pengetahuan tentang sentrifugal

Main Topics
•
•
•
•
•

Introduction
Impeller Design
Diffuser Design
Performance
Examples

2

Introduction
• Slightly less efficient than axial-flow compressors
• Easier to manufacture
• Single stage can produce a pressure ration of 5 times that of a
single stage axial-flow compressor
• Application: ground-vehicle, power plants, auxiliary power units
• Similar parts as a pump, i.e. the impeller, the diffuser, and the
volute
• Main difference: enthalpy in place of pressure-head term
• Static enthalpy (h) and total (stagnation) enthalpy (ho)

4

EULER EQUATION
Torque T = m (Cθ2r2 – Cθ1r1)
Power P = Tω
= m (U2Cθ2 – U1Cθ1)

5

Introduction
• Isentropic Stagnation State
2

V
h0 = h +
2

7

Introduction
• For an ideal gas with constant specific heat
2

V
h0 = h +
2

kRT  T0 
V = 2( h0 − h ) = 2C p ( T0 − T ) = 2
 − 1
k −1 T 
2
c = kRT
2

8

Introduction
• For an ideal gas with constant specific heat
2

2c  T0 
V =
 − 1
k −1 T 
2
V
2  T0 
2
=M =
 − 1
2
c
k −1 T 
T0
k −1 2
= 1+
M
T
2
2

9

Introduction
• For an isentropic process

 T0 
 
T 

k ( k −1 )

 T0 
 
T 

p0
= ,
p

p0  ( k − 1) 2 
= 1 +
M 
p 
2


ρ 0  ( k − 1) 2 
= 1 +
M 
ρ 
2


1 ( k −1 )

ρ0
=
ρ

k ( k −1 )

1 ( k −1 )

10

Introduction
• For the critical state (M=1)
*

T
2
=
T0 k + 1
*

p  2 
=

p0  k + 1 

ρ  2 
=
ρ 0  k + 1

*

k ( k −1 )

1 ( k −1 )

11

Introduction

E = η m ( h03 − h01 ) = U 2Vt 2′
p03
h

p02

02
i’

p3
03

p2

i

3

p1
2

V12

01
1

2
s

13

Introduction
• The Specific Shaft Work into the Compressor

The specific shaft work = E ηm

ηm = 0.96

14

Introduction
• Compressor Efficiency:
– The ratio of the useful increase of fluid energy divided by the
actual energy input to the fluid
– The useful energy input is the work of an ideal, or isentropic,
compression to the actual final pressure P3

15

Introduction

Ei = hi − h01 = C pT01 [ Ti T01 − 1]
 p 
 03 
= C pT01 

 p01 


( k −1)

k


− 1


16

Introduction
• The Compressor Efficiency

Ei Ti − T01
ηc = =
E T03 − T01
• No external work or heat associated with the diffuser
flow, i.e.

h02 = h03 ,

T02 = T03
17

Introduction
• The Overall Pressure Ratio

p03  U 2Vt 2'η c 

= 1 +
p01  C pT01η m 



k k −1

• The compressor efficiency from experimental data
• Slip exists in compressor impeller

Vt 2 ' = µ sVt 2
18

Introduction
• The Slip Coefficient (Stanitz Equation)

0.63π
µ s = 1−
nB



1


 1 − ϕ 2 cot β 2 



• More relations in Appendix E
• But, Stanitz equation is more accurate for the practical
range of vane angle; i.e.

45 < β 2 < 90
0

0
19

Introduction
• Total pressure ratio from:
–
–
–
–

Ideal velocity triangle at the impeller exit
The number of vanes
The inlet total temperature
The stage and mechanical efficiencies

• Mechanical efficiency accounts for
– Frictional losses associated with bearing, seal, and disk
friction
– Reappears as enthalpy in the outflow gas
20

Impeller Design
• The impeller design starts with a number of
unshrouded blades (Pfleiderer)
• Flow is assumed axial at the inlet
• Favorable to have large tangential velocity at outlet
(Vt2’)
• Vanes are curved near the rim of the impeller ( β2 <90o)
• But, they are bent near the leading edge to conform to
the direction of the relative velocity Vrb1 at the inlet
21

Impeller Design
• The angle β1 varies over the leading edge, since V1
remains constant while U1 (and r) varies (V1 assumes
uniform at inlet)
• At D1S, the relative velocity Vrb1=(V12+U12)0.5 and the
corresponding relative Mach number MR1S are highest
• For a fixed set of, N, m,Po1, and To1, the relative Mach
number has its minimum where β1S is approximately
32o (Shepherd, 1956)
22

Impeller Design
• Choose a relative Mach number at the inlet
Vrb1S = M R1S a1
Acoustic Speed :
Static Temperature :

a1 = kRT1
T1 =

Absolute inlet Mach no :

T01
1 + ( k − 1) M 12 2
V1
M 1 = = M R1S sin β1S
a1
23

Impeller Design
• Calculation of V1 and U1S

V1 = Vrb1S sin 32

0

U1S = Vrb1S cos 32

0

• Calculation of the shroud diameter

2U1S
D1S =
N
24

Impeller Design
• Calculation of the hub diameter by applying the mass
flow equation to the impeller inlet

D1H

 2

4m 

=  D1S −

πρ1V1 



1

2

• Calculation of density from the equation of state of a
perfect gas

p1
ρ1 =
RT1

25

Impeller Design
• Calculation of static temperature and static pressure

T01
T1 =
2
1 + ( k − 1) M 1 2


p01
p1 = 

2
1 + ( k − 1) M 1 2 

k ( k −1)

26

Impeller Design
• The fluid angle at the hub

β1H

 V1
= tan 
U
 1H
−1






• The vane speed at the hub

U 1H

ND1H
=
2
27

Impeller Design
• The outlet diameter D2
&
Inlet flow rate: Q1 = m ρ1
Output head H:

H = Ei g
1

Dimensional specific speed:

Ns =

NQ1 2
H

3

4

1

D2 =

DS Q1 2
H

1

4

(DS from Table 3 in appendix A)
28

Impeller Design
• The ideal and actual tangential velocities
From Table 3 in appendix A :

ηC

η m Ei
The Energy transfer :
E=
ηC
The actual tangential velocity : Vt 2 ' = E U 2
Vt 2 '
( µ s = 0.85 − 0.9)
The ideal tangential velocity : Vt 2 =
µs
29

Impeller Design
• The vane angle and the number of vanes

Vrb 2t = U 2 − Vt 2

( 0.23 ≤ ϕ 2 ≤ 0.35)

Vrb 2 n = ϕ 2U 2
Vrb 2 n
β 2 = tan
Vrb 2t
−1

0.63π
µs = 1 −
nB



1


 1 − ϕ cot β 
2
2 


30

Impeller Design
• The impeller efficiency

1 −η I
χ=
1 − ηC

( 0.5 ≤ χ ≤ 0.6)

Ti′ − T01
ηI =
T02 − T01
31

Impeller Design
• The static temperature T2 is used to determine density
at the impeller exit
2
2′

V
T2 = T02 −
2C p

&
m
b2 =
2πρ 2 r2V2 n
32

Impeller Design
• The optimal design parameters by Ferguson (1963)
and Whitfield (1990) from Table 5.1
• Table 5.1 Should be used to check calculated results
for acceptability during or after the design process

33

Diffuser Design
• A vaneless diffuser allows reduction of the exit Mach number
• The vaneless portion may have a width as large as 6 percent of
the impeller diameter
• Effects a rise in static pressure
• Angular momentum is conserved and the fluid path is
approximately a logarithmic spiral
• Diffuser vanes are set with the diffuser axes tangent to the
spiral paths with an angle of divergence between them not
exceeding 12o
34

Diffuser Design
• Vanes are preferred where size limitations matter
• Vaneless diffuser is more efficient
• Number of diffuser vanes should be less than the number of
impeller vanes to:
– Ensure uniformness of flow
– High diffuser efficiency in the range of φ2 recommended

36

Diffuser Design
• The mass flow rate at any r (in the vaneless diffuser)

( r2 ≤ r ≤ r3 )
Vr = Vn
&
m = 2πrbρVn
37

Diffuser Design
• For constant diffuser width b

ρrVn = constant
ρrVn = ρ 2 r2Vn 2
• The angular momentum is conserved in the vaneless space

rVt = r2Vt 2′
38

Diffuser Design
• Typically, the flow leaving the impeller is supersonic

M 2′ > 1
• Typically, the flow leaving the vaneless diffuser is subsonic

M 3 < 1.0
39

Diffuser Design
• Denote * for the properties at the radial position at which M=1
(The absolute gas angle, α, is the angle between V and Vr)

Vr = Vn = V cos α
• The continuity equation

ρrV cos α = ρ r V cos α
* *

*

*

40

Diffuser Design
• The angular momentum equation

rV sin α = r V sin α
*

*

*

• Dividing momentum by continuity relations

tan α tan α
=
*
ρ
ρ

*

41

Diffuser Design
• Assuming an isentropic flow in the vaneless region

T  ρ 
= *
*
ρ 
T
 
• For M=1

k −1

,

T0
T=
k −1 2
1+
M
2

2T0
T =
k +1
*

42

Diffuser Design
• Substituting in the density relation

ρ  2  k − 1 2 
=
M 
1 +
*
ρ
2

 k +1

1 ( k −1)

• Substituting in the absolute gas angle relation

 2  k − 1 2 
tan α = tan α 
M 
1 +
2

 k +1

−1 ( k −1)

*

43

Diffuser Design
• The angle α* is evaluated by

α = α 2′
M = M 2′
r sin α
V V a
T 
= *=
= M * 
*
r sin α
V
aa
T 
*

*

1

2

 2  k − 1 2 
r sin α
=M
M 
1 +
r sin α
2

 k +1
*

*

−1 2

44

Diffuser Design
• The radial position r* is determined by

 2  k − 1 2 
r sin α
= M 2′ 
M 2′ 
1 +
r2 sin α 2
2

 k +1 
*

*

−1 2

• The angle α3* is evaluated by

 2  k − 1 2 
tan α 3 = tan α 
M 3 
1 +
2

 k +1 

−1 ( k −1)

*

45

Diffuser Design
• Finally r3 is determined by

 2  k − 1 2 
r sin α
= M3
M 3 
1 +
r3 sin α 3
2

 k +1 
*

*

−1 2

• The volute is designed by the same methods outlined in
chapter 4

46

Performance
• Typical compressor characteristics

ηmax

C

B
A

p01
p02

η = cte.

Surge line

Choke line

N
= cte.
T01
&
m T01
p01

47

Performance
• The sharp fall of the constant-speed curves at higher mass
flows is due to choking in some component of the machine
• The low flows operation is limited by the phenomenon of surge
• Smooth operation occurs on the compressor map at some point
between the surge line and the choke line
• Chocking is associated with the attainment of a Mach number
of unity

48

Performance
• In the stationary passage of the inlet The sharp fall of the
constant-speed curves at higher mass flows is due to choking in
some component of the machine
• The low flows operation is limited by the phenomenon of surge
• Smooth operation occurs on the compressor map at some point
between the surge line and the choke line
• Chocking is associated with the attainment of a Mach number
of unity

49

a=

Performance
• In the stationary passage of the inlet or diffuser for a Mach
number of unity

a = kRT
• The temperature at this point

 ( k − 1) 2 
T = T0 1 +
M 
2


50

a=

Performance
• By setting M=1

 2 
T = T0 
= Tt

 k + 1
*

• The chocking (maximum) flow rate

1

 k 
&
m = At pt 
÷
 RTt 

2

51

a=

Performance
• The throat pressure (isentropic process)
k ( k −1)

 Tt 
pt = pin  ÷
 Tin 

• The chocked flow rate in impeller (use relative velocity instead
of absolute velocity)

2
rb1

2
1

V
U
h01 = h +
−
2
2

52

a=

Performance
• The critical temperature


U 2  2T01
T * = 1 +
= Tt
÷
 2C pT01 ÷( k + 1)



• The throat mass flow rate (isentropic process)
1

( k +1)
  2( k −1)

2
 2 
 k 
U
&
m = At p01 
1 +
÷
÷ 
RT01   k + 1  2C pT01 ÷




2

53

Performance
• The chocked mass flow rate in stationary components is
independent of impeller speed
• The point A in the characteristic curve represents a point of
normal operation
• An increase in flow resistance in the connected external flow
system results in decrease in
and increase in Vn 2
Vt 2
• Causes increase in head or pressure
• Further increase in external system produces a decrease in
impeller flow (beyond point C) and surge phenomena results
54

Performance
• The at some point in the impeller leads to change of direction of
Vrb 2
and an accompanying decrease in head.
• A temporary flow reversal in the impeller and the ensuing
buildup to the original flow condition is known as surging.
• Surging continues cyclically until the external resistance is
removed.
• Surging is an unstable and dangerous condition and must be
avoided by careful operational planning and system design.

55

TURBOMACHINERY BASICS
CENTRIFUGAL COMPRESSOR
Hasan Basri
Jurusan Teknik Mesin
Fakultas Teknik – Universitas Sriwijaya

Phone: 0711-580739, Fax: 0711-560062
Email: hasan_basri@unsri.ac.id
78

EULER EQUATION
Torque T = m (Cθ2r2 – Cθ1r1)
Power P = Tω
= m (U2Cθ2 – U1Cθ1)

80

LSD (Low Solidity Diffuser)

95

pengetahuan tentang sentrifugal

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