Environmental Engineering Reference
In-Depth Information
C.1
Moments
We now have solute concentration vs time data for both the input and output profiles. The
data can be integrated to calculate various moments as defined below.
C.1.1 Absolute moments
∞
m
0
=
c
(
t
)d
t
=
constant if system doesn't lose mass
0
∞
m
1
=
t
·
c
(
t
)d
t
0
∞
t
2
m
2
=
·
c
(
t
)d
t
.
0
We can normalize these values by dividing by
m
0
(
z
=
0)
m
1
m
0
=
µ
1
=
center of gravity of the profile
m
2
m
0
.
µ
2
=
C.1.2 Centralized moments (normalized)
We can also define moments where the time axis is shifted to the center of gravity.
∞
1
m
0
−
µ
1
)
2
c
(
t
)d
t
µ
2
=
⇒
(
t
measure of width of curve
0
∞
1
m
0
−
µ
1
)
3
c
(
t
)d
t
µ
3
=
(
t
⇒
measure of asymmetry
0
∞
∞
1
m
0
1
m
0
=
−
µ
1
)
c
(
t
)d
t
=
µ
1
−
µ
1
µ
1
=
(
t
c
(
t
)d
t
0
.
0
0
m
0
We now introduce the use of Laplace transforms to illustrate how we can use the solute
concentration and various derivatives in the Laplace domain to obtain equations for the
various moments.
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