Supporting Material: Model Description

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SUPPORTING MATERIALS Detailed description of the Multi-Species model and default parameterization Thomas M. Cahill, Ian Cousins and Donald Mackay* Canadian Environmental Modelling Centre Trent University 1600 West Bank Drive Peterborough, Ontario, K9J 7B8, CANADA * To whom correspondence may be addressed, [email protected] INTRODUCTION The multiple chemical species model presented here is an extensively modified version of a previous multimedia model as presented in Mackay et al [1]. Familiarity with this previous model would significantly aid in the comprehension of the current model. While we summarize the entire model, particular attention is paid to the differences between the previous and the current model. The fugacity system The model was created using the fugacity framework, so a review of the fugacity system is useful for understanding the model mechanics. A thorough review of the development and application of fugacity models to environmental situations is presented in Mackay [2]. In the fugacity system, the chemical concentration in a medium (C, mol/m3) equals the fugacity of chemical in the medium (f, in units of Pascals) multiplied by the fugacity capacity, or Z-value, of the medium for that chemical (Z, mol/m3·Pa). Therefore, C equals Z·f. The Z-value quantifies the relative affinity of the chemical for a particular medium. The Z-value for most non-interacting chemicals in the gas phase is 1/RT, where R is the gas constant (8.314 Pa·m3/mol·K) and T is temperature (K). For other phases, such as water, the Z-values are derived from a known Z-value and widely available dimensionless partition coefficients. The dimensionless partition coefficients, such as the air-water (KAW) or octanol-water (KOW) partition coefficients, are the ratios of concentrations in two media, which directly equate to a ratio of Zvalues at equilibrium. For example, the KAW equals CA/CW, which also equals ZA/ZW. Processes that govern chemical transport and reaction are quantified by fugacity rate coefficients or D-values (D, mol/h·Pa). The D-values share common units despite describing widely divergent processes. The chemical flux (N, mol/h) that is either transported or reacted is calculated as the product of the D-value and the chemical fugacity in a medium, so N is D·f. There are three classes of processes quantified by D-values, namely advection, diffusion and reaction. The simplest chemical transport process is when the chemical is transported within a moving media by advection. The D-value that quantifies chemical transport by advection (Da) is the product of the medium flow rate (Gi, m3/h) and the fugacity capacity of the medium for the chemical, so Da is Gi·Zi. The chemical flux is expressed as the product of the D-value and the

S2 fugacity of the chemical within the moving medium, so N is Da·f (which also equals Gi·Zi·fi) or Gi·Ci. The second class of chemical transport parameters is diffusion. Diffusion is typically used to describe the chemical movement between media, such as a chemical migrating from the soil into the air. The general form for the D-value that describes chemical diffusion is: Dd = (Bi·A·Zi)/∆y

(S1)

where B is diffusivity of the chemical in the medium (m2/h), A is the area over which the diffusion occurs (m2), Z is the fugacity capacity of the diffusion medium (mol/m3·Pa) and ∆y is the distance, or path-length, over which the diffusion occurs (m). The chemical flux transported by diffusion is again Dd·f. Another method of formulating intermedia transport, which is the method used in the current model, is to combine the diffusivity (B) and path length (∆y) terms into a single term called a mass transfer coefficient (U) that has units of m/h. Engineers represent mass transfer coefficients with “k”, which, unfortunately, is the same symbol that chemists have assigned to reaction rate constants. To avoid confusion, mass transfer coefficients are abbreviated herein as “U” and reaction rate constants are designated “k”. The mass transfer coefficient represents a “velocity” at which a chemical moves and it is typically defined for a specific transport distance. The mass transfer coefficients can represent both molecular and eddy (or turbulent) diffusion. Using the mass transfer coefficient formulation for intermedia transport processes results in the D-value being defined as: Dd = Ui·A·Zi

(S2)

The mass transfer coefficient formulation of the intermedia transport D-values is the preferred method in previous models [1, 2], so this formulation is used in the current model. Chemical reaction is the last class of processes that is described by D-values. The Dvalue that describes first-order and pseudo first-order reactions is Vi·Zi·k, where Vi is the volume of the compartment (m3), Zi is the fugacity capacity of the compartment and k is the first order rate constant for the reaction (h-1). All chemical reactions in the model are treated as first order or pseudo first order and follow the same structure. The flux of chemical lost by reaction is thus: N = mi·k = Vi·Ci·k = Vi·Zi·k·fi = Dr·fi

(S3)

Chemical reaction data in the literature is often presented as a half-life (t1/2, h) rather than a rate constant. The rate constant can be calculated as 0.693/t1/2. Reaction data is also sometimes presented as a lifetime (τ, h), which is simply the reciprocal of the reaction rate constant (i.e. τ = 1/k). While most of the reactions are assumed to be first order, some reaction are distinctly second order in structure. In particular, the atmospheric reactions involving hydroxyl radicals are clearly second order. The program user must multiply the second order rate constants for these reactions with the hydroxyl radical concentrations to obtain a pseudo first order rate constant to use in the model. If multiple processes remove a chemical from a compartment, then the individual Dvalues can be converted into a single overall D-value (Dt). If these processes act in parallel, then

S3 Dt is the sum of the individual process D-values (i.e. Dt = D1 + D2). If the processes act in series (e.g. “process A” occurs before “process B”), then the individual D-values sum as reciprocals (i.e. 1/Dt = 1/D1 + 1/D2). The ability to combine multiple transport parameters into a single term permits considerable simplification of the modeling equations. Model Environment The model environment consists of four bulk media, namely air, water, soil and sediment (Figure S1). Aerosols, suspended sediments and aquatic biota are included as subcompartments within the bulk air and water compartments, respectively. These subcompartments are assumed to be at equilibrium with the bulk compartment in which they reside, so they share a common fugacity. Since there is equilibrium partitioning between the fish and the water, the concentration of chemical in the fish includes only gill exchange with the water and it does consider food chain effects.

Air

Aerosols

Soil Water Aquatic Biota Suspended Sediment Bottom Sediment Figure 1. Schematic diagram of the model environment. The current model allows the program user to adjust the volume and surface areas of each compartment in the model, thus the model can be adapted to simulate specific conditions as desired. The volume and interfacial areas of the bulk media are therefore model inputs along with the volume fractions of the secondary compartments. Table S1 gives the typical volume fractions of the subcompartments that have been used in previous models. The current model uses these values as default conditions.

S4 Table S1. Volume fractions of the subcompartments in the bulk media [1]. Compartment Subcompartment Designation Air Gas phase FAG Aerosol FAA

Volume fraction ≈1 2x10-11

Water

Water Suspended sediments Aquatic biota

FWW FWS FWB

≈1 5x10-6 1x10-6

Soil (earth)

Gas phase Water Solid material

FEG FEW FES

0.2 0.3 0.5

Sediment

Water Solid material

FSW FSS

0.8 0.2

The fugacity capacities of the bulk media are determined by the volume fractions and fugacity capacities of the subcompartments, or phases, that comprise the bulk media. Table S2 summarizes the equations used to calculate the Z-value for the subcompartments and the bulk media. Table S2. Fugacity capacity calculation for the subcompartments and bulk media (Adapted from [1]). Note that R is the gas constant (8.314 Pa·m3/mol·K); T is the temperature (K); ρ is the density of the media (kg/L); ϕ is the organic carbon fraction of the medium (g/g); KOC is the organic carbon partition coefficient (L/kg); L is the lipid fraction of the fish; and W is the water fraction of the fish. The “F” terms represent the volume fractions of the subcompartments within a bulk medium (See Table S1). Symbol Equation Fugacity capacities of the subcompartments Air Z1 = 1/RT Water (pure) Z2 = Z1/KAW Octanol and lipids Z3 = Z2·KOW Soil solids Z4 = Z2·ρ4·ϕ4·KOC Sediment solids Z5 = Z2·ρ5·ϕ5·KOC Suspended sediment solids Z6 = Z2·ρ6·ϕ6·KOC Aquatic biota (fish) Z7 = Z3·L + Z2·W or (BCF)Z2 Aerosols Z8 = Z1·KQA Bulk Media Z-values Bulk air Bulk water Bulk soil Bulk sediment

ZA ZW ZE ZS

= Z1·FAG + Z8·FAA = Z2·FWW + Z6·FWS + Z7·FWB = Z1·FEG + Z2·FEW + Z4·FES = Z2·FSW + Z5·FSS

S5

The fugacity capacity of pure air, water and lipid are determined directly from the airwater (KAW) and octanol-water (KOW) partition coefficients. The fugacity capacity of solid material, in both soil and sediment, is determined from the organic carbon fraction (ϕi, g/g) and the density of the solid phase (ρi, kg/L). Partitioning into the organic matter is related to chemical partitioning into lipids by [3]: KOC = 0.41·KOW

(S4)

Note that KOC has units of L/kg while KOW is unitless, so the “0.41” conversion factor has units of L/kg. The fugacity capacity of a fish is simulated as the lipid fraction of the fish (L) multiplied by the fugacity capacity of lipid plus the water fraction of the fish (W) multiplied by the fugacity capacity of water, thus: Zfish = (L)(Zoctanol) + (W)(Zwater)

(S5)

Alternatively, an empirical bioconcentration factor (BCF) can be used to predict the chemical concentrations in the fish. This would provide a more accurate estimate of the concentrations since the empirical BCF value would include the effects of metabolism on the body burden of the chemical. The program allows the user to input an empirical BCF value to override the default partitioning algorithm. In this case, the fugacity capacity of the fish is given by: Zfish = (BCF)(Zwater)

(S6)

Lastly, partitioning between aerosols and the gas phase is simulated using the aerosol-air partition coefficient (KQA) where: KQA = 6x106/PsL

(S7)

Where PsL is the vapor pressure (Pa) of the liquid or the subcooled vapor-pressure if the chemical exists as a solid under ambient environmental conditions. Ionic species that are effectively non-volatile are modeled by assigning them an arbitrarily miniscule air-water partition coefficient (e.g. 10-15) so that the chemical is essentially absent in the gas phase. The model calculates the fugacity capacities of the subcompartments within the bulk media using the above equations. However, the model also contains a “manual override” whereby the program user can input a fugacity capacity for a particular subcompartment that overwrites the default calculations. This manual override is important for chemicals with unusual partitioning properties, such as binding to mineral matter. The fugacity capacity override can therefore be used to create an empirical model even when the mechanism of partitioning is not known.

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Transport and transformation processes The model includes three classes of chemical transport and transformation processes, namely advection, diffusion and reaction. Advection Advection is responsible for both chemical gain and loss from the system. It is simply parameterized as D-value that is the product of the bulk medium flow rate (Gi, m3/h) and the fugacity capacity of the bulk media (Zi). In the current model, all the bulk media, namely air, water, soil and sediment, may have advective gains and losses that are defined by the program user. The subcompartments, such as fish and aerosols, advect along with the bulk media in which they reside. The media flow rates are therefore required model inputs, although advection can always be turned off by setting the media flow rate to zero. If chemicals are introduced to the model system by advection, then a background concentration in the advection media need to be defined in addition to the media flow rates. The advection terms are chemical specific based on the fugacity capacity of the advection media. This model differs from previous model in allowing the soil and the sediment compartment to have advection terms, which may seem unusual since these media are typically stationary. The reason for allowing stationary media to advect is so that model can be parameterized to simulate a Lagrangian system in addition to the Eulerian system typical of the previous models. For example, a parcel of air can be simulated through time by keeping the air parcel intact and advecting the soil, water and sediment underneath the parcel of air. Thus, the air does not move as time passes, but the world moves around it. This is valuable for simulating a pulse emission in either the air or water compartments and thus expands the capabilities of the model. Diffusion and intermedia transport parameters There are many different processes that redistribute the chemical between the compartments within the system. The input parameters for the intermedia transport processes are mass transfer coefficients, which are summarized in Table S3 and generally conform to the parameterization from previous models [1].

S7 Table S3. Mass transfer coefficients used in the model. Parameterization follows Mackay et al [1] unless otherwise noted. Mass transfer coefficients Symbol m/h Air side of the air-water MTC U1 50a Water side of the air-water MTC U2 0.1a Rainfall rate (wet deposition) U3 1.1×10-4 (or 97 cm/year)b Aerosol settling velocity U4 10 Air phase diffusion MTC in soil U5 0.02 Water phase diffusion MTC in soil U6 10-5 Soil-air boundary layer MTC U7 25 (est.) Sediment-water interface MTC U8 10-4 Sediment deposition rate U9 5×10-7 Sediment resuspension rate U10 2×10-7 Water runoff from soil U11 = U3(1 - Fevap) = 4×10-5 c Soil solids erosion from soil U12 10-8 Total aerosol deposition U13 (U4·FAA + U3·S·FAA)d (dry and scavenged by rain) = 6.41×10-10 m/h a Intermediate value of the typical range for air-side mass transfer coefficients (10-100 m/h) and water side mass transfer coefficients (0.02 to 0.2 m/h) presented in Schwarzenbach et al [5]. b Based on the global average rainfall rate [4]. c The water runoff from the terrestrial environment is calculated from the rainfall rate multiplied by the fraction of the water that does not evaporate. An average rainfall evaporation fraction (Fevap) of 0.64 from Baumgartner and Reichel [4] is used, so the rate of surface runoff is U3(1Fevap) or 0.36·U3. d The total aerosol deposition is the sum of two terms, namely the aerosols that deposit by dry deposition and the aerosols that are scavenged by rainfall. The scavenging efficiency of rainfall (S) was set to 200,000 [2]. The notable exceptions are the rainfall rate, which was updated with the global average rainfall rate from Baumgartner and Reichel [4] and the water runoff from the terrestrial environment, which is calculated using the rainfall rate and the fraction of rainfall that evaporates from the terrestrial surface. Therefore, the mass transfer coefficient that describes water runoff is calculated as: Water runoff (in m/h) = (rainfall rate)(1- fraction of rain evaporating)

(S8)

This water runoff mass transfer coefficient is then applied to the terrestrial surface area to obtain a volumetric flow of water from the land surface. Unlike the EQC model [1], the current model maintains a water balance in the system. The surface water flow leaving system equals surface water flow entering the system plus the product of the region area, rainfall rate, and the fraction of rainfall that does not evaporate, or Gout = Gin + (region area)(rainfall rate)(1- fraction of rain evaporating)

(S9)

S8 Currently, we do not consider the volume of water lost to groundwater to be significant, so it is not included in the model. Percolation to groundwater will be included in future models. While a water balance is not particularly important for hydrophobic chemicals, it is critical for highly hydrophilic and ionic chemicals that reside predominantly in the aqueous phase. Once the mass transfer coefficients have been defined and surface areas between the bulk compartments have been inputted, then it is possible to define the fugacity rate coefficients (Dvalues) that quantify the rate of chemical transport between the media. The D-values are chemical specific, so each chemical species will have its own set of D-values. Often, the overall D-value that describes the chemical transport between compartments consists of multiple processes. Table S4 describes the D-values that are used in the model. Once again, the D-values used in this model are similar in structure to previous models [1]. Table S4. Calculation of the fugacity rate coefficients for intermedia transport of chemicals (adapted from [1]). The “A” terms are the surface area (m2), the “U” terms are mass transfer coefficients (see Table S3) and the “Z” terms are the fugacity capacities of the media (see Table S2). The subscripts are: A is air; W is water; S is sediment; and E is soil/earth. Environmental Media From To Process Equation Air Water Absorption D1 = AW/(1/U1Z1 + 1/U2Z2) Rain dissolution D2 = AWU3Z2 Aerosol deposition D3 = AWU13Z8 Total: DAW = D1 + D2 + D3 Water

Air

Volatilization

DWA = D1

Soil

Air

Air boundary layer Air diffusion in soil Water diffusion in soil Total:

D4 = AEU7Z1 D5 = AEU5Z1 D6 = AEU6Z2 DEA = 1/[1/D4 + 1/(D5 + D6)]

Air

Soil

Rain dissolution Aerosol deposition Total:

D7 = AEU3Z2 D8 = AEU13Z8 DAE = DEA + D7 + D8

Water

Sediment

Diffusion Deposition Total:

D9 = AWU8Z2 D10 = AWU9Z6 DWS = D9 + D10

Sediment

Water

Resuspension Total:

D11 = AWU10Z5 DSW = D9 + D11

Soil

Water

Water runoff Soil solid erosion Total:

D12 = AEU11Z2 D13 = AEU12Z4 DEW = D12 + D13

S9

Chemical reactions The model considers four inter-converting chemical species (designated α, β, γ, δ) and a group of “other” chemical species (designated ω). The current model allows any chemical species to convert into any other chemical species. This inter-converting scheme allows backwards reactions and cycling of chemical species. The “other” chemical class is a mixture of chemicals formed from non-unity product yields between the main species or subsequent degradation of the primary species beyond the species of interest. Since this class is a mixture of different chemicals with different properties, the model is unable to simulate chemicals transformed into the “other” class. Thus, the “other” class is accounting tool to ensure mass balance closure. Chemical reactions can take place in each of the bulk media, so the reaction rates are needed for each media. Empirical reaction rates in soil, sediment and natural surface waters represent bulk reaction rates and can be used directly. Reaction rates in air are generally determined as gas-phase, so the default model considers only the gas phase reactions and assumes that the entire mass of chemical in the air is subject to the gas phase reaction rate. However, some non-volatile chemicals may be sorbed to particles and are inaccessible to gas phase reactions or some chemical may react much faster on particles than in the gash phase. In both these cases, it is wise to weigh the bulk reaction rate by fraction of chemical in the atmosphere that is subjected to each type of reaction. If, MT = total mass of chemical in the bulk air, mQ = mass of chemical associated with aerosols mA = mass of chemical in gas phase kQ = reaction rate on aerosol kA = gas phase reaction rate Then the bulk air reaction rate is: kT = (mA/MT)·kA + (mQ/MT)·kQ

(S10)

This bulk reaction rate calculation should be conducted for chemicals that are mostly present on particles or have a very fast reaction rate on particles. Since the model was programmed as a general model that could be configured to simulate many different situations, there are a large number of potential reactions pathways. Each chemical species can be transformed into any of the other three primary species as well as the “other” products in each of the four media, thus any given chemical species can have up to 16 different reactions occurring in the model environment. Since the model considers four primary species, the model is programmed to handle 64 different chemical reactions simultaneously. This represents the most general form of the model that uses all the reaction rates and all the chemical species, but most applications will not use all the reaction rates (i.e. most of the backward reactions are not needed for organic chemicals) or all the chemical species. Currently, the model can only describe reactions with a molar product ratio of unity or less. The model cannot account for reactions that have product yields of primary chemical species that exceed one. (e.g. one mole of alpha forms two moles of beta or one mole of alpha

S10 forms one mole beta and one mole of gamma.) In essence, the model treats the molecule as having one “labeled” atom that it follows. Reactions with product yields less than unity are modeled by diverting the extra moles of the reactant into “other” products. For example, one mole of alpha forms 0.8 moles of beta plus 0.2 moles of omega. This is simulated using two competing reactions with rate constants αβk and αωk, each of which forms exactly one mole of the product. The total rate of loss of the reactant is then: ktotal =

αβ

k + αωk

and the product yield of beta is αβk/(αβk + reactions with a product yield less than one.

(S11) αω

k). This approach allows for the simulation of

Chemical inputs The model has three modes of chemical input. The first is a single pulse of chemical that is added to a particular compartment at the beginning of the simulation. These types of simulations are useful for predicting the chemical concentrations over time resulting from a discrete chemical application or spill. Any chemical species can be added to any compartment, so the model can start off with a mixture of different chemical species in different media. The second type of chemical input is the continuous input. The continuous chemical input adds a little chemical each hour to a particular compartment. Examples of these types of simulations would be a constant emission to air from a manufacturing facility or a wastewater outfall into a river. The model allows the continuous input to cease after a period of time specified by the program user, so it can predict the clearance of the chemical from the system. Once again, any chemical species can be added to any compartment by continuous inputs. The last form of chemical input is by advection. A background concentration of the chemical species is specified in each media and thus the chemical is introduced by advection. These simulations are advantageous when an upwind source of chemical is the dominant source to the system. The model can handle all three input types in a single simulation, although interpretation of the results gets more complicated because the relative importance of the three input types may not be clear. Model designations The complexity of the model makes it advantageous to clearly define the terms and symbols. The volume of the environmental media (V) and the advection rates (G) are subscripted with a single letter that describes which bulk media is being referenced. The subscripts are as follows: A = air, W = water, S = sediment, E = soil/earth, and i for the general case. For example, VW is the volume of the water compartment while GA is the air flow rate. The fugacity (f), mass (m) and fugacity capacities (Z) are chemical and media specific. The chemical species are designated with a superscripted Greek letter (α, β, γ, δ, ω or χ for the general case) before the parameter letter (f, m or Z). A single subscripted letter after the parameter letter denotes the environmental media to which the parameter is being applied. For example, βZW represents the fugacity capacity of species beta in water and αfE is the fugacity of species alpha in the soil. In addition to the fugacity of each species in each medium, the fugacity of the background advection medium that enters the system needs to be identified. In this case, a

S11 subscripted “a” before the parameter letter denotes the background fugacity of the advecting media, so αafW represents the background fugacity of species alpha in the water entering into the system. The transport and transformation parameters are defined by the class of process to which they belong: Reaction: The reaction D-values are indicated by a double Greek superscript. The first symbol denotes the species that is being lost while the second symbol indicates the species that is formed. The subscript letter after the “D” symbol describes the bulk media in which the reaction takes place. For example, βδDW is the D-value for the transformation of species β into species δ in the water compartment. Advection: The advection D-values are indicated by subscripted “a” before the “D”. A single superscripted Greek symbol identifies the chemical species advecting. Lastly, a subscripted letter after the “D” denotes which media is advecting. For example, γaDA represents the advection of chemical species gamma in air. Water advection is a special case since the volume of water entering the system is different from the water flow leaving the system. The difference in flow rates arises from rainfall augmenting the water flow from the system. The D-values that describes water advection must be further identified as “in” and “out”, so they are subscripted as χaDW(in) and χaDW(out). Diffusion/intermedia transport: The general intermedia transport values are described in Table S4. This notation is simply augmented by a chemical species superscript. The first subscripted letter indicates the source compartment of the chemical while the second indicates the destination compartment of the chemical. Therefore αDAW describes the transport of species alpha from air to water.

MODELLING EQUATIONS The model was programmed using differential equations and it was solved using numerical integration. For each chemical species, there is a mass balance equation for each of the primary media, namely air, soil, water and sediment, thus resulting in a total of 16 mass balance equations that need to be simultaneously solved. The Euler method of numerical integration was used to solve the differential equations. In this approach, the rate of change (mol/h) of chemical over time (due to reaction for example) in a given compartment is given by: dm/dt = - m·k ≈ ∆m/∆t

(S12)

when ∆t is an integration time step small enough that ∆m is small compared to m (e.g. less than a few percent). Therefore,

S12 ∆m = - m·k·(∆t) = - V·C·k·(∆t)

(S13)

Since the model is written in the fugacity framework, the chemical mass in the media is replaced by the chemical fugacity (f), the fugacity capacity of the media (Z), and the volume of the media (V), of which the later two are constant over time. The equation can then be re-written as: Z·V·(∆f) = - V·Z·k·f·(∆t) = - Dr·f·(∆t)

where Dr is a reaction D-value that equals V·Z·k

(S14)

Therefore, the change in fugacity in a compartment during a small integration time step (∆t) is: ∆f = (- Dr·f·(∆t))/(Z·V)

(S15)

The above example of the numerical integration method had only one chemical loss process, which was reaction, but other processes (such as advection and intermedia transport) can also be included in the right side of the equation 12. The process is the same regardless of the number of processes that remove the chemical from a compartment. It should be noted that chemical loss from one compartment is often a gain of chemical in another compartment. For example, the degradation of chemical species alpha in air may form chemical species beta. Thus, the amount (moles) of alpha reacted in air during a ∆t time step is then added to the mass balance of beta during the same ∆t time step. The model simulation period is then divided into a large number of small ∆t time steps. The model is programmed as a loop that 1) calculates the initial fugacity in each compartment from the chemical mass in the compartment, 2) calculates the change in fugacity during a ∆t time step, 3) adjusts the fugacity of each compartment by the change of fugacity during the time step, and 4) recalculates the mass of each chemical in each compartment. Specifically, the numerical integration process is: 1) Calculate the initial fugacity of the compartment from the initial mass, volume and fugacity capacity of the compartment. Thus, χ

fi = χmi/(Vi·χZi)

(S16)

where χfi is the fugacity of chemical species χ in media i; χmi is the initial mass of the chemical species; Vi is the volume of compartment i; and χZi is the fugacity capacity of species χ in media i. 2) Once the initial fugacity of each chemical species has been determined in each media, then the change in fugacity that results from advection, intermedia transport and reaction during the integration time step is calculated. The general formula for calculating the change in fugacity (dχfi) of a compartment during the integration time step (dt) is: d(χfi)/dt = [1/(Vi·χZi)]·[Σ(transport and reactions gains) – Σ(transport and reactions losses)] (S17)

S13 The equations that describe the change in fugacity during the integration time step are described below. From a mathematical perspective, the model environment can be viewed as consisting of 4 “layers” that correspond to the four primary chemical species. The mass balance equation for a given chemical in a given media is linked to the other compartments by both transport parameters (which redistribute the chemical within the layer) and transformation parameters (which move the chemical between layers). For example, chemical species alpha in soil may be transported to the atmosphere by volatilization, transported to water by runoff, and transformed into beta by reaction. In this example, the volatilization and runoff transport parameters simply redistribute chemical mass in the “alpha” layer while the transformation process moves chemical mass from the “alpha” layer to the “beta” layer. To make the equations as transparent as possible, the following equations are split into three lines by the function of the parameters on the lines. The first line represents the differential statement (d(χfi)/dt), volume and the fugacity capacity of the compartment that is being modified. The second line represents the intermedia transport and advection processes that redistribute or remove, as in the case of advection, the chemical within its current layer. The third line describes the transformation processes that move the chemical between different layers (i.e. species to species transformation). The equations that describe the change in fugacity in the various compartments during an integration time step are: (S18) Chemical species alpha in air: α α d( fA)/dt = [1/(VA· ZA)]· [(αfW·αDWA) + (αfE·αDEA) – (αfA·αDAW) – (αfA·αDAE) – (αfA·αaDA) + (αafA·αaDA) + (βfA·βαDA) + (γfA·γαDA) + (δfA·δαDA) – (αfA·αβDA) – (αfA·αγDA) – (αfA·αδDA) – (αfA·αωDA)] Chemical species alpha in water: (S19) α α d( fW)/dt = [1/(VW· ZW)]· [(αfA·αDAW) + (αfE·αDEW) + (αfS·αDSW) – (αfW·αDWA) –(αfW·αDWS) –(αfW·αaDW(out)) +(αafW·αaDW(in)) + (βfW·βαDW) + (γfW·γαDW) + (δfW·δαDW) – (αfW·αβDW) – (αfW·αγDW) – (αfW·αδDW) – (αfW·αωDW)] Chemical species alpha in soil: d(αfE)/dt = [1/(VE·αZE)]· [(αfA·αDAE) – (αfE·αDEA) – (αfE·αDEW) – (αfE·αaDE) + (αafE·αaDE) + (βfE·βαDE) + (γfE·γαDE) + (δfE·δαDE) – (αfE·αβDE) – (αfE·αγDE) – (αfE·αδDE) – (αfE·αωDE)]

(S20)

Chemical species alpha in sediment: d(αfS)/dt = [1/(VS·αZS)]· [(αfW·αDWS) – (αfS·αDSW) – (αfS·αaDS) + (αafS·αaDS) + (βfS·βαDS) + (γfS·γαDS) + (δfS·δαDS) – (αfS·αβDS) – (αfS·αγDS) – (αfS·αδDS) – (αfS·αωDS)

(S21)

Chemical species beta in air: (S22) d(βfA)/dt = [1/(VA·βZA)]· [(βfW·βDWA) + (βfE·βDEA) – (βfA·βDAW) – (βfA·βDAE) – (βfA·βaDA) + (βafA·βaDA) + (αfA·αβDA) + (γfA·γβDA) + (δfA·δβDA) – (βfA·βαDA) – (βfA·βγDA) – (βfA·βδDA) – (βfA·βωDA)]

S14 Chemical species beta in water: (S23) β β d( fW)/dt = [1/(VW· ZW)]· [(βfA·βDAW) + (βfE·βDEW) + (βfS·βDSW) – (βfW·βDWA) – (βfW·βDWS) – (βfW·βaDW(out)) + (βafW·βaDW(in)) + (αfW·αβDW) + (γfW·γβDW) + (δfW·δβDW) – (βfW·βαDW) – (βfW·βγDW) – (βfW·βδDW) – (βfW·βωDW)] Chemical species beta in soil: d(βfE)/dt = [1/(VE·βZE)]· [(βfA·βDAE) – (βfE·βDEA) – (βfE·βDEW) – (βfE·βaDE) + (βafE·βaDE) + (αfE·αβDE) + (γfE·γβDE) + (δfE·δβDE) – (βfE·βαDE) – (βfE·βγDE) – (βfE·βδDE) – (βfE·βωDE)]

(S24)

Chemical species beta in sediment: d(βfS)/dt = [1/(VS·βZS)]· [(βfW·βDWS) – (βfS·βDSW) – (βfS·βaDS) + (βafS·βaDS) + (αfS·αβDS) + (γfS·γβDS) + (δfS·δβDS) – (βfS·βαDS) – (βfS·βγDS) – (βfS·βδDS) – (βfS·βωDS)]

(S25)

Chemical species gamma in air: d(γfA)/dt = [1/(VA·γZA)]· [(γfW·γDWA) + (γfE·γDEA) – (γfA·γDAW) – (γfA·γDAE) – (γfA·γaDA) + (γafA·γaDA) + (αfA·αγDA) + (βfA·βγDA) + (δfA·δγDA) – (γfA·γαDA) – (γfA·γβDA) – (γfA·γδDA) – (γfA·γωDA)]

(S26)

Chemical species gamma in water: (S27) γ γ d( fW)/dt = [1/(VW· ZW)]· [(γfA·γDAW) + (γfE·γDEW) + (γfS·γDSW) – (γfW·γDWA) – (γfW·γDWS) – (γfW·γaDW(out)) + (γafW·γaDW(in)) + (αfW·αγDW) + (βfW·βγDW) + (δfW·δγDW) – (γfW·γαDW) – (γfW·γβDW) – (γfW·γδDW) – (γfW·γωDW)] Chemical species gamma in soil: d(γfE)/dt = [1/(VE·γZE)]· [(γfA·γDAE) – (γfE·γDEA) – (γfE·γDEW) – (γfE·γaDE) + (γafE·γaDE) + (αfE·αγDE) + (βfE·βγDE) + (δfE·δγDE) – (γfE·γαDE) – (γfE·γβDE) – (γfE·γδDE) – (γfE·γωDE)]

(S28)

Chemical species gamma in sediment: d(γfS)/dt = [1/(VS·γZS)]· [(γfW·γDWS) – (γfS·γDSW) – (γfS·γaDS) + (γafS·γaDS) + (αfS·αγDS) + (βfS·βγDS) + (δfS·δγDS) – (γfS·γαDS) – (γfS·γβDS) – (γfS·γδDS) – (γfS·γωDS)]

(S29)

Chemical species delta in air: d(δfA)/dt = [1/(VA·δZA)]· [(δfW·δDWA) + (δfE·δDEA) – (δfA·δDAW) – (δfA·δDAE) – (δfA·δaDA) + (δafA·δaDA) + (αfA·αδDA) + (βfA·βδDA) + (γfA·γδDA) – (δfA·δαDA) – (δfA·δβDA) – (δfA·δγDA) – (δfA·δωDA)]

(S30)

Chemical species delta in water: (S31) d(δfW)/dt = [1/(VW·δZW)]· [(δfA·δDAW) + (δfE·δDEW) + (δfS·δDSW) – (δfW·δDWA) – (δfW·δDWS) – (δfW·δaDW(out)) + (δafW·δaDW(in)) + (αfW·αδDW) + (βfW·βδDW) + (γfW·γδDW) – (δfW·δαDW) – (δfW·δβDW) – (δfW·δγDW) – (δfW·δωDW)]

S15 Chemical species delta in soil: d(δfE)/dt = [1/(VE·δZE)]· [(δfA·δDAE) – (δfE·δDEA) – (δfE·δDEW) – (δfE·δaDE) + (δafE·δaDE) + (αfE·αδDE) + (βfE·βδDE) + (γfE·γδDE) – (δfE·δαDE) – (δfE·δβDE) – (δfE·δγDE) – (δfE·δωDE)]

(S32)

Chemical species delta in sediment: d(δfS)/dt = [1/(VS·δZS)]· [(δfW·δDWS) – (δfS·δDSW) – (δfS·δaDS) + (δafS·δaDS) + (αfS·αδDS) + (βfS·βδDS) + (γfS·γδDS) – (δfS·δαDS) – (δfS·δβDS) – (δfS·δγDS) – (δfS·δωDS)]

(S33)

3) After the integration time step, the initial fugacity of the compartment is adjusted by the change in fugacity during integration time step. Thus, χ

fi = χfi + d(χfi)

(S34)

4) Once the fugacity of a chemical species is known in a compartment, then the mass of the chemical species is then recalculated by: χ

mi = χfi·Vi·χZi

(S35)

Once the mass in the compartment is known, then the integration loop starts over again. By default, the integration time step is one hour, although it can be decreased to any value less than one hour. The shorter integration time steps are required if there are fast processes in the model that require high temporal resolution. The shorter time steps also increase the amount of computer time to run the model, so small time steps on the order of minutes or seconds adds considerable computational time and should only be used when needed. Model output The output of the model is the concentration of each of the chemical species as a function of time. The concentrations are reported as grams of chemical per cubic meter of bulk media, although the units can be converted into other more common concentrations units (i.e. ng/g of soil or ng/L of water) using the density of the media and some volumetric conversion factors (i.e. 1 m3 = 1000 L). In addition, the model also gives the total moles of chemical in each compartment as well as total moles of chemical that entered or left the system by advection. Lastly, the model monitors the moles of chemical lost to “other” products, which is useful for mass balance calculations. Although the model is a dynamic model, it can be used to conduct steady-state simulations. In these simulations, a continuous chemical input is enters into the model and the model is run for extended periods of time until chemical concentrations cease to change. The concentrations in the media are then exported once steady-state has been achieved.

S16 REFERENCES IN SUPPORTING MATERIALS 1) Mackay D, Di Guardo A, Paterson S, Cowan CE. 1996. Evaluating the environmental fate of a variety of types of chemicals using the EQC Model. Environ Toxicol Chem 15:1627-1637. 2) Mackay D. 2001. Multimedia Environmental Models: The Fugacity Approach, 2nd ed. Lewis Publishers, Boca Raton, FL, USA. 3) Karickhoff SW. 1981. Semi-empirical estimation of sorption of hydrophobic pollutants on natural sediments and soils. Chemosphere 10:833-846. 4) Baumgartner A, Reichel E. 1975. The World Water Balance: Mean Annual Global, Continental and Maritime Precipitation, Evaporation and Run-off. Elsevier Scientific Publishing Co. Amsterdam, The Netherlands. 5) Schwarzenbach RP, Gschwend PM, Imboden DM. 1993. Environmental Organic Chemistry. John Wiley & Sons, Inc. New York, NY, USA.