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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