Fall velocity of multi-shaped clasts

Journal of Volcanology and Geothermal Research 289 (2014) 130–139

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Fall velocity of multi-shaped clasts Jacobus P. Le Roux ⁎ Departamento de Geología, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile/Andean Geothermal Center of Excellence, Plaza Ercilla 803, Santiago, Chile

a r t i c l e

i n f o

Article history: Received 23 June 2014 Accepted 6 November 2014 Available online 15 November 2014 Keywords: Settling velocity Settling tube Gravity sorting Sediment transport Pyroclast dynamics

a b s t r a c t Accurate settling velocity predictions of differently shaped micro- or macroclasts are required in many branches of science and engineering. Here, a single, dimensionally correct equation is presented that yields a significant improvement on previous settling formulas for a wide range of clast shapes. For smooth or irregular clasts with known axial dimensions, a partially polynomial equation based on the logarithmic values of dimensionless sizes and settling velocities is presented, in which the values of only one coefficient and one exponent need to be adapted for different shapes, irrespective of the Reynolds number. For irregular, natural clasts with unknown axial dimensions, a polynomial equation of the same form is applied, but with different coefficients. Comparison of the predicted and measured settling velocities of 8 different shape classes as well as natural grains with unknown axial dimensions in liquids, representing a total of 390 experimental data points, shows a mean percentage error of −0.83% and a combined R2 value of 0.998. The settling data of 169 differently shaped particles of pumice, glass and feldspar falling in air were also analyzed, which demonstrates that the proposed equation is also valid for these conditions. Two additional shape classes were identified in the latter data set, although the resultant equations are less accurate than for liquids. An Excel spreadsheet is provided to facilitate the calculation of fall velocities for grains settling individually and in groups, or alternatively to determine the equivalent sieve size from the settling velocity, which can be used to calibrate settling tubes. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The settling velocity of micro- and macroclasts has wide applications in science and engineering, including mineral processing (Concha and Almendra, 1979; Ofori-Sarpong and Amankwah, 2011), tailings segregation (Talmon et al., 2014), materials engineering (Purmajidian and Akhlaghi, 2014), powder technology (Bullard and Garboczi, 2013), atmospheric science (Chen and Fryrear, 2001; Siewert et al., 2014), and food engineering (Koyama and Kitamura, 2014). In the life sciences it has been applied, inter alia, to evolutionary microbiology (Briguglio and Hohenegger, 2011) and phytoplankton distribution (Greenwood and Craig, 2014), whereas in earth sciences it has been used to predict sediment transport (Le Roux, 1997a, 1998, 2001, 2005), as well as to model the fluid dynamics of magma chambers (Gutiérrez and Parada, 2010) and the aerodynamic behavior of volcanic particles, e.g. in dilute pyroclastic density currents (Choux et al., 2004; Dellino et al., 2004; Alfano et al., 2011; Douillet et al., 2014). In the geothermal energy field, the settling velocity of particles also has practical uses (Bonham, 1994). Some high temperature geothermal brine resources, e.g. in the Imperial Valley of California, contain large amounts of geothermal energy, but may not be suitable for the commercial production of electricity because of a high dissolved silica content that precipitates on the walls of pipes within the power production ⁎ Tel.: +56 22 9784123. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.jvolgeores.2014.11.001 0377-0273/© 2014 Elsevier B.V. All rights reserved.

plants. In the past, some power plants used flash crystallizer/reactor clarifier technology to precipitate the silica into sludge that is removed from the reactor clarifiers and pressed into a filter cake for disposal. However, there is still some precipitation growing on the walls of vessels and pipes, which tends to break loose in particulate scales to join the silica particles in the geothermal brine flowing through the plant. This material, which intermingles with the particulate silica in the sludge and is eventually incorporated into the sludge filter cake, contains hazardous substances such as As, Pb, Cu, Zi and Ag, thus restricting its disposal as simple landfill. A solution to this problem is to separate the toxic scale particulates from the non-toxic particulate silica by vertical separation, effected by means of a continuous decantation process in which a liquid slurry of silica and scale particulates is pumped upward at a velocity greater than the terminal settling velocity of the silica and less than the terminal settling velocity of the toxic scale. This process minimizes toxic waste disposal problems and enables a large amount of particulate silica waste to be disposed of as ordinary landfill (Bonham, 1994). However, in order to fine-tune the separation process, it is necessary to accurately predict the settling velocities of the silica and toxic particulates. Here, a single equation is proposed that gives accurate results from the Stokes range (quartz spheres less than 0.005 cm in diameter settling in water at 20 °C) to quartz spheres up to at least 22 cm settling under the same conditions. In addition, the same equation can be used to accurately predict the group settling velocity of natural clay- to gravel-sized sediments by simply adapting the coefficients.

J.P. Le Roux / Journal of Volcanology and Geothermal Research 289 (2014) 130–139

Cheng (1997) derived an equation for the settling velocity of natural grains based on the data of Zhu and Cheng (1993):

2. Previous work Numerous papers have been written on the settling velocity of differently shaped clasts (e.g. Stokes, 1851; Wadell, 1932; Rubey, 1933; Rouse, 1936; Corey, 1949; Janke, 1965, 1966; Gibbs et al., 1971; Warg, 1973; Komar and Reimers, 1978; Komar, 1980; Baba and Komar, 1981a,b; Dietrich, 1982; Le Roux, 1992, 1996; Cheng, 1997; Le Roux, 1997a,b, 2002a,b, 2004; Ferguson and Church, 2004). The most widely used single equations have probably been those of Dietrich (1982), Cheng (1997) and Ferguson and Church (2004), although none of these are very accurate over a wide range of clast shapes and sizes. Le Roux (1992) proposed a set of 5 equations for spheres of any size and density settling in fluids of different densities and viscosities, which are accurate to within 2% for the very well constrained data set of Gibbs et al. (1971). The obtained sphere settling velocity was subsequently incorporated into an equation for clasts with shapes varying from spheroids to ellipsoids, discs and cylinders (Le Roux, 2004). Nevertheless, the use of 5 different equations to calculate sphere settling velocities remains awkward. In this paper, the settling equations of Dietrich (1982), Cheng (1997) and Ferguson and Church (2004) are compared with the results of the present study. Dietrich (1982) defined a dimensionless sphere size as

Ddn
¼

ργ gDn

3

ρν2

;

ð1Þ

where ργ is the submerged grain density (the grain density ρs minus the fluid density ρ), g is the acceleration due to gravity, Dn is the nominal
 grain diameter, and ν is the kinematic viscosity μρ , μ being the dynamic viscosity. This author also defined a dimensionless settling velocity as

U dwn
¼

ρU wn 3 : ργ gν

131

ð2Þ

where Uwn is the actual settling velocity. His settling equation for spheres was given as 2

log10 U dwn
¼ −3:76715 þ 1:92944ð log10 Ddn
Þ−0:09815ð log10 Ddn
Þ 3 4 −0:00575ð log10 Ddn
Þ þ 0:00056ð log10 Ddn
Þ :

ð3Þ

ν U wp ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 = 2 25 þ 1:2D
2 −5 Dn

;

ð6Þ

where D
¼ Dn

 1 ργ g =3 ν2

:

ð7Þ

Ferguson and Church (2004) defined the following equation to determine the settling velocity of particles: 2

U wp ¼

ργ gDn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : C 1 ν þ 0:75C 2 ργ gDn 3

ð8Þ

For smooth spheres, C1 = 18 and C2 = 0.4, for natural grains C1 = 18 and C2 = 1 if sieve diameters are used, and for other shapes or where nominal diameters are used for natural grains, C1 = 20 and C2 = 1.1. 3. Methodology Water and air are both Newtonian fluids, i.e. they deform continuously and permanently when subjected to external forces. It is therefore convenient to use a non-dimensional approach, which means that dimensionally correct equations that are accurate for Newtonian liquids will also be accurate for Newtonian gases, provided that all the significant variables are taken into account (Middleton and Southard, 1984). The variables playing a role in particle settling velocity include the particle diameter and density, the fluid density and dynamic viscosity, as well as the particle shape (ψ), where various expressions or combinations of the three orthogonal axes can be used. The particle and fluid density can be combined into a submerged particle density, which leaves only the variables Uw, g, Dn, ργ, μ, and ψ. Considering only spheres for the moment, ψ can be left out, so that the remaining variables can be combined into a dimensionless nominal grain diameter given by Ddn

sffiffiffiffiffiffiffiffiffiffiffi ρgργ 3 : ¼ Dn μ2

ð9Þ

In the same way, a dimensionless settling velocity can be defined for spheres by

According to Dietrich (1982), Eq. (3) should not be used for values of Ddn⁎ smaller than 0.05 or greater than 5 × 109. For clasts of any size, shape, density and roundness, Dietrich proposed the following equation:

U dwn ¼ U wn

 ð1þ3:5−PÞ 2:5 CSF tanhð logDdn
−4:6Þ U dwp ¼ 0:65− 2:83 2:3 2 log U
þ log 1−1−CS F −ð1−CS F Þ tanhð log10 Ddn
−4:6Þþ0:3ð0:5−CS F Þð1−CS F Þ ð log10 Ddn
−4:6Þ ; 10½ 10 dwn ð ð 0:85 ÞÞ

The settling of particles is controlled by shear stress and pressure exerted by the fluid on its surface, called the viscous drag and pressure or form drag, respectively (Middleton and Southard, 1984). In the case of small spheres settling at very low Reynolds numbers, given by

ð4Þ Rew ¼ where P is the Powers roundness scale and CSF is the Corey (1949) shape factor given by Ds ffi; CSF ¼ pffiffiffiffiffiffiffiffiffi Di Dl

ρDU wn ; μ

ð10Þ

ð11Þ

Stokes' law of settling (Stokes, 1851) can be used to precisely calculate the fall velocity:

ð5Þ

Ds, Di and Dl being the short, intermediate and long orthogonal grain axial diameters, respectively. For smooth clasts, P = 6, whereas for natural grains, Dietrich proposed that P = 3.5 and the CSF = 0.7.

sffiffiffiffiffiffiffiffiffiffiffi 2 ρ 3 : μgργ

2

U wn ¼

Dn gργ : 18μ

ð12Þ

The upper limit of Stokes' law of settling is not clear: there is a gradual deviation of the observed settling velocity from the latter, beginning at a Reynolds number which most have considered to be 1, although

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J.P. Le Roux / Journal of Volcanology and Geothermal Research 289 (2014) 130–139

Eq. (12) at this value already overestimates the settling velocity by about 12% (Middleton and Southard, 1984). Le Roux (1992, 2005) considered this value to be around 0.1, but as the deviation is gradual, it is an open guess as to the Rew that should be used to switch between equations. For higher Reynolds numbers, fluid accelerations, flow separation and turbulence produced by the settling particles also come into play, so that theoretical predictions are no longer practical. For example, Newton's impact law of settling, where U wn ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ργ gD ; 3C d ρ

ð13Þ

is valid, but the dimensionless drag force Cd depends on Uwn so that it cannot be used directly to predict the settling velocity. Empirical plots (e.g. Middleton and Southard, 1984) show that Cd declines between Rew numbers of 10−2 to about 103 and remains more or less constant between 103–105.5, at which point it drops sharply. However, even in the range where it is supposed to remain constant, experimental results indicate enough variation to affect accurate prediction. For example, a sphere with a density of 7.9 g cm−3 and diameter of 3.8 cm has a recorded settling velocity of 258.6 cm s−1 in water at 20 °C (Table A in accompanying Excel spreadsheet). Eq. (13) in this case would calculate Cd as 0.51, quite different from the “constant” value of 0.44 for this range (Dellino et al., 2005). For non-spherical clasts, Cd deviates even more from its value for equivalent spheres, so that experimental work is required. A single settling equation for differently shaped, smooth clasts was proposed by Le Roux (2004):   z

D −1 ; U wp ¼ −U wn 0:572 1− s Dl

ð14Þ

where Uwp is the settling velocity of spherical as well as non-spherical clasts and z is an exponent that varies with the clast shape and settling Reynolds number. However, the value of Uwn in Eq. (14) had to be calculated using one of the series of equations proposed by Le Roux (1992). To combine these different equations for Uwn into a single equation, the sphere settling data of Gibbs et al. (1971) were re-examined here, while for very small spheres settling at Reynolds numbers less than 0.1, Stokes' law (Eq. (12)) was used to calculate the settling velocity. First, the sphere sizes and experimentally observed settling velocities were non-dimensionalized using Eqs. (9) and (10), respectively. Plotting these against each other for sphere settling data and examining a variety of different possible relationships showed that their log10 values define a 5th degree polynomial curve (Fig. 1a), which is given by: 5

4

3

log10 U dwn ¼ 0:0015ð log10 Ddn Þ þ 0:0155ð log10 Ddn Þ −0:086ð log10 Ddn Þ 2 −0:2288ð log10 Ddn Þ þ 1:9052ð log10 Ddn Þ−1:2456:

ð15Þ The obtained value of log10Udwn was then inserted into the following equation giving the settling velocity of differently shaped clasts: 1

0 U wp

B log10 U dwn C   z

C B10 C 0:52 1− Ds s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ −yB −1 ;  C B 2 Dl A @3 ρ

ð16Þ

μgργ

where y and z vary with the specific clast shapes. To define the latter, the hydrodynamic shape classification of Le Roux (2004) was used, with the addition of one new clast shape. In order to evaluate the best values for y and z in Eq. (16), different data sets were examined and the coefficient and exponent were adapted to find the minimum mean percentage error (MPE), lowest

Fig. 1. a) Plot of log10Ddn (dimensionless grain-size) against log10Udwn (dimensionless settling velocity) for spheres (data from Gibbs et al., 1971). b) Plot of log10Ddn against log10Udwn for irregular grains with unknown axial dimensions. Data from Zhu and Cheng (1993) and Ferguson and Church (2004).

absolute percentage error (MAPE), highest R2 value, and a 1:1 relationship between the predicted and observed or calculated (Stokes) settling velocities. The mean percentage error was determined by MPE ¼

∑ni¼1 ½100ðA−OÞ=O ; n

ð17Þ

where A is the predicted value and O the observed/calculated value. 4. Results 4.1. Settling velocity of individual clasts with known axial dimensions in liquids First, this study focused on the settling velocities in liquids of individual spheres, spheroids, oblate and prolate spheroids, ellipsoids, discoidal ellipsoids, discs, and rods. For the total of 35 spheres studied by Gibbs et al. (1971), y was found to be 0.99835, DDsl and z both being 1, whereas for spheroids, the settling velocities of 16 clasts measured by Williams (1966) and Stringham et al. (1969) yielded values of y = 0.9809 and z = 0.64. In the case of oblate spheroids, the experiments of Stringham et al. (1969) provided a data set of 36 clasts, to which was added one oblate spheroid from the data of Baba and Komar (1981a). Here, the best values of y and z are 0.8564 and 1.46, respectively. The data of Williams (1966), Stringham et al. (1969), Komar and Reimers (1978), Komar (1980), and Baba and Komar (1981a,b) were used for prolate spheroids, providing a set of 43 clasts with y = 0.9899 and z = 1.69. For ellipsoids, the experimental results of Komar and Reimers (1978) and Baba and Komar (1981a) were examined. These two data sets are useful in that the same basic shapes (ellipsoids) are represented, but in the case of the former authors, smooth basaltic pebbles were used, whereas the latter authors examined irregular beach glass fragments that varied from sharply angular shards to frosted, wellrounded grains. Comparing these two data sets, the relative effect of

J.P. Le Roux / Journal of Volcanology and Geothermal Research 289 (2014) 130–139

grain roundness can therefore be determined. Among the 55 pebbles used by Komar and Reimers (1978), all except 5 can be classified as ellipsoids, 1 of which is a prolate spheroid and 4 are here re-classified as discoidal ellipsoids. These pebbles were analyzed separately within their specific shape classes. The Baba and Komar (1981a) data set, on the other hand, contains 72 clasts of which 62 are ellipsoids, 1 is a prolate spheroid, 1 is an oblate spheroid, and 8 are discoidal ellipsoids. As before, these clasts were examined within their respective shape classes. The nominal diameters of the pebbles used by Komar and Reimers (1978) were calculated from their measured orthogonal axial dimensions by Dn ¼

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Ds Di Dl ;

ð18Þ

whereas in the case of Baba and Komar (1981a), the measured weight of the grains (M) was used:

Dn ¼

sffiffiffiffiffiffiffiffi 3 6M : πρs

ð19Þ

133

indicates that grains behave differently during group settling than during individual settling, the 3 coarsest sizes were eliminated from this data set. A second set of data employed here, that of Zhu and Cheng (1993), presents a similar problem in the sense that the authors did not state exactly how their sizes were measured. Their reported sizes of, for example, 0.0001, 0.0005 and 0.001 cm, do not represent sieve sizes traditionally used in western countries. On the other hand, sizes as small as 0.0001 cm suggest that the grains were not measured individually, implying that some sort of screening was employed. This is supported by the fact that the settling velocities predicted by Eq. (15) plot as a polynomial curve against the observed settling velocities and not in a straight line as would be expected for individual grain settling. However, it is not clear whether the releasing or retaining sieve sizes were reported and what the φ intervals were between sieves. Therefore, to make these data compatible with those of Ferguson and Church (2004), the reported sizes of Zhu and Cheng (1993) were incrementally changed by the same percentage until the resultant regression curve equation yielded the best correlation with the observed cloud settling velocities of Ferguson and Church (2004). This factor turned out to be a decrease of 14%, which is considered to be equivalent to the mean sieve diameter. As before, the grain sizes and settling velocities were here nondimensionalized and their log10 values plotted against each other (Fig. 1b), with a polynomial curve given by

Examining the two data sets, it was found that the same coefficient and exponent can be used for both smooth and irregular ellipsoids, namely y = 0.9911 and z = 4.1. This demonstrates that grain roundness plays a minor role in settling velocity and that clast size, density and shape are the determinant elements. Data extracted from Komar and Reimers (1978) and Baba and Komar (1981a) yielded 12 clasts that can be classified as discoidal ellipsoids, in which case y = 1.0598 and z = 2.94 give the best results. Williams (1966) and Stringham et al. (1969) provided the settling data of 23 discs, with y = 0.9486 and z = 0.52, whereas data for 51 clasts classified as rods were obtained from Stringham et al. (1969) and Komar (1980). Here, y = 1.1571 and z = 0.6.

Because the axial ratios of the individual grains are not known, only the left-hand bracket of Eq. (16) is necessary, where log10Udwn is in this case given by Eq. (20) and y = 1.0026.

4.2. Settling velocity of groups of particles with unknown axial dimensions

5. Comparison of results with previous equations

For natural grains with unknown axial dimensions where the sizes are derived from sieve analyses, a number of problems arise. First, it is not possible to know the DDsl ratios of individual particles, so that the right-hand bracket of Eq. (16) cannot be applied. As the average grain shape may also change with an increase in grain size, a “mean” DDsl ratio would not necessarily give good results. Second, the sizes can be reported using the releasing sieve, the retaining sieve, the mean size between the two sieves, or a nominal size calculated from the sieve size. Not all authors specify which of these were employed in their analyses. Third, because of the range of grain sizes between the releasing and retaining sieves, not all will settle at the same velocity, so that a decision has to be taken whether to measure the velocity of the grains that arrive first, last or in the middle of the settling cloud. Finally, many grains will collide during settling and will also be affected by turbulence generated by other grains, thus altering their individual settling behavior and reducing their fall velocity. This effect is clearly illustrated by the data of Ferguson and Church (2004), who performed experiments with river sand settling in water at 23–24 °C. Their 12 different grain size fractions were separated using quarter-phi sieve classes and the reported sizes represent the mean value between the releasing and retaining sieves. For this reason, Ferguson and Church (2004) timed the settling of the center of the cloud of grains within each sieve size range, performing about 50 experiments for each mean grain size. However, for the coarsest 3 grain sizes they measured the settling velocity of 47 individual, compact grains. A plot of their measured settling velocities against the velocities predicted by Eq. (15), shows a smooth polynomial curve for the settling cloud experiments and a straight line for the individual particles, with a displaced knick-point between the two data sets. As this clearly

5.1. Settling velocity of individual particles with known axial dimensions

5

4

3

log10 U wn ¼ 0:0195ð log10 Ddn Þ −0:0075ð log10 Ddn Þ −0:1679ð log10 Ddn Þ 2 −0:1936ð log10 Ddn Þ þ 1:9606ð log10 Ddn Þ−1:2582:

ð20Þ

Tables A–K, comparing the settling velocity predictions of Dietrich (1982), Cheng (1997) and Ferguson and Church (2004) for different clast shapes, are found in the accompanying Excel spreadsheet, “Settling Velocity Calculator”. Table 2 summarizes the statistical data for the different authors. In the case of settling spheres, the experimental data of Gibbs et al. (1971) were used for comparative purposes. In Fig. 2a, the settling velocities for spheres predicted by the different equations are compared with the settling velocities measured by Gibbs et al. (1971) and as theoretically calculated by Stokes (1851). In the case of Dietrich (1982), the values calculated with Stokes' law were left out for the statistics shown in Table 2, as they become increasingly discrepant with smaller clast

Table 1 Hydrodynamic classification of grain shapes based on axial ratios (Ds = short axis, Di = intermediate axis, Dl is long axis) and values of y and z to use for different shapes. Shape (smooth)

Ds/Di

Di/Dl

y

z

Sphere Spheroid Oblate spheroid Prolate spheroid Ellipsoid Discoidal ellipsoid Disc Rod Ellipsoidal rod Blade

1.0 0.9–1.0 0.35–0.9 0.9–1.0 0.35–0.9 0–0.35 0–0.35 0.9–1.0 0.35–0.9 0–0.35

1.0 0.9–1.0 0.9–1.0 0.35–0.9 0.35–0.9 0.35–0.9 0.9–1.0 0–0.35 0–0.35 0–0.35

0.99835 0.9809 0.8564 0.9899 0.9911 1.0598 0.9486 1.1571 1.387 2.5661

1 0.64 1.46 1.69 4.1 2.94 0.52 0.6 0.02 0.01

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J.P. Le Roux / Journal of Volcanology and Geothermal Research 289 (2014) 130–139

Table 2 Comparison of mean percentage error (MPE), maximum absolute error (MAPE) and correlation coefficient (R2) for different authors and clast shapes. Author(s)

Dietrich (1982)

Cheng (1997)

Ferguson and Church (2004)

This paper

Spheres MPE MAP R2

2.63 9.18 0.9996

−26.54 42.16 0.9992

8.68 24.05 0.9994

−0.06 6.67 0.9998

Spheroids MPE MAPE R2

2.14 8.65 0.9976

−26.84 36.68 0.9793

18.54 55.44 0.9754

0.89 7.53 0.9985

Oblate spheroids MPE 10.05 MAP 19.09 2 0.9947 R

4.86 22.85 0.9730

11.36 35.27 0.9479

4.2 18.13 0.9895

Prolate spheroids MPE 10.36 MAP 27.12 2 0.9945 R

−4.35 40.16 0.9268

2.22 40.84 0.9268

−1.37 19.19 0.9957

−3.34 24.46 0.9673

7.89 35.93 0.9632

23.07 54.91 0.9618

2.31 22.3 0.9698

Discoidal ellipsoids MPE −11.09 MAP 24.83 2 0.9455 R

20.96 44.68 0.9240

37.15 64.36 0.9144

1.75 18.78 0.9555

Discs MPE MAP R2

74.99 96.94 0.9858

90.49 116.33 0.9823

−3.97 20.72 0.9887

29.92 78.01 0.9730

44.2 102.62 0.9806

−6.7 19.9 0.9967 −0.44 37.21 0.7705

Ellipsoids MPE MAP R2

Rods MPE MAP R2

22.2 42.72 0.9806

Ellipsoidal rods MPE 20.35 MAP 75.59 2 0.7469 R

4.95 78.01 –

15.04 102.62 –

Blades MPE MAP R2

– – –

– – –

3.48 27.42 0.8859

−17.75 34.65 0.9951

−2.71 22.89 0.9876

0.37 20.97 0.9968

– – –

Natural grains MPE −7.72 MAP 61.4 2 0.9906 R

sizes, as shown in Table A. If these had to be taken into account the MPE and MAPE of Dietrich (1982) would be −6.51% and 53.33%, respectively. Table 2 shows that Eq. (16) has the highest R2 value and the lowest MPE and MAPE. It also has a 1:1 relationship between measured/calculated and predicted settling velocities. As concerns the settling velocity of spheroids (Table B), the data sets of Williams (1966) and Stringham et al. (1969) were compared with the different equations, here also using Eq. (3) of Dietrich (1982) as it gives much better results than Eq. (4) of the same author. Fig. 2b shows that Cheng (1997) underestimates the observed settling velocities, whereas Ferguson and Church (2004) overestimate the latter. Experiments conducted by Stringham et al. (1969) were used for oblate spheroids (Table C1). The nominal sphere diameter Dn was calculated from the given weight of the clasts, using Eq. (19). For this shape, Fentie et al. (2004) compared the settling velocity equation of Cheng (1997) with those of Sha (1956), Concharov (reported in Cheng, 1997), Dietrich (1982), Rubey-Watson (reported in Dingman, 1984),

Van Rijn (1989), and Zhang (1989), using the data sets of Raudkivi (1990) and Van Rijn (1997), finding that the best overall formula is that of Cheng (1997). In Fig. 2c, comparisons are made between the equations of Dietrich (1982), Cheng (1997), and Ferguson and Church (2004), together with Eq. (16). In this case, all four trend-lines plot very close together, with Eq. (16) and Cheng (1997) on top of each other, whereas both Dietrich (1982) and Ferguson and Church (2004) somewhat overestimate the observed settling velocities. Although Eq. (16) and Cheng (1997) both show a 1:1 relationship between the observed and predicted settling velocities, the results of the latter author are considerably more scattered than those of Eq. (16) as shown by their R2 values (Table 2). Fig. 2d compares the settling velocities observed for prolate spheroids (Table D), with data from Williams (1966), Stringham et al. (1969), Komar (1980), and Baba and Komar (1981a). Here Eq. (4) of Dietrich (1982) overestimates the observed settling velocity, whereas both Cheng (1997) and Ferguson and Church (2004) underestimate it. For smooth and irregular ellipsoids (Table E1) the different equations were compared using the data sets of Komar and Reimers (1978) and Baba and Komar (1981a). Fig. 2e shows that Dietrich (1982) underestimates the velocities, whereas both Cheng (1997) and Ferguson and Church (2004) overestimate them. Data for discoidal ellipsoids (Table F1) were extracted from the experimental results of Komar and Reimers (1978) and Baba and Komar (1981a). In Fig. 2f, it can be seen that Dietrich (1982) underestimates the observed settling velocities, whereas both Cheng (1997) and Ferguson and Church (2004) overestimate them. In Fig. 3a, the different equations are compared for discs (Table G) from the data sets of Williams (1966) and Stringham et al. (1969). In this case Dietrich (1982) can only resolve 1 of the 23 observed settling velocities (where it underestimates the value by 11.51%), because the

F to be positive. value of the CSF must be larger than 0.15 for 1− 1−CS 0:85 Therefore, only Eq. (16), Cheng (1997) and Ferguson and Church (2004) are shown. The last authors plot close together, but greatly overestimate the observed settling velocities. Finally, the data sets of Stringham et al. (1969) and Komar (1980) were used for the comparison of rods (Table H; Fig. 3b). Cheng (1997) and Dietrich (1982) here plot on top of each other, both overestimating the observed settling velocity, as do Ferguson and Church (2004). In summary, whereas Eq. (16) always has a 1:1 ratio between observed and predicted settling velocities, the other equations sometimes overestimate and at other times underestimate the observed velocities. Eq. (3) of Dietrich (1982), for example, performs well in the case of spheres and spheroids, (except in the Stokes range of settling), but his Eq. (4) overestimates the settling velocities of oblate and prolate spheroids as well as rods, while underestimating the velocity of ellipsoids and discoidal ellipsoids. It also cannot resolve discs which have a CSF of 0.15 or less. Cheng (1997), on the other hand, works well in the case of oblate and prolate spheroids and ellipsoids (somewhat overestimating the latter), but underestimates the velocity of spheres and spheroids while overestimating discoidal ellipsoids, discs and rods. Ferguson and Church (2004) overestimate the velocity of all shapes except prolate spheroids, where their equation gives good results. Of the 4 equations, Eq. (16) yields the highest R2 value in all cases except oblate spheroids, where Dietrich (1982) performs better. In addition, Eq. (16) yields the lowest MPE and MAPE for all shapes. Fig. 4 is a combined plot of the settling velocity in liquids of individual clasts of all shapes. R2 in this case is 0.9979. 5.2. Settling velocity of groups of particles with unknown axial dimensions For natural grains, Zhu and Cheng (1993) examined 56 sediment samples ranging in mean reported sizes between 0.0001 and 1 cm, which were here recalculated, reducing the values by 14% to get the equivalent sieve size, as explained in Section 4.2. Fig. 5 compares the experimentally observed settling velocities of these authors together with

J.P. Le Roux / Journal of Volcanology and Geothermal Research 289 (2014) 130–139

135

Fig. 2. Observed settling velocity of differently shaped particles plotted against settling velocity predicted by different equations. Black dots: Eq. (16); red squares: Dietrich (1982); green triangles: Cheng (1997); blue diamonds: Ferguson and Church (2004). a) Spheres (data from Gibbs et al., 1971). b) Spheroids (data from Williams, 1966; Stringham et al., 1969). c) Oblate spheroids (data from Stringham et al., 1969). d) Prolate spheroids (data from Williams, 1966; Stringham et al., 1969; Komar, 1980; Baba and Komar, 1981a). The trendline of Cheng (1997) coincides with that of Eq. (16). e) Smooth and irregular ellipsoids (data from Komar and Reimers, 1978; Baba and Komar, 1981a). f) Discoidal ellipsoids (data from Komar and Reimers 1978; Baba and Komar, 1981a). (For interpretation of the references to colors in this figure legend, the reader is referred to the web version of this article.)

those of Ferguson and Church (2004) with the fall velocities predicted by Eqs. (4), (6) (8), and (16) as adapted for natural grains. Ferguson and Church (2004) compared their results with the settling equations of Dietrich (1982), Cheng (1997), and Ahrens (2000), finding a significant improvement. Fig. 5 shows that Eq. (16) and Ferguson and Church (2004) plot close together, whereas Dietrich (1982) overestimates and Cheng (1997) underestimates the observed settling velocities. Eq. (16) has the highest R2 and lowest MAPE and MPE values of the compared equations. 5.3. Fall velocity of volcanic clasts in air As already mentioned in Section 3, the fact that water and air are both Newtonian fluids means that correctly dimensioned equations

developed from experiments in one medium are just as applicable to the other. This concept is widely applied in science and engineering, for example in scale modeling. Because the relationship between the dimensionless clast size and dimensionless settling velocity is given by Eq. (16), which also considers clast shape, it is only necessary to know the density, size and shape of a volcanic bomb, for example, together with the density and viscosity of the air or gas, in order to calculate its fall velocity. Wilson and Huang (1979) provided a set of measured fall velocities of differently shaped pumice, glass and feldspar clasts in air, for which a density of 0.00122 g cm−3 and dynamic viscosity of 0.00018 g cm−1 s−1 were assumed here. The three oblate spheroids in their data set overlap with the range of Reynolds numbers in the data set for liquids and coincide very well with the latter (Fig. 6a), with a maximum absolute error

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Fig. 3. Observed settling velocities of differently shaped grains plotted against settling velocities predicted by various equations. Black dots: Eq. (16); green triangles: Cheng (1997); blue diamonds: Ferguson and Church (2004). a) Discs (data from Williams, 1966; Stringham et al., 1969). b) Rods (data from Stringham et al., 1969; Komar, 1980). (For interpretation of the references to colors in this figure legend, the reader is referred to the web version of this article.)

of 15.54% predicted by Eq. (16). For the 57 clasts classified as ellipsoids, their observed velocities plotted against the velocities predicted by Eq. (16), fall on a linear extension of the liquid settling data, although the points are somewhat more scattered (Fig. 6b). However, considering the fact that densities were assumed to be constant for different size classes and the difficulty in accurately timing short, high-velocity events, this is to be expected. These additional data indicate that Eq. (16) can be used with confidence for ellipsoids up to Reynolds numbers exceeding 80. For the 95 clasts classified as discoidal ellipsoids, the data of Wilson and Huang (1979) also plot along the linear extension of the liquid settling data (Fig. 6c), although considerable scattering is manifested. Two additional clast shapes, for which no information could be obtained from liquids, are also represented in the Wilson and Huang (1979) data, namely ellipsoidal rods and blades (Table 1). For ellipsoidal rods, the fall velocity data of 9 pumice clasts (Table I; Fig. 7a) indicate that y = − 1.387 and z = 0.02, with R2 = 0.7705. Finally, although only 5 blades were identified in pumice and feldspar (Table J; Fig. 7b), the value of R2 is 0.8859, which suggests that the obtained values of −2.5661 for y and 0.01 for z can be used as a good approximation. Dellino et al. (2005) also formulated a model based on their experiments with pumice clasts and provided a graph (their Fig. 4) plotting the terminal fall velocity in air against the nominal diameter of differently shaped clasts with mean densities of 0.75 g cm− 3 and 2.0 g cm−3, respectively. Again, an air density of 0.00122 g cm−3 and a dynamic viscosity of 0.00018 g cm-1 s-1 are assumed here. Table 3 shows how this model compares with the equation of Dietrich (1982) for spheres and Eq. (16). Considering that the resolution of Fig. 4 of Dellino et al. (2005) is poor and the settling velocities therefore

Fig. 4. Settling velocities predicted by Eq. (16) for all particles and clasts with known axial dimensions and different shapes.

approximate, both Dietrich (1982) and Eq. (16) yield reasonable results for the range of clast sizes considered, with differences not exceeding 10.9% in the case of Eq. (16). Non-spherical clasts were not compared, as the shape factor used by Dellino et al. (2005) incorporates not only the three orthogonal clast axes, but also the maximum projection area, maximum projection perimeter, sphericity, and circularity, which cannot be compared with the classification in Table 1 of this paper without having the actual axis data. 6. Grain size determined from group settling velocity Many laboratories use settling tubes instead of sieves to determine sediment size distributions. Employing the data from Zhu and Cheng (1993) and Ferguson and Church (2004), the following equation can be used to calculate the equivalent mean sieve size: 5

4

log10 Ddv ¼ 0:0039ð log10 U dwv Þ þ 0:0413ð log10 U dwv Þ þ 0:152ð log10 U dwv Þ 2 þ0:2489ð log10 U dwv Þ þ 0:7021ð log10 U dwv Þ þ 0:7129;

3

ð21Þ where Udwv is calculated from Eq. (10). Subsequently, the equivalent sieve size Dv is derived from 10 log10 Ddv Dv ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  : 3

ρgργ

ð22Þ

μ2

The nominal grain size can be found by increasing the sieve size by 10% (Raudkivi, 1990; Ferguson and Church, 2004).

Fig. 5. Observed settling velocity of natural, mean sieve-size grains with unknown axial dimensions compared to settling velocities predicted by different equations. Black dots: Eq. (10); red squares: Dietrich (1982); green triangles: Cheng (1997); blue diamonds: Ferguson and Church (2004). Data from Zhu and Cheng (1993) and Ferguson and Church (2004).

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a

b

Fig. 7. Observed settling velocity of volcanic clasts in air plotted against settling velocity predicted by Eq. (16). a) Ellipsoidal rods. b) Blades. Data from Wilson and Huang (1979).

c

Fig. 6. Observed settling velocity of differently shaped particles settling in liquids and air plotted against settling velocity predicted by Eq. (16). Air settling data of volcanic clasts (pumice, glass, feldspar) from Wilson and Huang (1979). a) Oblate spheroids. b) Ellipsoids. c) Discoidal ellipsoids.

7. Conclusions Eq. (16) is presented here to predict the settling velocity of differently shaped and rounded clasts of which the axial dimensions are known, as well as irregular, natural grains with unknown axial dimensions, where log10 U wd ¼ a log10 Dd 5 þ b log10 Dd 4 þ c log10 Dd 3 þ d log10 Dd 2 þ e log10 Dd þ f . For smooth as well as irregular grains with known axial dimensions, the coefficients are: a = 0.0015, b = 0.0155, c = − 0.086, d = − 0.2288, e = 1.9052, f = − 1.2456. For different particle shapes, the values of y and z in Eq. (16) can be obtained from Table 1, which also shows the shape classification. In the case of irregular grains where the axial dimensions are unknown, a = 0.0195, b = − 0.0075, c = − 0.1679, d = − 0.1936, e = 1.9606, and f = −1.2582. Only the value of the left-hand bracket needs to be

determined, and once this has been calculated from Eq. (20), it should be multiplied by 1.0026 to obtain the mean sieve size (Uwv). An Excel spreadsheet (Setting Velocity Calculator) is provided to facilitate the calculations. This spreadsheet will also be available on the web page of the Andean Geothermal Centre of Excellence (http://www.cega.ing. uchile.cl/) and will be updated from time to time with new data if necessary. One of the advantages of using Eq. (16) is that the Reynolds number is not required, as for example in the equations of Le Roux (2004). Because Rew increases directly with Uw (Eq. (11)), this meant that a preliminary settling velocity had to be calculated first to determine the possible Reynolds number, which was then used to select the appropriate equation. This process had to be repeated to check whether the settling velocity given by the chosen equation still coincided with its correct range of Reynolds numbers. In Eq. (16), the values of y and z are linked to the grain shape, which directly affects Uw and consequently Rew. The effect of Rew is therefore integrated into the equation via the y and z values. Fig. 8 plots the ratio between the observed and predicted settling velocities of all the different clast shapes (including natural grains with unknown axial dimensions) against the Reynolds number. This indicates that Eq. (16) is valid for a wide range of Reynolds numbers (from less than 10−6 to more than 105) and that its accuracy is largely independent of the latter (i.e. a trend-line would pass vertically through the data points at an observed/predicted settling velocity ratio of about 1). The smaller horizontal dispersion at very low and very high Rew numbers is most likely due to the fact that fewer samples were analyzed under these extreme conditions, with a smaller chance of having anomalous values due to experimental errors, and not to a higher inherent accuracy. Acknowledgments This paper was written under the auspices of CONICYT–FONDAP Project 15090013, “Andean Geothermal Center of Excellence (CEGA)”.

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Table 3 Comparison of pumice settling velocities in air according to the empirical model of Dellino et al. (2005) and Dietrich (1982), and Eq. (16). Clast density ρs: 0.75 g cm−3

Clast density ρs: 2.00 g cm−3

Dn (cm)

Uw modeled by Dellino et al. (m s−1)

Uw calculated by Dietrich (m s−1)

Uw calculated by Eq. (16) (m s−1)

Dn (cm)

Uw modeled by Dellino et al. (m s−1)

Uw calculated by Dietrich (m s−1)

Uw calculated by Eq. (16) (m s−1)

0.1 0.5 1.0 2.0

4.00 10.00 15.00 21.80

3.47 11.16 15.69 20.90

3.62 11.09 15.20 20.17

0.1 0.5 1.0 2.0

6.40 16.20 24.10 36.00

6.44 18.34 25.02 32.94

6.69 17.99 24.10 32.20

Fig. 8. Ratio of observed settling velocity to predicted settling velocity plotted against settling Reynolds number. The fact that there is no relationship shows that the accuracy of Uw as predicted by Eq. (16) does not depend on Rew.

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