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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
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