Introduction: motivation and objectives Chemical kinetics. Finite rate chemical reactions. Fast chemical reactions: chemical equilibrium Coexistence of slow and fast chemical reactions Numerical methods A geochemical example Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes Alfredo Berm´ udez de Castro and Luz M. Garc´ ıa-Garc´ ıa Departamento de Matem´ atica Aplicada, Universidade de Santiago de Compostela. Instituto Espa˜ nol de Oceanograf´ ıa. A Coru˜ na Industrial and Environmental Mathematical Day Instituto de Matem´ aticas de la Universidad de Sevilla 8 de junio de 2012 Alfredo Berm´ udez de Castro and Luz M. Garc´ ıa-Garc´ ıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
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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
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
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,
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
Antes del llenado
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
Comienzo del llenado
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
Primeras fases del lago
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
El lago en los medios
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
El lago en los medios
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
El lago en los medios
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
The chemical composition of the wall rocks
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)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
The chemical composition of the wall rocks
C.N. Alpers, 1994, C. Blodau., 2006 y L.E. Eary., 1999
Pyrite
O2
O2
Fe3+Fe2+
SO2−
4
H+
Fe2+(aq)
+ 1/4O2(aq) +H+(aq)−→ Fe3+
(aq)+ 1/2H2O(ac)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
The chemical composition of the wall rocks
C.N. Alpers, 1994, C. Blodau., 2006 y L.E. Eary., 1999
Pyrite
O2
O2
Fe3+Fe2+
SO2−
4
H+
FeS2(s) + 14Fe3+(aq)
+ 8H2O −→ 15Fe2+(aq)
+ 2SO2−4(aq)
+ 16H+(aq)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
The chemical composition of the wall rocks
C.N. Alpers, 1994, C. Blodau., 2006 y L.E. Eary., 1999
Pyrite
O2
O2
Microorganisms
Microorganisms
Fe3+Fe2+
SO2−
4
Fe(OH)3
H+
Fe3+(aq)
+ 3H2O ←→ Fe(OH)3(s) + 3H+(aq)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
The chemical composition of the wall rocks
C.N. Alpers, 1994, C. Blodau., 2006 y L.E. Eary., 1999
Silicates
H+
Cu2+
K+
Fe2+Al3+Ca2+
Mn2+
Fe3+
Al(OH)3gypsum
Fe(OH)3
Mn(OOH)
Silicates+H+ −→ Heavy metals
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
The chemical composition of the wall rocks
C.N. Alpers, 1994, C. Blodau., 2006 y L.E. Eary., 1999
SO2−
4
Fe2+L
Complexes
Hydrolysis products
Fe2+
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
The magnitude and geochemistry of the water sources
Balance between the clean (river, rain water) and the polluted watersources (subterranean, infiltration water).
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
The magnitude and geochemistry of the water sources
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
The magnitude and geochemistry of the water sources
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.
Limnological behavior (vertical circulation)
Solar radiation
Wind
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
Summer (vertical stratification)
Thermocline
ρ ↓
ρ ↑
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
Winter (vertical mixing)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Geochemical models for water quality
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
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 )
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
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 )
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
Finite rate chemical reactions
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
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).
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
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,∞).
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
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,∞).
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
Proof of the existence and uniqueness of solution
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
Proof of the existence and uniqueness of solution
Steps of the proof:
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
Proof of the existence and uniqueness of solution
Steps of the proof:1 w is continuous differentiable with respect to y → locally
Lipschitz-continuous → there is a unique local solution (Picard’sTheorem).
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
Proof of the existence and uniqueness of solution
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
Proof of the existence and uniqueness of solution
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.
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).
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
Proof of the existence and uniqueness of solution
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.
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).
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Finite rate chemical reactions. The mass action lawProblem statement. Existence and uniqueness of solution
Proof of the existence and uniqueness of solution
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.
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).
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 ].
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
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
.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Gibbs free energy minimization
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).
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Gibbs free energy minimization
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Gibbs free energy minimization
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 .
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
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 .
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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 )
.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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∗
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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∗
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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
Mass conservation is automatically satisfied. Only the positivity constraintsremain.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
The equilibrium state is achieved by solving the one-dimensional constrainedoptimization problem:
min(λi−νi)ξ+ni,init≥0
g(ξ)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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
)
.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
Equilibrium equation
Ke(θ) =N∏
i=1
yi(λi−νi)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
Equilibrium equation
Ke(θ) =N∏
i=1
yi(λi−νi)
yi = ρni ni = ni,init + (λi − νi)ξ
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
Equilibrium equation
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
Equilibrium equations
Kej (θ) =
N∏
i=1
yi(λ
2j−1i
−ν2j−1i
)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
Equilibrium equations
Kej (θ) =
N∏
i=1
yi(λ
2j−1i
−ν2j−1i
)
yi = ρnini = ni,init +
∑Jl=1(λ
2l−1i − ν2l−1
i )ξl
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionGibbs free energy minimizationNon-linear system of algebraic equations
Chemical equilibrium. Non-linear system of algebraic equations
Equilibrium equations
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
Coexistence of slow and fast chemical reactions
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)))
.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
Coexistence of slow and fast chemical reactions
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
)
.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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
]
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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 ).
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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.
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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.
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 ].
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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))
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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!!!
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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 <∞.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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 <∞.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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 <∞.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
The limit model
We recall that
limn→∞
{gεn (·,yεn(·))} = 0 weakly-* in L∞(0, T ;RN )
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
The limit model
We recall that
limn→∞
{gεn (·,yεn(·))} = 0 weakly-* in L∞(0, T ;RN )
Then we have,
g(t,y(t)) = 0.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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 .
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
The limit model
LIMIT PROBLEM
There exists y ∈W 1,1(0, T ;RN ) and p ∈M(0, T ;RJ ) solution to the limitproblem
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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).
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
Reduction method I
By taking the time derivative of y1 = ξ(t,y2)
y1′(t) =
∂ξ
∂y2y2′(t) +
∂ξ
∂t
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
Reduction method I
By taking the time derivative of y1 = ξ(t,y2)
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
Reduction method I
By taking the time derivative of y1 = ξ(t,y2)
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
Problem approximation. The limit modelA particular case: solubility reactionsMethods to reduce the chemical problem
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Reduced problem (by reduction method II): 19 ODEs and a NLSE (19equations + 19 unknowns)
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionConsidered chemical reactionsProblem settingNumerical results
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
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionConsidered chemical reactionsProblem settingNumerical results
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionConsidered chemical reactionsProblem settingNumerical results
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.
Integration time
0 2 4 6 8 100
5
10
15x 10
10
Years after the beginning of flooding
Vol
ume
(dm
3 )
A
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionConsidered chemical reactionsProblem settingNumerical results
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 .
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionConsidered chemical reactionsProblem settingNumerical results
A geochemical example: Numerical results
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
Years after the beginning of flooding
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
Years after the beginning of flooding
Con
cent
ratio
n in
mg/
l
A
All chem. reacChem. reac. no prec.No chem. reac.
Legal limit: 2mg/l
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionConsidered chemical reactionsProblem settingNumerical results
A geochemical example: Numerical results
Aluminium
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
Years after the beginning of flooding
Con
cent
ratio
n in
mg/
l
A
All chem. reacChem. reac. no prec.No chem. reac.
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
Years after the beginning of flooding
Con
cent
ratio
n in
mol
/l
A
All chem. reacChem. reac. no prec.No chem. reac.
Legal limit: 2mg/l
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionConsidered chemical reactionsProblem settingNumerical results
A geochemical example: Numerical results
Manganese
0 1 2 3 4 5 6 7 8 9 102
2.5
3
3.5
4
4.5
5
5.5
Years after the beginning of flooding
Con
cent
ratio
n in
mg/
l
A
All chem. reacChem. reac. no prec.No chem. reac.
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
Years after the beginning of flooding
A
All chem. reacChem. reac. no prec.No chem. reac.
Legal limit: 5.5-9
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes
Introduction: motivation and objectivesChemical kinetics. Finite rate chemical reactions.
Fast chemical reactions: chemical equilibriumCoexistence of slow and fast chemical reactions
Numerical methodsA geochemical example
IntroductionConsidered chemical reactionsProblem settingNumerical results
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.
Alfredo Bermudez de Castro and Luz M. Garcıa-Garcıa Mathematical Modelling in Geochemistry. Application to Water Quality Problems in Open Pit Lakes