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 ).
®
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