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Flocs in Shear and Strain Flows
Journal of Colloid and Interface Science 225, 273–284 (2000) doi:10.1006/jcis.1999.6671, available online at http://www.idealibrary.com on
Flocs in Shear and Strain Flows Stefan Blaser Institute of Hydromechanics and Water Resources Management, Swiss Federal Institute of Technology, CH-8093 Z¨urich, Switzerland E-mail: [email protected] Received March 18, 1999; accepted December 13, 1999
However, to date the dynamics of aggregated particles and their interaction with the fluid motion are not fully understood. In the present work therefore we study the behavior of single flocs by using optical methods. Preflocculated ferric hydroxide flocs were subjected to well-defined flow conditions such as either simple shear or two-dimensional straining flow and their motion was recorded with a CCD camera. Appropriate types of apparatus were designed to produce the flow fields on a microscopic scale. By means of digital image analysis, quantitative information on deformation and orientation of the flocs can be extracted from the images. A simple model is established in which a floc is represented by a solid ellipsoid. This will allow us to compare the measured time-evolution of orientation with an analytical solution. In addition, the forces needed to break the flocs apart can be estimated.
Preflocculated ferric hydroxide flocs were subjected to either a simple shear flow or a two-dimensional straining flow, and their motion was optically observed. Digital image analysis was applied to extract information on orientation and deformation from the digitized frames. It was found that the simple shear flow led to a rotation of the flocs whose motion can be understood from the behavior of a solid ellipsoid. In the extensional flow, no continuous rotation occurred and flocs were broken apart along the axis of straining. The rupture forces estimated from an ellipsoid model were found to be in the range of 0.1 N/m2 . °C 2000 Academic Press Key Words: flocs; shear flow; strain flow; digital image analysis; orientation; deformation; break-up; rupture forces.
1. INTRODUCTION
The process of flocculation results in the aggregation of small particles suspended in a fluid to larger clusters (“flocs”) and thus favors the separation of solid and liquid phases. Particle aggregation occurs in many industrial and engineering applications such as water and waste water treatment (1, 2) and the manufacture of paper (3). In order to facilitate aggregation, the fluid is mixed, which usually leads to turbulent flow conditions. However, if the hydrodynamic forces are too strong, the forming flocs may be broken apart (4–6). Despite the vast literature on flocculation, investigations based on the direct observation of the disintegration process are rather limited. In Table 1 some relevant references are compiled. Basically, two break-up mechanisms have been found: large-scale splitting (break-up into a few fragments of comparable size) (8, 12, 13) and fine-particle erosion (gradual shearing off of small fragments from the floc surface) (7–10). Kao and Mason (7) were among the first to draw attention to the fact that shear and straining flows are profoundly different in the dispersion of particles. They carried out experiments with non-cohering aggregates in silicone oil. They found that in a simple shear field individual spheres gradually disengaged from the aggregate and continued to circle around it, whereas in the pure straining field an aggregate disintegrated much more rapidly with the spheres dispersed along the axis of straining.
2. MATERIALS AND METHODS
2.1. Couette Cell and Taylor Four-Roll Mill As a preliminary remark, I illustrate the necessity of flow cells with small dimensions by the following consideration. Suppose that the flow field is characterized by the kinematic viscosity ν, the length scale L, and the scale G associated with the velocity gradient. In order to have a well-defined fluid motion, the flow must be kept laminar, which implies that the Reynolds number, given by Re = G L 2 /ν, does not exceed a critical value Rec . On the other hand, a floc will only deform substantially if the forces exerted by the fluid on the particle’s surface are large enough. As these forces are related to the stress tensor, this condition involves a large-scale G. Hence the length scale L has to satisfy the inequality r ν Rec , [1] d
273
1
The strip was produced by W. Schmidheiny, Institute of Polymers, ETHZ. 0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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STEFAN BLASER
TABLE 1 Literature Overview on Floc Break-up Investigated in Different Flow Fields and for Different Flocculated Systems Floc system Ref. no.
Flow
Medium
Particles
Flocculants
7 8 9 10 11 12 13
Shear, strain Strain Shear Flow through grid Contraction Grid-flow Jet
Silicone oil Water + sucrose Silicone oil Water Water + glycerol Water Water
PMMA spheres None Polystyrene Kaolin Latex Starch Kaolin
None Iron(III) Water Polymer Al2 (SO4 )3 + polymer Al2 (SO4 )3 Iron(III) + polymer
four cylinders as is depicted in Fig. 1. Another cylinder could be adjusted to tighten the strip. This geometrical arrangement utilized by other investigators (14, 15) to study the turbulent plane Couette flow has the advantage that the mean flow vanishes as two plane walls move in opposite directions but with the same magnitude of velocity. The test section where the flocs could be observed was 100 mm in length, 1 mm in width, and 10 mm in height. The strip was driven through friction by one of the two larger cylinders (16 mm in diameter), which was mechanically coupled to a step motor. The number of revolutions could be adjusted in the range 0–1,200 rpm resulting in a maximum speed of either wall of 10 cm/s. The strip was transparent, which allowed sideways illumination of the test section by a light sheet. The entire assembly was placed in a rectangular Plexiglas tank, which was filled with water through several feed pipes. The four-roll mill invented by Taylor (16) is commonly used to produce a two-dimensional straining flow (17–20). Higdon (21) compared various design criteria for this device, in particular the ratio of cylinder diameter to cylinder spacing. He concluded that a ratio of 0.625 yields the best fit between the circular surface of a cylinder and the hyperbolic streamlines characteristic of the pure extensional flow.
FIG. 1. Plan and side views of the cell to produce a plane Couette flow.
Figure 2 shows a sketch of the apparatus used in the experiments. Four cylinders (3.12 mm in diameter and 10 mm in height) were mounted at the corners of a square the sides of which were 5 mm. They were enclosed in a rectangular tank, which could be fed with water through two pipes. Four gear wheels driven by a step motor were arranged such that the rollers rotated in the directions indicated by arrows in Fig. 2. 2.2. Data Acquisition The observation area of the Couette and Taylor cell was illuminated by a 5 W argon ion laser (Coherent Innova 305), which had 2.0 W of the total power concentrated in the 514.5 nm line and 1.5 W in the 488 nm line. An optical system consisting of three lenses and two mirrors was used to generate a light sheet having the desired location and width of its beam-waist. Each of the two flow cells could be mounted on a three-axis translating table with micrometers for positioning in either of the three directions. Movement along the vertical axis was used to position the light sheet at the desired height of the apparatus. In order to prevent a blurred imaging of the moving particles, the illumination time was controlled by a Bragg cell (IntraAction
FIG. 2. Plan and side views of the Taylor four-roll mill to generate a twodimensional straining flow.
FLOCS IN SHEAR AND STRAIN FLOWS
Corp., Model AOM-40). This acousto-optic device deflected the light beam toward the flow cell at the appropriate time intervals. An aperture was used to block the undiffracted laser beam. The flocs were observed by means of a stereoscopic microscope (Nikon SMZ-2T) having a zooming ratio of 1 : 6.3 and a total magnification of up to 200. The free working distance (distance between the front lens of the objective and the object in focus) was 100 mm, allowing focus on any desired point between the top and bottom plate of the flow cells. The microscope was equipped with a relay lens (1×/16) and a C-mount adapter for mounting a CCD-camera. The motion of the flocs was recorded by a 30 Hz CCD camera (IMAC XC77, EIA norm) having a spatial resolution of 480 × 640 pixels. The output of the camera was first recorded by a U-Matic video-recorder on an analog tape, which had the advantage that time-sequences of up to 60 min could be stored. After completion of the experiments, the video tape was replayed and selected sequences were digitized by means of a frame grabber (Sound & Motion J3000 Turbo Channel Interface), which applied a dynamical JPEG compression to store the data temporarily on an extended RAM of a DECα 3000 AXP workstation. Afterward, the sequence was written on hard disk. Since the CCD camera was run in the interlaced full frame mode, phase and width of the light pulse had to be chosen such that the illumination time fell within the overlapping part of the integration time of the two fields. This synchronization was achieved by means of an electronic device (22). Furthermore, a consecutive number in a binary code was inserted on each image, which allowed checking of the sequential correctness of the pictures during processing.
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The suspension was stirred by means of a magnetic stirrer and a Teflon coated cylindrical stirring bar (31.5 mm in length and 6 mm in diameter). A stock solution of ferric chloride with [Fe] = 1.79 × 10−2 M was prepared. A 6-mL amount of this solution (corresponding to 30 mg Fe/L) was added to the suspension and mixed for 2 min at a stirring rate of 1,000 rpm (“rapid mixing”). During this phase, the pH value monitored by a pH meter (WTW pH 538) slightly decreased from the value 7.0 to 6.7. Afterward, the suspension was stirred for 5 min at 100 rpm resulting in the formation of visible brown-orange flocs. After switching off the magnetic stirrer, the flocs settled and after about 10 min, they covered the entire bottom of the beaker. In order to stabilize the fluid motion (in particular in the four-roll mill), it was necessary to increase the viscosity of the fluid. For this purpose a 5 wt% sucrose solution was prepared by dissolving 25 g of sucrose in 475 mL of mineral water. Single-sweep flocs were transferred to a beaker containing 200 mL of the sucrose solution. At first these floated on the surface, but they settled to the bottom of the beaker after a few minutes. In order to prevent a break-up of the flocs before they were subjected to the flow, a careful handling was required, in particular, when they were transferred from the beaker to the flow apparatus. For this purpose a dropper was used which consisted of a cylindrical glass tube (with an inner diameter of 4 mm and a length of 150 mm) on which a suction bulb was fixed. Since the top of the dropper did not taper as it is for instance the case with Pasteur pipettes, the mechanical load on the flocs could be reduced to that extent that they were not destroyed during the process of suction and blowing-out of fluid. 2.4. Image Analysis
2.3. Floc Preparation The following procedure was adopted at room temperature (22◦ C) to prepare sweep flocs consisting of precipitated hydrolysis products of ferric chloride and polydisperse silica particles (4 µm in diameter [median] and with a density of 2.5 × 103 kg/m3 ): 20 mg SiO2 (Sikron B600, Sihelco AG, Birsfelden, CH) was added to a 250-mL beaker containing 200 mL of mineral water with an ionic strength of 10−2 M.
In order to quantify the optical observations, parameters describing the orientation and shape of a floc were extracted from the digitized images. For this purpose a Matlab routine was written to calculate the desired quantities. The orientation α was defined as the angle between a fixed axis and the axis of the least moment of inertia. Moreover, seven length scales were used to characterize the shape (see Fig. 3): the equal-area radius Rp , the semi-axes of the best-fit ellipse
FIG. 3. (a) Digitized image of a floc. (b) Area of the circle with radius Rp equals the projected area of the floc. The orientation α is defined as the angle between the x axis and the axis of the least moment of inertia. The ellipse whose second moment equals that of the particle determines the semi-axes a and b. (c) The length scales la and lb define the smallest bounding rectangle enclosing the particle that is aligned with its orientation. ra and rb are the maximum chords along the direction of a and b, respectively.
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TABLE 2 Formulae for Calculating the Geometric Parameters Description
Formula
Projected area
Ap =
Equal-area radius
Rp =
X
µ2,0 − µ0,2
( p, q) order central moment
p
Ap /π Ã ! 1 X X (x¯ , y¯ ) = x, y Ap P X P µ p,q = (x − x¯ ) p (y − y¯ )q P
Minimum moment of inertia
¶ µ 1 2µ1,1 α = arctan 2 µ2,0 − µ0,2 X α Imin = [(x − x¯ ) cos α − (y − y¯ ) sin α]2
Maximum moment of inertia
α Imax =
Orientation
Semimajor axis Semiminor axis
µ1,1
α
1
P
Coordinates of center of mass
TABLE 3 Ellipse Tilt Angle α for Various Cases of the Signs of the Second Moments (Taken from (23))
P X
[(x − x¯ ) sin α + (y − y¯ ) cos α]2
P
¡ α ¢3/8 ¡ α ¢−1/8 Imin a = (4/π)1/4 Imax ¡ ¢ ¡ ¢−1/8 1/4 α 3/8 α Imin Imax b = (4/π)
Zero
Zero
0
Zero
Positive
+45◦
Zero Positive Negative Positive Positive Negative Negative
Negative Zero Zero Positive Negative Positive Negative
−45◦ 0 −90◦ (1/2) arctan ξ (1/2) arctan ξ (1/2) arctan ξ + 90◦ (1/2) arctan ξ − 90◦
ξ≡
2µ1,1 µ2,0 − µ0,2
(0 < α < 45◦ ) (−45◦ < α < 0) (45◦ < α < 90◦ ) (−90◦ < α < −45◦ )
Table 3 the quantity α is always the angle between the x axis and a. 3. RESULTS AND DISCUSSION
Note: The following shorthand notation for summations over all pixels beP Nx P N y P longing to a particle has been adopted: P (·) := x=1 y=1 I (x, y)(·), where N x × N y is the size of the image. The charactristic function I (x, y) describes the binary image and is defined as I (x, y) = 1 if the pixel belongs to the particle and I (x, y) = 0 for pixels in the background.
a and b, the length and width of smallest bounding rectangle la and lb , and the maximum chords ra and rb . These length scales however, give only a rough estimation of the shape as it is highly irregular and can vary substantially with each floc even when they are composed of the same material. Note that as the particle is moving it is not illuminated in a constant way, which leads to an artificial fluctuation in the length scales. In order to overcome this deficiency all length scales have been non-dimensionalized with Rp . This procedure is justified if the motion of the floc is two-dimensional such that the projected area is constant in time. In order to determine these geometric parameters a binary image is created by setting a threshold on the gray level. Table 2 summarizes all formulae needed for calculation. The tilt angle α defined in Table 2 may be with respect to either the semimajor axis a or the semiminor axis b. To resolve this ambiguity 90◦ has to be added or subtracted depending on the sign of the second moments µ2,0 − µ0,2 and µ1,1 . With the convention given in
3.1. Flocs in the Simple Shear Flow The shear rate G of the simple shear flow whose velocity is given by u = Gy was varied between 40 and 70 s−1 . A sample of 19 flocs with a total of 228 frames was analyzed. Figure 4 shows one sequence of a rotating and deforming floc. Clearly, the rotation is not uniform: the floc tends to be aligned with the streamlines, whereas the time when it is perpendicular to them is much shorter. Moreover, we see that the transverse bending is rather pronounced, reminiscent of a hinge. Apparently, the floc was composed of two smaller ones which were connected during the growing process. Only 4 out of the 19 flocs displayed a substantial deformation as in Fig. 4, while the other flocs performed a solid-body rotation. The flocs showed no significant deformation in the longitudinal direction indicating that they were more easily bent than stretched by the fluid motion. The flocs were rather asymmetrical in shape as we deduce from the histogram for the ratio of the two semi-axes depicted in Fig. 5a: on average, a was about twice as long as b. Also the histograms for lb /la and rb /ra reveal this asymmetry. The histogram for the angle α given in Fig. 5b confirms the observation that the rotation of the flocs was not uniform in time:
FIG. 4. Rotating and deforming ferric hydroxide floc in a simple shear flow (first row). The second row shows the calculated best-fit ellipses tilted with α. Time is increasing from left to right in steps of 1/30 s.
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FLOCS IN SHEAR AND STRAIN FLOWS
This procedure leads to Fig. 6a. If the flocs displayed no deformation and Rp was constant then a ∗ (i, j) = 1, i.e., all values would lie on the unit circle. In order to quantify the deviation from this ideal case we fit an ellipse on the measured values: Ã
x(i, j) y(i, j)
!
! ¶ Ã cos αe −sin αe x¯ e + = y¯ e sin αe cos αe ! Ã ! Ã εx (i, j) ae cos α(i, j) + . × be sin α(i, j) ε y (i, j) µ
[5]
In Eq. [5] ae and be are the semi-axes of the fitted ellipse, (x¯ e , y¯ e ) denote the coordinates of its center of mass, and αe is the angle between the x axis and ae . Furthermore, (εx (i, j), ε y (i, j)) are the residuals which include the deviation from the ellipse. Given the data points (x(i, j), y(i, j)) we can use the same formula as for the flocs to determine (x¯ e , y¯ e ) and αe . However, the semi-axes ae and be have to be calculated in a different way. A linear regression model can be found by multiplying equation Eq. [5] either by cos αe or sin αe and subtracting the resulting
FIG. 5. (a) Histogram for the ratio hbi/hai, which is the value of b/a averaged over a sequence displaying the same floc. (b) Histogram for the angle α on the basis of the analysis of 19 flocs with a total of 228 frames.
it is more likely that α was monitored around 0◦ or 180◦ than around 90◦ . 3.1.1. Deformation. In the following we want to investigate the deformation of the flocs in a more quantitative manner. Let a(i, j) be the value of a associated to floc i and frame number j. In our case i ∈ {1, 2, . . . , 19} and j ∈ {1, 2, . . . , Ni }, where Ni is the number of frames showing the same floc i. We now plot the normalized values a ∗ (i, j): " a ∗ (i, j) =
Ni 1 X a(i, j) Ni j=1 Rp (i, j)
#−1
a(i, j) , Rp (i, j)
[2]
in the x y plane such that the coordinates reads x(i, j) = a ∗ (i, j) cos α(i, j), ∗
y(i, j) = a (i, j) sin α(i, j),
[3] [4]
where α(i, j) represents the orientation measured for floc i and frame number j.
FIG. 6. The normalized length scales a ∗ and b∗ plotted in the x y plane (a and b, respectively). The measured values are represented by little circles. The solid lines are the fitted ellipse with parameters given in Table 4, whereas the dashed lines indicate the direction of ae and be .
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TABLE 4 Parameters of the Fitted Ellipse for the Six Length Scales
a∗ la∗ ra∗ b∗ lb∗ rb∗
Finally, knowledge of the values for T and t0 enables us to calculate the ratio of the two semi-axes. However, since the tangent jumps at odd multiples of π/2 it is better to cast Eq. [6] into the following form:
x¯ e
y¯ e
αe
ae
be
0.05 0.05 0.04 −0.04 −0.04 −0.04
0.04 0.04 0.03 0.04 0.04 0.04
6.4◦ 6.6◦ 5.4◦ 1.3◦ 2.6◦ 2.2◦
1.02 ± 0.02 1.03 ± 0.02 1.02 ± 0.02 1.04 ± 0.01 1.04 ± 0.02 1.04 ± 0.02
0.93 ± 0.02 0.92 ± 0.02 0.96 ± 0.03 0.98 ± 0.01 0.98 ± 0.02 0.98 ± 0.01
Note: The straight-line model also provides the standard deviation in the estimates of ae and be .
equations. The parameters ae and be are then determined by the method of least squares. If we proceed with the other five length scales b, la , lb , ra , and rb in the same way we find the values for the fitted parameters given in Table 4. All length scales show a similar behavior, but the scattering around the fitted ellipse is for la∗ and lb∗ and in particular for ra∗ and rb∗ much more pronounced than for a ∗ and b∗ . From Table 4 we infer that on average the elongation of the flocs culminates at an angle of α ≈ 6◦ , whereas their compression is maximum at about α ≈ 2◦ . Remember that the eigenvectors of the rate-of-strain tensor are inclined ±45◦ to the x axis. It seems that due to the rotation there is a certain retardation until the floc experiences the stretching and compression of the fluid. However, the deformation of the analyzed flocs is rather small as the ratio be /ae ≈ 0.9 differs only slightly from unity.
¶ µ ¶ a 2π (t − t0 ) 2π (t − t0 ) sin α = − sin cos α. cos T b T µ
[8]
Again we make use of the linear regression to estimate the value for a/b. An example comparing the measured and fitted data is shown in Fig. 7b. One might ask whether the fitted values correlate with the values extracted from the digitized images. In the following, fitted quantities will be indicated by the subscript F, whereas data provided by the image analysis are supplied with the subscript I.
3.1.2. Orientation. We now want to investigate whether a suitable model is able to relate the time-evolution of α with the parameters a, b, and G. It is well known (24) that a rigid ellipsoid with semi-axes a, b, and c can rotate in a planar orbit according to µ ¶¸ · 2π(t − t0 ) a , α(t) = −arctan tan b T
[6]
where T = 2π(a 2 + b2 )/abG is the period of the motion. In order to fit the measured data points α(i, j) of each sequence with Eq. [6], note that the phase velocity α(t) ˙ averaged in time is related with T through the following equation: 1 T
Z 0
T
α(t) ˙ dt =
1 2π . [α(T ) − α(0)] = T T
[7]
On the other hand, if α(i, j) is plotted against the time t, then the slope of the straight line given by linear regression is nothing but the time-averaged phase velocity, and by means of Eq. [7], T can be immediately determined (see Fig. 7a). Owing to the periodicity of the motion, the time-shift t0 , which arises in Eq. [6], can be determined by means of the relation: α(t0 − kT /4) = kπ/2, k ∈ Z. As is shown in Fig. 7a, the discrete data points are interpolated using a cubic spline fit.
FIG. 7. Fit of the measured time-evolution of the orientation. (a) Values for α measured at different times are indicated by dots. The intermediate points represented by the solid line are given by a cubic spline interpolation. The dashed line results from a least-squares straight line fit to the data. (b) Comparison of the measured values (dots) with the fitted curve given by Eq. [6].
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FLOCS IN SHEAR AND STRAIN FLOWS
3.2. Flocs in the Four-Roll Mill
FIG. 8. Comparison of the fitted and measured data bF /aF and bI /aI . The solid line is the regression line if bF /aF is fitted on bI /aI , whereas the dashed line represents the regression line if the two quantities are interchanged. The vertical error bars represent the standard deviation of bI /aI . The horizontal error bars are supplied by the linear regression model. Observe that the large horizontal error bar at bI /aI = 0.5 originates from a sample of only four points.
Figure 8 shows a comparison of bF /aF with bI /aI , where the latter value was calculated by averaging the measured ratio of the semi-axes over a sequence of the same floc. The correlation coefficient is r = 0.75, indicating that the two quantities are correlated. For further details consult Table 5. Although there are few outliers arising in Fig. 8, it is most striking that such a simple model given by Eq. [6] may explain quite well the observed time-evolution of the orientation. This is in particular remarkable as this equation is valid for a solid impermeable ellipsoid, whereas the flocs have actually to be considered as porous bodies which can be deformed. Since the applied shear rate G is known, we can also calculate the period of motion by means of TI = 2π (aI2 + bI2 )/aI bI G, where we have assumed that the ratio of the semi-axes can be estimated by bI /aI . We infer from Fig. 9, which compares the values of TI with TF , that the correlation between the two quantities is satisfactory. This statement is also confirmed by the values compiled in Table 5.
3.2.1. Flow field. To begin, we consider the velocity field produced by the Taylor four-roll mill. For low Reynolds number the trajectories of single particles were found to be hyperbolic. However, if the angular velocity Ä of the rollers is greater than 30 rad/s, effects originating mainly from the presence of the top and bottom plates become dominant, which leads to a vertical component of velocity also at the horizontal mid-plane. Thus the critical Reynolds number is given by Re = 88, where Re = ÄRS/ν with R = 1.56 mm being the roller radius and S = 1.88 mm the gap width between the rollers. Lagnado and Leal (19) found a smaller critical Reynolds number Re = 37. However, the dimensions of their four-roll mill were different. In particular, the ratio R/S was 1.7, whereas in our case it was 0.83. In order to estimate the strain rate E for a given Ä we suppose that the velocity field can be described by u = E x, v = −E y and that it coincides at the p point x = y = S/2 + R(1 − 2−1/2 ) with the speed of the roller: E x 2 + y 2 = ÄR. Solving this equation for E leads to E=
√ 2Ä S/R + 2 −
√ = 0.79 · Ä. 2
[9]
Taylor (16) proposed an experimental method to determine E. The trajectory of a small tracer particle which moves along the x axis is given by xp (t) = xp0 exp Et. If the time t which the particle needs to cover the distance |xp (t) − xp0 | is measured,
TABLE 5 Linear Regression for (bF /aF , bI /aI ) and (TF , TI ) Independent variable
Slope
Fig. 8
bF /aF bI /aI
0.66 ± 0.30 0.86 ± 0.40
Fig. 9
TF TI
0.77 ± 0.29 (0.08 ± 0.07) s 0.85 ± 0.30 (0.01 ± 0.09) s
y intercept 0.16 ± 0.15 0.07 ± 0.19
Correlation Confidence coefficient interval 0.75
[0.45, 0.90]
0.81
[0.57, 0.92]
Note: The third and fourth column give the slope and the y intercept of the regression line. Note also that the correlation coefficient r in the fifth column is nothing but the geometric mean of the two values for the slope. The sixth column is the confidence interval for r determined according to the method described in (25).
FIG. 9. Comparison of TF with TI . Fitting of TF on TI results in the solid line; the dashed line represents the regression line if the two quantities are interchanged. The vertical bars are calculated by applying Gauss’ law of the propagation of errors assuming that the standard deviation of G is 5 s−1 .
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STEFAN BLASER
FIG. 10. Break-up of ferric hydroxide flocs in a two-dimensional extensional flow. Time is increasing from left to right in steps of 1/30 s. The first row shows the process of erosion, whereas the final two rows display the splitting of a floc.
then the strain rate can be calculated by means of E = t −1 ln
xp . xp0
[10]
Explicitly, for Ä = 25.1 rad/s a particle was observed with xp0 = 0.13 mm and xp (t = 3/30 s) = 0.99 mm, resulting in E = 20.3 s−1 . On the other hand, we find from Eq. [9] the value E = 19.9 s−1 , which is in satisfactory agreement. 3.2.2. Break-up of flocs. As in the case of the simple shear flow, the experiments were performed with ferric hydroxide flocs in a 5 wt% sucrose solution. The angular speed was kept constant at Ä = 25.1 rad/s corresponding to a strain rate of E = 20 s−1 . Figure 10 shows three examples of flocs which were broken apart by the fluid motion when they approached the stagnation point. Both breaking modes, erosion of a small particle from the floc surface and splitting of the floc into equal-sized fragments, could be monitored. However, not every floc, even some very large ones, were subject to fragmentation. Most of the flocs were either stretched along the axis of incoming fluid or aligned with this axis without any deformation. This is mainly due to the fact that the flocs have a limited residence time near the stagnation point, and thus the fluid has only a short amount of time to act on the surface of the particles. In order to gain quantitative information on orientation and deformation the Matlab program was applied to a sample of 22 flocs which were broken apart. As one can see from Fig. 11 the ratio b/a is shifted toward smaller values with increasing time, which indicates that the flocs tend to be stretched. Almost all flocs display a decrease in the ratio of the semi-axes, which is quite substantial in a few cases. There are some flocs which get first more spherical-like with time before they are stretched: this is related to the observation that as they approach the stagnation point the fluid motion tries to accelerate the rear of the flocs toward the front part, such that they are compressed.
To investigate the time-evolution of the orientation we plot a line for each floc in the x y plane such that it makes an angle of α with the x axis and that its length corresponds to 2a/Rp . Applying this procedure to the flocs when they became visible the first time and for t = 0 leads to Figs. 12a and b. It can be clearly seen that the fluid tries to align and stretch the flocs along its axis of stretching. Yet, when a smaller particle is eroded from the floc surface the angle α can be considerably larger than 0◦ . See for instance the top row in Fig. 10 for which case α lies between −82◦ and −23◦ . All analyzed flocs were broken into two smaller parts. Figure 13a shows the values for Rp,1 /Rp,2 , where the subscripts 1 and 2 refer to the two fragments. Note that the data are sorted such that Rp,1 ≤ Rp,2 . We see that large-scale fragmentation (Rp,1 ≈ Rp,2 ) outweighs surface erosion (Rp,1 ¿ Rp,2 ). However, one has to bear in mind that the statistics may be biased by the
FIG. 11. Time-evolution of ba0 /ab0 for each of the 22 flocs. b0 /a0 is the value of b/a at t = 0, which indicates the time just before the flocs is broken apart. Note that the observation time varies with each floc.
FLOCS IN SHEAR AND STRAIN FLOWS
281
on the surface, and M a dimensionless 3 × 3 matrix, which is a function of α and the ratios a/c and b/c. (The reader is referred to (26) or (27, Sect. 3.3) for the detailed values of M, which can be expressed in terms of elliptic integrals.) As an illustrative example we consider an ellipsoid with semiaxes 2.5 : 1.5 : 1 which has its a-axis aligned along the axis of fluid stretching. To facilitate the illustrations, f is decomposed into the normal and tangential components, which are given by f n = f·n and f t = |f− f n n|, respectively. Figure 14 shows that the normal component is extreme along the axes of fluid compression and stretching (y and x axes), whereas the tangential component is large near the stretching axis (x axis) but vanishes at the piercing points of the ellipsoid’s surface with the x, y, and z axes. In order to characterize the hydrodynamic load by a single quantity (denoted in the following by f h ), the force distribution f is integrated over half of the ellipsoid’s surface which is parametrized by x = (a cos φ sin θ , b sin φ sin θ , c cos θ ) with φ ∈ [−π/2, π/2] and θ ∈ [0, π ] (see also inset in Fig. 16).
FIG. 12. Orientation and length of the analyzed flocs (a) at the time when they entered the straining flow and (b) at the beginning of breakage (t = 0). The length of each line whose ends are marked with black dots is given by 2a/Rp . Note that the origin of the coordinate system coincides with the center of mass of each floc.
fact that the latter breaking mode was harder to detect. Finally, we deduce from Fig. 13b that the fragments displayed no tendency of being spherical as the mean of b/a is at about 0.6. 3.2.3. Estimation of rupture force. We now want to estimate the rupture forces which were needed to split the flocs subjected to the straining flow. For this purpose the following ad hoc model is adopted: each floc which was seen to be broken apart is represented by a solid ellipsoid with semi-axes a, b, and c. The values for a and b are obtained from the particle image analysis at the onset of breakage, whereas c, which is unknown, is assumed to lie in the range b ≤ c ≤ a. In addition, the orientation of the ellipsoid is equated by the measured value for α. In an early paper, Jeffery (26) gave a general formula for the hydrodynamic force f per unit area on the surface of a small ellipsoid which is subjected to a linear flow field. For the present case of strain flow, f is of the form f = µEMn,
[11]
where µ is the viscosity of the fluid, n the unit outward normal
FIG. 13. Histograms (a) for the ratio of the equal-size radius of the fragments, Rp,1 /Rp,2 , and (b) for the ratio of the two semi-axes b/a. The data are based on a sample of 44 fragments.
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FIG. 16. Rupture force f hr of individual flocs with size Rp in twodimensional straining flow. The error bars are given by the uncertainty of c. Their length is correlated with the asymmetry in the shape of the flocs.
FIG. 14. Force per unit area exerted by the straining flow u = (E x, −E y, 0) on the surface of a solid ellipsoid with semi-axes 2.5 : 1.5 : 1. (a) Normal and (b) tangential components of f. The scale gives the non-dimensional values of f n /(µE) and f t /(µE), respectively.
Considering, for example, a sphere, a = b = c, one finds that f h = qµc2 E with q = 5π. If the semi-axes are different a 6= b 6= c, then q is a function of a/c, b/c, and α (see Fig. 15). Note that integration of f over the whole surface equals zero, as
FIG. 15. Contour lines for the integral force f h on half of the ellipsoid’s surface. The orientation is chosen to be α = 0, i.e., the a axis is aligned with the stretching axis of the fluid motion. The contour levels give the values of f h /(µc2 E).
we assume in our model that the ellipsoid moves “freely” with the fluid motion, i.e., there is no resultant force acting on the particle. If we assume that the floc is split into two fragments, then a simple criterion for breakage is given by the condition f h ≥ f b , where f b is the integrated internal binding forces which occur along the dividing plane. Based on the measured values for a, b, and α, the hydrodynamic load f hr can be calculated for each floc which was observed to be broken apart by the straining flow. (In order to emphasize that the flocs were ruptured, the superscript “r” is added.) Figure 16 shows f hr as a function of the measured equal-area radius Rp . By dividing f hr by the half area of the ellipsoids and taking the average over the whole sample, we find ¿
À f hr = (5.8 ± 0.5) · 10−2 Nm−2 , 2S
[12]
where S is the total surface area of the ellipsoid. In the present experiment the strain rate was increased until flocs were seen to be broken apart. The value given in Eq. [12], which is on the order of µE, gives therefore a rough estimate for the hydrodynamic forces which are needed to induce breakage of iron flocs of size Rp < 0.4 mm. Although Fig. 16 shows a certain tendency ( f hr is greater for larger flocs), it does not confirm that f hr correlates with Rp . This is supported by the fact that flocs of comparable size were observed which were unaffected by the straining motion. Thus the inner structure must be considered as inhomogeneous and the mechanical strength is variable. The flocs used in the experiments were produced by adding Fe(III) salt in a concentration well above its solubility in water. Hence, as the primary silica particles were enmeshed by the precipitating ferric hydroxide, they were unlikely to be in contact and do not contribute to the internal strength. Furthermore, in contrast to the aggregation by double-layer compression, the
283
FLOCS IN SHEAR AND STRAIN FLOWS
ionic strength will have little effect on the binding forces of this kind of sweep floc. However, the hydrolysis scheme of Fe(III) salt and the morphology of the precipitates are complex (28, 29) and not fully understood. Judging by the optical observation (see Fig. 10), the flocs tended to be of a compact structure, such that several bonds were broken apart before the fragments were separated. Other investigations on the size–density relationship of iron flocs suggest that the fractal dimension Df is in the range 1.9–2.78 (30, 31). Because of the internal inhomogeneity the binding force f b cannot be taken simply as a function, for example of the floc size Rp , but instead has to be treated as a random variable; i.e., given a floc of a definite size, there is a certain probability that the floc breaks or not. In this context the investigations performed by Yeung and Pelton (32) are noteworthy. The authors determined directly the rupture strength of single flocs by means of two suction pipettes. While one pipette was kept stationary, the other one deflected elastically in response to axial forces such that from the known stiffness and its deflection the tensile strength on the flocs could be determined. The measured values showed no correlation with the floc sizes when plotted on logarithmic scales.2 Very little data on rupture forces is currently available. Quigley (8) also investigated ferric hydroxide flocs in a straining flow. He found that rupture occurred for µE ≥ 0.1 N/m2 , which is of the same order as Eq. [12]. Yeung and Pelton (32) investigated flocs which consisted of calcium carbonate and polymers. They displayed a much higher rupture force: 30–2,000 N/m2 . However, the conglomerates were 6–40 µm in size and therefore much smaller than the sweep flocs investigated in the present experiment. 3.2.4. Effects of porosity. It is clear that a floc is not impermeable for a fluid and one could therefore ask to what extent porosity effects the estimated values above. For this purpose we consider the Stokes drag FD of a porous sphere with radius a which is subjected to the constant flow u0 . A widely used approach of tackling this problem is solving the creeping flow equations for the flow exterior to the aggregate and applying either Darcy’s or Brinkman’s equation for the interior flow coupled with continuity equations. (See (33) for a more detailed description.) Assuming that inertia effects can be neglected and that the porosity of the aggregate is homogeneous, the drag can be written as FD = 6πµau0 MD , where the factor MD depends on the permeability k of the sphere. Furthermore, if we assume that the porous medium is composed of ˆ the Carman-Kozeny formula gives small spheres with radius a, a relationship between k and the void fraction ε of the aggregate (see for instance (34)). Figure 17 shows the Stokes multiplier MD as a function of ε and the ratio a/aˆ for different model equations. It can be seen that for low and intermediate void fractions the difference of the Stokes drag between a permeable and impermeable sphere is
2
D −1
In particular, the scaling law f b ∝ Rp f
is not applicable to their data.
FIG. 17. Drag multiplier MD as a function of the void fraction ε and the ratio a/aˆ for several models: Darcy equation with Saffman boundary condition (35) (solid line); Darcy equation with Jones boundary condition (36) (dashed line); and Brinkman equation (37) dotted line).
minute. Only at a very high porosity, ε > 0.95, does MD differ substantially from one. Aggregates which consist of monodisperse spheres will in general show low void fractions: for closed packing, e.g., we have ε = 0.26, while for random packings 0.34 ≤ ε ≤ 0.45 (38). Ferric hydroxide flocs, however, can display a much larger porosity. In general, the water content of flocs increases with increasing size (39). For a floc size of 100 µm the void fraction is about ε = 0.95, whereas it increases to ε = 0.999 at a size of about 800 µm (39). Judging from this analysis, we conclude that the influence of porosity is just at the edge of becoming important to get the correct load on the surface as the size of flocs investigated here lie in the range 200–800 µm (see Fig. 16). 4. CONCLUSIONS
Flocs which are immersed in a simple shear flow perform a periodic but not uniform motion. Comparison with the planar motion of a solid ellipsoid reveals that for given a, b, and shear rate, the time-evolution of the measured orientation can be predicted to satisfactory agreement. Also, the measured and theoretically calculated periods of motion were correlated. It is remarkable that the shape of the flocs projected on the x y plane determines the dynamical behavior: their motion is independent of their shape in the x z and yz planes. The bending of the flocs is more pronounced than the stretching. The semi-axis a peaks when it is almost aligned with the streamlines. However, because of the presence of vorticity, a floc is kept in rotation and thus the load on its surface fluctuates with time. Thus it can constantly escape a stretching along the directions given by the eigenvectors of the rate-of-strain tensor. In this context it is noteworthy to mention that highly viscous drops with viscosity µd do not necessarily disintegrate in a shear flow, even at high shear rates. Taylor (16) showed that if the viscosity ratio µd /µ is larger than 4 and if the drop begins with
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STEFAN BLASER
a spherical shape, it will not break up in a simple shear flow, independent of the magnitude of the shear rate: increasing the shear rate has only the effect that the drop fluid circulates faster, but the state of its deformation remains unchanged. In contrast to the simple shear flow, flocs in a two-dimensional straining flow do not rotate continuously and are stretched and aligned along the axis of outgoing fluid. If the strain rate is large enough they may be broken apart. However, flocs of comparable size have different tendencies toward disintegration. This is in part due to the fact that their residence time near the stagnation point varies and hence the fluid has not always “enough” time to break up all bonds within the flocs. In addition, the inner structure, and with it the mechanical stability, is different for each floc. The ellipsoid model allows the estimation of the force distribution on the surface of a particle. For the hydroxide flocs a lower bound for the rupture forces was found to be on the order of 0.1 N/m2 . Furthermore, only if the void fraction of the flocs is particularly high will porosity have dominant effects on the load exerted by the fluid on the particle surface. ACKNOWLEDGMENTS This study was supported by a grant of the Swiss National Science Foundation (Grant No 21-40514.94 and 20-46680.96) and was performed in collaboration with Dr. A. Gyr, Dr. A. M¨uller, Prof. Dr. W. Gujer, and Prof. Dr. M. Boller. The author is grateful to Emily Ogenyi for improving the English.
REFERENCES 1. Gujer, W., and Boller, M., Prog. Wat. Technol. 10, 741 (1978). 2. Boller, M., Wat. Sci. Technol. 27, 167 (1993). 3. Deng, M., and Dodson, C. T. J., “Paper: An engineered stochastic structure.” TAPPI Press, Atlanta, 1994. 4. Sonntag, R. C., and Russel, W. B., J. Colloid Interface Sci. 113, 399 (1986). 5. Williams, R. A., Peng, S. J., and Naylor, A., Powder Tech. 73, 75 (1992).
6. Peng, S. J., and Williams, R. A., J. Colloid Interface Sci. 166, 321 (1994). 7. Kao, S. V., and Mason. S. G., Nature 253, 619 (1975). 8. Quigley, J. E., “Strength properties of liquid-borne flocculated matter.” M.S. thesis, University of Delaware, Newark, 1977. 9. Smith, P. G., and Van de Ven, T. G. M., Colloids Surf. 15, 191 (1985). 10. Glasgow, L. A., and Liu, X., AIChE J. 37, 1411 (1991). 11. Higashitani, K., Inada, N., and Ochi, T., Colloids Surfaces A 56, 13 (1991). 12. Bache, D. H., Rasool, E., Ali, A., and McGilligan, J. F., J. Water SRT–Aqua 44, 83 (1995). 13. Liu, S. X., and Glasgow, L. A., Water, Air, Soil Pollut. 95, 257 (1997). 14. Daviaud, F., Hegseth, J., and Berg´e, P., Phys. Rev. Lett. 69, 2511 (1992). 15. Malerud, S., M˚aløy, K. J., and Goldburg, W. I., Phys. Fluids 7, 1949 (1995). 16. Taylor, G. I., Proc. R. Soc. A 146, 501 (1934). 17. Bentley, B. J., and Leal, L. G., J. Fluid Mech. 167, 241 (1986). 18. Stone, H. A., and Leal, L. G., J. Fluid Mech. 206, 223 (1989). 19. Lagnado, R. R., and Leal, L. G., Exp. Fluids 9, 25 (1990). 20. Szeri, A. J., Milliken, W. J., and Leal, L. G., J. Fluid Mech. 237, 33 (1992). 21. Higdon, J. J. L., Phys. Fluids A 5, 274 (1993). 22. Virant, M., “Anwendung des dreidimensionalen ‘Particle-TrackingVelocimetry’ auf die Untersuchung von Dispersionsvorg¨angen in Kanalstr¨omungen.” Ph.D. Thesis No. 11678, ETH-Z¨urich, 1996. [in German] 23. Teague, M. R., J. Opt. Soc. Am. 70, 920 (1980). 24. Van de Ven, T. G. M., “Colloidal Hydrodynamics.” Academic Press, London, 1989. 25. Kreyszig, E., “Statistische Methoden und ihre Anwendungen.” Vandenhoeck & Ruprecht, G¨ottingen, 1975. [in German] 26. Jeffery, G. B., Proc. R. Soc. A 102, 161 (1922). 27. Kim, S., and Karrila, S. J., “Microhydrodynamics: Principles and selected applications.” Butterwoth-Heinemann, Boston, 1991. 28. Flynn, C. M., Chem. Rev. 84, 31 (1984). 29. Stumm, W., and Morgan, J. J., “Aquatic chemistry.” Wiley, New York, 1996. 30. Lagvankar, A. L., and Gemmel, R. S., J. AWWA 60, 1040 (1968). 31. Tambo, N., and Watanabe, Y., Wat. Res. 13, 409 (1979). 32. Yeung, A. K. C., and Pelton, R., J. Colloid Interface Sci. 184, 579 (1996). 33. Blaser, S., “The hydrodynamical effect of vorticity and strain on the mechanical stability of flocs.” Ph.D. Thesis No. 12851, ETH-Z¨urich, 1998. 34. Dagan, G., “Flow and transport in porous formations.” Springer-Verlag, Berlin, 1989. 35. Saffman, P. G., Stud. Appl. Math. 50, 93 (1971). 36. Jones, I. P., Proc. Camb. Phil. Soc. 73, 231 (1973). 37. Brinkman, H. C., Appl. Sci. Res. A 1, 27 (1947). 38. Coelho, D., Thovert, J.-F., and Adler, P. M., Phys. Rev. E 55, 1959 (1997). 39. Boller, M., and Blaser, S., Wat. Sci. Technol. 37, 9 (1998).
Flocs in Shear and Strain Flows Stefan Blaser Institute of Hydromechanics and Water Resources Management, Swiss Federal Institute of Technology, CH-8093 Z¨urich, Switzerland E-mail: [email protected] Received March 18, 1999; accepted December 13, 1999
However, to date the dynamics of aggregated particles and their interaction with the fluid motion are not fully understood. In the present work therefore we study the behavior of single flocs by using optical methods. Preflocculated ferric hydroxide flocs were subjected to well-defined flow conditions such as either simple shear or two-dimensional straining flow and their motion was recorded with a CCD camera. Appropriate types of apparatus were designed to produce the flow fields on a microscopic scale. By means of digital image analysis, quantitative information on deformation and orientation of the flocs can be extracted from the images. A simple model is established in which a floc is represented by a solid ellipsoid. This will allow us to compare the measured time-evolution of orientation with an analytical solution. In addition, the forces needed to break the flocs apart can be estimated.
Preflocculated ferric hydroxide flocs were subjected to either a simple shear flow or a two-dimensional straining flow, and their motion was optically observed. Digital image analysis was applied to extract information on orientation and deformation from the digitized frames. It was found that the simple shear flow led to a rotation of the flocs whose motion can be understood from the behavior of a solid ellipsoid. In the extensional flow, no continuous rotation occurred and flocs were broken apart along the axis of straining. The rupture forces estimated from an ellipsoid model were found to be in the range of 0.1 N/m2 . °C 2000 Academic Press Key Words: flocs; shear flow; strain flow; digital image analysis; orientation; deformation; break-up; rupture forces.
1. INTRODUCTION
The process of flocculation results in the aggregation of small particles suspended in a fluid to larger clusters (“flocs”) and thus favors the separation of solid and liquid phases. Particle aggregation occurs in many industrial and engineering applications such as water and waste water treatment (1, 2) and the manufacture of paper (3). In order to facilitate aggregation, the fluid is mixed, which usually leads to turbulent flow conditions. However, if the hydrodynamic forces are too strong, the forming flocs may be broken apart (4–6). Despite the vast literature on flocculation, investigations based on the direct observation of the disintegration process are rather limited. In Table 1 some relevant references are compiled. Basically, two break-up mechanisms have been found: large-scale splitting (break-up into a few fragments of comparable size) (8, 12, 13) and fine-particle erosion (gradual shearing off of small fragments from the floc surface) (7–10). Kao and Mason (7) were among the first to draw attention to the fact that shear and straining flows are profoundly different in the dispersion of particles. They carried out experiments with non-cohering aggregates in silicone oil. They found that in a simple shear field individual spheres gradually disengaged from the aggregate and continued to circle around it, whereas in the pure straining field an aggregate disintegrated much more rapidly with the spheres dispersed along the axis of straining.
2. MATERIALS AND METHODS
2.1. Couette Cell and Taylor Four-Roll Mill As a preliminary remark, I illustrate the necessity of flow cells with small dimensions by the following consideration. Suppose that the flow field is characterized by the kinematic viscosity ν, the length scale L, and the scale G associated with the velocity gradient. In order to have a well-defined fluid motion, the flow must be kept laminar, which implies that the Reynolds number, given by Re = G L 2 /ν, does not exceed a critical value Rec . On the other hand, a floc will only deform substantially if the forces exerted by the fluid on the particle’s surface are large enough. As these forces are related to the stress tensor, this condition involves a large-scale G. Hence the length scale L has to satisfy the inequality r ν Rec , [1] d
273
1
The strip was produced by W. Schmidheiny, Institute of Polymers, ETHZ. 0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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STEFAN BLASER
TABLE 1 Literature Overview on Floc Break-up Investigated in Different Flow Fields and for Different Flocculated Systems Floc system Ref. no.
Flow
Medium
Particles
Flocculants
7 8 9 10 11 12 13
Shear, strain Strain Shear Flow through grid Contraction Grid-flow Jet
Silicone oil Water + sucrose Silicone oil Water Water + glycerol Water Water
PMMA spheres None Polystyrene Kaolin Latex Starch Kaolin
None Iron(III) Water Polymer Al2 (SO4 )3 + polymer Al2 (SO4 )3 Iron(III) + polymer
four cylinders as is depicted in Fig. 1. Another cylinder could be adjusted to tighten the strip. This geometrical arrangement utilized by other investigators (14, 15) to study the turbulent plane Couette flow has the advantage that the mean flow vanishes as two plane walls move in opposite directions but with the same magnitude of velocity. The test section where the flocs could be observed was 100 mm in length, 1 mm in width, and 10 mm in height. The strip was driven through friction by one of the two larger cylinders (16 mm in diameter), which was mechanically coupled to a step motor. The number of revolutions could be adjusted in the range 0–1,200 rpm resulting in a maximum speed of either wall of 10 cm/s. The strip was transparent, which allowed sideways illumination of the test section by a light sheet. The entire assembly was placed in a rectangular Plexiglas tank, which was filled with water through several feed pipes. The four-roll mill invented by Taylor (16) is commonly used to produce a two-dimensional straining flow (17–20). Higdon (21) compared various design criteria for this device, in particular the ratio of cylinder diameter to cylinder spacing. He concluded that a ratio of 0.625 yields the best fit between the circular surface of a cylinder and the hyperbolic streamlines characteristic of the pure extensional flow.
FIG. 1. Plan and side views of the cell to produce a plane Couette flow.
Figure 2 shows a sketch of the apparatus used in the experiments. Four cylinders (3.12 mm in diameter and 10 mm in height) were mounted at the corners of a square the sides of which were 5 mm. They were enclosed in a rectangular tank, which could be fed with water through two pipes. Four gear wheels driven by a step motor were arranged such that the rollers rotated in the directions indicated by arrows in Fig. 2. 2.2. Data Acquisition The observation area of the Couette and Taylor cell was illuminated by a 5 W argon ion laser (Coherent Innova 305), which had 2.0 W of the total power concentrated in the 514.5 nm line and 1.5 W in the 488 nm line. An optical system consisting of three lenses and two mirrors was used to generate a light sheet having the desired location and width of its beam-waist. Each of the two flow cells could be mounted on a three-axis translating table with micrometers for positioning in either of the three directions. Movement along the vertical axis was used to position the light sheet at the desired height of the apparatus. In order to prevent a blurred imaging of the moving particles, the illumination time was controlled by a Bragg cell (IntraAction
FIG. 2. Plan and side views of the Taylor four-roll mill to generate a twodimensional straining flow.
FLOCS IN SHEAR AND STRAIN FLOWS
Corp., Model AOM-40). This acousto-optic device deflected the light beam toward the flow cell at the appropriate time intervals. An aperture was used to block the undiffracted laser beam. The flocs were observed by means of a stereoscopic microscope (Nikon SMZ-2T) having a zooming ratio of 1 : 6.3 and a total magnification of up to 200. The free working distance (distance between the front lens of the objective and the object in focus) was 100 mm, allowing focus on any desired point between the top and bottom plate of the flow cells. The microscope was equipped with a relay lens (1×/16) and a C-mount adapter for mounting a CCD-camera. The motion of the flocs was recorded by a 30 Hz CCD camera (IMAC XC77, EIA norm) having a spatial resolution of 480 × 640 pixels. The output of the camera was first recorded by a U-Matic video-recorder on an analog tape, which had the advantage that time-sequences of up to 60 min could be stored. After completion of the experiments, the video tape was replayed and selected sequences were digitized by means of a frame grabber (Sound & Motion J3000 Turbo Channel Interface), which applied a dynamical JPEG compression to store the data temporarily on an extended RAM of a DECα 3000 AXP workstation. Afterward, the sequence was written on hard disk. Since the CCD camera was run in the interlaced full frame mode, phase and width of the light pulse had to be chosen such that the illumination time fell within the overlapping part of the integration time of the two fields. This synchronization was achieved by means of an electronic device (22). Furthermore, a consecutive number in a binary code was inserted on each image, which allowed checking of the sequential correctness of the pictures during processing.
275
The suspension was stirred by means of a magnetic stirrer and a Teflon coated cylindrical stirring bar (31.5 mm in length and 6 mm in diameter). A stock solution of ferric chloride with [Fe] = 1.79 × 10−2 M was prepared. A 6-mL amount of this solution (corresponding to 30 mg Fe/L) was added to the suspension and mixed for 2 min at a stirring rate of 1,000 rpm (“rapid mixing”). During this phase, the pH value monitored by a pH meter (WTW pH 538) slightly decreased from the value 7.0 to 6.7. Afterward, the suspension was stirred for 5 min at 100 rpm resulting in the formation of visible brown-orange flocs. After switching off the magnetic stirrer, the flocs settled and after about 10 min, they covered the entire bottom of the beaker. In order to stabilize the fluid motion (in particular in the four-roll mill), it was necessary to increase the viscosity of the fluid. For this purpose a 5 wt% sucrose solution was prepared by dissolving 25 g of sucrose in 475 mL of mineral water. Single-sweep flocs were transferred to a beaker containing 200 mL of the sucrose solution. At first these floated on the surface, but they settled to the bottom of the beaker after a few minutes. In order to prevent a break-up of the flocs before they were subjected to the flow, a careful handling was required, in particular, when they were transferred from the beaker to the flow apparatus. For this purpose a dropper was used which consisted of a cylindrical glass tube (with an inner diameter of 4 mm and a length of 150 mm) on which a suction bulb was fixed. Since the top of the dropper did not taper as it is for instance the case with Pasteur pipettes, the mechanical load on the flocs could be reduced to that extent that they were not destroyed during the process of suction and blowing-out of fluid. 2.4. Image Analysis
2.3. Floc Preparation The following procedure was adopted at room temperature (22◦ C) to prepare sweep flocs consisting of precipitated hydrolysis products of ferric chloride and polydisperse silica particles (4 µm in diameter [median] and with a density of 2.5 × 103 kg/m3 ): 20 mg SiO2 (Sikron B600, Sihelco AG, Birsfelden, CH) was added to a 250-mL beaker containing 200 mL of mineral water with an ionic strength of 10−2 M.
In order to quantify the optical observations, parameters describing the orientation and shape of a floc were extracted from the digitized images. For this purpose a Matlab routine was written to calculate the desired quantities. The orientation α was defined as the angle between a fixed axis and the axis of the least moment of inertia. Moreover, seven length scales were used to characterize the shape (see Fig. 3): the equal-area radius Rp , the semi-axes of the best-fit ellipse
FIG. 3. (a) Digitized image of a floc. (b) Area of the circle with radius Rp equals the projected area of the floc. The orientation α is defined as the angle between the x axis and the axis of the least moment of inertia. The ellipse whose second moment equals that of the particle determines the semi-axes a and b. (c) The length scales la and lb define the smallest bounding rectangle enclosing the particle that is aligned with its orientation. ra and rb are the maximum chords along the direction of a and b, respectively.
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TABLE 2 Formulae for Calculating the Geometric Parameters Description
Formula
Projected area
Ap =
Equal-area radius
Rp =
X
µ2,0 − µ0,2
( p, q) order central moment
p
Ap /π Ã ! 1 X X (x¯ , y¯ ) = x, y Ap P X P µ p,q = (x − x¯ ) p (y − y¯ )q P
Minimum moment of inertia
¶ µ 1 2µ1,1 α = arctan 2 µ2,0 − µ0,2 X α Imin = [(x − x¯ ) cos α − (y − y¯ ) sin α]2
Maximum moment of inertia
α Imax =
Orientation
Semimajor axis Semiminor axis
µ1,1
α
1
P
Coordinates of center of mass
TABLE 3 Ellipse Tilt Angle α for Various Cases of the Signs of the Second Moments (Taken from (23))
P X
[(x − x¯ ) sin α + (y − y¯ ) cos α]2
P
¡ α ¢3/8 ¡ α ¢−1/8 Imin a = (4/π)1/4 Imax ¡ ¢ ¡ ¢−1/8 1/4 α 3/8 α Imin Imax b = (4/π)
Zero
Zero
0
Zero
Positive
+45◦
Zero Positive Negative Positive Positive Negative Negative
Negative Zero Zero Positive Negative Positive Negative
−45◦ 0 −90◦ (1/2) arctan ξ (1/2) arctan ξ (1/2) arctan ξ + 90◦ (1/2) arctan ξ − 90◦
ξ≡
2µ1,1 µ2,0 − µ0,2
(0 < α < 45◦ ) (−45◦ < α < 0) (45◦ < α < 90◦ ) (−90◦ < α < −45◦ )
Table 3 the quantity α is always the angle between the x axis and a. 3. RESULTS AND DISCUSSION
Note: The following shorthand notation for summations over all pixels beP Nx P N y P longing to a particle has been adopted: P (·) := x=1 y=1 I (x, y)(·), where N x × N y is the size of the image. The charactristic function I (x, y) describes the binary image and is defined as I (x, y) = 1 if the pixel belongs to the particle and I (x, y) = 0 for pixels in the background.
a and b, the length and width of smallest bounding rectangle la and lb , and the maximum chords ra and rb . These length scales however, give only a rough estimation of the shape as it is highly irregular and can vary substantially with each floc even when they are composed of the same material. Note that as the particle is moving it is not illuminated in a constant way, which leads to an artificial fluctuation in the length scales. In order to overcome this deficiency all length scales have been non-dimensionalized with Rp . This procedure is justified if the motion of the floc is two-dimensional such that the projected area is constant in time. In order to determine these geometric parameters a binary image is created by setting a threshold on the gray level. Table 2 summarizes all formulae needed for calculation. The tilt angle α defined in Table 2 may be with respect to either the semimajor axis a or the semiminor axis b. To resolve this ambiguity 90◦ has to be added or subtracted depending on the sign of the second moments µ2,0 − µ0,2 and µ1,1 . With the convention given in
3.1. Flocs in the Simple Shear Flow The shear rate G of the simple shear flow whose velocity is given by u = Gy was varied between 40 and 70 s−1 . A sample of 19 flocs with a total of 228 frames was analyzed. Figure 4 shows one sequence of a rotating and deforming floc. Clearly, the rotation is not uniform: the floc tends to be aligned with the streamlines, whereas the time when it is perpendicular to them is much shorter. Moreover, we see that the transverse bending is rather pronounced, reminiscent of a hinge. Apparently, the floc was composed of two smaller ones which were connected during the growing process. Only 4 out of the 19 flocs displayed a substantial deformation as in Fig. 4, while the other flocs performed a solid-body rotation. The flocs showed no significant deformation in the longitudinal direction indicating that they were more easily bent than stretched by the fluid motion. The flocs were rather asymmetrical in shape as we deduce from the histogram for the ratio of the two semi-axes depicted in Fig. 5a: on average, a was about twice as long as b. Also the histograms for lb /la and rb /ra reveal this asymmetry. The histogram for the angle α given in Fig. 5b confirms the observation that the rotation of the flocs was not uniform in time:
FIG. 4. Rotating and deforming ferric hydroxide floc in a simple shear flow (first row). The second row shows the calculated best-fit ellipses tilted with α. Time is increasing from left to right in steps of 1/30 s.
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FLOCS IN SHEAR AND STRAIN FLOWS
This procedure leads to Fig. 6a. If the flocs displayed no deformation and Rp was constant then a ∗ (i, j) = 1, i.e., all values would lie on the unit circle. In order to quantify the deviation from this ideal case we fit an ellipse on the measured values: Ã
x(i, j) y(i, j)
!
! ¶ Ã cos αe −sin αe x¯ e + = y¯ e sin αe cos αe ! Ã ! Ã εx (i, j) ae cos α(i, j) + . × be sin α(i, j) ε y (i, j) µ
[5]
In Eq. [5] ae and be are the semi-axes of the fitted ellipse, (x¯ e , y¯ e ) denote the coordinates of its center of mass, and αe is the angle between the x axis and ae . Furthermore, (εx (i, j), ε y (i, j)) are the residuals which include the deviation from the ellipse. Given the data points (x(i, j), y(i, j)) we can use the same formula as for the flocs to determine (x¯ e , y¯ e ) and αe . However, the semi-axes ae and be have to be calculated in a different way. A linear regression model can be found by multiplying equation Eq. [5] either by cos αe or sin αe and subtracting the resulting
FIG. 5. (a) Histogram for the ratio hbi/hai, which is the value of b/a averaged over a sequence displaying the same floc. (b) Histogram for the angle α on the basis of the analysis of 19 flocs with a total of 228 frames.
it is more likely that α was monitored around 0◦ or 180◦ than around 90◦ . 3.1.1. Deformation. In the following we want to investigate the deformation of the flocs in a more quantitative manner. Let a(i, j) be the value of a associated to floc i and frame number j. In our case i ∈ {1, 2, . . . , 19} and j ∈ {1, 2, . . . , Ni }, where Ni is the number of frames showing the same floc i. We now plot the normalized values a ∗ (i, j): " a ∗ (i, j) =
Ni 1 X a(i, j) Ni j=1 Rp (i, j)
#−1
a(i, j) , Rp (i, j)
[2]
in the x y plane such that the coordinates reads x(i, j) = a ∗ (i, j) cos α(i, j), ∗
y(i, j) = a (i, j) sin α(i, j),
[3] [4]
where α(i, j) represents the orientation measured for floc i and frame number j.
FIG. 6. The normalized length scales a ∗ and b∗ plotted in the x y plane (a and b, respectively). The measured values are represented by little circles. The solid lines are the fitted ellipse with parameters given in Table 4, whereas the dashed lines indicate the direction of ae and be .
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STEFAN BLASER
TABLE 4 Parameters of the Fitted Ellipse for the Six Length Scales
a∗ la∗ ra∗ b∗ lb∗ rb∗
Finally, knowledge of the values for T and t0 enables us to calculate the ratio of the two semi-axes. However, since the tangent jumps at odd multiples of π/2 it is better to cast Eq. [6] into the following form:
x¯ e
y¯ e
αe
ae
be
0.05 0.05 0.04 −0.04 −0.04 −0.04
0.04 0.04 0.03 0.04 0.04 0.04
6.4◦ 6.6◦ 5.4◦ 1.3◦ 2.6◦ 2.2◦
1.02 ± 0.02 1.03 ± 0.02 1.02 ± 0.02 1.04 ± 0.01 1.04 ± 0.02 1.04 ± 0.02
0.93 ± 0.02 0.92 ± 0.02 0.96 ± 0.03 0.98 ± 0.01 0.98 ± 0.02 0.98 ± 0.01
Note: The straight-line model also provides the standard deviation in the estimates of ae and be .
equations. The parameters ae and be are then determined by the method of least squares. If we proceed with the other five length scales b, la , lb , ra , and rb in the same way we find the values for the fitted parameters given in Table 4. All length scales show a similar behavior, but the scattering around the fitted ellipse is for la∗ and lb∗ and in particular for ra∗ and rb∗ much more pronounced than for a ∗ and b∗ . From Table 4 we infer that on average the elongation of the flocs culminates at an angle of α ≈ 6◦ , whereas their compression is maximum at about α ≈ 2◦ . Remember that the eigenvectors of the rate-of-strain tensor are inclined ±45◦ to the x axis. It seems that due to the rotation there is a certain retardation until the floc experiences the stretching and compression of the fluid. However, the deformation of the analyzed flocs is rather small as the ratio be /ae ≈ 0.9 differs only slightly from unity.
¶ µ ¶ a 2π (t − t0 ) 2π (t − t0 ) sin α = − sin cos α. cos T b T µ
[8]
Again we make use of the linear regression to estimate the value for a/b. An example comparing the measured and fitted data is shown in Fig. 7b. One might ask whether the fitted values correlate with the values extracted from the digitized images. In the following, fitted quantities will be indicated by the subscript F, whereas data provided by the image analysis are supplied with the subscript I.
3.1.2. Orientation. We now want to investigate whether a suitable model is able to relate the time-evolution of α with the parameters a, b, and G. It is well known (24) that a rigid ellipsoid with semi-axes a, b, and c can rotate in a planar orbit according to µ ¶¸ · 2π(t − t0 ) a , α(t) = −arctan tan b T
[6]
where T = 2π(a 2 + b2 )/abG is the period of the motion. In order to fit the measured data points α(i, j) of each sequence with Eq. [6], note that the phase velocity α(t) ˙ averaged in time is related with T through the following equation: 1 T
Z 0
T
α(t) ˙ dt =
1 2π . [α(T ) − α(0)] = T T
[7]
On the other hand, if α(i, j) is plotted against the time t, then the slope of the straight line given by linear regression is nothing but the time-averaged phase velocity, and by means of Eq. [7], T can be immediately determined (see Fig. 7a). Owing to the periodicity of the motion, the time-shift t0 , which arises in Eq. [6], can be determined by means of the relation: α(t0 − kT /4) = kπ/2, k ∈ Z. As is shown in Fig. 7a, the discrete data points are interpolated using a cubic spline fit.
FIG. 7. Fit of the measured time-evolution of the orientation. (a) Values for α measured at different times are indicated by dots. The intermediate points represented by the solid line are given by a cubic spline interpolation. The dashed line results from a least-squares straight line fit to the data. (b) Comparison of the measured values (dots) with the fitted curve given by Eq. [6].
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FLOCS IN SHEAR AND STRAIN FLOWS
3.2. Flocs in the Four-Roll Mill
FIG. 8. Comparison of the fitted and measured data bF /aF and bI /aI . The solid line is the regression line if bF /aF is fitted on bI /aI , whereas the dashed line represents the regression line if the two quantities are interchanged. The vertical error bars represent the standard deviation of bI /aI . The horizontal error bars are supplied by the linear regression model. Observe that the large horizontal error bar at bI /aI = 0.5 originates from a sample of only four points.
Figure 8 shows a comparison of bF /aF with bI /aI , where the latter value was calculated by averaging the measured ratio of the semi-axes over a sequence of the same floc. The correlation coefficient is r = 0.75, indicating that the two quantities are correlated. For further details consult Table 5. Although there are few outliers arising in Fig. 8, it is most striking that such a simple model given by Eq. [6] may explain quite well the observed time-evolution of the orientation. This is in particular remarkable as this equation is valid for a solid impermeable ellipsoid, whereas the flocs have actually to be considered as porous bodies which can be deformed. Since the applied shear rate G is known, we can also calculate the period of motion by means of TI = 2π (aI2 + bI2 )/aI bI G, where we have assumed that the ratio of the semi-axes can be estimated by bI /aI . We infer from Fig. 9, which compares the values of TI with TF , that the correlation between the two quantities is satisfactory. This statement is also confirmed by the values compiled in Table 5.
3.2.1. Flow field. To begin, we consider the velocity field produced by the Taylor four-roll mill. For low Reynolds number the trajectories of single particles were found to be hyperbolic. However, if the angular velocity Ä of the rollers is greater than 30 rad/s, effects originating mainly from the presence of the top and bottom plates become dominant, which leads to a vertical component of velocity also at the horizontal mid-plane. Thus the critical Reynolds number is given by Re = 88, where Re = ÄRS/ν with R = 1.56 mm being the roller radius and S = 1.88 mm the gap width between the rollers. Lagnado and Leal (19) found a smaller critical Reynolds number Re = 37. However, the dimensions of their four-roll mill were different. In particular, the ratio R/S was 1.7, whereas in our case it was 0.83. In order to estimate the strain rate E for a given Ä we suppose that the velocity field can be described by u = E x, v = −E y and that it coincides at the p point x = y = S/2 + R(1 − 2−1/2 ) with the speed of the roller: E x 2 + y 2 = ÄR. Solving this equation for E leads to E=
√ 2Ä S/R + 2 −
√ = 0.79 · Ä. 2
[9]
Taylor (16) proposed an experimental method to determine E. The trajectory of a small tracer particle which moves along the x axis is given by xp (t) = xp0 exp Et. If the time t which the particle needs to cover the distance |xp (t) − xp0 | is measured,
TABLE 5 Linear Regression for (bF /aF , bI /aI ) and (TF , TI ) Independent variable
Slope
Fig. 8
bF /aF bI /aI
0.66 ± 0.30 0.86 ± 0.40
Fig. 9
TF TI
0.77 ± 0.29 (0.08 ± 0.07) s 0.85 ± 0.30 (0.01 ± 0.09) s
y intercept 0.16 ± 0.15 0.07 ± 0.19
Correlation Confidence coefficient interval 0.75
[0.45, 0.90]
0.81
[0.57, 0.92]
Note: The third and fourth column give the slope and the y intercept of the regression line. Note also that the correlation coefficient r in the fifth column is nothing but the geometric mean of the two values for the slope. The sixth column is the confidence interval for r determined according to the method described in (25).
FIG. 9. Comparison of TF with TI . Fitting of TF on TI results in the solid line; the dashed line represents the regression line if the two quantities are interchanged. The vertical bars are calculated by applying Gauss’ law of the propagation of errors assuming that the standard deviation of G is 5 s−1 .
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STEFAN BLASER
FIG. 10. Break-up of ferric hydroxide flocs in a two-dimensional extensional flow. Time is increasing from left to right in steps of 1/30 s. The first row shows the process of erosion, whereas the final two rows display the splitting of a floc.
then the strain rate can be calculated by means of E = t −1 ln
xp . xp0
[10]
Explicitly, for Ä = 25.1 rad/s a particle was observed with xp0 = 0.13 mm and xp (t = 3/30 s) = 0.99 mm, resulting in E = 20.3 s−1 . On the other hand, we find from Eq. [9] the value E = 19.9 s−1 , which is in satisfactory agreement. 3.2.2. Break-up of flocs. As in the case of the simple shear flow, the experiments were performed with ferric hydroxide flocs in a 5 wt% sucrose solution. The angular speed was kept constant at Ä = 25.1 rad/s corresponding to a strain rate of E = 20 s−1 . Figure 10 shows three examples of flocs which were broken apart by the fluid motion when they approached the stagnation point. Both breaking modes, erosion of a small particle from the floc surface and splitting of the floc into equal-sized fragments, could be monitored. However, not every floc, even some very large ones, were subject to fragmentation. Most of the flocs were either stretched along the axis of incoming fluid or aligned with this axis without any deformation. This is mainly due to the fact that the flocs have a limited residence time near the stagnation point, and thus the fluid has only a short amount of time to act on the surface of the particles. In order to gain quantitative information on orientation and deformation the Matlab program was applied to a sample of 22 flocs which were broken apart. As one can see from Fig. 11 the ratio b/a is shifted toward smaller values with increasing time, which indicates that the flocs tend to be stretched. Almost all flocs display a decrease in the ratio of the semi-axes, which is quite substantial in a few cases. There are some flocs which get first more spherical-like with time before they are stretched: this is related to the observation that as they approach the stagnation point the fluid motion tries to accelerate the rear of the flocs toward the front part, such that they are compressed.
To investigate the time-evolution of the orientation we plot a line for each floc in the x y plane such that it makes an angle of α with the x axis and that its length corresponds to 2a/Rp . Applying this procedure to the flocs when they became visible the first time and for t = 0 leads to Figs. 12a and b. It can be clearly seen that the fluid tries to align and stretch the flocs along its axis of stretching. Yet, when a smaller particle is eroded from the floc surface the angle α can be considerably larger than 0◦ . See for instance the top row in Fig. 10 for which case α lies between −82◦ and −23◦ . All analyzed flocs were broken into two smaller parts. Figure 13a shows the values for Rp,1 /Rp,2 , where the subscripts 1 and 2 refer to the two fragments. Note that the data are sorted such that Rp,1 ≤ Rp,2 . We see that large-scale fragmentation (Rp,1 ≈ Rp,2 ) outweighs surface erosion (Rp,1 ¿ Rp,2 ). However, one has to bear in mind that the statistics may be biased by the
FIG. 11. Time-evolution of ba0 /ab0 for each of the 22 flocs. b0 /a0 is the value of b/a at t = 0, which indicates the time just before the flocs is broken apart. Note that the observation time varies with each floc.
FLOCS IN SHEAR AND STRAIN FLOWS
281
on the surface, and M a dimensionless 3 × 3 matrix, which is a function of α and the ratios a/c and b/c. (The reader is referred to (26) or (27, Sect. 3.3) for the detailed values of M, which can be expressed in terms of elliptic integrals.) As an illustrative example we consider an ellipsoid with semiaxes 2.5 : 1.5 : 1 which has its a-axis aligned along the axis of fluid stretching. To facilitate the illustrations, f is decomposed into the normal and tangential components, which are given by f n = f·n and f t = |f− f n n|, respectively. Figure 14 shows that the normal component is extreme along the axes of fluid compression and stretching (y and x axes), whereas the tangential component is large near the stretching axis (x axis) but vanishes at the piercing points of the ellipsoid’s surface with the x, y, and z axes. In order to characterize the hydrodynamic load by a single quantity (denoted in the following by f h ), the force distribution f is integrated over half of the ellipsoid’s surface which is parametrized by x = (a cos φ sin θ , b sin φ sin θ , c cos θ ) with φ ∈ [−π/2, π/2] and θ ∈ [0, π ] (see also inset in Fig. 16).
FIG. 12. Orientation and length of the analyzed flocs (a) at the time when they entered the straining flow and (b) at the beginning of breakage (t = 0). The length of each line whose ends are marked with black dots is given by 2a/Rp . Note that the origin of the coordinate system coincides with the center of mass of each floc.
fact that the latter breaking mode was harder to detect. Finally, we deduce from Fig. 13b that the fragments displayed no tendency of being spherical as the mean of b/a is at about 0.6. 3.2.3. Estimation of rupture force. We now want to estimate the rupture forces which were needed to split the flocs subjected to the straining flow. For this purpose the following ad hoc model is adopted: each floc which was seen to be broken apart is represented by a solid ellipsoid with semi-axes a, b, and c. The values for a and b are obtained from the particle image analysis at the onset of breakage, whereas c, which is unknown, is assumed to lie in the range b ≤ c ≤ a. In addition, the orientation of the ellipsoid is equated by the measured value for α. In an early paper, Jeffery (26) gave a general formula for the hydrodynamic force f per unit area on the surface of a small ellipsoid which is subjected to a linear flow field. For the present case of strain flow, f is of the form f = µEMn,
[11]
where µ is the viscosity of the fluid, n the unit outward normal
FIG. 13. Histograms (a) for the ratio of the equal-size radius of the fragments, Rp,1 /Rp,2 , and (b) for the ratio of the two semi-axes b/a. The data are based on a sample of 44 fragments.
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STEFAN BLASER
FIG. 16. Rupture force f hr of individual flocs with size Rp in twodimensional straining flow. The error bars are given by the uncertainty of c. Their length is correlated with the asymmetry in the shape of the flocs.
FIG. 14. Force per unit area exerted by the straining flow u = (E x, −E y, 0) on the surface of a solid ellipsoid with semi-axes 2.5 : 1.5 : 1. (a) Normal and (b) tangential components of f. The scale gives the non-dimensional values of f n /(µE) and f t /(µE), respectively.
Considering, for example, a sphere, a = b = c, one finds that f h = qµc2 E with q = 5π. If the semi-axes are different a 6= b 6= c, then q is a function of a/c, b/c, and α (see Fig. 15). Note that integration of f over the whole surface equals zero, as
FIG. 15. Contour lines for the integral force f h on half of the ellipsoid’s surface. The orientation is chosen to be α = 0, i.e., the a axis is aligned with the stretching axis of the fluid motion. The contour levels give the values of f h /(µc2 E).
we assume in our model that the ellipsoid moves “freely” with the fluid motion, i.e., there is no resultant force acting on the particle. If we assume that the floc is split into two fragments, then a simple criterion for breakage is given by the condition f h ≥ f b , where f b is the integrated internal binding forces which occur along the dividing plane. Based on the measured values for a, b, and α, the hydrodynamic load f hr can be calculated for each floc which was observed to be broken apart by the straining flow. (In order to emphasize that the flocs were ruptured, the superscript “r” is added.) Figure 16 shows f hr as a function of the measured equal-area radius Rp . By dividing f hr by the half area of the ellipsoids and taking the average over the whole sample, we find ¿
À f hr = (5.8 ± 0.5) · 10−2 Nm−2 , 2S
[12]
where S is the total surface area of the ellipsoid. In the present experiment the strain rate was increased until flocs were seen to be broken apart. The value given in Eq. [12], which is on the order of µE, gives therefore a rough estimate for the hydrodynamic forces which are needed to induce breakage of iron flocs of size Rp < 0.4 mm. Although Fig. 16 shows a certain tendency ( f hr is greater for larger flocs), it does not confirm that f hr correlates with Rp . This is supported by the fact that flocs of comparable size were observed which were unaffected by the straining motion. Thus the inner structure must be considered as inhomogeneous and the mechanical strength is variable. The flocs used in the experiments were produced by adding Fe(III) salt in a concentration well above its solubility in water. Hence, as the primary silica particles were enmeshed by the precipitating ferric hydroxide, they were unlikely to be in contact and do not contribute to the internal strength. Furthermore, in contrast to the aggregation by double-layer compression, the
283
FLOCS IN SHEAR AND STRAIN FLOWS
ionic strength will have little effect on the binding forces of this kind of sweep floc. However, the hydrolysis scheme of Fe(III) salt and the morphology of the precipitates are complex (28, 29) and not fully understood. Judging by the optical observation (see Fig. 10), the flocs tended to be of a compact structure, such that several bonds were broken apart before the fragments were separated. Other investigations on the size–density relationship of iron flocs suggest that the fractal dimension Df is in the range 1.9–2.78 (30, 31). Because of the internal inhomogeneity the binding force f b cannot be taken simply as a function, for example of the floc size Rp , but instead has to be treated as a random variable; i.e., given a floc of a definite size, there is a certain probability that the floc breaks or not. In this context the investigations performed by Yeung and Pelton (32) are noteworthy. The authors determined directly the rupture strength of single flocs by means of two suction pipettes. While one pipette was kept stationary, the other one deflected elastically in response to axial forces such that from the known stiffness and its deflection the tensile strength on the flocs could be determined. The measured values showed no correlation with the floc sizes when plotted on logarithmic scales.2 Very little data on rupture forces is currently available. Quigley (8) also investigated ferric hydroxide flocs in a straining flow. He found that rupture occurred for µE ≥ 0.1 N/m2 , which is of the same order as Eq. [12]. Yeung and Pelton (32) investigated flocs which consisted of calcium carbonate and polymers. They displayed a much higher rupture force: 30–2,000 N/m2 . However, the conglomerates were 6–40 µm in size and therefore much smaller than the sweep flocs investigated in the present experiment. 3.2.4. Effects of porosity. It is clear that a floc is not impermeable for a fluid and one could therefore ask to what extent porosity effects the estimated values above. For this purpose we consider the Stokes drag FD of a porous sphere with radius a which is subjected to the constant flow u0 . A widely used approach of tackling this problem is solving the creeping flow equations for the flow exterior to the aggregate and applying either Darcy’s or Brinkman’s equation for the interior flow coupled with continuity equations. (See (33) for a more detailed description.) Assuming that inertia effects can be neglected and that the porosity of the aggregate is homogeneous, the drag can be written as FD = 6πµau0 MD , where the factor MD depends on the permeability k of the sphere. Furthermore, if we assume that the porous medium is composed of ˆ the Carman-Kozeny formula gives small spheres with radius a, a relationship between k and the void fraction ε of the aggregate (see for instance (34)). Figure 17 shows the Stokes multiplier MD as a function of ε and the ratio a/aˆ for different model equations. It can be seen that for low and intermediate void fractions the difference of the Stokes drag between a permeable and impermeable sphere is
2
D −1
In particular, the scaling law f b ∝ Rp f
is not applicable to their data.
FIG. 17. Drag multiplier MD as a function of the void fraction ε and the ratio a/aˆ for several models: Darcy equation with Saffman boundary condition (35) (solid line); Darcy equation with Jones boundary condition (36) (dashed line); and Brinkman equation (37) dotted line).
minute. Only at a very high porosity, ε > 0.95, does MD differ substantially from one. Aggregates which consist of monodisperse spheres will in general show low void fractions: for closed packing, e.g., we have ε = 0.26, while for random packings 0.34 ≤ ε ≤ 0.45 (38). Ferric hydroxide flocs, however, can display a much larger porosity. In general, the water content of flocs increases with increasing size (39). For a floc size of 100 µm the void fraction is about ε = 0.95, whereas it increases to ε = 0.999 at a size of about 800 µm (39). Judging from this analysis, we conclude that the influence of porosity is just at the edge of becoming important to get the correct load on the surface as the size of flocs investigated here lie in the range 200–800 µm (see Fig. 16). 4. CONCLUSIONS
Flocs which are immersed in a simple shear flow perform a periodic but not uniform motion. Comparison with the planar motion of a solid ellipsoid reveals that for given a, b, and shear rate, the time-evolution of the measured orientation can be predicted to satisfactory agreement. Also, the measured and theoretically calculated periods of motion were correlated. It is remarkable that the shape of the flocs projected on the x y plane determines the dynamical behavior: their motion is independent of their shape in the x z and yz planes. The bending of the flocs is more pronounced than the stretching. The semi-axis a peaks when it is almost aligned with the streamlines. However, because of the presence of vorticity, a floc is kept in rotation and thus the load on its surface fluctuates with time. Thus it can constantly escape a stretching along the directions given by the eigenvectors of the rate-of-strain tensor. In this context it is noteworthy to mention that highly viscous drops with viscosity µd do not necessarily disintegrate in a shear flow, even at high shear rates. Taylor (16) showed that if the viscosity ratio µd /µ is larger than 4 and if the drop begins with
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STEFAN BLASER
a spherical shape, it will not break up in a simple shear flow, independent of the magnitude of the shear rate: increasing the shear rate has only the effect that the drop fluid circulates faster, but the state of its deformation remains unchanged. In contrast to the simple shear flow, flocs in a two-dimensional straining flow do not rotate continuously and are stretched and aligned along the axis of outgoing fluid. If the strain rate is large enough they may be broken apart. However, flocs of comparable size have different tendencies toward disintegration. This is in part due to the fact that their residence time near the stagnation point varies and hence the fluid has not always “enough” time to break up all bonds within the flocs. In addition, the inner structure, and with it the mechanical stability, is different for each floc. The ellipsoid model allows the estimation of the force distribution on the surface of a particle. For the hydroxide flocs a lower bound for the rupture forces was found to be on the order of 0.1 N/m2 . Furthermore, only if the void fraction of the flocs is particularly high will porosity have dominant effects on the load exerted by the fluid on the particle surface. ACKNOWLEDGMENTS This study was supported by a grant of the Swiss National Science Foundation (Grant No 21-40514.94 and 20-46680.96) and was performed in collaboration with Dr. A. Gyr, Dr. A. M¨uller, Prof. Dr. W. Gujer, and Prof. Dr. M. Boller. The author is grateful to Emily Ogenyi for improving the English.
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