, with the appearance of a shoulder on the CLD. As the number of facets increases, the uniform polyhedron CLD tends toward a sphere CLD. Any kind of particle can be simulated using the technique, as basic shapes can be combined. For example twinning has been modeled by combining two flat parallelepipeds. The first parallelepiped is large and has a length a, a width b and a thickness c such as a = 2b, c = b/10. The second parallelepiped has the same width and thickness than the first one. In Fig. 10a the relative importance of the second parallelepiped has been progressively increased by varying a from a = 5c to a = 10c. The angle between crystals has been kept constant (90◦ ). The corresponding CLDs exhibit two main peaks: one around cl r = 1 with cl r = s/b and a second one, near cl r ≈ 0.15 corresponding to the thickness of the agglomerate. In Fig. 10b the angle between the two crystals has been changed gradually between 10◦ and 90◦ . The main differences between these CLDs can be found in the first peak (cl r ≈ 0.15). The peaks’ intensity decreases with the angle. At that point it can be said that a CLD carries information both on the size and shape of the particle. 4.2. Non-ideality of the laser beam The effect of the non ideality of the beam on the CLD of spheres is shown in Fig. 11. The distance D between the focal point and the sphere center of diameter L = 1 varies between 0 and 3. The number of dilations applied is 0 for D ∈ [0, 1[, 1 for D ∈ [1, 2[, 2 for D ∈ [2, 3[ and 3 for D = 3. Short chords arise due to the noise introduced by blurring.
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Fig. 6. CLDs of (a) “elongated” and (b) flat ellipsoids. Averages over 500 random positions in space.
Fig. 7. CLDs of a sharp-corner cube (- - -) and round-corner cubes A (—) and B (
The maximal chord length which is detected decreases with increasing D. Depending upon the probe electronics (fine or coarse) and the chord range settings, these small chords might be discarded in the FBRM signal conditioning step. The average CLD for a cube of side length 1 moving randomly between 1 and 3 is compared in Fig. 12 to the ideal CLD. As for spheres short chords appear, which decreases the mean chord length from 0.874 to 0.753. If short chords smaller than
). The distances between flat faces are equal to a.
0.1 are discarded, the mean chord length increases slightly up to 0.762.
4.3. Population CLD The aim of this section is to examine how the CLD of a specific type of particle is transformed when a size
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Fig. 8. CLDs of sharp-corner parallelepipeds; (- - -) a = b = c (cube), (thin continous line —) a = b < c, (thick continuous line
5
6
4
5
2
CLD
4
3
CLD
CLD
Disdyakistriacontahedron (120 facets)
Pentakisdodecahedron (60 facets)
Icosahedron (20 facets)
3 2
1
1 0
0 0
0.2
0.4
0.6
0.8
Reduced chord length
1
0
0.2
0.4
0.6
0.8
) a = b > c.
1
Reduced chord length
8 7 6 5 4 3 2 1 0 0
0.2
0.4
0.6
0.8
1
Reduced chord length
Fig. 9. CLDs of some uniform polyhedra.
distribution (PSD) is taken into account. For simplification beam broadening is not considered. As a first example we will compare the population CLDs for spheres and ellipsoids. Finding the appropriate reference to compare them is a difficulty. Here the equivalent spherical diameter has been selected as the characteristic parameter for calculating the CLDs of populations of ellipsoids. One thousand particles with a standard distribution assuming different values for the standard deviation have been considered to build each CLD.
The volume of an ellipsoid is V = 43 abc
(7)
and the diameter of the equivalent sphere is 3 6V L= (8) In Fig. 13 the average equivalent diameter has been set to 50 m and the CLDs are plotted for two values for the standard deviation , 1 m and 10 m, assuming truncated normal size
M.-N. Pons et al. / Chemical Engineering Science 61 (2006) 3962 – 3973
Fig. 10. CLDs of twinned crystals: (a) Size of the secondary parallelepiped increases a = 5c (—), a = 10c ( ), 30◦ ( ), 60◦ (- - - -) and 90◦ (–). increases: 10◦ (
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), (b) Angle between the parallelepipeds
has demonstrated how the j th moment of the CLD (j ) is proportional to the (j + 1)th moment of the PSD (mj ): j = U T S j mj +1 ,
Fig. 11. Reduced mean () and maximal () chord length for a sphere in function of its reduced distance to the focal point (L/D).
(9)
where U is the scanning speed of the laser beam, T its scanning depth, Sj a constant which depends only on the particle shape. For a sphere, S0 = 1, S1 = /4, S2 = 2/3 and S3 = 3/16. Table 2 gives the theoretical values of the first ratios j mj +1 Sj = . (10) j −1 mj Sj −1 and the values obtained for the simulated CLDs of populations of spheres. An excellent agreement is obtained, which validates the procedure to simulate the population CLDs. In Table 3, S1 /S0 , S2 /S1 and S3 /S2 have been calculated for populations of cubes and 20-facets, 60-facets and 120-facets uniform polyhedra for a standard deviation of = 1. Except for cubes, it could be expected that the mean chord length of these shapes would be about 30% smaller than the mean particle size. 4.4. Analysis of gibbsite particles
Fig. 12. Cube CLD with beam broadening effect (—) and without (
).
distributions (no negative values). As the standard deviation increases, the population CLD differs more and more from the CLD of a single particle and information on the shape is gradually lost. It is interesting to examine how the mean particle size and the mean chord length of a population are related. Wynn (2003)
In Fig. 14, the mean chord lengths of gibbsite particles have been plotted versus the sieve median (Dsieve ). The sieve median is the arithmetic mean between the limits of the sieve range. (Dsieve ) = 20 m and (Dsieve ) = 180 m have been arbitrarily selected for the representation of the lower (< 32 m) and upper (> 160 m) size classes. For comparison the mean Deq (OM) and Ds (BS-SEM) have also been reported in Table 4. The runs with coarse settings have been duplicated by a second operator for repeatability checking. Excluding the lower (< 32 m) and upper sieve classes (> 160 m) a linear relation was found
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100 CLD (#/channel)
CLD (#/channel)
400 300 200 100 0
20
40 20 0
100
20
(b)
40 60 80 Chord length (µm)
100
100 CLD (#/channel)
CLD (#/channel)
40 60 80 Chord length (µm)
180 160 140 120 100 80 60 40 20 0
80 60 40 20 0
0
20
40 60 80 100 120 Chord length (µm)
0
(e)
20
40 60 80 100 120 Chord length (µm)
0
20
40 60 80 100 120 Chord length (µm)
0
20
40 60 80 100 120 Chord length (µm)
(d)
CLD (#/channel)
(c)
CLD (#/channel)
60
0 0
(a)
90 80 70 60 50 40 30 20 10 0
80
(f)
80 70 60 50 40 30 20 10 0
Fig. 13. Comparison of the CLDs of spheres (a and b), elongated ellipsoids (c and d) and flat ellipsoids (e and f) for two standard deviations: 1 m (a, c and e) and 10 m (b,d and f).
Table 2 Comparison of the ratios Sj /Sj −1 for j =1–3 and for two standard deviations of the PSD for spheres j
j /j −1
mj +1 /mj
Sj /Sj −1 (Simulated)
Sj /Sj −1 (Theory)
1 2 3 1 2 3
1 1 1 10 10 10
39.28 42.53 44.29 39.27 44.21 47.65
50.01 50.04 50.06 52 53.85 57.57
0.785 0.850 0.885 0.753 0.821 0.857
0.785 0.848 0.884 0.785 0.848 0.884
Table 3 S1 /S0 , S2 /S1 and S3 /S2 for cubes, icosahedra, pentakisdodecahedra and disdyakistriacontahedra Polyhedra Cube Icosahedron Pentakisdodecahedron Disdyakistriacontahedron
S1 /S0 0.87 0.67 0.74 0.73
S2 /S1 1.02 0.76 0.81 0.80
S3 /S2 1.08 0.79 0.85 0.83
between (Dsieve ) and smean with a coefficient of correlation of 0.97. For particles > 70 m smean is smaller than the mean size. It is the contrary for the smallest size classes. Various
Fig. 14. Comparison of the mean chord length (no weight): coarse settings, operator 1 (), operator 2 (), fine settings (◦), the equivalent diameter by optical microscopy (Deq ) () and the sieve median size.
weightings are proposed for the FBRM distributions. They have been applied to obtain the data summarized in Fig. 15, where the distribution mode has also been plotted. Linear relations can be obtained for each of them with the sieve mean size, although the cube weighted mean produces a low coefficient of correlation (0.72). Similar results have been reported by Heath et al. (2002) for aluminum particles. As shown on the images in Fig. 16, intact gibbsite particles may be approximated as polyhedra, such as icosahedra or
M.-N. Pons et al. / Chemical Engineering Science 61 (2006) 3962 – 3973
pentakisdodecahedra. However, the gibbsite samples contains broken particles, which are more similar to parallelepipeds. By analysis of OM and BS-SEM images it can be concluded that the smaller size classes contain more fragments than larger size classes: they are characterized by a larger circularity and elongation and a lower robustness (Table 4). The ratios S1 /S0 have been evaluated from the mean chord length and the median sieve diameter. They decrease as the particle size increases. One reason could be the higher proportion of icosahedra-like
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particles in the upper size classes as their theoretical S1 /S0 , is 0.67, when it is 0.87 for cubes. Another reason could be that for the smallest particles a large proportion of small chords are eliminated by the binning between 2 and 2000 m. Fig. 17 compares the experimental CLDs of two size classes obtained with the coarse and the fine settings. For the 32–50 m sieve fraction the CLD mode is lower with the fine settings and a secondary peak appears around 8 m. This peak corresponds to the numerous parallelepiped-like fragments present in that size class. For the 125–140 m sieve fraction, the CLD modes almost coincide but the fine settings reveals a shoulder similar to the one observed on the icosahedron CLD (Fig. 9). 5. Conclusions
Fig. 15. Comparison of the mean chord lengths given by various weightings and the mode with the sieve median size.
A procedure for the calculation of the chord length distribution of any rigid and opaque 3D object was developed by combination of a scene-rendering software and an image analysis software. It was successfully applied to simple convex shapes and to their combinations used to represent crystal agglomerates. For the individual particle its CLD is a shape fingerprint. However in the case of a population of particles, a loss of information is observed depending upon the spread of the particle size distribution. Another loss of information is due to the fact that short chords are discarded during the FBRM signal treatment step. A loss of short chords may be desirable in order to reduce the effect of noise due to the decrease in laser power with distance and/or to filter out the signal of possibly present fines (Ruf et al., 2000). On the other hand the information on short chords is important to detect the presence of elongated
Fig. 16. SE-SEM (a and b) and BS-SEM (c and d) images of 32–50 m (a and c) and 140–160 m (b and d) particles.
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Table 4 Size and shape characteristics of the gibbsite fractions Sieve fraction (m) < 32 32–50 50–71 71–90 90–112 112–125
Deq (m)
C
E
R
52 9 64 8 80 12 98 13 120 15
1.24 0.18 1.12 0.06 1.12 0.05 1.11 0.05 1.13 0.12
1.15 0.09 1.19 0.11 1.18 0.12 1.15 0.09 1.16 0.09
0.76 0.05 0.77 0.04 0.76 0.04 0.78 0.04 0.77 0.04
Ds (m)
Cs
Es
Rs
21.4 37.5
1.64 1.69
1.71 1.57
0.67 0.68
47.9
1.65
1.41
0.71
60.0
1.43
1.4
0.72
70.3
1.48
1.32
0.73
98.2
1.49
1.27
0.75
105
1.44
1.30
0.74
122
1.27
1.21
0.76
142
1.37
1.22
0.76
S1 /S0
1.59 1.12 0.91 0.78 0.75
125–140
0.72 140–160 > 160
157 15 189 19
1.07 0.04 1.07 0.14
1.15 0.08 1.15 0.09
0.78 0.03 0.78 0.04
0.66
For OM visualization, number-averaged equivalent diameter Deq , circularity C, elongation E and robustness R. Standard deviations in italics. For BS-SEM visualization, Ds -averaged equivalent diameter Ds , circularity Cs , elongation Es and robustness Rs . Experimental values of S1 /S0 .
the mean particle size (Hobbel et al., 1991) when Tadayyon and Rohani (1998) give values around 0.8. The main advantage of the FBRM lies in its capability to deliver combined information on shape, size and concentration, on-line and in real-time. The procedure described in the present work helps to understand the relations between particle size, shape and chord length distributions in order to make best use of the FBRM signal. Notations
Fig. 17. Comparison of the CLDs of two gibbsite sieve fractions: (a) 32–50 m, (b) 125–140 m for the fine (—) and the coarse ( ) settings.
objects. Their discarding increases also the mean chord length. In literature, the comparison with the particle mean size obtained by other techniques (sieving, laser diffraction) has given rise to very different conclusions (Wynn, 2003). Some authors have found a ratio of 2.5 between the mean chord length and
a A b c cl mean D Dsieve k ka kv L Lmax mj n ne nr P q qp s S Sj T U
size parameter for ellipsoids and parallelepipeds area size parameter for ellipsoids and parallelepipeds size parameter for ellipsoids and parallelepipeds mean chord length distance to focal point median sieve size shape parameter in Eq. (1) surface shape factor volume shape factor characteristic size maximal characteristic size j th moment of the particle size distribution particle size distribution number of morphological erosions number of particles Crofton perimeter chord length distribution chord length distribution of a population of particles chord length surface constant in Eq. (9) scanning depth of the laser beam scanning speed of the laser beam
M.-N. Pons et al. / Chemical Engineering Science 61 (2006) 3962 – 3973
V zcam
volume position of the camera
Greek letters j
j th moment of the chord length distribution standard deviation
Acknowledgments The authors are thankful for financial support in part from the CNRS/UIUC cooperation program, a CAREER award to Eberhard Morgenroth from the National Science Foundation under grant No. BES-0134104, and The WaterCAMPWS, a Science and Technology Center of Advanced Materials for the Purification of Water with Systems under the National Science Foundation agreement number CTS-0120978. We also thank K. Meddi-Berrada for his contribution to the experimental work, and G. Tran for his help with Pov-ray. References Alfano, J.C., Carter, P.W., Dunham, A.J., Nowak, M.J., Tubergen, K.R., 2000. Polyelectrolyte-induced aggregation of microcrystalline cellulose: reversibility and shear effects. Journal of Colloid and Interface Science 223, 244–254. Belaroui, K., Pons, M.N., Vivier, H., 2002. Morphological characterisation of gibbsite and alumina. Powder Technology 127, 246–256. Clark, N.N., Turton, R., 1988. Chord length distributions related to bubble size distributions in multiphase flow. International Journal of Multiphase Flow 14, 413–424. De Clercq, B., Lant, P.A., Vanrolleghem, P.A., 2004. Focused beam reflectance technique for in situ particle sizing in wastewater treatment settling tanks. Journal of Chemical Technology and Biotechnology 76, 610–618. Dunham, A.J., Tubergen, K.R., Govoni, S.T., Alfano, J.C., 2000. The effects of dissolved and colloidal substances on flocculation of mechanical pulps. Journal of Pulp and Paper Science 26, 95–101. Dunham, A.J., Sherman, L., Alfano, J.C., 2002. Effect of dissolved and colloidal substances on drainage properties of mechanical pulp suspensions. Journal of Pulp and Paper Science 28, 298–304. Fuente, E., Blanco, A., Negro, C., San Pio, I., Tijero, J., 2003. Monitoring flocculation of fillers in papermaking. Paper Technology 44, 41–50. Heath, A.R., Fawell, P.D., Bahri, P.A., Swift, J.D., 2002. Estimating average particle size by focused beam reflectance measurement (FBRM). Particle and Particle Systems Characterization 19, 84–95.
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Hukkanen, E.J., Braatz, R.D., 2003. Measurement of particle size distribution in suspension polymerization using in situ laser backscattering. Sensors and Actuators B Chemical 96, 451–459. Jeffers, P., Raposo, S., Lima-Costa, M.E., Connolly, P., Glennon, B., Kieran, P.M., 2003. Focussed beam reflectance measurement (FBRM) monitoring of particle size and morphology in suspension cultures of Morinda citrifolia and Centaurea calcitrapa. Biotechnology Letters 25, 2023–2028. Law, D.J., Bale, A.J., Jones, S.E., 1997. Adaptation of focused beam reflectance measurement to in situ particle sizing in estuaries and coastal waters. Marine Geology 140, 47–59. Li, M., Wilkinson, D., 2005. Determination of non-spherical particle size distribution from chord length measurements. Part 1: Theoretical analysis. Chemical Engineering Science 60, 3251–3265. Liu, W., Clark, N.N., 1995. Relationships between distributions of chord lengths and distributions of bubble sizes including their statistical parameters. International Journal of Multiphase Flow 21, 1073–1089. Liu, W., Clark, N.N., Karamavruc, A.I., 1998. Relationship between bubble size distributions and chord-length distribution in heterogeneously bubbling systems. Chemical Engineering Science 52, 1267–1276. Marrot, B., Pons, M.N., Biscans, B., 2000. Identification of impact attrition mechanisms in solution by morphological analysis. Chemical Engineering Journal 79, 123–131. Owen, A.T., Fawell, P.D., Swift, J.D., Farrow, J.B., 2002. The impact of polyacrylamide flocculant solution age on flocculation performance. International Journal of Mineral Processing 67, 123–144. Pearson, A.P., Glennon, B., Kieran, P.M., 2003. Comparison of morphological characteristics of Streptomyces natalensis by image analysis and focused beam reflectance measurement. Biotechnology Progress 19, 1342–1347. Pons, M.N., Vivier, H., Dodds, J.A., 1997. Particle shape characterization using morphological descriptors. Powder Technology 14, 272–277. Pons, M.N., Plagnieux, V., Vivier, H., Audet, D., 2005. Comparison of methods for the characterisation by image analysis of crystalline agglomerates: the case of gibbsite. Powder Technolology 157, 57–66. Richmond, W.R., Jones, R.L., Fawell, P.D., 1988. The relationship between -particle aggregation and rheology in mixed silica-titania suspensions. Chemical Engineering Journal 71, 67–75. Ruf, A., Worlitschek, J., Mazzotti, M., 2000. Modeling and experimental analysis of PSD measurements through FBRM. Particle and Particle Systems Characterization 17, 167–169. Sheynin, S.A., Tuzilov, A.V., 2001. Explicit formulae for polyhedra moments. Pattern Recognition Letters 22, 1103–1109. Swift, J.D., Simic, K., Johnston, R.R.M., Farwell, P.D., Farrow, J.B., 2004. A study of the polymer flocculation reaction in a linear pipe with a focused beam reflectance measurement probe. International Journal of Mineral Processing 73, 103–118. Wynn, E.J.W., 2003. Relationship between particle-size and chord-length distributions in focused beam reflectance measurement: stability of direct inversion and weighting. Powder Technology 133, 125–133.
Modeling of chord length distributions Marie-Noëlle Pons a,∗ , Kim Milferstedt b , Eberhard Morgenroth b,c a Laboratoire des Sciences du Génie Chimique, CNRS–ENSIC–INPL, 1, rue Grandville, BP 20451, F-54001 Nancy cedex, France b Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 N. Mathews Avenue, Urbana, IL 61801, USA c Department of Animal Sciences, University of Illinois at Urbana-Champaign, 205 N. Mathews Avenue, Urbana, IL 61801, USA
Received 24 May 2005; received in revised form 13 January 2006; accepted 27 January 2006 Available online 20 March 2006
Abstract A procedure for the calculation of chord length distributions (CLD) of populations of rigid and opaque particles of any size and shape distribution is given. It combines the capabilities of a virtual reality renderer to create 2D projections of particles and of an image analysis software which determine their chord lengths. The procedure has been validated on simple shapes (spheres, ellipsoids, parallelepipeds, cuboids, uniform polyhedra) that can be combined to simulate agglomerates or twinned crystals. The procedure has been used to discuss the experimental results obtained on gibbsite particles in different size ranges and to compare the mean chord length to the average particle size. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Crystallization; FBRM; Imaging; Particle; Polyhedra; Powder technology
1. Introduction In the focused beam reflectance method (FBRM), a laser beam, which is rotating at a high speed, is projected through a sapphire window and can be focused either inside or outside the probe into a suspension of particles. Due to the rotation speed (ranging from 2 to 6 m/s) the measurement is not affected significantly by the displacement of the particles. When a particle is hit by the beam, it reflects the light back toward the probe, where it is detected. This retro-propagation stops when the beam does not hit the particle anymore. The time of retrodiffusion of the light is measured by the probe and corresponds to a chord length, as the speed of the beam is known. The main advantage of the FBRM probe is that it can be placed in situ in a reactor or a pipe to monitor in real-time processes such as crystallization or flocculation (either induced (Alfano et al., 2000; Owen et al., 2002; Richmond et al., 1988; Swift et al., 2004) or natural as reported in Law et al., 1997). Applications in bioreactors (Jeffers et al., 2003; Pearson et al., 2003) have been reported as well as in the field of pulp and paper (Dunham et al., 2000, 2002; Fuente et al., 2003). ∗ Corresponding author. Tel.: +33 3 83 17 52 77; fax: +33 3 83 17 53 26.
E-mail address: [email protected] (M.-N. Pons). 0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.01.036
Bubble size distributions have been obtained with this device, especially by Clark and coworkers (Clark and Turton, 1988; Liu and Clark, 1995; Liu et al., 1998). Recently applications in the field of water treatment have been reported to investigate settling processes in clarifiers (De Clercq et al., 2004). The FBRM device provides a chord length distribution (CLD), which is not a particle size distribution (PSD). The back-calculation of PSD from CLD has been the topic of many papers and discussions (Heath et al., 2002; Hukkanen and Braatz, 2003; Ruf et al., 2000; Wynn, 2003). It is necessary to understand correctly the effects of particle properties (shape, size, transparency, etc.) and of the instrument settings on measured CLD to make full use of the information given by the probe. The research objective for many researchers has been to obtain the equivalent sphere size distribution from an experimental chord length distribution. Some have considered 2D-ellipses. Ruf et al. (2000) have investigated more complex shapes such as ellipsoids and rounded-corners parallelepipeds, that can be represented by x k y k z k (1) + + =1 a b c with a, b and c being the half-lengths of the three axes and k a constant 1.
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In a recent contribution, Li and Wilkinson (2005) has performed extensive theoretical work on the development of analytical solutions for PSD-CLD models for spherical and ellipsoidal particles. But particles presenting concavities such as twinned crystals and agglomerates cannot be modeled by such a function. These objects are frequently found in crystallization processes. Another key issue is the effect of the position of the focal point. Beyond the focal point the laser beam broadens and looses power. Our purpose has been to use a scene-rendering software to represent 3D rigid and opaque particles of any convex or concave shape. 2D-projections of these objects on a plane are images of how the particle is “seen” by the probe. For each image the chord length distribution is calculated with the help of an image analysis software. This general scheme is applied to a set of random positions to finally obtain the average CLD of a particle of given shape and size. Finally, the CLD of a particle population is determined by combining individual particle CLDs weighted using the PSD. The procedure has first been applied to simple shapes (ellipsoids, cuboids) for comparison with literature data and then assemblies of parallelepipeds and uniform polyhedra, simulating crystalline aggregates such as gibbsite particles. 2. Materials Gibbsite particles (HS 90, Alcan-Pechiney, Gardanne, France) have been classified by sieving in seven size classes: < 32, 32–50, 50–71, 71–90, 90–112, 112–125, 125–140, 140–160, > 160 m. A subsample from each class was examined by optical microscopy (OM) (Dialux 20, Leitz, Wetzlar, Germany) fitted with a Sony 3CCD camera. Particles were deposited on a glass slide, immersed in a drop of linseed oil and covered by a cover-slip. Images were grabbed using Visilog 5 (Noesis, Les Ulis, France) and analyzed according to procedures previously described (Belaroui et al., 2002). A second sub-sample of each class was embedded in resin and polished: the polished section was covered with a thin layer of graphite prior to examination on a TSM 330 (Jeol, Croissy-surSeine, France) scanning electron microscope fitted with a back-scattered electron detector (BS-SEM). Some images of gibbsite powder deposited on carbon tape and metal-plated have been obtained on same device with a secondary electron detector. SEM images were grabbed using a Spirit system (PGT, Rocky Hill, NJ, USA) and analyzed according to procedures previously described using Visilog 5 (Pons et al., 2005). For both types of visualization the particle size was defined by equivalent diameters: Deq of the projected area for OM and Ds of the section for BS-SEM. The shape was described by the elongation (E), the circularity (C), the index of concavity (I C) and the robustness (R) (Belaroui et al., 2002; Pons et al., 2005). E is calculated as the ratio of the largest to the smallest Feret diameter and expresses the slenderness of the convex bounding polygon to the projected particle or to its section. The circularity is equal to P 2 /(4 · A) where P and A are respectively the Crofton perimeter and the area (projected for OM or of the section for BS-SEM). I C is the ratio of A to the
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√ surface of the convex bounding polygon. R = (2 · ne )/ A, where ne is the number of morphological erosions to observe a complete disappearance of the object of interest, describes the slenderness of the object. More details on its calculations can be found in Pons et al. (1997). CLD measurements were made using a Lasentec D600L F/C FBRM (Lasentec, Richmond, WA, USA). Measurements were performed using the coarse electronics setting except when specified otherwise. The focal point was set at −20 m. Suspensions of each particle size class were analyzed with the FBRM probe in a 500 mL beaker (tall form, diameter 75 mm) using the stand provided by Lasentec. The speed of the four angled bladed impeller was set to 400 rpm. Data were collected using 90 log-channels over the range 2 to 2000 m (scanning speed set to 4 m/s). A measurement duration of 30 s was used, with each distribution being the average of 15 consecutive measurements. The count axis on the experimental chord length distributions represents the number of counts per second. 3. Methods 3.1. Single particle CLD The basic particle, centered at the origin of the coordinates system, is modeled using the software Persistence of Vision (POV-ray) 3.5 (POV-Team, Williamstown, Victoria, Australia). The software offers a range of primary particles such as sphere, ellipsoids, parallelepipeds, cylinders, cuboids. They can be combined to simulate complex shapes. Basic polyhedra are included in POV-ray but more complex uniform polyhedra (icosahedron with 20 facets, pentakisdodecahedron with 60 facets and disdyakistriacontahedron with 120 facets), can be built as triangular mesh objects: their coordinates have been obtained with kaleido (Dr. Z. Har’el, Department of Mathematics, Technion University, Haifa, Israel) available on the [email protected] site. Polyhedron volumes (V ) have been calculated by dividing the polyhedron into n tetrahedra, each of them defined by one of the polyhedron facets and the coordinate system origin (Sheynin and Tuzilov, 2001). Polyhedron surfaces (S) have been calculated as the sum of the facet surfaces. Table 1 summarizes their volume and surface shape factors ka and kv , respectively: ka = S/L2 ,
(2)
kv = V /L3 ,
(3)
Table 1 Geometrical characteristics of the sphere and some uniform polyhedra. L is the diameter of the sphere having the same volume as the solid Solid
L
ka
kv
Lmax
Sphere Cube Icosahedron Pentakisdodecahedron Disdyakistriacontahedron
1 1.238 0.848 0.934 0.918
3.14 (=) 3.89 3.34 3.21 3.48
0.52 (=/6) 0.52 0.52 0.53 0.53
1 1.732 1.000 1.000 1.000
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where L is the characteristic size of the particle, i.e., the diameter of the sphere having the same volume as the solid. Also their largest diagonal the largest distance between two vertices (Lmax, ) is given. Twinned crystals were simulated by associations of parallelepipeds (Fig. 1a). The opaque particle is rotated in space with respect of the three axes Ox, Oy and Oz to obtain nr random positions. For each of them an image (512 × 384 pixels) is created, which contains the projection of the particle in a plane parallel to xOy, at zcam = −3, where the virtual camera is located (Fig. 1b). The following procedure is applied for image analysis using Visilog 5 (Noesis, Les Ulis, France):
Fig. 1. 3D view of a (a) twinned crystal and (b) of its projection (grey-level image) on a plane.
• thresholding of the 8-bits (i.e., 256 gray levels) intensity image, at a fixed threshold of 128, to get the particle binary silhouette (Fig. 2a); • determination of the bounding box of the silhouette (Fig. 2b); • creation of a chord image, which contains a single line with a width of 1 pixel. The number of lines to be tested corresponds to the height of the silhouette, i.e., the number of lines of the bounding box (Fig. 2c); • logical operation (AND) between the particle binary silhouette and the tested line in order to calculate the overlap between the two images: the result is an image which contains
(a)
(b)
(c)
(d) Fig. 2. Steps of image treatment: (a) binary image, (b) bounding box, (c) line image, (d) chord.
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Fig. 3. Effect of the broadening of the laser beam on the apparent size of particles depending upon their distance to the focal point.
the chord(s) (Fig. 2d). Several chords may exist if the particle is concave. The chord length(s) is (are) calculated as the area of the line fragment(s). The normalized CLD of a single particle is q(s, L) where q(s, L) is the fraction of chords between s and ∞ s + ds for a particle with a characteristic size L, assuming 0 q(s, L) ds = 1 . It is the average of the nr CLDs corresponding to the nr random positions. 3.2. Population CLD A population of particles of identical shape but different characteristic size L is considered. Its size distribution (PSD) is n(L) where n(L) dL represents the number of particles of characteristic size between L and (L + dL). The global CLD is obtained as a weighted sum of the single particle CLDs (Ruf et al., 2000). ds ∞ qp (s) ds = q(s, L) · n(L) · L · dL, (4) m1 0 ∞ where m1 = 0 L · n(L) · dL.
(equivalent to 7.7 × 10−3 unit length) independent of the distance to the particle. Experimentally a broadening is expected as shown in Fig. 3. On the one hand the particle will appear larger when it is far from the focal point. On the other hand, as simultaneously the power of the beam decreases, a “blurring” of the particle contour occurs: it will be noisier and more difficult to detect. The blurring effect has been simulated by modifying the position of the focal point of the virtual camera, without changing the distance between the camera and the particle (Fig. 4a). If that distance had been changed, the perspective would have been modified, with a change in the apparent size of the particle. The intensity of blurring can be changed by adjusting the virtual camera aperture: with a low aperture almost everything is in focus. The threshold to get the binary image is kept at 128. The broadening of the laser beam has been simulated by applying to the silhouette a dilation step, whose size is function of the distance between the particle and the focal point (Fig. 4b). 4. Results 4.1. Single particle
3.3. Focal point The CLDs that have been calculated so far (Eq. (4)) considering that the laser beam is ideal: it has a thickness of 1 pixel
Fig. 5 compares the simulated CLD, obtained for a ideal laser beam (i.e., without taking into consideration the distance between the object and the focal point), with the theoretical
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Fig. 4. Simulation of the (a) blurring effect and (b) the beam broadening.
Fig. 5. Theoretical (- - - -) and simulated (—) CLDs of a sphere.
CLD given by q(s, dp ) =
s
dp dp2 − s 2
for 0 < s < dp ,
(5)
where dp is the diameter of the sphere. The simulated CLD is noisier than the theoretical curve, due to the discrete nature of images. Ellipsoids are described by x 2 y 2 z 2 (6) + + = 1, a b c where a, b and c are the half-lengths of the three axes. In Fig. 6 the average CLDs of two types of ellipsoids, the “elongated” one with c = 2a = 2b and the “flat” one with b = c = 2a, obtained over nr = 500 random positions in space are given. They are within 5% of those obtained for 1000 random positions in space. These curves are similar to the ones presented by Ruf et al. (2000). The reduced chord length, cl r , is here defined as cl r = s/c. In Fig. 7 the CLDs of a sharp-corner cube and of two roundcorner cubes with different degrees of roundness have been plotted. The cube CLD exhibits a local maximum at cl r = 1, with√cl r = s/a. The largest chord would be obtained for cl r = 3, which corresponds to the internal diagonal. It has not been detected due to the very low probability to observe it. This largest chord decreases when the roundness increases. Simultaneously the secondary maxima tend to disappear. This reflects the change that would be observed for cubic crystals,
such as those of sodium chloride, when they are subject to abrasion (Marrot et al., 2000). Fig. 8 compares the CLDs for a cube (side lengths a = b = c) and two square-based parallelepipeds, a flat one (a = b, c = a/2) and an elongated one (a = b, c = 2a). As in the case of the elongated and flattened cuboids described by Ruf et al. (2000), peaks arise, which correspond to diagonals. Any large elongation of the shape in one direction will result in a relatively broad CLD, with a larger contribution of the shorter chords (related to the particle “thickness”) with respect to the larger ones (related to the particle “length”). The effect of faceting has been evaluated using uniform polyhedra (Fig. 9). When the number of facets increases, the overall shape tends towards a sphere. The differences start to be clearly visible for the icosahedron, with the appearance of a shoulder on the CLD. As the number of facets increases, the uniform polyhedron CLD tends toward a sphere CLD. Any kind of particle can be simulated using the technique, as basic shapes can be combined. For example twinning has been modeled by combining two flat parallelepipeds. The first parallelepiped is large and has a length a, a width b and a thickness c such as a = 2b, c = b/10. The second parallelepiped has the same width and thickness than the first one. In Fig. 10a the relative importance of the second parallelepiped has been progressively increased by varying a from a = 5c to a = 10c. The angle between crystals has been kept constant (90◦ ). The corresponding CLDs exhibit two main peaks: one around cl r = 1 with cl r = s/b and a second one, near cl r ≈ 0.15 corresponding to the thickness of the agglomerate. In Fig. 10b the angle between the two crystals has been changed gradually between 10◦ and 90◦ . The main differences between these CLDs can be found in the first peak (cl r ≈ 0.15). The peaks’ intensity decreases with the angle. At that point it can be said that a CLD carries information both on the size and shape of the particle. 4.2. Non-ideality of the laser beam The effect of the non ideality of the beam on the CLD of spheres is shown in Fig. 11. The distance D between the focal point and the sphere center of diameter L = 1 varies between 0 and 3. The number of dilations applied is 0 for D ∈ [0, 1[, 1 for D ∈ [1, 2[, 2 for D ∈ [2, 3[ and 3 for D = 3. Short chords arise due to the noise introduced by blurring.
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Fig. 6. CLDs of (a) “elongated” and (b) flat ellipsoids. Averages over 500 random positions in space.
Fig. 7. CLDs of a sharp-corner cube (- - -) and round-corner cubes A (—) and B (
The maximal chord length which is detected decreases with increasing D. Depending upon the probe electronics (fine or coarse) and the chord range settings, these small chords might be discarded in the FBRM signal conditioning step. The average CLD for a cube of side length 1 moving randomly between 1 and 3 is compared in Fig. 12 to the ideal CLD. As for spheres short chords appear, which decreases the mean chord length from 0.874 to 0.753. If short chords smaller than
). The distances between flat faces are equal to a.
0.1 are discarded, the mean chord length increases slightly up to 0.762.
4.3. Population CLD The aim of this section is to examine how the CLD of a specific type of particle is transformed when a size
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Fig. 8. CLDs of sharp-corner parallelepipeds; (- - -) a = b = c (cube), (thin continous line —) a = b < c, (thick continuous line
5
6
4
5
2
CLD
4
3
CLD
CLD
Disdyakistriacontahedron (120 facets)
Pentakisdodecahedron (60 facets)
Icosahedron (20 facets)
3 2
1
1 0
0 0
0.2
0.4
0.6
0.8
Reduced chord length
1
0
0.2
0.4
0.6
0.8
) a = b > c.
1
Reduced chord length
8 7 6 5 4 3 2 1 0 0
0.2
0.4
0.6
0.8
1
Reduced chord length
Fig. 9. CLDs of some uniform polyhedra.
distribution (PSD) is taken into account. For simplification beam broadening is not considered. As a first example we will compare the population CLDs for spheres and ellipsoids. Finding the appropriate reference to compare them is a difficulty. Here the equivalent spherical diameter has been selected as the characteristic parameter for calculating the CLDs of populations of ellipsoids. One thousand particles with a standard distribution assuming different values for the standard deviation have been considered to build each CLD.
The volume of an ellipsoid is V = 43 abc
(7)
and the diameter of the equivalent sphere is 3 6V L= (8) In Fig. 13 the average equivalent diameter has been set to 50 m and the CLDs are plotted for two values for the standard deviation , 1 m and 10 m, assuming truncated normal size
M.-N. Pons et al. / Chemical Engineering Science 61 (2006) 3962 – 3973
Fig. 10. CLDs of twinned crystals: (a) Size of the secondary parallelepiped increases a = 5c (—), a = 10c ( ), 30◦ ( ), 60◦ (- - - -) and 90◦ (–). increases: 10◦ (
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), (b) Angle between the parallelepipeds
has demonstrated how the j th moment of the CLD (j ) is proportional to the (j + 1)th moment of the PSD (mj ): j = U T S j mj +1 ,
Fig. 11. Reduced mean () and maximal () chord length for a sphere in function of its reduced distance to the focal point (L/D).
(9)
where U is the scanning speed of the laser beam, T its scanning depth, Sj a constant which depends only on the particle shape. For a sphere, S0 = 1, S1 = /4, S2 = 2/3 and S3 = 3/16. Table 2 gives the theoretical values of the first ratios j mj +1 Sj = . (10) j −1 mj Sj −1 and the values obtained for the simulated CLDs of populations of spheres. An excellent agreement is obtained, which validates the procedure to simulate the population CLDs. In Table 3, S1 /S0 , S2 /S1 and S3 /S2 have been calculated for populations of cubes and 20-facets, 60-facets and 120-facets uniform polyhedra for a standard deviation of = 1. Except for cubes, it could be expected that the mean chord length of these shapes would be about 30% smaller than the mean particle size. 4.4. Analysis of gibbsite particles
Fig. 12. Cube CLD with beam broadening effect (—) and without (
).
distributions (no negative values). As the standard deviation increases, the population CLD differs more and more from the CLD of a single particle and information on the shape is gradually lost. It is interesting to examine how the mean particle size and the mean chord length of a population are related. Wynn (2003)
In Fig. 14, the mean chord lengths of gibbsite particles have been plotted versus the sieve median (Dsieve ). The sieve median is the arithmetic mean between the limits of the sieve range. (Dsieve ) = 20 m and (Dsieve ) = 180 m have been arbitrarily selected for the representation of the lower (< 32 m) and upper (> 160 m) size classes. For comparison the mean Deq (OM) and Ds (BS-SEM) have also been reported in Table 4. The runs with coarse settings have been duplicated by a second operator for repeatability checking. Excluding the lower (< 32 m) and upper sieve classes (> 160 m) a linear relation was found
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100 CLD (#/channel)
CLD (#/channel)
400 300 200 100 0
20
40 20 0
100
20
(b)
40 60 80 Chord length (µm)
100
100 CLD (#/channel)
CLD (#/channel)
40 60 80 Chord length (µm)
180 160 140 120 100 80 60 40 20 0
80 60 40 20 0
0
20
40 60 80 100 120 Chord length (µm)
0
(e)
20
40 60 80 100 120 Chord length (µm)
0
20
40 60 80 100 120 Chord length (µm)
0
20
40 60 80 100 120 Chord length (µm)
(d)
CLD (#/channel)
(c)
CLD (#/channel)
60
0 0
(a)
90 80 70 60 50 40 30 20 10 0
80
(f)
80 70 60 50 40 30 20 10 0
Fig. 13. Comparison of the CLDs of spheres (a and b), elongated ellipsoids (c and d) and flat ellipsoids (e and f) for two standard deviations: 1 m (a, c and e) and 10 m (b,d and f).
Table 2 Comparison of the ratios Sj /Sj −1 for j =1–3 and for two standard deviations of the PSD for spheres j
j /j −1
mj +1 /mj
Sj /Sj −1 (Simulated)
Sj /Sj −1 (Theory)
1 2 3 1 2 3
1 1 1 10 10 10
39.28 42.53 44.29 39.27 44.21 47.65
50.01 50.04 50.06 52 53.85 57.57
0.785 0.850 0.885 0.753 0.821 0.857
0.785 0.848 0.884 0.785 0.848 0.884
Table 3 S1 /S0 , S2 /S1 and S3 /S2 for cubes, icosahedra, pentakisdodecahedra and disdyakistriacontahedra Polyhedra Cube Icosahedron Pentakisdodecahedron Disdyakistriacontahedron
S1 /S0 0.87 0.67 0.74 0.73
S2 /S1 1.02 0.76 0.81 0.80
S3 /S2 1.08 0.79 0.85 0.83
between (Dsieve ) and smean with a coefficient of correlation of 0.97. For particles > 70 m smean is smaller than the mean size. It is the contrary for the smallest size classes. Various
Fig. 14. Comparison of the mean chord length (no weight): coarse settings, operator 1 (), operator 2 (), fine settings (◦), the equivalent diameter by optical microscopy (Deq ) () and the sieve median size.
weightings are proposed for the FBRM distributions. They have been applied to obtain the data summarized in Fig. 15, where the distribution mode has also been plotted. Linear relations can be obtained for each of them with the sieve mean size, although the cube weighted mean produces a low coefficient of correlation (0.72). Similar results have been reported by Heath et al. (2002) for aluminum particles. As shown on the images in Fig. 16, intact gibbsite particles may be approximated as polyhedra, such as icosahedra or
M.-N. Pons et al. / Chemical Engineering Science 61 (2006) 3962 – 3973
pentakisdodecahedra. However, the gibbsite samples contains broken particles, which are more similar to parallelepipeds. By analysis of OM and BS-SEM images it can be concluded that the smaller size classes contain more fragments than larger size classes: they are characterized by a larger circularity and elongation and a lower robustness (Table 4). The ratios S1 /S0 have been evaluated from the mean chord length and the median sieve diameter. They decrease as the particle size increases. One reason could be the higher proportion of icosahedra-like
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particles in the upper size classes as their theoretical S1 /S0 , is 0.67, when it is 0.87 for cubes. Another reason could be that for the smallest particles a large proportion of small chords are eliminated by the binning between 2 and 2000 m. Fig. 17 compares the experimental CLDs of two size classes obtained with the coarse and the fine settings. For the 32–50 m sieve fraction the CLD mode is lower with the fine settings and a secondary peak appears around 8 m. This peak corresponds to the numerous parallelepiped-like fragments present in that size class. For the 125–140 m sieve fraction, the CLD modes almost coincide but the fine settings reveals a shoulder similar to the one observed on the icosahedron CLD (Fig. 9). 5. Conclusions
Fig. 15. Comparison of the mean chord lengths given by various weightings and the mode with the sieve median size.
A procedure for the calculation of the chord length distribution of any rigid and opaque 3D object was developed by combination of a scene-rendering software and an image analysis software. It was successfully applied to simple convex shapes and to their combinations used to represent crystal agglomerates. For the individual particle its CLD is a shape fingerprint. However in the case of a population of particles, a loss of information is observed depending upon the spread of the particle size distribution. Another loss of information is due to the fact that short chords are discarded during the FBRM signal treatment step. A loss of short chords may be desirable in order to reduce the effect of noise due to the decrease in laser power with distance and/or to filter out the signal of possibly present fines (Ruf et al., 2000). On the other hand the information on short chords is important to detect the presence of elongated
Fig. 16. SE-SEM (a and b) and BS-SEM (c and d) images of 32–50 m (a and c) and 140–160 m (b and d) particles.
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M.-N. Pons et al. / Chemical Engineering Science 61 (2006) 3962 – 3973
Table 4 Size and shape characteristics of the gibbsite fractions Sieve fraction (m) < 32 32–50 50–71 71–90 90–112 112–125
Deq (m)
C
E
R
52 9 64 8 80 12 98 13 120 15
1.24 0.18 1.12 0.06 1.12 0.05 1.11 0.05 1.13 0.12
1.15 0.09 1.19 0.11 1.18 0.12 1.15 0.09 1.16 0.09
0.76 0.05 0.77 0.04 0.76 0.04 0.78 0.04 0.77 0.04
Ds (m)
Cs
Es
Rs
21.4 37.5
1.64 1.69
1.71 1.57
0.67 0.68
47.9
1.65
1.41
0.71
60.0
1.43
1.4
0.72
70.3
1.48
1.32
0.73
98.2
1.49
1.27
0.75
105
1.44
1.30
0.74
122
1.27
1.21
0.76
142
1.37
1.22
0.76
S1 /S0
1.59 1.12 0.91 0.78 0.75
125–140
0.72 140–160 > 160
157 15 189 19
1.07 0.04 1.07 0.14
1.15 0.08 1.15 0.09
0.78 0.03 0.78 0.04
0.66
For OM visualization, number-averaged equivalent diameter Deq , circularity C, elongation E and robustness R. Standard deviations in italics. For BS-SEM visualization, Ds -averaged equivalent diameter Ds , circularity Cs , elongation Es and robustness Rs . Experimental values of S1 /S0 .
the mean particle size (Hobbel et al., 1991) when Tadayyon and Rohani (1998) give values around 0.8. The main advantage of the FBRM lies in its capability to deliver combined information on shape, size and concentration, on-line and in real-time. The procedure described in the present work helps to understand the relations between particle size, shape and chord length distributions in order to make best use of the FBRM signal. Notations
Fig. 17. Comparison of the CLDs of two gibbsite sieve fractions: (a) 32–50 m, (b) 125–140 m for the fine (—) and the coarse ( ) settings.
objects. Their discarding increases also the mean chord length. In literature, the comparison with the particle mean size obtained by other techniques (sieving, laser diffraction) has given rise to very different conclusions (Wynn, 2003). Some authors have found a ratio of 2.5 between the mean chord length and
a A b c cl mean D Dsieve k ka kv L Lmax mj n ne nr P q qp s S Sj T U
size parameter for ellipsoids and parallelepipeds area size parameter for ellipsoids and parallelepipeds size parameter for ellipsoids and parallelepipeds mean chord length distance to focal point median sieve size shape parameter in Eq. (1) surface shape factor volume shape factor characteristic size maximal characteristic size j th moment of the particle size distribution particle size distribution number of morphological erosions number of particles Crofton perimeter chord length distribution chord length distribution of a population of particles chord length surface constant in Eq. (9) scanning depth of the laser beam scanning speed of the laser beam
M.-N. Pons et al. / Chemical Engineering Science 61 (2006) 3962 – 3973
V zcam
volume position of the camera
Greek letters j
j th moment of the chord length distribution standard deviation
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