EPA/600/R-17/326 | September 2017| www.epa.gov
United States
Environmental Protection
Agency
Water Quality Assessment
Simulation Program (WASP8):
Upgrades to the Advanced
Toxicant Module for Simulating
Dissolved Chemicals,
Nanomaterials, and Solids
Office of Research and Development
National Exposure Research Laboratory
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EPA/600/R-17/326 | September 2017| www.epa.gov
United States
Environmental Protection
Agency
SEPA
Water Quality Assessment
Simulation Program (WASP8):
Upgrades to the Advanced
Toxicant Module for Simulating
Dissolved Chemicals,
Nanomaterials, and Solids
CSS 10.04 Emerging Materials
Project Leads: William Boyes and Dermont Bouchard
Task 3: Develop Predictive Modeling and Tools
Task Leads: Dermont Bouchard and Paul Harten
QA Category: Level 4 (Category B)
Computational Exposure Division
Athens, GA
Office of Research and Development
National Exposure Research Laboratory
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Authors
Robert B. Ambrose. Jr., Emeritus, USEPA, Office of Research and Development. Athens. GA
Brian Avant, ORISE, USEPA, Office of Research and Development, Athens, GA
Yanlai Han, Ph.D., ORISE, USEPA, Office of Research and Development, Athens, GA
Christopher D. Knightcs, Ph.D., USEPA, Office of Research and Development, Athens, GA
Tim Wool, USEPA, Region 4, Atlanta, GA
Disclaimer
This document has been reviewed in accordance with U.S. Environmental Protection Agency
policy and approved for publication. Mention of trade names or commercial products does not
constitute endorsement or recommendation for use.
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Table of Contents
1.0 Introduction 1
2.0 The Advanced Toxicant Module 2
3.0 Solids Transport 3
3.1. Introduction 3
3.2. Theory 3
3.2.1. Solids Systems 3
3.3. Implementation 3
3.4. Water Body Compartments 4
3.4.1. Stream Sedimentation Regimes 4
3.5. Descriptive Solids Transport 4
3.6. Process Based Solids Transport 5
3.7 Biotic Solids Production and Dissolution 7
3.8. Solids Burial 7
3.9. Analytical Solution for Settling 8
3.10. Example 9
4.0 Light 11
4.1. Introduction 11
4.2. Theory 11
4.2.1 Light Attenuation 11
4.3. Implementation 11
4.3.1 Input Total Radiation 11
4.3.2. Light Attenuation Above Water Surface 12
4.3.3. Light Attenuation Below the Water Surface 12
5.0 Particle Attachment 14
5.1. Introduction 14
5.2. Theory 14
5.2.1. Equilibrium Sorption 14
5.2.2. Kinetic Sorption 15
5.2.3. Nanomaterial Heteroaggregation 15
5.3. Implementation 15
5.3.1. Equilibrium Sorption 15
5.3.2. Kinetic Sorption 16
5.3.3. Nanomaterial Heteroaggregation 16
6.0 Nanomaterial Reactions 17
6.1. Introduction 17
6.2. Nanomaterial Reactions 17
6.2.1. General Reactions Module 17
6.2.2. Effect of Temperature 18
6.2.3. Phase Multiplication Factor 18
6.2.4. Segment Multiplication Factor 18
6.2.5. Monod Kinetics 18
6.2.6. Environmental Kinetics 18
6.3. Nanomaterial Photo transformation 18
6.4. Implementation 19
6.5. Examples 19
6.5.1. Silver Nanoparticles Dissolution 19
6.5.2. Calculation of Nanomaterial Phototransformation Rate Constants 19
6.5.3. Nanomaterial Parallel Reaction 21
6.5.4. Segment Multiplication Factor 21
6.5.5. Phase Multiplication Factor 21
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7.0 Dissolved Chemicals Reactions 23
7.1. Introduction 23
7.2. Simulation for Oxidation, Reduction and Biodegradation 23
7.2.1. Generic Reaction Module 23
7.2.2. Effect of Temperature 23
7.2.3. Phase Multiplication Factor 23
7.2.4. Segment Multiplication Factor 23
7.2.5. Monod Kinetics 24
7.2.6. Environmental Kinetics 24
7.2.7. Second Order Kinetics 24
7.2.8. Biodegradation 24
7.3. Dissolved Chemicals Phototransformation 24
7.4. Examples 24
7.4.1. First-Order Biodegradation Influenced by Temperature 24
7.4.2. Second-Order Reaction 25
8.0 References 26
9.0 APPENDIX A-1
9.1 Solids QA/QC A-l
9.1.1. Scenario 1 - Constant Boundary Conditions, Initial Concentration = 0, No Settling A-l
9.1.2. Scenario 2 - Initial Conditions, Boundary Condition = 0, No Settling A-2
9.1.3. Scenario 3 - Initial Concentration. With and Without Streatnflow. Settling A-2
9.1.4. Scenario 4 - Resuspension A-3
9.2. Light QA/QC A-4
9.2.1. Option 0, Calculated Diel Light A-5
9.2.2. Option 1, User Input Diel Light A-6
9.2.3. Option 2, User Input Daily Light, Calculated Diel Light A-l
9 3 Particle Attachment QA/QC A-9
9.3.1. Equilibrium Sorption A-9
9.3.2. Kinetic Sorption A-9
9.3.3. Nanomaterial Heteroaggregation A-11
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is! of Figures
Figure 1. Stream Sedimentation Regimes 4
Figure 2 Noncohesive erosion and resuspension 6
Figure 3. Solids Concentration in the Surface Water 9
Figure 4. Solids Concentration in the Surface Sediments 10
Figure 5. Solids Concentration in the Subsurface Sediments 10
Figure 6. Sediment Burial Rates 10
Figure 7 Possible Nanomaterial Reaction Pathways. Chem indicates dissolved chemicals (e.g., organics, metal ion). . . . 17
Figure 8. (a) [Ag] experimental data and WASP8 simulation results, (b) [Ag+]released experimental data and WASP8
simulation results 20
Figure 9. Nanomaterial parellel reaction pathways 21
Figure 10. Nanomaterial parallel reaction simulation results using WASP8 21
Figure 11. WASP8 simulation results 22
Figure 12. WASP8 output of biodegradation rate at different temperatures, fitted with analytical solution 25
Figure 13. Second-order reaction simulation results by WASP8 25
Figure 14. Comparison of WASP Simulation to Analytical Solution for Scenario 1 A-l
Figure 15. Comparison of WASP Simulations to Analytical Solutions for Scenario 2 A-2
Figure 16. Comparison of WASP Simulations to Analytical Solutions for Scenario 3, With Streamflow A-2
Figure 17. Comparison of WASP Simulations to Analytical Solutions for Scenario 3, Without Streamflow A-3
Figure 18. Comparison of WASP Simulations to Analytical Solutions for Scenario 4, Solids in Sediment Layer A-4
Figure 19. Comparison of WASP Simulations to Analytical Solutions for Scenario 4, Solids in Water Column A-4
Figure 20. Comparison of Analytical Solution and WASP Simulation for Sorption Kinetics of Scenario 1. Chem 1
represents the freely dissolved chemical concentration, and Chem 2 represents the chemical concentration sorbed
to the suspended solid A-10
Figure 21. Comparison of Analytical Solution and WASP Simulation for Sorption Kinetics of Scenario 2. Chem 1
represents the freely dissolved chemical concentration, and Chem 2 represents the chemical concentration sorbed
on the suspended solid A-10
Figure 22. Comparison of Analytical Solution and WASP Simulated Nanoparticle Concentration over Time
for Scenario 1 A-12
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is! of Table:
Table 1. Particle size classification 3
Table 2. Particle densities 3
Table 3. Channel Geometry of WASP Segments 9
Table 4. Solids Demonstration Parameters 9
Table 5. WASP Default Light Extinction Coefficients for the Wavelength Bands 13
Table 6. Division of Wavelengths by Wave Class 13
Table 7. Division of Wavelengths into Wave Bands, with Fractions Given by Latitude 13
Table 8. Equilibrium Sorption Parameters 14
Table 9. Data retrieved from publication and WASP8 simulated results 20
Table 10. Wavelength-dependent reaction rate of each wavelength band 20
Table 11. Parameters used in Analytical Equations A-l
Table 12. Channel Geometry of WASP Segment A-l
Table 13. Physical properties of solids A-2
Table 14. Channel Geometry of WASP Segments A-4
Table 15. Attenuation Fractions Used for Above Surface Parameters A-4
Table 16. Average Ke Outputs for Each Attenuation Parameter by Light Group A-5
Table 17. WASP Outputs by Wavelength Group for Option 0 Using Athens, GA Coordinates A-5
Table 18. Comparison of WASP Outputs to Analytical Solutions for Above Surface Light Attenuation-Option 0 .... A-5
Table 19. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation - Option 0 .... A-6
Table 20. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation-Option 0 .... A-6
Table 21. Comparison of WASP Outputs to Analytical Solutions for Above Surface Light Attenuation-Option 1 .... A-6
Table 22. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation-Option 1 .... A-7
Table 23. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation-Option 1 .... A-7
Table 24. WASP Outputs by Wavelength Group for Option 2 Using Athens, GA Coordinates A-7
Table 25. Comparison of WASP Outputs to Analytical Solutions for Above Surface Light Attenuation-Option 2 .... A-8
Table 26. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation-Option 2 .... A-8
Table 27. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation-Option 2 .... A-8
Table 28. Variables Related to Equilibrium Sorption A-9
Table 29. Parameter Values and Results for each Scenario A-9
Table 30. Sorption Kinetic Constants and Analytical Errors of Two Scenarios A-9
Table 31. Heteroaggregation Kinetics Parameters A-11
Table 32. Channel Geometry of WASP Segment A-11
Table 33. Calculated Rate of Collision by Mechanism A-11
Table 34. Parameters for Different Cases of a for Brownian Motion Scenario A-12
Table 35. Parameters for Different Cases of a for Brownian Motion and Fluid Motion Scenario A-12
Table 36. Calculated and Simulated Nanoparticle Concentrations Using Different Alphas A-13
Table 37. Analytic Solutions for Heteroaggregation Constants Using A-13
Table 38. Nanoparticle Concentrations Using Different Alphas A-13
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1.0
The Water Quality Analysis Simulation Program (WASP)
is a dynamic, spatially-resolved, differential mass balance
fate and transport modeling framework. WASP is used to
develop models to simulate concentrations of environmental
contaminants in surface waters and sediments. As a modeling
framework, it allows users to construct the model design
that is appropriate for the system of interest, in one, two, or
three dimensions. WASP allows for time-varying processes
of advection, dispersion, point and diffuse mass loading,
boundary conditions and boundary exchange. WASP can be
linked to hydrodynamic and sediment transport models, or
the hydrodynamic algorithms within the WASP framework
can be used. WASP is one of the most widely used water
quality models in the USA and throughout the world. It has
been applied in development of Total Maximum Daily Loads
(TMDLs)1,2; simulation of nutrients in Tampa Bay, FL3; and
remediation strategies for mercury in the Sudbury River,
MA4-5.
WASP is an enhancement of the original version developed
in the 1980s 6 S. Over the years, it has undergone many
improvements and enhancements. In July 2017, WASP8
(WASP, v.8.1) was released. It was a complete overhaul
and recoding that moved from Fortran77 to Fortran95 to
take advantage of many updated features. This release also
incorporated a new WASP interface, a new post-processor
(WRDB), and the ability to run on a PC or Mac.
WASP8 contains two modules: the Advanced Eutrophication
module and the Advanced Toxicant module. These provide
the tools to simulate different environmental contaminants
of concern. Advanced Eutrophication simulates conventional
pollutants (e.g., dissolved oxygen, nitrogen, algae). Over
the years, much of WASP's development focused on the
eutrophication module, details that are not part of this
document. Similarly, the overall workings of WASP, such as
how to construct a WASP model or how to simulate flow, are
not included. The focus here is on the Advanced Toxicant
module's recent advances.
WASP's Toxicant module (WASP TOXI, WASP7 and earlier)
originally focused on dissolved organic contaminants, with
the capacity to model metals in a descriptive manner and
simulate solids. The new Advanced Toxicant module in
WASP8 was completely rewritten to advance it from being
able to simulate three chemicals and three solids (generally,
sands, fines, and particulate organic matter). WASP8 now
permits simulation of seven state variables: dissolved
chemicals, nanomaterials, solids, dissolved organic carbon,
temperature, salinity, and bacteria. For each variable, the
number of how many can be simulated is pre-set (e.g., up
to 10 chemicals); due to the new architecture, these can be
increased with small adjustments in the source code, by
request.
As part of the upgrade to WASP8, and the incorporation of
nanomaterials as a new state variable, we present here an
overview of the theory and application of:
Solids Transport
Light and Phototransformations
Particle Attachment
Nanomaterial Transformation Reactions
Chemical Transformation Reactions
Each topic is discussed by providing an introduction; the
theory of the topic with equations used to describe the
processes; implementation of this module within WASP; and
specific examples in some cases. Quality assurance/quality
control (QA/QC) tests to verify that incorporation of these
processes into WASP match analytical results are presented
in Appendix A.
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The WASP8 Advanced Toxicant Module is structured to
provide flexibility in constructing the processes that govern
the contaminant(s) of interest. The user selects the type of
state variables and number of each. The type of contaminants
that WASP8 can simulate, ranging from simple to more
complex, are shown below.
Metals
o Copper, Lead, Zinc, Cadmium
o Arsenic, Tin, Selenium, Chromium
Mercury
o Elemental, Divalent, Methyl (Explicit Mercury Model
to Be Released)
Organics
oMTBE, PCB Homolog
o Petroleum, BTEX, PAHs, Chlorinated Solvents,
VOCs
o Pesticides, Organic Acids
Nanomaterials
o Carbon Nanotubes, Graphene Oxide, Titanium
Dioxide, Silver Sulfide
WASP8's Advanced Toxicant Module is a noticeable advance
over WASP7. The types and numbers of state variables now
include:
Dissolved Chemicals
(n =
= 1 to
10+)
Nanomaterials
(n =
= 1 to
10+)
Solids
(n =
= 1 to
10+)
Dissolved Organic Carbon
(n =
= 1 to
5+)
Bacteria
(n =
= 1 to
5+)
Temperature
(n =
= 1)
Salinity
(n =
= 1)
The maximum number of variables for each class of state
variable is set as a limit in the code, however, it can be
updated by adjusting and recompiling the source code.
We discuss the three main state variables of dissolved
chemicals, nanoparticles, and solids. Solids are particularly
important for simulating toxicants because they provide
a surface for attachment. Their transport is governed by
associated particles. Light governs phototransformations of
dissolved chemicals and nanoparticles, so we incorporate the
new way that WASP handles light and phototransformations.
Simulation of Dissolved Organic Carbon, Bacteria,
Temperature, and Salinity are not incorporated here, as it is
beyond the scope of this effort. Some processes governing
chemicals are also left out (e.g., volatilization). Details on
how to simulate these will be incorporated as develop the
user's guide documents that is part of the WASP executable.
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3.0
Solids Transport
3.1. Introduction
Suspended and benthic solids are important components
of water quality. Excess suspended solids concentrations
can harm fish directly through direct mortality, or by
reducing their growth rate and resistance to disease. High
concentrations increase light attenuation and surface heating;
the consequential reductions in light affect algal growth rates
and abundance of food available to fish. Excess silts can
blanket benthic spawning areas and damage invertebrates.
Organic deposits can reduce dissolved oxygen levels, causing
an imbalance in natural biota.
Solids affect conventional water quality through sorption of
nutrients. Sorption reduces dissolved ammonia (NH4) and
phosphate (P04) fractions, reducing nitrification and algal
uptake and growth. Particulate nutrient fractions are removed
from the water column by deposition and returned by erosion
and resuspension.
Solids also affect the fate of potential toxicants, including
organic chemicals and nanomaterials. Sorption reduces
their dissolved fraction and bioavailability, and deposition
removes them from the water column, attenuating some peak
loading events. Net deposition stores chemicals in sediments
for long periods. Pore water diffusion and resuspension
return chemicals to the water column between loading events.
Large flood events scour significant amounts of sediment
and chemical from the upper sediment to the water column.
Burial below bioturbation depth potentially sequesters
chemicals from biota.
The solids module is an independent set of routines for
the solids state variable. It has its own set of associated
Constants, Parameters, and Time Functions and is
implemented as a unit within each of these modules.
Like other state variables. Solids are transported between
segments by advection and dispersion. They can also settle
through the water column; deposit to the surface benthic
(sediment) layer; erode and resuspend back to the water
column; and bury to lower benthic layers. These processes
are described below. Using the Solids Option in the
Constants section. Solids Transport group, you can choose
the descriptive option (0) or one of the process-based solids
transport options (1 or 2) for each Solid system.
3.2. Theory
3.2.1. Solids Systems
WASP8 can simulate up to 10 different Solids systems,
each representing a discrete size range and density. You
must choose how many solids types to simulate, and then
specify their characteristic sizes and densities. Table 1 gives
characteristic size ranges for different classes of solids.
Table 2 gives typical densities.
Table 1. Particle size classification9
Common
Size Range
Wentworth Name
Name
1 to 100 nm
nanoparticle
nanoparticle
< 1 jjm
colloid
mud
1.0-3.9 |jm
clay
mud
3.9 - 62.5 jjm
silt
mud
62.5- 125 |jm
very fine sand
sand
125 - 250 |jm
fine sand
sand
0.25 - 0.5 mm
medium sand
sand
0.5 - 1 mm
coarse sand
sand
1 - 2 mm
very coarse sand
sand
2-4 mm
granule
gravel
4-64 mm
pebble
gravel
64 - 256 mm
cobble
gravel
> 256 mm
boulder
gravel
Table 2. Particle densities10
Substance Density [g/mL]
Organic matter (dry weight) 1.27
Siliceous minerals 2.65
Garnet sands 4.0
3.3. Implementation
Solids are specified in the Systems section. Each row is an
independent model system. Enter new rows by clicking the
'Insert' button or by setting the cursor to the bottom row
and pressing the keyboard's down arrow. To specify a solids
variable, double-click a cell in the System Type column and
select "SOLID"; a default name is provided in the System
Name column. You can specify a more descriptive name by
double-clicking a cell in that column.
Particle densities are also specified in the Systems section.
The Density column is preset to 1.0 g/mL, the nominal
density of water. WASP8 will reset the particle density to
2.65 g/mL if you do not specify an alternate density.
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The characteristic particle diameter for each Solid is specified
in the Constants section Solids Transport group. WASP8 will
assign a default value of 0.025 mm, which is characteristic of
silt, if you do not specify particle diameter.
3.4. Water Body Compartments
WASP8 model networks are composed of spatially-discrete
segments, or compartments. Detailed network segmentation
is best generated with special WASP builder software linked
to GIS platforms such as BASINS (https://www.epa.gov/
exposure-assessment-models/basins). Simple networks can
be specified directly in the WASP8 user interface.
Each segment is represented by a row in the Segments section
of the interface. Segments are specified in the Segment Type
column and consist of four types:
Surface Water
Subsurface Water
Surface Benthic
Subsurface Benthic
The Transport Mode determines how advective transport
through each segment is calculated. This is covered in the
WASP8 Advective Flow document.
For networks with vertical discretization, map segments
vertically, using the Segment Below column. The default
setting is "None," indicating there is no model segment
immediately below the current one. To specify a segment
below, double-click in the cell and select the proper segment
from a pick list. WASP8 will internally map the segments in
vertical columns.
The water column - The water column is composed of
Surface Water and Subsurface Water segments, linked by
advective flow paths and dispersive exchanges.
The sediment bed - Sediment beds are layers composed of
Surface Benthic and Subsurface Benthic segments, arranged
in vertical stacks beneath a water column segment. Each
segment is defined by its bulk density, porosity, cohesiveness
and organic content. These properties are not specified
directly in WASP8, but are a product of the simulated
individual solids systems with their properties.
The initial total solids concentration in a benthic segment
[mg/L or g/m3] sets the reference bulk density [g/mL] and
porosity [Lw/L]. As described in Section 3.8, the solids mass
balance preserves reference bulk densities and porosities
for benthic segments when individual solids are added or
removed. Initial concentrations are specified in the Segments
section Initial Conditions group.
A benthic segment is considered "non-cohesive" or
"cohesive" based on whether the fraction of clay and silt-size
solids (those less than 0.10 mm) exceeds the specified critical
fraction. You can specify the "Critical cohesive sediment
fraction, above which bed acts cohesively" in the Constants
section Solids Transport group; the default value is 0.2. In
cohesive beds, clay and silt particles are eroded as a unit, but
eroded separately in non-cohesive beds.
3.4.1. Stream Sedimentation Regimes
The transport of solids in surface waters is governed to a
large degree by particle size and stream velocity (or
bottom shear stress).
1000
500
. 300
& 200
CO
g 100
o
EROSION
3.16
0.284
o
o
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Resuspension - The Solids Resuspension Velocity, w
[m/d], should be specified for resuspension from each
surface bcnthic segment. WASP multiplies wR by the solids
concentration in the segment [g/m3] to obtain the solids flux
[g/m2-d] to the segment above.
, ; . ;'»s Based Soli* isport
If Solids Option 1 or 2 is chosen. WASP8 will use a set
of solids constants, along with process-based equations
to calculate dynamic settling, deposition, erosion, and
resuspension velocities. While settling is a function
of particle si/c and density, deposition, erosion, and
resuspension arc functions of bottom shear stress. Erosion
and resuspension also depend on whether the sediment
bed is acting cohesively or noncohesively. Solids burial is
calculated internally based on mass balance calculations for
total solids within the bcnthic segments, which is described
in Section 3.8.
Bottom Shear Stress - Flowing water exerts a shear stress. %b
[N/m2], on the bcnthic surface layer. WASP8 uses the Darcy-
Weisbach expression for the grain-related bottom shear stress
(skin friction) that is a function of the average water velocity,
u [m/s], and water density, p [kg/m3]:
Tfa =
Pw/M
Equation 1
/is the Darcy-Wcisbach friction factor, estimated by:
log2 (12-?-)
%
Equation 2
where H is water depth |m|. D5g is median sediment grain
si/c |m|. and ks is the equivalent roughness height |m|.
calculated as 3D5g or 0.01 H, whichever is larger. Note that for
a bed of medium sand (0.5 nun), or finer, in streams greater
than 5 cm deep./assumes a constant value of 0.0253. With
values of pw close to 998 kg/m3, the bottom shear stress
simplifies to:
Tb = 3.16u2 Equation 3
Settling - Settling is the movement of solids down through
the water column. WASP8 calculates the settling velocity, ws
[m/s], for each solid using the van Rijn (1984) method. This
set of equations is based on mean particle diameter, Ds |ni|.
particle density. ps [kg/m'|, water density, pw [kg/m3], and
absolute viscosity, ft [kg/m-s]:
4₯d '
Rd
18
- It + O.OlRf,
1.1
D < 100nm
100nm < D < 1000fim
D > 1000jim
Equation 4
where Rd is the sediment particle dcnsinictric Reynolds
number:
*W 9'
i"! Pw
Equation 5
where g is the acceleration of gravity, 9.807 [m/sec2]. For
Ds < 100 (.nil (very fine sands and smaller), the van Rijn
expression reduces to Stokes' Law:
Ds r
: ir^ps
¦Pw)
Equation 7
For L\ > 1000 (.mi (very coarse sands and larger), and particle
density of 2650 kg/m3, the van Rijn expression simplifies to:
ws = 4.42 5^0^
Equation 8
In the model initialization phase. WASP8 calculates the
characteristic settling velocity for each simulated solid using
the input particle densities and diameters, along with nominal
values for water viscosity (0.001 kg/m-s) and water density
(1000 kg/m3).
Deposition - Deposition is the movement of solids from
the water column to the surficial bcnthic (or sediment) bed.
In noncohcsivc deposition, the settling of individual solids
particles is attenuated by the shear stress from water flow.
WASP8 calculates the deposition velocity. wD [m/s], for each
solid as the product of its settling velocity, ws, and probability
of deposition upon contact with bed. aD.
wD = wsaD
Equation 9
where an is a function of bottom shear stress, x., as well as
D y b?
the lower and upper critical shear stress thresholds, %cD1 and
%cD2. Using a formulation by Krone (1963), aD is equal to
1 for xb < tcD1, and equal to 0 for %b > %cD2. Within the critical
shear stress range. aD varies from 1 to 0 as bottom shear
stress rises from %cD1 to %cD2 in a roughly linear fashion:
a = (jEcmJbV0
^cDZ-tcDl'
Equation 10
where %cD1 and xcD2 arc in [N/m2], and yD is a dinicnsionlcss
exponent. For the default g/: of 1.0, the interpolation function
is linear.
These three constants arc input for each solid in the Constants
section. Solids Transport group. The lower critical shear
stress for deposition is generally considered to be close to
0.0 N/m2, while the upper critical shear stress for deposition
is in the range of 0.01 - 0.2 N/m2, depending on particle si/c.
For Solids Options 1 and 2, the default values for xcD1 and xcD2
arc set to 0.0 and 0.2 N/m2. For Solids Option 0, they are set
to 10 and 20 N/m2 so that, under all reasonable conditions,
deposition is set to the specified settling velocity.
Noncohesive Erosion - Noncohcsivc erosion is the
detachment of solids particles from the surface bcnthic
sediment into a mobile boundary layer. Resuspension is
the transport of the solids particles from the mobile layer
into the water column (Figure 2). In noncohcsivc bcnthic
segments, all solids particles are subject to noncohcsivc
erosion and resuspension. In cohesive bcnthic segments, only
sands and larger particles (> 0.1 mm diameter) arc subject to
noncohcsivc erosion and resuspension.
and
»' = »&-')
Equation 6
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Components of noncohesive sediment transport:
Erosion from surface sediment: Es, WE
Deposition from the boundary layer: Ds, Ws
Entrainment from the boundary layer, W2
Net resuspension from sediment to water, WRS
Bedload through boundary layer: gbl
Surface Water Column
A A
Water-Bedload
Boundary Layer
ubl
Surface Sediment Layer
-f
WE | E,
\°
Ws I Ds
9bl
Figure 2 - Noncohesive erosion and resuspension
In WASP8, erosion velocity and flux are calculated for
each particle size class using either the van Rijn or Roberts
formulation (Solids option 1 or 2). These are based on particle
diameter and density, the bottom shear stress, and the critical
shear stress for erosion.
The van Rijn erosion algorithm (Solids Option 1) calculates
a non-dimensional quantity, E, which is the ratio of the gross
erosion to gross deposition rate. The erosion velocity, then, is
the product of E and the settling velocity:
wE = Ews
The van Rijn non-dimensional E is:
Equation 11
E = 0.015 y
El
b ks
' Equation 12
where yE is a user-specified multiplier that defaults to 1.0
Ds is the median particle size [m], k is the roughness height
[in], Rj is the sediment particle densimetric Reynolds number
(defined above), // is a user-specified exponent that defaults to
1.5, and x is the non-dimensional shear stress:
tcE
T* = 0
*6 ^ rc
Tb < tcE
Equation 13
Equation 14
where %i is the bottom shear stress [N/m2] and x /; is the
critical shear stress for erosion:
^ce yrF (Ps pw)gDsQCE
Equation 15
where "t is a user-specified multiplier that defaults to 1.0
and QcIl is the non-dimensional Shields parameter, which is
calculated by the Brownlie (1981) fit to the Shields curve:
QcE = 0.22 R^°'6 + 0.06 X 10~7,7 Rd°6 Equation 16
The critical velocity for erosion, ucE [m/s], is the velocity that
produces zc£:
ucE V8W Pwf Equation 17
In WASPS, calibrate van Rijn noncohesive erosion by
specifying values for the following constants in the Solids
Transport group: shear stress exponent for noncohesive
resuspension, h (default = 1.5); critical shear stress multiplier
for noncohesive resuspension, f (default = 1.0), and
shear stress multiplier for noncohesive resuspension yB
(default = 1.0).
Eroded solids in the mobile boundary layer may be
transported along the sediment bed by bed load, or to the
water column by resuspension.
The Roberts erosion algorithm (Solids Option 2) calculates
erosion velocity ir/; [m/s] for each particle size class as a
function of bottom shear stress xb [N/m2] and bulk density p..
fkg/m'|:
wE = YeAPb
Equation 18
where jl is a user-specified multiplier that defaults to
1.0. The fitting coefficients A, in. and n were determined
experimentally for different particle sizes from fine silt
(less than 5.7 mn) to coarse sand (greater than 1.25 mm).
In WASP8, you can calibrate the Roberts erosion rate
by specifying a value for the shear stress multiplier for
noncohesive resuspension, yE (default = 1.0) in the Solids
Transport group.
Noncohesive Resuspension - Noncohesive resuspension is
the transport of the solids particles from the mobile layer
or from the surface benthic segment into the water column
(Figure 2). Eroded particles move in the boundaiy layer as
bed load below the critical shear stress for resuspension, pcRS
[N/m2].
T-cRs ~~ 0-1-
(4 IVS100/D,)2
Equation 19
pw/1000
where wt is the settling velocity [m/s], pw is the water density
[kg/m3], and D, is the non-dimensional particle diameter,
given by
D* =
'Ps~Pw\
D.
Equation 20
When bottom shear stress exceeds rcfiS, particles are entrained
from the mobile boundaiy layer and resuspension begins. The
net resuspension velocity is given by:
Wr fRSwE Equation 21
\vhcrc /,;; is the fraction of the noncohesive erosion that is
entrained to suspension given by:
/as 0
(n(u,/ws)- ln(u,cRs/ws)
ln(4)- ln{u,cRs/ws)
Tb < rcRs Equation 22
Tb > tcRs Equation 23
u, > 4ws
Irs '
/rs = 1 u» 4ws Equation 24
where u, is shear velocity [m/s], itHM is critical shear velocity
for resuspension [m/s], and w is particle settling velocity
[m/s]. The shear velocity and critical shear velocity are given
by:
W* Pw
U*cRs ~ y/^cRs/Pw
Equation 25
Equation 26
-------
Noncohesive Bed Load - Bed load is the transport of
noncohesive solids particles downstream through the mobile
layer Bed load begins when the bottom shear stress exceeds
the critical shear stress for erosion. x%E. Most eroded particles
arc redeposited back to the surface sediment layer. The bed
load flux per unit width, gu [g/m-s] is given by the van Rijn
expression:
9bl=ablPsuh(^f2M: Equation27
where abl is a fitted coefficient, u is stream velocity [m/s], h is
depth | in |, h is a fitted exponent, and A/ is given by:
M = ¦
(P Pw") 9 Dso Equation 28
where ucE is the critical velocity for erosion, given in the
previous section.
van Rijn calibrated the bed load flux equation to measured
transport data (van Rijn. 2007), yielding aM = 15 and
r| = 1.5. In WASP8, aM is given by:
abi 15 * vBLmult
Equation 29
where vBLmult is the calibration multiplier for bed load flux
(default = 1.0) and h is set to vRNonCohExp, the shear stress
exponent for noncohesive resuspension (default = 1.5). Both
arc specified in the Constants section. Solids Transport group.
Cohesive Resuspension - Cohesive erosion is the detachment
and transfer of a thin layer of cohesive sediment from the
surface bcnthic sediment to the water column. All cohesive
solids in the eroded layer arc transferred at the erosion
velocity, wR [m/s],
A commonly-used expression for coliesive erosion flux
[g/m2-s] is the following excess shear stress power law
formulation (Lick ct al., 1994):
Ecoh. ~ fcohM't't
Equation 30
where Mis the shear stress multiplier [g/m2-s], n is the shear
stress exponent, / ^ is the fraction of the surface bed that is
cohesive, and t, is the excess shear stress [N/m2]:
Tb~ tcE
T* = 0
Tb rcE
rb < rcE
Equation 31
Equation 32
where xb is the bottom shear stress [N/m2] and %cE is the
critical shear stress for erosion [N/m2].
The set of cohesive constants can be specified for each solid
in the Constants section. Solids Transport group. The shear
stress multiplier varies between 0.1- 100 [g/m2-s], with a
default value of 5. The shear stress exponent varies between
1.6 - 4, with a default value of 3. The critical sliear stress for
erosion varies between 0.5 - 8 [N/m2], with a default value
of 2.
The shear stress multiplier, exponent, and the critical shear
stress for erosion can vary spatially in a water body. Input
different values for these for each surface bcnthic segment
in the Parameter Data, Solids group. If a nonzero value is
specified for a segment, that value is used instead of the
constant.
tic Solids Production and Dissolution
Biotic solids include living algae and non-living detritus.
The WASP8 cutraphication model simulates the growth,
settling and death of phvtoplankton and macro algae, with the
subsequent production, settling, and dissolution of detritus.
The total dry weight of these biotic solids components is
added to the inorganic solids concentrations to produce total
suspended solids in water column segments.
In the WASP8 toxicant model, one or more solids variables
can be characterized as biotic. For solid "i" the net
production rate Rprod{ [g/m3-d] is given by:
Rprod,i {Rp.seg + Rp,t)%rod Equation 33
where R is the spatially-variable Biotic Solids Net
p.seg r J
Production Rate [g/m3-d] specified in the Parameters section.
Solids group; R is the time-variable Biotic Solids Net
Production Rate [g/m3-d] specified in the Time Functions
section, and 9pwd is the temperature correction coefficient,
specified in the Constants section. Solids Transport group.
Similarly, the dissolution rate constant kdjssI [d 1 ] is given by:
kdiss,i = (kd.sctj + iiss Equation 34
where k is the spatially-variable Biotic Solids Dissolution
Rate Constant [d 1 ] specified in the Parameters section. Solids
group; kdt is the time-variable Biotic Solids Dissolution Rate
Constant [d 1 ] specified in the Time Functions section, and
Qdjss is the temperature correction coclTicicnt. specified in the
Constants section. Solids Transport group.
The dissolution rate Rdjssj [g/m3-d] is the product of its
dissolution rate constant [d 1 ] and its concentration [g/m3]:
R.
diss,i '
kdissj^i
Equation 35
The inorganic residue of biotic solid "i" dissolution will be
added to solid "j " if the user specifics the Ash Dry Weight
Residue and the Residue Solid Identification Number in
the Constants section. Solids Transport group. The organic
carbon fraction of biotic solid "i" dissolution will be added
to DOC "k" if the user specifics the Organic Carbon Fraction
and Dissolution Product DOC Identification Number in the
Constants section, Solids Transport group.
3.8, Solids Burial
Sediments below each water segment can be represented in
WASP by using one or more layers. The initial total solids
concentration in each bcnthic segment [mg/L or g/m3] sets
its reference bulk density [g/mL| and porosity [Lw/L], Driven
by deposition, erosion (including resuspension and bed
load), growth, and dissolution fluxes, WASPS conducts a
solids mass balance in these layers. Two options appear in
the Datasct screen - Static (constant volumes) and Dynamic
(constant densities). We recommend the Dynamic option
-------
(described below) in which you must also set the benthic
time step DTB in the Datasct screen; the default value
is 1 day.
Surface Benthic Layer - The surface bcnthic layer is active.
When solids are deposited, it accumulates volume and depth.
When solids arc eroded, it loses volume and depth. Except as
noted below, the initial reference bulk density and porosity
arc maintained.
If there arc no subsurface bcnthic layers, deposition
causes the surface layer to accumulate depth and volume
indefinitely, meaning there is no net burial. Erosion reduces
the depth and volume until it reaches 5% of the initial values
and no further erosion is allowed.
If there arc underlying subsurface bcnthic layers, the surface
layer volume and depth arc reset to initial reference values
for each bcnthic time step. The reset correspond to net burial
(Vm and dB]) or net erosion (V£J and dE1).
Under dcpositional conditions, the surface bcnthic
segment buries VM and dM to the first subsurface bcnthic
segment at each bcnthic time step. The solids and pollutant
concentrations in the buried volume arc also passed
downward. The burial velocity, wB] |m/s|, from the surface
bcnthic segment is
w isj.
B1 dtb Equation 36
Note that burial velocity is reported in the model output in
units of [cm/yr]. The burial fluxes from the surface layer for
solids and pollutants arc:
ffll.fe Ck,lwBl
Equation 37
where C,, is the concentration of constituent in the
k,l
surface bcnthic layer [g/m3], and FgI k is the burial flux
[g/m2-s] from the surface bcnthic layer.
Under erosion conditions, the surface bcnthic segment
recruits VE] and d£] from the first subsurface bcnthic
segment at each bcnthic time step. The solids and pollutant
concentrations in the subsurface volume. CkB2, arc also
passed 'upward' (the surface bcnthic segment is moving
downward). In subsurface bcnthic layers, total solids arc
usually packed more tightly and have higher bulk densities
and lower porosities. To preserve mass and volume balance,
the surface layer bulk density and porosity arc recalculated
at each bcnthic time step. If erosion continues over time,
the bulk density and porosity approach the values in the
subsurface layer.
Subsurface Benthic Layers - Subsurface bcnthic layers
arc passive. Each bcnthic time step, mass is transported
downward or upward through the subsurface layers,
depending on whether the surface bcnthic segment
experiences deposition or erosion conditions.
Under deposition, the first subsurface segment receives
solids from the surface layer at each bcnthic time step. If
the subsurface segment has a higher bulk density, the buried
volume and depth (VB1 and dB]) arc compressed to VB2 and
d and pore water is squeezed upward. The compressed
volume VB2 is passed downward to the next bcnthic layer or
out of the system. Solids and pollutant concentrations in the
subsurface volume. CkB2, arc also passed downward to the
next lower bcnthic segment through the bed. maintaining the
initial bulk density and porosity.
Under erosion conditions, the surface segment receives
volume VE1 from the first subsurface layer at each bcnthic
time step. Solids within this volume arc also transferred
upward. In turn, lower subsurface bcnthic layers transfer
eroded volume and solids concentrations to their overlying
bcnthic layers. Erosion reduces the depth and volume of the
bottom layer until it reaches 5% of its initial value; no further
erosion of that layer is allowed. Erosion then reduces the
depth and volume of the next lowest bcnthic layer until it,
too. reaches 5% of its initial value. If erosion continues, all
bcnthic layers will eventually reach 5% of their initial values.
At this point, no more erosion is allowed.
alyticai Solution for Settling
WASP8 normally uses a backward difference numerical
solution technique. For each state variable, the concentration
at the beginning of a time step. Cg, is used in transport and
transformation equations. The time step. DT, is adjusted to
maintain stability.
For coarse silts and sands, however, high settling velocities
could cause WASP8 to use increasingly smaller time steps.
When settling removes more than 0.1% of a solid from a
water column segment during the normal calculation time
step, WASP8 uses the alternative analytical solution, which
calculates C* the average solid concentration during the time
step, applied in the solid settling and advcction loss for the
time step.
Balancing loading, advcction, and settling, the analytical
steady-state solution for a solid under prevailing
conditions is:
^ss -
Q+V^
Equation 38
where L is the total loading of solid "i" [g/d] including
external loadings, advcction in. and resuspension; Q is the
advcctivc flow [mVd|, Vis the segment volume [m3], d is
the segment depth |m|. and ws is the settling or deposition
velocity |m/d|. During the time step, the concentration will
move from Cg toward Css.
For convenience, this equation can be rearranged:
C = L
ss vxks Equation 39
whereXh is the overall loss rate constant due to outflow plus
settling | l/d|:
V _ Q 1 ws
A l> p " I
V d
The first order attenuation equation is:
= L-{Q + Vjr)c
V d.C
dt
Equation 40
Equation 41
-------
The solution for C as a function of time is:
Table 4. Solids Demonstration Parameters
C = C0 + (1 ¦
0 VXks v
Xks
C = C0 + Css(l-e-x*st)
Integrating this equation over DT gives:
Equation 42
Equation 43
Variable
Description
Units Value
f0 Cdt [ (C0 Css) e Xks 1 + Csst] Dg Equation 44
evaluating the right-hand side at I DT minus I 0 and
rearranging terms gives:
Jo Cdt CSSDT + (C0 Css )(1 e Xkst ) Equation 45
The average concentration during DT is:
C* = = Css + XksDT (Co - Css )(1 - e Xks DT ) Equation 46
It is convenient to define A' , which varies from close to 1
term7
(for small DT or Afa) to 0 (for large DT or Afa):
_ i-e-XksC-
^term
so that:
Xks DT
Equation 47
CnXt,
. Css(l Xterm )
Equation 48
The average concentration C* varies from Cg at small time
steps or loss rates to (for large time steps or loss rates.
3.10. Example
This section describes how interaction of solids behaves
between the water column and benthic segments. A simple
pond system consisting of a water column, a surface benthic
layer, and a subsurface benthic layer model the transport of
solids. Table 3 describes the geometry of the segments.
Table 3. Channel Geometry of WASP Segments
Segment
Volume
(m3)
Depth (m)
Water Column 10,000
Surface Sediments 200
Subsurface
Sediments 500
1
0.02
0.05
The WASP model simulates flow, as well as silt and sand
loads, with simple process-based settling and resuspension.
Dynamic bed compaction was used in the sediment layer at a
time step of one day. Table 4 describes parameters used and
their values.
Flow through the Water
Column
Silt Load into the Water
Column
Sand Load into the
Surface Sediment
Cross-sectional Area of
Bed
Silt radius
Silt Settling Velocity
Silt Resuspension Velocity
Bed Compaction Time
Step
m3/d
kg/d
kg/d
m2
mm
m/d
m/d
10,000
60
40
10,000
0.004
5.E-01
5.E-05
The model is simulated until all three segments have reached
steady state, with the subsurface sediment layer taking the
longest time. Figures 3-5 show the concentrations of total
solids, silt, and sand in each segment layer. Sand is not
present in the water column because the load is entered
directly into the surface sediment layer. Silt concentrations
reach steady state in the water column segment relatively
quickly, compared to sediment layers.
Pond Surface Water
Total Solids
50000
100000
Time (Day)
150000
Figure 3. Solids Concentration in the Surface Water
9
-------
Total solids and sand concentrations are plotted on the left
y axis and silt concentrations are plotted on the right. Sand
concentration decreases rapidly as silt is deposited from the
water column. The dynamic bed compaction option allows
the surface benthic layer to accumulate volume and depth
until the bed compaction time step is reached, which then
resets the volume and depth of the surface benthic segment
while burying excess volume and depth to the segment
below. Concentration of total solids remains constant after
minor volume adjustments in the first few time steps.
Sediment composition of the subsurface benthic layer
changes more slowly than surface layers as a function of
the bed compaction time step. Overall, sand concentration
decreases as mass from the surface sediments is buried.
Steady state is eventually reached in all three layers.
Burial or removal from the subsurface benthic layer is lost
from the system and equals the rate of mass being moved
downward from the surface benthic layer. Figure 6 shows
the burial rates of the surface and subsurface benthic layers.
Pond Surface Sediments
Burial velocity
Total Solids
Silt
Sand
50000
100000
150000
Time (Day)
Figure 4. Solids Concentration in the Surface Sediments
Pond Subsurface Sediments
O) o -
e r
Total Solids
Silt
Sand
0
50000
100000 150000
Time (Day)
Figure 5. Solids Concentration in the Subsurface
Sediments
o
o
Surface Sediments
Subsurface Sediments
0
50000
100000
150000
Time (Day)
Figure 6. Sediment Burial Rates
10
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4.0
Light
4.1. Introduction
Solar radiation can result in phototransformation of
contaminants and constituents in surface water systems.
Chemicals, metals, and nanomaterials all undergo
photochemical reactions, which can result in the degradation
and/or transformation of contaminants. Solar radiation
can deactivate pathogens and viruses, and be used for
photosynthesis by microorganisms. It is therefore important
to capture transmission of light from the water surface and
through the water column.
4.2, Theory
Several factors affect the amount of solar radiation within
the water column. The amount of light reaching the surface
water is affected by clouds and shading of the water, and
the amount of radiation is dependent on the time of day and
time of year1113. Light is comprised of a range of different
wavelengths. Total radiation is copmrised of different
fractions of the different wavelengths. Each wavelength is
attenuated as it travels through the water column, and the
extent depends on the wavelength as well as factors such
as suspended solids and dissolved organic carbon. How
WASP addresses light reaching the surface, how it divides
up wavelength bands, and how it attenuates light through the
water column is detailed in this section.
4.2.1 Light Attenuation
Within the water column, light is attenuated with depth
following the Beer-Lambert equation 115:
lz loe Ke,AZ Equation 49
where / is the light intensity (W/m2) at depth z, Ig is the
light intensity (W/m2) at the water surface Ke t is the light
extinction coefficient [1/m], 1 is the wavelength index, and z
is depth below the surface [m].
The light extinction coefficients are calculated internally as a
function of background water, algal chlorophyll a, DOC, and
solids:
^chi,,-. (chl)n 1 + KDOCjDOC + SS Equation 50
where Chi is algal chlorophyll concentration ||ig/L|. a. is a
wavelength-specific exponent for chlorophyll, DOC is total
dissolved organic carbon [mg/L], TSS is total suspended
solids [mg/L], /\u, [1/m] is the light attenuation coefficient
of water for wavelength X, K(hj. [1/m] is the light attenuation
coefficient due to chlorophyll for wavelength ^Kdoc,,
[1/m] is the light attenuation coefficient due to chlorophyll
for wavelength X, and Krss, [1/m] is the light attenuation
coefficient due to suspended solids for wavelength X.
4,3, Implementation
4.3.1 Input Total Radiation
The first aspect of using the light module within WASP8 is to
incorporate incident radiation that reaches the surface of the
water body. Total solar radiation reaching the water surface
can be specified by user input light and internally-calculated
diel light. This option is specified in the Constants section,
Light group; the default is 0, Calculated diel light.
Option 0, Calculated diel light
Light option 0 is driven by internally-calculated light, based
on latitude, longitude, day of year and time of day. The
latitude of the water body is input to the Constants section,
General group. Simulated day and time are kept internally
based on the simulation start day and time and the calculation
time step. Calculated light represents clear sky radiation, / kar,
in W/m2.
Option 1, User Input Diel Light
Light Option 1 is driven by as many as four specified time
series of hourly (or less) surface light fluxes. WASP8 assumes
that these input functions represent the total spectrum of solar
radiation [W/m2]. At each time step, the model reads the total
radiation and applies it to each surface segment.
The appropriate solar radiation time function for each
segment (1 - 4) is specified in the Parameter Data section,
Environmental group. The solar radiation time functions are
entered in the Time Functions section. If the data represent
a fraction of the total (i.e., visible or PAR light), or if the
data are expressed in alternate units, the user must enter an
appropriate scale factor in the "Solar Radiation Multiplier
[unitless or W/m2]" located in the Parameter Data section,
Environmental group. To convert from PAR light to total
radiation, the multiplier is 2.155.
Option 2, User Input Daily Light, Calculated Diel Light
Total daily lightflux - Light Option 2 is driven by as many
as four specified time series of total daily surface light
fluxes [W/m2]. On each new simulation day, the model reads
the total daily radiation for that day, ITot. At each time step
through the diel cycle, the model calculates and applies a
portion of the daily total radiation to each surface segment.
-------
The appropriate solar radiation time function for each
segment (1 - 4) is specified in the Parameter Data section.
Environmental group. The solar radiation time functions
arc entered in the Time Functions section. If the data
arc expressed in alternate units, the user must enter an
appropriate scale factor in the "Solar Radiation Multiplier
limitless or W/m2]" located in the Parameter Data section.
Environmental group.
Default diel light distribution - The default diel cycle is
based on latitude, longitude, day of year and time of day.
The model calculates the total clear-sky radiation for the
day (IckarDa) and the clear-sky radiation for each time step
through the day (.IclearSky) The instantaneous solar radiation
flux is the input total daily flux (ITJ times the diel ratio for
that time step:
I hot (/ Equation 51
\*ClearDay)
The latitude and longitude of the water body is input to the
Constants section. General group. The simulation day and
time arc kept internally based on the simulation start day and
time and the calculation time step.
Alternate diel light distribution - An alternative user-
controlled diel option is applied if a daylight fraction time
function f is specified. In this case, total daily radiation is
distributed through daylight hours (between dawn and dusk)
using a half-sine curve with a maximum value at noon.
j _ hotn
max ~ 2 fday Equation 52
/ _ Imax sin(nTday) Equation 53
where ITot is the total daily radiation [W/m2],^is the fraction
of day that is daylight (0.2 - 0.8) and rday is normalized time
between sunrise and sunset, expressed as fraction of the
daylight interval. Sunrise is 1, noon is 0.5, and sunset is 1.
1=0 during nighttime hours.
The "Fraction Daily Light (fraction)" is input to the Time
Functions section. The simulation day and time arc kept
internally based on the simulation start day and time and the
calculation time step.
4.3.2. Light Attenuation Above Water Surface
Input solar radiation can be attenuated by cloud cover,
canopy shading, ice cover, and water surface reflectance.
Cloud Cover
Internally-generated light (option 1) represents clear sky
radiation. This can be attenuated by user-specified cloud
cover | fraction of sky],
I = 0.65 I dear (1 ~ Cloud)2 Equation 54
The appropriate cloud cover time function (1 - 4) is
specified for each segment in the Parameter Data section.
Environmental group. The cloud cover time functions
arc entered in the Time Functions section as a fraction
(0.0 - 1.0). Options 1 and 2 assume cloud cover is already
accounted for in the surface light flux time scries.
Canopy Shading
Near-surface light can be diminished by vegetative shading.
Light beneath the canopy must be attenuated by user-
specified vegetative shading |fraction of light intercepted].
Is = /(I Shade} Equation 55
The appropriate canopy shading time function (1 - 4) is
specified for each segment in the Parameter Data section.
Environmental group. The canopy shading time functions
arc entered in the Time Functions section as a fraction
(0.0-1.0).
Water Surface Reflectance
A fraction of light reaching the water surface. Is, is reflected.
Light at the top of the water column. Ig, is reduced by the
fraction of light reflected:
I0 = «1 " Pw) Equation 56
The default reflectance (Pw) is 0.06 and is automatically
implemented for surface water segments. The user can input
an alternate value in the Constants section. Light group.
Attenuation in Ice
When ice forms on the water surface, the surface light./,
is attenuated by reflectance of the ice surface (albedo, a),
surface absorption (P.), and light extinction through the ice
thickness, h..
y i
I0 = /s( 1 a)(l - Pi)e"Ylhi Equation 57
The default values for albedo, surface absorption, and ice
extinction coefficient (y.) arc 0; these values can be updated
in the Constants section. Water temperature group.
Surface ice cover fractions can be input in the Parameter Data
section. Environmental group and the Time function group.
Together, they give the fraction surface areas covered by ice.
Pice = SGiceTFice Equation 58
Ice thickness is calculated by the temperature module. If
water temperature and ice arc not being simulated, then the
"Minimum ice thickness before ice formation is allowed" is
applied as an average thickness. This is input in the Constants
section. General group.
4.3.3. Light Attenuation Below the Water Surface
As discussed previously, within the water column, light is
attenuated with depth following the Beer-Lambert equation:
Iz = I0e~Ke*z Equation 59
where Ke; is the light extinction coefficient [1/m], 1 is the
wavelength index, and z is depth below the surface |m|.
-------
Table 5. WASP Default Light Extinction Coefficients for the Wavelength Bands
Water
Chlorophyll
DOC
Solids
Index
Color
[m1]
[m1 (mQ/L)"1]
[m1 (mg/L)1]
[m1 (mg/L) 1]
1
UVB med
0.151
0.103
6.22
0.34
2
UVB high
0.109
0.0816
5.40
0.34
3
UVA low
0.0805
0.069
4.59
0.34
4
UVA med
0.0512
0.057
3.40
0.34
5
UVA high
0.0340
0.053
2.54
0.34
6
violet
0.0169
0.039
1.266
0.34
7
blue
0.0166
0.0262
0.514
0.34
8
green
0.0475
0.0143
0.289
0.34
9
yellow-orange
0.217
0.0063
0.115
0.34
10
red
1.007
0.0065
0.0
0.34
11
infrared
2.07
0.0
0.0
0.34
The light extinction coefficients are calculated internally as
a function of background water, algal chlorophyll a ||ig/L|.
chlorophyll exponent [unitless], DOC [mg/L], and solids
[mg/L]:
Ke,z = kw,a + WcW)»» + kdoc,^>OC + Ksotid ATSS Equation 60
Table 5 provides the default coefficients by wavelength.
The chlorophyll exponent by wavelength defaults to 1. The
coefficients for total ultraviolet and total visible light are
calculated internally as the weighted sum of their component
wavebands (1 - 5 for ultraviolet, 6-10 for visible).
The default coefficients for visible light are listed in Table 6.
Division of Wavelengths by Wave Class. These can be
changed in the Constants group. Light section. If multiple
fractions of DOC are simulated, the user can enter a set of
light extinction coefficients for each DOC fraction.
Total solar radiation is divided into three classes:
The classes are used in the heat balance equations in the
WASP8 Temperature module, and in the bacterial death
equations.
Table 7 provides how WASP divides ultrbiolet and visible
solar radiation into 10 wave bands.
These wavebands allow each band to have its own light
attenuation at it penetrates the water column. The radiation of
each wavelength band then drives photochemical reactions.
WASP8 uses time series of total radiation, along with
waveband fractions, to deliver solar radiation to the water
surface. The total radiation can be input by the user or
calculated internally by the model. Waveband fractions can
be specified or default values used.
Table 6. Division of Wavelengths by Wave Class
Wave
Wavelengths,
Fraction of
Class
Color
nm
Total
1 ultraviolet 295 - 379 0.036
2 visible 380 - 749 0.464
3 infrared 750 - 2500 0.500
Table 7. Division of Wavelengths into Wave Bands, with Fractions Given by Latitude
Wavelengths
Latitude
Index
Color
[nm]
0° N
10° N
20° N
30° N
40° N
50° N
60° N
1
UVB med
295 - 304
0.00015
0.00015
0.00013
0.00011
0.00008
0.00006
0.00004
2
UVB high
305-314
0.00142
0.00139
0.00132
0.00120
0.00104
0.00085
0.00067
3
UVA low
315-334
0.00845
0.00839
0.00825
0.00801
0.00766
0.00721
0.00681
4
UVA med
335 - 354
0.01141
0.01137
0.01126
0.01108
0.01082
0.01052
0.01054
5
UVA high
355 - 379
0.01723
0.01718
0.01706
0.01686
0.01655
0.01619
0.01630
6
violet
380 - 449
0.07626
0.07617
0.07593
0.07550
0.07482
0.07394
0.07443
7
blue
450 - 494
0.06664
0.06663
0.06659
0.06652
0.06639
0.06616
0.06644
8
green
495 - 569
0.10386
0.10388
0.10394
0.10402
0.10406
0.10390
0.10285
9
yellow-
570-619
0.06546
0.06549
0.06556
0.06566
0.06576
0.06568
0.06422
orange
10
red
620 - 749
0.14914
0.14934
0.14995
0.15106
0.15282
0.15550
0.15769
-------
5.0
Particle Attachment
5.1. Introduction
Sorption is the association of aqueous species with a solid
material16. Surface waters are abundant with suspended
solids (e.g., silt, clay and particulate organic matter), and
sorption affects the fate transport of contaminants in surface
waters. Sorption involves sorption and desorption r 1S.
Sorption is the association of a contaminant with the surface
of a solid particle. Desorption is the reverse and describes
dissociation of a sorbed molecule and its return to the
aqueous or gaseous phase.
Depending on system assumptions, sorption can be simulated
by an equilibrium or kinetic model. If an equilibrium
model is used, it is assumed that sorption is fast and occurs
instantaneously. If a kinetic model is used, the processes are
simulated as two competing reactions.
Previous versions of WASP included equilibrium sorption
only, but WASP8 includes kinetic sorption, as well as
heteroaggregation of nanomaterials.
5.2. Theory
5.2.1. Equilibrium Sorption
Sorption reactions are usually fast, relative to enviromnental
processes, and equilibrium may be assumed. For
environmentally relevant concentrations (less than 105 M or
one-half water solubility), equilibrium sorption is linear with
dissolved chemical concentration19 or:
Cf = Kd iCjv Equation 61
where ('/ is chemical concentration in the solid phase
and fis chemical concentration in the aqueous phase.
Table 8 provides a full list of parameter descriptions used in
this section.
Table 8. Equilibrium Sorption Parameters
Symbol
Q
C,w
cw
rs.
L' j
C'S.
L' j
Definition
Total Chemical / Concentration
Dissolved Chemical /
Concentration
Dissolved Chemical /
Concentration in Water
cw= ( .: /n
Concentration of Sorbed
Chemical / on Solid j
Concentration of Sorbed
Chemical / on Solid j
CjSj = CjSj/Sj
Units
m9chem'
mg
/ L
l
/L
mg /L
^nhem w,
mg /L
^chem
mg /kg
"^chem "-'s
Symbol
sj
SJ
S'j
Kd,ij
fd,i
fs,ij
Definition
Solid j Concentration
Solid j Concentration
S = s x 1 o-6
Solid j Concentration in Water
Porosity or Volume Water
per Volume
Partition Coefficient of
Chemical / on Solid j
Fraction of Chemical / in
Dissolved Phase
Fraction of Chemical / Sorbed
to Solid j
Units
mgs/L
kgs/L
kg /L
°s w
L /L
w
L /kg
W "-'S
At equilibrium, distribution between the aqueous phase
and the solid phase is determined by partition coefficients.
When multiple solid phases are present (e.g., sand, silt,
clay, organic matter), the total mass of chemical associated
with each phase is controlled by Kd where i represents the
specific chemical and j is the solid phase of interest (e.g.
K.. , K.. ... K,. , , and A.' ). Complexation with DOC
a,l saner a, t si If a, i clay a,i organic-' r
is handled in a similar manner, using the same equations
and associated Kd jDOC. All of these relationships are solved
simultaneously, assuming instantaneous distribution among
the phases (equilibrium assumption). The fraction of mass
associated with each phase is given by:
jrd.i
n+ Zf=1Kd,ijSj
and
fS,lJ _
Kd.ijSj
Equation 62
Equation 63
71+ 2j=i Kd.ijSj
where fd j is the fraction of chemical I concentration present
in the freely dissolved form, fs jd is the fraction of chemical
i sorbed to solid S.. If complexation with DOC is simulated,
the fraction in the aqueous phase includes the freely dissolved
form and all complexed with DOC. These functions are
determined in time and space throughout a simulation from
the partition coefficients, internally calculated porosities, and
simulated sediment concentrations.
In addition to the assumption of instantaneous equilibrium,
the assumption of reversibility is also implicit in the use
of these equations. Laboratory data for very hydrophobic
chemicals suggest, however, that a hysteresis exists, with
desorption being much slower than adsorption. They
also suggest this effect may be the result of intraparticle
kinetics in which the chemical is slowly incorporated into
components of the sorbant.19
-------
Site-specific values for partition coefficients can be obtained
from laboratory experiments or field data; literature values
may be used in lieu of site-specific values. WASP can also
simulate equilibrium partitioning to DOC state variables,
but is not included in current documentation. A more
comprehensive discussion of chemical processes will be
released in the WASP8 manual.
5.2.2. Kinetic Sorption
When the equilibrium assumption is not applicable, sorption
can be simulated kinetically which incorporates a forward
reaction and competing reverse reaction. Because the system
is changing dynamically, separate state variables must be
incorporated to represent freely-dissolved chemical, Chem1
(mg/L), and chemical sorbed to the solid particles, Chem2
(mg/L). The sorption process is described by the following
differential equation:
= -kfor[Chem^lS] + krev[Chem2] Equation 64
where [5] is the suspend solid concentration in the
aqueous phase (mg/L), kfor is the forward, sorption rate
constant (L/mg-d), and ktrr is the reverse, desorption rate
constant (1/d).
[Chem2] is the contaminant concentration sorbed on the solid,
which can be expressed as:
[Chem2] = [C/iem-Jo [Chemx] Equation 65
where \('heml\0 represents the initial contaminant
concentration (mg/L) in surface waters.
Therefore, Equation 1 can be derived into the following
differential equation:
"tC^"1'1 = -kforlChem^lS] + kr([Chem0] - [C/iem,]) Equation 66
Initial contaminant concentration (CHEMg) and initial solid
concentration (Sj are set in the Segment section, Initial
Conditions group.
5.2.3. Nanomaterial Heteroaggregation
The architecture of the WASP has been redesigned to allow
simulation of nanomaterials.
A new state variable class NANOC, for nanomaterials,
is included in WASP8 which uses the kinetic process
of heteroaggregation2"-21 to simulate attachment
of a nanomaterial to suspended particulate matter.
Heteroaggregation is the process by which nanoparticles
collide and stick to particulate matter, based on three separate
collision processes. The overall heteroaggregation rate is
defined by:
kfiet,ij &kcoii ijNj Equation 67
where
a: The collision efficiency or the probability that
a nanoparticle will stick to a suspended solid
particle in the event of a collision. This is a
unitless parameter that can range between 0-1.
kcoii y ¦ The rate of collision between two particles in
units of volume per day.
N,SI>VI: The number of suspended particulate matter per
volume.
ka,iujis defined by:
kcoii,ij = + ic(rMp j + rSPMj)3 + rc(rPji + rSPMJ)2\v?fti -
3 i^water ^NP.i^SPM.i 3 ' ' ' ' 1
Equation 68
which consists of three components:
,, ^ 2kBTwa,t(rNpi+TSpM:i)2 Equation 30a
Browman Motion =
3f^water rNP,irSPM,j
Fluid Motion = jG(rWPi + rSPMjf Equation 30b
Differential Settling = n{rNP:i + rSPMJ)2\v^pti - vssj\ Equation 30c
Parameter descriptions and values used in this analysis
are presented in Table 31. The rate of collision between
nanoparticles and particulate matter is dependent on three
processes: Brownian motion (perikinetic aggregation), fluid
motion (orthokinetic aggregation), and differential settling.
Settling velocity is calculated using Stokes' law:
J)article _ 2 ^(Pparticle~Pwater) _2
vset ~ 9d fiwater rparticie Equation 69
Assuming a spherical particle, is calculated by:
s?
N? = 1 Equation 70
PSPM,j^nrSPM,]
The heteroaggregation process is described by the following
differential equation:
Arwwc
^ = ~l{fletNj/VCSj Equation 71
The collision efficiency is a system dependent parameter that
must be measured or estimated, is calculated internally by
WASP. These equations assume that the particles collide with
nanoparticles at the same rate as hard spheres.
Implementation
In the Constant Group section, Chemical Kinetic Sorption
group, reaction products must be specified to simulate
contaminant transport. This is implemented by checking the
'Chemical sorbed to Solid (i), [ID#]' for the desired chemical
(ID#) and solid (i) state variables, and entering a value
pointing to the sorbed chemical.
5.3.1. Equilibrium Sorption
To assign a partition coefficient of a dissolved chemical to a
solid, check the 'Partition Coefficient of chemical to Solid(l),
[L/kg], CHEMl' constant and enter a coefficient in the
15
-------
value column. Unlike kinetic sorption and heteroaggregation
which handle phases as different state variables, equilibrium
sorption uses initial total concentration of a single state
variable to partition dissolved and sorbed chemical
concentrations internally in WASP8, and outputs them as
different time series.
5.3.2. Kinetic Sorption
To assign sorption {kfJ and desorption (kre) rates to a
chemical, check the 'Sorption rate constant to Solid (1), [L/
kg-d], CHEMl,' and 'Desorption rate constant from Solid (1),
[1/d], CHEMl,' and enter rates in the value column.
5.3.3. Nanomaterial Heteroaggregation
In the Systems section, users can add nano chemical
(NANOC) state variables to the model with unique names and
densities. Users must add a nano state variable for each phase
of a specific nano chemical i (e.g., aqueous phase, sorbed
phase to solid j, etc.). Nano state variables that represent
heteroaggregated phases assume the density, diameter, and
settling rate of the solid that it is sorbed to. Concentrations
can be added to the system as initial conditions (IC) in the
Segments section, Initial Conditions tab, or as boundary
conditions (BC) in the Boundaries and Loads section. IC and
BC are entered as mg/L. Loads are entered as kg/d.
Under the Parameter Data Group section, Nano Chemical
group, nano chemical collision efficiency (a) and nano
chemical settling velocity can be entered for each segment.
Input values will only display the first four decimal places,
but WASP8 retains the value even if it is too small to display.
For small values, the Scale Factor ensures that the entered
values can be seen.
Under the Constant Group section, Nano Chemical Kinetic
Sorption group, reaction products must be specified to
simulate contaminant transport. This is implemented by
checking the 'Nano chemical sorbed to Solid (i), [ID#]' for
the desired nanochemical (ID#) and solid (i) state variables,
and entering a value pointing to the sorbed nano chemical.
Particle diameter,s which are used in the heteroaggregation
equation, must be specified in the Nano Chemical
Partitioning group.
-------
Nanomaterial Reactions
6.1. Introduction
Nanomaterials are routinely defined as materials sized
between 1 nm to 100 nm, with properties not found in bulk
samples of the same materials.22 24 Engineered nanomaterials
(ENMs) have been applied in all areas of our daily lives,
and their production has increased appreciably in recent
years.25"31 Such rapid expansion of ENM production
increases the likelihood of ENPs being released into the
environment. Besides the heteroaggregation process in
surface waters, ENMs may undergo transformation reactions
including photochemically-driven reactions, sulfidation,
oxidation, and dissolution. For example, graphene oxide
undergoes phototransformation under simulated sunlight
radiation, resulting in reduced graphene oxide and polycyclic
aromatic hydrocarbons (PAHs).32,33 Dissolution,34 36
oxidation37-38 and sulfidation34-39"41 are common reactions to
metal nanomaterials.
To simulate the fate of nanomaterials in aquatic
environments, WASP8 incorporates algorithms to describe
ENM reactions. WASP8's Advanced Toxicant Module
simulates 10 state variables for nanomaterials and dissolved
chemicals, respectively. This limit can be adjusted upward
to handle any number of state variables. WASP8 users may
simulate concentrations of multiple nanomaterials, their
heteroaggregated forms, and their reaction products in
surface waters and sediments. Nanomaterials are simulated
using the WASP nanomaterial state variables; dissolved
chemicals (e.g., organics, metal ions) are simulated using
chemical state variables. For internal calculations, WASP8
assumes a spherical shape for nanomaterials. Another
assumption is that a nanomaterial keeps uniform size and
morphology both before and after reactions.
Figure 7 Possible Nanomaterial Reaction Pathways. Chem
indicates dissolved chemicals (e.g., organics, metal ion)
shows different possible reaction pathways for nanomaterials
and chemicals.
Nano, ^Nano2
Nano! ~ Chem1
Nano, ^ Nancb + Chemj
Nano | + Chem | Nano2 + Chem2
Figure 7 Possible Nanomaterial Reaction Pathways.
Chem indicates dissolved chemicals (e.g., organics,
metal ion)
Nanomaterial Reactions
6.2.1. General Reactions Module
The WASP8 nanomaterial reaction module employs a general
reaction to simulate nanomaterial reactions which consider a
series of factors. The general form reaction rate constant in
WASP8 is structured as
^reaction ~~ ^rate^temp^phase-^seg^-conc ^env Equation 72
where k is the overall reaction rate constant (d1),
reaction v 7'
k t is the base reaction rate constant (d1) at 20°C, A'
rate v 7 ' temp
is the temperature correction factor, X. is the phase
A 7 phase L
multiplication factor, Xseg is the segment type mulplication
factor, X is the Monod Equation term, andX is the
7 cone L 7 env
environmental factor structured similarly to Monod kinetics.
k , is the first-order reaction rate constant at 20°C that is
rate
input into WASP8. The other terms default to 1 in Equation
72 if no other terms are entered. The default reaction for
nanomaterial follows first-order kinetics. If the reaction
rate is influenced by other terms or WASP8 users intend to
simulate second-order reaction, users turn on the other terms
and input their values.
Nanomaterial reactions are essentially surface area-
dependent, because only nanomaterial surfaces are available.
The first-order nanomaterial surface reaction is expressed as
dm-Np K A _ _
dt ~ksaanp Equation 73
where mNp is the mass of nanomaterials (|ig). kSA is the
surface-area-normalized reaction rate (|ig/m2-d). and ,1 v/, is
the total surface area of nanomaterials (m2).
Because a key assumption in WASP8 is a nanomaterial keeps
uniform size during the reaction, AN|, is
Anp = {4nr2) = (47rr2) Equation 74
where is nanomaterial density (g/m3), is the volume
of a single nanoparticle (m3), is the total quantity of
nanomaterials, and r is the radius of a single nanoparticle
(m), assuming a spherical structure.
The nanomaterial surface area-dependent reaction can be
written as
dJT = -kSA (4^r2) = -kSA = - (|is±) mNP = -kmNP
etc pArpf_rrr3j pNpr \pNPrJ
Equation 75
where k is first-order reaction rate constant (d1).
Since we assume nanomaterial size remains uniform during
reactions, we use a mass-dependent reaction. Another reason
-------
is that when users create a nanomaterial state variable
in WASP, it is convenient to input nanomaterial mass
(or concentration) rather than nanomaterial surface area.
6.2.2. Effect of Temperature
The temperature corrector factor must be turned on in
WASP8 to capture the effect of temperature. The van't Hoff-
Arrhenius relationship adjusts the value of the reaction rate
constant to reflect the effect of temperature:
%temp = d^T~20^ Equation 76
where 0 is the temperature-activity coefficient, and T is
temperature (°C). 0 value can be set in WASP8, based on
different reaction types.42 For example, typical values for 0
vary from 1.020 - 1.10 for a biodegradation reaction.
6.2.3. Phase Multiplication Factor
X. is the phase multiplication factorused to account for
phase L L
processes happening in different phases. For example, if a
certain nanomaterial is sorbed on a solid, or the surface is
coated with DOC, the physicochemical properties might
differ from pristine nanomaterial, and reaction rate might
change. Also, if silver nanoparticle (AgNP) surface is
associated with DOC in surface waters, the AgNP dissolution
rate may be reduced.
Phase multiplication factor is expressed as
Xphase ~~ fnano^nano "I" fdoc^doc + fsolids-solid Equation 77
where f , f, and f ... are the mass fractions of nanomaterial
J nemo J doc J solid
in its pristine form, nanomaterial coated with DOC, and
nanomaterial sorbed on solid phase, respectively. Their
relationship is
fnano fdoc fsolid 1 Equation 78
x , x. and x ,, are phase multiplication factor for its
nano7 doc solid L L
pristine form, nanomaterial complexed with DOC, and
nanomaterial sorbed on solid phase. The default values for
each of these three rate multipliers is 1. The use offsoljd is
specifically for sorbed nanomaterials. When a nanomaterial
heteroaggregates with a solid phase, it is accounted for by
using a separate state variable, and all reactions are structured
for that variable.
6.2.4. Segment Multiplication Factor
Like most pollutant fate models for surface waters, WASP8
breaks the modeled region into different segments along
the river mainstream. Reaction rates for nanomaterial or
dissolved chemicals may be different in different segments,
and WASP8 users can adjust reaction rates in different
segments by using the segment multiplication factor option.
Segment multiplication factors include water column (xwc),
surface scdimcnt(.r;). and subsurface sediment (x J.
An example would be a river divided into five segments
along the mainstream, where AgNPs are released in the
first segment and dissolution rates for AgNPs in these five
segments are 0.0010, 0.0013, 0.0006, 0.03, and 0.00058 d"1,
respectively, in the water column. In WASP8, users can
set a dissolution rate in "Constant Group," and use a phase
multiplication factor to adjust dissolution rates in different
segments. For this example, users set a dissolution rate as
0.0010 in "Constant Group" for water column and segment
multiplication factors from segments 1 - 5 as 1, 1.3, 0.6, 30,
and 0.58, respectively.
If users want different reaction rates in the sediment
in different segments, they can follow the approach
described above.
6.2.5. Monod Kinetics
X is the factor that accounts for Monod-type kinetics as a
cone J A
function of the nanomaterial structure. This is structured in
WASP as follows
V" _
conc ~ Km+Nano Equation 79
where Km is the Monod half-saturation coefficient (mg/L).
This is structured so that the base reaction rate constant, k ,,
' rate7
must be adjusted so the maximum reaction rate is .
6.2.6. Environmental Kinetics
Similar to Monod kinetics, environmental factors can be
incorporated into the reaction rate using X . This can
A 0 env
incorporate oxidation with oxygen, for example. This is
structured in WASP as
£
^env ~ k +E Equation 80
where E is the environmental concentration of interest
.x
(mg/L) (e.g., oxygen, sulfate) and Ke is the half-saturation
coefficient for this environmental parameter. The use of this
functionality requires an E-option switch of 1, 2, or 3. If the
switch is 1, thenX = 1. If the switch is 2, then it is modeled
' env 7
as second order andX = E .If the switch is 3, Equation 81
env x 7 ±
is used.
nomaterial Phototransformation
Nanomaterials can undergo phototransformation. For
example, lab studies show graphene oxide (GO) undergoes
phototransformation under simulated sunlight radiation,
resulting in products that include reduced GO (rGO) and
poly cyclic aromatic hydrocarbons (PAHs). Nanomaterial
phototransformation follows first-order kinetic reaction as:
dN _ ,
~ ^KpfLotoN Equation 81
where kphoto is the phototransformation rate constant (d1), and
N is nanomaterial concentration (ng/L).
Formations of new nanomaterials and dissolved chemicals
can be simulated using first-order kinetics. The reaction
yield coefficient (g/g) in WASP8 allocates how much of each
daughter product (nanomaterials and dissolved chemicals) is
formed.
In WASP8, the spectrum of sunlight is divided into 11
specific wavelength bands, as shown in Tables 3-5. The
division is owing to the fact that different wavelengths drive
different environmental processes.43-48 Since infrared light is
generally not photoreactive, we only consider contributions
-------
of ultraviolet and visible lights to the phototransformation
rate constants. WASP8 calculates the phototransformation
rate constant for each wavelength first, then lumps these
10 together. The lumped photoreaction rate constant is
expressed as:
kphoto ~ X/=[(kphoto./j Equation 82
where k, is the overal phototransformation rate constant
photo 1
(d1) and kphoto; is the phototransformation rate constant (d1)
for each specific wavelength, X. An example is available to
show how to calculate k, in WASP8.
photo
piementatioii
First-order kinetic rate constants and other correctors can
be set in the Nano Chemical Photolysis and Nano Chemical
Decay sections. Besides setting the wavelength-dependent
reaction rate constant for Nano Chemical Photolysis, the solar
radiation intensity data should use WASP's light routines.
imples
6.5.1. Silver Nanoparticles Dissolution
Silver nanoparticles (AgNPs) are one of the most widely
used nanomaterials. AgNPs dissolution is the heterogeneous
reaction of AgNPs surface with oxygen in aquatic
environments, and AgNPs dissolution is a size-controlled
process. Most studies suggest that solubility of AgNPs
increases as their size decreases.49 51
Several studies simulate silver ion release using first-
order reaction kinetics, with the model fitting well with
experimental data.52 54 Peretyazhko et al. studied size-
controlled dissolution of AgNPs at neutral and acidic pH
conditions. They found that AgNPs increase in size after
dissolution, which follows first-order reaction kinetics.54
Zhang et al. indicate that dissolved oxygen (DO) significantly
influences AgNPs aggregation kinetics, and that aggregation
rate increases when DO is present compared to those without
DO.55 DO and proton concentrations can affect silver ion
dissolution kinetics based on the heterogeneous reaction that
occurs on AgNPs surfaces in the aquatic environment:
Ag(s) +lo2{aq) + 2H^aq) 2Ag£aq) + H20 Equation 83
The consumption of [02] and [H+] from AgNPs in the aquatic
environment is negligible.53 Besides first-order reaction,
Martin, et al. found that silver ion release matches second-
order kinetics for protein-coated AgNPs.56
WASP8 can simulate first-order and second-order kinetic
reactions. We also assume that the nanomaterial maintains
a uniform size throughout the reaction process. It has been
suggested that even though nanomaterial size changes after
reaction, reaction can still match first-order or second-order
reaction kinetics.
Here, we demonstrate the simulation of AgNP dissolution.
Experimental dissolution data are obtained from Zhang et
al.'s publication.53 In WASP8, AgNPs are simulated using
the nanomaterial state variable, and dissolved silver ion as a
chemical state variable. Dissolution rate is set in "Constant
Group" "Nanodecay" section. Parameters like nanomaterial
density and diameter can be set in the "System" and
"Constant Group." Heteroaggregation coefficient can be set
in "Parameter."
The equation used to model AgNPs dissolution derived in
Zhang et al.'s publication can be simplified as:
[Ag+]reieased = [Ag]initial^ - exp(-at)] Equation 84
where [Ag+]released is the aqueous silver ion(|ig/L). [Ag]imtM
is the fitted initial aqueous AgNP concentration (ug/L).
and a is dissolution rate constant (h1). Experimental data
indicated that
\^g\total ~ \^g ]released [A*?] Equation 85
where [Ag] is AgNP concentration at a given time (ug/L).
and [Ag]total is the sum of [Ag] and |Ag |lck.ascil concentrations
when t is zero.
We choose two sets of data from Zhang et al. 's publication.53
The first data set for AgNP dissolution kinetics is that size of
AgNPs is 40 nm, | Ag|tii is 300 ug/L. The second has AgNP
size of 80 nm and [Ag]totalis 600 ug/L. Two types of AgNPs
in WASP8 are created as two nanomaterial state variables,
and two chemical state variables are created that correspond
to dissolved silver ion released from each type. First-order
reaction kinetics simulate [Ag+]release and [Ag] in aqueous
solution. The reaction rate constant and [Ag]imM are available
in Zhang et al.'s publication.
Data retrieved from Zhang et al. 's paper and WASP simulated
results are presented in Table 9. Figure 8 shows experimental
data and WASP8 simulation results. Simulation results are
similar to Zhang et al. 's publication: for the first eight hours,
results show that overall R2 values for 40 nm-300 ug/L and
80 nm - 600 ug/L are 0.98 and 0.99, respectively; between
8-96 hours, R2 values are 0.97 and 0.95, respectively. R2
values are lower between 96 - 300 hour - 0.71 and 0.62,
respectively ~ possibly due to aggregation effects.
In Peretyazhko et al.'s paper,54 the following equation is used
to model [Ag+]released in aqueous solution:
[Ag Jreleased [Ag I total I ^ 6XJ.) ( /ct)] Equation 86
where [Ag+]released is silver ion concentration in aqueous
solution at a given time, and |Ag |ijia| is total dissolved silver
concentration when no further silver ion concentration
increases in aqueous solution during the reaction, and k
is dissolution rate. Their research also gets satisfactory
agreement between experimental data and model fitting.
Because they didn't provide experimental data, we made
no effort to fit it, however, the example shows that WASP8
is able to simulate first-order and second-order dissolution
reactions.
Besides AgNPs dissolution, new updates of WASP8 can
simulate the fate of AgNP in surface waters, which are
summarized in Dale et al.57
-------
Table 9. Data retrieved from publication and WASP8 simulated results
40 nm - 300 uL/L 80 nm - 600 uL AgNPs
Dissolved silver ion data Dissolved silver ion Dissolved silver ion data Dissolved silverion
Time (hour) from Zhang et al's paper simulated by WASP8 from Zhang et al's paper simulated by WASP8
0
12
0.00
12
0.00
2
18
6.41
12
4.18
4
21
12.16
12
7.97
8
27
21.96
18
14.53
12
30
29.84
18
19.93
24
39
45.42
30
31.04
48
51
57.78
36
40.69
72
54
61.15
42
43.69
96
60
62.07
48
44.62
144
63
62.39
48
45.00
192
66
62.41
48
45.03
240
66
62.41
42
45.04
288
69
62.41
48
45.04
336
63
62.41
48
45.04
600
" * A ± A.
500
(a)
ttO
3
400
300
200
, H
M 100
<
~ AgNPs 40 nm - 300 ug/L (publication data)
~ AgNPs 80 nm - 600 ug/L (Publication data)
--- WASP8 simulation
120
oo 100
3
c
o
C
80
60
40
20
~ AgNPs 40 nm - 300 ug/L (publication data) /li
~ AgNPs 80 nm - 600 ug/L (Publication data)
WASP8 simulation
_ ...» ~
~
~
/ A _4 A k. A
¦fT
f
A
100
300
400
200
Time (hour)
Figure 8. (a) [Ag] experimental data and WASP8 simulation results, (b) [Ag+
simulation results.
100 200 300
Time (hour)
. experimental data and WASP8
eased ¦
400
6.5.2. Calculation of Nanomaterial Phototransformation
Rate Constants
This example shows how WASP8 internally calculates
nanomaterial phototransformation rate constants. WASP8
divides the light spectrum into 10 specific wavelengths. The
wavelength-dependent reaction rate constant k"ob/I" (d'nr/W)
is determined. k"olJ&') is the reaction rate constant due to the
irradiation intensity of I" (W/m2), which is the light intensity
of a specific wavelength band. Example wavelength-
dependent reaction rate constants and wavelength-dependent
reaction rates are listed in Table 10.
With the wavelength-dependent reaction rate concentrations
determined, the phototransformation rate constant for each
wavelength band is calculated as follows:
knh
kobs.A= -^rWt Equation 87
Table 10. Wavelength-dependent reaction rate of each
wavelength band
Wavelength (d1 (W/m2)1)
620-749 nm 0.0001
570 - 619 nm
0.0001
495 - 569 nm
0.0001
450 - 494 nm
0.0001
380 - 449 nm
0.0002
355 - 379 nm
0.0002
335 - 354 nm
0.0002
315 - 334 nm
0.0004
305 - 314 nm
0.0003
295 - 304 nm
0.0003
20
-------
where Iavr (W/m2) is average light intensity in water column,
and kobs / (d1) is the phototransfonnation rate constant of each __
wavelength band.
The overall phototransfonnation rate constant is expressed as
^obs, overall
_ vio
_ 2,A=
iA=1 k0bs,A.
Equation 88
Iav / is average light intensity in water column which WASP8
calculates by attenuating the sunlight radiation intensity
at water surface (Ig). Calculation of average light intensity
and light attenuation is documented in Chapter 4. Hourly
sunlight intensity on earth surface data in the U.S. can be
retrieved from the North American Land Data Assimilation
Systems (NLDAS) database. Surface solar radiation varies
over the course of the day. Time varying solar radiation data
(e.g., hourly) can be input using "Time Functions." WASP8
can internally calculate the hourly GO phototransfonnation
rate and nanomaterial phototransfonnation and new
nanomaterial production.
6.5.3. Nanomaterial Parallel Reaction
The nanomaterial, graphene oxide, undergoes
phototransfonnation and generates daughter products:
reduced graphene oxide and PAHs. This is a parallel reaction
whose reaction pathways are shown in Figure 9.
Nano
Nanoo
Chem
Figure 9. Nanomaterial parallel reaction pathways.
For illustration we assumed that initial Nano concentration
is 10 |ig/L and the overall phototransfonnation rate
constant is 0.1 d1. Because Nano, and Chem are generated
simultaneously, reaction yield (g/g) is introduced to quantify
how much of each product is generated. In this example,
reaction yield for Nano, production is 0.9 (T;). and reaction
yield for Chem production is 0.1(7,).
Based on the reaction scheme, reaction rates can be
written as:
dlNanOi]
dt
= k[Nano^\
d[Nano2\
dt
-Y1k[Nano1
Equation 89
Equation 90
d[Chem\
dt
= Y2k[Nano1]
Equation 91
WASP8 simulation results are presented in Figure 10. This
example shows that WASP8 tracks the degradation of the
parent nanomaterial and fonnation of reaction products.
6.5.4. Segment Multiplication Factor
When conditions change along the course of a water body, it
may be useful to incorporate different reaction rates through
= 8 ¦ ~
12
11
H
10
00
9
3
8
C
o
7
4-»
6
TO
imm
¦M
5
c
a;
4
-------
100
no
3. 80
C
O
2 60
+¦»
c
Q)
C
o
40
c
o
'Z 20
Q)
>
0
AgNPs 40 nm - 300 ug/L
-AgNPs without DOC coating
-AgNPs with DOC coating
100
200
Time (hour)
Figure 11. WASP8 simulation results.
300
100
3. 80
£
O
c
400
60
40
20
AgNPs 80 nm - 600 ug/L
AgNPs without DOC coating
AgNPs with DOC coating
100
200
Time (hour)
300
400
22
-------
Dissolved Chemicals Reactions
Introduction
Distribution and concentrations of dissolved contaminants
in surface waters are significantly influenced by interactions
between contaminants and the physical and chemical
components of aquatic environments. These fate processes
must be fully assessed during the evaluation of contaminant
fate and transport to accurately simulate the behavior of
contaminants in surface waters. Oxidation-reduction (redox)
reactions, biodegradation and photochemical reactions
account for a range of chemical reactions that occur in
surface waters (e.g., rivers, lakes, sediments). Redox
reactions involve oxidation and reduction, which occur
with the exchange of electrons between reacting chemical
species. Electrons are lost in oxidation, and gained in
reduction. Organic contaminants of concern such as aromatic
hydrocarbons, aldehydes, ketones, phenols, hydroquinones,
and aliphatics are susceptible to oxidation. Typical reductive
transformation of environmental contaminants includes
dehalogenation of chlorinated aliphatic or aromatic
contaminants and reduction of nitroaromatic compounds.
Typical environmental oxidants include oxygen, ozone,
chlorine dioxide, ferrate, and chromate; typical environmental
reductants include low molecular weight organics, dithionite,
sulfides (and polysulfides), Fe(II) at mineral surfaces, and
zero-valent iron. Biodegradation refers to the conversion to
mineralized end products (e.g., C02, H20 and salts) through
metabolism by living organisms, an important process
for removing organic contaminants in surface waters and
sediment. Phototransformation refers to a chemical reaction
initiated by the absorption of energy in the form of light;
most phototransformations occurring in surface water is
driven by sunlight radiation.
WASP8 includes biodegradation, oxidation, and reduction
and photochemical reactions. The previous WASP version
(WASP7) can simulate these four chemical reactions for
up to three chemical state variables, but WASP8allows
simulation of up to 10 variables and handles more
complicated situations.
nutation for Oxidation, Reduction and
Biodegradation
7.2.1. Generic Reaction Module
Oxidation, reduction, and biodegradation reaction uses a
generic transformation reaction and is constructed as:
^reaction ^ratj^ Lemp^seg^phase^vionodj.onc'^monodjmv
Equation 94
where k is the overall reaction rate constant (d1) that
reaction v 7
WASP8 calculates, k , is first-order reaction rate constant
' rate
(d1) at 20°C, Xtemp is the temperature correction factor,
X is the segment multiplication factor, X, is the phase
seg o l 7 phase A
multiplication factor, X . is Monod equation and X
1 ' monod_conc 1 monod_
is Monod environmental reactant corrector, k , is first-
env rate
order reaction rate constant at 20°C that needs to be input
into WASP8. Default values are 1 for the other five terms in
Equation 96 if reaction rate is not affected by them. If the
reaction rate is influenced by other terms, WASP8 users must
activate them and input the correct values.
7.2.2. Effect of Temperature
X, is the temperature correction factor when temperature
temp A A
affects the reaction rate, expressed as
Xtemp = 0(T~W'> Equation 95
where 0 is temperature-activity coefficient, and T is
temperature (°C).
7.2.3. Phase Multiplication Factor
X, is used when the reaction only occurs for specific
phase J 1
phases. For example, if a contaminant in aqueous
environment is sorbed on a suspended solid, its reaction
rate may be different than that in dissolved form. Phase
multiplication factor is thus designed and expressed as
Xphase ~ fdiss^diss fdoc^doc fsolid^solid Equation 96
where f. , f. , and f ,. are the mass fractions of the
diss' doc' solid
contaminant in its dissolved form, DOC-complexed form,
and the contaminant sorbed on solid phase, respectively;
x. , x. , and x ,. are phase multiplication factors for the
diss' doc' solid A A
contaminant in its dissolved form, DOC-complexed form,
and sorbed form.
Mass fractions of the contaminant in three different forms are
determined by their partition coefficients, and their relation is
fdiss + fd oc fsolid 1 Equation 97
Default values for three phase multiplication factors are 1 in
WASP8.
7.2.4. Segment Multiplication Factor
Similarly to nanomaterial reaction, segment multiplication
factor adjusts chemical reaction rates in different segments
in the modeled region. Methods are described in a previous
chapter.
23
-------
7.2.5. Monod Kinetics
X is the factor that accounts for Monod-type kinetics as a
cone J L
function of the nanomaterial structure. This is structured in
WASP as follows:
Y _Km
* cone Km+C Equation 98
where Km is the Monod half-saturation coefficient (mg/L).
This is structured so that the base reaction rate constant, k ,,
? rate7
must be adjusted so the maximum reaction rate is k /K .
J rate m
7.2.6. Environmental Kinetics
Similarly to Monod kinetics, environmental factors can be
incorporated into the reaction rate using This can be
used to incorporate oxidation with oxygen, for example and
is structured in WASP as
v _ Ex
env ~~ Ke+Ex Equation 99
where E is the environmental concentration of interest
*
(mg/L) (e.g., oxygen, sulfate) and Kf is the half-saturation
coefficient for this environmental parameter. The use of this
functionality requires an E-option switch of 1, 2, or 3. If the
switch is 1, then this is not used, and A = 1. If the switch is
7 7 env
2, then it is modeled as first order, and A = /'. If the switch
' 7 env x
is 3, then A = E/K +E Equation 80 is used.
' env x e x 1
7.2.7. Second Order Kinetics
For chemical reactions, second order kinetics can also be
implemented. For this case, the stoichiometry of the reaction
is important.
k
nC1 ¦₯ mC2 i C3 + j C4 Equation 100
where n, m,and j arc molar quantities input in the new
constant group: Chemical Second Order Rxn. Chemical
reactant and product identifiers arc also input, along with
reaction rate constant, temperature correction constants, and
pha.se efficiencies. The reaction is of the form:
d.Cx mi
= kC2 nC j Equation 101
The reaction yields [g/g] include the consumption of C] and
production of C} and C4. These arc calculated from the molar
stoichiometry and molecular weights.
7.2.8. Biodegradation
When degradation of dissolved chemical is through
biodegradation reaction, Monod kinetics can be used to
simulate biodegradation. and A . is expressed as
7 monodjeone L
follows:
^monodsonc ~^~Tc ^ Equation 102
where Xis the biomass concentration (mg/L), C is the
limiting substrate concentration (mg/L), Ks is the half-
saturation coclTicicnt (mg/L). The way this is structured is
such that the ba.se reaction rate constant. kmte, needs to be
adjusted so that the maximum reaction rate is k /K. This
J rate s
allows the Monod relation to collapse to first order at low
substrate concentrations. When this occurs. C « which
is common in surface waters, the reaction becomes
kreaction ^rate ^ ^ ^ra£e ^ ^ krateX Equation 103
where k , is the maximum substrate utilization rate (d1) in
rate v 7
Monod equation, and k' is first-order kinetic reaction rate
constant (d1)-
In biodegradation module. WASP8 offers users two
alternatives to model biodegradation: Users can cither
directly input first-order reaction rate constant, or input all
the parameters in Monod equation into WASP8.
Dissolved oxygen, nitrate or other environmental reactants
can influence enzyme metabolism activity and. thus,
biodegradation. The effect of possible environmental
reactants on the biokinctic is accounted for by one correction
factor, expressed asX . . in the form of one saturation
' A monoa_env~
term:
Y - E
Amonod_env Kenv+E Equation 104
where E is the environmental reactant concentration (mg/L),
and K is the half-saturation coclTicicnt of environmental
env
reactant (mg/L).
.solved Chemicals Phototransforrnation
The dissolved chemical phototransformation module is
the same as the nanoniatcrial phototransforniation niodule:
reaction follows first-order kinetic reaction and the light
spectrum is divided into 10 specific wavelengths. WASP8
calculates a phototransformation rate for each wavelength
first, and then aggregates them.
Dissolved chemical phototransforniation is expressed as
follows:
dC _ , _
dt ~Kphoto^ Equation 105
where kphoto is phototransforniation rate (d1) that WASP8
calculates, and C is chemical concentration (mg/L). The
overall photorcaction rate constant is expressed as
kphoto ~ Sa=i(kphoto,a) Equation 106
where kphoto is the overal phototransformation rate constant
(d"1), and kphotol is phototransforniation rate constant for
specific wavelength (d1).
rniples
7.4.1. First-Order Biodegradation Influenced by
Temperature
When temperature influences reaction rate, the temperature
corrector must be turned on. WASP8's default setting is
that when temperature is 20°C, reaction rate is not affected.
In this example, we examine biodegradation rates at 10°C,
20°C, and 32°C. We assume the reaction rate is 0.2 d1 at
20°C, which can be set in "Constant Group." Initial dissolved
-------
chemical concentration is 10 mg/L. At a temperature of 20°C,
the temperature corrector is not needed, but at temperatures
of 10°C and 30°C, users must consider temperature's effect.
In general, when temperature influences on reaction rate, the
reaction rate constant is expressed as
7.4.2. Second-Order Reaction
WASP8 simulates second-order kinetic reaction by turning
on the second-order kinetics option. Here, we use hydrogen
peroxide (H202) decomposition reaction as an example. H202
can decompose into water and oxygen as follows:
k k y k
^reaction ^rate^temp ar
ag(T-20)
Equation 107
For biodegradation, typical values for 0 vary from 1.020 to
1.10. In WASP8, we set 0 as 1.020 and temperature (T) can
be set in "Constant Group."
The dissolved chemical biodegradation rate is
dC
= kr
Equation 108
Its analytical solution is
Ct Cinit iai 0 cxp(fcr (, a c tio n t)) Equation 109
where (\ is dissolved chemical concentration (mg/L)
at a given time, and C. mal is initial dissolved chemical
concentration (mg/L). Simulation results are shown in the
following figure.
2H2Oz ' 2H20 + 02 Equation 110
Suppose initial H202 contraction is 20 mg/L, and reaction rate
k is 0.02 L/Onmold1).
Consumption of H,0, and formation of 0, are expressed as
^ [H'2 0'2 I
dt
k[H202]\H202]
^ = -2k[H202\[H202\
Equation 111
Equation 112
If WASP8 users only want to simulate the consumption
of H202 and formation of 02, they create two chemical
state variables, and then input the initial concentrations,
molecular weights of these two state variables and second-
order reaction rates into WASP8. Next, turn on second-
order functionality and provide concentration of the second
reactant, H202. In these two differential equations, H202 and
02 are in the unit of mmol/L. In WASP8, the default unit of
chemical state variable is "mg/L." WASP8 simulation results
are shown in the following figure.
_ 12
_J
cuo
J.10
c
.2 8
c
01
u
c
o
u
E
QJ
6 ¦
4 ¦
2 ¦
~ WASP output at 18°C
A WASP output at 20°C
O WASP output at 32°C
Analytical solution
- X
0 5 10 15
Time (day)
Figure 12. WASP8 output of biodegradation rate at
different temperatures, fitted with analytical solution
20
25
20
no
E,
c 15
o
+¦»
fO
^ 10
a)
u
c
O 5
~ Hydrogen peroxide
Oxygen
10 20
Time (day)
30
Figure 13. Second-order reaction simulation results
by WASP8.
25
-------
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55. Zhang, W.; Yao, Y.; Li, K.; Huang, Y.; Chen, Y., Influence of dissolved oxygen on aggregation kinetics of citrate-coated
silver nanoparticles. Environmental Pollution 2011,159 (12), 3757-3762.
56. Martin, M. N.; Allen, A. J.; MacCuspie, R. I.; Hackley, V A., Dissolution, agglomerate morphology, and stability limits of
protein-coated silver nanoparticles. Langmuir 2014,30 (38), 11442-11452.
57. Dale, A. L.; Lowry, G. V.; Casman, E. A., Modeling nanosilver transformations in freshwater sediments. Environ. Sci.
Technol 2013, 47 (22), 12920-12928.
I
-------
9.0
APPENDIX
9.1 Solids QA/QC
Table 11. Parameters used in Analytical Equations
Variable Description Units
Oiu Flow in m3/d
0.
z'in
Flow out
m3/d
jWC
Water Column Volume
m3
Vs
Sediment Layer Volume
m3
V
s
Settling Velocity
m/d
vr
Resuspension Velocity
m/d
Cross-sectional Area of Bed
m2
swc
J
Concentration of Solid j in
Water Column
mg/L
Ss
J
Concentration of Solid j in
Sediments
mg/L
s
°J
Initial Concentration of
Solid j
mg/L
v wc
BC. j
Boundary Condition
Concentration of Solid j
mg/L
t
Time
d
A two-segment WASP model (consisting of a water column
and sediment layer below) simulated solids processes for
comparing analytical solutions to WASP simulation results.
Table 12 shows the channel geometry of the segments.
Length
100 m
100 m
Depth
10 m
0.05 m
Width
100 m
100 m
To ensure quality assurance of the solids module, we tested
boundary condition (BC) and initial condition (IC) of
solids concentrations. Stokes' settling, and resuspension.
For both BC and IC scenarios, we simulated a single solid
with streamflow moving through the segment. Settling was
simulated with 10 solids simultaneously, varying streamflow
and settling velocities. Resuspension was simulated using a
single solid with no flow.
9.1.1. Scenario 1 - Constant Boundary Conditions,
Initial Concentration = 0, No Settling
Using a solid boundary condition of 10 mg/L, we modeled
the increase of concentration over time. With no initial solid
concentration, and a steady state streamflow, the analytical
equation for this problem is defined as:
wc
yWC = QinSBCCj - QoutSZ.i Equation 113
where:
= 0 mg/L
SBC,j = 10 m§/L
Qin= Qout = 172,800 m3/d
Vwc = 100,000 m3
At steady state, Q=Qm=Q0Ut We solve for S"'tuj. where the
concentration flowing out equals the concentration in the
water column (S"'r to get:
sZ,i = sZC,i ~ Equation U4
Figure 14 shows the comparison of WASP simulation to
analytical solution. WASP outputs were set to a 0.1/d time
step, about two and a half hours.
Solids
o
02
E
c
o
c
a>
C CN
o
O
o
Analytical Solution
v WASP Simulation
0
2
6
8
10
4
Time (Day)
Figure 14. Comparison of WASP Simulation to Analytical
Solution for Scenario 1
Table 12. Channel Geometry of WASP Segment
Segment Water Column Sediments
Volume 100,000 m3 500 m3
-------
9.1.2. Scenario 2 - Initial Conditions, Boundary
Condition = 0, No Settling
Similarly to the BC scenario, an initial solid concentration
modeled the decrease of concentrations over time. With no
BC input and a stead}' state flow, the analytical equation for
this problem is defined as:
where:
AS^C
yWC _tj ¦
At
' QoutS,
out.j
Equation 115
Soj= 10 mg/L
SBCJ = 0 m§/L
Qin= Qout = 172,800 m3/d
Vwc = 100,000 m3
Solving for Sfc we get:
, Qt ^
Sj = So,J e yWC Equation 116
Figure 15 shows the comparison of WASP simulation to
analytical solution.
Solids
CD
C
o
TO
C
0
o
c
o
O
OO -
CD
CM
O -
Analytical Solution
v WASP Simulation
"T
2
mmmmmmmmmammmmmm
¦ i
6
~T~
8
10
Time (Day)
Figure 15. Comparison of WASP Simulations to
Analytical Solutions for Scenario 2
9.1.3. Scenario 3 - Initial Concentration, With and
Without Streamflow, Settling
We tested settling scenarios with and without streamflow.
Because settling can be simulated with multiple solids
independently, we were able to simulate 10 solids
simultaneously settling. For the first scenario, we simulated
10 solids, all with different settling velocities and all with
an initial concentration of 10 mg/L, and no streamflow.
Descriptive settling rates were applied to each solid using:
particle
^set
= -9g
(Pparticle Pwater) 2
Hwater particle Equation 117
Table 13 lists properties for the 10 solids used in scenario 3.
Table 13. Physical properties of solids
Solid
Density
(mg/L)
Particle
Radius
(mm)
Settling
Velocity
(m/d)
1
2.65
8.97E-04
0.25
2
2.65
1.27E-03
0.5
3
2.65
1.79E-03
1
4
2.65
4.01 E-03
5
5
2.65
8.97E-03
25
6
2.65
1.27E-02
50
7
2.65
1.79E-02
100
8
2.65
2.20E-02
150
9
2.65
2.54E-02
200
10
2.65
2.84E-02
250
Settling can be described with the mass balance equation:
AS^C
yWC
At
- ewe
V out"3 out ,j
- n A
vsbecL^j
Equation 118
where:
Soy = 10 mg/L for solids 1-10
vs = 0.25, 0.5, 1, 5, 25, 50, 100, 150, 200, 250 m/d
Abed - 10000 m2
Qin= Qout = 0; 172,800 m3/d
Vwc = 100,000 m3
Solving for to get:
Q+vs>l«,ed
Syc= Sffie yWC Equation 119
Figure 16 shows the output comparison of all 10 solids
modeled in WASP when 0 =0 = 172.800 mVd.
^ in ^ out 7
-------
Analytical Solution 25 m/day
v WASP Simulation 25 m/day
Analytical Solution 5 m/day
WASP Simulation 5 m/day
Analytical Solution 1 m/day
v WASP Simulation 1 m/day
Analytical Solution 0.5 m/day
WASP Simulation 0.5 m/day
Analytical Solution 0.25 m/day
v WASP Simulation 0.25 m/day
Solids Settling
1.0 1.5
Time (Day)
Analytical Solution 250 m/day
v WASP Simulation 250 m/day
Analytical Solution 200 m/day
WASP Simulation 200 m/day
Analytical Solution 150 m/day
v WASP Simulation 150 m/day
Analytical Solution 100 m/day
WASP Simulation 100 m/day
Analytical Solution 50 m/day
v WASP Simulation 50 m/day
Figure 16. Comparison of WASP
Simulations to Analytical Solutions for
Scenario 3, With Streamflow
Solids Settling
.fc
o
o
Analytical Solution
WASP Simulation
Analytical Solution
WASP Simulation
Analytical Solution
v WASP Simulation
Analytical Solution
WASP Simulation
Analytical Solution
v WASP Simulation
25 m/day
25 m/day
5 m/day
5 m/day
1 m/day
1 m/day
0.5 m/day
0.5 m/day
0.25 m/day
0.25 m/day
0.0
0.2
0.4
1.0
o
O
Analytical Solution
S7 WASP Simulation
Analytical Solution
WASP Simulation
Analytical Solution
WASP Simulation
Analytical Solution
WASP Simulation
Analytical Solution
v WASP Simulation
250 m/day
250 m/day
200 m/day
200 m/day
150 m/day
150 m/day
100 m/day
100 m/day
50 m/day
50 m/day
40 60
Time (Day)
In the scenario where Q =0 the solution can be simplified:
Figure 17. Comparison of WASP
Simulations to Analytical Solutions for
Scenario 3, Without Streamflow
urr t/irr ( VsAbedt\
Sy = Sq^-e vwc
Resuspension from the sediment layer can be described with
the mass balance equation:
Equation 120
Figure 17 shows the output comparison of all 10 solids
modeled in WASP when Q= 0 inVd.
9.1.4. Scenario 4 - Resuspension
Simulating resuspension analytically is more complicated
because all solids concentrations in sediments affect the
concentration of an individual solid that is being resuspended.
We therefore modeled a single solid being resuspended from
the sediment layer into the water column. This demonstrates
the processes in WASP8 and tests if code is working properly.
c AS? c
V--VrAbedSjS
where:
0 = 0
y= 500 m3
We simplify and differentiate to solve:
"rAbedSjt
Sf = Slje' vs
Figure 18. Shows the solid concentration in
sediments over time.
Equation 121
Equation 122
-------
Solids in the Sediments
Solids in the Water Column
CD
C
o
-t'
TO
i_
C
-------
Attenuation parameters below the water surface were tested
with the average Ke for each light group, an output calculated
internally by WASP, and applied to the Beer-Lambert
equation (Equation 21). The average Ke outputs for each
attenuation parameter, by light group, is listed in Table 16.
9.2.1. Option 0, Calculated Diel Light
Using coordinates from Athens, GA (33.9519, -83.3576), the
model is simulated for a single day to get outputs for UV,
PAR, and IR at the top and bottom of each segment. Using
default settings with no attenuators turned on, the output light
for a single day in each segment is described by wavelength
group in Table 17.
Table 16. Average K Outputs for Each Attenuation
Parameter by Light Group
Chlorophyll
0.168
0.289
2.07
Solids
2.551
2.789
4.57
DOC Global
10.962
2.689
2.07
DOC (i)
8.394
1.189
2.07
Frac of Day
0.051
0.289
2.07
Light
Extinction
0.051
0.5
2.07
Chlorophyll
0.167
0.164
2.07
Solids
0.051
0.164
2.07
DOC Global
0.034
0.164
2.07
DOC (i)
5.154
0.764
2.07
Frac of Day
0.051
0.164
2.07
Table 17. WASP Outputs by Wavelength Group for
201.42 217.05
73.57 79.28
93.79 16.86
152.22 27.39
55.60 10.00
93.72 16.86
152.22 27.39
55.60 10.00
79.93 2.13
129.83 3.46
47.42 1.26
Table 18 shows the comparison of WASP simulations and
analytical solutions for light attenuation parameters, above
and at the water surface in percent error. Table 19 and Table
20 show the comparison between WASP simulations and
analytical solutions for light attenuation parameters, below
the water surface in percent error.
Segment
Time
Water Surface
Reflection
UV PAR IR
Canopy Shading
UV PAR IR
Cloud Cover
UV PAR IR
UV
Ice Cover
PAR IR
9:00
0.00
0.00
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Top of Surface
12:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
15:00
0.01
0.00
0.00
0.01
0.00
0.00
0.01
0.00
0.00
0.01
0.00
0.00
9:00
0.00
0.00
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Bottom of
Surface
12:00
-0.01
0.00
0.00
-0.01
0.00
0.00
-0.01
0.00
0.00
-0.01
0.00
0.00
15:00
0.00
0.00
0.00
-0.01
0.00
0.00
-0.01
0.00
0.00
-0.01
0.00
0.00
9:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Top of
Subsurface
12:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
15:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
9:00
0.00
0.00
-0.01
-0.01
0.00
-0.02
0.00
0.00
0.02
0.01
0.00
-0.01
Bottom of
Subsurface
12:00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.01
-0.01
0.00
0.01
15:00
0.00
0.00
0.07
0.02
0.00
0.04
0.01
0.00
0.02
0.02
0.00
0.00
Segment Parameter KeUV KePAR KelR
^'9ht 0.051 0.5 2.07
Extinction
Subsurface
IZ.UU 14.44
15:00 5.27
9:00 8.22
Option 0 Using Athens, GA Coordinates
Segment Time UV PAR IR
9:00 9.62 124.02 133.68
Top of
Surface
Bottom of
Surface
Top of
Subsurface
Bottom of
Subsurface
12:00
15.63
15:00
5.71
9:00
8.89
12:00
14.44
15:00
5.27
9:00
8.89
12:00
14.44
15:00
5.27
9:00
8.22
12:00
13.4
15:00
4.88
Table 18. Comparison of WASP Outputs to Analytical Solutions for Above Surface Light Attenuation - Option 0
-------
Table 19. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation - Option 0
Light Extinction
Chlorophyll
Segment
Date
uv
PAR
IR
uv
PAR
IR
9:00
-0.05
0.00
0.00
-0.04
-0.02
0.00
Bottom of
Surface
12:00
-0.05
0.00
0.00
-0.03
-0.02
0.00
15:00
-0.04
0.00
0.00
-0.03
-0.02
0.00
Top of
Subsurface
9:00
0.00
0.00
0.00
0.00
0.00
0.00
12:00
0.00
0.00
0.00
0.00
0.00
0.00
15:00
0.00
0.00
0.00
0.00
0.00
0.00
9:00
-0.01
0.00
0.01
0.01
0.02
0.01
Bottom of
Subsurface
12:00
-0.01
0.00
0.00
-0.01
0.02
0.00
15:00
-0.01
0.00
-0.03
0.00
0.02
-0.03
Table 20. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation - Option 0
Segment
Date
UV
Solids
PAR
IR
DOC Global
UV PAR IR
uv
DOC (i)
PAR
IR
9:00
-0.07
-0.02
-0.01
0.04
-0.02
0.00
0.01
-0.02
0.00
Bottom of
Surface
12:00
-0.09
-0.02
-0.01
0.03
-0.02
0.00
0.02
-0.02
0.00
15:00
-0.05
-0.01
-0.02
0.02
-0.03
0.00
0.06
-0.02
0.00
9:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Top of
Subsurface
12:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
15:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
9:00
0.04
0.02
0.21
-0.07
0.02
0.01
0.04
0.02
0.01
Bottom of
Subsurface
12:00
0.05
0.02
0.12
-0.05
0.02
0.00
0.03
0.02
0.00
15:00
0.03
0.00
0.39
-0.03
0.02
-0.03
-0.03
0.02
-0.03
Table 21. Comparison of WASP Outputs to Analytical Solutions for Above Surface Light Attenuation - Option 1
Water Surface Reflection
Canopy Shading
Ice
Segment
uv
PAR
IR
UV
PAR
IR
UV
PAR
IR
Top of Surface
0.00
0.00
0.00
-0.01
0.00
0.00
0.02
0.00
0.00
Bottom of Surface
-0.06
-0.02
0.00
-0.05
-0.02
0.01
-0.07
-0.01
0.00
Top of Subsurface
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Bottom of
Subsurface
0.02
0.02
0.04
-0.01
0.02
0.04
-0.02
0.01
0.09
-------
9.2.2. Option 1, User Input Diel Light
Table 21 shows the comparison of WASP simulations and
analytical solutions for light attenuation parameters, above
and at the water surface in percent error.
Table 22 and Table 23 show the comparison of WASP
simulations and analytical solutions for light attenuation
parameters, below the water surface in percent error.
9.2.3. Option 2, User Input Daily Light, Calculated
Diel Light
Using coordinates from Athens, GA (33.9519, -83.3576), the
model is simulated for a single day to get outputs for UV,
PAR, and IR at the top and bottom of each segment. Using
default settings with no attenuators turned on, the output light
for a single day in each segment is described by wavelength
group in Table 24.
Table 22. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation - Option 1
Light Extinction
Chlorophyll
Segment
UV PAR
IR
UV
PAR
IR
Top of Surface 0.00 0.00 0.00 0.00 0.00 0.00
Bottom of Surface 0.00 0.00 0.00 0.00 0.00 0.00
Top of Subsurface 0.00 0.00 0.00 0.00 0.00 0.00
Bottom of Subsurface 0.00 0.00 0.00 0.00 0.00 0.01
Table 23. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation - Option 1
Solids
DOC Global
DOC (i)
Segment
UV
PAR
IR
UV PAR IR
UV PAR
IR
Top of Surface
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Bottom of Surface
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Top of Subsurface
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Bottom of Subsurface
0.00
0.00
0.01
-0.01
0.00
0.01
-0.01
0.00
0.01
Table 24. WASP Outputs by Wavelength Group for Option 2 Using Athens, GA Coordinates
Segment
Time
UV
PAR
IR
9:00
8.58
110.62
119.20
Top of Surface
12:00
13.94
179.66
193.60
15:00
5.09
65.62
70.72
9:00
7.93
83.59
15.04
Bottom of Surface
12:00
12.88
135.77
24.43
15:00
4.70
49.59
8.92
9:00
7.93
83.59
15.04
Top of Subsurface
12:00
12.88
135.77
24.43
15:00
4.70
49.59
8.92
9:00
7.34
71.30
1.90
Bottom of Subsurface
12:00
11.91
115.80
3.08
15:00
4.35
42.30
1.13
-------
Table 25 shows the comparison of WASP simulations and
analytical solutions for light attenuation parameters above
and at the water surface in percent error.
Table 26 and Table 27 show the comparison of WASP
simulations and analytical solutions for light attenuation
parameters, below the water surface in percent error.
Table 25. Comparison of WASP Outputs to Analytical Solutions for Above Surface Light Attenuation - Option 2
Water Surface
Reflection
Canopy Shading
Ice Cover
Segment
Time
UV PAR IR
UV
PAR IR
UV PAR IR
9:00
0.00
0.00
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
Top of Surface
12:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
15:00
0.01
0.00
0.00
0.01
0.00
0.00
0.01
0.00
0.00
9:00
0.00
0.00
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
Bottom of Surface
12:00
-0.01
0.00
0.00
-0.01
0.00
0.00
-0.01
0.00
0.00
15:00
0.00
0.00
0.00
-0.01
0.00
0.00
-0.01
0.00
0.00
9:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Top of Subsurface
12:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
15:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
9:00
0.00
0.00
-0.01
-0.01
0.00
-0.02
0.01
0.00
-0.01
Bottom of Subsurface
12:00
0.00
0.00
0.00
0.00
0.00
0.01
-0.01
0.00
0.01
15:00
0.00
0.00
0.07
0.02
0.00
0.04
0.02
0.00
0.00
Table 26. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation - Option 2
Light Extinction
Chlorophyll
Fraction of Day
Segment
Date
UV PAR IR
UV PAR IR
UV PAR IR
9:00
-0.05
0.00
0.00
-0.02
-0.02
0.00
-0.05
-0.02
0.00
Bottom of Surface
12:00
-0.04
0.00
0.00
-0.03
-0.02
0.00
-0.04
-0.02
0.00
15:00
-0.05
0.00
0.00
-0.03
-0.02
0.00
-0.05
-0.02
0.00
9:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Top of Subsurface
12:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
15:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
9:00
0.00
0.00
-0.03
0.00
0.02
-0.03
0.00
0.02
-0.03
Bottom of Subsurface
12:00
-0.01
0.00
0.00
0.00
0.02
0.00
-0.01
0.02
0.00
15:00
0.00
0.00
-0.03
-0.01
0.02
-0.03
0.00
0.02
-0.03
Table 27. Comparison of WASP Outputs to Analytical Solutions for Below Surface Light Attenuation - Option 2
Solids
DOC Global
DOC (i)
Segment
Date
UV PAR
IR
UV PAR IR
UV PAR
IR
9:00
-0.06
-0.02
-0.04
-0.01
-0.02
0.00
0.03
-0.02
0.00
Bottom of Surface
12:00
-0.03
-0.02
-0.03
0.04
-0.02
0.00
0.04
-0.02
0.00
15:00
-0.01
-0.02
0.02
0.05
-0.02
0.00
0.02
-0.02
0.00
9:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Top of Subsurface
12:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
15:00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
9:00
0.02
0.02
-0.30
-0.03
0.03
-0.03
0.00
0.02
-0.03
Bottom of Subsurface
12:00
0.03
0.01
0.03
-0.07
0.01
0.00
-0.06
0.01
0.00
15:00
0.02
0.03
-0.30
-0.03
0.02
-0.03
-0.01
0.02
-0.03
-------
Table 28. Variables Related to Equilibrium Sorption
9.3. Particle Attachment QA/QC
Variable
Coj
Kd
SJ
CjW
Qs
Description Units
Initial Concentration of Chemical /' mg/L
Partition Coefficient
Concentration of Solid j
Concentration of Dissolved
Chemical /'
Concentration of Sorbed
Chemical /'
Volume of Segment
L/kg
mg/L
mg/L
mg/L
m3
9.3.1. Equilibrium Sorption
Four scenarios tested the equilibrium sorption algorithm
in WASP8 and steady state WASP output concentrations
were then compared to analytic solutions. Variables used in
this section are defined in Table 28. WASP outputs sorbed
chemicals in mg of chemical i per kg of total solids, but units
have been converted for consistency.
Parameter values as well as output results and percent error
for each scenario are shown in Table 29.
9.3.2. Kinetic Sorption
Kinetic sorption algorithms were tested by constructing
a WASP model with a single water segment. Sorption
process were assumed to occur between chemical and solid
in the water segment, with no flow or transport of solids.
Two scenarios were tested and the parameters are listed in
Table 30.
We tested sorption kinetics in WASP8 with two different
scenarios (Table 30). Analytical solutions of these test cases
were simulated for comparison and results are presented in
Figure 20.
Table 29. Parameter Values and Results for each Scenario
Scenario Parameter
Variable Units Value WASP output % Error
Initial Concentration of Chemical i
C0,
mg/L
10
Partition Coefficient
Kd
L/kg
10
Concentration of Solid i
Sj
mg/L
500
Concentration of Dissolved Chemical /
QW
mg/L
9.950
9.952
-0.018
Concentration of Sorbed Chemical /
Cis
mg/L
0.050
0.050
-0.020
Volume of Segment
V
m3
100,000
Initial Concentration of Chemical /
C0,
mg/L
10
Partition Coefficient
Kd
L/kg
10
Concentration of Solid /
Sj
mg/L
250
Concentration of Dissolved Chemical /
Cjw
mg/L
9.975
9.976
-0.009
Concentration of Sorbed Chemical /
Cis
mg/L
0.025
0.025
-0.010
Volume of Segment
V
m3
100,000
Initial Concentration of Chemical /
C0,
mg/L
10
Partition Coefficient
Kd
L/kg
100
Concentration of Solid /
Sj
mg/L
250
Concentration of Dissolved Chemical /
Cjw
mg/L
9.756
9.757
-0.009
Concentration of Sorbed Chemical /
Cis
mg/L
0.244
0.244
-0.010
Volume of Segment
V
m3
100,000
Table 30. Sorption Kinetic Constants and Analytical Errors of Two Scenarios
Scenario Parameter Variable Units Value % Error
1 Initial chemical concentration Cqw mg/L 5 0
Suspended solid concentration Cs mg/L 5
Sorption rate constant kfor L/mg-d 0.05
Desorption rate constant krev d-1 0.005
2 Initial chemical concentration Cqw mg/L 15 0
-------
Table 30. Sorption Kinetic Constants and Analytical Errors of Two Scenarios (continued)
Scenario Parameter
Suspended solid concentration
Sorption rate constant
Desorption rate constant
Variable
cs
kfor
Units
mg/L
L/mg-d
d-1
Value
5
0.01
0.005
% Error
Chem 1
Chem 2
ID
CD
C
o
'+->
CO
aj cm
o
c
o
O _
Analytical Solution
P Simulation
CO
£
c
o
4'
03
-i'
C
(D
O
C
o
O
-
CO -
CM
1 1 1 1
0 2 4 6 8 10
Time (Day)
Analytical Solution
v WASP Simulation
T
2 4 6 8
Time (Day)
10
Figure 20. Comparison of Analytical Solution and WASP Simulation for Sorption Kinetics of Scenario 1. Chem 1
represents the freely dissolved chemical concentration, and Chem 2 represents the chemical concentration sorbed
to the suspended solid.
CD
E
c.
o
-4>
CO
-411
c
0)
o
c
o
o
Lf)
-
CO -
CM -
Chem 1
Analytical Solution
Chem 2
P Simulation
CD
E
c
o
-t>
c
0
o
c
o
O
CO -
CM
0 5 10
20
30
Analytical Solution
P Simulation
30
Time (Day)
Time (Day)
Figure 21. Comparison of Analytical Solution and WASP Simulation for Sorption Kinetics of Scenario 2. Chem 1
represents the freely dissolved chemical concentration, and Chem 2 represents the chemical concentration sorbed
on the suspended solid.
-------
Variable
Description
Units
Value
kb
Boltzmann Constant
m2 g s-2 K-1
1.38E-20
T
water
Absolute Temperature of Water
K
288.15
G
Shear Rate
s-1
2.00E-05
^ water
Dynamic Viscosity of Water
g/m/s
1.13
g
Gravitational Acceleration on Earth
m/s2
9.81
SjSPM
Initial/Boundary Concentration of Solid j
mg/L
100
CNP
Initial/Boundary Concentration of Nanoparticle i
mg/L
20
PSPM.j
Density of Solid j
g/cm3
2.65
P water
Water Density
g/cm3
1.00
Pnpj
Density of Nanoparticle i
g/cm3
1.30
rNP,i
Radius of Nanoparticle i
nm
100
rSPM,j
Radius of Solid j
mm
8.00E-03
V tNP
set,i
Settling Velocities of Nanoparticle i
m/d
4.99E-04
v SPM
set.)
Settling Velocities of Solid j
m/d
17.55
Q,n
Inflow
m3/s
Varies
Qout
Outflow
m3/s
Varies
Table 31. Heteroaggregation Kinetics Parameters
9.3.3. Nanomaterial Heteroaggregation
To test WASP8's heteroaggregation routines, we solved
and simulated scenarios for all three components of the ij
parameter. Variables used in this section are defined in
Table 31.
A WASP model consisting of a single segment simulated
the heteroaggregation process between concentrations of
nanoparticle i and solid j suspended in the water column.
Table 32. Channel Geometry of WASP Segment
Channel Geometry
Volume 100,000 m3
Length 100 m
Depth 10 m
Width 100 m
Table 33 Shows calculated values for k and its individual
components. For each scenario we varied a as 0.1, 0.01,
0.001, and 1E-6.
Table 33. Calculated Rate of Collision by Mechanism
Component
Calculated Value
Units
Brownian
1.66E-11
m3/d
Fluid
1.23E-15
m3/d
Settling
3.62E-9
m3/d
^ colli]
3.63E-09
m3/d
We used a concentration of 100 mg/L for and 20
mg/L for ('.v/' for all three scenarios. The first used these
concentrations as initial conditions; for scenarios two
and three, these concentrations are boundary conditions.
Scenario 1 used only Brownian motion and was solved
dynamically. The second and third scenarios added
components of k to the first scenario. Scenario 2
1 coll,ij
incorporates Brownian motion and fluid motion. Scenario 3
incorporates Brownian motion, fluid motion, and differential
settling. The second and third scenarios were solved at
steady state.
A-ll
-------
Scenario 1 - Brownian motion only
Scenario 1 looks solely at the effects of the Brownian motion
component of \olir It is solved dynamically because the
steady state solution would result in zero concentrations.
Initial concentrations of 100 mg/L and 20 mg/L are used
for SfM and Cfp, respectively. The scenario is solved
analytically using the mass balance equation:
ACNP
v^t= QinCZ ~ QoutC?" - khetiijcrv Equation 125
We assumed no settling or flows in or out of the system.
Therefore, Qin = 0 and Qout = 0, leaving:
AC:
l ~k, ¦ C
NP
At xhet,ij
From tliis we can differentiate and solve:
Ciwp = Cg[ e~kheUit
Equation 126
Equation 127
Table 34 shows calculated khet t - for different a's, using
an analytic solution. Figure 22 compares the analytic
solution and WASP simulation of free nanoparticle and
heteroaggregated nanoparticle (Nano-Solid) concentrations,
over time when a = 0.01.
Table 34. Parameters for Different Cases of a for
Brownian Motion Scenario
iHHIHii Case 1 Hass! Case 3 Hasfe4i II
kheU. 2.92E-02 2.92E-03 2.92E-04 2.92E-07
0.1
0.01
0.001 1.00E-06
1.76E+10 1.76E+10 1.76E+101.76E+10
Scenario 2 - Brownian and fluid motion only
Scenario 2 incorporates Brownian motion and fluid
motion; because the two cannot be isolated, we solved
them simultaneously. A constant inflow (Qm) and outflow
(Qout) °' 0-2 n| Vs was added to the WASP model. Solid and
nanoparticle transport were not included in this scenario.
Table 35 shows calculated khetjj for different alphas using an
analytic solution. Solids concentration is set to 100 mg/L.
Table 35. Parameters for Different Cases of a for
Brownian Motion and Fluid Motion Scenario
Parameter Case 1 Case 2 Case 3 Case 4
kheUj 2.92E-02 2.92E-03 2.92E-04 2.92E-07
a 0.1 0.01 0.001 1.00E-06
SfV 100 100 100 100
NJASPM 1.76E+10 1.76E+10 1.76E+10 1.76E+10
WASP's simulated and analytically solved nanomaterial
concentrations for all four alphas are compared in Table 36.
Table 36. Calculated and Simulated Nanoparticle
Concentrations Using Different Alphas
Nano Cone
[mg/L] a = 0.1 a = 0.01 a = 0.001 a = 1E-6
Analytic 17 11 1967 19 97 20.00
Solution
\A/AQp
vvrtor
Simulated
% Error 0.00 0.00 0.00 0.00
Free Nano
Nano-Solid
c
o
'4»
CO
c
(1)
o
c
o
o
o
CO -
CO -
CM -
0 500
Analytical Solution
v WASP Simulation
1500
Time (Day)
2500
CD
E
c:
o
'
03
c
d)
o
c
o
o
CO -
CD -
CM -
Analytical Solution
WASP Simulation
T
T"
1500
Time (Day)
2500
Figure 22. Comparison of Analytical Solution and WASP Simulated Nanoparticle Concentration over Time for Scenario 1
A-12
-------
Scenario 3 - Brownian motion, fluid motion, and
differential settling.
We add settling into k . Again, we cannot isolate dynamic
settling from other components, so we simulate all three
simultaneously. Scenario 3 uses Brownian motion fluid
motion, and differential settling.
Using the steady state equation:
ASff
V-
_ n cSPM _ n cSPM _ SPM cS
^in^in.j Vout^j wset,j°j
Equation 128
££ xi/t-m,; -x.uu.l~ j uset,juj
we solve for S,si']'1. By incorporating settling we get a
steady state solids concentration of approximately 9 mg/L.
Following steps from the previous scenario, we solve for the
steady state CNP concentration:
cr =
QinCg
Qout^~khet,ijV
Equation 129
khet j. constants for different alphas were solved analytically.
Parameter values for each case are presented in Table 37.
Table 37. Analytic Solutions for Heteroaggregation
Constants Using
Parameter
knet,ij
a
SjSPM
NjSPM
Case 1
0.573
0.1
8.96
Case 2 Case 3 Case 4
5.73E-02 5.73E-03 5.73E-06
0.01
8.96
0.001
8.96
1.00E-06
8.96
1.58E+09 1.58E+09 1.58E+09 1.58E+09
Calculated and simulated free nanoparticle concentrations
for all four alphas are shown in Table 38. WASP outputs are
within 0.02 percent of the analytic solution.
Table 38. Nanoparticle Concentrations Using
Different Alphas
Nano Cone
[mg/L] a=0.1 a=0.01
Analytic
Solution
4.63
WASP
Simulated
4.63
% Error -0.02
15.02
15.02
-0.01
a=0.001 a=1E-6
19.36 20.00
19.36 20.00
0.00 0.00
-------
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