energy per unit mass e, which is euke pe uV^2 /2 gz(Fig.

5–53). This yields

(5–54)

Substituting the left-hand side of Eq. 5–53 into Eq. 5–54, the general form of

the energy equation that applies to fixed, moving, or deforming control vol-

umes becomes

(5–55)

which can be stated as

Here V

→

rV

→


V

→

CSis the fluid velocity relative to the control surface, and

the product r(V

→

r· n

→) dArepresents the mass flow rate through area element

dAinto or out of the control volume. Again noting that n→is the outer normal of

dA, the quantity V

→

r· n

→and thus mass flow is positive for outflow and negative

for inflow.

Substituting the surface integral for the rate of pressure work from Eq. 5–51

into Eq. 5–55 and combining it with the surface integral on the right give

(5–56)

This is a very convenient form for the energy equation since pressure work is

now combined with the energy of the fluid crossing the control surface and

we no longer have to deal with pressure work.

The term P/rPvwflowis the flow work,which is the work associated

with pushing a fluid into or out of a control volume per unit mass. Note that

the fluid velocity at a solid surface is equal to the velocity of the solid surface

because of the no-slip condition and is zero for nonmoving surfaces. As a

result, the pressure work along the portions of the control surface that coincide

with nonmoving solid surfaces is zero. Therefore, pressure work for fixed con-

trol volumes can exist only along the imaginary part of the control surface

where the fluid enters and leaves the control volume (i.e., inlets and outlets).

This equation is not in a convenient form for solving practical engineering

problems because of the integrals, and thus it is desirable to rewrite it in terms

of average velocities and mass flow rates through inlets and outlets. If P/re

is nearly uniform across an inlet or outlet, we can simply take it outside the

integral. Noting that is the mass flow rate across an inlet

or outlet, the rate of inflow or outflow of energy through the inlet or outlet can

be approximated as m.(P/re). Then the energy equation becomes (Fig. 5–54)

Q (5–57)

#

net,in
W

#

shaft,net out

d

dt

(^)

CV
er dVa
out
m

a
P
r
eb
a
in
m

a
P
r
eb
m



Ac
r 1 V
S
r#n
S
2 dAc
Q

net,in
W

shaft,net out
d
dt
(^)

CV
er dV

CS
a
P
r
ebr 1 V
S
r#n
S
2 dA
°
The net rate of energy
transfer into a CV by
heat and work transfer
¢°
The time rate of
change of the energy
content of the CV
¢°
The net flow rate of
energy out of the control
surface by mass flow
¢
Q

net,in
W

shaft,net out
W

pressure,net out
d
dt
(^)

CV
er¬dV

CS
er 1 V
S
r#n
S
2 dA
dEsys
dt

d
dt
(^)

CV
er¬dV

CS
er 1 V
S
r#n
S 2 A
254 | Thermodynamics
=+brdV
B = Eb = eb = e
dBsys
dt
V
d
dt
CV

br(^ r^ ·^ n^ )^ dA
CS


=+erdV
dEsys
dt
V
d
dt
CV
er( (^) r · n ) dA
CS


→
→ →
FIGURE 5–53
The conservation of energy equation is
obtained by replacing an extensive
property Bin the Reynolds transport
theorem by energy Eand its associated
intensive property bby e(Ref. 3).
Wshaft, net,in
mout,
energyout
energyout
⋅
min,
energyin
⋅
min,
energyin
⋅
energyout m⋅out,
m⋅out,
⋅
Q⋅net,in
In
In
Out
Out
Out
Fixed
control
volume
FIGURE 5–54
In a typical engineering problem, the
control volume may contain many
inlets and outlets; energy flows in at
each inlet, and energy flows out at
each outlet. Energy also enters the
control volume through net heat
transfer and net shaft work.