Biomedical Engineering Reference
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ð
w
ρΔ
x
ð
2
dA
Þ
ð
0
ð
W
1
Þj
t
1
Δt
Δ
-
d
-
area 1
lim
Δ
w
ρ
Δ
ð
3
:
20
Þ
5
52
t
Δ
t
t
-
area 1
The previous equation uses the equalities for
0
Δ
x
lim
Δ
-
t
5
t
-
Δ
d
-
to change the quantities into vector form. Combining
Equations 3.17 and 3.20
into
Equation 3.16
,
dA
5
ð
ð
ð
dW
dt
5
@
d
-
d
-
-
-
w
ρ
dV
w
ρ
w
ρ
ð
3
:
21
Þ
1
U
1
U
@
t
V
area 3
area 1
The entire system of interest consists of areas 1, 2, and 3, and we can make the assumption
that there is no change in flow within region 2 during the time interval of
t
(this is why
we choose to overlap the systems from the two time intervals). Therefore,
-
is zero for
area 2, and we can combine the two area integrals in
Equation 3.21
into a general form,
where “area” is equal to area 1 plus area 3.
dW
Δ
ð
ð
dt
5
@
d
-
-
w
ρ
dV
w
ρ
ð
3
:
22
Þ
1
U
@
t
area
V
When developing the formulation for the time rate of change of a system property, we
took the limit of the system as time approached zero. This forces the relationship to be
valid at the instant when the system and the control volume completely overlap. The first
term of
Equation 3.22
is the time rate of change of any arbitrary system property (
W
). The
second term in
Equation 3.22
is the time rate of change of the inherent property within the
volume of interest (
w
). The third term in
Equation 3.22
is the flux of the property out of
the surface of interest or into the surface of interest. From this relationship, all of the con-
servation laws can be derived by substituting the appropriate system property and inher-
ent property, which were described above.
In Chapter 2, we defined conservation of mass as
d
ðρ
V
Þ
m
system
5
m
in
2
m
out
2
ð
2
:
1
Þ
dt
In a more concise form, mass balance can be stated as
dm
dt
system
5
0
ð
3
:
23
Þ
The mass of a system can be defined as
ð
ð
m
system
dm
system
ρ
dV
5
5
m
system
V
2
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