Fluid mechanics Fluid mechanics Fluid mechanics Fluid mechanics

Fluid mechanics Fluid mechanics Fluid mechanics

1. CE1120-Elementary Fluid Mechanics
and Thermodynamics
Dr. LashithaRathnayake
Department of Civil Engineering,
University of Peradeniya,
[email protected]
Lecture4
1

2. Problem2
2
??????
5
=300 ????????????
??????
5
=200 ????????????
??????=85 ??????/??????
1 2
CS
1
CS
2
CS
3
CV
A horizontalpipeline carrying oil of ρ = 685 kg/m
3
at Q = 85 l/s reduces in diameter using
a standard, symmetrical taper section. Given, gauge pressures, p
1
= 10 kPa, p
2
= 8 kPa, and
D
1
= 300 mm, D
2
= 200 mm, find the force exerted by oil on the reducer section, assuming
steady 1D flow.
Select so that fluid crosses control
surfaces at right angles
x
y

3. Forces
•Body forces:
•Flow is horizontal, so weight of fluid acting in z-direction.•Surface forces:
•CS1: p1 = const, pressure force = p
1
A
1
→
•CS2: p2 = const, pressure force = p
2
A
2
←
•CS3: pressure (p
w
) and shear (τ
w
) forces (unknown)
•Let F= force exerted by reducer section on the fluid in CV(due to Pressure and shear)
3
CS
1
CS
2
CS
3
CV

4. •No change of momentum in y-direction, so we expect F to act along x-axis.
•Applying steady flow force-mom eqnfor fluid body in CV
4
∑??????=??????̇
???
−??????̇
??
∑??????=????????????(??????
6
-??????
5
)
→ ??????
5
??????
5
−??????
6
??????
6
−??????=????????????(??????
6
-??????
5
)

5. Problem 3
•A vertical pipeline with a reducer section
and flow downwards (85 l/s).
•p
1
= 10 kPa, p
2
= 8 kPa, ρ = 685 kg/m
3
•D
1
= 300 mm, D
2
= 200 mm,
•Assume volume of the reducer section is 5
liters
5
??????
5
=300 ????????????
??????
5
=200 ????????????
??????=85 ??????/??????
1
2
CS
1
CS
2
CS
3
CV
W

6. 6
Elemental CV

7. 7
→?F
q
=??????Q(V
6?
−V
5?
)
−??????????????????+??????A+
??
6
dA− ??????????????????????????????=??????????????????(??????+????????????−??????)
????????????????????????=????????????/????????????
Force due to weight
= ??????????????????????????????????????????????????????= ??????????????????????????????
Subtract uniform value p
from all sides and apply
force-mom eqnin
streamwise (s) direction
??????
??????
??????
??????+????????????
??????+????????????
??????+????????????
1
2
??????
????
=??????+????????????/2??????=0 (????????????????????????????????????????????????)
Elemental CV

8. 8
→?F
v
=??????Q(V
6?
−V
5?
)
????????????
????????????
+
??????????????????
??????
+ ???????????? =0Dividing by ρg,
????????????+????????????????????????+????????????????????????=0
Euler’s Equation
−????????????.??????− ??????????????????????????????=??????????????????????????????Neglecting products of small quantities
−??????????????????+??????A+
??
6
dA− ??????????????????????????????=??????????????????(??????+????????????−??????)
Swiss mathematician Leonhard Euler (1707–83).

9. 9
????????????
????????????
+
??????????????????
??????
+ ???????????? =0
?
????????????
????????????
6
5
+ ?
??????????????????
??????
6
5
+ ?????????????
6
5
=0
For incompressible fluid flow (ρ=const) ), integrating between any
two points (1) & (2) on a streamline
(1)
(2)

10. 10
?
?
+
?
.
6?
+ ?????? =??????????????????????????????, along a streamline
Bernoulli’s Equation (B.E.)
?
????????????
????????????
6
5
+ ?
??????????????????
??????
6
5
+ ?????????????
6
5
=0
This result is usually known as Bernoulli’s equation or the Bernoulli equation in honour
of a Swiss mathematician, Daniel Bernoulli (1700–82) who in 1738 published one of the
first books on fluid flow.
??????
6
−??????
5
????????????
+
??????
6
6
−??????
5
6
2??????
+??????
6
−??????
5
=0
??????
5
????????????
+
??????
5
6
2??????
+??????
5
=
??????
6
????????????
+
??????
6
6
2??????
+??????
6

11. Bernoulli’s Equation (B.E.)
11
??????
????????????
+
??????
6
2??????
+ ?????? =??????????????????????????????
Along a
streamline
Assumptions:
-Steady flow
-Inviscid (frictionless) fluid flow
-Incompressible fluid
-No energy added (eg, pump) or removed (eg, turbine) between pts. (1) & (2)

12. Note:
12
Each term in the B.E. has dimensions [L] or units ‘m’ and gives energy per unit weight of fluid.
??????
????????????
+
??????
6
2??????
+ ?????? =??????????????????????????????
pressure energy
per unit weight
kinetic energy
per unit weight
potential energy
per unit weight
energy per unit weight->
‘Head’ = Height of
equivalent static
column of that fluid
Pressure HeadVelocity Head
Elevation Head
(Potential Head)
B.E. → Total energy per unit weight is constant along a streamline.
Conservation of Energy

13. 13
Total Energy (Head) Line (is horizontal for
frictionless flow)
z = 0 (datum)
Piezometric Head Line (Hydraulic gradient)
??????
????????????

??????+
??????
????????????
??????
??????
6
2??????

14. Problem4
•Steady flow of a liquid of ρ = 685 kg/m
3
at Q = 85 l/s through a level
reducing pipe. Given, D
1
= 300 mm, D
2
= 200 mm, and p
1
= 10 kPa,
find p
2
.
14
??????
5
=300 ????????????
??????
5
=200 ????????????
??????=85 ??????/??????
1 2
CS
1
CS
2
CS
3
CV

15. •For mass continuity:
??????=??????
5
??????
5
=??????
6
??????
6
=0.085
V
1
=1.2 m/s
V
2
=2.7 m/s
15
??????
5
=300 ????????????
??????
5
=200 ????????????
??????=85 ??????/??????
1 2
CS
1
CS
2
CS
3
CV
Assume fluid is inviscid (so, frictionless) and incompressible
Applying B.E. between pts (1) and (2):
??????
5
????????????
+
??????
5
6
2??????
+??????
5
=
??????
6
????????????
+
??????
6
6
2??????
+??????
6

16. 16
??????
5
=300 ????????????
??????
5
=200 ????????????
??????=85 ??????/??????
1 2
CS
1
CS
2
CS
3
CV
Assume fluid is inviscid (so,
frictionless) and incompressible
Applying B.E. between pts (1)
and (2):
??????
5
????????????
+
??????
5
6
2??????
+??????
5
=
??????
6
????????????
+
??????
6
6
2??????
+??????
6
??????
5
????????????
+
??????
5
6
2??????
=
??????
6
????????????
+
??????
6
6
2??????
Z
1
=Z
2
, level pipe
??????
6
=??????
5
+
5
6
??????(??????
5
6
−??????
6
6
)= 10×10
3
+ 0.5×685×(1.2
2
-2.7
2
)= 8kPa

17. Problem5
A vertical pipeline with a reducer section and flow downwards.
Given, D
1
= 300 mm, D
2
= 200 mm, and p
1
= 10 kPa, find p
2
.
From previous example V
1
= 1.2 m/s, V
2
= 2.7 m/s
17
??????
5
=300 ????????????
??????
5
=200 ????????????
??????=85 ??????/??????
1
2
CS
1
CS
2
CS
3
CV
W
Applying B. E. between (1) & (2):
Assume fluid is inviscid and incompressible
??????
5
????????????
+
??????
5
6
2??????
+??????
5
=
??????
6
????????????
+
??????
6
6
2??????
+??????
6
??????
6
=??????
5
+
5
6
??????(??????
5
6
−??????
6
6
)+????????????(??????
5
−??????
6
)
= 10×10
3
+ 0.5×680×(1.2
2
-2.7
2
)+685×9.81(0.15)= 9kPa
150 mm

18. 18