fluid_mechanics

5.1 Conservation of Mass—The Continuity Equation 197
V plane =
971 km/hr
Control volume
m • fuel in
(a)
V plane =
971 km/hr
W 1 =
971 km/hr
Section (1)
Section (2)
V 2 = 1050 km/hr
W 2 = 1050 + 971 =
2021 km/hr
(b) F I G U R E E5.6
SOLUTION
The control volume, which moves with the airplane 1see Fig.
E5.6b2, surrounds the engine and its contents and includes all fluids
involved at an instant. The application of Eq. 5.16 to these
contents of the control volume yields
0 1flow relative to moving control
volume is considered steady on a
time-average basis2
Assuming one-dimensional flow, we evaluate the surface integral
in Eq. 1 and get
or
0
0t
cv
r dV cs
rW nˆ dA 0
m # fuel
in
m # fuel
in
r 1 A 1 W 1 r 2 A 2 W 2 0
r 2 A 2 W 2 r 1 A 1 W 1
We consider the intake velocity, , relative to the moving control
volume, as being equal in magnitude to the speed of the airplane,
971 kmhr. The exhaust velocity, W 2 , also needs to be
measured relative to the moving control volume. Since a fixed
W 1
(1)
(2)
observer noted that the exhaust gases were moving away from the
engine at a speed of 1050 kmhr, the speed of the exhaust gases
relative to the moving control volume, is determined as follows
by using Eq. 5.14
or
and is shown in Fig. E5.6b.
From Eq. 2,
m # fuel
in
m # fuel
in
V 2 W 2 V plane
W 2 ,
W 2 V 2 V plane 1050 kmhr 1971 kmhr2
2021 kmhr
10.515 kgm 3 210.558 m 2 212021 kmhr2 11000 mkm2
10.736 kgm 3 210.80 m 2 21971 kmhr211000 mkm2
1580,800 571,7002 kghr
9100 kghr
(Ans)
COMMENT Note that the fuel flowrate was obtained as the
difference of two large, nearly equal numbers. Precise values of
and are needed to obtain a modestly accurate value of m # W 2
W 1 fuel.
in
E XAMPLE 5.7
Conservation of Mass—Relative Velocity
GIVEN Water enters a rotating lawn sprinkler through its base
at the steady rate of 1000 ml/s as sketched in Fig. E5.7. The exit
area of each of the two nozzles is 30 mm 2 .
Control volume
A 2 = 30 mm 2
Section (2)
FIND Determine the average speed of the water leaving the
nozzle, relative to the nozzle, if
(a) the rotary sprinkler head is stationary,
(b) the sprinkler head rotates at 600 rpm, and
(c) the sprinkler head accelerates from 0 to 600 rpm.
Section (3)
Section (1)
Q
W 2
Sprinkler head
F I G U R E E5.7
Q = 1000 ml/s