Environmental Engineering Reference
In-Depth Information
n
ϕ
f
ϕ
f
V
B
!
FIGURE 3.1
Domain volume V with boundary B, normal vector
!
, and flux
ϕ
f
.
Figure 3.1 shows a randomly shaped volume V of an open-flow system with a
surrounding boundary B. There is a supply flux of
f
(
f
being mass density, momentum
density, or volumetric energy density) into the system and an exit flux of
f
out of the
system.
Based on the property
f
, the general conservation equation can be written as
∂
f
∂
−r
ϕ
!
+
s
f
t
=
ð
Eq
:
3
:
2
Þ
!
f
is the flux of
f
(a vector), and
s
f
is a
source term for
f
(a scalar quantity). The system is at steady state when (
Here,
f
is a continuous scalar unit per volume,
t) = 0.
The conservation equations derived from Equation (3.2) are described in more
detail in the following sections.
∂
f
/
∂
3.2 CONSERVATION OF MASS
Matter is conserved in physical systems and in chemical reactors, the only exception
being nuclear conversion processes, which are not dealt with here. Overall mass con-
servation in an open system can be given by the mass balance in macroscopic form
(over a control volume):
dm
dt
=
φ
m
,
in
−
φ
m
,
out
=
−
Δ
φ
m
ð
Eq
:
3
:
3
Þ
Formulated in differential form and using Equation (3.2) as the basis, overall mass
conservation can be expressed as
!
+0
!
=0
∂
ρ
∂
,
∂
ρ
∂
t
=
−r
ρ
t
+
r
ρ
ð
Eq
:
3
:
4
Þ
In this equation, also called the continuity equation, with
f
=
ρ
, the system density
!
!
, the total mass flux.
and
ϕ
f
=
ρ
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