Environmental Engineering Reference
In-Depth Information
, N should result in total
mass conservation, given by Equation (3.4). This condition is fulfilled, as the chemical
source terms for all components present in the system sum up to zero:
The summation of Equation (3.7) over all species i = 1,
…
X
N
i
=1
ω
:
=0
ð
Eq
:
3
:
18
Þ
Moreover, the total density of the reacting phase is the sum of the individual
partial densities in the mixture, and the total mass flux is the sum of the partial mass
fluxes:
X
and
X
N
i
=1
ρ
i
=
N
i
=1
ρ
i
v
!
=
!
ρ
ρ
ð
Eq
:
3
:
19
Þ
With the mass fraction Y
i
of component i, this can be written as
X
N
Y
i
= 1 and
X
N
Y
i
v
!
=
!
ð
Eq
:
3
:
20
Þ
i
=1
i
=1
This equation defines the (density-weighted) mean velocity
!
of the flow. The
difference between the partial velocity and the mean velocity is defined as the diffu-
sion velocity, V
!
=v
!
!
, and the mass flux due to diffusion as j
!
=
V
!
. Note that the
−
ρ
following holds:
X
j
!
=
X
N
N
i
=1
ρ
=
v
!
!
!
!
=0
−
ρ
ρ
−
ρ
ð
Eq
:
3
:
21
Þ
i
=1
Using the aforementioned definitions, then Equation (3.17) can be rewritten as
j
!
=
+
∂
ρ
Y
i
∂
:
!
Y
i
+
r
ρ
r
ω
i
for i = 1,
…
,N
ð
Eq
:
3
:
22
Þ
t
Example 3.1 Mass balance calculation for a well-stirred chemical reactor
Both ethanol and acetic acid can be derived from biomass, so also the product of
esterification, ethyl acetate, can be produced in a biorefinery complex. The reac-
tion equation for the esterification reaction is
:
:
C
2
H
5
OH 1
ðÞ
+CH
3
COOH 2
ðÞ$
CH
3
COOC
2
H
5
3
ðÞ
+H
2
O4
ðÞ ð
RX
3
1
Þ
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