= 0 and / = 0 where 
is the
turbulent time average of the velocity in the -direction [m/s];
Modeling Density Currents in Circular
Clarifiers
Scott A. Wells
David M. LaLiberte
Key words:
sedimentation, mathematical modeling, clarifiers,
density currents, buoyancy
Abstract
Design of sedimentation tanks for solid-liquid
separation is often dependent on assuming ideal flow conditions.
But the geometry of the tank and density currents as a result
of temperature and suspended solids influences the fluid mechanics
of the tank and can result in significant deviations from ideal
flow.
A two-dimensional radial flow model was proposed
which incorporated the effect of density currents resulting from
temperature and suspended solids differentials within the tank.
The numerical model predicted the steady-state, layer-averaged
radial flow and layer depth.
The model solution and field data showed that the momentum and suspended solids of the inflow caused a density current that moved along the clarifier bottom. In the winter, this density current rose as a result of warm inflow water after the initial momentum had decreased and the suspended solids had settled.
Introduction
Design of sedimentation tanks for water and
wastewater treatment processes are often based on the assumption
of uniform unidirectional flow through the tank as shown in Figure
1. Dick (1982) showed though that many full-scale sedimentation
tanks do not follow ideal flow behavior. Because of uncertainties
in the hydrodynamics of clarifiers, designers typically use safety
factors to account for this non-ideal flow behavior (Abdel-Gawad
and McCorquodale, 1984).
Non-ideal flow behavior can be the result of
the following (DeVantier and Larock, 1987; Tay and Heinke, 1983;
Wells, 1990):
Steady-state models of currents in clarifiers have been developed for rectangular clarifiers by Ostendorf (1986) and for circular clarifiers by Abel-Gawad and McCorquodale (1984) and Shamber and Larock (1981). In these cases, the effect of density differences due to temperature and suspended solids stratification were ignored. Later, DeVantier and Larock (1987) used a modified form of Rodi's (1980) k- turbulence model to predict currents in a circular clarifier at steady-state. The effect of buoyancy on tank hydrodynamics was considered of secondary importance and was ignored in their final model formulation. McCorquodale (1997) summarized the state-of-the-art in modeling clarifier hydrodynamics and showed that currently available clarifier models do not account for the impact of temperature differences within the tank caused by atmospheric heating or cooling. Hence, in this study, the relative importance of the impact of temperature differentials within a tank on clarifier hydrodynamics were evaluated.
Mathematical Modeling of the Two-Layer Flow
A hydrodynamic model of the density currents
in a 2-layer system can be characterized by the continuity and
momentum equations for each layer assuming a radial coordinate
system as shown in Figure 2. The turbulent time-averaged Navier-Stokes
equations for radial coordinates (Sabersky, et al. 1989)
were simplified making the following assumptions:
1. axisymmetric flow, i. e.,
= 0 and / = 0 where is the
turbulent time average of the velocity in the -direction [m/s];
2. molecular terms were negligible compared to the turbulent flux terms;
3. flow was incompressible;
4. fluid density changes were neglected with the exception of body force term (the Boussinesq approximation);
5. 
; where 
is the time averaged atmospheric pressure [kPa].
The resulting general mass continuity equation
in cylindrical coordinates was then:
(1)
where w: vertical velocity [m/s]
u: radial velocity [m/s]
r: radial coordinate [m]
z: vertical coordinate [m]
and overbars represent the turbulent time-average.
The resulting r-momentum equation in cylindrical
coordinates was:
(2)
where t: time [s]
: time average fluid
pressure [kPa]
: turbulent fluctuating
vertical velocity [m/s]
: turbulent fluctuating
radial velocity [m/s]
: fluid density [kg/m3]
The z-momentum equation simplified to the hydrostatic
equation by assuming negligible vertical accelerations:
(3)
where g: acceleration due to gravity [m/s2]
h: total depth of the tank [m]
Assuming a flow regime with an inlet baffle
as in Figure 3, Equations 1-3 were simplified as follows (following
the approach of Schijf and Schonfeld, 1953):
The resulting continuity and r-momentum equations
in cylindrical coordinates for each layer after simplification
were:
Layer 1 (h
z
h2)
:
Continuity Equation 
(4)
Momentum Equation: 
(5)
Layer 2 (h2
z
0):
Continuity Equation:
(6)
Momentum Equation: 
(7)
After layer averaging in the z-direction, assuming
steady-state, and simplifying, the layer equations (Equations
4-7) become:
Layer 1:
(8)
Layer 2:
(9)
where 1,2: densities in the upper and lower layers, respectively [kg/m3]
Q1,Q2: flow rates in the upper and lower layers, respectively, where Q1=(1-Dv)Qo and Q2=DvQo [m3/s]
Qo: tank inflow rate [m3/s]
Dv: vertical dilution factor at baffle entrance (if =1, no dilution)=(Qo+Qentrained)/Qo [-]
h1,h2: upper and lower layer heights, respectively [m]
b,i,s: shear stress at the bottom, interface, and at the surface, respectively
[kPa]
The surface shear stress was assumed to be
zero (implying no wind effects or surface ice formation) and the
other shear stresses were calculated using (Grubert, 1989)
(10)
(11)
where fi: interfacial friction factor (=0.01) [-]
f: bottom Darcy-Weisbach friction factor (=0.01)
[-]
Numerical Solution and Results
These equations were solved simultaneously
for h1 and h2 by using a fourth order Runge-Kutta
method. Experimental data used in solving the above equations
was taken from a study by LaLiberte (1990). This study was performed
at Bend, Oregon at the Bend Wastewater Control Plant on its uncovered
secondary clarifier during a period of strong winter cooling.
Temperature and suspended solids data were taken as a function
of depth and radial position within the tank. These data showed
that an inflow entered the tank with a suspended solids about
2000 mg/l and a temperature over 1oC warmer than the
tank surface. After the initial momentum of the inflow decreased
and solids settled, the inflow rose as a buoyant plume. The resulting
temperature distribution is shown in Figure 4. Data from this
study in Bend Oregon that were used in the numerical model predictions
are summarized in Table 1.
Table 1. Experimental data from LaLiberte (1990) for the Bend, Oregon Wastewater Control Plant secondary clarifer. Data were taken during a period of low flow and high surface cooling from 2-4 am on 3/5/89.
| Parameter | Value from field study |
| Flow rate, m3/s | 0.03 |
| Tank depth at centerwell, m | 4.6 |
| Tank depth at periphery, m | 3.5 |
| Inlet baffle radius, m | 2.1 |
| Inlet baffle depth, m | 2.4 |
| Tank radius, m | 12.2 |
| Area of tank, m2 | 467 |
| Tank volume, m3 | 1900 |
| Inflow temperature, oC | 13.7 |
| Outflow temperature, oC | 12.5 |
| Reduction in suspended solids from inflow to outflow, mg/l | 2000 mg/l |
Model predictions of the layer height are shown
in Figure 5 for the Bend, Oregon secondary clarifier with an inlet
baffle. For this simulation the suspended solids were assumed
to settle out linearly with distance along the tank. A more comprehensive
model of the sedimentation process (like the models summarized
by McCorquodale, 1997) would be necessary to predict this variation
in suspended solids. These results though are meant to illustrate
the temperature impact on the flow field. Figure 5 contrasts the
layer height with and without suspended solids for a temperature
loss of 1.2oC over the tank. The suspended solids keep
the warmer, but still denser inflow, near the bottom. But as the
solids settle as the momentum of the radial flow diminishes, temperature
then influences the trajectory of the plume. If there were no
suspended solids in the inflow, temperature becomes more important
in determining the location of the interface. Hence, the shape
of the interface is a function of the flow rate, inflow suspended
solids, temperature difference across the tank. Temperature data
shown in Figure 4 also illustrate the effect of buoyancy on the
rising of the flow near the middle of the tank.
Summary and Conclusions
A mathematical model of the turbulent flow
in a circular tank was developed assuming a radial coordinate
system, steady-state, and averaging over each vertical layer.
The effect of longitudinal density differences between the vertical
layers as a result of temperature and suspended solids differences
was also incorporated into the mathematical model. The mathematical
model showed that
The mathematical model agreed qualitatively
with field data showing that temperature differentials within
the clarifier can cause the inflow to rise after both the initial
momentum of the inflow has decreased and the suspended solids
have settled.
This work shows that ideal flow in a clarifier
(as shown in Figure 1) may not be realistic for conditions where
inlet baffles direct the suspended solids laden flow downward.
More complex mathematical models of these processes have been
developed that model this flow field. But during conditions where
temperature differentials exist within clarifiers, such as during
periods of winter cooling, the effect of temperature can also
be significant and should therefore be included in models of these
processes.
References
Abdel-Gawad, S. M. and McCorquodale, J. A. (1984) "Hydrodynamics of Circular Primary Clarifiers," Canadian J. Civil Engr. ,11, 299-307.
DeVantier, B. A. and Larock, B. E. (1987) Modeling Sediment-Induced Density Currents in Sedimentation Basins, " J. Hydr. Engr., ASCE, 113(1), 80-94.
Dick, R. I. (1982) "Sedimentation Since Camp," J Society of Civil Engrs, 68, 199-235.
Grubert, J. P. (1989) "Interfacial Mixing in Stratified Channel Flows," J. Hydr. Engr. 115(7), 887-905.
LaLiberte, D. M. (1990) "Density Currents in Circular Wastewater Treatment Tanks," M. S. Thesis, Department of Civil Engineering, Portland State University, Portland, Oregon.
McCorquodale, J. A. (1997) "Mathematical Modeling of Settling Tanks," Proceedings American Filtration and Separations Society, 6th Education Conference, Houston, TX, S21-S216.
Ostendorf, D. W. (1986) "Hydraulics of Rectangular Clarifiers, J. of Envir. Engr., ASCE, 112(5), 939-953.
Rodi, W. (1980) Turbulence Models and Their Application in Hydraulics, IAHR Publication, Delft, Netherlands.
Sabersky, R. H.; Acosta, A. J.; Hauptmann, E. G. (1989) Fluid Flow A First Course in Fluid Mechanics, MacMillian Publishing Co., New York.
Schijf, J. B. and Schonfeld, J. C. (1953) "Theoretical Considerations on the Motion of Salt and Fresh Water," Proceed. Minn. Inter. Hydraulics Convention, Sept, 321-333.
Shamber, D. R. and Larock, B. E. (1981) "Numerical Analysis of Flow in Sedimentation Basins," J. Hydr. Div., ASCE, 107(HY5), 575-591.
Tay, A. J. and Heinke, G. W. (1983) "Velocity and Suspended Solids Distribution in Settling Tanks," J. Water Pollution Control Fed., 55, 261-269.
Wells, S. A. (1990) "Effect of Winter Heat Loss on Treatment Plant Efficiency," J. Water Pollution Control Fed., 26(1),34-39.
Wells, S. and LaLiberte, D. (in-print)"Density Currents in Circular Tanks," Water Envir. Research.
Figure 1. Uniform unidirectional flow in a baffled clarifier.
Figure 2. Coordinate
system for radial flow in a circular clarifier.
Figure 3. Definition sketch for flow model
in a baffled clarifier.
Figure 4. Temperature profile across the secondary
sedimentation tank at the Bend treatment plant on 3/5/89 at 2:46
am (Wells and LaLiberte, in-print).
Figure
5. Model predictions of interface layer height as a function of
radial position for Q=0.03 m3/s, Dv=1.5,
for the Bend secondary clarifier comparing model predicted interface
height with and without suspended solids and temperature effects.