quantities are as follows. T
i ,k
total concentration of component i, i ¼ 1 to N
C
, in cell k for k ¼ 1 to N
cells
T
k
the vector of total concentration of components in cell k ¼
T
1 ,
k
, T
2 ,
k
, …, T
N
C
, k
t
C
i ,k
aqueous concentration of species i, for i ¼ 1 to N
S
, in cell k
C
k
the vector of aqueous concentrations of species in cell k ¼
C
1 ,
k
, C
2 ,
k
, …, C
N
S
, k
t
T
i ,in
total aqueous concentration of component i in solution flowing into cell 1 at flow rate v
t, i ¼ 1 to N
C
T
in
the vector of total aqueous concentrations of components in the solution flowing into cell 1 at
flow rate v t,
T
1 ,
in
, …, T
N
C
, in
t
F
i ,k
the inflow rate of the total aqueous concentration of component i into cell k [mol s
¹ 1
] F
k
the vector of inflow rates of total aqueous concen- trations for all components into cell k,
F
k
¼ F
1 ,
k
, …, F
N
C
, k
t
h
k
volume of cell k [l] All concentrations are in mol l
¹ 1
; in the case of the solid phase, P
k
, this is a bulk concentration in the sense of being the number of moles in the cell divided by the volume
of the cell. We note that lower case ‘c’ is used in Ref.
19
for C
i ,k
.
2.1 The general form of the mass balance equations and transport equations
The conservation of mass for the chemical subsystem can be described in terms of total component concentrations:
T
k
¼ AC
k
þ P
k
b k ¼ 1
, …, N
cells
, 2
which will lead to a set of N
C
mass balance equations for each cell. Eqn 2 indicates that the total component con-
centrations are combinations of dissolved species and the solid phase with the stoichiometric coefficients as weights.
These coefficients appear in the rows of the N
C
3 N
S
matrix A and in the N
C
vector b see Refs. 20 or 21 and the example below. We assume that no solid calcite is
carried into cell 1 by the inflow; hence, T
in
is related to the concentrations of the dissolved species in the inflow,
which we could write as C
i ,in
, by T
in
¼ AC
in
, i.e. the form of eqn 2 with P
in
¼ 0. At this point, it should be noted that
total concentrations of components have to take positive values to be physically meaningful, with the exception of
H
þ
, for which the mass balance equation is based on the proton condition and is therefore not required to be
positive.
12,17
For our N
cell
¼ 2 model of Fig. 1, the cell coupling
requirement 1 implies flow v
2
transports solutes from cell 2 into cell 1. Hence, we can express the mass transport of the
total concentrations of components as follows: F
1
t ¼
v t
T
in
t ¹
AC
1
t þ
v
1
t A
C
2
t ¹
C
1
t 3
¼ v
t T
in
t ¹
v t
þ v
1
t AC
1
t þ
v
1
t AC
2
t F
2
t ¼
v
1
t AC
1
t þ
v
2
t AC
2
t ¹
v
3
t AC
2
t ¼
v
1
t AC
1
t ¹
C
2
t :
Combining these mass transport expressions with a conti- nuity equation yields the mass conservation equation for
the physical subsystem, which can be described by a set of N
cells
N
C
differential equations: h
k
dT
k
t =
dt ¼ F
k
t for k ¼ 1 to N
cells
: 4
Since F
i ,k
is linear in C
i ,k
, eqn 4 can be written as dT
k
t =
dt ¼ X
N
cells
k 9 ¼ 1
V
k ,
k 9
t AC
k 9
t þ
V
k ,
t T
in
t for k ¼ 1 to N
cells
, ð
5Þ where, in general, the coefficients in the N
cells
3 N
cells
þ 1
matrix V
k ,k9
t describe the inter-cell flows. In this paper, with N
cells
¼ 2, these flows are
V
1 ,
V
1 ,
1
V
1 ,
2
V
2 ,
V
2 ,
1
V
2 ,
2
¼ v
t =
h
1
¹ v
t þ
v
1
t =
h
1
v
1
t =
h
1
v
1
t =
h
2
¹ v
1
t =
h
2
, ð
6Þ From eqns 2 and 5, we have 2N
cells
N
C
equations for the N
cells
N
C
þ N
S
þ N
P
variables describing the total compo- nent concentrations for N
C
components, plus the concentra- tions of all dissolved species and the solid phase in the N
cells
cells. For a complete model description, we require addi- tional equilibrium conditions for the dissolved species and
the solid phase in each cell, which provide N
S
þ N
P
¹ N
C
equations for each cell, closing the system of model equations.
2.2 Specific form of the homogeneous and heterogeneous reaction equations
