Environmental Engineering Reference
In-Depth Information
3.3 CONSERVATION OF ENERGY
The conservation of energy, a scalar quantity, is given by the following general
balance equation:
Accumulation rate of energy = rate of energy supply
rate of energy release
+ rate of energy production
−
ð
Eq
:
3
:
23
Þ
The different forms of energy considered in the balance are heat, kinetic energy
(KE), and potential energy (PE) due to the presence of fields of gravity, electricity,
and magnetism.
3.3.1 Energy Balance for Systems without Chemical Reactions
The first law of thermodynamics forms the basis of the principle of conservation of
energy and for a closed system (a good treatise of closed and open systems can be
found in Moran and Shapiro, 2010) this is simply expressed as
Þ
where U is the internal energy of the system, KE (=1/2 mv
2
) the kinetic energy, PE
the potential energy due to the action of a field (for gravitation PE = mgz, with z the
height in this field), Q the heat supply (positive if added to the system, negative if
removed), and W the work done by the system (positive if the system performs work,
negative if work is exerted on the system by some net force action). The terms at the left-
hand side of Equation (3.23) are equal to
Δ
U+
Δ
KE +
Δ
PE =
Q
−
W
ð
Eq
:
3
:
24
E, the total energy change of the system.
For an open (flow) system with a characteristic control volume as indicated in
Figure 3.2, the macroscopic energy balance can now be written as follows:
Δ
dE
cv
dt
φ
m
,
in
u
in
+
v
in
−
φ
m
,
out
u
out
+
v
out
=
Q
−
W
+
2
+gz
in
+gz
out
ð
Eq
:
3
:
25
Þ
2
φ
m, in
(
u
in
+
v
in
/
2 +
g
z
in
)
System boundary =
control volume
W
v
in
Q
z
in
v
out
φ
m
, out
(
u
out
+
v
out
/
2 +
g
z
out
)
z
out
FIGURE 3.2
Energy balance illustration for an open-flow system.
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