1. Turbomachinery Lecture 4a Pi Theorem Pipe Flow Similarity
Flow, Head, Power Coefficients
Specific Speed

2. Introduction to Dimensional Analysis
Thus far course has shown elementary fluid mechanics – now one can appreciate Dimensional Analysis
Dimensional Analysis
Identifies significant parameters in a process not completely understood.
Useful in analyzing experimental data.
Permits investigation of full size machine by testing smaller version
Predicts consequences of off-design operation
Useful in preliminary design studies for sizing machine for optimal performance
Useful in sizing pumps & blowers based on performance maps
Geometric similarity: assumes all linear dimensions are in constant proportion, all angular dimensions are same

3. Dimensional Analysis Buckingham -Theorem
Basic Premise
Physical process involving dimensional parameters, Q's and f(Q) is unknown.
Q1 = f(Q2,Q3,...Qn)
Group the n variables into a smaller number of dimensionless
groups, each having 2 or more variables
Physical process can be expressed as:
1 = g(2, 3,...n-k)

4. Dimensional Analysis Buckingham -Theorem
Each  is a product of the primary variables, Q's raised to various exponents so that 's are dimensionless.
where
n = no. primary variables
k = no. physical dimensions [L,M,T]
n-k = no. 's

5. Dimensional Analysis Dimensional analysis requires
postulation of proper primary variables
judgement, foresight, good luck
Dimensional analysis cannot
give form of 1 = g(2, 2,...n-k)
prevent omission of significant Q’s
exclude an insignificant Q’s

6. Dimensional Analysis Basic Units Mass M Length L Time T
Force is related to basic units by F=ma
Force ML/T2

7. Example: Pressure Drop in Pipe P = f(V,,,l,d,)
Pick V, , d as the 3 Q’s which will be used with each of the remaining
Q’s to form the = 4  terms.
Pick M, L, T as the 3 primary dimensions
Result

8. Example: Pressure Drop in Pipe P = f(V,,,l,d,)
Therefore
Moody Diagram
turbulent
laminar
smooth
What happens when there are several length scales: D, L, …?

9. Dimensional Analysis of Turbomachines Primary Variables - Q’s

10. Background: Head, Power, and Viscosity Q’s
Head - work per unit mass
- fluid dynamic equivalent to enthalpy
Recalling Gibbs Equation:
So head in "feet" is clearly erroneous.

11. Background: Head, Power, and Viscosity Q’s
Power - Work per unit time
- Mass Flow Rate Work per unit Mass

12. Background: Head, Power, and Viscosity Q’s
Newtonian Fluid: Shear stress  Velocity gradient
Viscosity is  - with units:

13. Dimensional Analysis of Turbomachines
Since there are 10 Q's & 3 Dimensions we can identify 7 's.
Each  contains 4 Q's, Q1, Q2, Q3, and Qn.
The parameters chosen for 1, 2 & 3 were chosen carefully.
Task is to find exponents of primary variables to make dimensionless groups.

14. Dimensional Analysis of Turbomachines
The system of equations is:

15. Dimensional Analysis of Turbomachines
Each  has 3 linear equations:
M 0a + 0b + 1c +0 = 0  c = 0
L 0a + 1b - 3c + 3 = 0  b = -3
T -a + 0b + 0c - 1 =0  a = -1

16. Dimensional Analysis of Turbomachines
M 0a + 0b + 1c + 0 = 0  c = 0
L 0a + 1b - 3c + 2 =  b = -2
T -1a + 0b + 0c - 2 = 0  a = -2

17. Dimensional Analysis of Turbomachines
Aside: What is meaning of H=head?
Hydraulic engineers express pressure in terms of head
Static pressure at any point in a liquid at rest is, relative to pressure acting on free surface, proportional to vertical distance from point to free surface.

18. Dimensional Analysis of Turbomachines
M 0a + 0b + 1c + 1 = 0  c = -1
L 0a + 1b - 3c + 2 = 0  b = -5
T -1a + 0b + 0c - 3 = 0  a = -3

19. Dimensional Analysis of Turbomachines
M 0a + 0b + 1c + 1 = 0  c = -1
L 0a + 1b - 3c - 1 = 0  b = -2
T -1a + 0b + 0c - 1 = 0  a = -1

20. Dimensional Analysis of Turbomachines
M 0a + 0b + 1c + 0 = 0  c = 0
L 0a + 1b - 3c + 1 = 0  b = -1
T -1a + 0b + 0c - 1 = 0  a = -1

21. Dimensional Analysis of Turbomachines
M 0a + 0b + 1c + 0 = 0  c =
L 0a + 1b - 3c + 1 = 0  b = -1
T -1a + 0b + 0c + 0 = 0  a = 0

22. Turbomachinery Non-Dimensional Parameters
Derived 7 s from 10 Qs in first part of class
Now ready to
- develop physical significance of s
- relate to traditional parameters
- discuss general similitude

23. Flow Coefficient

24. Head Coefficient

25. Hydraulic Pump Performance
Geometric similarity:
all linear dimensions are in constant proportion,
all angular dimensions are same
Performance curves are invariant if no flow separation or cavitation
BEP= best efficiency point [max] or operating point

26. Head Curve

27. Example: Changing Level of Performance for a Given Design
Pressure rise

28. Example: Changing Level of Performance for a Given Design Same fan but different size / speed

29. Scaling for Performance[limited by M, Re effects]

30. Example

31. Example

32. Define New Variable: Vary More Than One Parameter
later

33. Similarity – Compressible Flow - Engine

34. Similarity – Compressible Flow

35. Nondimensional Parameters