Note that the settHng velocity is negative (i.e. downwards)
Spherical particle
For a spherical particle, the cross-sectional area and volume of the particle are, respectively:
4 = f<'
s ^ s
And the settling velocity becomes:
14g^s I Ps _ 11 Spherical particle Remarks
Note that, on Earth, the buoyant force is proportional to the liquid density p and to the gravity accel- eration g. The buoyancy is larger in denser liquids: e.g. a swimmer floats better in the water of the Dead Sea than in fresh water. In gravitationless water (e.g. waterfall) the buoyant force is zero.
Comments
1. Drag coefficient of spherical particles
Considering the flow around a sphere, the flow is everywhere laminar at very small Reynolds numbers (Re < 1). The flow of a viscous incompressible fluid around a sphere was solved by Stokes (1845, 1851). His results imply that the drag coefficient equals:
CA = — Laminar flow around a spherical particle (Stokes' law) 24 Re
which is defined in terms of the particle size and settling velocity: Re = pw^dJix. Stokes' law is valid fori?^< 0.1.
In practice, the re-analysis of a large number of experimental data with spherical particles that were unaffected by sidewall effects yielded (Brown and Lawler, 2003):
Q = 24 ^ ^^^^ ^^0.681) + 0^407 Re<2X 10^
^ Re^ l + (8710/i?e)
At large Reynolds numbers {Re > 1000), the flow aroimd the spherical particle is turbulent and the drag coefficient is nearly constant:
Q « 0.5 Turbulent flow around a spherical particle (1 X 10^ < i?^ < 1 X 10^) 2. Drag coefficient of natural sediment particles
For natural sand and gravel particles, experimental values of drag coefficient were measured by Engelund and Hansen (1967) (Table 7.4). Their data are best fitted by:
Q = — + 1.5 24
^ Re
based on experimental values obtained for sand and gravel (Re < 1 X 10"^) Another empirical correlation was recently proposed (Cheng, 1997):
r. x2/3 \"^
Q = 7A_ + 1 Re)
based on experimental values for natural sediment particles {Re < 1 X 10"^)
7.3 Particle fall velocity 163 3. George Gabriel Stokes (1819-1903) was a British physicist and mathematician, known for his
studies of the behaviour of viscous fluids (Stokes, 1845, 1851).
4. Terminal settling velocity of spherical particles
Gibbs et al. (1911) provided an empirical formula for the settling velocity of spherical particle (50 |ULm < ds< 5 mm). Their results can be expressed in SI units as:
w=\0 -30v + ^900;^^ + gd^js - 1)(0.003869 + 2.480^, )
0.011607+ 7.4405J, Spherical particles where v is the kinematic viscosity of the fluid (i.e. v = ix/p).
Brown and Lawler (2003) proposed an empirical formula for spherical particles obtained with experiments unaffected by sidewall effects:
W « 3
l ^ M s - l )
dl+
W_3
0.317 d.
0.449 .114
Spherical particles {Re<2X 10^)
dtill.S + J*^""^) Spherical particles ' ^ gpL{s - 1) 0.0258J.'-''' + 2.81J*'-''' + 18Jf'4' + 405 {Re < A X lO') where d* is the dimensionless particle number defined as:
d* = dA (s-l)g 5. Terminal settling velocity of sand particles
For naturally worn quartz sands (0.063-1 mm sizes) settling in water, Jimenez and Madsen (2003) proposed a simple formula:
1
V(S - l)g^s 0.954 + 5.12 Worn quartz particles (0.063 < d^<\ mm) {dJAv)4{s - \)gd.
where d^ is the nominal diameter or equivalent volume sphere diameter.
6. Note that the temperature affects the fall velocity as the fluid viscosity changes with temperature.
Discussion: settling velocity of sediment particles
Observed values of terminal settling velocity are reported in Table 7.4. First, note that large-size particles fall faster than small particles. For example, the terminal fall velocity of a 0.01 mm particle (i.e. silt) is about 0.004 cm/s while a 10 mm gravel settle at about 34cm/s. Practically, fine particles (e.g. clay and silt) settle in a laminar flow motion {w^djv < 1) while large particles (e.g. gravel and boulder) fall in a turbulent flow motion {w^djv > 1000).
Further, at the limits, the fall velocity and the particle size satisfy:
W^ oc d^ Laminar flow motion {w^djv < 1)
^o "^ V^s Turbulent flow motion (w^djv > 1000)
(7.11a) (7.11b)
Application
The size distribution of a sandy mixture is recorded using a settling tube in which the settling time in water over a known settling distance is measured. The results are:
Settling rate (cm/s) Mass (g)
Settling rate (cm/s) Mass (g)
32 0.4 2.29 44.80
16 1.20 2 48.04
10.7 4.31
1.78 49.07
8 7.00 1.6 49.60
5.33 15.98 1.45 49.81
4 31.60
1.3 49.87
3.2 38.12
1.23 49.95
2.7 42.59
1.14 50.00
Deduce the median grain size and the sorting coefficient S — ^d^^ld^^ , Solution
First, we deduce the median settling velocity (^0)50 and the settling velocities (Wo)io and (>Vo)9o for which 10% and 90% by weight of the material settle faster, respectively. The equivalent sedimenta- tion diameters can then be determined.
For the measured data, it yields {WQ)^Q = 4.56 cm/s, (Wo)io = 10.0 cm/s and (^0)90 = 2.27 cm/s.
The equivalent sedimentation diameter may be deduced from equation (7.7):
4 g ( s - l ) ^ in which the drag coefficient may be estimated using equation (7.9):
Q = ^-— + 1.5 For sand and gravels 24a pkol^s
For a known settling velocity, the equivalent sedimentation diameter may be derived from the above equations by iterative calculations or using Table 7.3.
The equivalent sedimentation diameters for the test are d^^ = 0.31 mm, dio = 0.18 mm, dg^ = 0.83 mm and the sorting coefficient isS = 2.15.
Remarks
Fine particles settle more slowly than heavy particles. As a result, the grain sizes diQ and dg^ are deduced, respectively, from the settling velocities (>Vo)9o and (WJIQ for which 90% and 10% by weight of the material settle faster, respectively.
7.3.3 Effect of sediment concentration
The settling velocity of a single particle is modified by the presence of surrounding particles.
Experiments have shown that thick homogeneous suspensions have a slower fall velocity than that of a single particle. Furthermore, the fall velocity of the suspension decreases with increas- ing volumetric sediment concentration. This effect, called hindered settling, results from the interaction between the downward fluid motion induced by each particle on the surrounding fluid and the return flow (i.e. upward fluid motion) following the passage of a particle. As a particle settles down, a volume of fluid equal to the particle volume is displaced upwards. In thick sedi- ment suspension, the drag on each particle tends to oppose to the upward fluid displacement.
7.4 Angle of repose 165
Notes
1. The fall velocity of a suspension w^ may be estimated as:
W 3 = ( 1 - 2 . 1 5 C 3 ) ( 1 • 0.75C,^-^^K Fall velocity of a fluid-sediment suspension
where WQ is the terminal velocity of a single particle and C^ is the volumetric sediment concen- tration. Van Rijn (1993) recommended this formula, derived from experimental work for sedi- ment concentration up to 35%.
It must be noted that a very dense cloud of particles settling in clear water may fall faster than an individual particle. A very dense cloud tends to behave as a large particle rather than as a suspension.
7.3.4 Effect of turbulence on the settling velocity
In turbulent flov^s, several researchers discussed the effects of turbulence upon the sediment settling velocity. Graf (1971) presented a comprehensive reviev^ of the effects of turbulence on suspended- solid particles. Nielsen (1993) suggested that the fall velocity of sediment particles increases or decreases depending upon turbulence intensity, the particle density, and the characteristic length scale and time scale of the turbulence. Although the subject is not yet fully understood, it is agreed that turbulence may drastically affect the particle settling motion.
7.4 ANGLE OF REPOSE
Considering the stability of a single particle in a horizontal plane, the threshold condition (for motion) is achieved w^hen the centre of gravity of the particle is vertically above the point of contact. The critical angle at v^hich motion occurs is called the angle of repose cf)^.
The angle of repose is a function of the particle shape and, on a flat surface, it increases v^ith angularity. Typical examples are show^n in Fig. 7.5. For sediment particles, the angle of repose ranges usually from 26° to 42°. For sands, cfy^ is typically between 26° and 34°. Van Rijn (1993) recommended to use more conservative values (i.e. larger values) for the design of stable chan- nels (Table 7.5).
Note
For a two-dimensional polygon, the angle of repose equals 180° divided by the number of sides of the polygon. For example: (^s = 60° for an equilateral triangle and </>s = 0 for a circle.
Cylinder
VI
Square
^s=45°
Angle of repose
Triangle Four spheres
</>s= 19.46°
Five spheres (/)s= 35.26°
Fig. 7.5 Examples of angle of repose.
Table 7.5 Angle of repose for stable channel design
d^ (mm) (j)^: rounded particles (degrees) (f)^: angular particles (degrees) Comments
(1) (2) (3) (4)
<1 5 10 50
>100 30 32 35 37 40
35 37 40 42 45
Gravel Gravel Gravel
Cobble and boulder Note: Silicate material, see van Rijn (1993).
7.5 LABORATORY MEASUREMENTS 7.5.1 Particle size distribution
In laboratory, particle size distributions may be determined by direct measurements or indirect methods. Direct measurements include immersion and displacement volume measurements, direct measurements of particle diameter, semi-direct measurements of particle sizes using sieves.
Indirect methods of particle size measurements relate fall velocity measurements to particle size. Most common methods are the visual accumulation tube (VAT), the bottom withdrawal tube (BWT) and the pipette. The former (VAT) is used only for sands. The last two methods are used only for silts and clays.
7.5.2 Concentration of suspended sediments
Suspended sediment concentrations may be measured from 'representative' samples of the sediment-laden flow. The sampling techniques may be instantaneous sampling, point sampling or depth-integrated sampling. Graf (1971) and Julien (1995) reviewed various techniques.
7.6 EXERCISES Bed forms
What is the difference between ripples and dunes? Can dune formation occur with wind-blown sands?
In what direction do antidunes migrate? Can antidunes be observed with wind-blown sands?
In a natural stream, can dunes form in supercritical flows?
Considering a natural stream, the water discharge is 6.4m^/s and the observed flow depth is 4.2 m. What is the most likely t5^e of bed form with a movable bed?
In a natural stream, the flow velocity is 4.1 m/s and the observed flow depth is 0.8 m. What is the most likely type of bed form with a movable bed? In what direction will the bed forms migrate (i.e. upstream or downstream)?
Considering a 2.3 m wide creek, the water discharge is 1.5 m^/s and the observed flow depth is 0.35 m. What is the most likely type of bed form with a movable bed? In what direction will the bed forms migrate (i.e. upstream or downstream)?
7.6 Exercises 167
Sediment properties
The dry density of a sand mixture is 1655 kg/m^. Calculate (and give units) (a) the sand mixture porosity and (b) the wet density of the mixture.
The characteristic grain sizes of a sediment mixture are diQ = OAmm, ^50 = 0.55 mm, dgo = 1.1 mm. Indicate the type of sediment material. Calculate (a) the sedimentological size parameters </> corresponding to diQ, d^o and dgo, (b) the sorting coefficient and (c) the dry sedi- ment mixture density.
Settling velocity
Considering a spherical particle (density p^) settling in still water, list all the forces acting on the particle at equilibrium. Write the motion equation for the settling particle at equilibrium.
Deduce the analytical expression of the settling velocity.
For a 0.03 mm diameter sphere, calculate the settling velocity in still water at 20°C. (a) Use Fig. 7.4 to calculate the drag coeflBcient. (b) Compare your result with the correlation of Brown and Lawler (2003).
Solution: (a) WQ = 0.80mm/s (Fig. 7.4) and (b) WQ = 0.86mm/s (Brown and Lawler's correlation).
Considering a 0.9 mm sphere of density Ps = 1800 kg/m^ settling in water, calculate the settling velocity. Use Fig. 7.4 to calculate the drag coefficient.
Solution: w^ = 0.041 m/s.
Considering a 1.1 mm sediment particle settling in water at 20°C, calculate the settling velocity using (a) the formula of Gibbs et al. (1971) and (b) the semi-analytical expression derived by Chanson (1999):
-30v + J900v^ + gd^(s - 1)(0.003869 + 2.4S0dJ
w^ = 10 ^ ^^^ '- Gibbs etal. (1971) 0.011607 + 7.4405^3
^o = - |-7 ^^ sr(s - 1) Chanson (1999)
' 24)Li ^ '3 I I + 1 - ^
Calculate the fall velocity of a 0.95 mm quartz particle using the correlation of Jimenez and Madsen(2003).
Solution: w^ = 0.1 m/s.
The size distribution of a sandy mixture is recorded using a settling tube in which the settling time in water over a known settling distance is measured. The results are:
Settling rate (cm/s) Mass (g)
Settling rate (cm/s) Mass (g)
35 1.22 2.39 89.80
12.0 2.50 2.1 93.01
10.0 9.20 1.92 96.03
8.0 15.10
1.8 96.90
5.0 25.98
1.65 98.01
4.0 55.60
1.5 99.28
3.0 72.81
1.43 99.90
2.8 81.09
1.34 100.00