FUNDAMENTALS ON EQUILIBRIUM CONCENTRATION CURVES...
37
FUNDAMENTALS ON EQUILIBRIUM
CONCENTRATION CURVES
FOR SEDIMENTATION AND
DIFFUSION OF PARTICLES IN FLUIDS
Harry Edmar Schulz
José Eduardo Alamy Filho
Swami Marcondes Villela
Department of Hydraulics and Sanitary Engineering,
School of Engineering at São Carlos, University of São Paulo, Brazil
(Convênio FINEP 23.01.0606.00/Laboratório de Hidráulica Ambiental-CRHEA/SHS/EESC/USP)
Abstract
Solutions for the equilibrium sediment concentration profiles, defined as the situation in which the settling movement
of sediment particles is compensated by turbulent diffusion in channel flows, are presented and compared with the
classical Rouse profile. In the Rouse solution, the settling velocity is considered independent of the sediment concentration,
a very restrictive situation which leads to unrealistic results for the sediment concentration near the bottom of the flow.
In this paper, the use of the mass conservation principle permits to obtain a more realistic solution for the equilibrium
profiles. It is shown that it is possible to extend the obtained profiles until the bottom of the flow (which simplifies
further calculations). Further, considering the result of the momentum equation (potential flow) around a sphere, a
nonlinear approximation between the settling velocity and the volumetric concentration of sediment is obtained. It is
discussed that nonlinear effects must be considered to obtain more realistic equilibrium profiles, exposing a situation
very different to that of the Rouse profiles. The new profiles do not present the mentioned physical limitations.
Key words: sediment concentration profiles, settling velocity, equilibrium condition.
Introduction: The Traditional Low
Concentration Case
In the literature, the 2D steady-state turbulent advectiondiffusion equation is used to estimate the distribution of
suspended sediment along fluid flows. The equation is
given by:
u

∂C
∂C ∂ 
∂C
+w
=
− c 'u '  +
 Dx
∂x
∂z ∂x 
∂x

 ∂  ∂C
 ∂
− c ' w ' +
( w sC)
+
 Dz
 ∂z  ∂z
 ∂z
(1)
In equation (1) x and z are the longitudinal (horizontal)
and vertical coordinates respectively [L]; u , w are
longitudinal and vertical mean velocities [LT–1]; u ′, w ′ are
the longitudinal and vertical velocity fluctuations [LT–1]; C
is the mean suspended sediment concentration [ML–3]; c ′ is
the fluctuation of suspended sediment concentration [ML–
3
]; c ′u ′, c ′w′ are statistical correlations between
concentration and velocity fluctuations [ML–2T–1]; Di is
the molecular diffusion coefficient for the i direction (i =
x,z) [L2T–1]; and ws is the sediment settling velocity [LT–1].
Equilibrium situations in channel flows imply that the
concentration profile does not vary along the flow direction
(longitudinal derivatives vanish, or ∂/∂x = 0). In such
flows, mean vertical velocities can be neglected in
comparison to the mean longitudinal velocity ( w = 0 ).
This situation, a particular case for channel flows, is adopted
as the initial condition for numerical simulations of complex
flows. Zedler and Street (2001), for example, used the
Rouse equilibrium profile to initialize an LES simulation
for alluvial channels.
The products c ′u ′, c ′w′ are usually quantified
through the eddy diffusivity concept and assuming
proportionality between momentum and mass transport,
leading to equation (2)
−c ' w ' = D tz
∂ C  νt  ∂ C
= 
∂ z  σc  ∂ z
(2)
Minerva, 1(1): 37-43
38
SCHULZ, ALAMY FILHO & VILLELA
Dtz is the eddy diffusivity [L2T–1];ν t is the eddy
viscosity [L2T–1] and σ c is an empirical proportionality
constant (usually σ c = 1 to 0.74). It is known that Dtz >>
Dz far from the channel bottom, so that Dz is neglected.
For equilibrium situations, equations (1) and (2) permit
to obtain the simplified equation (3):
 νt  ∂ C
+ w sC = 0
 
 σc  ∂ z
(3)
For such simple flows, the eddy viscosity follows
the parabolic distribution derived from the logarithmic
velocity profile, as shown in equation (4):
 z
ν t = κ ⋅ u * ⋅z ⋅  1 − 
 h
(4)
where κ is the von Kárman constant (0.4 for clear water);
u* is the shear velocity [LT–1] and h is the flow depth [L].
Equations (3) and (4) lead to equation (5):
∂φ
∂Z
=−
 σ 
Z (1 − Z )
ws ⋅ φ ⋅  c 
 κ⋅u*
(5)
Exploring the High Concentration Case
where φ is the nondimensional volumetric sediment
concentration, c /ρs, which varies from 0 to 1; Z is the
nondimensional distance to the bottom, z/h, also varying
from 0 to 1; and ρs is the sediment density [ML–3]. Integrating
this expression for constant ws lead to the Rouse equilibrium
profile, widely diffused into literature, given by equation
(6):
φ  a   Z − 1  
= 
⋅

φa  a − 1   Z  
L w0
Using the mass conservation law for settling particles
In real cases, the settling velocity decreases while the
particle approaches the bed, where higher particle concentrations
are reached. Considering only vertical displacements, the
integral mass conservation law for settling particles leads to
a linear dependence between the settling velocity and the
volumetric concentration. To obtain this equation, the schemes
of Figures 1a and 1b may be followed.
From the integral mass conservation equation we
have:
(6)
where a is a reference level above the bed; φa is the reference
concentration at Z = a; L is given by L = ∂c / (κ.u*) and wo
is the constant settling velocity [LT–1], usually given by
the Stokes solution for a settling sphere. Equation (6) shows
that the evaluation of the volumetric concentration requires
the reference value at some point (a) along the water depth.
This point is generally assumed near the bottom, being a =
0.05, for example, a value adopted in the literature. The
Rouse solution is a classical equation used in the study of
suspended sediment in channel flows. Julien (1995) affirms
that, despite of its shortcomings and simplifications, the
Rouse expression conducts to qualitatively good results,
Minerva, 1(1): 37-43
being considered accurate mainly for lower concentrations.
Indeed, the hypothesis of constant settling velocities may
be considered valid only for very low concentrations, where
interactions among particles are ignored. According to
Einstein & Chien (1955), Rouse equation predicts well
concentration values for φ < 0.04, but deviates progressively
for higher concentrations. In real flows, however, much
higher concentrations are present, so that particle interactions
cannot be neglected. Thus, the constant settling velocity
is a very limiting hypothesis. An interesting consequence
of this assumption is the prediction of infinite concentration
values at Z → 0, a gross deviation from the daily observations.
To overcome this problem, the Rouse expression is used
only above the reference level. However, it also may show
deviations close to the surface, where concentrations close
to null are usually predicted. In this case, experimental
results for fine particles studies show that turbulent mixing
brings sediment closer to the water surface, resulting in
somewhat higher concentration values in these regions.
The purpose of this paper is to furnish better theoretical
approaches for the equilibrium concentration profile, without
some of the limitations observed in the Rouse profile.
Physical considerations are followed to obtain better
evaluations of the settling velocity.
aV1 = AV2 or φ V1 = (1 − φ)V2
(7)
It is known (see Schulz, 1990, for example) that
a/A = φ, that is, the ratio between the areas defined in
figure 1a equals the volumetric concentration. The relative
velocity (wo) between a single spherical particle and the
fluid is given by the Stokes law. In the present case, the
following relation is straightforward:
w 0 = V1 + V2
(8)
From equations (7) and (8) it follows that:
w s = w 0 (1 − φ ) (because V1 = ws)
(9)
FUNDAMENTALS ON EQUILIBRIUM CONCENTRATION CURVES...
Using equation (9) into equation (5) and integrating
the result leads to:
φ
=
φa
 a   Z − 1  
 a − 1  ⋅  Z  
 


L ⋅w o
 a   Z − 1  
1 − φa + φa ⋅ 
⋅

 a − 1   Z  
L⋅ w o
(10)
Equation (10) is still very elegant. It may be said
that the Rouse equation (equation 6) is a simplification
of this more general result. A difference between both
predictions is the fact that the normalized profile φ/φa of
equation (10) depends on φa, while equation (6) furnishes
only one profile for any φa. However, both predictions
become closer for φa?0. Concentration profiles for different
values of φa and Lwo are presented in Figures 2a to 2d,
using equations (6) and (10). In all cases, the reference
level is a = 0.05. At the reference level (Z = a), equations
(6) and (10) are adjusted to reproduce the reference
concentration φa.. For points above the reference level,
the concentration values calculated by equation (10) are
higher than that provided by the Rouse equation. This
indicates that the particle interaction effects propagate
along all the flow depth. On the other hand, for points
located under the reference level, equation (10) conducts
to adequate finite values (φ ≤ 1.0), while equation (6)
leads to unreal values tending to infinite for Z 
→ 0. So,
the profiles generated using equation (9) (the mass
conservation principle for the particle movement) denotes
that particle interactions decisively affect the sediment
distribution along all the fluid depth and that this equation
must be used to attain more realistic results. Figures 3 a
and b illustrate the concentration profiles behavior given
Figure 1
39
by equations (6) and (10) for different values of the product
L.wo and φa = 0.8. It can be seen that for higher L.wo the
concentration values are lower along the fluid depth (above
the reference level a). In other words, the effect of the
bottom propagates more intensively for lower L.wo values.
This result is qualitatively coherent with the influence of
the physical phenomena which take part on particles
sedimentation. Lower L.wo values imply in:
a) lower L values, related to higher u* values and,
consequently, higher ressuspension of particles (or
turbulent diffusion), or
b) lower wo values, which imply in slow settling and,
consequently, more time with the particles in the bulk
liquid.
An approximation using the momentum conservation
equation (movement equation) for relative vertical motion
between settling particles
Although equation (9) incorporates the mass
conservation principle for particle settling, it does not consider
any relative motion among particles. In other words, all
particles maintain the mean distance between each other
during the downward movement. The information that,
for example, one particle near the bottom can be at rest
while others above it move trough the fluid was not
considered. To verify how this condition can modify the
settling velocity and the equilibrium concentration profile,
some approximations are presented here. The sketch of
Figure 4a is first considered, which permits to define φ.
In this figure, Ro is the radium of a spherical particle and
z is the radial distance with origin at the centre of the
sphere. It is imposed that z corresponds to the equivalent
radium of the volumetric region around the sphere which
guarantees equation (11).
(a) Ratio between the horizontal area occupied by particles and the total area of the flow;
(b) sedimentation (V1) and upward flow (V2).
Minerva, 1(1): 37-43
40
SCHULZ, ALAMY FILHO & VILLELA
4
3
π R 03
 R0 
3
φ=
=

4
3
 z 
πz
3
(11)
Equation (11) may be understood as a definition of
the volumetric density φ or as a definition of the distance
z. Figure 4b shows a particle in rest while fluid and neighbor
particles move in relation to it, mainly in the vertical direction.
At any distance above the particle, the flow is retarded
(the flow lines diverge). To obtain a first evaluation of
this retarding effect on the flow and, consequently, on the
neighbor particles, the solution of the potential motion
about a sphere is used here (as described by Prandtl &
Tietjens, 1934). Considering that a second particle at the
position α z (Figure 4b) follows exactly the fluid movement
(an ideal situation), the velocity of this second particle is
given by equation (12).
density (equations 14 and 15) than the linear equation
presented before. The Stokes solution is still valid for
low φ; while for high φ the zero velocity is attained “faster”
than predicted by the linear model (equation 9). The present
result is mainly theoretical, but many empirical studies in
the literature show the settling velocity as a nonlinear function
of the sediment concentration. Among them are, for example,
Fischerström (1967), Merkel (1971a, b), Larsen (1977),
Rölle (1990), Takacs et al. (1990), Krebs (1991), Zhou
& McCorquodale (1992) and Baldock et al. (2004). Figure
5a shows a comparison among the different equations
mentioned in this paper.
Using equation (15) into equation (5) and integrating
the result leads to:
1
Z=
1
1  1
1 
 1 − a   φ   1 − φa   L⋅w o L⋅w o  1−φ −1−φa 
1+ 
e

 ⋅ 
 ⋅
 a    1 − φ   φa  
(16)
 R 
w s = w P 1 −  0 
  α z 
3

 or

3

 R0  
w s = w P 1 − β 
 

 z  
(12)
α and β = 1/α3 are constants, related to the distance between
two particles which guarantees equation (11). wP is the
velocity “at infinitum”, that is, the velocity without the
retarding effect. In this study, wp follows from equation
(9), in the form:
w P = w o (1 − φ )
(13)
From equations (11), (12) and (13) we obtain equation
(14):
w s = w o [1 − φ][1 − βφ]
(14)
In the following discussion, a simplified case is
considered, where α = β = 1. Equation (14) then simplifies
to:
w s = w o [1 − φ] 2
(15)
It is necessary to stress that this is only an approximate
solution. A particular value of α was used together with
the direct application of the potential flow solution. However,
an important result is that the equation for the settling
velocity is a still more complex function of the volumetric
Minerva, 1(1): 37-43
Once more, a simple equation is obtained. However,
in this prediction, it is not possible to isolate the normalized
profile φ/φa. Figure 5 presents the concentration behavior
for φa = 0.8 and different values of Lwo. The comparison
with Figure 3 shows that the nonlinear effect of the concentration
on the settling velocity conduces to equilibrium profiles much
more sensitive to the value of Lwo. So, such effects must be
considered when the concept of equilibrium profile is used.
Conclusions
The Rouse model for equilibrium concentration profiles
in channel flows leads to unreal values of concentration near
the bottom of the flow (a volumetric concentration higher
than unity constitutes a physically incoherent result). The
constant settling velocity hypothesis is the factor that inhibits
to attain better results. A more adequate alternative consists
in the use of the mass conservation principle to calculate
the settling velocity, which permits to obtain more realistic
sediment concentrations along all the fluid depth.
Considering the movement equation in the form of
the solution of potential flow around a sphere, it was possible
to show that a nonlinear dependence exists between the
settling velocity and the volumetric sedimentation
concentration. Equilibrium profiles based on the obtained
nonlinear equation show that these profiles may be very
sensitive to the settling velocity function (that is, its
dependence on concentration).
For procedures where the equilibrium profile concept
is needed, it is suggested that the use of profiles based on
constant settling velocities be avoided. At least the results
based on the mass conservation principle must be considered.
FUNDAMENTALS ON EQUILIBRIUM CONCENTRATION CURVES...
41
Figure 2 (a) Relative concentration profiles (φ/φa) for different φa values and Lw0 = 0.5;
(b) usual nondimensional profiles for φa = 0.8 and Lw0 = 0.5; (c) relative concentration profiles (φ/φa)
for different φa values and Lw0 = 1.0; (d) usual nondimensional profiles for φa = 0.8 and Lw0 = 1.0.
1
1
(a)
Lw0 = 0.5
(b)
0.8
0.8
1.0
1.0
Z
0.6
Z
0.6
Lw0 = 0.5
0.4
1.5
0.4
1.5
0.2
2.0
0.2
2.0
0
5.0
0
5.0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Figure 3 Equilibrium profiles for φa = 0.8, a = 0.05 and different values of Lw0
(a) rouse profiles (equation 6); (b) equation (10), more sensitive to changes in the Lw0 value.
Minerva, 1(1): 37-43
42
SCHULZ, ALAMY FILHO & VILLELA
Figure 4
Constant velocity
1
1
(Rouse model)
0.8
0.8
0.6
0.6
Lw0 = 0.5
1.0
1.5
2.0
Z
ws/w0
(a) Definition of φ and; (b) flow lines above a sphere in rest.
Equation (9)
0.4
0.4
Equation (15)
0.2
5.0
0.2
(a)
0
0
0
0.2
0.4
0.6
0.8
1
(b)
0
0.2
0.4
0.6
0.8
1
Figure 5 (a) Different settling velocities used in this study;
(b) equilibrium concentration profiles using equation (16) with φa = 0,8 at a = 0.05.
Acknowledgements
To FIPAI, FINEP, CAPES, FAPESP and CNPq,
Brazilian research support institutions, which collaborate
in different aspects of the maintenance of the studies of
the research program on erosion, sedimentation and longtime reservoirs behavior.
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