Upload
hahanh
View
255
Download
11
Embed Size (px)
Citation preview
Mekanika Fluida II
Fluida Tekompresi
Referensi Definisi
1. Fluida terkompresi statis 2. Fluida terkompresi dinamis
Compressible Flow
Natural gas well
Tall Mountains
Compressible fluid
Fluida gas disebut compressible karena densitasnya bervariasi terhadap suhu dan tekanan
=P M /RT
Dalam perubahan elevasi yang kecil (contoh :
tangki, pipa, dll), kita dapat mengabaikan efek
perubahan tekanan terhadap elevasi.
Namun dalam kasus umum :
o
o
RT
zzMgPP
Tfor
)(exp
:constT
1212
gdz
dP
Linear Temperature Gradient
)( 00 zzTT
z
z
p
pzzT
dz
R
Mg
p
dp
00)( 00
RMg
T
zzTpzp
0
000
)()(
Persamaan di Atmosfer
Asumsi linear
RMg
T
zzTpzp
0
000
)()(
Asumsi konstan
0
0)(
0)(RT
zzMg
epzp
Contoh Kasus 1
Suhu udara di dekat permukaan bumi akan turun
sekitar 5 C setiap 1000 m elevasi. Jika suhu udara
di permukaan tanah 15 C dan tekanannya 760 mm
Hg, berapakah tekanan udara di puncak G. Ciremai
3800 m? Asusmsikan perilakunya mengikuti gas
ideal.
1st
LAW OF THERMODYNAMICS
System
e (J/kg)
Boundary
Surroundings
• System (gas) composed of molecules moving in random motion
• Energy of molecular motion is internal energy per unit mass, e, of system
• Only two ways e can be increased (or decreased):
1. Heat, dq, added to (or removed from) system
2. Work, dw, is done on (or by) system
THOUGHT EXPERIMENT #1
• Do not allow size of balloon to change (hold volume constant)
• Turn on a heat lamp • Heat (or q) is added to the system
• How does e (internal energy per unit mass)
inside the balloon change?
THOUGHT EXPERIMENT #2
• *You* take balloon and squeeze it down to a small size
• When volume varies work is done • Who did the work on the balloon?
• How does e (internal energy per unit mass)
inside the balloon change? • Where did this increased energy come from?
1st
LAW OF THERMODYNAMICS
• System (gas) composed of molecules moving in random motion • Energy of all molecular motion is called internal energy per unit mass,
e, of system
• Only two ways e can be increased (or decreased): 1. Heat, dq, added to (or removed from) system 2. Work, dw, is done on (or by) system
SYSTEM
(unit mass of gas)
Boundary
SURROUNDINGS
dq
wqde dd
e (J/kg)
1st
LAW IN MORE USEFUL FORM
• 1st Law: de = dq + dw – Find more useful
expression for dw, in terms of p and (or v = 1/)
• When volume varies → work is done
• Work done on balloon, volume ↓ • Work done by balloon, volume ↑
pdvqde
wqde
pdvw
sdAppsdAw
spdA
AA
d
dd
d
d
ΔW
distanceforceΔW
Change in
Volume (-)
ENTHALPY: A USEFUL QUANTITY
vdpdhq
vdpdedhdeq
pdvdeq
vdppdvdedh
RTepveh
d
d
d
Define a new quantity
called enthalpy, h:
(recall ideal gas law: pv = RT)
Differentiate
Substitute into 1st law
(from previous slide)
Another version of 1st law
that uses enthalpy, h:
HEAT ADDITION AND SPECIFIC HEAT
• Addition of dq will cause a small change in temperature dT of system
• Specific heat is heat added per unit change in temperature of system
• Different materials have different specific heats
– Balloon filled with He, N2, Ar, water, lead, uranium, etc…
• ALSO, for a fixed dq, resulting dT depends on type of process…
Kkg
J
dT
qc
d
dq
dT
SPECIFIC HEAT: CONSTANT PRESSURE
• Addition of dq will cause a small change in temperature dT of system
• System pressure remains constant
Tch
dTcdh
dTcq
dT
qc
p
p
p
p
d
d
pressureconstant
dq
dT
Kkg
J
dT
qc
d
SPECIFIC HEAT: CONSTANT VOLUME
• Addition of dq will cause a small change in temperature dT of system
• System volume remains constant
Kkg
J
dT
qc
d
dq
dT
Tce
dTcde
dTcq
dT
qc
v
v
v
v
d
d
olumeconstant v
HEAT ADDITION AND SPECIFIC HEAT
• Addition of dq will cause a small change in temperature dT of system
• Specific heat is heat added per unit change in temperature of system
Tch
dTcdh
dTcq
dT
qc
p
p
p
p
d
d
pressureconstant
• However, for a fixed dq, resulting dT depends on type of process:
Tce
dTcde
dTcq
dT
qc
v
v
v
v
d
d
olumeconstant v
Kkg
J
dT
qc
d
v
p
c
c
Specific heat ratio
For air, = 1.4
Constant Pressure Constant Volume
ISENTROPIC FLOW
• Goal: Relate Thermodynamics to Compressible Flow • Adiabatic Process: No heat is added or removed from system
– dq = 0 – Note: Temperature can still change because of changing density
• Reversible Process: No friction (or other dissipative effects)
• Isentropic Process: (1) Adiabatic + (2) Reversible – (1) No heat exchange + (2) no frictional losses – Relevant for compressible flows only – Provides important relationships among thermodynamic variables
at two different points along a streamline
1
1
2
1
2
1
2
T
T
p
p = ratio of specific heats
= cp/cv
air=1.4
DERIVATION: ENERGY EQUATION
022
0
0
0
0
0
2
1
2
212
2
1
2
1
VVhh
VdVdh
VdVdh
VdVvdh
VdVdp
vdpdhq
q
wqde
V
V
h
h
d
d
ddEnergy can neither be created nor destroyed
Start with 1st law
Adiabatic, dq=0
1st law in terms of enthalpy
Recall Euler’s equation
Combine
Integrate
Result: frictionless + adiabatic flow
ENERGY EQUATION SUMMARY
• Energy can neither be created nor destroyed; can only change physical form – Same idea as 1st law of thermodynamics
constant2
222
2
22
2
11
Vh
Vh
Vh
constant2
222
2
22
2
11
VTc
VTc
VTc
p
pp
Energy equation for frictionless,
adiabatic flow (isentropic)
h = enthalpy = e+p/= e+RT
h = cpT for an ideal gas
Also energy equation for
frictionless, adiabatic flow
Relates T and V at two different
points along a streamline
GOVERNING EQUATIONS STEADY AND INVISCID FLOW
2
22
2
11
2211
2
1
2
1VpVp
VAVA
222
111
2
22
2
11
1
2
1
2
1
2
1
222111
2
1
2
1
RTp
RTp
VTcVTc
T
T
p
p
VAVA
pp
• Incompressible flow of fluid along a
streamline or in a stream tube of
varying area
• Most important variables: p and V
• T and are constants throughout flow
• Compressible, isentropic
(adiabatic and frictionless)
flow along a streamline or in a
stream tube of varying area
• T, p, , and V are all variables
continuity
Bernoulli
continuity
isentropic
energy
equation of state
at any point
EXAMPLE: SPEED OF SOUND
• Sound waves travel through air at a finite speed • Sound speed (information speed) has an important role in
aerodynamics • Combine conservation of mass, Euler’s equation and isentropic
relations:
RTp
a
a
VM
• Speed of sound, a, in a perfect gas depends only on temperature of gas
• Mach number = flow velocity normalizes by speed of sound
– If M < 1 flow is subsonic
– If M = 1 flow is sonic
– If M > flow is supersonic
• If M < 0.3 flow may be considered incompressible
d
dpa 2
KEY TERMS: CAN YOU DEFINE THEM?
• Streamline • Stream tube
• Steady flow • Unsteady flow
• Viscid flow • Inviscid flow
• Compressible flow • Incompressible flow
• Laminar flow • Turbulent flow
• Constant pressure process • Constant volume process
• Adiabatic
• Reversible
• Isentropic
• Enthalpy
MEASUREMENT OF AIRSPEED: SUBSONIC COMPRESSIBLE FLOW
• If M > 0.3, flow is compressible (density changes are important) • Need to introduce energy equation and isentropic relations
2
1
1
0
1
2
1
1
0
0
2
11
2
11
21
2
1
MT
T
Tc
V
T
T
TcVTc
p
pp
11
2
1
1
0
12
1
1
0
2
11
2
11
M
Mp
p
cp: specific heat at constant pressure
M1=V1/a1
air=1.4
MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW
• So, how do we use these results to measure airspeed
111
2
111
2
11
2
11
2
1
10
22
1
1
10
2
12
1
1
1
0
2
12
1
1
1
02
1
s
scal
p
ppaV
p
ppaV
p
paV
p
pM
p0 and p1 give
Flight Mach number
Mach meter
M1=V1/a1
Actual Flight Speed
Actual Flight Speed
using pressure difference
What is T1 and a1?
Again use sea-level conditions
Ts, as, ps (a1=340.3 m/s)
EXAMPLE: TOTAL TEMPERATURE
• A rocket is flying at Mach 6 through a portion of the atmosphere where the static temperature is 200 K
• What temperature does the nose of the rocket ‘feel’?
• T0 = 200(1+ 0.2(36)) = 1,640 K!
2
1
1
0
2
11 M
T
T
Total temperature
Static temperature Vehicle flight
Mach number
MEASUREMENT OF AIRSPEED:
SUPERSONIC FLOW
• What can happen in supersonic flows?
• Supersonic flows (M > 1) are qualitatively and quantitatively different from subsonic flows (M < 1)
HOW AND WHY DOES A SHOCK WAVE FORM?
• Think of a as ‘information speed’ and M=V/a as ratio of flow speed to information speed
• If M < 1 information available throughout flow field
• If M > 1 information confined to some region of flow field
MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW
1
21
124
1 2
1
1
2
1
2
1
2
1
02
M
M
M
p
p
Notice how different this expression is from previous expressions
You will learn a lot more about shock wave in compressible flow course
SUMMARY OF AIR SPEED MEASUREMENT
• Subsonic,
incompressible
• Subsonic, compressible
• Supersonic
1
21
124
1 2
1
1
2
1
2
1
2
1
02
M
M
M
p
p
111
21
10
22
s
scal
p
ppaV
s
e
ppV
02
2/6/2007 BIEN 301 – Winter 2006-
2007
Compressible Flow
– Mach Regimes
• Ma < 0.3 Incompressible Flow • 0.3 < Ma < 0.8 Subsonic Flow • 0.8 < Ma < 1.2 Transonic Flow • 1.2 < Ma < 3.0 Supersonic Flow • 3.0 > Ma Hypersonic Flow
– These are only guides, the individual flow scenarios affect how shock waves might develop.
MORE ON SUPERSONIC FLOWS
V
dVM
A
dA
V
dV
A
dA
a
VdV
V
dV
A
dA
dp
VdVd
VdVdp
V
dV
A
dAd
AV
1
0
0
0
constantlnlnVlnAln
constant
2
2
Isentropic flow in a streamtube
Differentiate
Euler’s Equation
Since flow is isentropic
a2=dp/d
Area-Velocity Relation
CONSEQUENCES OF AREA-VELOCITY RELATION
V
dVM
A
dA12
• IF Flow is Subsonic (M < 1) – For V to increase (dV positive) area must decrease (dA negative) – Note that this is consistent with Euler’s equation for dV and dp
• IF Flow is Supersonic (M > 1)
– For V to increase (dV positive) area must increase (dA positive)
• IF Flow is Sonic (M = 1) – M = 1 occurs at a minimum area of cross-section – Minimum area is called a throat (dA/A = 0)
TRENDS: CONTRACTION
M1 < 1
M1 > 1
V2 > V1
V2 < V1
1: INLET 2: OUTLET
TRENDS: EXPANSION
M1 < 1
M1 > 1
V2 < V1
V2 > V1
1: INLET 2: OUTLET
PUT IT TOGETHER: C-D NOZZLE
1: INLET 2: OUTLET
MORE ON SUPERSONIC FLOWS
• A converging-diverging, with a minimum area throat, is necessary to produce a supersonic flow from rest
Supersonic wind tunnel section
GOVERNING EQUATIONS STEADY AND INVISCID FLOW
2
22
2
11
2211
2
1
2
1VpVp
VAVA
222
111
2
22
2
11
1
2
1
2
1
2
1
222111
2
1
2
1
RTp
RTp
VTcVTc
T
T
p
p
VAVA
pp
• Incompressible flow of fluid along a
streamline or in a stream tube of
varying area
• Most important variables: p and V
• T and are constants throughout flow
• Compressible, isentropic
(adiabatic and frictionless)
flow along a streamline or in a
stream tube of varying area
• T, p, , and V are all variables
continuity
Bernoulli
continuity
isentropic
energy
equation of state
at any point
Kondisi Isentropic
v
p
C
C
Pconstant
P
1
1y
P
P
T
T1
11
1
12
1
1
12
11
11
RT
zgMTT
RT
zgMPP
Aliran Steady
Batas kompresibilitas
Pertimbangan Termodinamik
• Persamaan gas ideal
• Proses Reversibel
• Entropi
• Entalpi
• Kalor spesifik
05.0
Studi Kasus 1
Suatu gas di-ekspansi dari 5 bar ke 1 bar dengan mengikuti persamaan pV1.2=C. Suhu awal 200 C. Hitung perubahan entropi spesifik yang terjadi!
Jika =1.4 dan R =287 J/kg K.
Solusi Kasus 1
Studi Kasus 2
Tabung pitot dipasang untuk mengukur aliran gas dalam pipa yang bertekanan 105 kPa. Beda tekanan yang terukur 20 kPa dan suhunya 20 C. Hitung besarnya kecepatan aliran gas!
Jika =1.4 dan R =287 J/kg K.
Solusi Kasus 2
Questions?