Mechanics of Fluids

−
Energy loss to friction/time
Mass/time
= pC
ϱ
General energy equation for steady flow of any fluid 105
1
+
2 u2 C + gzC
Energy added by pump
∴
Time
= Mass
�
pC − pA
+
Time ϱ
u2 C − u2 �
A
+ g(zC − zA)
2
+ Energy loss to friction
Time
= Volume
�
pC − pA +
Time
1
2 ϱ
�
u 2 C − u2 �
�
A + ϱg(zC − zA)
+ Energy loss to friction
Time
= π
4 (0.225 m)2 1.35 m · s −1
�
35 000 −[13560 × 9.81(−0.150)]
+ 1
2 × 1000(3.0382 − 1.35 2 ) + 1000 × 9.81 × 5.6
�
N · m −2
+ 2.5 kW
= π
4 (0.225)2 1.35{35 000 + 19 950 + 3702 + 54 900} N · m · s −1
+ 2.5 kW = 8.6 kW
∴ Overall efficiency of pump = 8.6/12.7 = 67.7%
Notice that pA, a vacuum pressure, is negative. ✷
3.5.5 Energy transformation in a constant-density fluid
The concept of head, that is, energy divided by weight of a constant-density
fluid, is of great value in allowing a geometrical representation of energy
changes. We recall from Section 2.2.1 that steady pressures not greatly in
excess of atmospheric pressure may be measured by the rise of liquid in a
glass tube. We may therefore imagine, for example, the system depicted in
Fig. 3.7, in which such piezometer tubes are connected at certain points to
a pipe conveying liquid from a large reservoir. At a point where the (gauge)
pressure in the pipe is p the liquid will rise in the piezometer tube to a
height p/ϱg.
At points in the reservoir far from the outlet the velocity of the liquid is
so small as to be negligible. At such a point 1 at a depth h1 below the free
surface, the pressure is therefore given by the hydrostatic relation p1 = ϱgh1,
so the sum of the three terms in Bernoulli’s expression is
ϱgh1/ϱg + 0 2 /2g + z1 = h1 + z1 = H