Incorporation of aqueous reaction and sorption kinetics andbiodegradation into TOUGHREACT Page: 2 of 6
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-2-
N, NP Nq
TO = c +YVk ck + YVjC +Y'VnC (
k=1 m=1 n=1
j =1...Nc
where superscript 0 represents time zero; c are
concentrations (chemical reactions are always solved
per kg of water, and concentration units used here are
mol/kg which is close to mol/1 when water density is
close to 1 kg/l); subscripts j, k, m, and n are the
indices of basis species, aqueous complexes, minerals
at equilibrium and minerals under kinetic constraints,
respectively; Ne, NX, Np, and Nq are the number of the
corresponding species and minerals; vkj, vmj, and vs1
are stoichiometric coefficients of the basis species in
the aqueous complexes, equilibrium and kinetic
minerals, respectively. After a time step At, the total
concentration of basis species j is given by
N, NP
T = c + ZVkjck + VjCM +
k=1 m-1 (3)
Nqvnj -rnAt j=..Nc
where r is the kinetic rate of mineral dissolution
(negative for precipitation, units used here are moles
per kg water per time).
Now let's add aqueous kinetic reactions to the
system, according to mass conservation, we haveNa
T 0 T -Y r(4)
where 1 is aqueous kinetic reaction index, Na is total
number of aqueous kinetic reactions, and r, is the
kinetic rate which is in terms of generation of one
mol of product species such as S042- per unit time
Therefore, for product species the stoichometric
coefficient v1 are positive, for reactant species they
are negative. For reaction (1), v1 for S042- and H' are
1, for HS- is -1, and for 202(aq) is -2
The set of nonlinear chemical reactions is solved by
the Newton-Raphson iterative method. The use of
this method requires lumping all the terms in the
right-hand side in a single term (or residual function
which is zero in the limit of convergence), and we
denote this term by F
N,
FJ =T, - T -Z vrAt=0 (5)By substituting Eqs. (2) and (3) into Eq. (5), we
obtainN,
F= cj+Zvkjck+
Nk=i
N,
-IvrAt - U
=AN
m=-N9
ZvnjrnAt
n-1(6)
0 j =1...Nc
where
N,
U = c +I V C"
k=1(6a)
NP
+ZV=nj C
rn-iAccording to mass action equations, concentrations
of aqueous complexes ck can be expressed as
functions of concentrations of the basis species cj.
Kinetic rates r~ and r, are functions of cj. No explicit
expressions relate equilibrium mineral concentrations
cm, to cj. Therefore, Np additional mass action
equations (one per mineral) are needed. Details on
solution of the nonlinear-system of equations by
Newton-Raphson iterative method are given in (Xu et
al., 1999).
RATE EXPRESSIONS
Following the expression of Curtis (2003) and adding
multiple mechanisms (or pathways), a general rate
law is used,
Mecki x (yil C x'
j=1 k=1 KMi,k +i,k
YiI N (7)
rP =A X K 11 hP+ Ci,
C.
where r is the reaction rate of the ith reaction, Mech
is number of mechanisms or pathways and s is the
mechanism counter, k, is a rate constant, (often
denoted vmax in the Monod formulation), y, is the
activity coefficient of species j, C, is the
concentration of species j, v,,, is a stoichiometric
coefficient, N is the number of reacting species in the
forward rate term (called product terms), N, is the
number of Monod factors (Monod terms), C,,k is the
concentration of the kth Monod species, C,, is the
concentration of the pth inhibiting species, KM,,k is the
kth monod half-velocity coefficient of the ith species,
NP is the number of inhibition factors (inhibition
terms), Kj,, is the pth inhibition constant. Equation
(7) accounts for multiple mechanisms, multiple
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Xu, Tianfu. Incorporation of aqueous reaction and sorption kinetics andbiodegradation into TOUGHREACT, article, April 17, 2006; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc884849/m1/2/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.