4. 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
8. 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
9. 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
10. 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
11. 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
15. 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
16. Introduction
Ei = hi − h01 = C pT01 [ Ti T01 − 1]
p
03
= C pT01
p01
( k −1)
k
− 1
16
17. 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
18. 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
19. 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
20. 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
21. 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
22. 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
23. 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
24. 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
25. 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
26. 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
27. 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
28. 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
29. 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
30. 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
32. 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
33. 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
34. 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
36. 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
37. Diffuser Design
• The mass flow rate at any r (in the vaneless diffuser)
( r2 ≤ r ≤ r3 )
Vr = Vn
&
m = 2πrbρVn
37
38. 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
39. 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
40. 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
41. 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
42. 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
43. 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
44. 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
45. 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
46. 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
48. 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
49. 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
50. 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
51. 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
52. 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
53. 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
54. 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
55. 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