The transport of large boulders in a furrow from a mining area to a nearby pond was considered. The furrow is filled with a mixture of water and soil particles flowing down to the pond at a very high velocity. Due to operating constraints, the slope of the furrow is reduced progressively. A formula is derived, relating the slope of the furrow and the composition of the fluid to the maximum size and shape of the transported boulders. The characteristics of the boulders carried all

the way down to the pond may then be determined

Charpin, J.P.F.

de Hill, P.

Lombe, M.

Myers, T.G.

English

Mathematics in Industry Information Service Eprints Archive

TRANSPORTATION OF A WATER BASED
SLURRY IN AN OPEN FURROW, LAUNDER
OR STREAM
J.P.F. Charpin ∗, T.G. Myers †, M. Lombe ‡ and P. de Hill §.
Abstract
The transport of large boulders in a furrow from a mining area to a
nearby pond was considered. The furrow is filled with a mixture of
water and soil particles flowing down to the pond at a very high ve-
locity. Due to operating constraints, the slope of the furrow is reduced
progressively. A formula is derived, relating the slope of the furrow
and the composition of the fluid to the maximum size and shape of the
transported boulders. The characteristics of the boulders carried all
the way down to the pond may then be determined.
1 Introduction
A mineral processing plant produces vast amounts of waste material. This is
made up of rocks of all sizes, ranging from very fine pebbles, with a diameter
less than a millimetre, to rocks, with a diameter up to twenty centimetres. In
the set-up presented at the study group, the waste is washed away with high
pressure water and flows downhill to a pond three kilometres and approxi-
mately three hundred metres lower. As it flows the slurry carves a channel,
typically one metre by one metre. Since the mining area, where the material
∗Department of Mathematics and Applied Mathematics, University of Cape Town,
Private Bag, Rondebosch 7701, South Africa. e-mail: jcharpin@maths.uct.ac.za
†Department of Mathematics and Applied Mathematics, University of Cape Town,
Private Bag, Rondebosch 7701,University of Cape Town, South Africa. e-mail:
myers@maths.uct.ac.za
‡Department of Mathematics and Applied Mathematics, University of Cape Town,
Private Bag, Rondebosch 7701,University of Cape Town, South Africa. e-mail:
mlombe@maths.uct.ac.za
§Lynx Geosystems South Africa, P.O. Box 22022, Helderkruin 1733, South Africa.
e-mail: pierredh@lynxgeo.com
23
24 J.P.F. Charpin, T.G. Myers, M. Lombe and P.de Hill
is produced, is slowly moved away from the pond, the average angle of the
furrow to the horizontal decreases progressively. As the angle decreases the
possibility of larger rocks becoming stuck in the channel increases.
Every month, about one million tons of material is produced, most of
it consisting of particles with a diameter less than a millimetre. They are
washed away by water at the ratio 1000kg of water for 200kg to 400kg of
material. Considering the size of the channel created by the flow, the flux
of material leads to an average mixture velocity in the channel between
v = 1m·s−1 and v = 2m·s−1. In these conditions, the flow is clearly turbu-
lent, the Reynolds number is well above Re = 106.
Due to the high velocity, small particles should be carried away easily
by the flow. This study therefore focuses on the largest boulders, with a
diameter that may reach up to 20cm. In the following section, we determine
a criterion for boulders to be washed away. The results depend on the
Shields number, the ratio of shear stress and gravity, the two main forces
applied on the boulder and the shape of the boulder. Numerical results are
presented in Section 3.
2 Moving large boulders
g
θ v ~2 m/s
stress
Shear
Figure 1: Typical configuration for large boulders.
Transportation of a Water Based Slurry 25
Figure 1 shows a typical configuration for the transport of large boulders.
The channel is inclined at an angle θ to the horizontal and the boulder lies
at the bottom of it. The potential movement of the boulder is governed by
two main forces:
1. Shear stress is applied on the upwards facing surfaces and pushes the
boulder in the downstream direction.
2. Effective gravity, the difference between gravity and the Archimedes
force.
The ratio between these two forces is summarised in a non-dimensional
parameter called the Shields number:
S =
τ
∆ρgD
, (2.1)
where:
• τ is the shear stress applied by the flow on the boulder,
• ∆ρ = ρb − ρf is the difference between the density of the boulder, ρb,
and the density of the fluid, ρf ,
• g denotes gravity,
• D is the diameter of the boulder.
Critical values of the Shields number may be found in tables. Below the
critical value, gravity dominates and the boulder will remain motionless on
the ground. When the Shields number exceeds the critical value, shear stress
is the dominant force and movement is triggered. To calculate the minimum
angle to the horizontal required to wash away boulders, the shear stress in
Equation 2.1 will be expressed as a function of the flow characteristics,
geometry of the channel, slope, particle concentration, density and dynamic
viscosity, and the resulting expression will be combined with the critical
value of the shear stress.
2.1 Expression for the Shields number
The average velocity of the particle and water mixture is such that the flow
is highly turbulent. In this case, an appropriate expression for the shear
stress applied to the boulder, τ , is, see Patel (1999):
τ =
f
8
ρfu
2 , (2.2)
where
26 J.P.F. Charpin, T.G. Myers, M. Lombe and P.de Hill
• f is the friction factor. For the type of turbulent flow considered in this
study, according to Patel (1999), the friction factor may be expressed
as:
f =
0.292
Re1/4
where Re =
ρfuL
µf
(2.3)
and Re is the Reynolds Number, L is a typical length of the flow, ρf
and µf are the density and dynamic (or shear) viscosity of the mixture
respectively. According to experimental observations, L = 1m is ap-
propriate here. The density and dynamic viscosity vary significantly
with the concentration of soil particles in the fluid mixture. The den-
sity of the mixture increases with particle volumetric concentration, c:
ρf = ρw + c(ρb − ρw) , (2.4)
where ρw and ρb denote the density of pure water and soil respectively,
see Abraham et al (2001) and Guy et al (1990). The dynamic viscosity
may be approximated by:
µf = µw
(
1 +
0.75c
0.605 − c
)2
, (2.5)
where µw is the dynamic viscosity of pure water, see Coussot (1997).
Equation (2.5) is valid for very concentrated solutions, up to c = 0.45.
The average soil volumetric concentration in the fluid may be esti-
mated around c ≈ 0.1, therefore formula (2.5) is entirely appropriate
here.
• u is the velocity of the fluid. According to experimental observations,
this may be estimated around 2 m·s−1. The velocity may be related
to the slope of the channel using for example the Glaucker-Manning
formula, see Chanson (1999), valid for fully rough turbulent flows:
u =
1
n
R
2/3
H
√
sin θ , (2.6)
where
 n is the Manning coefficient. A typical value for earth surface rough-
ness is n = 0.025. For gravels, the value is n = 0.029. The value
for this parameter will therefore be taken as n = 0.027.
 RH denotes the hydraulic radius,
Transportation of a Water Based Slurry 27
 θ is the slope of the channel.
Combining Equations (2.2-2.6), the right hand side of Equation (2.1)
may be rewritten:
S =
0.292
8gD
(ρw + c(ρb − ρw))3/4
(ρb − ρw)(1− c)
(µw
L
)1/4√
1 +
0.75c
0.605 − cR
7/3
H
(sin θ)7/8
n7/4
.(2.7)
The slope of the channel may then be expressed as:
θ = arcsin
[
n2
R
4/3
H
(
8gDS
0.292
(ρb − ρw)(1− c)
)8/7
×
(
1
(ρw + c(ρb − ρw))3
L
µw
√
1 + 0.75c/(0.605 − c)
)2/7 .(2.8)
The Shields number, S, remains the only unknown quantity on the right
hand side of Equation (2.8): the critical value of this parameter is required
to determine when large boulders will move. It depends on the boulder, flow
and fluid characteristics. This will now be detailed.
2.2 Critical Shields number
The value of the critical Shields number corresponding to the present setting
may be found in Julien (1994):
Sc = 0.06 tan φs. (2.9)
This holds when the nondimensional particle parameter, d∗, satisfies:
d∗ = D 3
√
ρf (ρb − ρf )g
µ2f
> 50 .
The angle φs is called the angle of repose. This is the critical angle of the
channel floor at which a particle would move of its own accord, i.e. with
no fluid motion, see Julien (1994). The angle of repose is zero for a sphere
on its own and increases with the angularity of the boulder. Equation (2.9)
is valid if the bottom of the channel is entirely covered with large boulders
of diameter D. In the present situation the channel is likely composed of
smaller particles, in which case Equation (2.9) will over estimate the critical
Shields number.
28 J.P.F. Charpin, T.G. Myers, M. Lombe and P.de Hill
The minimum slope of the channel required to transport large boulders
may then be expressed as a function of the angle of repose and Equation
(2.8) becomes:
θ = arcsin
[
n2
R
4/3
H
(1.64gD tanφs(ρb − ρw)(1 − c))8/7
×
(
1
(ρw + c(ρb − ρw))3
L
µw
√
1 + 0.75c/(0.605 − c)
)2/7 .(2.10)
This shows that the angle of the slope, θ, required to move a large boulder
increases with the angle of repose, φ and the boulder diameter, D. This is
confirmed by the observations at the end of the channel (and common sense).
A straightforward analysis of the formula shows that the angle θ decreases
when the volumetric particle concentration in the flow increases. Note the
volumetric concentration is the only parameter that may be adjusted in
practice. A more quantitative analysis of the formula will now be performed.
3 Numerical results
Figure 2 shows the angle of the slope necessary to move boulders with a
diameter varying up to d = 20cm and angle of repose up to φ = 50o.
As could be expected, the required angle of the slope for small particles is
extremely low, whatever their angularity. When the diameter of the boulder
increases, the effect of the angle of repose appears more clearly: low values of
this parameter (corresponding to almost spherical boulders) do not require a
high channel inclination. The more angular the boulder gets, i.e. the larger
the angle of repose, the steeper the slope of the channel required to move it.
In the configuration presented at the study group, the channel is 3 kilo-
metres long and the decrease in altitude is 300metres. The corresponding
average angle of the slope is therefore θ0 = 5.7
o. In this configuration with
a volumetric concentration c ≈ 0.1, boulders with a diameter d = 20cm may
only be transported if their angle of repose is lower than φ ≈ 11o i.e. they
must be close to spherical but again, the critical value of the shear stress is
over estimated and real values could be significantly higher. If the diameter
is d = 10 cm, the allowed angle of repose increases to φ ≈ 18o. However, due
to the operating conditions, the volumetric concentration of particles in the
fluid may vary significantly: the waste is piled up at the top of the hill and a
Transportation of a Water Based Slurry 29
0
2
4
6
8
10
0 0.05 0.1 0.15 0.2
An
gl
e 
of
 th
e 
slo
pe
 (d
eg
ree
s)
Boulder diameter (m)
φ=5
φ=10o
φ=15o
θ=5.7o
o
o
oφ=40 φ=30o φ=25o φ=20oφ=50
Figure 2: Slope angle, θ, required to move a boulder as a function of its
diameter and angle of repose, φ, in a slurry with the particle volumetric
concentration c = 0.1.
much larger quantity of material may suddenly be washed away, triggering
movement for more angular boulders. For the channel slope θ0 = 5.7
oand
volumetric concentration c = 0.3, a 20 centimetre diameter boulder may
have an angle of repose up to φ ≈ 16o. For a 10 centimetre diameter boul-
der, the angle increases to φ ≈ 26o and the boulder will still be washed
away. These values may be read from Figure 3. Once movement has been
triggered, the boulder’s inertia allows it to keep on rolling down the hill,
even if the starting conditions are no longer satisfied. The boulder may
then either roll down to the pond or settle at the bottom of the channel for
a while, until movement conditions are once again met, due to the varying
volumetric concentration.
Practically, the model states that boulders with low angularity may reach
the pond. This agrees with the experimental observations: the largest boul-
ders reaching the pond are all rather round.
30 J.P.F. Charpin, T.G. Myers, M. Lombe and P.de Hill
0
2
4
6
8
10
0 0.05 0.1 0.15 0.2
An
gl
e 
of
 th
e 
slo
pe
 (d
eg
ree
s)
Boulder diameter (m)
o
φ=5o
φ=10o
φ=15o
φ=20
θ=5.7
φ=25oφ=30ooφ=50 oφ=40
o
Figure 3: Slope angle, θ, required to move a boulder as a function of its
diameter and angle of repose, φ, in a slurry with the particle volumetric
concentration c = 0.3.
4 Conclusion
Turbulent flow in an open channel was studied to derive a formula relating
the slope of a channel to the characteristics of the boulders the flow may wash
away. This study is based on the critical value of the Shields number, which
determines when the boulders to be washed away will start rolling down the
channel. The study did not focus on the later course of the boulders: the
condition derived here only indicates when movement is triggered. Inertia
then allows boulders to continue moving even when the criterion is not
verified. A geometrical study of the channel shape could complete the study
and ensure that the boulders roll as far as possible
Acknowledgements
The contributions of A. Fitt, University of Southampton, UK, N. Fowkes,
University of Western Australia, Perth, Australia, K. Pereira and R. Sar-
racino, both from the University of Witwatersrand, Johannesburg, South
Transportation of a Water Based Slurry 31
Africa, are here kindly acknowledged. All contributors would like to thank
Mr. P. de Hill from Lynx Geosystems for introducing the problem and as-
sisting in answering questions. T.G. Myers acknowledges support of this
work under the National Research Foundation of South Africa grant num-
ber 2053289. J.P.F. Charpin acknowledges the support of the Claude Leon
Harris Foundation.
References
Abraham, A. D., Li, G., Krishnan, C., and Atkinson, J. F. (2001). A
sediment transport equation for interrill overland flow on rough surfaces,
Earth Surface Processes and Landforms, 26, 1443-1459.
Chanson, H. (1999). The Hydraulics of Open Channel Flow, Chapter 4,
John Wiley & Sons, New York.
Coussot, P. (1997). Mudflow Rheology and Dynamics, Chapter 3, Balkema,
Rotterdam.
Guy, B. T., Dickinson, W. T., Rudra, R. P. and Wall, G. J. (1990). Hy-
draulics of sediment-laden sheetflow and the influence of simulated rainfall,
Earth Surface Processes and Landforms, 15, 101-118.
Julien, P. Y. (1994). Erosion and Sedimentation, Chapter 7, Cambridge
University Press, Cambridge.
Patel, B. R. (1999). Internal Flows. In Fundamentals of Fluid Mechanics,
Chapter 5.2, J. A. Schetz and A. E. Fuhs, Ed., John Wiley & Sons, New
York.


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