Environmental Engineering Reference
In-Depth Information
where
k
denotes the reaction parameter, which is a characteristic for the 'speed' of
the reaction.
K
S
is a 'half reaction concentration', which means that for given
c
org,
s
the
reaction rate takes half of its maximum value.
f
is a parameter relating the mass
of organic matter to the sulphate mass. In
f
the stoichiometric relation in the
sulphate reduction process has to be taken into account. The bulk density and
porosity also play a role if organic matter in the solid phase reacts with sulphate
in the fluid phase.
The system (
9.18
) of two ordinary differential equations has no analytical
solution. Results can only be obtained by numerical methods. The system can be
treated using the MATLAB
®
bvp4c
function for the solution of boundary value
problems (bvp). The function is designed for solving problems which can be
formulated in the following notation (see MATLAB
help on 'bvp'):
®
y
0
¼ fðx; y; pÞ
0
¼ bcðyðaÞ; yðbÞ; pÞ
where the first line characterizes the differential equation telling that the first
derivative
y
0
of the unknown variable
y
is a function of
y
itself, of the independent
variables
x
and of a parameter set
p
. As always in MATLAB
,
y
as well as
p
can be
vectors. The second line characterizes the boundary conditions telling that these are
specified at the boundaries
x ¼ a
and
x ¼ b
.
The first differential equation in the system (
9.18
) is of first order and can easily
be brought into the required form:
®
@
@z
c
org;s
¼
1
w
kc
org
;
s
c
S
K
S
þc
S
(9.19)
The second equation has a second derivative and can be the brought into the
necessary form by a simple trick. Another variable is introduced:
c
S
0
, which is the
first derivative of the unknown function
c
S
. The second-order differential equation
can thus be written as a system of two first order systems:
@
@z
c
S
¼ c
S
0
@
@z
c
S
0
¼
fkc
org
;
s
c
S
K
S
þc
S
1
D
u
@
@z
c
S
0
þ
(9.20)
Note that for the entire formulation of the problem there are three unknown
functions:
c
s;org
; c
S
and
c
S
0
. The vector
y
in the MATLAB
notation has three
components, which have to be taken into account in the formulation of the system
of differential equations and of the boundary conditions. The following m-code
excerpt shows how the MATLAB
®
routine for boundary value problems can be
utilized to solve the system (
9.19
) and (
9.20
).
®