The Mass Balance Equation Flux in = Flux out

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

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)

Concentration Format dMW/dt = 1 - 0.001.MW - 0.004.MW - 0.002.MW - 0.003.MW dMW/dt = 1 - (0.001 + 0.004+ 0.002+ 0.003).MW 0 = 1 - (0.001 + 0.004+ 0.002+ 0.003).MW 1 = (0.001 + 0.004+ 0.002+ 0.003).MW MW = 1/(0.001 + 0.004+ 0.002+ 0.003) = 1/0.01 CW = 0.01/100,000,000 = 1.10-10 mol/m3

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)

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

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.

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

LEVEL I

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)

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

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 (1/day) kR = Sum(Mi.ki) / Mi Reaction Residence time: tREACTION = 1/kR

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

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

LEVEL III

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

Time Dependent Fate Models / Level IV