Thermodynamics Equation Sheet

CLOSED SYSTEMS OPEN SYSTEMS

Cycle Process General Form Steady State Steady Flow (SSSF) Uniform State Uniform Flow

(USUF)

Firs

t Law

(I) = = +

!" + !! ! + !!2 + ! = !" + !! ! + !!2 + ! +!" + ! + !!2 + ! = ! + !!2 + ! +

!" + !! ! + !!2 + ! = ! ! + !!2 + ! ! ! + !!2 + ! + !! ! + !!2 + ! +!"

(I 1) (I 2) (I 3) (I 4) (I 5)

Seco

nd L

aw (I

I)

Kelvin-Plank Statement Clausius Statement 0

+ ! ! ! ! !" !! !! + !! !! !" (II 2) (II 4) !"! = !"! + !"## 0 !"! = !" + !"## 0 (II 1) (II 3) (II 5) (II 6) (II 7) = + (III 1) IDEAL GASES = (III 2) (IG 1) =

(IG 2) = !! (IG 3) = !! (IG 4) ! ! = ! ln !!!! + ln !!!! (IG 5) ! ! = ! ln !!!! ln !!!! (IG 6) ! = ! +

For Reversible Adiabtic Processes: (IG 7) ! = const,where = !!!! (IG 7a) !!!! = !!!! !!! (IG 7b) !!!! = !!!! !!!!

For Reversible SSSF Processes: = +!! !! !!2 + ! ! (III 3) Miscellaneous:

= = ! + !"

FIRSTLAWClosedsystemQ W U KE PE ' ' ' Forpistoncase: 2

1W pdV

Isentropic: 0Q , 0W z Isovolumetric: 0W , 0Q z Others: 0Q z , 0W z Isobaric: 2 2 2 1 2 2 1 1

1 1W pdV p dV pV pV p V pV

2 1 2 2 2 1 1 1 2 1Q U U W U p V U pV H H Opensystem

cv cvQ W H KE PE ' ' ' Turbine

0cvQ , 2 1cvW m h h 1 2

1 2

t

s

h h

h hK Compressorandpump

0cvQ , 2 1cvW m h h 2 1

/

2 1

sc p

h h

h hK Forpump:(incompressibleliquid) 2 22 1 1 2 1

1 1int, int,

pcvs

rev rev

WWh h vdp v dp v p p

m m

| Heatexchange

0cvW , 2 1cvQ m h h

SECONDLAWTds du pdv Tds dh vdp EntropychangeIncompressible: 2

2 1

1

lnT

s s cT

Idealgas: 2 2

1 1

2 22 1

1 1

( ) ln ( ) lnT T

v pT T

v pdT dTs s c T R c T R

T v T p

Internallyreversibleclosedsystem: 2int,1

revQ TdS EntropybalanceClosedsystem

2

2 11

boundary

QS S

T

G V Ifwithnoheattransfer: 2 1m s sV Opensystem

jcvi i e e cv

j i ej

QdSm s m s

dt TV

Steadystate:0 j i i e e cvj i ej

Qm s m s

TV

IsentropicprocessIdealgasmodelForaironly: 2 2

1 1

r

r

p p

p p or 2 2

1 1

r

r

v v

v v andthenuseTableA22

ConstantspecificheatsUsedincoldairstandardanalysis

12 2 1 2

1 1 2 1

kk k

kp v T

p v T

UU

EXERGYClosedsystemDefinition

0 0 0 0 0( ) ( ) ( )E U U p V V T S S KE PE 0 0E U p V T S KE PE' ' ' ' ' '

Balance ^ `2 02 1 0 2 1 01

1q w db

TE E E E E Q W p V V T

TG V

RatebalanceSteadystate: 00 1 j d

j j

TQ W E

T

OpensystemDefinition

2

0 0 0( )2

f

Ve h h T s s gz Ratebalance

00 1 j cv i fi e fe dj i ej

TQ W m e m e E

T

ExergeticEfficienciesTurbines

1 2

/cv

f f

W m

e eH Compressorsandpumps 2 1/f fcve eW mH HeatexchangerNomix: 4 31 2c f fh f fm e em e eH Mix(1hot+2cold>3): 3 21 3c f fh f fm e em e eH

IDEALGASm

pV nRT RTM

V R

p pv T RTm M 8.314 / kR kJ mol K 0.287 /airR kJ kg K

CYCLESTOREMEMBERClosedsystemOttoDieselDualOpensystemRankineCogenerationBraytonCombineThermalefficiency

1net out

in in

W Q

Q QK Becarefulwithheatlossandsomeprocesseswithbothheatandwork

PROBLEMSOLVINGPROCEDURE1. Useisentropic,isobaric,orisovolumetricrelationstofindpropertiesateachpoint.2. Doenergyconservationatheatexchangerstorelatem andheat&work.3. ForRankincycle,verifythestatusofeachpointafterturbine.(mixture/superheated

vapor)4. Startfromoriginalequations.5. Becarefulwithheatloss.6. Doublecheckunits.