Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

5–2 ■ FLOW WORK AND THE ENERGY

OF A FLOWING FLUID

Unlike closed systems, control volumes involve mass flow across their

boundaries, and some work is required to push the mass into or out of the

control volume. This work is known as the flow work,or flow energy,and

is necessary for maintaining a continuous flow through a control volume.

To obtain a relation for flow work, consider a fluid element of volume V

as shown in Fig. 5–11. The fluid immediately upstream forces this fluid ele-

ment to enter the control volume; thus, it can be regarded as an imaginary

piston. The fluid element can be chosen to be sufficiently small so that it

has uniform properties throughout.

If the fluid pressure is Pand the cross-sectional area of the fluid element

is A(Fig. 5–12), the force applied on the fluid element by the imaginary

piston is

FPA (5–22)

226 | Thermodynamics

where AjetpD^2 jet/4 is the cross-sectional area of the jet, which is constant.

Noting that the density of water is constant, the mass of water in the tank at

any time is

(3)

where AtankpD^2 tank/4 is the base area of the cylindrical tank. Substituting

Eqs. 2 and 3 into the mass balance relation (Eq. 1) gives

Canceling the densities and other common terms and separating the vari-

ables give

Integrating from t0 at which hh 0 to ttat which hh 2 gives

Substituting, the time of discharge is

Therefore, half of the tank is emptied in 12.6 min after the discharge hole is

unplugged.

Discussion Using the same relation with h 2 0 gives t43.1 min for the

discharge of the entire amount of water in the tank. Therefore, emptying

the bottom half of the tank takes much longer than emptying the top half.

This is due to the decrease in the average discharge velocity of water with

decreasing h.

t

2 4 ft
 2 2 ft

2 32.2/2 ft/s^2

¬ a

3 12 in

0.5 in

b

2

757 s12.6 min

t

0

dt
¬

Dtank^2

Djet^222 g

(^)

h 2
h 0
dh
2 h
S t
2 h 0
2 h 2
2 g/2
¬a
Dtank
Djet
b
2
dt
Dtank^2
Djet^2
dh
22 gh

r 22 ghAjet
d(rAtankh)
dt
→
r 22 gh(pDjet^2 /4)
r(pDtank^2 /4)dh
dt
mCVrVrAtankh
Imaginary
piston
CV
A
V
P
m
L
F
FIGURE 5–11
Schematic for flow work.
SEE TUTORIAL CH. 5, SEC. 2 ON THE DVD.
INTERACTIVE
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