THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law...

22
THERMODYNAMICS REVIEW Thermodynamics required for compressible flow: First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State Ideal Gas Law

Transcript of THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law...

Page 1: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

THERMODYNAMICS REVIEW

Thermodynamics required for compressible flow:

First Law Energy Equation

Second LawAdiabatic, isentropic, constant entropy process Entropy Calculation

Equation of StateIdeal Gas Law

Page 2: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

First Law of ThermodynamicsExamples:Observations:

wheela brakingbearing wheelain friction togetherhandsyou rubbing

∑ ∑∝

=

WQheat more friction more

heat into ed transferrbecan work

δδ

Experiments:

( ) 0δWδQ

δWδQ

Cycle afor LawFirst m kg .427calorie 1

lbft 778BTU 1

δWCδQ

WCQ

f

=−

=

==

=

=

∫∫∫

∫∫∑∑

W

The First Law is a Fundamental observationof nature, an axiom, which can not be provedbut has never been found to be violated

Page 3: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

First Law of Thermodynamics

( )( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( ) WQδWδQEE

U(T).PEKE forms, allin energy as E Define property. a therefore

andpath oft independen is δWδQ

δWδQδWδQ

0δWδQδWδQ C and B Processes

0δWδQδWδQ CA Cycle

0δWδQδWδQ BA Cycle

0δWδQ

Cycle afor LawFirst δW δQ

1

221

CB

CB

CA

BA

−=−=−

++

−=−

=−−−

=−−−⇒

=−−−⇒

=−

=

∫∫∫

∫∫∫∫∫∫

∫∫∫

AB

C

δWdEδQW∆EQ

Processes afor LawFirst

+=+=

1

2

Page 4: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

( )

shaft

2

shaft1

2

1111

2

111

22211shaftout flowin flowshaft

1111final 1initial 11in flow

Wρgh)2

Vpv(umQ

W)ρgh2

Vvpm(u)ρgh2

Vvpm(uQ

ρgh2

Vu(T)PEKEu(T)E

vp mvp mWWWWW

vp mVpVVppdVW

LawFirstW∆EQ

++++∆=

++++−+++=

++=++=

−+=++=

==−==

+=

Steady Flow Energy EquationOpen, steady flow thermodynamic system - a region in space

QshaftW

p1p2

v2

v1 h2

V

V

h1

Page 5: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Steady Flow Energy EquationOpen, steady flow thermodynamic system -a region in space

2Vh

2VhQ

flow icnonadiabatan for 2

Vh2

Vh

flow adiabatican for hpvu since

forcesbody without 0,Q 0,W nozzleor diffuser channel, adiabatican for

Wh) g ρ2

V vp(umQ

22

2

21

1

22

2

21

1

shaft

2

+=++

+=+

=+

==

++++∆=

Page 6: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

IDEAL GAS CALCULATION

( )

( )

3

22

ooo

o

o

air

atmospheregage

oo3

3

3o3o

O

atmospheregage

.5047ftV/inft 144psia 514

R459.69F124Rlbmlbf/ ft 53.336lbm 1.2p

T R mV

R lbm

lbfft 53.33628.97

lbmole / R lbm / lbf 1545.15R

psia 514psia 14.7psia 500ppp

kg 9.28K273.16C24/kgm kPa .259813

m 1.2kPa 597RTpVm

/kgm kPa .25981332

K /kmolem kPaor K kJ/kmole 8.314R

kPa 597kPa 97kPa 500ppp

2

=

×+××

==

==

=+=+=

=+×

×==

==

=+=+=

psia. 14.7 is pressure cAtmospheripsia. 500 of pressure gage a and F124at air of lbm 1.2 of volume the isWhat

kPa 97 is pressure cAtmospherikPa. 500 of pressure gage a and C24at oxygen of m 1.2 of mass the isWhat

o

o3

Page 7: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Ideal Gas Law

TnRpV

TRpv

WeightMolecular nm WeightMolecular molesmass

Kkmole

m kPaorKkmole

kJ8.314R

lbmoleRlbf/lbm 1545.15 R

weightmolecular RR

KR, re, temperatuabsolute - T kPapsia, pressure, absolute - p

mRTpV RTpv

LAW GAS (PERFECT) IDEAL

*

*

o

3

o*

o*

*

oo

=

=

×=×=

=

=

=

==

2

1

2

1

2

1

2

1

2211

23

TT

pp

TT

vv

LAW CHARLES

vpvp LAW BOLYES

)0 and atm (1 STP at gas of /molemolecules 106.023

liters. 22.4 gasany of mole (1) One

LAW SAVOGADRO'

=

=

×=×

×

=

Co

Page 8: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

tCoefficien JT=

∂∂

hpT

h=constant

% Error in assuming water is an ideal gas

Page 9: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

PRINCIPAL OF CORRERSPONDING STATESCOMPRESSIBILITY FACTOR Z

P

criticalR

criticalR

TTT

ppP

=

=

Z is about the same forall gases at the same reduced temperatureand the same reduced pressure where:

mRTpVZ

RTpvZ

=

=

Page 10: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

SPECIFIC HEAT

( ) ( )

v

p

vp

vp

vp

vp

constvv

constpp

cR1

1γγRc

ONLYGASIDEALFORunitssamewithRcc

RdTdTcdTcRdTdudh

RTuhRTpv

pvuh

dTc∆udTc∆hGasIdeal

dTTc∆udTTc∆h

Tuc

Thc

=−γ

−=

=−

+=+=

+==+=

==

==

∂∂

=

∂∂

=

∫ ∫∫∫

==

Page 11: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

( )( )dTTcu

dTTch

v

p

∫∫

=

=

IDEAL GAS IMPROVEMENTS

Page 12: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Kelvin Planck Statement of the Second LawIt is impossible to construct an engine which, operating in a cycle, will produce no

other effect than the extraction of heat from a single reservoir and the performance of an equivalent amount of work.

Clausius Statement of the Second Law

It is impossible to have a system operating in a cycle which transfers heat from a cooler to a hotter body without work being done on the system by the surroundings

Reversible Heat Engine Reversible Refrigerator

Page 13: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Carnot Power Cycle

14 process3,2 process expansion, adiabatic Reversible43 process2,1 process

fer,heat trans emperatureconstant t Reversible

→→

→→

inQW

InputRequiredEffectDesiredEfficiency ==

H

LHCYCLE Q

QQη −=

Page 14: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Carnot Principles

1. No engine operating between two heat reservoirs, each having a fixed temperature, can be more efficient than a reversible engine operating between the same reservoirs.

Carnotactual ηη ≤

2. All reversible engines operating between two heat reservoirs, each having its own fixed temperature, have the same efficiency.

3. The efficiency of any reversible engine operating between two reservoirs is independent of the nature of the working fluid and depends only on the temperature of the reservoirs.

4. An absolute temperature scale can be defined in a manner independent of the thermometric material.

2

1

2

1

TT

QQ

=

Page 15: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

5-15

FIGURE 5-47Proof of the firstCarnot principle.

Page 16: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Thermodynamic Temperature Scale

h

lhl

l

h

l

h

313221

3

2

2

1

3

1

313

132

3

221

2

1

313

1

1

3

31

TTQQand

TT

QQ

if,onlysatisfiedbecanequationthis)T,f(T)T,f(T)T,f(T

ng,substitutiQQ

QQ

QQ

identity,by

)T,f(TQQ)T,f(T

QQ)T,f(T

QQ

schematicsenginefrom

)T,function(TQQ

QQ1η

)T,function(Tη

==

×=

=

===

=

−=

=

Page 17: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

0

0

=

=

processesreversible T

dQTQδ

Page 18: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Clasius Inequality

0TδQ

0TQ

TQ

TT1

QQ1

TTT

QQQ

Cycle Reversible afor

l

l

h

h

h

l

h

l

h

lh

h

lh

=

=−

−=−

−=

∫ ≤ 0TQδ

0TδQ

0TQ

TQ

TT1

QQ1

TTT

QQQ

CycleleIrreversiban for

l

l

h

h

h

l

h

l

h

lh

h

lh

⟨−

−⟨−

−⟨

Page 19: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Entropy Definition and Change

∫∫

∫∫∫

=

=≥

≤+

≤−+

≤+=

TdsδQTδQds

TδQds

TδQds

0dsTδQ

0SSTδQ

ENTROPY S PROPERTYA DEFINE

0TδQ

TδQ

TδQ

process leirreversiban andreversible a of composed cycle afor

InequalityClasius0TδQ

revirrev

1

irrev2

1

irrev2

12

1

irrev2

2

rev1

1

irrev21

2reversible

irreversible

Page 20: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

Entropy Change of an Ideal Gas

=−

−=

=

=

−=

=+−−=

+=−−=++=

+=

1

2

1

2p12 p

plnR TT lncss

pdpR

TdTcds

pdpRdp

Tv

TdTc

Tdh , gas idealan For

dpTv

Tdhds

vdp-dhTds pdvvdppdvdhTds

pdvduTds into ngSubstitutivdppdvdhdu vdppdvdudh

pvuh

p

p

+

=−

+=

=

=

+=

+==

+=

1

2

1

2v12 v

vlnR TT lncss

Law Second Tdsδq LawFirstδwduδq

vdvR

TdTcds

vR

Tp

dTcdu :gas idealan For T

pdvTduds

pdvduTds

v

v

Page 21: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

( )

( )

( )

( )

entropyconstant also is constant pv process, adiabatic

0cc

cc

cc

cc

cpplnss

cc1cpplnss

pplncc

ppln1css

pplncc

pplncss

ppRln

TTlncss

pp

vv

TT

RTpv from substitutevv

pp

vpvpconstantpv

vp

v

p

v

v

v

p

p1

212

vpp1

212

1

2vp

1

2p12

1

2vp

1

1

2p12

1

2

1

2p12

1

1

2

1

2

1

1

2

2

1

1

2

2211

=

=

−−−

=−

−−

γ−γ

=−

−−

γ−γ

=−

−−

=−

=−

=

=

=

=

=

=

γ

γ−γ

γ−γ

−γ

γ

γγ

γ

Isentropic Adiabatic Process

IdealGas0∆s,Isentropic

Adiabaticconstantpvconstantplnvln

gintegratin

0p

dp1

1_v

dv11

1

0vdpRcpdv1

Rc

0pdvvdpRcpdv

Rc

Rvdp

RpdvdT,

RpvTgasidealanfor

0pdvdTc0δWdU

0δQprocess AdiabaticLawFirstδWdUδQ

vv

vv

v

==

=+γ

=

−γ

+

+

−γ

=+

+

=++

+==

=+=+

=+=

γ

Page 22: THERMODYNAMICS REVIEW First Law Second Law Adiabatic ... · First Law Energy Equation Second Law Adiabatic, isentropic, constant entropy process Entropy Calculation Equation of State

ADIABATIC PROCESSQ=0, 0s constant,pv =∆=γ

γ−γ

−γ

γ

γγ

γ

=

=

=

=

=

=

1

1

2

1

2

1

1

2

2

1

1

2

2122

pp

vv

TT

RTpv substitutevv

pp

vpvpconstantpv