3D imaging studies of rigid-fiber sedimentation

David W. Vahey*a, Emilio J. Tozzi

b, C. Tim Scott

a, Daniel J. Klingenberg

c

aUSDA Forest Products Laboratory, 1 Gifford Pinchot Drive, Madison, WI, 53726;

bDept. of Chemical Engineering and Material Science, University of California,

Davis, CA 95616 cDept. of Chemical Engineering and Rheology Research Center, University of Wisconsin,

Madison, WI, 53706

ABSTRACT

Fibers are industrially important particles that experience coupling between rotational and translational motion during

sedimentation. This leads to helical trajectories that have yet to be accurately predicted or measured. Sedimentation

experiments and hydrodynamic analysis were performed on 11 copper “fibers” of average length 10.3 mm and diameter

0.20 mm. Each fiber contained three linear but non-coplanar segments. Fiber dimensions were measured by imaging

their 2D projections on three planes. The fibers were sequentially released into silicone oil contained in a transparent

cylinder of square cross section. Identical, synchronized cameras were mounted to a moveable platform and imaged the

cylinder from orthogonal directions. The cameras were fixed in position during the time that a fiber remained in the field

of view. Subsequently, the cameras were controllably moved to the next lower field of view. The trajectories of

descending fibers were followed over distances up to 250 mm. Custom software was written to extract fiber orientation

and trajectory from the 3D images. Fibers with similar terminal velocity often had significantly different terminal

angular velocities. Both were well-predicted by theory. The radius of the helical trajectory was hard to predict when

angular velocity was high, probably reflecting uncertainties in fiber shape, initial velocity, and fluid conditions

associated with launch. Nevertheless, lateral excursion of fibers during sedimentation was reasonably predicted by fiber

curl and asymmetry, suggesting the possibility of sorting fibers according to their shape.

Keywords: Sedimentation, 3D imaging, fibers, orthogonal cameras, terminal velocity, angular velocity, helical

trajectory, 3D image processing, hydrodynamic theory, fiber curl, fiber asymmetry.

1. INTRODUCTION

Fibers are industrially important particles that experience coupling between rotational and translational motion during

sedimentation in viscous fluids. Our work expands hydrodynamic theory to account for their sedimentation, and it uses

3D imaging of their trajectory and orientation to verify the theory. Experimental results support the possibility of using

sedimentation to sort fibers according to their shape.1

Settling of particles in fluids is relevant to many industrial processors, including mixers,2 hydrocyclones and decanters.3

While terminal velocities are a common focus, particles of complex shape usually acquire horizontal and angular

velocity components that greatly influence their settling trajectories. For example, non-planar fibers acquire angular

velocity about their center of mass during sedimentation. The same fibers follow a helical trajectory.4 At steady state, the

time required to negotiate a single pitch of the helix is predicted to be the same as that required for 360° fiber rotation

about its center of mass. This time can differ significantly among fibers, even for those with similar terminal velocities.

Experimental studies of settling fibers are complicated by the need to track orientations and trajectories in three

dimensions over times long enough to approach steady-state behavior. Another challenge is to relate observations to

theories from which dependences of hydrodynamic properties on fiber shape can be predicted.5 In this work, we report

measurements of terminal velocity, angular velocity, and helical trajectory of rigid, sedimenting asymmetric fibers using

3D imaging.6 We discuss the measurements in terms of simple shape factors that correlate with experimental

observation and demonstrate the potential for fiber sorting based on shape.

Three-Dimensional Imaging, Interaction, and Measurement, edited by J. Angelo Beraldin, et al.,Proc. of SPIE-IS&T Electronic Imaging, SPIE Vol. 7864, 786405 · © 2011 SPIE-IS&T

CCC code: 0277-786X/11/$18 · doi: 10.1117/12.872128

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2. METHOD

2.1 Orthogonal Cameras

In many applications of 3D imaging of solid objects, the concern is with spatial relationships between multiple objects in

the field of view. Stereoscopic images of individual solid objects are not generally able to provide full 3D coordinate

information of the object’s complete surface, but only of the surface in view of both cameras. In the case of fibers and

curvilinear objects in general, with one object in the field of view, this limitation is overcome by the use of two

orthogonally positioned cameras.7 One camera provides coordinate information for the xz projection of the object. The

other provides coordinate information for the yz projection. Of course, z information is overdetermined and care must be

taken to know relative vertical displacements for the two cameras if errors are to be minimized.8

Figure 1 shows the camera configuration for the experiments. The cameras are identical Pulnix TM-1325 models (JAI,

San Jose, CA) with a 2/3-inch, 8-bit progressive scan format and 1392(H) × 1040(V) spatial resolution. The cameras are

mounted to a platform controlled to within 0.0001 inch vertical by a linear motor (Compumotor L20; Parker Hannifin

Corp.). The camera lenses are identical Toyo Optics TV Zoom lenses (formerly Toyo Optics USA, Irvine CA) 12.5-75

F1.8 with macro capabilities. Magnification was determined by the desire to match the horizontal window of the

containment cell to the camera field of view. For this purpose, and in view of the short range involved, extender tubes

were employed.

2.2 The Containment Cell

The containment cell was of square cross section, with 50.8-mm (2-inch) interior dimensions on each side. The length

was 45.7 cm (18 inches). The walls were transparent Poly(methyl methacrylate) (PMMA) of thickness 5.56 mm and

refractive index 1.4914 (acrylic). The cell was filled with silicone oil of density 971 kg/m3 and viscosity of 1.04 Pa·sec

(Polydimethylsiloxane, trimethylsiloxy terminated, United Chemical Technologies, Inc., Bristol PA). This material was

chosen for its long-term stability in the lab environment, and because it limited the terminal velocity of the fibers used in

the experiments to about 1 mm/sec. Its refractive index was 1.4034. This was sufficiently close to the refractive index of

the PMMA walls that the error in vertical fiber coordinates made from considering the refractive index of both walls and

oil to be that of the oil was negligible.

Figure 1. Photograph of the experimental station.

2.3 Illumination

Back lighting of descending fibers was provided by a Fostec fiber optic light plate of active dimensions 5.6 cm × 26 cm

(Schott North America, Elmsford, NY). The long dimension determined the range of the trajectory measurement. The

source of illumination was tungsten-halogen (Dolan Jenner Fiber-Lite DC-950, Lancaster SC). The light plate was

centered symmetrically behind the containment cell to provide the same degree of illumination to both cameras. This

produced an approximately 45° angle of incidence of light rays on the cell. White diffusers (papers) were mounted to

redirect some reflected light back toward the container. The fibers showed as dark objects against a brighter background.

Loss of background brightness near the end of the vertical range of illumination was not a factor.

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2.4 Fibers

Silver-coated copper fibers of density 8940 kg/m3 were prepared from 0.20-mm diameter wire by cutting to length and

manually bending to shape. The two classes of shapes used are represented in Figure 2. All fibers were non-coplanar.

Some fibers had three segments of substantially different lengths, while others had at least two segments of

approximately the same length. The fiber shapes were characterized by imaging from three perspectives, as shown in

Figure 3. The dimensions obtained from these images were used in predictive models.6 Table 1 shows the measured

segment lengths of the 11 fibers used in the study. Fiber IDs are from prior publications and don’t signify noteworthy

groupings in the context of the present study.6,8 For each fiber, highlighted cells indicate segments of approximately

equal length. Total lengths, segment lengths, and segment angles were chosen to provide a variety of shapes, sizes, and

degrees of symmetry. The choices ultimately provided a wide range of well-distributed angular velocities during

sedimentation.

Figure 2. The two major classes of fiber shapes used: Left), three segments of different lengths; right) at least two

segments of approximately equal length.

The last two columns of Table 1 give shape metrics, defined later, that may be used to correlate with sedimentation

measurements in the hope of predicting sedimentation properties for arbitrarily shaped fibers. These shape measures,

curl and asymmetry, are simple geometric constructs and not necessarily optimal for prediction5, but they are easy to

visualize and one of them, curl, has been used widely in the paper industry.9 Note that each shape class in Figure 2 is

represented by fibers that have both high and low curl and asymmetry values.

Figure 3. Fiber shape characterization. Each fiber is considered as lying in two planes that make an angle with

respect to each other. Images are taken normal to the two planes, and a third image is taken to capture the angle .

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Table 1. Lengths and shape measures of fibers used in sedimentation measurements. Rows with highlights indicate

fibers having at least two segments equal within 0.2 mm.

Fiber

ID

Length

(mm)

First

Segment

(mm)

Middle

(mm)

Last

Segment

(mm)

Curl Asymmetry

C4 11.4 4.2 2.7 4.4 1.53 0.04

C5 6.9 4.2 1.2 1.5 0.34 1.32

C7 10.7 7.0 1.5 2.3 0.27 1.58

C8 9.7 5.8 1.2 2.7 0.34 0.95

D1 9.9 1.6 1.6 6.6 0.20 2.00

D2 11.3 3.9 3.2 4.1 0.22 0.05

E1 10.3 5.9 2.2 2.2 0.27 1.10

E2 11.1 3.9 3.0 4.2 0.16 0.06

E3 10.8 2.4 5.6 2.7 0.12 0.06

E4 10.4 3.7 3.6 3.1 0.34 0.11

E5 11.0 3.7 3.6 3.6 0.59 0.02

2.5 Experiment

Fibers were held by tweezers and released just below the liquid surface. No attempt was made to anticipate steady-state

conditions by providing an initial velocity or orientation. We recognized that some time and distance would be

consumed approaching steady state. The cameras were triggered at 0.6 Hz, corresponding to 1.7-mm (typical) vertical

fiber motion between frames. About 18 frames were grabbed before it was necessary to lower the cameras to the next

field of view. With practice, this could sometimes be accomplished without dropping a single frame. Pulnix camera

signals from each frame were sent to an X64-CL Express framegrabber (Dalsa Coreco, Saint-Laurent, Quebec, Canada).

Streampix software (v.3.20.0, Norpix, Montreal, Canada) displayed and stored the synchronized images as bit-mapped

files along with a time stamp.

Figure 4 shows vertically cropped, orthogonal views of descending fiber E5, containing approximately equal-length

segments oriented at 90° with respect to each other. The selected frames were separated in time by 20 seconds. Slightly

more than half the trial is represented by the sequence i-vii. By image iii, 53 seconds from start, the middle segment is

essentially horizontal and rotation about the center of mass is apparent. However, later analysis showed that steady-state

rotation did not commence until about 100 seconds from start.

For most fibers, terminal velocities were within ±5% of 1.0 mm/sec. The main exception was fiber C4, the highest curl

fiber in Table 1, which descended at 1.28 mm/sec. Fiber C7, the most asymmetric fiber in Table 1, was predicted to have

a terminal velocity of 1.16 mm/sec, but was slowed to 1.01 mm/sec after approaching within 6 mm of the container

wall. In contrast with most terminal velocities, angular velocities were generally quite different in their magnitude and

onset.

No attempt was made to recover fibers from the bottom of the containment cell. Silicone oil, once thermally or

mechanically disturbed, takes a long time to dissipate resulting currents. Initially fibers were made from iron wires, and a

magnet was used to recover them from the bottom of the containment cell. However, we discovered that the earth’s

magnetic field was influencing their sedimentation. Copper fibers coated with silver were used thereafter.

2.6 Data Analysis

The goal of data analysis is to determine the xyz coordinates of the center of mass of each fiber in each synchronized set

of frames and to determine the angular orientation of the fiber relative to its center of mass. To this end, an interactive

program written in Visual Basic was employed. The fiber shape determined as described in Section 2.4 is superimposed

as a “virtual fiber” on the experimental image of the sedimenting fiber. The virtual fiber is then translated, rotated, and

enlarged or reduced until it coincides with the image of the sedimenting fiber. The position and orientation of the fiber in

the experimental image are thus determined by the position and orientation of the virtual fiber. This operation is repeated

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for each image to determine the fiber center-of-mass position (x, y, and z) and orientation (via the Euler parameters p0,

p1, p2, and p3) as functions of time.5, 10

Camera 1: xz view Camera 2: yz view

Figure 4. Fiber E5 as seen by orthogonal cameras as a function of time, t, after release. i) t = 13 sec; ii) t = 33

sec; iii) t = 53 sec; iv) t = 73 sec; v) t = 93 sec; vi) t = 113 sec; vii) t = 133 sec.

Gross changes in vertical position that occur when the cameras move to change the vertical field of view can in principal

be handled by calibration of the linear drive. However, sufficient accuracy and a greater comfort level are achieved by

mounting transparent rulers to the containment cell walls opposite the cameras. The rulers are also helpful in focusing

the camera at the center of the containment cell.

When the complete data for a fiber are available, the xyz coordinates of the center of mass as a function of time describe

the fiber’s trajectory. Experimental velocity versus time can be calculated from the trajectory, as can angular velocity

and the radius of the smallest cylinder containing the trajectory (henceforth, helical radius). The Euler parameters

describe the evolution of the fiber’s angular velocity from the perspective of the rotating fiber.

2.7 Predictions

Model predictions in broad agreement with experimental results include both a numerical simulation based on the work

of Brenner,11,12 and an analytical solution based on estimates for hydrodynamic resistances.4 The numerical simulation

is capable of predicting transient fiber position and orientation during sedimentation, whereas the analytical approach

delivers steady-state values for velocity, angular rotation, and helical radius. Both treatments deal with arbitrarily shaped

fibers and are not based on curl or asymmetry inputs. Details of this work are described by Tozzi6 and Tozzi et al.8

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3. RESULTS

3.1 Terminal Velocity

For the present fibers, the difference between terminal velocities calculated numerically and analytically averaged less

than 1%. The disparity between experimental and numerical average velocities was 4%, with experimental velocities

tending to be lower. If fiber C7 is excluded from the analysis because of an apparent “wall effect,” the disparity

decreases to 3%. Average velocities over the full sedimentation interval were used in this comparison because it was

difficult to distinguish the transition to terminal velocity experimentally. There is no certainty that a terminal velocity is

actually reached in the experiments, though one is reached in the simulation. For example, fiber E5 shown in Figure 4

was found to accelerate very slightly over the course of its 4-minute sedimentation, from a velocity of 1.004 mm/sec to a

velocity of 1.042, According to a logarithmic fit to the data, a velocity matching the 1.10 mm/sec terminal velocity

determined numerically for fiber E5 would have required a sedimentation time of 2 hours! However, numerical

simulation suggested that terminal velocity should be approached within 0.1% in only 25 seconds. Some assumptions

inherent in the simulation are likely being violated; for example, thermal gradients in the silicone oil could result in

viscosity changes that make the terminal velocity vary as the fiber descends.

3.2 Angular Velocity

Figure 5 shows the correlation between experimental and numerically simulated terminal angular velocity. Values are

obtained from rotation of the fiber about its center of mass, as in Figure 5a, or as rotation of the center of mass about the

gravity axis, as in Figure 5b. The correlation between simulated and experimental angular velocity values in Figure 5a is

excellent. The standard error of estimate, 0.12°/sec is associated with 3% error for the four fastest rotating fibers and an

average 12% error over all. Fiber C7, which approached a wall of the containment cell, was not an outlier with regard to

angular velocity and is included in the analysis. It is labeled in Figure 5 using an open circle.

Figure 5b shows the correlation between experimental and simulated angular velocity as determined by the helical

trajectory. The correlation is marred by outliers that correspond to the three fibers having the highest simulated angular

velocity. Experimental angular velocity measured from the helical trajectory is lower in all three cases. Center of mass

rotation about the gravity axis can’t keep up with the fiber’s rotation about its center of mass when that rotation exceeds

about 1°/sec.

(a) (b)

Figure 5. Experimentally determined angular velocity versus numerically simulated angular velocity a), using

Euler parameters to calculate the fiber’s rotation about its center of mass, and b), using least-squares analysis to fit

the xy projection of the fiber’s center of mass to a circle.

3.3 Helical Radius

One of the three outliers above corresponds to fiber E5 shown in Figure 4. Figure 6a is a projection of the E5 trajectory

on the xy plane. The small, red (in colored versions) grouping of points in the simulation shows the helical radius

associated with the trajectory. It doesn’t appear until 100 seconds have passed. Once begun, the calculated radius is too

small to be well-resolved in the figure, only 14 m. There is no clear evidence of helical motion in the experimental

trajectory, the dotted curve in Figure 6a, though data for times t > 100sec have been fit with a circular arc having a radius

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of 1.56 mm. This larger experimental radius is in contrast to the smaller simulated radius of only 14 m. This size

relationship is characteristic of all three fibers associated with outliers in Figure 5b, though the disparity is not as great.

For contrast, Figure 6b shows the helical projection for fiber C8. This is one of eight fibers in Figure 5 showing close

agreement between angular velocity measured from rotation about the center of mass and from the helical trajectory. It

has a radius predicted from numerical simulation to be 6.1 mm, whereas the corresponding experimental radius was 5.0

mm.

3.4 Radial Excursion

Although the simulated trajectory in Figure 6a doesn’t match the experimental trajectory in detail, both trajectories do

show that excursions from the starting position are small; that is, in the range 1-2 mm. In Figure 6b, the excursions are

6–10 mm. A simple procedure could be set up to sort these fibers on the basis of their radial excursion from an initial

starting position. If one ignores complications associated with wall effects, length and rigidity of the fibers, a cylindrical

bin of radius 3 mm centered beneath their starting position at an appropriate depth would effectively separate them.

Figure 7 shows the time evolution of radial excursion for all 11 fibers used in the study. If the experimental and

simulated excursions (shown in Figures 7a and 7b, respectively) at 200 seconds are correlated, the r2 is 0.919. The

equation predicting experimental excursions, Rexp, from simulated values, Rsim, is

mmRR Sim 14.3698.0exp . (1)

The standard error of estimate is 1.52 mm. The practical impact of this finding is to limit the number of sorting bins. For

the present fibers, three concentric bins of radius 6.4, 10.7 and 20.2 mm would capture four of four fibers with simulated

helical radius under 5.0 mm, five of five fibers with simulated helical radius between 5.0 mm and 12.2 mm, and two of

two fibers with still larger simulated radius. Any different arrangement or greater number of sorting bins would result in

sorting errors.

(a) (b)

Figure 6. Simulated (Sim) and Experimental (Exp) trajectories of two fibers. a), Fiber E5, showing disparity

between simulated and experimental trajectories. b), Fiber C8, showing relative agreement between simulated and

experimental trajectories. Note the much greater excursions from the starting point in the case of fiber C8.

4. ANALYSIS

4.1 Fiber Sorting According to Shape

Tozzi et al. have done a detailed study of the influence of 17 different fiber-shape metrics on simulated intrinsic

viscosity.5 Present interest is in the relationship between fiber shape and experimental velocity, angular velocity, and

helical radius. Curl and asymmetry, shape metrics introduced in Table 1, have been selected for their simplicity.

Influence of total fiber length is also considered. A data set limited to 11 fibers and 3 correlating variables doesn’t justify

SPIE-IS&T/ Vol. 7864 786405-7

greater complexity. However, the results are interesting and may serve to stimulate further research on optimal shape

metrics for fiber-based industries.

(a) (b)

Figure 7. Radial excursions from a common starting point for all eleven fibers used in the study, as a function of

time. a), Experimental results; b), Numerically simulated results. Note: Fiber C5 sedimentation was terminated at

125 sec. Excursion at later times was extrapolated from experimental angular velocity and radius at 125 sec.

Figure 8 shows definitions of curl and asymmetry that were applied to the fibers of Table 1. Curl calculation is readily

applicable to fibers of general shape and has been used widely in the paper industry.9 Asymmetry calculation is readily

applied to the three-segment fibers of the present study. However, it is not clear how to transfer the calculation to fibers

of general shape. The idea in Figure 8b is to model the fiber with three connecting line segments and use the formula

shown. The use of multiple straight line segments has been used to describe fiber “kink”, in the paper industry.13

However, the number of segments is not limited to three and the calculation depends on the angles at which the segments

meet, rather than the length of the segments. Tozzi et al. have considered curl and kink indices, and found that they

didn’t correlate particularly well with fiber intrinsic viscosity.5 The likelihood exists that there are better shape metrics

for predicting sedimentation characteristics than curl and asymmetry, though they may not be as readily visualized. Note

that the curl and asymmetry of a straight fiber are both zero according to the definitions in Figure 8.

(a) (b)

Figure 8. Calculations of curl and asymmetry applied to a fiber of general shape. a), curl; b), asymmetry.

4.2 Relation of Curl to Sedimentation Velocity

Figure 9a shows the correlation between the average experimental sedimentation velocity and the curl index. The r2

value of 0.84 is clearly enhanced by the presence of fiber C4 with a curl index of 1.53, several times larger than the next

highest curl index represented. The standard error of estimate is 0.04 mm/sec, corresponding to 4% of the average

velocity. The possible influences of fiber length and asymmetry on the correlation were also considered using multiple

linear regression. Neither improved the predictive capabilities of the curl index alone. The fibers labeled E4 and C8 have

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no obvious features that explain their appearance as outliers in Figure 9a. Fiber C7, noted for its close approach to the

container wall, was not an outlier.

The correlation between curl and velocity is physically appealing in that the curl gap length approximately represents the

horizontal projection of the descending fiber’s total length. This portion experiences the main hydrodynamic resistance.

A shorter gap indicates less hydrodynamic resistance, greater velocity and higher curl index.

4.3 Relation of Curl, Asymmetry, and Fiber Length to Angular Velocity

Figure 9b shows the predicted versus experimental angular velocity (absolute value) based on the empirical equation

from multiple regression analysis:

ACLL 139.049.298.1301.1 , (2)

where is angular velocity in °/sec, L is fiber length, L is average fiber length, 10.31 mm, C is curl, and A is

asymmetry. Only the absolute value of angular velocity is plotted in Figure 9b. The P-values for the three variables are

0.029, for LL , 0.0001, for C , and 0.013 for A1 . Curl remains the dominant shape factor determining angular

velocity. Correlation with C alone results in a correlation coefficient r2 of 0.833 and a standard error of estimate of

0.32 °/sec. However, when length and anisotropy are included in multiple regression analysis, the correlation coefficient

improves to 0.939 and the standard error reduces to 0.22 °/sec.

(a) (b)

Figure 9. (a), Correlation between curl and the average experimental velocity during fiber sedimentation. (b),

Correlation of curl, asymmetry and fiber length to absolute angular velocity using multiple regression analysis.

4.4 Relation of Curl, Asymmetry, and Fiber Length to Radial Excursion

Because of the uncertainty in measuring the (small) helical radius of fibers with high angular velocity, radial excursion

of all fibers is used as a stand-in. Radial excursion is relevant to the practical application of sorting fibers. Figure 10

shows the predicted versus experimental radial excursion at 200 seconds based on the following empirical equation from

multiple regression analysis:

ACLLR 1.1035.14.300.31 , (3)

where R is the radial excursion, in millimeters, and other variables are defined as in Eq. (2). The P-values for the

correlating variables in Eq. (3) are 0.023, for LL , 0.485, for C1 , and 0.005 for A , indicating that asymmetry is

the dominant shape factor influencing radial excursion. Note the complimentary dependences of angular velocity and

radial excursion on curl and anisotropy: Curl is most significant in Eq. (2), and anisotropy appears as a reciprocal. The

situation is reversed in Eq. (3). The results suggest AC as an appropriate shape factor for predicting angular velocity,

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while CA is appropriate for predicting radial excursion. In fact, these combination shape factors correlate with the

indicated data, having r2 values of 0.738 and 0.361, respectively. The lower r2 between CA and radial excursion must

reflect, at least in part, the approximation inherent in using radial excursion as a stand-in for helical radius.

Figure 10. Correlation of curl, asymmetry, and fiber length to radial excursion using multiple regression analysis.

5. DISCUSSION

This work demonstrates that sedimentation of complex-shaped fibers is a 3-dimensional phenomenon. The nature of the

sedimentation, and in particular its 3D aspects, are strongly influenced by the shape of the fibers. The influence is

demonstrated here by correlations between sedimentation variables and simple shape factors. The success of these shape

factors gives hope that complex interactions between fibers and surrounding fluids can be understood and controlled

well enough to tailor large distributions of fibers emphasizing preferred shapes. Such distributions could help improve

the outcomes of industrial processes such as composite manufacture and papermaking.

At the present time, a number of instrument manufacturers make equipment for 2D imaging and subsequent analysis of

pulp fibers in water suspension (OpTest, Hawkesbury, ON Canada; Lorentzen & Wettre USA, Alpharetta, GA; Metso

Automation, Helsinki, Finland; TechPap North America, Norcross, GA). Several of these companies use the images to

calculate curl9 and kink indices,13 along with fiber length. At least two strongly promote curl as important to the tensile

properties of paper.14,15 The present sedimentation experiments have shown that high curl promotes high angular velocity

about the particle center of mass. If this holds in the papermachine environment, it supports fiber entanglements leading

ultimately to greater fiber bonding and paper strength. Asymmetric fibers, noted here for large radial excursions in

sedimentation, may serve to disrupt the tendencies of cellulose fibers to flocculate. This would promote more uniform

mass distribution in paper, another component of strength.16 A modeling framework to evaluate these possibilities has

been provided by the work of Switzer and Klingenberg.17,18

A disadvantage of the commercial fiber analyzers cited above is that they are limited to two-dimensional fiber images.

Their curl calculation is based on projected fiber length and projected gap on the focal plane of a single camera. The

instruments usually confine the fibers to a thin volume, so that all parts of the fiber are captured in the image. Curl

measured in this way is approximate in that the fiber may be distorted to fit into the thin volume.14,15 It may recover its

former shape when leaving the measurement field of view. The prediction of the fiber’s hydrodynamic properties from

its shape may be compromised as a result.

If one attempts to measure the fiber’s projection with a single camera, the nature of the errors can be inferred by

examination of Figure 6. Looking at the two fibers’ experimental excursion along the y axis only, Figure 6a indicates

that fiber E5’s net excursion is 1.5 mm. This is less than the actual xy excursion of 2.5 mm. However, Figure 6b

indicates that that fiber C8 has a net excursion of 0 mm along the y axis at the end of its descent, less than that of fiber

E5. This determination is opposite that provided by the full 3-dimensional measurement. It could lead to erroneous

conclusions about the dependences of fiber sedimentation on shape. For this reason, 3D imaging of fibers is preferred

whenever possible. Efforts should be made to incorporate 3D imaging of fibers into practical measurement systems.

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6. CONCLUSION

This work demonstrates that 3D imaging using two orthogonal cameras is a good way to study rigid-fiber sedimentation.

The results give credence to the accuracy of theoretical and numerical methods for predicting sedimentation properties,

so that in the future they may be more confidently applied to cases where experiments are difficult to perform. At the

same time, the results have found areas of disagreement between experiment and theory that warrant further study. Why,

in the experiment, is it difficult to identify the transition to terminal velocity? Why does close approach to a container

wall appear to influence some sedimentation properties more than others? Why does the trajectory of descending fibers

apparently cease to be helical when their angular velocity exceeds a certain value? Efforts to answer these questions can

add to our existing knowledge of rigid fiber sedimentation and further our understanding of the potential for fiber-sorting

according to shape using sedimentation dynamics.

7. ACKNOWLEDGMENTS

This project was supported in part by the National Research Initiative of the USDA Cooperative State Research,

Education and Extension Service, grant number 2006-35504-17401.

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Phys. Fluids, to be published

[9] Page, D. H., Seth, R. S., Jordan, B. D. and Barbe, M. C., “Curl, crimps, kinks and microcompressions in pulp fibres:

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