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.

wheren = no. primary variablesk = no. physical dimensions [L,M,T]n-k = no. 's

1 1 1 1

1 1 2 3a b c x

nQ Q Q Q

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 7 - 3 = 4 terms. Pick M, L, T as the 3 primary dimensions

Q1 Velocity V LT-1 Q2 Density ML-3 Q3 Pipe diameter D L Q4 Pressure drop P ML-1T-2

1 VabdcP (L/T)a(M/L3)b(L)c(M/LT2) Mb+1La-3b+c-1T-a-2 2 Vabdc (L/T)a(M/L3)b(L)c (M/LT) Mb+1La-3b+c-1T-a-1 3 l (L/T)a(M/L3)b(L)c L MbLa-3b+cT-a 4 (L/T)a(M/L3)b(L)c L MbLa-3b+cT-a

Result

121

2

P

V

1

2 ReVd

3 d

4 d

8

Example: Pressure Drop in Pipe P = f(V,,,l,d,)

Vd

Re

d

2( , , )

12

P VdCp f

d dV

Therefore

dV

P2

21

laminar

turbulent

smooth

Moody Diagram

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.

dP dPTdS dh dH

3

2

f f

m m

f f

m m

ft lb ft lbBTUJdh

BTU lb lb

lb ft lbdP ft

ft lb lb

2 2 2

3 3 2

/ / /

/ /

P F A M L T L L

M L M L T

p

But if Hg

units of H in feet

11

Background: Head, Power, and Viscosity Q’s

• Power - Work per unit time

- Mass Flow Rate Work per unit Mass

3 2 2

3 2 3

dW dx pPower pA p Q Q

dt dt

Power Q H

M L L MLPower

L T T T

12

Background: Head, Power, and Viscosity Q’s

• Viscosity

Newtonian Fluid: Shear stress Velocity gradient

• Viscosity is - with units:

0

o

y

u

y

20

2 2

//

F L FT ML Mu L T L FT LTy L

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:

107

37

27

17

96

36

26

16

86

35

25

15

74

34

24

14

63

33

23

13

52

32

22

12

41

31

21

11

QQQQ

QQQQ

QQQQ

QQQQ

QQQQ

QQQQ

QQQQ

cba

cba

cba

cba

cba

cba

cba

1

2

3

Q N

Q D

Q

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

0001331

1

LTMTLLMLT

QDNcba

cba

1 3

QFlowcoefficient

ND

16

Dimensional Analysis of Turbomachines

M 0a + 0b + 1c + 0 = 0 c = 0

L 0a + 1b - 3c + 2 = 0 b = -2

T -1a + 0b + 0c - 2 = 0 a = -2

0002231

2

LTMTLLMLT

HDNcba

cba

2 2 2

gHEnergy transfer or head coefficient

N D

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.

0pH head z metersg

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

0003231

3

LTMTLMLMLT

PDNcba

cba

3 3 5

PPower coefficient

N D

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

0001131

4

LTMTLMLMLT

DNcba

cba

4 2

1 = =

ReND UD

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

000131

5

LTMTLLMLT

aDNcba

cba

5 5or more conventionally a ND

ND a

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

00031

6

LTMLLMLT

DNcba

cba

6

7Similarly

D

k

D

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

1 3

2

1 3

x

x x

Q

ND

U ND A D Q C A

C A CQ

ND UA U

[ ]

[ ]R

Nomenclature

U wheel speed

C V velocity absolute frame

W V velocity relative frame

24

Head Coefficient

2 2 2

2 2 2 2

02 2 2 2 2

gH

N DgH h

N D UpgH

N D N D

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

Head Curve

27

Example: Changing Level of Performance for a Given Design

1 3

03 2 2

[ ]

0 100 0 1500 0 100

.

80 1.0 90

1.2 102.6 1.2 / [ ]

800.452 0.055

177

o

o

Given test fan with operating range in air

Q mps p kPa P kW

with design point properly matched

Given

Q mps p kPa P kW

D m N s kg m air

Then

pQ P

ND N D

3 50.0279

0.89T

N D

Net power input to flow QgHEfficiency

Power input to shaft P

Pressure rise

28

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

1 3

1 1

3 3

3 3

2 20

80 1.0 90

1.2 1003 1.2 / [ ]

30 . 2.5 1800 188.5 2 / 60 sec

0.074 / 0.0023 /

0.452[2945] 1325 / 79,252

o

Old fan

Q mps p kPa P kW

D m N s kg m air

New fan

D in ft N rpm s RPM

g lbf ft slugs ft

Q ND ft s cfm

p N D

2

3 5 6

0.55 510.8 28.1 / 0.20

0.0279 1.504 41,950 / 76.3 56

0.89T

lbf ft psi

P N D lbf ft s HP kW

29

Scaling for Performance[limited by M, Re effects]

3 3

3

3

pm

m m p p

p pp m

m m

T p T m

m model p prototype

QQ

N D N D

N DQ Q fan or pump law

N D

30

Example

3

3

3

1.5 0.696 55 /

[ 1030 / ],

98 4

10 , 1000

[ 998 / ]

[1030][9.81[1.5][55][0.696] 579

p T p p

p

m m

p T

Consider turbine with

H m Q m s

operating in warm sea water kg m with

N rpm D m

Devise test model with P kW N rpm in fresh

water kg m

P gHQ

1/5 1/53 31030 10 98

0.111998 579 1000

0.444

p pm m

p m p m

m

kW

ND P

D P N

D m

31

Example

2 2 2 2

3 33

8 9

0.446 10001.9

4.0 99

0.446 100055 0.770 /

4.0 99

Re 1 10 Re 1 10

pmm p

p m

m mm p

p p

m p

NDH H m

D N

D NQ Q m s

D N

32

Define New Variable: Vary More Than One Parameter

2 20

3

1/ 2 1/ 2

3/ 4 3/ 4

0

s

p N D defines N

Q ND defines N

NQN Specific speed

p

Used in Cordier diagram

later

33

Similarity – Compressible Flow - Engine

02 01 01 02 01 02

02 01 01 02

( , , , , , , , , )

( , , , , , , )

p f D N m p T T

p RT

p f D N m p RT RT

3 2 22 00 0 0 0 0

0

0022

00 0

2

0 00 0 022 2 2 2

0 0 0 0 01

p p p

Flow coefficient

m m m maND a D RT DND DND

mRT m RT

p Dp RT D

Head coefficient

c T c T cp T TND

N D N D a RT T T

0

0 00

p

ph c T

34

Similarity – Compressible Flow

02 01 01 02

0102 02

01 01 01 01

( , , , , , , )

,Re, ,

p f D N m p RT RT

m Tp T Nf

p T p T

2

0 00 0 022 2 2 2

0 0 0 0 01

0 00 02 2 2 2

0 0 0 0

1

p p p

p p

Head coefficient

c T c T cp T TND

N D N D a RT T T

also

c T c Tp TND ND

N D N D a ND NDR T T

01

01

m T

p

02

01

p

p

01

N

T

35

Nondimensional Parameters