Hussian Z., AbdullahM.Z., AlimuddinZ. Basic Fluid Mechanics and Hydraulic Machines
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50 Basic Fluids Mechanics and Hydraulic Machines
Fig. 2.23
To
determine the inlet angle PI' consider the inlet velocity triangle ABD. The
velocity
of
the fluid relative to vane at inlet
in
Vrl
must
be
tangential to the vane
and make
an
angle
PI
with the direction
of
motion
CD
VI
sin U
I
BC
(VI
cos
U,
-u)
Substituting V = 36 m/s' u =
15
mls and a = 30°
I'
I
36x
0.5
tanP
I
=
(36xO.866-15)
=1.13
:.
P
I
=48°
To
determine outlet angle P
2
,
as
V
2
has
no
component
in
the direction
of
motion,
outlet angle
P
2
in
given by
u
cos P
2
=
V,
but V
r2
= 0.85 V
ri
r2
and
from
the inlet triangle
CD
V
=--
rl
sinp,
Vrl
= 24.2 mls
36x
sin 30
sin 48
V
r2
= 0.85 x
Vrl
= 0.85 x 24.2 = 20.57
Fluid
Flow
51
u
15
cos
[32
= V
r2
= 20.57 = 0.729
[32
= 43°.
(b) Since the
jet
strikes series
of
vanes perhaps mounted on the periphery
of
a
wheel so that each vane moves on its place
is
taken by the next
in
the series,
the average length
of
the
jet
does not alter and whole flow from the nozzle
of
diameter d
is
deflected by the vanes.
Mass flow rate
of
the fluid from the nozzle
• 1t 1t
m = p x
"4
d
2
X
VI
= 1000 x
"4
(0.1)2 x
36=
282.8 kgls
Velocity component
of
VI
in
direction
of
X =
VI
cos a
l
V
IX
= 36 x 0.866 = 31.17 mls
V = V cos a = V cos 90° = 0
2x 2 2 2
Force on the direction
of
motion
.
F = m
(V
lx
- V
2
)
= 282.8 (31.17) = 8814 N
F
= 8814 N
E.2.13 Example
of
a stationary deflector
A deflector turns a sheet
of
water through an angle
of
30° as shown
in
Fig. 2.24.
What force
is
necessary to hold the
deflector
in
place
if
mass flow rate is
32 kg/s?
y
~Vl
L,
2
em
x 4
em
a = 300
Fig. 2.24
52
Basic Fluids Mechanics and Hydraulic Machines
Solution
Mass flow rate in = 32 kg/s
m
32
The velocity
VI
=
pxarea
= 1000xO.02xO.04 = 40 mls
According to Bemoullis equation
if
there is no pressure drop in the deflector then
velocity V
I = V
2'
The force produced in the x-direction
Fx
= Mass flow rate x change
of
velocity
. .
Fx = m
(V
I
-V
2
)
= m (VI - V
2
cos
a)
= 32 (40 -
40
x cos 30)
= 32 (40 - 34.6) = 172.8 N
Fy
= m
(VI
- V
2y
) = 32 (40 - 40 sin 30)
= 32 (40 - 40 x 0.5) = 640 N
E.2.14 Example
of
a moving deflector
A deflector shown in Fig. 2.25 moves to the right at 3 mls while the nozzle remains
stationary. Determine the force components needed to move the deflector. The
jet
velocity is 8 mls. The nozzle area is 2 cm x 40 cm.
2 em x
40
em
V
=8m
/
s~
=
, -
I p 1
v,
u=3m/s~
Fig. 2.25
Solution
Inlet triangle
of
velocity is a straight line as shown in Fig. 2.25(a).
The outlet triangle
of
velocity is shown in Fig. 2.25(b).
v, = 8 m/s
-----~
u = 3 m/s
'L
V" = 5 m/s
__
~
inlet diagram
(a)
Fluid Flow 53
outlet diagram
(b)
Fig. 2.25(a) & (b) Inlet and outlet velocity diagrams,
Bernoullis equation can
be
used to
show
that the velocity
of
the
sheet
V'I = V
r1
=
5 m/s as observed from the deilector.
Mass
flow rate is calculated
on
velocity V'I = 5 m/s
.
m =
pAV
rl
= 1000 x 0.02 x 0.4 x 5 =
40
kg/s
Force in the x-direction
F = Mass flow rate x
change
of
velocity
y
.
F = m [(V - V
)]
x
r1
x
r2x
.
= m [(V - V
cos
30
°
)]
rl x
r2
F =
40
[5
- 5 x 0.866] = 26.8 N
x
. .
F = - m V = -
In
V sin
30
= -
40
x 5 x 0.5
y r2y r2y
=
lOON
54 Basic Fluids Mechanics and Hydraulic Machines
Problems
P.2.1 A pitot-static tube is used to measure air velocity.
If
the manometer connected to
the instrument indicates the difference in pressure head between the tappings
of
4
mm
of
water, calculate the air velocity assuming the coefficients
of
pitot tube to
be unity. Density
of
air
l.2
Kg/m3.
P.2.2 A nozzle is attached to a pipe as shown in Fig. 2.26.
The
inside diameter
of
the
pipe is
100
mm
while the nozzle
jet
diameter is 50 mm.
If
the pressure at inlet is
500 kPa, determine the
jet
velocity.
Fig. 2.26
P.2.3 Water flows from upper reservoir to lower reservoir while passing through a turbine
as shown in Fig. 2.27. Determine the power
gene~ated
by the turbine.
1-4
<--
.........
....-
..........
-1
-----
.197 m
Fig. 2.27
P.2.4 A large artery in a person's body can be approximated by a tube
of
diameter 9
mm
and length 0.35 m. Assume that the blood has a viscosity
of
approximately
4
X 10-
3
and sp. gravity
of
1.0 and that the pressure at the beginning
of
artery
is equal to
120
mm
ofHg.
If
the flow were steady (it is not in reality) with velocity
equal to
0.2 mis, determine Reynold's number and pressure at the end
of
the
artery
if
it is oriented (a) horizontal (b) vertically up (flow upwards).
P.2.S Glycerine
of
viscosity 0.9 N-s/m
2
and density 1260 kg/m3 is pumped along a
horizontal pipe 6.5 m long,
of
diameter 0.01 m at a flow rate
of
1.81/min. Determine
Fluid Flow 55
Reynold's member and pressure drop using Hagen
Poiseullie equation given below
1281
f.l
Q
~p=
4
nd
where
1 = length
of
pipe
in
m
/l = viscosity in
N-s/m
2
Q = flow rate in m
3
/s
d = diameter
of
the pipe
in
m.
P.2.6 Water flows from a lake as shown in Fig. 2.28 at a rate
of
0.112 m
3
/s
. Is the
device inside the building a pump or a turbine? Explain and determine the power
of
the device. Neglect all minor losses and assume the friction factor f = 0.025
Iy
2
pipeline the hydraulic losses
is
calculated by the formula h
f
=
f-
- , where f
is
2gd
friction factor,
1 is length
of
pipe, Y the velocity and d the diameter
of
pipe.
Elevation 160 m
\7
"WJ,...
!!:"1
. . - /
//7////////;-
90mlong
0.12 m diameter
Fig. 2.28
P.2.7 A manometer positioned inside a cylinder
is
sh')wn in the Fig. 2.29 and reads
4 cms
of
water. Estimate the velocity assuming
in
viscid flow. The temperature
of
air is 20 °
C.
P.2.S Air flows into the atmosphere from a nozzle and strikes a vertical plate as shown
in Fig.
2.30. A horizontal force
of
12N
is
required to hold the plate in place.
Determine the reading on the pressure gauge. Assume the flow to be incompressible
and frictionless.
56
Basic Fluids Mechanics and Hydraulic Machines
Cy
linder
~
~
Fig. 2.29
p
_J
t
---.
Plate
Area 0.
01
m
2
A,ea[ml~
Fig. 2.30
P.2.9 For the flow shown in Fig. 2.31 estimate the pressure
PI
and velocity
VI'
if
V 2 = 20 mls and H = 5 cm.
V
1
----
~
Fig. 2.31
Fluid
Flow
57
P.2.IO
A wind tunnel shown
in
Fig. 2.32
is
designed to draw air from the atmosphere and
produce a
~elocity
of
100 m/s
in
the test section. The fan is located downstream
of
the test section. What pressure
is
to be expected
in
the test section
if
the
atmospheric pressure and temperature 92 kPa and
20
DC.
Fig. 2.32
P.2.II
A vacuum cleaner
is
capable
of
creating a vacuum
of2
kPa inside the hose. What
maximum velocity would be expected
in
the hose?
"This page is Intentionally Left Blank"
CHAPTER
- 3
Thermal
and
Hydropower
Plants
Overall
view
of
Two 640 MW each steam
power
station
at Maastrich Netherlands
built
with
Dutch company.
Courtesy:
Black - Durr, Germany.