21352493-thermodynamic-formulas.pdf

8/15/2019 21352493-Thermodynamic-Formulas.pdf

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Thermodynamics

MTX 220 Formules

Chapter 2 – Concepts & DefinitionsFormule Units

Pressure F

P A

=Pa

• Units21 1 / Pa N m=

51 10 0.1bar Pa Mpa= =

1 101325atm Pa=

Specific VolumeV

vm

=3

/m kg

Density

m

V ρ =

1

v ρ =

3/kg m

Static Pressure Variation

P gh ρ ∆ =

,↑= − ↓= +

Pa

Absolute Temperature( ) ( ) 273.15T K T C = ° +

Chapter 3 – Properties of a Pure SubstanceFormule Units


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Quality

vapor

tot

m x

m=

(vapour mass fraction)

1 liquid

tot

m x

m− =

(Liqui mass fraction)

Specific Volume

f fg v v xv= + 3 /m kg

Avera!e Specific Volume

(1 ) f g v x v xv= − + (only t"o p#ase

mi$ture)

3 /m kg

%eal &!as la"

c P P


3/17

Displacement /or+2 2

1 1! Fdx PdV = =∫ ∫

%nte!ration2

2 1

1

( )! PdV P V V = = −

∫

Specific /or+

! "

m=

("or+ per unit mass)/ kg

Po"er (rate of "or+)! FV PV T ω = = =0 0 !

• VelocityV r ω = /rad #

• TorqueT Fr = Nm

Polytropic Process

( 1)n ≠1 1 2 2

n n n PV Con#t PV PV = = =

n Pv C =

• Polytropic '$ponent2

1

1

2

ln

ln

P

P nV

V

=

• n*11 1 2 2

PV Con#t PV P V = = =

Polytropic Process /or+

1 2 2 2 1 1

1( ) 1

1! PV PV n

n= − ≠

−

• n*12

1 2 2 21

ln V

! PV V

=

Aiabatic Process0$ =


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onuction 2eat Transfer

dT $ kA

dx= −0

.

k

*conuctivity!

onvection 2eat Transfer

$ hA T = ∆0 .

h

*convection coefficient

!

-aiation 2eat Transfer 4 4( ) # amb

$ A T T εσ = −0 !

Terminolo!y

$

* #eat

1 2$

* #eat transferre urin! t#e process bet"een state 1 an state 3

$

* rate of #eat transfer

!

* "or+

1 2!

* "or+ one urin! t#e c#an!e from state 1 to state 3

! 0

* rate of "or+ * Po"er4 1 /*1 56s

Chapter # – The First $a% of Thermo'namicsFormule Units


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Total 'ner!y( ) ( ) % & K% P% d% d& d K% d P% = + + → = + +

'ner!y2 1 1 2 1 2

d% $ ! % % $ ! δ δ = − → − = −

7inetic 'ner!y20.5 K% mV =

Potential 'ner!y2 1 2 1( ) P% mgZ P% P% mg Z Z = → − = −

%nternal 'ner!yliq vap liq f vap g & & & mu m u m u= + → = +

Specific %nternal 'ner!y

of

Saturate Steam(t"o8p#ase mass

avera!e)

(1 ) f g

f fg

u x u xu

u u xu

= − +

= +

/k kg

Total 'ner!y2 2

2 12 1 2 1 1 2 1 2

( )( )

2

m V V & & mg Z Z $ !

−− + + − = −

Specific 'ner!y20.5' u V gZ = + +

'nt#alpy

( & PV = +Specific 'nt#alpy

h u Pv= + /k kg

For %eal asses( ) Pv RT and u f T = =

• 'nt#alpyh u Pv u RT = + = +

• - onstant( ) ( )u f t h f T

= → =Specific 'nt#alpy for

Saturation State

(t"o8p#ase mass

avera!e)

(1 ) f g

f fg

h x h xh

h h xh

= − +

= +

/k kg


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Specific 2eat at

onstant Volume 1 1

v

v v v

$ & uC

m T m T T

δ δ δ

δ δ δ

= = =

( ) ( )' i v ' i

u u C T T

→ − = −Specific 2eat at

onstant Pressure 1 1

p

p p p

$ ( hC

m T m T T

δ δ δ

δ δ δ

= = =

( ) (' i p ' i

h h C T T → − = −

Solis 0 Liquis %ncompressible. so v*constant

c pC C C = =

(Tables A49 0 A4:)

2 1 2 1( )u u C T T − = −

2 1 2 1 2 1( )h h u u v P P − = − + −

%eal ash u Pv u RT = + = +

2 1 2 1( )

vu u C T T − ≅ −

2 1 2 1( ) ph h C T T − ≅ −

'ner!y -ate( ) % $ ! rat' in out = − = + −00 0

2 1 1 2 1 2 ( ) % % $ ! chang' in out → − = − = + −


7/17

Chapter ( – First)$a% *nal'sis for * Control +olumeFormule Units

Volume Flo" -ate

V V dA AV = =

∫

0

(usin! avera!e velocity)

;ass Flo" -ate

V m VdA AV A

v ρ ρ = = =∫ 0

(usin! avera!e values)/kg #

Po"er

p! mC T =0 0 V

v! mC T =0 0 V

V m

v= 00

!

Flo" /or+ -ate flo"! PV mPv= =0 0 0

Flo" Direction From #i!#er P to lo"er P unless si!nificant 7' or P'

• Total

'nt#alpy21

2tot h h V gZ = + +

%nstantaneous

Process• ontinuity

'quation . .C V i '

m m m= −

∑ ∑0 0 0

• 'ner!y

'quation. . . . . .C V C V C V i tot i ' tot

% $ ! m h m h= − + −∑ ∑00 0 0 0 First La"

( )2 21 1( )2 2i i i ' ' 'd%

$ m h V gZ m h V gZ ! dt

→ + + + = + + + −∑ ∑0 00 0

Steay State

Process

A steay8state #as no stora!e effects. "it# all properties

constant "it# time

•


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• 'ner!y

'quation. . . .C V i tot i C V ' tot '$ m h ! m h+ = +∑ ∑0 00 0

(in * out) First La"

( )

2 21 1( )2 2i i i ' ' '

$ m h V gZ ! m h V gZ → + + + = + + +

∑ ∑0 00 0

• Specific

2eat

Transfer . .C V

$q

m=

0

0

/k kg

• Specific

/or+ . .C V !

"m

= 0

0

/k kg

• SS Sin!leFlo" 'q4

tot i tot 'q h " h+ = + (in * out)

Transient Process #an!e in mass (stora!e) suc# as fillin! or emptyin! of acontainer4

• ontinuity

'quation 2 1 i 'm m m m− = −∑ ∑

• 'ner!y

'quation 2 1 . . .C V C V i tot i ' tot ' % % $ ! m h m h− = − + −∑ ∑

( ) ( )2 22 1 2 2 2 2 1 1 1 11 12 2 % % m u V gZ m u V gZ − = + + − + +

( ) ( )2 2. 2 2 2 2 1 1 2 1 . .. .

1 12 2C V i tot i ' tot ' C V C V

$ m h m h m u V gZ m u V gZ ! + = + + + − + + − ∑ ∑

Chapter , – The Secon $a% of Thermo'namicsFormule Units

All

,! $

can also be rates

,! $00


9/17


10/17

• -eal

-efri!erator ) ) R%F Carnot R%F R%F ( )

$ T

! T T β β = ≤ =

−

Absolute Temp4 ) )

( (

T $

T $=

Chapter - – .ntrop'Formule Units

%nequality of lausis

0$T

δ ≤∫ =

'ntropy

r'v

$d*

T

δ ≡

/k kgK

#an!e of 'ntropy 2

2 1

1 r'v

$* *

T

δ − = ∫

/k kgK

Specific 'ntropy(1 ) f g # x # x#= − +

f fg # # x#= +

/k kgK

'ntropy #an!e


11/17

• arnot ycle %sot#ermal 2eat Transfer>2

1 22 1

1

1

( (

$* * $

T T δ − = =∫

-eversible Aiabatic (%sentropic Process)>

r'v

$d*

T

δ =

-eversible %sot#ermal Process>

4

3 44 3

3 r'v )

$$* *

T T

δ − = = ∫

-eversible Aiabatic (%sentropic Process)> 'ntropy

ecrease in

process 98: * t#e entropy increase

in process 1834

• -eversible

2eat8Transfer

Process

2 2

1 22 1

1 1

1 1 fg fg

r'v

hq$ # # # $

m T mT T T

δ δ

− = = = = = ∫ ∫

ibbs 'quations Td# du Pdv= +

Td# dh vdP = −

'ntropy eneration

g'n

$d* *

T

δ δ = +

irr g'n! PdV T * δ δ = −

2 2

2 1 1 2

1 1

g'n

$* * d* *

T

δ − = = +∫ ∫

'ntropy ?alance 'q4 %ntrop+ in out g'n= + − +V


12/17

Principle of t#e

%ncrease of 'ntropy . . 0n't c m #urr g'nd* d* d* * δ = + = ≥∑

'ntropy #an!e

• Solis 0

Liquis2

2 1

1

lnT

# # cT

− =

-eversible Process>

0 g'nd# =

Aiabatic Process>

0dq =

• %eal as onstant Volume>2

22 1 0

11

lndT v

# # Cv RvT

− = +∫

onstant Pressure>2

22 1 0

11

lndT P

# # Cp R P T

− = −∫

onstant Specific 2eat>2 2

2 1 01 1

ln lnT v

# # Cv RT v

− = +

2 22 1 0

1 1

ln lnT P

# # Cp RT P

− = −

Stanar 'ntropy

0

00

T p

T T

C # dT

T = ∫

/k kgK

#an!e in Stanar

'ntropy ( )0 0 22 1 2 11

lnT T

P # # # # R

P − = − − /k kgK


13/17

%eal as Uner!oin!

an %sentropic Process2 2

2 1 01 1

0 ln lnT P

# # Cp RT P

− = = −

0

2 2

1 1

RCpT P

T P

→ =

but

0 0

0 0

1 p v

p p

C C R k

C C k

− −= =

.

0

0

p

v

C k

C =

* ratio of

specific #eats

1

2 1 2 1

1 2 1 2

,

k k

T v P v

T v P v

−

⇒ = =

Special case of polytropic process "#ere + * n>

k Pv con#t =

-eversible Polytropic

Process for %eal as 1 1 2 2n n n PV con#t PV PV = = =

1 1

2 1 2 2 1

1 2 1 1 2

,

nn nn P V T P V

P V T P V

− −

→ = = =

• /or+ 2 22 2 1 1 2 1

1 2

1 1

( )

1 1n

PV PV mR T T dV ! PdV con#t

V n n

− −= = = =

− −∫ ∫

• Values for n %sobaric process>

%sot#ermal Process>

%sentropic Process>


14/17

%soc#ronic Process>

,n v con#t = ∞ =

Chapter / – Secon)$a% *nal'sis for a Control +olumeFormule Unit

s

3n La" '$presse as a

#an!e of 'ntropy . .c m g'n

d* $*

dt T = +∑

00

'ntropy ?alance 'q4rat'of chang' in out g'n'ratio= + − +

. . . .C V C V i i ' ' g'n

d* $m # m # *

dt T → = − + +∑ ∑ ∑

000 0

"#ere

. . . . ...

C V c v A A , ,* #dV m # m # m # ρ = = = + +∫

an

. . ...

g'n g'n g'n A g'n ,* # dV * * ρ = = + +∫ 0 0 00

Steay State Process

. . 0C V d*

dt =

. .

. .

C V ' ' i i g'n

C V

$m # m # * T → − = +∑ ∑ ∑

0

00 0

• ontinuity eq4

i 'm m m= =0 0

. .

. .

( ) C V ' i g'nC V

$m # # *

T ⇒ − = +∑

000


15/17

• Aiabatic

process ' i g'n i # # # #= + ≥

Transient Process

( ) . .. .

C V i i ' ' g'nC V

$d m# m # m # *

dt T

= − + +∑ ∑ ∑ 0

00 0

( ) . .2 2 1 1 1 2. .0

t

C V i i ' ' g'nC V

$m # m # m # m # dt *

T → − = − + +∑ ∑ ∫

00

-eversible Steay State

Process• %f Process

-eversible 0

Aiabatic

' i # #=

'

' i

i

h h vdP − = ∫

( )2 2

2 2

( )2

( )2

i 'i ' i '

'

i 'i '

i

V V " h h g Z Z

V V vdP g Z Z

−= − + + −

−= − + + −∫

• %f Process is

-eversible an

%sot#ermal( ) . .. .

. .

1 C V ' i C V

C V

$m # # $

T T − = =∑ 00

or

( ) . .C V ' i$

T # # qm

− = =0

0

( ) ( )'

' i ' i

i

T # # h h vdP → − = − − ∫

• %ncompressible

Flui( ) ( )

2 2

02

' i' i ' i

V V v P P g Z Z

−− + + − =

?ernoulli 'q4


16/17

• -eversible

Polytropic

Process

for %eal as

'

n n

i

" vdP and Pv con#t C = − = =∫

( ) ( )

1

1 1

' '

ni i

' ' i i ' i

dP " vdP C P

n nR P v Pv T T

n n

= − = −

= − − = − −− −

∫ ∫

• %sot#ermal

Process (n*1) ln' '

'i i

ii i

P dP " vdP C Pv

P P = − = − = −∫ ∫

Principle of t#e

%ncrease of 'ntropy . . 0n't C V #urr g'nd* d* d*

* dt dt dt = + = ≥∑ 0

'fficiency

• Turbine

a i '

# i '#

" h h

" h hη

−= =

−

Turbine "or+ is out

• ompressor

(Pump) # i '#

a i '

" h h

" h hη −= = −

ompressor "or+ is in

• oole

ompressorT "

"η =

•


17/17

21352493-thermodynamic-formulas.pdf

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