2. The
Mass Balance Equation
Flux in = Flux out

3. Tool : Use steady-state mass-balance model
Question : What is the concentration of chemical X in the water (fish kills?)
Tool : Use steady-state mass-balance model
Volatilisation
Lake
Emission
Outflow
CW=?
Reaction
Lake Volume = 100,000,000 m3
Lake Surface Area = 1,000,000 m2
Sedimentation

4. dMW/dt = E - kV.MW - kS.MW - kO.MW - kR.MW
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
MW : Mass in Water (moles)
t : time (days)
E : Emission (mol/day)
kV: Volatilization Rate Constant (1/day)
kS: Sedimentation Rate Constant (1/day)
kO: Outflow Rate Constant (1/day)
kR.: Reaction Rate Constant (1/day)

5. Concentration Format
dMW/dt = MW MW MW MW
dMW/dt = 1 - ( ).MW
0 = 1 - ( ).MW
1 = ( ).MW
MW = 1/( ) = 1/0.01
CW = 0.01/100,000,000 = mol/m3

6. d(VW ZW.fW )/dt = E - DV.fW - DS.fW - DO.fW - DR.fW
Fugacity Format
d(VW ZW.fW )/dt = E - DV.fW - DS.fW - DO.fW - DR.fW
VW ZW.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
VW : Volume of Water (m3)
ZW : Fugacity Capacity in water (mol/M3.Pa)
fW : Fugacity in Water (Pa)
t : time (days)
E : Emission (mol/day)
DV: Transport Parameter for Volatilization (mol/Pa. day)
DS: Transport parameter fro Sedimentation (mol/Pa.day)
DO: Transport Parameter for Outflow (mol/Pa.day)
kR.: Transport Parameter for Reaction (mol/Pa.day)

7. Steady-state mass-balance model: 2 Media
Volatilisation
Emission
Outflow
Settling
CW=?
Reaction
Resuspension
CS=?
Burial

8. 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.

9. 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

10. LEVEL I

11. Mass Balance Total Mass = Sum (Ci.Vi) Total Mass = Sum (fi.Zi.Vi)
At Equilibrium : fi are equal
Total Mass = M = f.Sum(Zi.Vi)
f = M/Sum (Zi.Vi)

12. LEVEL II
GA.CA
GA.CBA E
GW.CBW
GW.CW

13. 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)

14. Reaction Rate Constant for Environment:
Fraction of Mass of Chemical reacting per unit of time : kR (1/day)
kR = Sum(Mi.ki) / Mi
Reaction Residence time: tREACTION = 1/kR

15. Removal Rate Constant for Environment:
Fraction of Mass of Chemical removed per unit of time by advection: kA 1/day
kA = Sum(Gi.Ci) / Vi.Ci
tADVECTION = 1/kA

16. Total Residence Time in Environment:
ktotal = kA + kR = E/M
tRESIDENCE = 1/kTOTAL = 1/kA + 1/kR
1/tRESIDENCE = 1/tADVECTION + 1/tREACTION

17. LEVEL III

18. 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

19. Time Dependent Fate Models / Level IV