10.3.1 Rate of Work Done by a Control Volume and Energy
Balance in Integral Form for a Control Volume
The term _
W in Eq. (191) has a positive numerical value when work is done by the
control volume on the surroundings. This includes the possible positive useful work
output, called shaft work. The rate of work done by the control volume is typically
subdivided into the following three classifications:
_
W ¼ _
W shaft þ _
W surface þ _
W resistive
ð193AÞ
where
_
W surface ¼ _
W normal þ _
W shear
¼ À
Z
cs
^ n Á Àpd þ s
h
i
Á ~ VdA
0
@
1
A ¼
Z
cs
p ~ V Á d ~ A À
Z
cs
^ n Á s
À
Á Á ~ VdA
ð193BÞ
Note, in this consideration of ^ n Á s
À
Á Á ~ V to the surface of the control volume, cs, we
have the following special results:
Areas of the cv corresponding to stationary solid surfaces where ~ V ¼ 0; thus _
W shear ¼ 0
Areas of the control volume at inlet and outlet are perpendicular to velocity; thus _
W shear ¼ 0
It follows, therefore, that
R
cs
^ n Á s
À
Á Á ~ VdA ¼ 0, and correspondingly,
_
W ¼ _
W shaft þ _
W normal þ _
W resistive ¼ _
W shaft þ
Z
cs
p ~ V Á d ~ A þ _
W resistive
ð198Þ
We now consider the form of the first law for a control volume by using
Eqs. (191) and (198). Noting that
Z
cs
p ~ V Á d ~ A þ
Z
cs
eq ~ V Á d ~ A ¼
Z
cs
q pv þ e
ð
Þ ~ Vd ~ A
and
e ¼ u þ
V
2
2
þ gz;
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10 A Theory of Heat as a Prelude …