Area, Mass Flow Functions C-D Nozzles
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• Area, Mass Flow Functions
• C-D Nozzles
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Compressible Flow Relationships
• Mass Flow parameter
0
0
m VA
dm VdA AdV VAd
dm dA dV d
m A V
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Compressible Flow Relationships
• Area-Mach number differential relation
• Area-Mach number integral relation
222
11
MdA dV dpM
A V M p
1
2 121 2 1
11 2
AM
A M
More on next chart
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Compressible Flow Relationships
• What does subscript * mean?– For all flow variables it means when that
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 1
11 2
AM
A M
More on next chart
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Compressible Flow Relations
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222
11
MdA dV dpM
A V M p
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222
11
MdA dV dpM
A V M p
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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
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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 nozzle
p atm T K A A
Subsonic A A M
p Tp atm T K
p T
a a RT mps
V mps
Supersonic A A M
pp
p
0
1 /2 1
0 *
0
1584 3.269 91.77
192 646.7
2
1
ss
s
Tatm T K
T
a a mps V mps
p Am VA if choked
RT
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Compressible Flow
• Mass Flow Parameters:
VRT
P
A
m
AVm
cos
cos
1/ 2
0
0cos
Tm 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/ 2
0 2
( , )
11
2
11
2ss
m VA
m p V pV RT M
A RT a RT
m T TpM f M
p A p T R
m T R MFP
p AM
m T RFP M M
p A
Note: FPo, FPs are similar, but different f[M] powers
0
( , ).79
F MFP p
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Compressible Flow Relationships• Mass flow parameters
00 1
0 2 12
01
0 2 12
11
2
11
2
m T R MFP
p AM
p A Mm
RTM
How to get more mass flow, i.e. greater thrust, more power?
1
2 10
0
, 1
2
1
if choked at throat M
p Am
RT
00
0
0
0
0
0
0
0
1
1716 /1.4
32.2
1.0888
RTmFP
p A g
m T gR
p A g
m T
p A
m T
p A
For air
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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
2
1
s ss s
s
Consider isentropic flow in C D nozzle
p atm T K A A
Supersonic A A M
p Tp atm T K
p T
a a mps V mps
p Am VA if choked
RT
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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 m
A
area to choke
p Am FP M kg
T R
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Static Pressure Mass Flow Parameter
• Defining: FP = Flow parameter=f(M)
• For Air
1/ 220 1
1cos 2s
RTmFP M M
PA g
01.0883coss
m TFP
PA
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Static Pressure Mass Flow Parameter
• Remember that
• can be inverted:
2/12
2
11
MMFPs
2/1
2
1
1211
sFPM
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Total Pressure Mass Flow Parameter• Introducing P0:
• No explicit solution for M
• FPs is single valued, FPo is not
1/ 2
00
0 0cos
Tm P gPM
A P RT T
1/ 2
0 00
0 0cos
RT Tm PFP M
P A g P T
1
2 12
0
11
2FP M M
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Total Pressure Mass Flow Parameter
• FP0 is double valued
• FP0 maximizes at M = 1.0
• Maximum FP0 is 0.5787 for = 1.4
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Mass Flow Parameters
Be careful: FPs single valued, FPo double values
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Other Parameters
• Example:
m = 50 lb/sec A = 200 sq.in.
P0 = 14.7 psia = 30
T0 = 519 R
00
0
0.4869cos
RTmFP
P A g
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Example Solution to Mass Flow Parameter
• Rearrange FP0 equation:
1 / 2 / 12
0
11
2calc guessM FP M
0.5997M