Environmental Fate Models

LEVEL I

Merits

To assess multi-media environmental concentrations

under more realistic conditions

To assess a chemicals’ persistence

To assess dominant processes

To screen substances for the purpose of ranking

Cwater

Steady-State & Equilibrium

GILL UPTAKE

GILL ELIMINATION

Steady-State Flux Equation:

“Mass Balance Equation”

dMF/dt = DWF.fW - DFW.fF = 0

DWF.fW = DFW.fF

fF/fW = DWF/DFW = 1.0

CF/CW = fF.ZF/fW.ZW = ZF/ZW = KFW

Equilibrium

Steady-State Flux Equation:

“Mass Balance Equation”

d(CF.VF)/dt = kWF.VF.CW - kFW.VF.CF = 0

dCF/dt = kWF.CW - kFW.CF = 0

kWF.CW = kFW.CF

CF/CW = kWF/kFW = KFW

Equilibrium

Cwater

Steady-State & Equilibrium

GILL UPTAKE

GILL ELIMINATION

METABOLISM

Steady-State Flux Equation:

“Mass Balance Equation”

dMF/dt = DWF.fW - DFW.fF - DM.fF = 0

DWF.fW = DFW.fF + DM.fF

fF/fW = DWF/(DFW + DM) < 1.0

CF/CW = (ZF/ZW). DWF/(DFW + DM) < KFW

Steady-State

Steady-State Flux Equation:

“Mass Balance Equation”

dCF/dt = kWF.CW - kFW.CF - kM.CF = 0

kWF.CW = kFW.CF + kM.CF

CF/CW = kWF/(kFW + kM) < 1.0

CF/CW = kWF/(kFW + kM) < KFW

Steady-State

What is the difference between

Equilibrium & Steady-State?

Question : What is the concentration of chemical X in the water (fish kills?)

Tool : Use steady-state mass-balance model

Lake

CW=?

Volatilisation

Emission

Sedimentation

Reaction

Outflow

Concentration Format

dMW/dt = E - kV.MW - kS.MW - kO.MW - kR.MW

dMW/dt = E - (kV + kS+ kO+ kR).MW

0 = E - (kV + kS+ kO+ kR).MW

E = (kV + kS+ kO+ kR).MW

MW = E/(kV + kS+ kO+ kR) & CW = MW/VW

Fugacity Format

d(VW ZW.fW )/dt = E - DV.fW - DS.fW - DO.fW - DR.fW

dfW/dt = E - (DV + DS+ DO+ DR).fW

0 = E - (DV + DS+ DO+ DR).fW

E = (DV + DS+ DO+ DR).fW

fW = E/ (DV + DS+ DO+ DR) & CW = fW.ZW

Steady-state mass-balance model: 2 Media

Burial

CW=?

Volatilisation

Emission

Settling

Reaction

Outflow

CS=?

Resuspension

Water:

dMw/dt = Input + ksw.Ms - kw.Mw - kws.Mw = 0

Sediments:

dMs/dt = kws.Mw - kb.Ms - ksw.Ms = 0

From : Eq. 2

kws.Mw = kb.Ms + ksw.Ms

Ms = kws.Mw / (kb + ksw)

Substitute in eq. 1

Input + ksw.{kws.Mw / (kb + ksw)} = kw.Mw + kws.Mw

Input = kw.Mw + kws.Mw - ksw.{kws.Mw / (kb + ksw)}

In Fugacity Format

Water:

dMw/dt = Input + Dsw.fs - Dw.fw - Dws.fw = 0

Sediments:

dMs/dt = Dws.fw - Db.fs - Dsw.fs = 0

From : Eq. 2

Dws.fw = Db.fs + Dsw.fs

fs = Dws.fw / (Db + Dsw)

Substitute in eq. 1

Input + Dsw.{Dws.fw / (Db + Dsw)} = Dw.fw + Dws.fw

Input = Dw.fw + Dws.fw - Dsw.{Dws.fw / (Db + Dsw)}

Level III fugacity Model:

Steady-state in each compartment of the environment

Flux in = Flux out

Ei + Sum(Gi.CBi) + Sum(Dji.fj)= Sum(DRi + DAi + Dij.)fi

For each compartment, there is one equation & one unknown.

This set of equations can be solved by substitution and elimination, but this is quite a chore.

Use Computer

LEVEL I

Level II fugacity Model:

Steady-state over the ENTIRE environment

Flux in = Flux out

E + GA.CBA + GW.CBW = GA.CA + GW.CW

All Inputs = GA.CA + GW.CW

All Inputs = GA.fA .ZA + GW.fW .ZW

Assume equilibrium between media : fA= fW

All Inputs = (GA.ZA + GW.ZW) .f

f = All Inputs / (GA.ZA + GW.ZW)

f = All Inputs / Sum (all D values)

Reaction Rate Constant for Environment:

Fraction of Mass of Chemical reacting per unit of time

kR = Sum(Mi.ki) / Mtotal

tREACTION = 1/kR

Removal Rate Constant for Environment:

Fraction of Mass of Chemical removed per unit of time by advection

kA = Sum(Gi.Ci) / Mtotal

tADVECTION = 1/kA

Total Residence Time in Environment:

ktotal = kA + kR = E/M

tRESIDENCE = 1/kTOTAL = 1/kA + 1/kR

1/tRESIDENCE = 1/tADVECTION + 1/tREACTION

Application of the Models

•To assess concentrations in the environment

(if selecting appropriate environmental conditions)

•To assess chemical persistence in the environment

•To determine an environmental distribution profile

•To assess changes in concentrations over time.

Fugacity Models

Level 1 : Equilibrium

Level 2 : Equilibrium between compartments & Steady-state over entire environment

Level 3 : Steady-State between compartments

Level 4 : No steady-state or equilibrium / time dependent

Recipe for developing mass balance equations

1. Identify # of compartments

2. Identify relevant transport and transformation processes

3. It helps to make a conceptual diagram with arrows representing the relevant transport and transformation processes

4. Set up the differential equation for each compartment

5. Solve the differential equation(s) by assuming steady-state, i.e. Net flux is 0, dC/dt or df/dt is 0.

dXwater /dt = Input - Output

dXwater /dt = Input - (Flow x Cwater)

dXwater /dt = Input - (Flow . Xwater/V)

dXwater /dt = Input - ((Flow/V). Xwater)

dXwater /dt = Input - k. Xwater

k = rate constant (day-1)

Time Dependent Fate Models / Level IV

Analytical Solution

Integration:

Assuming Input is constant over time:

Xwater = (Input/k).(1- exp(-k.t))

Xwater = (1/0.01).(1- exp(-0.01.t))

Xwater = 100.(1- exp(-0.01.t))

Cwater = (0.0001).(1- exp(-0.01.t))

0

20

40

60

80

100

120

0 200 400 600 800 1000

Time (days)

Xw

(g

)

Xw ater (g)

Xw ater (g)

Numerical Integration:

No assumption regarding input overtime.

dXwater /dt = Input - k. Xwater

Xwater /t = Input - k. Xwater +

If t then

Xwater = (Input - k. Xwater).t

Split up time t in t by selecting t : t = 1

Start simulation with first time step:Then after the first time step

t = t = 1 d

Xwater = (1 - 0.01. Xwater).1

at t=0, Xwater = 0

Xwater = (1 - 0.01. 0).1 = 1

Xwater = 0 + 1 = 1

After the 2nd time stept = t = 2 d

Xwater = (1 - 0.01. Xwater).1

at t=1, Xwater = 1

Xwater = (1 - 0.01. 1).1 = 0.99

Xwater = 1 + 0.99 = 1.99

After the 3rd time stept = t = 3 d

Xwater = (1 - 0.01. Xwater).1

at t=2, Xwater = 1.99

Xwater = (1 - 0.01. 1.99).1 = 0.98

Xwater = 1.99 + 0.98 = 2.97

then repeat last two steps for t/t timesteps

Analytical Num. IntegrationTime Xwater Xwater

(days) (g) (g)0 0 01 0.995017 12 1.980133 1.993 2.955447 2.97014 3.921056 3.9403995 4.877058 4.9009956 5.823547 5.8519857 6.760618 6.7934658 7.688365 7.7255319 8.606881 8.648275

10 9.516258 9.561792

Mass of contaminant in water of lake vs time

0

20000

40000

60000

80000

100000

120000

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Time (days)

Mas

s in

Lak

e W

ater

(g

ram

s)

Steady-State:Xw = Input/V