Transcript
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3. Diagram T-s or h-s
Enthalpy, Entropy: Stateparameters
2 parameters can describe state
of gas
By convention, p-v, T-s, h-s
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Diag T-s for compression
Equal pressure linep2>p1
1-2 comp s constant,
Dh=Cp(T2-T1) 1-2 irreversible comp,
under same p,
Dh=Cp(T2-T1) 31243 friction heatqin
1221 due to n
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Diag T-s for compression
Total loss 32243 Compressor efficiency
(neglect Cp variation)
Usually use total T
2 1
'
2 1
c
T T
T T
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Equal pressure linep2>p1
1-2 expan s constant,
Dh=Cp(T1-T2) 1-2 irreversible comp,under same p,
Dh=Cp(T1-T
2)
31243 friction heat qin
1221 due to n
Diag T-s for expansion
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Total loss 32243 Turbine efficiency
(neglect Cp variation)
TT
TTT
21
'
21
Diag T-s for expansion
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Diagram h-s
Using diagram h-s shows directly
relations of energy exchanges Compression efficiency C=A/C
Expansion efficiency T=C/A
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4. Cycle and cycle efficiency
Thermal machines transfer heatenergy to mechanical energy
SubstanceAir
ExpansionMechanical work
Thermal cycle
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Carnots cycle
DA
AB, heating
q1 BC
CD, release
heat q2unavoidably
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Carnots cycle
Efficiency
T1orT2
t
T
T
q
qqt
1
2
1
21 1
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Generic cycle
Efficiency
At the same T1
and T2, Carnots
cycle has thehighest efficiency.
ABA
ABA
t
S
S
431
21
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1.3 Fundamental equations of
aerodynamics
1. Continuity equationr1A1v1=r2A2v2=qm
(1-29)
Vis speed
For incompressible,r1=r2A1v1=A2v2 (1-30)
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2. Energy equationA volume of air from
1-2 to 1-2 in dt
Adding heat qdmOutput work Wdm
dm=vArdtismass
flow at any sectionKinetic energy
change
22d
2
1
2
2 vvm
1.3 Fundamental equations of
aerodynamics
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2. Energy equation (Cont'd)
Internal energychange: dm(u2-u1)
Work to gas
p2v2A2dt-p1v1A1dt
Neglecting gravity,
thenW
vh
vhq
22
2
2
2
2
1
1
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2. Energy equation(Cont'd)
Ifq=W=0, then
or
22
2
2
2
2
1
1
vh
vh
constTv
cp
2
2
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2. Energy equation (Cont'd)
Means that when gasflowing in a tube (with
friction), enthalpy +
kinetic energy remains
unchanged if no work
gets into the act.
constTv
cp 2
2
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1.3 Fundamental equations of
aerodynamics (Cont'd)
3. Bernoullis equation Differential for of above equation
First law of thermodynamics
dWdvdhdq 221
r
1pddudqdq in fin dWdq
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3. Bernoullis equation (Cont'd)
Enthalpy definition
Then
Integration
dppddhpddhdurrr111
02
2
fdWdW
vd
dp
r
02
1 21
2
2
2
1 fWWvv
dp
r
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3. Bernoullis equation (Cont'd)
First term depends on the process
If no work (W=0) and isentropic (Wf=0)pr-const
0
2
1 21
2
2
2
1
f
WWvvdp
r
02
11
1
1
2
2
1
1
21
vv
p
pRT
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3. Bernoullis equation (Cont'd)
First term depends on the process
Isentropic and incompressible, W=0:
0
2
1 21
2
2
2
1
f
WWvvdp
r
rrr12
2
1
ppdp
constvpvp 22
2
11
2
22 rr
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3. Bernoullis equation (Cont'd)
Isentropic
0212
122
2
1 fWWvvdpr
02
11
1
1
2
2
1
1
21
vv
p
pRT
Generic
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1.3 Fundamental equations of
aerodynamics (Cont'd) 4. Sound speed and Mach number
Sound speed in fluid
In air, sound propagation is seen as adiabatic process, ie.
pr- const. Then
Ratio of specific heats
RGas constant
TAbsolute temperature
rd
dpc
Tc R
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4. Sound speed and M (Cont'd)
Mach Number
M>1Supersonic
M
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1.3 Fundamental equations of
aerodynamics (Cont'd)
5. Stagnation parameters of flow andaerodynamic functions
From above equations, flow kineticenergy (speed), enthalpy and pressure
potential energy can be converted from
one to others.
hv and p
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5. Stagnation parameters andfunctions (Cont'd)
If flow stagnates (v=0) as isentropicprocess, the kinetic energy is
converted totally to enthalpy. It is
called stagnation enthalpy, or totalenthalpy
(1-38)2
2
* vTch p
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5. Stagnation parameters andfunctions (Cont'd)
v: ordered movement
T: disordered movement
2
2* vTch p
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5. Stagnation parameters andfunctions (Cont'd)
Corresponding stagnation temperatureor total temperature:
(1-39)
Since
1-40
Rcp1
pc
vTT
2
2*
Tc R
2
*
211 M
TT
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Stagnation process
Its isentropic, so
T
Tvv
pp
*
*
1*
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5. Stagnation parameters and
functions (Cont'd) Since stagnation is isentropic, so:
(1-41)
(1-42)
p*Total Pressure
r*Total Density
12
*
2
11
M
p
p
1
1
2
*
2
11
r
rM
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5. Stagnation parameters and
functions (Cont'd)According to the 3 equations above, for a
given gas flow, the ratios of the totalparameters and steady parameters are
function of Mach number. When air flows isentropically in a tube
without energy added, the totalparameters (Enthalpy, temperature,
pressure and density) remain unchanged.
5 St ti t d
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5. Stagnation parameters and
functions (Cont'd)
Critical sound speed ccr(in tunnel)
v increases along the tunnel (ex. Laval nozzle)
When v increases, Tdecreases. Sound speed
cis function ofT, itgoes down.
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5. Stagnation parameters and
functions (Cont'd)
Critical sound speed ccr(in tunnel)
When v=c, ie M=1, this chas special meaning,
called critical sound speed ccr. This section is
called critical section and it is the smallest
section in the tunnel, also called throat.
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5. Stagnation parameters and
functions (Cont'd)
Critical sound speed ccr(in tunnel)
ccr is a parameter of the isentropic flow of
the tunnel. ccr is constant in this kind of flow.
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5. Stagnation parameters and
functions (Cont'd)
Definition of speed coefficient in asection
(1-43)
From (1-40) , we obtaincr
c
v
*
1
2
TTcr
2*
2
11 M
T
T
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5. Stagnation parameters and
functions (Cont'd)
And*
1
2TTcr
*2
1
2RTccr
*
1
2RTccr
C d C
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Cand Ccr
C Ccr
Apply to local tunnel
Depending on T T*
Speed ratio M
Relation see followingM
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5. Stagnation parameters and
functions (Cont'd)
Using
in , we obtain
and
2
222
c
cM cr
*2
12 RTccr
RTc 2
2*
2
11 M
T
T
2
2
2
2
11
2
1
M
M
2
2
2
1
11
1
2
M
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5. Stagnation parameters and
functions (Cont'd)
Change (1-40), (1-41) & (1-42) to
(1-46)
(1-47)
(1-48)
t, p and e three aerodynamic func
2
* 1
11)(
tT
T
12
* 1
11)(
p
p
p
1
1
2
*
1
11)(
r
re
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5. Stagnation parameters and
functions (Cont'd)
Flow density function
kg/svAqm r
1
1
2*
111
rr
*
1
2RTcv cr
*1
1
2*
1
2
1
11 RTv
rr
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5. Stagnation parameters and
functions (Cont'd)
Flow density function
(1-49) q() presents relative flow density in
section A to the critical section even
though the critical section does not exist. Ratio of the sections
1
1
21
1
1
11
2
1
)()(
rr
A
A
v
vq cr
cr
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5. Stagnation parameters and
functions (Cont'd)
Using flow density function and totalparameters, mass flow can be expressed:
1-50
where
Air 0.04042, gas 0.03968
*
* )(
T
AqpKqm
1
1
1
2
RK J
Kkg.
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5. Stagnation parameters and
functions (Cont'd)
At the critical section, q()=1.
*
*
T
Ap
Kqcr
m
1 3 F d t l ti f
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1.3 Fundamental equations of
aerodynamics
6. Equation of momentum
Based on second Newtons law
Momentum change of an object at a
period of time is equal to the applied
force
In aircraft engines
)( 12 vvqF m
1 3 F d t l ti f
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1.3 Fundamental equations of
aerodynamics
7. Equation of moment of momentum Similar with above equation, but
rotational movement
vmrdt
dFr
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1.3 Fundamental equations ofaerodynamics
8. Shock waves and expansion waves
Ex. The tail trace when a boat goes with
a high speed.
M>1M=1
8 S
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8. Shock waves and expansion
waves (Contd)
Or bridge pier when water flows.
Accumulation of disturbances
M>1M=1
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8. Shock waves and expansion
waves (Contd)
Intakes: Fig (a) normal shock wave, due to
intakes form; Fig (b) oblique shock wave
The anglebdepends on Mach number of the
flow and geometrical angle of the cone .
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8. Shock waves and expansion
waves (Contd)
When Mreduces orincreases,will increase until the wave
becomes a normal shock wave.
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8. Shock waves and expansion
waves (Contd)
When supersonic flow passes
through the shock wave, sharply
speed decreases, pressure and
temperature increase.
After normal wave, the flow is
certainly subsonic. But after obliqueshock wave, it is still supersonic.
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8. Shock waves and expansion
waves (Contd)
Strength of the shock wave is
described by pressure ratio of after
and before. It is only function ofM
for normal shock wave, the greater
M, the stronger the wave.
For oblique shock wave, the greaterMand , the stronger the wave.
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8. Shock waves and expansion
waves (Contd) Supersonic flow passing through the shock
wave is NOT isentropic process. Partial
mechanical energy Irreversibly changes to
heat, and total pressure decreases. This is shock wave loss, and usually total
pressure recoverys is used to present the
loss. It is function of the wave strength, the
stronger the wave, the greater the loss.
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8. Expansion waves (Contd)
When a supersonicair flows to a lowerpressure zone, thereare expansion wavesdue to air continuousexpansion.
In Fig, turbine
cascade passage. Inthe throat AA,critical section, flowbecomes supersonic.
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8. Expansion waves (Contd)
In downstream, it islow pressure zone.The flow accelerates,it passes through aseries of expansionwaves, and speedincreases,
temperature andpressure decrease.
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8. Expansion waves (Contd)
The flow changesalso the direction. Thebigger the turnedangle, the moreexpansion and flowparameters changemore.
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8. Expansion waves (Contd)
The turned angledepends on exit
pressure. The lower
the pressure, thebigger the angle.
If pressure increases,
expansion waves maydisappear and the
flow may be subsonic.
Summary
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Summary
1.1 First law of thermodynamics Gas, state parameters, gas constants, processes
and parameters
Enthalpy and first law
1.2 Second law of thermodynamics Entropy and second law
Cycle and efficiency
1.3 Aerodynamics fundamental equations
Fundamental equations Sound speed and Mach number
Stagnation parameters and aerodynamic functions
Shockwaves and expansion waves
top related