MODELING SOLUTE TRANSPORT USING
QUICK SCHEME
1
By Roger A. Falconer, Member, ASCE, and Suiqing Liu2
Details are given of the refinement and application of a
two-dimensional depth integrated numerical model to predict the depth
mean velocityfieldand the spatial concentration distribution in hydraulic basins, such as chlorine contact tanks. The model includes a refined
and computationally manageable third-order spatial finite-difference
representation of the terms describing the advective transport of a
solute, with the corresponding difference scheme being particularly
suited to modeling high solute gradients. The scheme is shown to yield
high accuracy in comparison with the more conventional second-order
central-difference representation, with the associated spurious wavetype distribution solute concentrations associated with high-solute gradients being considerably reduced. The model has been applied to a
laboratory hydraulic model study of plug flow through a site-specific
chlorine contact tank, with the numerical model results for various tank
configurations being compared with corresponding laboratory model
results. In most cases the numerical model predictions of the flow
through curves for a conservative tracer were in close agreement with
the corresponding laboratory model results, particularly in comparison
with the predictions obtained using a central difference representation.
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ABSTRACT:
INTRODUCTION
In recent years there have been a number of hydraulic model and field
studies reported in the literature where the main objective has been to
improve the flow through characteristics of hydraulic basins, such as
chlorine contact tanks, or to design new tanks with uniform flow through
conditions occurring from the onset, e.g., Sawyer and King (1969),
Kothandaraman et al. (1973), McNaughton and Gregory (1977), Hart
(1979), Hart and Vogiatzis (1982), and Falconer and Tebbutt (1986). In
most of these studies this objective has been attempted by designing the
tank so that plug flow conditions occur as closely as possible—with ideal
plug flow occurring when all fluid elements within the tank have the same
residence time. When fluid elements travel directly from the inlet to the
outlet and pass through a contact tank in a much shorter time than the
theoretical retention time, then this process is termed short-circuiting; fluid
elements remaining in the tank for periods greatly exceeding the theoretical retention time give rise to dead-space zones. When a chlorine contact
tank is found to possess short-circuiting, and/or dead-space zones, then it
is often necessary to increase the chlorine dosage in order to maintain the
effectiveness of the treatment. However, increasing the chlorine dosage is
unsatisfactory for two main reasons: (1) The suspicion that potentially
'Prof, of Water Engrg., Dept. of Civ. and Struct. Engrg., Univ. of Bradford,
Bradford,
West Yorkshire BD7 1DP, England.
2
Lect., Dept. of Envir. Engrg., Tong-Ji Univ., Shanghai, People's Republic of
China.
Note. Discussion open until July 1, 1988. To extend the closing date one month,
a written request must befiledwith the ASCE Manager of Journals. The manuscript
for this paper was submitted for review and possible publication on March 31, 1987.
This paper is part of the Journal of Environmental Engineering, Vol. 114, No. 1,
February, 1988. ©ASCE, ISSN 0733-9372/88/0001-0003/$1.00 + $.15 per page.
Paper No. 22160.
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J. Environ. Eng., 1988, 114(1): 3-20
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carcinogenic compounds may be formed (McNaughton and Gregory 1977);
and (2) the increased costs incurred for the same through flow. With these
considerations in mind, particularly the former, it is therefore not surprising to find that there has recently been an increasing interest in the
hydraulic design and operation of such contact tanks.
A traditional approach adopted by hydraulic and environmental engineers in investigating the hydraulic design characteristics of chlorine
contact tanks has been to construct a scaled laboratory model and to insert
instantaneously a plug of tracer (such as Rhodamine B) at the inlet, at a
known initial time. The tracer has generally been assumed to be conservative, at least in terms of the contact time within the tank, and the time
variation hydrograph of the tracer concentration has been recorded at the
outlet. This time variation of the tracer concentration at the outlet is often
known as the flow-through curve. Further details of this type of laboratory
model approach are given by Falconer and Tebbutt (1986). However, such
laboratory model studies are not ideal, since a Froude law scaling for
dynamic similarity between the prototype and the physical model results in
an overestimation of dispersion-diffusion processes and momentum transfer by bed-generated turbulence, and an underestimation of the effects of
bed friction (Mardapitta-Hadjipandeli 1985). These disparities between the
prototype and physical model are made significantly worse when a
vertically distorted model is used.
In view of the increasing interest in the hydraulic design characteristics
of chlorine contact tanks, and the scaling disadvantages of using laboratory
model studies to investigate the flow-through curves for such tanks, it
appears to be prudent to consider using mathematical models for such
studies, since scaling problems no longer arise. However, the use of
shallow water-wave equation-type models, such as those previously documented and used by Falconer (1984, 1986), also have limitations in such
applications. Firstly, the size of such contact tanks and the correspondingly small size of a typical finite-difference grid spacing, relative to the
thickness of the boundary layer, mean that the inclusion and modeling of
the turbulent Reynolds stresses becomes much more significant in the
capability of the model to predict accurately the corresponding velocity
field. This is presently part of a research program being undertaken by the
writers and is not described herein. The second limitation of such
mathematical models is the inability of the finite-difference technique—or
other practical numerical methods—to represent precisely the high-solute
gradients arising as the plug of solute is advected through the tank. The
traditional central difference representation of the advective-diffusion
equation, as used by Falconer (1984, 1986), can lead to a measurable
degree of numerical diffusion of the high-solute gradient, which can often
be greater than, or similar in magnitude to, the physical processes of
turbulent diffusion and longitudinal dispersion. Furthermore, the Fourier
decomposition of such high-solute gradients, particularly arising at the
instant of including the plane source plug of solute in the model, has many
very short waves that cannot be resolved into waves with a few points per
wavelength using the central-difference method. The corresponding spurious waves, of model wavelength 2AX, propagate slowly across the
computational domain, giving rise locally to negative solute values at
alternate grid points and an underestimation of the influence of dispersion
4
J. Environ. Eng., 1988, 114(1): 3-20
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(Leendertse 1970). However, these difficulties have been overcome in
similar modeling studies of estuarine pollutant transport by using higher
accuracy difference schemes of fourth-order and upwards (Holley and
Preissman 1977; Komatsu et al. 1985). For the type of study outlined in this
paper, where the plug source input is of a short time duration rather than
a continuous input as for coastal outfall pollution studies, the open
boundary conditions, in particular, are difficult to define. Furthermore,
such higher order schemes have the disadvantage of being particularly
complex in the computational sense. A compromise between these two
schemes, i.e., the central difference and the fourth and higher order
accuracy schemes, is the QUICK scheme (quadratic upstream interpolation for convective kinematics), see Leonard (1979). Details are therefore
given herein of a comparison of this scheme with other schemes for
one-dimensional plug flow, with no diffusion or dispersion, and the
subsequent application of the scheme to a hydraulic model study to
determine and improve the hydraulic conditions and the flow through
curves for the Elan chlorine contact tank, in Birmingham, England. The
QUICK scheme has the desirable advantages of: (1) High accuracy,
including third-order spatial accuracy; (2) inherent numerical advective
stability, i.e., no grid scale oscillations; (3) computational simplicity.
Encouraging predictions are shown for the flow through curves for the
Elan contact tank, indicating that this computationally manageable difference scheme makes numerical model studies of such studies as plug flow
through chlorine contact tanks more accurate and preferable to using more
traditional difference schemes.
CONSERVATION EQUATIONS
The governing differential equations used in the mathematical model
include the depth-integrated conservation equations of mass, momentum
in the two mutually perpendicular horizontal coordinate directions, and
solute transport. The resulting hydrodynamic model equations were derived from the three-dimensional continuity and Navier-Stokes equations
(Schlicting 1960), giving respectively:
<9n 8UH
dVH
dt
dx
dy
dUH . JdU2H
8UVH\
_TT dr\ , x^
1- R
+
+ 9H —- +
P
P
dt
\ dx
dy J a
dx
1 fdHa„ + dHx^] = Q
dy
" dx
dVH
|
JdUVH
hB
V
dt
\
_l(d_HTM
p V dx
_ SV^JH\
+
dx
+
_^ nTJ dr\
(2)
^x^
+ QH —- +
P
dy )
* dy
d_Holl\0
(3)
dy J
where -n = water surface elevation above datum; t = time; H = total fluid
depth; U, V = depth mean velocity components in x-, y-coordinate
J. Environ. Eng., 1988, 114(1): 3-20
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directions; p = momentum correction factor for nonuniform vertical
velocity profile (assumed to be 1.017 for seventh-power law profile); g =
gravitational acceleration; tbx, tby = bed shear stress components in the x,
y-coordinate directions; p = fluid density; and axx , ayy , 7xy , Tyx = direct
and lateral shear stress components in the x, y-coordinate directions.
The bed shear stress components have been represented in the model
using a quadratic friction law and the Darcy friction factor (Henderson
1966), giving
T . - ' * ™
V
pfV\V\
xby=^-^
and
M
(46)
where IVI = depth mean fluid speed; a n d / = Darcy-Weisbach friction
factor, approximated by the Colebrook-White equation in the form:
77—(m+^)
;
<5>
where ks = length parameter characteristic of bed roughness; and Re =
Reynolds number (= AUHlv, where v = the kinematic viscosity of the
fluid).
For the direct and lateral shear stress components, a simple turbulence
model was used, based on the Boussinesq approximation for the eddy
viscosity and an assumed seventh-power law velocity profile in the vertical
plane. Thus, for the x-direction momentum Eq. 2, the corresponding direct
and lateral shear stresses were expressed in the following form (Falconer
1986):
• <«>
*f)
\.\61fH\V\
8
°xX-
\A61fH\V\
—
—^
(7)
dy
dxj
The depth integrated conservation equation of solute transport was
derived from the three-dimensional form of the advective-diffusion equation giving
^xy ~~
dCH
8t
+
8
/dCUH
\ dx
+
d (
+
dCVH\
dy ) ~
dC
HD
d (
dC
dC
— HDXX
xx — + HDxy —
dx \
dx
dy
„
dC\
= 0
+HD
Vy{ -Vx
(8)
^)_
where C = depth mean solute concentration; and Dxx , Dxy , Dyx , Dyy =
depth mean dispersion-diffusion coefficients in x-, y-coordinate directions,
given by Preston (1985) as
n
(hU2 + ktV*)H
D
**~
2\v\jr-^
f
,
{9a)
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J. Environ. Eng., 1988, 114(1): 3-20
(k^2
{9b)
'----^OffirJ'
"c)
yy
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+ k U2)H
t
^JV^tJhp!L/
f
D
V J
2\V\J2
(/q - kt)UVH
V
;
where kt = longitudinal depth mean dispersion coefficient; and k, =
transverse turbulent diffusion coefficient. In the subsequent tests, the
values for the coefficients k, and k, were either equated to zero or obtained
by calibration from the experimental data.
NUMERICAL MODEL DETAILS
The finite-difference representation of the hydrodynamic Eqs. 1-3,
including Eqs. 4-7, was similar to that used previously and described in
some detail in Falconer (1984, 1986). Basically, the governing differential
equations were expressed in an alternating direction implicit difference
form, with the unknown variables in, U, and V being located at the center
and around the sides of a regular grid. The scheme was fully centered in
both time and space, with the advective accelerations and the direct and
lateral stress terms being fully centered by iteration. The treatment of the
cross-product advective accelerations were refined to allow for jet-type
flows, without grid-scale oscillations occurring, with full details of the
treatment of these terms being given in Falconer (1986).
The main novel aspect of the numerical model in this particular study
was the finite-difference representation of the advective-diffusion equation
and, in particular, the adaptation of the QUICK scheme to represent the
advective transport process. The QUICK scheme, originally developed by
Leonard (1979), involves the use of quadratic interpolation to estimate the
solute concentration value, on a space-staggered grid, at the same location
as the velocity components. This is a refinement of the traditional central
difference scheme, wherein linear interpolation is used to estimate the
solute concentration at the velocity location on the grid. The formulation
of the QUICK scheme is best illustrated using the constant space staggered
grid representation shown in Fig. 1, where the mean flow velocity
component is to the right as shown.
For the x-direction advective term of the advective-diffusion equation,
i.e., Eq. 8, quadratic interpolation for the corresponding derivative at the
grid point j + 2 gives, for example
8CUH
dx
J+2
CUH\J+2.5-CUH\J+U5
Ax
(1Qa)
where
Cj+2.5 = i(C}+3 + Cj+2) - i ( C , + 3 - 2C, + 2 + CJ+1)
(10ft)
CJ+1.5=i(Cj+2
(10c)
+ Cj+1)-UCj+2-2Cj+1
+ Cj)
Using this representation for the advective terms, with flow in either
direction and in both coordinate directions, the alternating direction
implicit finite-difference representation used in the numerical model, for
7
J. Environ. Eng., 1988, 114(1): 3-20
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\*h
j*1
j+lk
j+2[4. j+3
j+2
Distance
FIG. 1. Sketch of Finite Difference Grid Representation for Quadratic Interpolation
the first half time step, can be expressed as follows:
CH" + 0S
'j.k
^n, h
— CH"
^Mj,k
TTTJ" + O.S
Urt
j~0.5,k\^j,k
+ ^
-I— rrrH-n + 0.5 (r'n + 0.5 , rn + 0.5\
+ 4 ^ x lutlj+o.s,k(cj+i,k
+ cj,k
)
//^n + O.S + ,C /-.n + 0.5-1-1
j-l,kJJ —
LVHik+0.5(Cik+1
+ qtk) - VHik_0,5(C"j,k
— 7 T - (^^jU+0.5 V C" k+0
At
2AX'
+
rnnitO.S
At
8Ax
5
+
cik-Jl
— F H " J t _ 0 5 V C";fc_0.5)
//-'n + 0.5
™ + 0.5\
u n i + 0.5
C/^""1
/-n + 0.5\
HD;nk+0JCU+1-Clk)-HD"yy
yyj,
' (Cj.li ~ C", fc-l) + """x)>j + o.5,lt( ^7+0.5,4 + 0.5 ~~ Cj+0.5,S-0.5)
HD"
5 , ^ ( ^ - 0 . 5 , 1 + 0.5
' (Cj+0.5,k + 0.5
— HDyXj
k0
^ - 0 . 5 , t - 0 . 5 ) + "DyxJ
k+0.
C j - 0 . 5 , ^ + 0.5)
5(Cj+0 s k_0 s
— C-y_0 5 t _ 0 5)J
(11)
where j , fc = grid square locations in x-, ^-coordinate directions; n = time
step number; At = time step; AX = grid size and
V 2 Cj + 0 . 5 , Jc
CJ+lfk-2CJik
+ Cj-ltk;
ift/^O
J. Environ. Eng., 1988, 114(1): 3-20
•(12a)
• (126)
if t / > 0
if £ / < 0
(13a)
(13b)
Cj,k + i 2C^ k + CjJl_i;
Cj,k ~ 2Cj k+1 + Cjik+2;
if V>0
if V<0
(14a)
(14b)
Cj,k
if V>0
ifF<0
(15a)
(15b)
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V2Cj-0.5,k
V2Cj,k-0.s =
cj,k
2Cj,k-l + Cj,k-2'
•i ~~ 2CJtk + CJik + l;
Likewise, a similar finite-difference representation can be obtained for the
second half time step, with all terms in the x-coordinate direction and the
terms given by the QUICK scheme being expressed explicitly.
In modeling the boundary conditions for the difference equations for
each half time step, i.e., Eq. 11 for time step n —> n + (1/2), the
central-difference component of the advection terms and the dispersiondiffusion terms are treated in the manner described previously by Falconer
(1984). Thus, at closed-wall boundaries, the normal velocity and dispersion
coefficients are equated to zero, thereby allowing no solute flux through
the wall. For the open boundaries, linear extrapolation is used at outflow
boundaries, whereas the boundary conditions must be prescribed precisely
at inflow boundaries. In modeling closed-boundary conditions for the
QUICK computation, the solute gradient is first evaluated at the wall using
quadratic interpolation, with the concentration of the exterior adjacent grid
square then being obtained from the wall gradient.
NUMERICAL MODEL APPLICATION
One-Dimensional Test Reach
The refined numerical model, including the QUICK scheme, was first
tested and compared with other finite-difference schemes by applying it to
a one-dimensional test reach. The test reach consisted of a steady
unidirectional flow, with a pure plug source of conservative tracer being
advected only along the reach, i.e., both physical diffusion and dispersion
were equated to zero. The reach was assumed to be 20 km long, with the
grid spacing and time step being 200 m and 100 sec, respectively. The
constant velocity was assumed to be 0.5 m s _ 1 , giving a transport time
along the reach of 400 time steps, and the plug lengths considered were
SAX (1 km), 10AZ (2 km), and 30AZ (6 km).
In total, five schemes were considered including: (1) The backward
implicit scheme (scheme 1); (2) the central implicit or Crank-Nicolson
scheme (scheme 2); (3) the upwind implicit scheme (scheme 3); (4) the
QUICK scheme (scheme 4); and (5) the six-point scheme by Komata et al.
(1985) (scheme 5).
The corresponding numerical model predictions for the advected plug
can be seen in Fig. 2 for the case when the 1-km plug has just passed the
midpoint along the test reach. In interpreting these results, with particular
reference to scheme 4, i.e., the QUICK scheme, the backward implicit
scheme 1 can be seen to have a relatively high degree of numerical
diffusion, with the predicted peak concentration being approximately 57%
of the actual assumed concentration of 10 mgl - 1 . Although the numerical
diffusion is less marked for the central difference scheme 2, there is a
9
J. Environ. Eng., 1988, 114(1): 3-20
Scheme 5 (Six Point)
^ ::i
00
E
Scheme A (Quick)
a
^a.
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S. io-o
0-0
100,
U'Ur
Scheme 3 (Upwind)
o-oL
1
100,
Scheme 2 (Central)
0-0
.nL Scheme 1 (Backward)
100,
J
00
0
.
2-0
•
4-0
•
6-0
,_^T~~-K,
8-0
10-0 12-0 14-0 16-0 18-0 20-0
Distance from the starting point (KM)
FIG. 2. Predicted One-Dimensional Plug Source Concentration Distributions for
Five Finite Difference Schemes
distinct phase lag between the location of the physical and numerical peak
concentrations and the characteristic spurious wave-type oscillation in the
upstream concentration distribution can be clearly seen. Thus, as can be
seen from Fig. 2, the QUICK scheme 4 shows a marked improvement in
comparison with the central-difference scheme, particularly since the peak
concentration for the QUICK scheme is also better, at 82% of the true
peak. As expected, the upwind difference scheme 3, with its relatively high
artificial diffusion, exhibits high damping with a peak concentration of only
24% of the correct value. Finally, the six-point scheme 5 exhibited
marginally superior accuracy to the QUICK scheme, with a peak concentration of only 14% above the correct peak value, and with a reduced
upstream negative concentration. However, these relatively refined improvements in the accuracy between schemes 4 and 5 must be compared
with the advantages of relative simplicity, at least in terms of treatment of
the boundary conditions, and computer manageability of scheme 4,
relative to scheme 5. The results and comparisons shown in Fig. 2 were
consistent with the results obtained at other sites along the test reach and
for the wider plug sources.
Elan Chlorine Contact Tank
Following the development and application of the model to the onedimensional test reach, it was then applied to a scaled physical model
study of the Elan chlorine contact tank, in Birmingham, England. The
main objectives of the original physical model study (Falconer and Tebbutt
1986) were to investigate the flow patterns and flow-through curves for the
tank and to consider various tank modifications, with a view to reducing
the problems of short-circuiting and optimizing retention times.
The Elan chlorine contact tank is used by the Severn Trent Water
Authority to supply potable water to various parts of Birmingham,
England. The contact tank has a total plan surface area of 2,090 m2 and
consists of two independent component tanks, namely the north and south
tanks, with each receiving average daily through-flows of 180,000,000 L.
10
J. Environ. Eng., 1988, 114(1): 3-20
PI
Submerged weir
South tank
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457
Baffles
*E
North tank
,y
^Overflow weir
Submerged weir
Inlet chamber wall
Outlet to
distribution system
ta
:lnlet
=Supj
from filters
Plan
Submerged weir
Overflow weir
T
7
/
2-55m. approx.
234m./
V,
^
£ 1-^m.
^ -VSection X-X
Vertical scale
Horizontal scale
6~~"1 2~^m.
FIG. 3. Schematic Illustration of Elan Chlorine Contact Tank
The approximate volumes of the north and south component tanks are
4,900 m3 and 5,100 m 3 , respectively, with the main dimensions of both
tanks being shown in Fig. 3. The through-flow enters the component tanks
at the inlet, flows over the submerged weir, and then discharges over the
overflow weir at the downstream end. Prior to the physical model study,
the water authority had established that the flows through both component
tanks had shorter retention times than the theoretical times, with lithium
chloride tracer tests indicating retention times of only 17 min and 29 min,
for the north and south component tasks, respectively, as compared with
a theoretical value of 38 min for a daily through-flow of 180,000,000 L. For
reasons outlined in detail in Falconer and Tebbutt (1986), it was only
possible to make modifications to the two submerged upstream weirs (see
Fig. 3) by extending them across the tank and raising them to below the
elevation at which critical depth would occur, and/or modifying the baffle
arrangements.
A 1:30 Froudian-scale physical model was constructed, with velocity
field measurements, observations, and dye tracer studies being undertaken
for a series of different tank configurations. The velocity field was
measured using weighted drinking straws as drogues, and the flow-through
curves were established by first inserting 6-8.5 ml of Rhodamine B in the
form of a plug source just upstream of the inlet. At the start of dye
injection, the time was recorded and sampling of the outflow—just beyond
the outlet weir—was commenced by continuously pumping a small flow of
10 ml min - 1 through a Turner fluorometer. Having established the outflow
concentration distribution for a specific tank configuration, the resulting
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J. Environ. Eng., 1988, 114(1): 3-20
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distribution was then represented in a dimensionless form by plotting CICo
against J/TR, where Co = total concentration volume/tank volume; T =
time after dye injection; and TR = theoretical retention time, i.e., Vol/Q,
where Vol = tank volume and Q = through flow. Further tests were also
undertaken for vertically distorted models, with distortion ratios of 2:1 and
3:1, with full details of the laboratory model results given in Thayanity
(1984) and Falconer and Tebbutt (1986).
The main recommendations of the physical model study included: (1)
Extending the submerged weir in both component tanks so that it spanned
the width of each component tank; and (2) raising the height of the
submerged weirs from 1.56 m to 2.16 m, corresponding to model weir
heights of 52 mm and 72 mm, respectively. These recommendations were
implemented by the water authority, with subsequent field measurements
for the actual tanks showing marked improvements, with the fieldmeasured retention times being 34.0 min and 35.5 min for the north and
south component tanks, respectively.
In the mathematical model of the scaled physical model of the Elan
chlorine contact tank, a mesh of 53 x 16 and 55 x 16 regular grid squares
was used for the north and south component tanks, respectively. The
corresponding grid size (AX) was 0.055 m and the time step (At) was 1.0
sec, giving an average Courant number (AtvgH I AX) of 16.4. The
hydrodynamic model was run for steady through-flow conditions, with the
inflow to each model component tank being 0.535 I s - 1 . Discharges were
specified at both open boundaries for each tank, with the inflow boundary
being assumed to be a jet inflow through 1 and 2 grid squares for the north
and south tanks, respectively, and with the downstream outflow being
assumed to be uniformly distributed across the weir, i.e., V = 0 everywhere along the outflow boundary. For the open boundary conditions of
the solute transport model, a constant plug source of 5.0 mg 1 _1 was
included at the upper boundary for a period of 10 sec. This was in close
agreement with the plug input conditions in the physical model, with the
plug source being included over a period of 10 time steps. Finally, the
representation of the baffle arrangement and the submerged weirs in the
mathematical model are worthy of comment. In representing these structures accurately within the model, the local grid size was adjusted where
necessary to give the correct cross-sectional area of flow in both the x and
v-directions. Furthermore, the local finite-difference representation was
refined to give enhanced (second-order) accuracy for these irregular grid
squares, with the details of this grid scheme modification being described
in some detail by Falconer et al. (1986) in modeling tidal eddies around
Rattray Island, Australia.
Before the model was run in a predictive capacity, in line with the
objectives of the physical model study, the mathematical model was first
calibrated using the physical model results of the original weir and baffle
arrangement. In the calibration test runs, the main parameter's variations
included the bed roughness coefficient ks, and the longitudinal dispersion
and turbulent diffusion coefficients, kt and kt, respectively. The final
optimum values chosen for these parameters were ks = 0.7 mm, k, = 13.0
m2 S" 1 and k, = 6.0 m2 S" 1 . The value obtained for ks is in line with that
which might be expected for a glass-fiber physical model and requires no
further comment. However, both the dispersion and diffusion coefficients
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are relatively high in comparison with laboratory and field data obtained by
Elder (1959) and Fischer (1973), respectively, with their corresponding
values being: kt = 5.93 m2 S~' and kt = 0.15 m2 S~l. The larger values
obtained for the present study can be accounted for by the following
conditions: (1) There is a strong recirculation flow within the contact tank,
which is not accurately reproduced by the simple turbulence model; (2)
laminar flow exists over much of the plan surface area of the physical
model, giving rise to higher dispersion than that given in the mathematical
model by a turbulent logarithmic vertical velocity variation; and (3) such
physical model studies can be shown by dimensional analysis to overestimate dispersion and diffusion processes (Mardapitta-Hadjipandeli 1985).
The corresponding values for ks, kt, and kt, as obtained by calibration,
were then used in all of the subsequent mathematical model simulations for
the various physical model configurations considered.
The predictions obtained from the calibrated mathematical model, both
for the hydrodynamic and the solute transport processes, have generally
proved to be in good agreement with the corresponding laboratory model
results.
The velocity field predictions, as shown in Fig. 4 for the north component tank, were in close agreement with the physical model measurements,
except near the inlet and in the immediate vicinity of the submerged weir.
The numerically predicted velocities at the laboratory measuring sites have
been extrapolated from the full predicted velocity field and are shown in
comparison with the corresponding laboratory measurements in Fig. 5.
The example comparison shown in Fig. 5 highlights the discrepancy
between the results near the inlet and the submerged weir,' with this
discrepancy being partly accounted for by the non-negligible vertical
velocity components near both the inlet and the submerged weir. Although
the results are shown here for the north component tank and one
configuration only, the complete set of results for both tanks and for all the
laboratory model configurations cited by Falconer and Tebbutt (1986) have
shown a similar degree of agreement.
For the flow-through curves, the results using the QUICK scheme
showed a very close similarity with the measured results for the original
tank configuration, i.e., with the submerged weir being 52 mm high and not
completely spanning the width of each component tank. An example of the
resulting predicted concentration distribution is shown in Fig. 6 for the
north component tank only, with the predicted and measured flow-through
curves being shown for the north and south component tanks in Figs. 7 and
8, respectively. The ideal plug flow curve shown in Figs. 7 and 8 illustrates
the equivalent flow-through curve for one-dimensional flow through the
model tank in the presence of pure Taylorian diffusion and dispersion, i.e.,
diffusion and dispersion arising from a logarithmic vertical velocity profile
only (Falconer and Tebbutt 1985). As can be seen from Figs. 7 and 8, the
measured and predicted flow-through curves have similar characteristics,
with the time for the tracer to be first detected at the outlet and the time for
the peak concentration to occur being similar for the measured and
predicted results. When the upstream submerged weir was extended
across the tank and raised to the recommended model elevation of 72 mm,
the agreement between the measured and predicted flow-through curves
was not so encouraging, as can be seen in Figs. 9 and 10 for the north and
13
J. Environ. Eng., 1988, 114(1): 3-20
ELAN CONTACT CHLORINE TANK — NORTH MODEL
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TIME =
WmTOTOWOTTOmTOTOWmrowwmmw?
LENGTH SCALE —
5.83 MIN
BrommTOwm^^K«wwmK^^sm^rawwmww
VELOCITY - •
55 hfi
58
IVUS
AVERAGE DEPTH = 8 . B 8 3 M
ROUGHNESS KS
= 8.78
THROUGHFLOW = 0 . 5 3 5 L.-S
HEIGHT OF WALL = 8 . 8 5 2 H
m
FIG. 4. Predicted Velocity Distribution for Original Model North Tank Configuration
1
1
—
«__
^
—
—
—
1
1
^ ,
^
/
-
/
1
/
"
I
1
\
~-*.
~
"
'
^
/
s
**
s
—^
—*-
-~
-~
"
-
wv.«w™v»«™™«
L\V^VV^mH^\V\\\\WW\\V\WW
-
1
1
I
t
1
K
w ^
W W
.^,y
(<•)
V \ W W ^ W M \ * \ W V * V " » " " " " " " " » " " " " " ' " ' " " ' "
\\WWWVmVW.'.WW.W,W t V.W.W.WW. 1
LENGTH SCALE —
VELOCITY —•
55 IW
18 H1VS
W
FIG. 5. Comparison of Predicted and Measured Velocities at Measuring Sites for
Model North Tank: (a) Predicted Velocity Field; (b) Measured Velocity Field
south component tanks, respectively. However, this disparity between the
predicted and measuredflow-throughcurves for the maximum upstream
elevation considered was attributed to the strong vertical velocity components in the vicinity of the upstream weir and the diffusion and dispersion
coefficients, rather than any features of the QUICK scheme. At this
upstream weir elevation of 72 mm, theflowwas close to being critical over
the submerged weir, and visual observations showed a pronounced local
three-dimensional flow field. Furthermore, the diffusion and dispersion
coefficients were assumed to be the same for all of the weir elevations
14
J. Environ. Eng., 1988, 114(1): 3-20
ELAN CONTACT CHLORINE TANK —
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TIHS =
NORTH MODEL
1.59 MIN
ISO-CONCENTS
LENGTH SCALE —
55 HM
MEAN VELXITY » 17 M1VS
HEAN CL CON -1.33 ttCL
paM.
VEL DEVIATION » 16 m-8
a CON DEV =1.36 K M .
FIG. 6. Tracer Concentration Distribution for Original Model North Tank Configuration
considered in the model, whereas the reduction in the circulation strength,
arising as a result of raising the weir, was observed to effect these
processes. In designing new chlorine contact tanks, the hydraulic engineer
would aim to avoid pronounced three-dimensional flow features, and thus
this disparity in the results should not detract from the benefits of using
mathematical models in general to design hydraulically such chlorine
contact tanks. When such three-dimensional flow effects become significant, then a two-dimensional depth-averaged numerical model cannot be
justified, and a fully three-dimensional flow model must be used.
Finally, to illustrate the benefits of using the QUICK scheme in such a
mathematical model application compared with the more traditional central difference representation, a comparison is shown in Fig. 11 of the
predicted flow-through curve for the north component tank using both the
QUICK and central-difference schemes without dispersion and diffusion.
ELAN CONTACT CHLORINE TANK
NORTH MODEL
ideal plug Plow
measured
curve
predicted curve
1.0
1.5
2.0
RELATIVE TlflE RATIO T/TR
FIG. 7. Comparison of Predicted and Measured Flow through Curves for Original
Model North Tank Configuration
15
J. Environ. Eng., 1988, 114(1): 3-20
ELAN CONTACT CHLORINE TANK — SOUTH MODEL
ideal plug Plow
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measured
•
8.5
curve
• predicted curve
1.0
1.5
2.6
RELATIVE TIME RATIO T'TR
FIG. 8. Comparison of Predicted and Measured Flow through Curves for Original
Model South Tank Configuration
The dispersion and diffusion terms have been deliberately excluded in this
simulation, since, as can be seen from Fig. 11, the central-difference
scheme becomes unstable in modeling high-solute gradients in comparison
with the stable solutions obtained for the QUICK scheme. As can be seen
from Fig. 11, the QUICK scheme simulation exhibits a flow-through curve
that is similar in form to the previous results, although naturally the
numerical values are different as a result of the exclusion of dispersion and
diffusion. This result, in particular, highlights the advantages of using the
QUICK scheme, in comparison with the central-difference scheme, for
modeling high-solute concentration gradients, with the increased accuracy
of the scheme therefore appearing to outweigh significantly the increased
computational effort and additional finite-difference complexity.
ELAN CONTACT CHLORINE TANK
NORTH MODEL
ideal plug Flow
oeasured
curve
predicted curve
A
1.8
1.5
2.
RELATIVE TIME RATIO T/TR
FIG. 9. Comparison of Predicted and Measured Flow through Curves for Modified
Model North Tank Configuration
16
J. Environ. Eng., 1988, 114(1): 3-20
ELAN CONTACT CHLORINE TANK —
SOUTH MODEL
ideal plug Flow
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• measured
curve
predicted curve
8.5
1-e
1.5
2.0
RELATIVE TIME RATIO T/TR
FIG. 10. Comparison of Predicted and Measured Flow through Curves for Modified Model South Tank Configuration
Quick Scheme
Central Scheme
A
0-4
17V r
A-/I
H)\
I 12
H
\i
L„. J I .
Relative Time Ratio
t/TR
FIG. 11. Comparison of Flow through Curve for QUICK and Central Difference
Scheme Excluding Dispersion and Diffusion
CONCLUSIONS
Details are given of the refinement of an existing two-dimensional depth
averaged numerical model, with particular emphasis being placed on the
modeling of plug flow of a conservative tracer through a chlorine contact
tank. The main refinement of the numerical model is the use of the QUICK
scheme to represent the advection terms of the advective-diffusion equation, with this third-order accurate scheme involving the use of quadratic
interpolation for the spatial derivations, rather than the more traditional
linear representation associated with the central-difference scheme.
The QUICK scheme was first applied to the modeling of plug flow in a
steady one-dimensional open-channel flow. The scheme was compared
with four other schemes and, with the exception of the higher order
six-point explicit scheme, the QUICK scheme was shown to have favor17
J. Environ. Eng., 1988, 114(1): 3-20
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able properties, particularly with respect to modeling the relatively high
solute concentration gradients.
The two-dimensional mathematical model, including the QUICK
scheme, was then applied to the modeling of plug flow through a sitespecific chlorine contact tank, namely the Elan chlorine contact tank in
Birmingham, England. The mathematical model results were compared
with physical model results obtained from a previous study, with the
agreement between the predicted and laboratory measured velocities and
flow-through curves being particularly encouraging, except when a strong
local three-dimensional velocity field was exhibited near the upstream
submerged weir. The QUICK scheme again exhibited much improved
predictions of the flow-through curves, in comparison with the traditional
central difference scheme, and this improvement was particularly marked
when the simulations of flow through the contact tank were undertaken in
the absence of dispersion and diffusion.
The QUICK scheme has therefore been shown to be favorably attractive
for modeling relatively high solute gradients, with its marginally more
complex finite-difference form, relative to the central-difference scheme,
being outweighed in comparison with the increase in accuracy. The
mathematical model, including the QUICK scheme, is particularly suited
to the design of such hydraulic basins as chlorine contact tanks, where an
analysis of a tracer plug source through the tank allows the retention time
to be optimized.
ACKNOWLEDGMENTS
This research study was predominantly undertaken while the first writer
was a lecturer in civil engineering at the University of Birmingham and
while the second writer was on sabbatical from Tong-Ji University, China.
The writers therefore wish to acknowledge the support and encouragement
given by both universities and the helpful comments and guidance given by
Dr. Hugh Tebbutt, of Birmingham, and Professor Gu, of Tong-Ji.
The research study outlined herein was also funded by Binnie and
Partners, Consulting Engineers, England, and the writers wish to express
their appreciation of Binnie's financial support and, in particular, the
technical support and encouragement of Mr. Graham Thompson, Divisional Director.
The physical modeling study referred to herein was funded by the
Severn Trent Water Authority (Tame Division) and the writers wish to
express their thanks to the then Divisional Manager, Mr. R. Hattersley, for
the authority's support. The study was also greatly aided by the helpful
contribution of Mr. E. S. Kain, systems engineer of Tame Division.
The writers are indebted to the referees for their helpful comments and
suggestions.
APPENDIX I. REFERENCES
Elder, J. W. (1959). "The dispersion of marked fluid in turbulent shear flow." J.
FluidMech., 5(4), 544-560.
Falconer, R. A. (1984). "Temperature distributions in a tidal flow field." / .
Envir. Engrg., ASCE, 110(6), 1099-1116.
Falconer, R. A. (1986). "Water quality simulation study of natural harbor." J.
18
J. Environ. Eng., 1988, 114(1): 3-20
Downloaded from ascelibrary.org by Colorado State Univ Lbrs on 03/06/19. Copyright ASCE. For personal use only; all rights reserved.
Wtrway., Port, Coast. Oc. Engrg., ASCE, 112(1), 15-34.
Falconer, R. A., Wolanski, E., and Mardapitta-Hadjipandeli, L. (1986).
"Modeling tidal circulation in island's wake." J. Wtrway., Port, Coast. Oc.
Engrg., ASCE, 112(2), 234-259.
Falconer, R. A., and Tebbutt, T. H. Y. (1986). "A theoretical and hydraulic
model study of a chlorine contact tank." Proc. Inst. Civ. Eng., London,
England., 81(2), 255-276.
Fischer, H. B. (1973). "Longitudinal dispersion and turbulent mixing in open
channel flow." Annu. Rev. Fluid Mech., 5, 59-78.
Hart, F. L. (1979). "Improved hydraulic performance of chlorine contact
chambers." J. Water Pollut. Control Fed., 51(12), 2868-2875.
Hart, F. L., and Vogiatzis, Z. (1982). "Performance of modified chlorine contact
chamber." J. Envir. Engrg. Div., ASCE, 108(EE3), 549-561.
Henderson, F. M. (1966). Open channel flow. Macmillan Publishing Co., New
York, N.Y.
Holley, F. M., Jr., and Preissmann, A. (1977). "Accurate calculation of
transport in two-dimensions." J. Hydr. Div., ASCE, 103(HY11), 1259-1277.
Komatsu, T., et al. (1985). "Numerical calculation of pollutant transport in one
and two dimensions." / . Hydrosci. Hydraul. Eng., 3(2), 15-30.
Kothandaraman, V., Southerland, H. L., and Evans, R. L. (1973). "Performance characteristics of chlorine contact tanks." J. Water Pollut. Control Fed.,
45(4), 611-619.
Leonard, B. P. (1979). "A stable and accurate modelling procedure based on
quadratic upstream modelling." Comput. Methods Appl. Mech. Eng., 19,
59-98.
Leendertse, J. J. (1970). "A water-quality simulation model for well-mixed
estuaries and coastal seas: Vol. 1. principles of computation." RM-6230-RC,
The Rand Corp., Santa Monica, Calif., 1-71.
Mardapitta-Hadjipandeli, L. (1985). "Numerical modelling of tide-induced
circulation," thesis presented to the University of Birmingham, at Birmingham, England, in partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
McNaughton, J. G., and Gregory, R. (1977). "Disinfection by chlorination in
contact tanks." Technical Report No. TR.60, Water Research Centre, Medmenham, U.K., D e c , 1-9.
Preston, R. W. (1985). "The representation of dispersion in two-dimensional
water flow." Report No. TPRDILI2783IN84, Central Electricity Research
Laboratories, Leatherhead, England, May, 1-13.
Sawyer, C. M., and King, P. H. (1969). "The hydraulic performance of chlorine
contact tanks." Proceedings of the 24th Industrial Waste Conference, External
Series, Vol. 135, Purdue Univ., West Lafayette, Ind., May, 1151-1168.
Schlicting, H. (1960). Boundary layer theory. 4th Ed., McGraw-Hill Book Co.,
New York, N.Y.
Thayanithy, M. (1984). "Hydraulic design aspects of chlorine contact tanks,"
thesis presented to the University of Birmingham, at Birmingham, England, in
partial fulfillment of the requirements for the degree of Master of Science.
APPENDIX II. NOTATION
The following symbols are used in this paper.
C
Co
L*xx » *^xy > *-*yx > *^yy
f
9
depth mean concentration in tank;
average concentration for tank volume;
combined depth mean dispersion and diffusion
coefficients in the x, y-plane;
Darcy friction factor;
acceleration due to gravity;
19
J. Environ. Eng., 1988, 114(1): 3-20
H
j, k
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k,
ks
K
n
Q
Re
T
TR
t
U, V
• IV)
Vol
x,y
P
At
Ax
r\
v
a
xx > Vyy
T
fct > Jby
Try > Ty.v
= total depth of fluid column;
= finite difference coordinates referring to x, ylocations, respectively;
= longitudinal dispersion coefficient;
= bed roughness length parameter;
= turbulent diffusion coefficient;
= time step location;
= throughflow in tank;
= Reynolds number;
= time after dye injection;
= theoretical retention time;
= time;
= depth mean velocity components in x, y-directions, respectively;
= depth mean fluid speed;
= volume of tank;
= coordinate directions in horizontal plane;
= momentum correction factor;
= time step size;
= grid spacing;
= water surface elevation above datum;
= kinematic viscosity of fluid;
=
direct stress components in x, y-directions,
respectively;
=
De
d shear stress components in x, y-directions,
respectively; and
=
lateral shear stress components in x, y-directions, respectively.
20
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