For steady flow of an incompressible fluid in the streamtube
Flux of energy
ρ 2 e 2 dA 2 v 2
out of the
=
(2.28)
control volume
Similarly, at the inflow section (section 1),
Flux of energy
ρ1 e1 dA 1 v 1
into the
=
(2.29)
control volume
and
Time rate of
change of energy
=
0
(2.30)
in the control volume
Here e is the energy per unit mass. Accordingly, the total energy E in a control volume is:
E = ∀ ρe d∀
(2.31)
Unless one is concerned with thermal pollution, evaporation losses, or problems concerning the
formation of ice in rivers, the rate at which heat is added to the control volume can be neglected;
that is:
Q~0
-
(2.32)
The work done by the fluid in the control volume on its surroundings can be in the form of
pressure work Wp, shear work Wτ, or shaft work (mechanical work) Ws. For the streamtube
shown in Figure 2.3, no shaft work is involved (Ws = 0).
The rate at which the fluid pressure does work on the control volume surrounding through the
boundary dA1 in Figure 2.3 is:
- p1 dA 1 v 1
and on boundary dA2, the rate of doing pressure work is
p 2 dA 2 v 2
At the other boundaries of the streamtube, there is no pressure work because there is no fluid
motion normal to the boundary. Hence, for the streamtube
W p = p 2 dA 2 v 2 - p1 dA 1 v 1
(2.33)
2.9
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