Environmental Engineering Reference
In-Depth Information
Thus, the expression for the conversion of ethanol simplifies to a relation based
on species 1:
2
+k
2
c
1
,
0
ζ
k
1
c
1
,
0
1
2
1
R
1
=
−
ð
−
ζ
1
Þ
Substitution of this equation in the reactor equation gives
2
+k
2
c
1
,
0
ζ
1
2
0=
ζ
1
−
k
1
c
1
,
0
1
ð
−
ζ
1
Þ
τ
τ
It is left to the reader to further simplify this quadratic equation, with
ζ
1
explic-
itly expressed as a function of
τ
.
Example 3.2 Piston compression of syngas: Example of an unsteady-state
microscopic balance
A horizontal piston compresses synthesis gas within a cylinder at a constant veloc-
ity v. The initial gas density and compartment length are
ρ
0
and L
0
, respectively.
The gas inside the cylinder has a velocity decreasing from
u
x
= v at the piston head
to u = 0 m
s
−1
at the position x = L. Let the gas density only be variable in
time
.
Derive an expression for the dynamic density behavior,
ρ
(t).
Solution
This is an unsteady-state (time-dependent) problem, for which only one dimension
counts, the horizontal (x)-direction. The general continuity equation in this case
simplifies to
∂
ρ
∂
t
+
∂
=
∂
ρ
∂
ρ
∂
∂
x
ρ
ðÞ
u
t
+
x
u
=0
∂
Here,
u
= v(1
−
x/L), L = L
0
−
vt, and
ρ
=
ρ
(t).
Then,
∂
u
v
L
x
=
−
∂
Separation of variables leads to
=v
ð
t
0
ð
ρ
d
ρ
ρ
dt
L
0
−
ð
vt
Þ
ρ
0
from which the following solution is obtained:
=
)
ρ
ln
ρ
ρ
0
vt
L
0
L
0
L
0
−
−
ln 1
−
=
ρ
0
vt
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