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1
Turbomachinery Lecture 3
- Compressibility- Isentropic- Area, Mass Flow Functions- C-D Nozzle
2
Turbomachinery• Definition:
– A turbomachine transfers energy to or from a fluid moving continuously through a casing by dynamic action of a rotor and by flow conditioning of a stator.
• Works on a fluid to produce power or flow (and pressure rise)
• Adds energy to fluid................Pump or Compressor– Fan: pressure rise up to 1 lbf/in2
– Blower: pressure between 1 - 40 lbf/in2
– Compressor: pressure rise > 40 lbf/in2
• Extracts energy from fluid............Turbine
– Pressure changes due to motion of parts or displacement of boundaries
3
Compressible Flow• Density varies making continuity & momentum more
difficult to solve.
because varies with velocity.
• Also, can't integrate Bernoulli directly
• Compressible flow problems can be solved iteratively using continuity, state et. al.
cosm AC
.2
2
constCdP
C
α
4
• Example:m = 50 lb/sec A = 200 sq.in.P0 = 14.7 psia = 30
T0 = 519 R
• GuessC = 646.8 ft/sec
2
0
22
2
2 2
2
646.8 / sec519 . .2 32.174 778.16 .24
.sec
6008.8 / sec484.19deg
p
CT TC
ftT ft lbm ft lbf BTU
lbf BTU lbm R
ft RT R
Compressible Flow
5
Compressible Flow
• Pressure
/ 1
00
TP PT
3.5484.1914.7 11.529519
P psi
6
Compressible Flow• Density can now be found from state:
11.529 14453.349 484.19
0.06427 / .lbm cu ft
RTP
7
Compressible Flow• Mass Flow:
• note:
• 19%>
cos
0.06427 646.8 cos30 200 /144
50.00 / sec
m AC
m
m lb
0 0.0765 / .lbm cu ft
8
Compressible Flow
• Mach Number Functions:
– Easily calculated & clarify physics
• Mach number & acoustic speed are critical concepts!
9
Compressible Flow
0
1
p
dPIsentropically TdS dh
dPc dT RTP
dT dPT P
State P RTdP RdT RTd
TdTd
PdP
10
Compressible Flow• Using isentropic relation between pressure &
temperature derivatives:– Use adiabatic state law
PdPd
PdP
1
d
PdP
1
2dP P a RTd
1P CT
11
Compressible Flow• Using equation of state, acoustic speed in an
ideal gas is [from kinetic theory]:
• By definition Mach Number is:2
2
2
V dynamic pressurep static pressureV VM
a a V kinetic energyRT thermal energy
1716
287
Ta gRT
T
12
Compressible Flow
• Static & Total properties as functions of Mach number: 2
0 2Vh h
g
20
20
12
12
p
p
T VT gc T
T R VT c gRT
0 2112
TM
T
13
Compressible Flow – Critical Velocity
• What does subscript * mean? It means value of variable when M=1 [sonic]
• Vcr is only function of gas [] and stagnation props.
2
0
2 2 22 220 0
0
21
1 1 2 1 2 2 1 1
21
cr crcr
cr
Vh h
a V V RTa V V
V RT
14
Compressible Flow
• The relation between static & stagnation properties is isentropic. Then:
/ 120 11
2P MP
1/ 10 211
2M
15
Compressible Flow• The relation between compressible and Bernoulli [B-p.55]
12
0
2 2
2 40
2 2 22
2
1/ 12
1(1 ) 1 ( 1) / 2 ... &1 2
2/ 1 ...2 2
2 2 2 / 2
n
p p M
Binomial expansion for small x is x nx n n x n x M
For small M one gets p p M M
V V VBut since pM p pa p
The
2 24
0
2
0
21 ...2 4 2
0.3, 2.3% ( ).2
V Mexpanded isentropic equationbecomes p p M
Vfor M p is in error from Bernoulli p
16
Compressible Flow Relationships
• Mass Flow parameter [=0]
0
0
m VAdm VdA AdV VAddm dA dV dm A V
17
Compressible Flow Relationships• Area-Mach number differential relation
• Area-Mach number integral relation
22
2
11
MdA dV dpMA V M p
1
2 121 2 11
1 2A MA M
More on next chart
18
Compressible Flow Relationships• What does subscript * mean?
– For all flow variables it means value of variable when M=1 [sonic]
– For area A* this is reference area for choking flow [M=1]
• Note this area is a minimum or throat
1
2 121 2 11
1 2A MA M
More on next chart
19
Compressible Flow Relationships
Flow textbooks
-www.engr.uconn.edu/barbertj- Compressible
- Aero Calculator- calcbody2
20
Compressible Flow Relations
21
22
2
11
MdA dV dpMA V M p
Of interest
here
Of interesthere
22
22
2
11
MdA dV dpMA V M p
Over-expanded
23
24
25
26
Compressible Flow Examples
0 00 0
: 450 1890 1.5
3.671 6938 1.45 652.5
1.4 1716 450 1040
1.5 1040 1560
s s
s s
s s
Given T R p psf M
p Tp psf T R
p T
a a RT fps
V fps
27
Compressible Flow Examples01 01 *
*
0 0
*
0
10 300 / 6
/ 6 0.097
1.006 9.94 1.002 299.4
1.4 287 299.4 346.8
33.6
/ 6 3.368
63.13 0.
s ss s
s s
ss
Consider isentropic flow in C D nozzlep atm T K A A
Subsonic A A Mp Tp atm T Kp T
a a RT mps
V mps
Supersonic A A Mp pp
0
1 /2 1
0 *
0
1584 3.269 91.77
192 646.7
21
ss
s
Tatm T KT
a a mps V mps
p Am VA if choked
RT
28
Compressible Flow
• Mass Flow Parameters:
VRTP
Am
AVm
cos
cos
1/ 20
0cosTm V g
PA RT TgRT
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Compressible Flow Relationships• Mass flow parameters
0 0
0 0
00 1
0 2 12
1/ 20 2
( , )
112
112s
s
m VA
m p V pV RT MA RT a RTm T Tp M f Mp A p T R
m T R MFPp A
M
m T RFP M Mp A
Note: FPo, FPs are similar, but different f[M] powers
30
Compressible Flow Relationships• Mass flow parameters
00 1
0 2 12
01
0 2 12
112
112
m T R MFPp A
M
p A MmRT
M
How to get more mass flow, i.e. greater thrust, more power?
1
2 10
0
, 1
21
if choked at throat M
p AmRT
31
Compressible Flow Relationships• Mass flow parameters in English
units– m in lbm/sec– p0A in lbf [spatial dimensions
cancel]– T0 in degs. Rankine
00
0
0
0
0
0
0
0
1
1716 /1.432.2
1.0888
RTmFPp A g
m T gRp A g
m Tp A
m Tp A
For air
0
0
0
[ 1, 1.4] 0.5787cos0.5322
If choked FP Mp Am
T
20
020 0
2
sec / sec: dimsec
m T gR m lbm lbf ftNote FP gRT ensionlesslbfgp A gp A lbmft ftft
32
Compressible Flow Relationships
• Mass flow parameters in SI units– m in kg/sec– p0A in Newtons [spatial
dimensions cancel]– T0 in degs. Kelvin
00
0
0
0
0
0
cos
cos
14.318cos
RTmFPp A
m T Rp A
m Tp A
For air
0
0
0
[ 1, 1.4] 0.5787cos0.0404
If choked FP Mp Am
T
33
Compressible Flow Examples
01 01 *
*
0 0
1 /2 1
0 *
0
10 300 / 6
/ 6 3.368
63.13 0.1584 3.269 91.77
192 646.7
21
s ss s
s
Consider isentropic flow in C D nozzlep atm T K A A
Supersonic A A Mp Tp atm T Kp Ta a mps V mps
p Am VA if chokedRT
34
Example2
0 0
2*
*
00
0
2 ,
0.5 1 300
1.4, 0.5 1.340 1.49
( , ) 353.6 / sec
Air in duct of A m has flow such that
M p atm T K
AFor M A mA
area to choke
p Am FP M kgT R
35
Static Pressure Mass Flow Parameter
• Defining: FP = Flow parameter=f(M)
• For Air
• Can be inverted
1/ 220 11
cos 2sRTmFP M M
PA g
01.0883coss
m TFP
PA
2/1
2
11211
sFPM
36
Total Pressure Mass Flow Parameter• Introducing P0:
• No explicit solution for M • FPs is single valued, FPo is not• FPo max = 0.5787 for =1.4• FPo max always at M=1
1/ 20
00 0cos
Tm P gP MA P RT T
1/ 20 0
00 0cos
RT Tm PFP MP A g P T
12 12
011
2FP M M
37
Calculate FPo• From Previous Example:
m = 50 lb/sec A = 200 sq.in.P0 = 14.7 psia = 30
T0 = 519 R
• Rearrange FPo
00
0
0.4869cos
RTmFPP A g
1 /2/ 12
011 0.5997
2calc guessM FP M M
38
Mass Flow Parameters
Be careful: FPs single valued, FPo double values
39
Total Pressure Mass Flow Parameter
• Consider FPt:
• For fixed , a fixed value of
produces the same Mach number - regardless of the level of pressure, temperature or molecular weight (R).
12 120
00
11cos 2
RTmFP M MP A g
0
0 cosm RT
P A
40
Total Pressure Mass Flow Parameter
• Defines common flow parameters.
• Valid for flow with one gas.
• Corrected flow.
0
0 cosm RT
P A
0
0
m TP
0
0
/ 519/14.7
m T mP
41
Other Parameters[Covered in Lecture 4]
• Ideal gas equation for Mach number leads to speed parameters, also for a single gas.
• Speed parameter
• Corrected speed
0
NT
N
0 0 14.696 518.7
in inP T
42
Significance of Flow & Speed Parameters
• A device operating at:• same speed parameter and flow parameter has • same Mach numbers, velocity diagrams, flow angles
etc, • regardless of level of physical speed, pressure &
temperature.
43
Flow and Speed Parameters• Conditions: same gas, high Reynolds number, same
clearances, and same • Speed and Flow parameters are used for turbine
maps
0
5
10
15
20
25
30
35
1.0 1.5 2.0 2.5 3.0
Exp Ratio
MrtT
/P
N/rtT
44
Corrected Flow & Speed Parameters
• Corrected Flow and Corrected speed used for compressor maps
3
4
5
6
7
8
9
50 60 70 80 90 100 110
Corrected Flow lb/sec
Pre
ssur
e R
atio
Corrected Speed
45
Flow Parameter
• Again, Consider FP0:
• Unlike P, T & R; cannot be "corrected".
• Changing , changes relation between FP0 and Mach number!
12 120
00
11cos 2
RTmFP M MP A g
46
Flow ParameterGamma Effect On Continuity
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.0 1.2 1.4 1.6 1.8
Gamma
Mac
h N
umeb
er
Air
Helium
Butane
FPT = .560
Message: More complex gasses choke at a lower Mach number