The document summarizes an experiment that measured the advection and dispersion of dye in a simulated river system under room temperature and ice water conditions. Key findings include: 1) Dye dispersed faster and mixed homogenously in room temperature water, but stratified between temperature layers and dispersed more slowly in ice water. 2) Modeling of the data found best fit when the ice water flow rate was decreased to represent stratification, as the model assumed homogeneous mixing. 3) The experiment supported the hypothesis that residence time would be higher and dispersion lower in cold water, though dye stratified rather than mixing as initially predicted.

1. ENVS 440/540, 2014 Vol 5:11-20 Pretend Journal of Environmental Modeling
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Measurement and modeling of advection and dispersion in an experimental channel
under contrasting hydraulic conditions:
A submission for Assignment A5 in ENVS 440/540, CSUMB, Fall 2014
McBrady, A1
& Watson, F2
.
Division of Science & Environmental Policy, California State University Monterey Bay, Seaside, CA, USA.
1
Conducted the experiment, did the modeling, and wrote the methods, results, and conclusion.
2
Conceived the experiment and wrote the introduction, goals, and postulate.
Abstract
Measurements and modeling methods were used to assess the movement of a pollutant through a simulated river
system in terms of its advection and dispersion. We recorded the concentration of Rhodamine dye as it traveled
downstream in a flume structure in both a room temperature water run and an ice water run. The ADK simulation
model was utilized in R using the data values from the room temperature run as the reference. We compared the ice
water run to the reference and adjusted the parameters to increase the fit to observed reference data. The physical
movement of the dye responded unexpectedly in the ice water by stratifying between the temperature layers and not
homogeneously as predicted. The best fit to the model came from adding an additional free parameter to decrease
the flow rate of the dye to represent the stratification as the unaltered model was only based on homogeneous
vertical mixing. This model is a best-fit approximation of homogeneous pollution transport using the ADK model
but a more accurate model may exist that incorporates more parameters that include possible stratification of
pollutants in the system.
Introduction
The impact of water-borne pollutants is dependent on
the processes governing how pollutants are transported.
Of key interest is the distribution of residence times of
water moving through the system, and the interaction
of this with rates of pollutant decay over time. For
example, wetland design seeks to maximize residence
time to allow for maximum pollutant decay, and to
minimize ‘short-circuiting’, whereby some water
parcels are allowed to pass through with insufficient
residence time.
Pollutant transport in stream channels and wetlands is
often characterized as a one-dimensional advection-
dispersion-decay system, whereby spatial variation
assumed to only occur longitudinally along the
channel, flow is assumed to be homogeneous across the
channel cross-section, and the dominant transport-
related processes are assumed to be longitudinal
advection and dispersion, and some form of decay.
Within this context, parameters for advection rate,
dispersion rate, and instantaneous rate of decay govern
the residence time and net decay of the system. In turn,
these reflect physical drivers such as the volumetric
discharge, water depth, turbulence, and interaction with
decay-related processes.
To explore these ideas, we constructed a laboratory
flume, operated it with a variety of different physical
drivers, monitored the transport of a tracer dye moving
through the system, and fitted a theoretical 1D
advection-dispersion-decay model to the monitoring
data. In particular we sought to compare the physical
system between two contrasting physical
configurations, and to use the fitted model parameters
to measure the contrast in hydraulic terms.
The contrast that we explored running the system under
involved using two different water temperatures to
compare possible changes of pollutant dispersion in the
river model. The first used room temperature water
through the flume system as the control reference for
the model while the second used ice water as the
experimental factor.
We postulated that the pollutant dispersion would be
lower in a cold water environment than in room
temperature conditions and have a higher residence
time. The dispersion and advection of a liquid shares a
similar relationship to thermal expansion of gasses. As

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a gas is heated, that heat energy is transferred to the
molecules as kinetic energy which causes them to
distance themselves from eachother and increase the
volume of the gas. We believed under the same
pressure conditions that the rate of dispersion of a
pollutant released in colder water would therefore be
lower than its dispersion in warmer water because there
would be less available heat energy in the cold water to
drive expansion and would stay in system longer.
Overall approach
A flume and a digital model were created as proxy of a
natural river system in order to observe and measure
the concentration and behavior of an imitation pollutant
(dye) as it traveled downstream in contrasting water
temperatures. We recorded the concentration of the
pollutant through the flume using room temperature
water and used those measurements as the parameter
values in a digital model using R software (R Core
Team. 2014). The experimental data gathered from
running the flume but substituting the room
temperature water for ice water was then compared to
the reference model and adjusted to fit.
We expected that the pollutant would disperse faster
with the room temperature water than in the ice water.
We expected that the pollutant would mix
homogeneously with the room temperature water faster
than the ice water. We also expected that the
concentration of the pollutant flowing through the ice
water would be higher at the same point downstream as
the room temperature water due to the reduced
dispersion.
Experimental flume
A long acrylic flume (3.6 m long×0.0254 m wide) was
used for both temperature contrast tests (Fig. 1). The
steady flow of water from the left side (upsteam) to the
right side (downstream) of the flume was monitored by
Camelbak flow meters. The water was pumped into the
left side of the flume from a storage tank located below
the flume and then pumped out into a receptacle tank.
A drop of Rhodamine dye (0.21 kg/L stock soln.) was
added at the upstream end of the flume to simulate the
introduction of a pollutant and monitored as it flowed
in the system. Two concentration probes (U1 and U2)
were placed downstream from the dye drop position to
measure the pollutant concentration at different points
in the model river system which were recorded by data
loggers. Dechlorinator was periodically added to the
flume system as chlorine decays the dye and would
lower the accuracy of the concentration probes.
Figure 1. Setup of flume modeling the flow of pollutants
through a river system with the right side representing
downstream.
*The colored dye was not easily visible in the black and
white photograph and is highlighted by a line which traces
the top of the dye layer.
Experiments
We assessed the pollution transport in the system under
room temperature (23 °C) conditions as the control
reference and then used ice water (8 °C) as the
experimental factor to observe possible differences in
dispersion based on temperature.
The reference run had the concentration probes
positioned 1.33 m and 2.5 m downstream from the left
side of the flume and the cold run had the probes
placed at 1.33 m and 2.495 m (U1 and U2
respectively). The time for each run was recorded and
additional qualitative observations were made of the
comparative movement of the dye through the flume
system.

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Model
We used the CSUMB ADK Model (Daniels et al.
2014), or “advection, dispersion, decay” model to
recreate our observed data in an R simulation. The
ADK model is based on the partial differential ADE
(advection dispersion equation) to track the
concentration of a pollutant over time.
𝛅𝑪
𝛅𝒕
= −𝑼
𝛅𝑪
𝛅𝒙
+ 𝑬
𝛅2
𝑪
𝛅𝐱2
− 𝑲
where
𝛅𝑪
𝛅𝒕
is the concentration C (mg/L) of a pollutant
over time t (s), U (m/s) is the flow velocity,
𝜹𝑪
𝜹𝒙
is the
gradient concentration over space, E is the mixing rate
or dispersion coefficient, and K is the decay rate. The
ADK model uses a discrete approximation of this
differential equation to measure the change in
concentration values of Rhodamine dye over time from
the spatial and measured parameters. We created fitted
parameters to minimize the difference between the
concentration and flow observations and the model.
Through fitting the model to the observed data. All
measured values and model parameters for the
reference and cold runs are stated in Table 1.
Results
The ADK model reference run is shown in Figure 2
and the experimental results of the cold run model are
presented in Figure 3. The quantitative results
supported the postulate that residence time was higher
and the dispersion of the pollutant in the cold river
system was lower comparatively to its dispersion in a
warm river system, but not in the way that it was
predicted in the postulate. Specifically, the dye in the
reference run mixed homogenously while the dye in
the cold run did not mix vertically as predicted but was
transported through advection and stratified into a
Table 1. Summary of all variables in the ADK model in R simulating Pollution Transport
Variable name Type Symbol Units Reference Run Cold Run
Flume Length Measured Parameter Len_m m 3.6 3.6
Flume Width Measured Parameter Wid_m m 0.0254 0.0254
Water Depth in Flume Measured Parameter Dep_m m 0.0815 0.116
Flow Rate Measured Parameter Q_Ls L/s 1/30 0.035
Dye Residence Time Measured Parameter Duration_s s 300 489
Position of Dye
Addition
Probe 1 Concentration
Reading
Measured Parameter
Measured Parameter
x_dose_m
PPM_per_mV_U1
m
ppm/mV
0.39
0.0025
0.1
0.0025
Probe 2 Concentration
Reading
Probe 1 Position
Measured Parameter
Measured Parameter
PPM_per_mV_U1
x_obs_m
ppm/mV
m
0.0025
1.33
0.0025
1.33
Probe 2 Position Measured Parameter x_obs_m m 2.5 2.495
Flow Rate Multiplier Fitted Parameter Q_Multi N/A N/A 0.315
Dispersion Fitted Parameter Disp_m2s m2
/s 0.0005 0.00036
Probe 1 Calibration
Coefficient
Fitted Parameter Probe_fudge_U1 N/A N/A 11
Probe 2 Calibration
Coefficient
Background Dye
Fitted Parameter
Fitted Parameter
Probe_fudge_U2
background_obs_mgL
N/A
mg/L
N/A
0
9
1.5

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diagonal ribbon of dye sandwiched between the cold
and warm water layers (see Fig 1.).
The reference run measured higher dye concentration
at probe U1 and slightly lower concentrations further
downstream at U2. The drop of Rhodamine dye entered
the system at 0.43 m along the flume and had a 300 s
residence time in the river model. The water depth was
measured at 0.0815 m and the flow rate was measured
at 0.033 m/s. The dispersion rate was estimated at
0.0005 m2
/s through the flume. There were no
background dye concentrations observed in the
reference run.
The drop of Rhodamine dye entered the cold run at
0.15 m and had a 489 s residence time in the river
model. The water depth was measured at 0.116 m. The
flow rate was measured at 0.035 m/s, but then adjusted
to 0.011 m/s using a Flow Rate Multiplier to better fit
the model. The dispersion rate was estimated at
0.00036 m2
/s. To achieve better fit between the model
and observed data, probes U1 and U2 were recalibrated
with coefficient values of 11 and 9 (respectively). The
background dye concentration in the cold run was
increased to 1.5 mg/L to better fit the observed data to
the model.
Discussion
The postulate was supported in that the residence time
increased and dispersion decreased in the cold model
compared to the warmer reference run but not in the
ways that the postulate predicted. The ADK pollutant
transport model that we used was based on the
assumption that the dye would mix homogenously in
the flume. The reference run followed the expected
model and mixed vertically but did not in the cold run.
As the ice water moved through the system it traveled
along the bottom of the flume as cold water is denser
than warm water. The density of the cold water would
have been less than 0.99975 g/cm3
while the room
temperature water would be more than 0.99802 g/cm3
(USGS 2014).
Even though we gave the system a few minutes to
stabilize after the different temperature waters were
exchanged, some room temperature water remained in
the flume. After the room temperature dye was added,
it became wedged between the cold layer moving
forward quickly along the bottom and the standing
warmer layer on top. Similar to the behavior of warm
and cold fronts in terms of air current, the dye became
compressed between the layers and was pulled along
the top of the denser ice water as it moved forward
creating an unpredicted, stratified dye layer.
Since the dispersion and advection rates could not be
measured directly, they were found from fitting the
model to the observational data. With all other
parameters being relatively equal, the fitted parameters
represented the last unknown variables which were
indirectly revealed the closer the model fit to the
observed data. The best fit between the model and the
observed cold run data came from including a
multiplier that decreased the flow rate and coefficients
that increased the probe readings which are listed in
Table 1.
The model for the reference run had a fairly accurate fit
along the observed data points but the fit for the cold
run was more erratic as the concentration readings for
both probes consistently fluctuated (see Fig 2 & 3.). It
is possible that dye from previous experimental or
reference runs was left on the concentration probes that
obstructed the readings. This coincides with having to
increase the background dye concentration to better fit
the model to the observed experimental data. The
differences between the concentration readings in the
models could also be due to the stratified dye layer
being further away from the sensors which could have
made it difficult for the probes to gauge the
concentration accurately as there was a gap of water
separating the dye from the probes.
Figure 2. Concentration vs Time of the room temperature
Reference run with the U1 probe upstream (black) and U2
probe downstream (red) concentrations.

5. ENVS 440/540, 2014 Vol 5:11-20 Pretend Journal of Environmental Modeling
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Figure 3. Fitted Concentration vs Time of the ice water Cold
Run with the U1 probe upstream (black) and U2 probe
downstream (red) concentrations.
Conclusions
The postulate was quantitatively supported by the ADK
model as the dispersion of the pollutant was decreased
in the ice water by 28% and had an increased residence
time of 63%, but the overall movement of the pollutant
in the flume system behaved unexpectedly and formed
a stratified dye layer. The postulate was not
qualitatively supported by the model as it assumed
homogenous mixing and the cold run did not conform
to that model. Fitted parameters were added to the
existing model to mitigate the effects associated with
the observed lack of vertical mixing. Further research
can be done using different water temperatures to test if
this is supported in all cases. Additionally, future
models may be able to integrate new parameters that
address non-homogenous mixing as to increase the
overall accuracy and versatility of the model.
References
Daniels, M.E., Hogan, J.N., Smith, W.A., Oates, S.C., Miller,
M.A., Hardin, D., Shapiro, K., Los Huertos, M., Conrad,
P.A., Dominik, C., Watson, F.G. 2014. Estimating
Environmental Conditions Affecting Protozoal Pathogen
Removal in Surface Water Wetland Systems Using a Multi-
Scale, Model-Based Approach. Science of the Total
Environment, 493:1036-1046
R Core Team. 2014. R: A language and environment
for statistical computing. R Foundation for Statistical
Computing, Vienna, Austria.URLhttp://www.R-
project.org/.
USGS. 2014. Density and weight of water, at standard
sea-level atmospheric pressure. Water Density
http://water.usgs.gov/edu/density.html