Environmental Engineering Reference
In-Depth Information
C.2
Laplace transforms
∞
e
−
st
c
(
t
)d
t
c
=
Definition of Laplace transform
0
∞
d
c
d
s
=
t
)e
−
st
c
(
t
)d
t
(
−
.
0
We can generalize this to the
k
th derivative. If we take the limit as
s
→
0, we see that we
obtain the equation for the
k
th moment multiplied by (
−
1)
k
:
lim
s
→
0
d
k
c
d
s
k
∞
1)
k
t
k
c
(
t
)d
t
1)
k
m
k
.
=
(
−
=
(
−
0
The approach is to obtain the experimental measurements and calculate numerical values
for the various moments. The model equation is solved in the Laplace domain to obtain
equations for the various moments. Equating the moment equations to the numerical values
allows one to solve for various unknowns in the model equation(s).
In theory, we could generate several moment equations and solve for an equivalent
number of parameters. In practice, it is best to only use the first and second moments.
The outlet concentration profile usually has some “tailing” at longer times. These values
usually have the largest error in their values and can skew the calculated moment values
since the concentration is multiplied by
t
k
for the
k
th moment.
C.2.1 Example C.1: inert packed bed (see Figure C.2)
Problem:
Use the Laplace transform to model mass transfer in a cylindrical packed bed.
z
=
L
d
z
z
=0
r
Figure C.2
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