Environmental Engineering Reference
In-Depth Information
cðtÞ¼c
0
exp
ð
l
tÞ
(10.2)
There are two parameters: the initial concentration
c
0
and the decay constant
.
Note that the fitting procedure can be performed for any dependent variable
c
and
the independent variable
t
; both symbols do not necessarily represent concentration
and time in this chapter.
As the
polyfit
command works with polynomials, one may use the idea that the
exponential curve is a linear curve in logarithmic representation. Thus the
polyfit
command can be used for the logarithm of the concentration vector:
l
p = polyfit(tfit,log(cfit),1)
p =
-0.2619 -0.3387
The best fit for the logarithm is thus given by:
log
ðcÞ¼
0
:
2619
t
0
:
3387
(10.3)
or:
cðtÞ¼
exp
ð
0
:
3387
Þ
exp
ð
0
:
2619
t
Þ
(10.4)
¼
0
:
7127 exp
ð
0
:
2619
t
Þ
Comparison with formula (
10.2
) shows that the second line of (
10.4
) is the aimed
formulation with
c
0
¼
0.8236 and
l
¼
0.3487. In Fig.
10.4
the result is again
plotted together with the original data:
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
Fig. 10.4 Exponential fit for example batch experiment