The settling of consecutive spheres in viscoplastic fluids

Int. J. Miner. Process. 82 (2007) 106 – 115 www.elsevier.com/locate/ijminpro

The settling of consecutive spheres in viscoplastic fluids M.M. Gumulya a,1 , R.R. Horsley a,⁎, K.C. Wilson b,2 a

Department of Applied Chemistry, Curtin University of Technology, Perth, WA 6845, Australia b Department of Civil Engineering, Queen's University Kingston, Ontario Canada K7L3N6

Received 3 March 2006; received in revised form 2 August 2006; accepted 29 November 2006 Available online 16 January 2007

Abstract One of the factors contributing to the uncertainties involved in the estimation of particle settling velocity in viscoplastic fluids is the time-dependent effect where the viscous parameters of the fluid change as a particle flows through and shears the medium. These changes, particularly at low shear Reynolds numbers, are reflected in the settling velocity of a following sphere that is released some time after an initial one, with the following sphere having a significantly greater velocity. This study found that changes in both fall velocity and equivalent viscosity can be correlated satisfactorily by a power law equation to the dimensionless form of the time interval between releases, and the rheogram shape factor for the fluid. A collision of particles occurs in cases where the time interval between releases is small, after which the particles combine and travel at a terminal velocity. A new variable, β, which takes into account the different surficial stress of the combined spheres, was introduced to the correlation of Wilson et al. [Wilson, K.C., Horsley, R.R., Kealy, T., Reizes, J.A., Horsley, M.R., 2003. Direct prediction of fall velocities in nonNewtonian materials. Int. J. Miner. Process. 71, 17–30] β was found to depend on the rheogram shape factor for the fluid and the shear Reynolds number for the particle. The validity of this approach was supported by experimental data. © 2007 Elsevier B.V. All rights reserved. Keywords: Fall velocity; Non-Newtonian flow; Drag curve; Equivalent viscosity

1. Introduction Viscoplastic fluids are frequently encountered in industries, often in the form of slurries with a high concentration of fine particles. In addition, these slurries often contain a fraction of larger particles which naturally have a higher tendency to settle. In practice, it is important for engineers to be able to predict the ⁎ Corresponding author. Fax: +61 8 9266 2300. E-mail addresses: [email protected] (M.M. Gumulya), [email protected] (R.R. Horsley), [email protected] (K.C. Wilson). 1 Fax: +61 8 9266 2300. 2 Fax: +1 613 533 2128. 0301-7516/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2006.11.005

settling behaviour of these particles, as this factor often determines the performance of many slurry-based types of equipment such as agitators, sphere mills and long distance slurry pipelines. There have been an extensive number of studies done on the settling of particles in an undisturbed viscoplastic medium. Well-known studies include Valentik and Whitmore (1965), du Plessis and Ansley (1967) and Ansley and Smith (1967) who have published a large number of experimental data. The work was pioneered by Andres (1961) who concluded that for a particle to settle in a viscoplastic fluid, its buoyant weight has to exceed the yield stress of the fluid. Based on this principle, Andres postulated that YG, which is defined as the yield stress-tobuoyant weight ratio, has to be lower than 0.212 for a

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Fig. 1. The rheograms of 0.9% (ρf = 999.0 kg/m3), 1.0% (ρf = 998.9 kg/m3), and 1.15% (ρf = 998.74 kg/m3) Floxit 5250 L aqueous solutions.

spherical particle to settle. The applicability of this theory has been reviewed both experimentally and theoretically by various researchers, and since then various critical values of YG, ranging from 0.04 (Brookes and Whitmore, 1968) to 0.20 (Boardman and Whitmore, 1961) have been reported. It is suspected that one of the factors causing the widely differing values of critical YG is the inconsistent method used in determining the yield stress (Chhabra, 1993). Another approach that has been proposed is the adaptation of Newtonian calculations, i.e. the customary CD-Re graph, to non-Newtonian medium. Various correlations have been published, notably the correlations of Ansley and Smith (1967) and Atapattu et al. (1995). However, as both of the dimensionless terms are dependent on the terminal velocity of the particle, extensive iteration was required to calculate the particle settling velocity. Furthermore, the iterations are generally further complicated by the fact that the viscosity of a nonNewtonian fluid is dependent on the shear rate applied to it, thus lowering the chance for the iteration to converge. The requirement for iteration has been eliminated by the method suggested by Wilson et al. (2003), as well as Wilson and Horsley (2004). This was done by adapting the CD-Re graph to another set of dimensionless variables, one of which (shear Reynolds number) is not dependent on the particle terminal velocity. This allows the particle terminal velocity to be determined directly. Furthermore, Wilson et al. also recommended the use of ‘equivalent viscosity’ which was defined as the viscosity of a Newtonian fluid (with the same density as the medium) that would produce the same particle terminal velocity. This variable can be obtained directly from the rheogram of the fluid at a certain reference point.

The modelling of the settling behaviour of particles in a viscoplastic fluid is further complicated by the fact that it is greatly affected by the state of the fluid, i.e. whether it is disturbed (dynamic settling) or undisturbed (static settling), as noted by Chhabra (1993) and Atapattu et al. (1990, 1995). Throughout the years it has been discovered that a sphere that is released following an earlier one has a velocity that is higher than the first sphere (Horsley et al., 2004). Furthermore, Horsley showed that the difference in the velocities decreases as the time interval between the releases is increased. Hariharaputhiran et al. (1998), who conducted a similar experiment, concluded that the dependence of the settling velocity on the time interval is due to the local rupture of the polymer network in the solution as the first sphere settles. This theory is similar to the hypothesis of Cho et al. (1984) who conducted the experiment in viscoelastic fluids. Over time, the structure of the fluid is recovered by back-diffusion of the ruptured polymer molecules, resulting in the difference between the velocities of the released spheres to decrease as the time interval is increased. The aim of this paper is to continue the work done by Horsley et al. (2004), i.e. to present a more general model for the settling behaviours of two identical, vertically aligned spheres falling consecutively in a viscoplastic fluid. 2. Experimental The experiments were conducted in a long, straightwalled Perspex tube with square cross section and closed bottom. The dimensions of the tube were set such that wall effects can be neglected (Atapattu et al., 1990),

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Fig. 2. The velocity trace of 6.35 mm bronze spheres travelling in a 0.9% Floxit 5250 L solution, with 2.5 s time difference.

and the height of the column was sufficient to allow long enough residence time for particles to reach a terminal velocity. The tube walls were clear, in order to allow measurements to be made by optical sensing devices. A sphere dropper mechanism, which is fully controlled by a computer program so that it can drop two spheres at specific time intervals, is located at the top of the tube. The fluids used were aqueous solutions of Floxit 5250 L (supplied by Imdex Ltd, Perth, WA), which is a high molecular weight polymer, used widely in various industries as a viscosity-modifying additive. Fig. 1 shows the rheograms and densities of the solutions that were used in the experiment. As the spheres flow through the fluid, their positions are detected by optical sensing devices placed along the length of the tube. The data obtained by this measurement is then transmitted to a computer using custom written data-collection software. A time interval of 15 min between experiments was used to ensure the full recovery of the fluid structure. 3. Results and discussion 3.1. Velocity changes The typical velocity trace that is obtained along the column is shown in Fig. 2. As can be seen in this graph, the first sphere, which travels through the undisturbed fluid, reaches a constant (terminal) velocity shortly after it enters the column. On the other hand, the second sphere, which travels in-line with the first sphere a few seconds later, travels at a much higher velocity. Furthermore, the

velocity of the second sphere also increases as the gap between the first and the second sphere decreases. The second sphere eventually catches up with the first one, and the two spheres collide. After the collision, the two spheres attach, and hence travel at the same velocity. It has also been observed that just before the collision, i.e. when the gap between the first and the second sphere becomes very small, there is a slight acceleration in the first sphere velocity and deceleration in the second one (in Fig. 2, this happens between sensors number 9 and 11). This observation will become important in the modelling of the velocity profile of the second sphere. 3.2. Rheological considerations Fig. 1 shows the rheograms obtained for Floxit 5250 L solutions with various concentrations. From this diagram it can be seen that although the rheogram is essentially linear at high values of strain rate (above 65 s− 1), the shear stress slowly declines from the straight-line approximation as the strain rate is decreased. Over the course of the experiment, it has been found that the typical strain rates obtained for the falling sphere system lie in the lower strain-rate region (γ′ b 100 s− 1, see Tables A1 and A2 in Appendix), i.e. the curved region of the rheogram. As a result, it was concluded that the Casson model would give a better representation of the rheogram than the linear Bingham model. Like the Bingham model, the Casson model has two parameters. It can be expressed by the following equation: pffiffiffi pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiV s ¼ s C þ gC g

ð1Þ

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where τ is the shear stress, τC is the Casson yield stress and ηC is the Casson viscosity. The concept of μeq was introduced by Wilson et al. (2003) who defined it as the calculated viscosity of a Newtonian fluid (with the same density as the medium) that would produce the same terminal fall velocity for a particle of the same type: leq ¼ s=g V

ð2Þ

γ′ is defined as the representative shear rate, which can be determined directly from the rheogram of the fluid, or by applying Eq. (1) at a specified reference point, τ: s ¼ f s¯

ð3Þ

where ζ is a constant that is taken to be 0.3 (Wilson et al., 2003) and τ¯ is the mean surficial shear stress applied by the particle to the medium: s ¼ ðqs −qf ÞgD=6 ¯

ð4Þ

where ρs and ρf are the densities of the particle and the fluid respectively, g is the gravitational acceleration, and D is the diameter of the particle. The shear Reynolds number, Re⁎, can then be determined by the following equation: Re⁎ ¼ qf V ⁎ D=leq

ð5Þ

where V⁎ is the shear velocity, which is equivalent to: V⁎ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðqs −qf ÞgD=ð6qf Þ

ð6Þ

The rheogram shape ratio, α, which is a measure of the viscoplasticity of the material, can then be determined. This parameter is based on the area beneath the section of a rheogram, to the left of the representative

Fig. 3. Definition plot for rheometric quantities (Horsley et al., 2004).

109

shear rate, γ′, as shown in Fig. 3. α is obtained by taking the ratio of the total area under the rheogram to the triangle beneath the same rheogram section. This triangle represents the area that would be associated with the same reference point if the material were Newtonian (Wilson et al., 2003). The value of α varies from 1.0, for a Newtonian fluid, to 2.0, for a pure plastic material. For the material used in this experiment, the average value of α was found to be 1.63. 3.3. The velocity of the second sphere As has been noted, a sphere that is dropped a few seconds after an earlier sphere travels at a greater velocity than the first sphere (Fig. 2). This is due to the shearing effect that is produced as the first sphere falls. This shearing effect reduces the equivalent viscosity of the material. This results in the second sphere experiencing less resistance, and hence travelling faster. Moreover, as the gap between the first and the second sphere decreases, there is less time for the fluid “structure” to recover. As a result, the velocity of the second sphere keeps increasing and a terminal velocity is not reached. The dependence of the second sphere velocity on the time difference between the first and second spheres can be seen in Fig. 4. In this graph, it can be seen that as the time difference increases, the ratio between the second sphere velocity (V2) and the terminal velocity of the same sphere if it were travelling in an undisturbed medium (Vt2) reaches a limiting value of 1, which indicates a state where the structure of the fluid has already recovered. On the other hand, as the time difference decreases, V2/Vt2 increases. In fact, Fig. 4 also suggests that as the time difference approaches zero (i.e. when there is minimal distance between the two spheres), V2/Vt2 reaches infinity. The above argument is obviously not physically realizable, as has been seen in Fig. 2. This is because as the distance between the two spheres decreases, the second sphere will not only experience the resistance of the fluid, it will also experience resistance from the first sphere which travels at a much lower velocity. The first sphere, on the other hand, will experience a “push” from the second sphere, and hence its velocity will increase slightly. This prevents the second sphere reaching its potential maximum velocity. It is clear that the velocity of the second sphere is dependent on both the time difference between the two spheres and the rheometric properties (i.e. the “structure”) of the medium, so a correlation that relates these variables to the velocity needs to be constructed.

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Fig. 4. 7.94 mm bronze spheres in 1.15% Floxit 5250 L solution.

An initial step is to express the time difference between the two particles, t, in a dimensionless form, denoted by t⁎. This variable is expressed as the product of the representative shear rate γ′ and the time difference t. Since the disruption of the fluid structure, which causes the second sphere to travel faster that the first one, does not apply with Newtonian fluids (for which α = 1.0), the variable for “structure” in the correlation is based on α − 1. Thus the disturbance in the nonNewtonian fluid structure was assumed to be proportional to (α − 1)N, where N is a power constant that is to be found from the experimental data. Fig. 4 also suggests that the recovery in the fluid structure can be modelled by a power function. The disruption in the structure of the fluid influences the movement of the second sphere, making it fall faster than its terminal velocity Vt2 which would be obtained if it were falling through an undisturbed fluid. Thus, if the above assumption is true, then the following equation should be able to reflect the velocity of the second sphere:

evaluation method was used, and the results of this analysis are: C1 = 4.94, C2 = − 0.30 and N = 1.89. Thus Eq. (7) can be written:

j k V2 ¼ Vt2 1:0 þ C1 :ðg V:tÞC2 :ða−1ÞN

As has been mentioned before, the discrepancy in the velocity of the first and second spheres is due to the disturbance caused by the flow of the first sphere in the fluid structure. This implies that the change in the fluid viscosity is reflected by the change in the settling velocity of the particles. At low shear Reynolds numbers, which is the most frequently-encountered situation in this experiment (Re⁎ b 10), the particle settling velocity is effectively

ð7Þ

where C1 and C2 are constants. The constants C1, C2, and N were evaluated by using 249 experimental data points obtained from consecutive drops of spheres with the same diameter and density, in which case Vt1 = Vt2. A multiple linear regression

j k V2 ¼ Vt1 1:0 þ 4:94:ðg V:tÞ−0:30 :ða−1Þ1:89

ð8Þ

The results of Eq. (8) can be seen in Fig. 5, which shows that there is a high regression and proportionality between the calculated and experimental values. This suggests that the assumptions made above are valid, and that Eq. (8) can be used for this system. Of the 249 data points, only 30 points lie with an error of greater than 20%, of which the maximum is 56.4%. Higher uncertainties were found in cases where the time interval was very small. It is suspected that the major contributor to this error is the uncertainties involved when using the average velocity over the small distance between the sensors. Table A1 provides a selected sample of 45 of these data points. 3.4. The equivalent viscosity of a disturbed fluid

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Fig. 5. The calculated values of the second sphere velocity as a function of the experimental values.

inversely proportional to the equivalent viscosity (Wilson et al., 2003). Eq. (7) can be rewritten as:

departure from direct proportionality as Re⁎ increases. This increases the uncertainty of equation 10 for Re⁎ N 4.

j k lt2 =l2 1:0 þ C1 :ðg V:tÞC2 :ða−1ÞN

3.5. The terminal velocity of the combined spheres

ð9Þ

where μ2 is the equivalent viscosity of the fluid and μt2 is the equivalent viscosity that would be encountered if a similar particle drops in an undisturbed fluid. Similar to the case with the velocity correlation, the constants C1, C2 and N were evaluated by using 249 experimental data points obtained from consecutive drop of spheres with the same diameter and material, in which case Vt1 = Vt2. Eq. (9) can then be simplified to: j k l1 =l2 1:0 þ 4:019:ðg V:tÞ−0:306 :ða−1Þ−0:964

ð10Þ

Fig. 2 shows that after colliding, the two spheres combine and travel at the same velocity. Furthermore, a terminal velocity is soon reached which indicates undisturbed flow. As this was one of the assumptions taken in the development of the Wilson correlation (Wilson et al., 2003), it was expected that this correlation would give a suitable starting point for the modelling of the sphere velocities. For combined spheres, the equations for the fall velocity of a single sphere need to be modified. First of all, the submerged weight of the combined spheres (FW12) is the total weight of the two: FW12 ¼ ðp=6Þg½D31 ðqs1 −qf Þ þ D32 ðqs2 −qf Þ

The results of Eq. (10) were compared to the experimental data in Fig. 6, where it can be seen that there is reasonable regression and proportionality between the experimental and calculated values. The maximum error between the calculated equivalent viscosity μ2 and the experimental equivalent viscosity μ2 for cases where Re⁎ b 4 was found to be 50%. However, the error was found to be higher with increasing Reynolds numbers (up to 70% for 4 b Re⁎ b 15). This was as expected as the Vt /V⁎ and Re⁎ correlation (Wilson et al., 2003) shows an increasing

ð11Þ

Secondly, because of the path interference of the two spheres, the effective surface area of the combined spheres (A12) will be less than the total surface area of the two: A12 ¼ A1 þ b:A2

ð12Þ

where A1 and A2 are the surface area of the first and second spheres, and β is a variable that is less than unity,

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Fig. 6. The calculated values of the equivalent viscosity of the second sphere versus the experimental values.

and that takes into account the path interference of the two spheres as they combine. Therefore, the mean surficial shear stress caused by the combined spheres is:    s 12 ¼ g ðqs1 −qf ÞD31 þ ðqs2 −qf ÞD32 ¯ n h io ð13Þ = 24 ðD1 =2Þ2 þ bðD2 =2Þ2 For consecutive drop of spheres with the same material and diameter, Eq. (11) can be simplified to: s 12 ¼ ½2ðFW12 Þ=½A1 ð1 þ bÞ ¼ ¯ s 1 ½2=ð1 þ bÞ ¯

ð14Þ

Hence, by calculating the mean surficial stress of the combined spheres, the effective shear stress can be

calculated by applying Eq. (3). This can then be used to determine the shear Reynolds number (Eq. (5)), and hence the velocity ratio (Wilson et al., 2003). Based on the experimental results for consecutive drops of identical spheres, the average value of β was found to be 0.812 ± 28%. Using this average value, the settling velocities of the combined spheres were calculated, and the comparison with the experimental values can be seen in Fig. 7. This graph, whilst indicating that there is a large scatter between the experimental and calculated values (percentage error of up to 80%), shows that the calculated values for the combined sphere velocity is approximately proportional to the experimental values, which suggests that this method of calculation can be used for the modelling of the combined sphere velocity.

Fig. 7. The calculated values of the velocity of the combined spheres as a function of the experimental values, with β constant.

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Fig. 8. The calculated values of the velocity of the combined spheres as a function of the experimental values, with β as a function of Ω.

The distribution of error has a tendency to increase with α. While this may have been caused by various experimental errors, such as inaccuracy in the rheology measurement at low shear rates (thus higher values of α), it is suspected that the other contributor to error is the chosen value for β, which has so far been assumed to be constant. Thus, to improve this correlation, the variation of β with other variables needs to be examined. Upon examining the variation of β, it was found that it can be closely correlated to a combination of the dimensionless variables α and Re⁎ represented by Ω, which has the following form: X ¼ aRe⁎0:065

ð15Þ

The correlation for β is given by b ¼ −56:04X2 þ 214:32X−203:92

ð16Þ

The above correlation is then used in conjunction with Eq. (14), and the results of this calculation, in comparison with the experimental values, can be seen in Fig. 8 and Table A2. The accuracy of the correlation has been improved substantially, as evident in the high proportionality and regression between the calculated and experimental values. The maximum percentage error has been reduced to 40%, and no trends in the calculation errors have been detected. It is recommended that this correlation is used to estimate the settling velocity of combined spheres.

4. Conclusion In this study of spheres falling in viscoplastic fluids, it was seen that the second of two vertically-aligned spheres, which was released some time after the first

one, had a settling velocity significantly higher than that of the first sphere. Thus, in correlating the settling behaviour, it was found that additional considerations were needed, particularly in regard to time-dependent effects of the fluid on the settling velocity of the second sphere. A power law correlation was developed to express the ratio of the velocities of the two spheres to a dimensionless form of the time interval between releases, and the rheogram shape factor for the fluid. It was found that this correlation fits the experimental data with relatively high accuracy over a wide range of Floxit solutions. The accuracy of the method was found to be less at low values of time interval, perhaps due to the inherent uncertainty of time measurement in the experimental method. The difference in the velocity of the second sphere from the first sphere was associated with changes in the viscous parameters of the fluid, due to the fluid having been sheared by the first sphere. The difference in the settling velocity was found to reflect the change in equivalent viscosity, especially at low Reynolds numbers. This result allowed the behaviour of a disturbed fluid to be modelled using a correlation that is similar to the correlation developed for the settling velocities of spheres. Furthermore, in cases where the time interval is sufficiently small, a collision occurs between the first and the second spheres, after which they combine and travel at a terminal velocity. Based on the correlation of Wilson et al. (2003), a new variable, β, was introduced to take into account the different surficial stress of the combined spheres. This variable was evaluated experimentally and found to depend on the rheogram shape factor for the fluid and the shear Reynolds number for a sphere. The resulting correlation reflects the experimental data with reasonable accuracy, supporting the validity of this approach.

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Appendix A Table A1 Selected results for the correlation of the settling velocity of the second sphere % Floxit 5250 L

Sphere

D (mm)

t (s)

Vt1 (m/s)

Experimental V2 (m/s)

μ (Pa s)

α

Estimated V2 (m/s)

% error

0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15

Bronze Bronze Bronze Chrome Chrome Chrome Bronze Bronze Bronze Chrome Steel Steel Steel Bronze Bronze Bronze Chrome Chrome Bronze Bronze Bronze Chrome Chrome Chrome Steel Steel Steel Bronze Chrome Chrome Steel Steel Steel Bronze Bronze Bronze Chrome Steel Steel Steel Steel Steel Steel Chrome Chrome

6.35 6.35 6.35 6.35 6.35 6.35 7.94 7.94 7.94 7.94 7.94 7.94 7.94 6.35 6.35 6.35 6.35 6.35 7.94 7.94 7.94 7.94 7.94 7.94 7.94 7.94 7.94 6.35 6.35 6.35 6.35 6.35 6.35 7.94 7.94 7.94 8.00 7.94 7.94 7.94 9.53 9.53 9.53 13.00 13.00

2.97 3.03 4.01 1.14 2.72 4.51 1.45 2.32 3.37 3.41 1.39 2.44 3.37 1.03 2.49 4.36 1.90 3.07 1.00 2.11 4.18 1.37 2.16 4.17 1.10 3.24 4.09 2.50 4.00 11.00 3.00 4.00 5.00 5.50 6.00 60.00 7.00 3.00 5.00 6.00 2.50 3.50 5.00 15.00 20.00

0.174 0.177 0.180 0.130 0.126 0.124 0.305 0.314 0.236 0.250 0.229 0.240 0.236 0.082 0.079 0.079 0.053 0.053 0.152 0.153 0.158 0.127 0.121 0.118 0.112 0.110 0.110 0.071 0.046 0.046 0.041 0.041 0.041 0.145 0.145 0.145 0.111 0.096 0.096 0.096 0.165 0.165 0.165 0.416 0.416

0.313 0.289 0.311 0.240 0.235 0.179 0.567 0.463 0.388 0.372 0.391 0.414 0.388 0.210 0.19 0.173 0.122 0.114 0.376 0.321 0.308 0.292 0.250 0.229 0.276 0.236 0.245 0.178 0.103 0.095 0.110 0.107 0.105 0.338 0.350 0.205 0.215 0.219 0.203 0.208 0.403 0.410 0.339 0.705 0.707

0.817 0.805 0.784 1.000 1.036 1.057 0.620 0.593 0.889 0.692 0.759 0.713 0.726 1.984 2.050 2.069 2.711 2.690 1.551 1.538 1.489 1.641 1.735 1.775 1.836 1.876 1.875 2.329 3.169 3.169 3.444 3.444 3.444 1.640 1.640 1.640 1.933 2.185 2.185 2.185 1.704 1.704 1.704 0.978 0.978

1.74 1.74 1.74 1.77 1.77 1.77 1.71 1.70 1.75 1.72 1.73 1.73 1.73 1.84 1.84 1.84 1.86 1.86 1.82 1.82 1.82 1.82 1.83 1.83 1.83 1.84 1.84 1.85 1.87 1.87 1.88 1.88 1.88 1.82 1.82 1.82 1.84 1.85 1.85 1.85 1.83 1.83 1.83 1.78 1.78

0.302 0.303 0.297 0.273 0.236 0.218 0.531 0.510 0.412 0.405 0.428 0.408 0.389 0.219 0.183 0.166 0.139 0.128 0.378 0.334 0.306 0.303 0.272 0.241 0.290 0.238 0.229 0.166 0.108 0.092 0.105 0.100 0.096 0.274 0.271 0.208 0.212 0.216 0.199 0.193 0.356 0.337 0.320 0.619 0.602

−3.6 5.0 −4.5 13.7 0.2 21.6 −6.4 10.2 6.3 8.8 9.5 −1.4 0.4 4.1 −3.7 −4.2 14.0 12.2 0.5 4.2 −0.5 3.7 8.5 5.5 5.0 0.6 −6.4 −6.9 4.9 −3.4 −4.8 −7.0 −8.3 −18.8 −22.5 1.5 −1.5 −1.4 −2.2 −7.2 − 11.8 −17.6 −5.6 −12.2 −14.8

Table A2 Results for the correlation of the settling velocity of combined spheres % Floxit 5250L

Sphere

D (mm)

α

Re⁎

Experimental V12 (m/s)

Ω

β

Estimated V12 (m/s)

% error

0.90 0.90 0.90 0.90 1.00

Bronze Chrome Bronze Steel Bronze

6.35 6.35 7.94 7.94 6.35

1.67 1.75 1.60 1.70 1.77

4.063 2.154 9.042 4.025 2.058

0.279 0.213 0.517 0.371 0.136

1.833 1.838 1.839 1.864 1.854

0.671 0.699 0.710 0.882 0.820

0.299 0.173 0.527 0.287 0.173

−7.11 19.1 −1.9 22.6 −27.2

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Table A2 (continued ) % Floxit 5250L

Sphere

D (mm)

α

Re⁎

Experimental V12 (m/s)

Ω

β

Estimated V12 (m/s)

% error

1.00 1.00 1.00 1.00 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15

Chrome Bronze Chrome Steel Chrome Bronze Steel Chrome Bronze Steel Chrome Steel

6.35 7.94 7.94 7.94 6.35 6.35 6.35 8.00 8.00 8.00 10.00 10.00

1.82 1.70 1.77 1.78 1.84 1.81 1.85 1.81 1.75 1.82 1.72 1.73

1.228 4.842 2.611 2.363 0.896 1.379 0.847 1.715 3.345 1.587 5.560 5.068

0.095 0.247 0.209 0.183 0.091 0.130 0.088 0.186 0.239 0.177 0.338 0.283

1.840 1.883 1.882 1.880 1.831 1.849 1.828 1.876 1.891 1.873 1.919 1.919

0.719 0.963 0.957 0.950 0.635 0.782 0.610 0.941 0.985 0.919 1.000 1.000

0.106 0.345 0.209 0.192 0.082 0.125 0.078 0.150 0.270 0.140 0.387 0.362

−12.2 −39.6 0.0 − 4.6 9.4 4.0 12.2 19.1 −12.9 20.7 −14.5 −27.9

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