Unit 6 Dimensional Analysis.pdf Unit 5 Open Channel flow

Unit 5 Open Channel flowUnit 5 Open Channel flowUnit 5 Open Channel flowUnit 5 Open Channel flowUnit 5 Open Channel flowUnit 5 Open Channel flowUnit 5 Open Channel flowUnit 5 Open Channel flow

1. CE 205 : ENGG. FLUID MECHANICS Unit 6
Dimensional Analysis

2. CE 205 : ENGG. FLUID MECHANICS Unit 6
Dimensional Analysis
Dimensional Analysis is a method to analyze a physical phenomenon
When there is an equation showing a relation between a number of physical
quantities involved in a certain phenomenon, the two sides of the equation
must be dimensionally balanced.
Fundamental Unit and Derived Unit :
Units of Length (L), Mass (M), and Time (T) are called fundamental Unit
Those units which are based on more than one fundamental unit is called
derived unit Eg: Velocity (LT
-1
), Density (ML
-3
) etc.

3. CE 205 : ENGG. FLUID MECHANICS Unit 6
Dimensional Analysis
Dimensions of physical quantities commonly used in Fluid Mechanics
Physical quantity Dimensions
Mass M
Length L
Time T
Temperature T
Area L
2
Volume L
3
Velocity LT
-1
Speed LT
-1
Acceleration LT
-2

4. CE 205 : ENGG. FLUID MECHANICS Unit 6
Dimensional Analysis
Dimensions of physical quantities commonly used in Fluid Mechanics
Physical quantity Dimensions
Density ML
-3
Momentum MLT
-1
Force MLT
-2
Pressure ML
-1
T
-2
Angular Velocity T
-1
Angular accelerationT
-2
Weight MLT
-2
Discharge L
3
T
-1
Specific Weight ML
-2
T
-2
Stress ML
-1
T
-2

5. CE 205 : ENGG. FLUID MECHANICS Unit 6
Dimensional Analysis
Dimensions of physical quantities commonly used in Fluid Mechanics
Physical quantity Dimensions
Work ML
2
T
-2
Energy ML
2
T
-2
Power ML
2
T
-3
Torque ML
2
T
-2
Young’s Modulus ML
-1
T
2
DynamicViscosity ML
-1
T
-1
Kinematic Viscosity L
2
T
-1
Surface Tension MT
-2

6. CE 205 : ENGG. FLUID MECHANICS Unit 6
Dimensional Analysis
Law of Dimensional Homogeneity
If both sides of an equation indicate the same physical quantity, the
dimensions of the left hand side of the equation must be equal to the
dimensions of the right hand side of the equation
Example :
Check the dimensional homogeneity of the equation p = h

7. CE 205 : ENGG. FLUID MECHANICS Unit 6
Dimensional Analysis
Exercise 6.1:
Determine the dimensions of
(i) Momentum (ii) Power (iii) Dynamic viscosity (iv) Kinematic viscosity
Exercise 6.2:
Check the dimensional homogeneity of the following equations
(i)P = A x, P being the pressure force
(ii)v = C
v2gh

8. Check the dimensional homogeneity of the following equations.
Q: Discharge
C: Constant factor
H: Total head
d: diameter of pipe
g : acceleration due to gravity
SOLUTION
L
3
/T = 1 x L [1 x L/T
2
]
1/2
L
3/2
L
3
/T = L
(1+1/2+3/2)
/ T
2 x 1/2
L
3
/T = L
3
/T
Homogeneous equation

9. Check the dimensional homogeneity of the following equation.
h
f
: head loss due to friction
f: friction factor
l: length of pipe
v: velocity of flow in pipe
g: acceleration due to gravity
d: diameter of pipe
SOLUTION
LHS:
hf= Length
= L
RHS:
= (1 x 1 x length x velocity
2
)/ (1 x acceleration x length)
= (1 x 1 x L x L
2
/T
2
)/ (1 x L/T
2
L)
= L
3
/T
2
/ L
2
/T
2
= L, Therefore, equation has dimensional homogeneity

10. CE 205 : ENGG. FLUID MECHANICS Unit 6
Dimensional Analysis
Dimensional Analysis by Indicial Method
In any physical phenomena, the dependency of a parameter over another
is determined by dimensional analysis.
There are two methods for dimensional Analysis
1.Indicial Method (Rayleigh’s Method)
2.Buckingham’s -theorem

11. Exercise 6.3a:
In an pendulum the frequency of period is function of length (l) and value
of gravity (g). Determine a formula for this frequency

12. CE 205 : ENGG. FLUID MECHANICS Unit 6
Dimensional Analysis
Exercise 6.3:
Force acting on the propeller of an air craft depends upon the variables
v (velocity), (Dynamic viscosity ), (Mass Density ), D (Diameter of the
propeller ) and N (Speed of rotation of the propeller). Determine a formula
for this force
Exercise 6.4:
The efficiency of a fan depends on the density (), dynamic viscosity ()
of the fluid, the angular velocity (), diameter D of the rotor and the
discharge Q. Express in terms of dimensionless parameters.

13. Models and Similitude
CE 205 : ENGG. FLUID MECHANICS Unit 6

14. Models and Similitude
Unit 6
Studies are usually made on small scaled models to
predict the behavior of hydraulic structures.
Model studies are done based on the theory of
hydraulic similitude
Theory of hydraulic similitude involves the study of the
similarity of the hydraulic relationships between the
prototype and the model.
All significant hydraulic structures are now designed and
built after certain preliminary model studies have been
completed
CE 205 : ENGG. FLUID MECHANICS

15. Models and Similitude
Model studies are conducted for the following purposes
1.To determine the discharge coefficient of a large
measurement structure such as overflow spillways or a
weir
2.To develop an effective method for energy dissipation at
the outlet of a hydraulic structure.
3.To reduce energy loss at an intake structure or at a
transition section.
4.To develop an efficient, economic spillway or other type of
flood releasing structure for reservoir.
CE 205 : ENGG. FLUID MECHANICS Unit 6

16. Models and Similitude
Model studies are conducted for the following purposes
5.To determine an average time of travel in a temperature
control structures, for example in a cooling pond at a
power plant.
6.To establish the best cross section, location, and
dimensions of various structural components such as break
water, the docks, and locks in harbor and water way
design.
7.To determine the dynamic behaviors of a floating, semi –
immersibleand bottom structures in transportation or
offshore facilities
CE 205 : ENGG. FLUID MECHANICS Unit 6

17. Models and Similitude
Model studies are conducted for the following purposes
8.To determine the pattern of a flood wave travels through a
river channel.
9.To determine the effect of artificial structures such as
bends, levees, dikes, jetties and training walls on the
sedimentation movements in the channel reach.
10.To determine the direction and force of natural and
anthropogenic currents in channels or harbors and their
effect on navigation and marine life
CE 205 : ENGG. FLUID MECHANICS Unit 6

18. Models and Similitude
Dimensions of physical Quantities commonly used in Hydraulic Engineering.
CE 205 : ENGG. FLUID MECHANICS Unit 6

19. Models and Similitude
Principles of Hydraulic Similitude
Similarity between hydraulic models and prototypes
may be achieved in three basic forms
Geometric Similarity
Kinematic Similarity
Dynamic Similarity
CE 205 : ENGG. FLUID MECHANICS Unit 6

20. Models and Similitude
Geometric Similarity Similarity of form
Geometric Similarity is achieved by maintaining a fixed ratio for
all corresponding lengths between the model and prototype
The physical quantities involved in geometric similarity are
1. Length (L)
2. Area (A)
3. Volume ( Vol)
To keep the corresponding length in the
prototype (L
p) and the model (L
m), a constant
ratio (L
r) should exist between them.32
;.........;.....
r
m
p
r
m
p
r
m
p
L
Vol
Vol
L
A
A
L
L
L

CE 205 : ENGG. FLUID MECHANICS Unit 6

21. Models and Similitude
Exercise 6.5
A geometrically similar open channel model is constructed
with a 5:1 scale. If the model measures a discharge of 7.07 cfs,
what is the corresponding discharge in the prototype ?
CE 205 : ENGG. FLUID MECHANICS Unit 6

22. Models and Similitude
Kinematic Similarity Similarity in motion
Kinematic similarity between a model and the prototype is
attained if the homologous moving particles have the same
velocity ratio along geometrically similar paths.
The kinematic similarity involves
1. Scale of time (T
r)
2. Scale of lengths (L
r)m
p
r
T
T
T
CE 205 : ENGG. FLUID MECHANICS Unit 6

23. Models and Similitudem
p
V
V mm
pp
TL
TL
=mp
mp
TT
LL =r
r
T
L =m
p
a
a 2
2
mm
pp
TL
TL
=22
mp
mp
TT
LL =2
r
r
T
L =m
p
Q
Q mm
pp
TL
TL
3
3
=
mp
mp
TT
LL
3 =r
r
T
L
3 =
Velocity Ratio
Ratio of acceleration
Ratio of discharge
CE 205 : ENGG. FLUID MECHANICS Unit 6

24. Models and Similitude
Exercise 6.6
A 10 : 1 scale model is constructed to study the flow movement
in a cooling pond. The designed discharge from the power plant
is 200m
3
/sec, and the model can accommodate a maximum
flow rate of 0.1 m
3
/ sec. What is the appropriate time ratio.
CE 205 : ENGG. FLUID MECHANICS Unit 6

25. Models and Similitude
Dynamic Similarity
Similarity in forces
involved in motion
Dynamic similarity between a model and the prototype is attained
if the ratio of homologous forces is kept at a constant value.m
p
r
F
F
F
CE 205 : ENGG. FLUID MECHANICS Unit 6

26. Models and Similitudem
p
F
F mm
pp
aM
aM
=mmm
ppp
aVol
aVol
..
..

 =2
2
3
3
..
mm
pp
m
p
m
p
TL
TL
L
L

 =224
4
1
..
mpm
p
m
p
TTL
L


=24
..

rrrTL =
Force ratio
CE 205 : ENGG. FLUID MECHANICS Unit 6

27. Models and Similitudem
p
M
M mm
pp
aF
aF
=mp
mp
aa
FF =12
..

rrrLTF =
Mass ratiom
p
W
W mm
pp
LF
LF
.
.
= =rrLF.
Ratio of Worksm
p
P
P mm
pp
TW
TW
= =mpm
p
TTW
W1
.
Power Ratio
=r
rr
T
LF.
CE 205 : ENGG. FLUID MECHANICS Unit 6

28. Models and Similitude
Exercise 6.7
A 59700 watt pump is used to power a water supply system.
The model of the same material, constructed to study the
system has an 8 : 1 scale. If the velocity ratio is 2 : 1, what is
the power needed for the model pump ?.
CE 205 : ENGG. FLUID MECHANICS Unit 6