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?