Mathematical Modelling in Geochemistry. Application to ... · Introduction: motivation and objectives Chemical kinetics. Finite rate chemical reactions. Fast chemical reactions: chemical

Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.

Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions

Numerical methodsA geochemical example

Mathematical Modelling in Geochemistry. Application to Water

Quality Problems in Open Pit Lakes

Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa

Departamento de Matematica Aplicada, Universidade de Santiago de Compostela.Instituto Espanol de Oceanografıa. A Coruna

Industrial and Environmental Mathematical Day

Instituto de Matematicas de la Universidad de Sevilla

8 de junio de 2012

Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes

1 Introduction: motivation and objectivesGeochemical models for water quality

2 Chemical kinetics.Finite rate chemical reactions.Problem statement. Existence and uniqueness of solution.

3 Fast chemical reactions: chemical equilibriumIntroductionGibss free energy minimizationNon-linear system of algebraic equations

4 Coexistence of slow and fast chemical reactionsProblem approximation.The limit model.A particular case: solubility reactions.Methods to reduce the chemical problem.Numerical methodsApplication to a simple example

5 A geochemical exampleIntroductionConsidered chemical reactionsProblem settingNumerical results




Geochemical models for water quality

Introduction

There exist applications in which it is necessary to follow the concentration ofcertain reacting chemical species along the time.In these situation we need,

to be able to model chemical reactions that proceed at slowrates and also at fast rates,

to be able to handle problems in which chemical reactionsthat proceed at very different rates coexist,

to be able to solve numerically the stated problems,






Water quality prediction of a future pit lake

Environmental and landscape recovery strategy for themining area: formation of an artificial lake

Iron sulfides at the pitwalls ⇒ acidity andheavy metal release.

Possible connection to awater reservoir ⇒certain standards mustbe fulfilled

Prediction in advanceof the future waterquality






Antes del llenado






Comienzo del llenado






Primeras fases del lago






El lago en los medios


















The most relevant factors

J.M. Castro and J.N. Moore, 2000 and A. Davis et al., 1996

The chemical composition of the wall rocks

The magnitude and geochemistry of the water sources

The precipitation / evaporation rate

The limnology of the lake

The effect of biological activity







C.N. Alpers, 1994, C. Blodau., 2006 y L.E. Eary., 1999

Pyrite

O2

Fe2+

SO2−

4

H+

FeS2(s) + 7/2O2(aq) +H2O −→ Fe2+(aq)

+ 2SO2−4(aq)

+ 2H+(aq)








Pyrite

O2

O2

Fe3+Fe2+

SO2−

4

H+

Fe2+(aq)

+ 1/4O2(aq) +H+(aq)−→ Fe3+

(aq)+ 1/2H2O(ac)








Pyrite

O2

O2

Fe3+Fe2+

SO2−

4

H+

FeS2(s) + 14Fe3+(aq)

+ 8H2O −→ 15Fe2+(aq)

+ 2SO2−4(aq)

+ 16H+(aq)








Pyrite

O2

O2

Microorganisms

Microorganisms

Fe3+Fe2+

SO2−

4

Fe(OH)3

H+

Fe3+(aq)

+ 3H2O ←→ Fe(OH)3(s) + 3H+(aq)








Silicates

H+

Cu2+

K+

Fe2+Al3+Ca2+

Mn2+

Fe3+

Al(OH)3gypsum

Fe(OH)3

Mn(OOH)

Silicates+H+ −→ Heavy metals








SO2−

4

Fe2+L

Complexes

Hydrolysis products

Fe2+







Balance between the clean (river, rain water) and the polluted watersources (subterranean, infiltration water).








Precipitation/evaporation rate

Higher values are favorable: the lake fills faster + dilution.








Precipitation/evaporation rate

Higher values are favorable: the lake fills faster + dilution.

Limnological behavior (vertical circulation)

Solar radiation

Wind






Summer (vertical stratification)

Thermocline

ρ ↓

ρ ↑






Winter (vertical mixing)






Types of lakes

Holomictic Meromictic

Which factors affect meromixis?

Morphology: J.M. Castro and J.N. Moore, 2000

Geochemistry: B. Boehrer and M. Schultze, 2006






Types of lakes

Holomictic Meromictic

Which factors affect meromixis?

Morphology: J.M. Castro and J.N. Moore, 2000

Geochemistry: B. Boehrer and M. Schultze, 2006

Limnological behavior: effect on the water quality

Total mixing: the worst situation possible (theoretically)

Meromixis: it implies better water qualityPhenomena that increment pollution might occur ⇒ eventual mixing eventsof hazardous consequencies.

Each lake layer evolves in a different manner ⇒ highlights the importanceof the limnology





Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution

Chemical kinetics

When the time scale of the chemical reactions is similar to the time scaleof the problem

Let us consider a set of reacting chemical species S :

S = {E1, . . . , EN}

in a closed stirred tank. LetMi the molecular mass of species Ei andM(kg/kmol) the molecular mass of the mixture.

They are involved in a set of L chemical reactions

νl1E1 + ...+ νl

NEN → λl1E1 + ...+ λl

NEN , 1 ≤ l ≤ L,

νi and λi, i = 1, . . . , N are the stoichiometric coefficients.






Finite rate chemical reactions

In a closed stirred tank, the time evolution of the concentration, yi (kmol/m3),of the i-th chemical species Ei, i = 1, . . . , N, is given by the ODE

dyi(t)

dt=

L∑

l=1

(λli − νl

i)δl(t, y1, . . . , yN)

Expressions for the reaction velocity δl

Elementary reactions: δl = kl∏N

j=1 yj(t)νlj . Law of mass action (C.M.

Guldberg and P. Waage (1864-67)).

Most literature sources: δl = kl∏N

j=1 yj(t)αlj

In general: δl = h(t, y1, . . . , yN )







In a closed stirred tank, the time evolution of the concentration, yi (kmol/m3),of the i-th chemical species Ei, i = 1, . . . , N, is given by the ODE

dyi(t)

dt=

L∑

l=1

(λli − νl

i)δl(t, y1, . . . , yN)

Expressions for the reaction velocity δl

Elementary reactions: δl = kl∏N

j=1 yj(t)νlj . Law of mass action (C.M.

Guldberg and P. Waage (1864-67)).

Most literature sources: δl = kl∏N

j=1 yj(t)αlj

In general: δl = h(t, y1, . . . , yN )







kl is the rate constant. It is a function of the temperature θ through theArrhenius law

kl(θ) = Alexp(−Eal

Rθ

)

Al is the pre-exponential factor, Ealis the activation energy of the l-th

reaction and R is the universal constant for ideal gases.






The final problem

dyi(t)

dt=

L∑

l=1

(λli − νl

i)kl(θ)

N∏

j=1

yj(t)νlj , i = 1, . . . , N,

yi(0) = yinit,i, i = 1, . . . , N. (1)

The chemical reaction model is the Cauchy problem

C1

dy

dt(t) = w(t,y(t)),

y(0) = yinit,

yinit ≥ 0 and∑N

i=1Miyinit,i = ρ (mixture density, kg/m3).






Proof of the existence and uniqueness of solution

Assumptions

H1: Mass conservation

N∑

i=1

Miwi(t,y) = 0 ∀t ∈ [0, T ] ∀y ∈ (R+)N , being R+ = [0,∞).







Assumptions

H1: Mass conservation

N∑

i=1

Miwi(t,y) = 0 ∀t ∈ [0, T ] ∀y ∈ (R+)N , being R+ = [0,∞).

Indeed,

N∑

i=1

Miwi(t,y) =N∑

i=1

[L∑

l=1

Mi(λli − νl

i)kl(θ)N∏

j=1

yj(t)νlj

]

=

L∑

l=1

kl(θ)

[N∏

j=1

yj(t)νlj

(N∑

i=1

Mi(λli − νl

i)

)]

= 0

because∑N

i=1Miλli =

∑Ni=1Miν

li , l = 1, · · · , L.







H2: Splitting of w(t) in consumption (u) and production terms (v).

w(t,y) = u(t,y) + v(t,y),

where u and v are continuous in [0, T ]× RN and continuously

differentiable with respect to the y variable, for each t ∈ [0, T ]. Moreover,they satisfy

1 ui(t,y) = −yiUi(t,y), with

Ui(t,y) ≥ 0 ∀y ∈ (R+)N ,

ui corresponds to reactions l for which λli − νli < 0,

2 vi(t,y) ≥ 0 ∀t ∈ [0, T ] ∀y ∈ (R+)N ,

vi corresponds to reactions l for which λli − νli ≥ 0.







Steps of the proof:







Steps of the proof:1 w is continuous differentiable with respect to y → locally

Lipschitz-continuous → there is a unique local solution (Picard’sTheorem).









2 Since the positive part z+ = max{0,z} is a Lipschitz-continuous function,then

C2

dz

dt(t) = w(t,z+(t)),

z(0) = yinit,

has also a unique local solution. Any non-negative solution z(t) to C2 isalso a solution of C1.










C2

dz

dt(t) = w(t,z+(t)),

z(0) = yinit,


3 If z is any maximal solution to C2 in an interval I of the form [0, τ ] or[0, τ) for some τ ≤ T , then it is non-negative (from H2).










C2

dz

dt(t) = w(t,z+(t)),

z(0) = yinit,



4 Any maximal solution of C2 is a global solution (it is defined in [0, T ])(from H1). Any maximal solution to C2 is also a global solution to C1.










C2

dz

dt(t) = w(t,z+(t)),

z(0) = yinit,



4 Any maximal solution of C2 is a global solution (it is defined in [0, T ])(from H1). Any maximal solution to C2 is also a global solution to C1.

5 The Cauchy problem C1 has a unique global solution y in [0, T ] and,

0 ≤ yi(t) and

N∑

i=1

Miyi(t) =

N∑

i=1

Miyinit,i = ρ ∀t ∈ [0, T ].





IntroductionGibbs free energy minimizationNon-linear system of algebraic equations

Chemical equilibrium

When the time scale of the chemical reactions is much faster than thetime scale of the problem

Let us consider a set of chemical species S :

S = {E1, . . . , EN}

They are involved in a set of J couples of reversible chemical reactions (2Jchemical reactions)

νl1E1 + ...+ νl

NEN → λl1E1 + ...+ λl

NEN , 1 ≤ l ≤ 2J,

νli and λl

i, i = 1, . . . , N are the stoichiometric coefficients that satisfy

ν2j−1i = λ2j

i

λ2j−1i = ν2j

i , j = 1, . . . , J






Chemical equilibrium

Calculation of the equilibrium concentration. Two “equivalent” methods:

METHOD 1: By minimizing the Gibbs free energy of the system.

METHOD 2: By solving a non-linear system of algebraic equations basedon the equilibrium constants.

Some references ...

Smith, W.R. and Missen, R. W. (1991):Chemical Reaction EquilibriumAnalysis: Theory and Algorithms, Krieger Publishing, Malabar, FLA.

Morel, F.M.M. and Hering, J.G. (1993): Principles and applications ofAquatic Chemistry. John Wiley and Sons.

Bermudez, A. (2005): Continuum Thermomechanics. Birkhauser






Chemical equilibrium. Gibbs free energy minimization

Definitions

ρ: solution mass density (kg/m3).ni: number of kmol of the i-th species per kg solution (molinity).Xi: molar fraction (number of kmol of the i-th species per kmol of solution).

ni =yiρ

Xi =ni

∑Nj=1 nj

.







Definition of the Gibbs free energy of a system

The specific free energy of the solution (J/kg) is given by,

G(θ, n1, . . . , nN ) =N∑

i

niµi

where µi = µoi +Rθ lnXi is the molar free energy of the i-th species (J/kmol).







Let us denote by H the set of K chemical elements involved in the species:

H = {H1, . . . ,HK}

Let us assume the following formula for species Ei:

Ei = (H1)h1i . . . (HK)hKi, i = 1, · · · , N.







Definition of equilibrium state

For a given temperature θ, the system is in chemical equilibrium if and only ifthe Gibbs free energy G(θ, n1, . . . , nN ) attains a minimum with respect tovariables n1, · · · , nN subjected to the following constraints:

Mass conservation:∑N

i=1 hkini =∑N

i=1 hkini,init = ηk, k = 1, . . . ,K.ηk is the initial mass of the k-th chemical element Hk (katom/kgsolution) (conserved entity).

Positivity: ni ≥ 0, i = 1, . . . , N .






Chemical equilibrium. Non-linear system of algebraic equations

Simple case: equilibrium of one reversible chemical reaction

ν1E1 + ...+ νNEN1⇋2λ1E1 + ...+ λNEN .

Defineλi := ν2

i = λ1i ,

νi := λ2i = ν1

i

The forward and backward reaction velocities δ1 and δ2 are written as

δ1 = k1(θ)N∏

i=1

yνii and δ2 = k2(θ)

N∏

i=1

yλii .







The time evolution of the concentration of the i-th chemical species Ei,i = 1, . . . , N is given by

dyi(t)

dt= (λi − νi)δ

∗,

with

δ∗ = (k1

N∏

i=1

yνii − k2

N∏

i=1

yλii )

.







ρ: solution density (kg/m3 )ni: molinity (number of kmoles of the i-th species per kg of solution)yi (= niρ): concentration of the i-th species (kmol/m3 )

dyi(t)

dt= (λi − νi)δ

∗,1

(λi − νi)ρdni(t)

dt= δ∗

ξ(t) =1

ρ

∫ t

0

δ∗(s)ds








dyi(t)

dt= (λi − νi)δ

∗,1


dt= δ∗

By integrating this system of equations,

n1(t)− n1,init

(λ1 − ν1)= · · · =

nN (t)− nN,init

(λN − νN)= ξ(t),

ni(t) = ni,init + (λi − νi)ξ(t)

ξ(t) =1

ρ

∫ t

0

δ∗(s)ds








dyi(t)

dt= (λi − νi)δ

∗,1


dt= δ∗

By integrating this system of equations,

n1(t)− n1,init

(λ1 − ν1)= · · · =

nN (t)− nN,init

(λN − νN)= ξ(t),

ni(t) = ni,init + (λi − νi)ξ(t)where ξ(t) is the reaction extent

ξ(t) =1

ρ

∫ t

0

δ∗(s)ds







G(θ, n1, . . . , nN ) =N∑

i

niµini(t) = ni,init + (λi − νi)ξ(t)

The Gibbs free energy can be written as a function of ξ

g(ξ) := G(θ, n1,init + (λ1 − ν1)ξ, . . . , nN,init + (λN − νN)ξ).

Mass conservation is automatically satisfied. Only the positivity constraintsremain.







The equilibrium state is achieved by solving the one-dimensional constrainedoptimization problem:

min(λi−νi)ξ+ni,init≥0

g(ξ)







The equilibrium state is achieved by solving the one-dimensional constrainedoptimization problem:

min(λi−νi)ξ+ni,init≥0

g(ξ)

If the minimum is attained in the interior of the constrained set then

g′(ξ) =

N∑

i=1

∂G

∂ni(θ, n1, . . . , nN )

dni

dξ=

N∑

i=1

(λi − νi)µi(θ, n1, . . . , nN ) = 0.







By replacing

g′(ξ) =N∑

i=1

(λi − νi)µi(θ, n1, . . . , nN ) = 0 by µi = µoi +Rθ ln yi

and after some algebraic manipulations we reach to

−1

Rθ

N∑

i=1

(λi − νi)µoi = ln

( N∏

i=1

y(λi−νi)i

)

Taking the exponential, the equilibrium constant based on concentrations isdefined by

Ke(θ) := exp

(

−1

Rθ

N∑

i=1

(λi − νi)µoi

)

.







Equilibrium equation

Ke(θ) =N∏

i=1

yi(λi−νi)








Ke(θ) =N∏

i=1

yi(λi−νi)

yi = ρni ni = ni,init + (λi − νi)ξ








Ke(θ) =N∏

i=1

yi(λi−νi)

yi = ρni ni = ni,init + (λi − νi)ξ

Ke = ρ∑N

i=1(λi−νi)N∏

i=1

(ni,init + (λi − νi)ξ)λi−νi

By solving this equation for ξ the equilibrium composition is calculated.







Extension to a set of equilibrium reactions

νl1E1 + ...+ νl

NEN → λl1E1 + ...+ λl

NEN , 1 ≤ l ≤ 2J,

ν2j−1i = λ2j

i

λ2j−1i = ν2j

i , j = 1, . . . , J

The time evolution of the concentration of a chemical species

dyidt

=J∑

j=1

(λ2j−1i −ν2j−1

i )(δ2j−1−δ2j)







Extension to a set of equilibrium reactions

νl1E1 + ...+ νl

NEN → λl1E1 + ...+ λl

NEN , 1 ≤ l ≤ 2J,

ν2j−1i = λ2j

i

λ2j−1i = ν2j

i , j = 1, . . . , J

The time evolution of the concentration of a chemical species

dyidt

=J∑

j=1

(λ2j−1i −ν2j−1

i )(δ2j−1−δ2j)dyidt

=J∑

j=1

(λ2j−1i − ν2j−1

i )δ∗j







By replacing yi by ρni and integrating in t

ni = ni,init +J∑

j=1

(λ2j−1i − ν2j−1

i )ξj , i = 1, . . . , N

ξj is the reaction extent of the j-th couple of reversible reactions.







By replacing yi by ρni and integrating in t

ni = ni,init +J∑

j=1

(λ2j−1i − ν2j−1

i )ξj , i = 1, . . . , N

ξj is the reaction extent of the j-th couple of reversible reactions.

ξj(t) =1

ρ

∫ t

0

δ∗j (s)ds

The equations above imply the mass conservation







Equilibrium equations

Kej (θ) =

N∏

i=1

yi(λ

2j−1i

−ν2j−1i

)








Kej (θ) =

N∏

i=1

yi(λ

2j−1i

−ν2j−1i

)

yi = ρnini = ni,init +

∑Jl=1(λ

2l−1i − ν2l−1

i )ξl








Kej (θ) =

N∏

i=1

yi(λ

2j−1i

−ν2j−1i

)

yi = ρnini = ni,init +

∑Jl=1(λ

2l−1i − ν2l−1

i )ξl

Kej = ρ

∑Ni=1(λ

2j−1i

−ν2j−1i

)N∏

i=1

[ni,init +

J∑

j=1

(λ2j−1i − ν2j−1

i )ξj ]λ2j−1i

−ν2j−1i

By solving this system of equations for ξ1, . . . , ξJthe equilibrium composition is calculated.





Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem

Coexistence of slow and fast chemical reactions

The set of chemical reactions is:

νl1E1 + ...+ νl

NEN → λl1E1 + ...+ λl

NEN , 1 ≤ l ≤ L+ 2J,

The first L reactions are slow.The last 2J reactions are J couples of fast reversible reactions, satisfying

νL+2j−1i = λL+2j

i

λL+2j−1i = νL+2j

i







The problem to be solved is:

dyi(t)

dt=

L+2J∑

l=1

(λli − νl

i)δl(t,y(t)) =L∑

l=1

(λli − νl

i)δl(t,y(t))

+J∑

j=1

(λL+2j−1i − νL+2j−1

i )(

δL+2j−1(t,y(t))− δL+2j(t,y(t)))

.







The problem to be solved is:

dyi(t)

dt=

L+2J∑

l=1

(λli − νl

i)δl(t,y(t)) =L∑

l=1

(λli − νl

i)δl(t,y(t))

+J∑

j=1

(λL+2j−1i − νL+2j−1

i )(

δL+2j−1(t,y(t))− δL+2j(t,y(t)))

.

Assuming that all the chemical reactions are elementary

dyi(t)

dt=

L∑

l=1

(λli − νl

i)kl

N∏

i=1

yνli

i

+

J∑

j=1

(λL+2j−1i − νL+2j−1

i )(

kL+2j−1

N∏

i=1

yνL+2j−1i

i − kL+2j

N∏

i=1

yλL+2j−1i

i

)

.






The limit model

The limit of the multi-scaled problem is obtained by introducing Lagrangemultipliers

Multi-scale problem:slow+fast chemical reactions

What is fast or slow?PROBLEM SCALING

Definition of dimensionless variables

yi(t) =yi(t)

Yi(t)and t =

t

T

where Yi and T are the typical scales for the concentration and time in theproblem.






The limit model

Scaled problem: the original problem is written as a function of thedimensionless variables

Yi

T

dyi

dt=

L∑

l=1

(λli − νl

i)kL

N∏

i=1

yνli

i +

J∑

j=1

(λL+2j−1i − νL+2j−1

i )[

kL+2j−1

N∏

i=1

yνL+2j−1i

i − kL+2j

N∏

i=1

yλL+2j−1i

i

]






The limit model

Yi

T

dyi

dt=

L∑

l=1

(λli − νl

i)kl

N∏

i=1

yνli

i +

J∑

j=1

(λL+2j−1i − νL+2j−1

i )kL+2j

[ kL+2j−1

kL+2j

N∏

i=1

yνL+2j−1i

i −N∏

i=1

yλL+2j−1i

i

]

Under the assumptions

kl = O(1), l = 1, . . . , L,

kL+2j−1 = O(ε−1), j = 1, . . . , J,

kL+2j = O(ε−1), j = 1, . . . , J,

Kej (θ) =

kL+2j−1

kL+2j

= O(1), j = 1, . . . , J.

ε > 0 is a small parameter.Ratio of fast time scales to slow ones.






The limit model

The model can be written as

dy

dt(t) = f(t,y(t)) +

1

εAgε(t,y(t))

with

f : [0, T ]× RN → R

N ,

A the N × J matrix

Aij = λL+2j−1i − νL+2j−1

i .

gε : [0, T ]× RN → R

J .

In what follows we assume rank(A)=J.






The limit model

We analyze the limit as ε→ 0 of the model.IDEA: The limit model is a good approximation to the “exact” one.

dyε

dt(t) = f(t,yε(t)) +

1

εAgε(t,yε(t))

It has been proved that the solution of the problem for ε > 0 satisfies,

0 ≤ yε,i ≤ K, ∀t ∈ [0, T ],

Then, the sequence {yε} is bounded in L∞(O, T ;RN ) =⇒ there exists asubsequence {yεn} and y ∈ L∞(0, T ;RN) such that

limn→∞

{yεn} = y weakly-* in L∞(0, T ;RN ).






The limit model


By integrating from 0 to t,

yεn(t) = yinit +

∫ t

0

f(s,yεn(s))ds+1

εn

∫ t

0

Agεn (s,yεn(s))ds






The limit model


By integrating from 0 to t,

yεn(t) = yinit +

∫ t

0

f(s,yεn(s))ds+1

εn

∫ t

0

Agεn (s,yεn(s))ds

By multiplying the equation by εn...

limn→∞

∫ t

0

Agεn (s,yεn(s))ds = 0 ∀t ∈ [0, T ].






The limit model

gεn is uniformly bounded on bounded sets and yεn is bounded, then thesequence {gεn(·,yεn(·))} is bounded in L∞(0, T ;RJ ) ⇒ there existsq ∈ L∞(0, T ;RJ such that

limn→∞

gεn (·,yεn(·)) = q weakly-* in L∞(0, T ;RJ ),

implying that ∀t ∈ [0, T ]

∫ t

0

Agεn(s,yεn(s))ds =

∫ T

0

Agεn (s,yεn(s))X[0,t](s)ds→

∫ T

0

Aq(s)X[0,t](s)ds

=

∫ t

0

Aq(s)ds = 0,

and hence q = 0, because A is injective.






The limit model

We have proved

limn→∞

{gεn (·,yεn(·))} = 0 weakly-* in L∞(0, T ;RJ )

We are not able to prove

g(t,y(t)) = 0

limn→∞

f(t,yεn(t)) = f(t,y(t))






The limit model

We have proved

limn→∞

{gεn (·,yεn(·))} = 0 weakly-* in L∞(0, T ;RJ )

We are not able to prove

g(t,y(t)) = 0

limn→∞

f(t,yεn(t)) = f(t,y(t))

ADDITIONAL ASSUMPTIONS ARE REQUIRED!!!






The limit model

{dyε

dt} is bounded in L1(0, T ;RN).

Then yε is bounded in W 1,1(0, T ;RN ) implying that there exists asubsequence {yεn} converging pointwise to y in [0, T ] (Helly’s Theorem).Since f and g are continuous, from the Lebesgue’s dominated convergencetheorem,

limn→∞

f(t,yεn(t)) = f(t,y(t)) strongly in Lr(0, T ;RN ),

limn→∞

gεn(t,yεn(t)) = g(t,y(t)) strongly in Lr(0, T ;RJ ),

∀r <∞.






The limit model

{dyε



limn→∞


limn→∞


∀r <∞.






The limit model

{dyε



limn→∞


limn→∞


∀r <∞.






The limit model

We recall that

limn→∞

{gεn (·,yεn(·))} = 0 weakly-* in L∞(0, T ;RN )






The limit model

We recall that

limn→∞

{gεn (·,yεn(·))} = 0 weakly-* in L∞(0, T ;RN )

Then we have,

g(t,y(t)) = 0.






The limit model

From the boundedness hypothesis and since A is injective⇒

1εn

gεn (t,yεn) is bounded in L1(0, T ;RJ ).

Therefore, there exists p ∈M(0, T ;RJ ) such that

limn→∞

1

εngεn (.,yεn(.)) = p(.) weakly-* inM(0, T ;RJ ),

withM(0, T ;RJ ) =(C0([0, T ];RJ)

)′the space of Radon measures from [0, T ]

in RJ .






The limit model

LIMIT PROBLEM

There exists y ∈W 1,1(0, T ;RN ) and p ∈M(0, T ;RJ ) solution to the limitproblem

y′(t) = f(t,y(t)) +Ap(t),g(t,y(t)) = 0,y(0) = yinit.






The limit model

LIMIT PROBLEM

There exists y ∈W 1,1(0, T ;RN ) and p ∈M(0, T ;RJ ) solution to the limitproblem


fi(t,y(t)) =

L∑

l=1

(λli − νl

i)kl

N∏

i=1

yνli

i

gj(t,y(t)) = Kej

N∏

i=1

yνL+2j−1i

i −

N∏

i=1

yλL+2j−1i

i






Solubility reactions

HETEROGENEOUS REACTIONS

DISSOLVED SPECIESprecipitation

⇋dissolution

SOLID SPECIES

Set of chemical reactions

νl1E1 + ...+ νl

NEN → λl1E1 + ...+ λl

NEN , 1 ≤ l ≤ L+M,

The first L reactions are slow.The last M reactions are solubility reactions in equilibrium.Set of chemical species

S = {E1, . . . , EN ,

solid species︷︸︸︷

EN+1, . . . , EN+M}

However, since the concentration of the solid species are approximatelyconstant, they are included in the equilibrium constant.







dyidt

= fi(t,y(t))+

L+M∑

m=L+1

(λL+mi −νL+m

i )kL+m

[

Ksm

N∏

i=1

yνL+mi

i −N∏

i=1

yλL+mi

i

]+

In the limit as kL+m →∞ we have:[

Ksm

∏Ni=1 y

νL+mi

i −∏N

i=1 yλL+mi

i

]

→ 0

kL+m

[

Ksm

∏Ni=1 y

νL+mi

i −∏N

i=1 yλL+mi

i

]+

→ psm ≥ 0.

psm is a Lagrange multiplier associated to the constraint

Ksm

N∏

i=1

yνL+mi

i −N∏

i=1

yλL+mi

i ≤ 0.







Limit model: unilateral equilibrium

dyidt

(t) = fi(t,y(t)) +

M∑

m=1

(λL+mi − νL+m

i )psm(t), i = 1, ..., N








dyidt

(t) = fi(t,y(t)) +

M∑

m=1

(λL+mi − νL+m

i )psm(t), i = 1, ..., N

psm(t), m = 1, . . . ,M is a Lagrange multiplier associated to the m-thinequality constraint.

psm(t) ≥ 0gsm(t,y(t)) ≤ 0

psm(t) gsm(t,y(t)) = 0

.








dyidt

(t) = fi(t,y(t)) +

M∑

m=1

(λL+mi − νL+m

i )psm(t), i = 1, ..., N

psm(t), m = 1, . . . ,M is a Lagrange multiplier associated to the m-thinequality constraint.

psm(t) ≥ 0gsm(t,y(t)) ≤ 0

psm(t) gsm(t,y(t)) = 0

.where

gsm(t,y(t)) = Ksm(θ(t))

N∏

i=1

yi(t)νL+mi −

N∏

i=1

yi(t)λL+mi






Reduction methods

Let us assume there is no precipitation reactions


The problem can be directly solved for all the chemical species but reductiontechniques facilitate resolution.Two reduction methods are proposed:

Reduction method I

Reduction method II

Objective: to eliminate the Lagrange multipliers p associated toequilibrium reactions






Reduction methods

Quasi-Steady-State Approximation (QSSA) methods:

Introduced by the chemists in the sixties

Related to the Tikhonov’s Theorem

L. A. Segel, M. Slemrod, The Quasi-Steady-State Assumption: A CaseStudy in Perturbation, SIAM Review, Vol. 31 (3) (1989), pp. 446-477.

B. Sportisse, Vivien Mallet, Calcul Scientifique pour l’Environnement.Cours ENSTA, 2005.

A.N. Tikhonov, Systems of differential equations containing a smallparameter multiplying the derivative, Mat. Sb. 31 (1952) pp. 575-586






Reduction method I

νl1E1+...+νl

NEN → λl1E1+...+λl

NEN

l = 1, . . . , L+ 2J

The first L reactions are slow.The last 2J reactions are J couplesof fast reversible reactions.

We assume that we can find J chemical species Ei, i = 1, . . . , Jexclusively involved in the J equilibrium reactions whose concentrationscan be written as a function of the remaining ones taking part in the

equilibrium reactions

y =

y1

y2

y3

y1 ∈ RJ : exclusively involved in equilibrium (fast species).

y2 ∈ RN−J−D: the remaining species.

y3 ∈ RD: exclusively involved in finite rate (slow species).






Reduction method I

In the same way the vector of species concentration was split, the originalproblem can be split as well

y1′(t) = f

1(t,y1,y2,y3) +A1p(t),

y2′(t) = f

2(t,y1,y2,y3) +A2p(t),

y3′(t) = f

3(t,y1,y2,y3) +A3p(t).

f1 is a J dimensional null vector, f2 ∈ RN−J−D and f3 ∈ R

D.A1 ∈ MJ×J , A2 ∈M(N−J−D)×J and A3 is a D × J null matrix.

We replace y1 = ξ(t,y2) in the equations above

y1′(t) = A1p(t),

y2′(t) = f

2(t, ξ(t,y2),y2,y3) +A2p(t),

y3′(t) = f

3(t, ξ(t,y2),y2,y3). (2)






Reduction method I

By taking the time derivative of y1 = ξ(t,y2)

y1′(t) =

∂ξ

∂y2y2′(t) +

∂ξ

∂t






Reduction method I


y1′(t) =

∂ξ

∂y2y2′(t) +

∂ξ

∂t

Replacing terms in the above equations,

A1p(t) =∂ξ

∂y2

[f2(t, ξ(t,y2),y2,y3) +A2p(t)

]+

∂ξ

∂t






Reduction method I


y1′(t) =

∂ξ

∂y2y2′(t) +

∂ξ

∂t

Replacing terms in the above equations,

A1p(t) =∂ξ

∂y2

[f2(t, ξ(t,y2),y2,y3) +A2p(t)

]+

∂ξ

∂t

Rearranging the equations

[A1 −

∂ξ

∂y2A2

]p(t) =

∂ξ

∂y2f2(t, ξ(t,y2),y2,y3) +

∂ξ

∂t






Reduction method I

This is a linear system of equations whose unknowns are the Lagrangemultipliers

G(t,y2,y3)p(t) = b(t,y2,y3)

p = (G)−1b






Reduction method I

Reduced problem by Reduction method I:{

y2′(t) = f

2(t, ξ(t,y2),y2,y3) +A2(G)−1

b,

y3′(t) = f

3(t, ξ(t,y2),y2,y3). (3)

Advantages of Reduction Method I

The problem has been reduced by J ODEs with respect to the original one.

There is no longer Lagrange multipliers in the system.

Disadvantages of Reduction Method I

The previous processing of the problem is tedious.

The obtention of the exact expressions for the Lagrange multipliersrequires powerful symbolic solvers.





Application to a simple example

Numerical methods

We focus on the complete problem


Two steps in the numerical resolution of the problem:

1 Time discretization using an Euler implicit scheme

2 Iterative algorithm to solve the discrete problem






Numerical methods

Time discretizationUniform partitioning of the time interval of interest (0, tf ).

The approximate solution is calculated at each tn = n∆t, with ∆t =tfN

and0 ≤ n ≤ N .By considering an Euler implicit scheme

P∆t

y0 = yinit,

If n ≥ 0,

yn+1 = y

n +∆t[f(tn+1,yn+1) +Apn+1

],

g(tn+1,yn+1, θn+1) = 0,






Simple example: complete problem


reaction index (l) Chemical reaction Reaction velocity or equil. constant

1 Fe2+ +H2O −→ FeOH+ +H+ δ1 = 10−5

2 Fe2+ + 2H2O ⇋ Fe(OH)2 + 2H+ Ke1 = 10−20.5

3 Fe2+ + Cl− ⇋ FeCl+ Ke2 = 100.14

4 H20 ⇋ H+ +OH− Ke3 = 10−14

Set of chemical species

i Ei

1 Fe(OH)22 FeCl+

3 OH−

4 Fe2+

5 H+

6 Cl−

7 FeOH+







Equilibrium constraints

ge1 = −y1 +Ke

1y4y25

= 0,

ge2 = −y2 +Ke2y4y6 = 0,

ge3 = y3 −Ke

3

y5= 0.







Equilibrium constraints

ge1 = −y1 +Ke

1y4y25

= 0,

ge2 = −y2 +Ke2y4y6 = 0,

ge3 = y3 −Ke

3

y5= 0.

Complete problem

y′1(t) = pe1(t),

y′2(t) = pe2(t),

y′3(t) = pe3(t),

y′4(t) = −δ1 − pe1(t)− pe2(t),

y′5(t) = δ1 + 2pe1(t) + pe3(t),

y′6(t) = −p

e2(t),

y′7(t) = δ1,

ge1(t) = −y1 +Ke

1y4y25

= 0,

ge2(t) = −y2 +Ke2y4y6 = 0,

ge3(t) = y3 −Ke

3y5

= 0,

y(0) = yinit,






Simple example: Reduction method I



1 Fe2+ +H2O −→ FeOH+ +H+ δ1 = 10−5

2 Fe2+ + 2H2O ⇋ Fe(OH)2 + 2H+ Ke1 = 10−20.5

3 Fe2+ + Cl− ⇋ FeCl+ Ke2 = 100.14

4 H20 ⇋ H+ +OH− Ke3 = 10−14


i Ei

1 Fe(OH)22 FeCl+

3 OH−

4 Fe2+

5 H+

6 Cl−

7 FeOH+









1 Fe2+ +H2O −→ FeOH+ +H+ δ1 = 10−5

2 Fe2+ + 2H2O ⇋ Fe(OH)2 + 2H+ Ke1 = 10−20.5

3 Fe2+ + Cl− ⇋ FeCl+ Ke2 = 100.14

4 H20 ⇋ H+ +OH− Ke3 = 10−14


i Ei

1 Fe(OH)22 FeCl+

3 OH−

4 Fe2+

5 H+

6 Cl−

7 FeOH+

Splitting the set of species

y1 = (y1, y2, y3)

T ,

y2 = (y4, y5, y6)

T ,

y3 = (y7).







Gp = b

The Lagrange multipliers p are the unknowns.

with

G =

1 +Ke

1

y25

(1 + 4y4

y5

) Ke1

y25

2Ke1y4y35

y6Ke2 1 +Ke

2(y4 + y6) 02Ke

3

y25

0(1 +

Ke3

y25

)

b =

−Ke

1δ1y25

(1 + 2y4

y5

)

−y6Ke2δ1

−Ke

3δ1y25

.







The expressions of the Lagrange multipliers can be explicitly obtained but wedo not reproduce them. The reduced problem is given by

Reduced problem by applying Reduction Method I

y′4(t) = −δ1 − pe1(y

2)− pe2(y2),

y′5(t) = δ1 + 2pe1(y

2) + pe3(y2),

y′6(t) = −p

e2(y

2),y′7(t) = δ1,

y2(0) = y2init,

y3(0) = y3init,





IntroductionConsidered chemical reactionsProblem settingNumerical results

A geochemical example: Introduction

Research project between with the company Lignitos de Meirama S.A.,financed by the I+D+I Galician Government plan (2006-2007)

Prediction of the water quality of afuture pit lake in NW Spain.

The lake, when filled, will be connectedto a water reservoir. It has to satisfycertain water quality standards.

Will these legal limits be fulfilled?






A geochemical example: Introduction

We have to take into account


The magnitude and geochemistry of the water sources flowing into the pit.

The precipitation/evaporation ratio

The lake limnology

The biological activity...






A geochemical example: pyrite oxidation by O2

Pyrite

O2

Fe2+

SO2−

4

H+

FeS2(s) + 7/2O2(aq) +H2O −→ Fe2+(aq)

+ 2SO2−4(aq)

+ 2H+(aq)






A geochemical example: abiotic oxidation of Fe2+

Pyrite

O2

O2

Fe3+Fe2+

SO2−

4

H+

Fe2+(aq)

+ 1/4O2(aq) +H+(aq)−→ Fe3+

(aq)+ 1/2H2O(ac)






A geochemical example: pyrite oxidation by Fe3+

Pyrite

O2

O2

Fe3+Fe2+

SO2−

4

H+

FeS2(s) + 14Fe3+(aq)

+ 8H2O −→ 15Fe2+(aq)

+ 2SO2−4(aq)

+ 16H+(aq)






A geochemical example: Ferrihydrite precipitation

Pyrite

O2

O2

Fe3+Fe2+

SO2−

4

Fe(OH)3

H+

Fe3+(aq)

+ 3H2O ←→ Fe(OH)3(s) + 3H+(aq)






A geochemical example: Ferrihydrite precipitation

Pyrite

O2

O2

Microorganisms

Microorganisms

Fe3+Fe2+

SO2−

4

Fe(OH)3

H+

Fe3+(aq)

+ 3H2O ←→ Fe(OH)3(s) + 3H+(aq)






A geochemical example: Silicate degradation

Silicates

H+

Cu2+

K+

Fe2+Al3+Ca2+

Mn2+

Fe3+

Al(OH)3gypsum

Fe(OH)3

Mn(OOH)

Silicates +H+ −→ Heavy metals






A geochemical example: equilibrium reactions

SO2−

4

Fe2+L

Complexes

Hydrolysis products

Fe2+






A geochemical example: equilibrium reactions

The model for LIMEISA lake

Complete problem: 112 ODEs + 2 algebraic equations

Reduced problem (by reduction method II): 19 ODEs and a NLSE (19equations + 19 unknowns)






A geochemical example: problem setting

The lake is assumed to be a stirred tank

The problem is stated by considering

The chemical reactions before and much more... Show reactions

The entrance of different water sources. Show sources

Heat exchange with the atmosphere.Show problem












Initial conditions

Obtained by assuming that all the watersources are mixed proportionally to theirvolume during the first time interval forwhich they are available.












Initial conditions

Obtained by assuming that all the watersources are mixed proportionally to theirvolume during the first time interval forwhich they are available.

Integration time

0 2 4 6 8 100

5

10

15x 10

10

Years after the beginning of flooding

Vol

ume

(dm

3 )

A






A geochemical example: Numerical results

We show the results of a sensitivity test to analyze the effect of the slowchemical reactions and the solubility reactions

Red line: Results of a simulation in which neither slow nor solubilitychemical reactions are considered, just homogeneous equilibria.

Blue line: Slow and homogeneous reactions but no precipitation are takeninto account.

Green line: Complete geochemical problem.

Results are shown for species that are relevant in mining environments.Fe2+, Fe3+, Al3+, Alunite, Mn2+ and pH .







Ferrous ion

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5


Con

cent

ratio

n in

mg/

l

A

All chem. reacChem. reac. no prec.No chem. reac.

Legal limit: 2mg/l

Ferric ion

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5


Con

cent

ratio

n in

mg/

l

A


Legal limit: 2mg/l







Aluminium

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3


Con

cent

ratio

n in

mg/

l

A


Legal limit: 1mg/l

Alunite

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5x 10

−5


Con

cent

ratio

n in

mol

/l

A


Legal limit: 2mg/l







Manganese

0 1 2 3 4 5 6 7 8 9 102

2.5

3

3.5

4

4.5

5

5.5


Con

cent

ratio

n in

mg/

l

A


Legal limit: 2mg/l

pH

0 1 2 3 4 5 6 7 8 9 105.6

5.8

6

6.2

6.4

6.6

6.8


A


Legal limit: 5.5-9






References

A. Bermudez, L. Garcıa-Garcıa, Mathematical modeling in chemistry.Application to water quality problems, Appl. Numer. Math., Vol. 62 (4),2012, pp. 305-327.

D.N. Castendyk, J.G. Webster-Brown, Sensitivity analyses in pit lakeprediction, Martha mine, New Zealand 2: Geochemistry, water-rockreactions, and surface adsorption, Chem. Geol., 244 (2007), pp. 56-73.

J. Delgado, R. Juncosa, F. Padilla, P. Rodrıguez-Vellando, Predictivemodeling of the water quality of the future Meirama open pit lake(Cerceda, A Coruna), Macla, 10 (2008), pp. 122-125.

L. Garcıa-Garcıa, Numerical resolution of water quality models: applicationto the closure of open pit mines, Ph.D. thesis, University of Santiago deCompostela, 2010.

A. N. Tikhonov, A. B. Vasileva, and A. G. Sveshnikov, DifferentialEquations. Springer Verlag, Berlin, 1985.


Thank you for your attention!

R. idx.(l) Chemical reaction Eq. R.cte. type

16 H+ + SO2−4 ⇌ HSO−

4 Ke1 he

17 Na+ + SO2−4 ⇌ NaSO−

4 Ke2 he

18 K+ + SO2−4 ⇌ KSO−

4 Ke3 he

19 Mg2+ + H2O − H+⇌ MgOH+ Ke

4 he20 Mg2+ + SO2−

4 ⇌ MgSO4 Ke5 he

21 Ca2+ + H2O − H+⇌ CaOH+ Ke

6 he22 Ca2+ + H+ + SO

2−4 ⇌ CaHSO

+4 Ke

7 he23 Ca2+ + SO2−

4 ⇌ CaSO4(aq) Ke8 he

24 Cu2+ + H2O − H+⇌ CuOH+ Ke

9 he25 Cu2+ + 2H2O − 2H+

⇌ Cu(OH)2(aq) Ke10 he

26 Cu2+ + 3H2O − 3H+⇌ Cu(OH)−3 Ke

11 he27 Cu2+ + SO2−

4 ⇌ CuSO4(aq) Ke12 he

28 Fe2+ + H2O − H+⇌ FeOH+ Ke

13 he29 Fe2+ + 2H2O − 2H+

⇌ Fe(OH)2(aq) Ke14 he

30 Fe2+ + SO2−4 ⇌ FeSO4(aq) Ke

15 he31 Fe2+ + H+ + SO2−

4 ⇌ FeHSO+4 Ke

16 he32 Fe3+ + H2O − H+

⇌ FeOH2+ Ke17 he

33 Fe3+ + 2H2O − 2H+⇌ Fe(OH)

+2 Ke

18 he34 Fe3+ + 3H2O − 3H+

⇌ Fe(OH)3(aq) Ke19 he

35 Fe3+ + 4H2O − 4H+⇌ Fe(OH)−4 Ke

20 he36 Fe3+ + SO2−

4 ⇌ FeSO+4 Ke

21 he37 Fe3+ + H+ + SO2−

4 ⇌ FeHSO2+4 Ke

22 he


38 Fe3+ + 2SO2−4 ⇌ Fe(SO4)

−

2 Ke23 he

39 Al3+ + H2O − H+⇌ AlOH2+ Ke

24 he40 Al3+ + 2H2O − 2H+

⇌ Al(OH)+2 Ke25 he

41 Al3+ + 3H2O − 3H+⇌ Al(OH)3(aq) Ke

26 he42 Al3+ + 4H2O − 4H+

⇌ Al(OH)−4 Ke27 he

43 Al3+ + H+ + SO2−4 ⇌ AlHSO2+

4 Ke28 he

44 Al3+ + SO2−4 ⇌ AlSO

+4 Ke

29 he45 Al3+ + 2SO2−

4 ⇌ Al(SO4)−

2 Ke30 he

46 Mn2+ + H2O ⇌ MnOH+ + H+ Ke31 he

47 Mn2+ + SO2−4 ⇌ MnSO4 Ke

32 he48 CO2−

3 + 2H+⇌ CO2 + H2O (H2CO3) Ke

33 he49 H+ + CO2−

3 ⇌ HCO−

3 Ke34 he

50 Na+ + CO2−3 ⇌ NaCO−

3 Ke35 he

51 Na+ + HCO−

3 ⇌ NaHCO3 Ke36 he

52 Mg2+ + CO2−3 ⇌ MgCO3 Ke

37 he53 Mg2+ + HCO−

3 ⇌ MgHCO+3 Ke

38 he54 Ca2+ + CO2−

3 ⇌ CaCO3 Ke39 he

55 Ca2+ + HCO−

3 ⇌ CaHCO+3 Ke

40 he56 Cu2+ + CO2−

3 ⇌ CuCO3 Ke41 he

57 Fe2+ + HCO−

3 ⇌ FeHCO+3 Ke

42 he58 Fe2+ + CO2−

3 ⇌ FeCO3 Ke43 he

59 Mn2+ + HCO−

3 ⇌ MnHCO+3 Ke

44 he


60 Mn2+ + CO2−3 ⇌ MnCO3 Ke

45 he61 H+ + F−

⇌ HF Ke46 he

62 H+ + 2F−

⇌ HF−

2 Ke47 he

63 Na+ + F−

⇌ NaF Ke48 he

64 Mg2+ + F−

⇌ MgF+ Ke49 he

65 Ca2+ + F−

⇌ CaF+ Ke50 he

66 Cu2+ + F−

⇌ CuF+ Ke51 he

67 Fe2+ + F−

⇌ FeF+ Ke52 he

68 Fe3+ + F−

⇌ FeF2+ Ke53 he

69 Fe3+ + 2F−

⇌ FeF+2 Ke

54 he70 Fe3+ + 3F−

⇌ FeF3 Ke55 he

71 Al3+ + F−

⇌ AlF2+ Ke56 he

72 Al3+ + 2F−

⇌ AlF+2 Ke

57 he73 Al3+ + 3F−

⇌ AlF3 Ke58 he

74 Al3+ + 4F−

⇌ AlF−

4 Ke59 he

75 Mn2+ + F−

⇌ MnF+ Ke60 he

76 Cu2+ + Cl− ⇌ CuCl+ Ke61 he

77 Cu2+ + 2Cl− ⇌ CuCl2 Ke62 he

78 Cu2+ + 3Cl− ⇌ CuCl−3 Ke63 he

79 Cu2+ + 4Cl− ⇌ CuCl2−4 Ke64 he


88 Fe(OH)3 − 6L + 3H+⇌ Fe3

++ 3H2O Ks

1 mp89 KAl3(SO4)2(OH)6 + 6H+

⇌ K+ + 3Al3+ + 2SO2−4 + 6H2O Ks

2 mp90 MnOOH + 3H+

⇌ Mn2+ + 2H2O Ks3 mp

91 CaSO4.2H2O ⇌ Ca2+ + SO2−4 + 2H2O Ks

4 mp92 HFO − sOH + H+

⇌ HFO − sOH+2 Ka

1 ad93 HFO − sOH ⇌ HFO − sO− + H+ Ka

2 ad94 HFO − sOH + Ca2+

⇌ HFO − sOCa+ + H+ Ka3 ad

95 HFO − sOH + Cu2+⇌ HFO − sOCu+ + H+ Ka

4 ad96 HFO − sOH + Mn2+

⇌ HFO − sOMn+ + H+ Ka5 ad

97 HFO − sOH + Fe2+ ⇌ HFO − sOFe+ + H+ Ka6 ad

98 HFO − wOH + H+⇌ HFO − wOH

+2 Ka

7 ad99 HFO − wOH ⇌ HFO − wO− + H+ Ka

8 ad

The discharge of the 8 water sources is known.

0 12 24 36 48 60 72 840

50

100

150

200

250

300

350

400

Months after the beginning of flooding

Dis

char

ge (

l/s)

Discharge of the different water sources

Basin(sch.)Basin(gr.)subt.sch.cl.subt.gr.clrainsubt.sch.pol.subt.gr.poldump

The water quality was measured in the field.

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Pst

dy

dt(t) = f

y(t,y(t)) +Aepe(t) +As

ps(t) +Aa

pa(t) +

1

V (t)

(mφy,b(t,y(t))Sb(t)

+mϕy,a(t,y(t))Sa(t))+

1

V (t)

(A(t)q(t)

), (4)

ge(t,y(t), θ(t), I(t)) = 0, (5)

gs(t,y(t), θ(t), I(t)) ≤ 0, (6)

ps(t) ≥ 0, (7)

gs(t,y(t), θ(t), I(t))ps(t) = 0, (8)

ga(t,y(t), θ(t), I(t),P(t)) = 0, (9)

I(t) =1

2(cc) · y(t), (10)

mσ(y(t)) =0.1174I(t)1/2

2P(t), (11)

dθ

dt(t) =

1

V (t)(

ρ(t) + θ(t) ∂ρ(θ(t))∂θ(t)

)

[

(mρ(t)mθ(t)) · q(t)− ρ(t)θ(t)q0(t)− θ(t)

(

V (t)( N∑

i=1

Midyi(t)

dt

)

+ρ(t)( Ns∑

j=1

qj(t)− q0(t)))

+Sa(t)

CeQatm(t, θ(t))

]

, (12)

dV (t)

dt=

Ns∑

j=1

qj(t)− q0(t), (13)

y(0) = yinit, (14)

θ(0) = θinit, (15)

V (0) = Vinit, (16)

Number of chemical reactions of each kind1, 72, 4, 19, 12, 1

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