lecture notes on thermodynamics 2008

Lecture Notes on Thermodynamics Lecture Notes on Thermodynamics 20082008Chapter 10 Thermodynamics Chapter 10 Thermodynamics

RelationsRelationsProf. Man Y. Kim, Autumn 2008, Prof. Man Y. Kim, Autumn 2008, ⓒⓒ[email protected], Aerospace Engineering, Chonbuk National University, Korea [email protected], Aerospace Engineering, Chonbuk National University, Korea

2 Propulsion and Combustion Lab.

http://procom.chonbuk.ac.kr

Two Important Partial Derivative RelationsConsider a variable z which is a continuous function of x

and y : ,z f x y and

y x

z zdz dx dy

x y

If we take y and z as independent variables :

,x f y z and yz

x xdx dy dz

y z

(*)

(**)

Substitute eq.(**) into (*) :

y y yz x

z x z x zdz dy dz

x y y z x

1

y y yz x

z x z x zdy dz

x y y z x

Since there are only 2 independent variables,

Reciprocity relation

Cyclic relation

1

1

y y y y

x zz z zx

xx

1yxy z x z

yz x zzy

x zxy yx



Maxwell Relations

2 Gibbs equations in Chapter 6 :

du Tds Pdv

dh Tds vdP

Maxwell Relations : Four equations relating the properties P, v, T, and s for a simple compressible system of fixed chemical composition

Helmholtz free energy :

a u TsGibb’s free energy :

g h Ts da du Tds sdT

da sdT Pdv dg dh Tds sdT

dg sdT vdP

Since u, h, a, and g are total derivative ;

s v

s P

T v

T P

du Tds Pdv

dh Tds vdP

da sdT Pdv

dg sdT vd

T Pv s

T vP s

s Pv T

s vP

PT

yx

M Ndz Mdx Ndy

y x



Clapeyron Equation

Let’s consider the 3rd Maxwell relation ;

Clapeyron Equation : P, v, T 를 통해 증발엔탈피 ( ) 와 같은 상변화와 관계있는 엔탈피 변화를 구하는 관계식fgh

T v

s Pv T

상변화가 일어나는 동안 압력은 온도에만 의존하고 비체적에는 무관한 포화압력을 유지함 . 즉 , sat satP f T

v sat

P dPT dT

등온 액체 - 증기 상변화과정에 대해서 세번째 Maxwell 관계식을 적분하면 ;

fgg f g f

sat sat fg

sdP dPs s v v

dT dT v이 과정 동안 압력도 일정하게 유지되므로 ,

g g

fg fgff

dh Tds vdP dh Tds h Ts

따라서 ,

fg

sat fg

hdPdT Tv



Clapeyron-Clausius Equation

저압상태일 때 ;

Clapeyron-Clausius Equation : Clapeyron 방정식에 약간의 근사를 사용하여 액체 - 증기와 고체 - 증기의 상변화에 적용함 .

증기를 이상기체로 가정하면 ;

g ff g gv v v v

g

RTv

P따라서 ,

2 2

fgf fg

sat s

g

atsat fg

hdPdT T

Ph hdP dP dTTv dT RT P R

작은온도구간에 대하여 는 어떤 평균값으로 일정하므로 ,

2

1 1 2

1 1ln fg

sat sat

hPP R T T

fgh

윗 식에서 를 ( 승화엔탈피 ) 로 대치함으로서 고체 -증기 영역에서도 사용함 .

fgh igh



Relations between du, dh, ds, Cv and Cp (1/6)• Change of Internal Energy

du Tds Pdv

If,

, vv TT

u uu u T v dT dv

udu C dT

vT vdv

If, ,

v T

s ss s T v ds dT dv

T v

v TT v

s sT dT dv Pd

s sdu T dTv

T vT P dv

T v

since

Therefore,

v

v

T vT

s

CsT T

uT T

v v TP

PP

Finally,

vv

Pdu C dT T P dv

Tand

2 2

1 12 1

T v

vT v v

Pu u C dT T P dv

T



Relations between du, dh, ds, Cv and Cp (2/6)• Change of Internal Energy - Example



Relations between du, dh, ds, Cv and Cp (3/6)• Change of enthalpy

dh Tds vdP

If,

, pP TT

h hh h T P dT dP

hdh C dT

PT PdP

If, ,

P T

s ss s T P ds dT dP

T P

P P TT

s sT dT dP vd

s sdh T dT v

PP

TT PT dP

since

Therefore,

T

p

T P

P

CsT T

hv T v

PvT

TsP

Finally,

pP

vdh C dT v T dP

Tand

2 2

1 12 1

T P

pT P P

vh h C dT v T dP

T



Relations between du, dh, ds, Cv and Cp (4/6)• Change of Entropy

If,

, v

vv T

s ss s T v dT dv

T vC P

ds dT dvT T

Therefore, 2 2

1 12 1

T vv

T v v

C Ps s dT dv

T T

If, , p

P T P

Cs s vs s T P ds dT dP dT dP

T P T TTherefore,

2 2

1 12 1

T Pp

T P P

C vs s dT dP

T T



Relations between du, dh, ds, Cv and Cp (5/6)• Specific Heat (1/2)

2

21T

v

v

v v

v v

C PT

v TC CP P

ds dTv

dT TT

vT T

2

22p p

P PT

p

P

C Cv v C vT

P Tds dT dP

T T P T T T

Substitute (3’) into (3) ;

1 32 p v

P v

v PT C C dT T dP T dv

T TTake

, 3

v T

P PP P T v dP dT dv

T v

p vP v T v

P v P T v

P v

v P P PC C dT T dT dv T dv

T T v T

v P v P PT dT T dv

T T T v T

v PT dT

T T

p vP v

v PC C T

T T



Relations between du, dh, ds, Cv and Cp (6/6)

• Specific Heat (2/2)

2 2

p vP v P T

v P v P TC C T

T T T v

We know the cyclic relations as :

1v P T v P T

P T v P v PT v P T T v

where,

1

1P

T

vT

vP

: volume expansivity

: isothermal compressibility

Comments :

1 0

2 0

3 ,

p v

p v

p v

C C

C C asT

for incompressibleliquidandsolid v constant C C



Joule-Thomson Coefficient (1/2)

• Joule-Thomson Coefficient : 교축 (h=constant) 과정 중의 유체의 온도 변화

0

0

0JT JT

h

temperatureincreasesT

temperatureremainsconstantP

temperaturedecreases



Joule-Thomson Coefficient (2/2)

10 p JT

P h Pp

v T vdh C dT v T dP v T

T P C T


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lecture notes on thermodynamics 2008

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