Environmental Engineering Reference
In-Depth Information
According to the law of mass action (see sub-chapter 8.2), the equilibrium condition
for the given reaction is given by the equation:
c
c
n
c
c
a
n
a
c
b
n
b
¼ K
(8.5)
with a specific reaction-dependent equilibrium constant
K
.
The general solution procedure for the entire system is as such: solve the differen-
tial (
8.4
) in order to obtain the variables
A :¼ c
a
þ
n
a
n
b
n
c
c
c
as
functions of time and space! In a second step, determine for each location and time
instant the three unknown values
c
a
,
c
b
and
c
c
n
c
c
c
and
B :¼ c
b
þ
from the three known values
A
,
B
and
K
!
In order to perform this task, we utilize the resulting explicit formulae for
c
a
and
c
b
as functions of
c :¼ c
c
as well as
A
and
B
:
c
a
¼ A n
a
=nð Þc
c
,
c
b
¼ B n
b
=nð Þc
c
.
Thus we may re-write (
8.5
)as:
c
n
c
n
a
n
b
¼ K
(8.6)
n
a
n
b
A
n
c
c
B
n
c
c
or:
n
a
n
b
n
a
n
c
c
n
b
n
c
c
fðcÞ :¼ c
n
c
A
B
K ¼
0
(8.7)
The further task is to find the zero of a non-linear equation. The standard
procedure for such a computation is Newton's method (see Sidebar 8.1).
Sidebar 8.1: Newton
1
's Method
Newton's method is a mathematical standard method for the determination of
the zeros of a function. Let's introduce the method for functions of one
variable
f
(
x
) first before extending it to several dimensions.
Newton's method is an iterative method; i.e. starting from an initial guess,
a new approximation for the zero is computed in each iteration step.
The formula for the next approximation of the zero is:
fðxÞ
f
0
ðxÞ
x x
(continued)
1
Isaac Newton (1642-1727), English scientist and philosopher.
