Transcript 열역학 제10장 강의노트 - Propulsion and Combustion Laboratory

Lecture Notes on Thermodynamics 2008
Chapter 10 Thermodynamics Relations
Prof. Man Y. Kim, Autumn 2008, ⓒ[email protected], Aerospace Engineering, Chonbuk National University, Korea
Two Important Partial Derivative
Relations
Consider a variable z which is a continuous function of x and y :
 z 
 z 
z  f  x , y  and dz    dx    dy (*)
 x  y
 y  x
If we take y and z as independent
variables :
 x 
 x 
x  f  y , z  and dx    dy    dz
 z  y
 y  z
(**)
Substitute eq.(**) into (*) :
 z   x   z  
 x   z 
dz          dy      dz
 z  y  x  y
 x  y  y  z  y  x 
 z   x   z  
  x   z  
         dy  1       dz
 x  y  y  z  y  x 
  z  y  x  y 
Since there are only 2 independent variables,
1
 x   z 
 x 
    1  
 z  y  x  y
 z  y  z x y
Reciprocity
relation
 z 
 x   y   z 
 z   x 
                1
 x  y  y  z
 y  x
 y  z  z  x  x  y
Cyclic relation
http://procom.chonbuk.ac.
kr
2
Propulsion and
Combustion Lab.
Maxwell Relations
Maxwell Relations : Four equations relating the properties P, v, T, and s for a simple compressible system
of fixed chemical composition
2 Gibbs equations in Chapter 6 :
du  Tds  Pdv
dh  Tds  vdP
a  u  Ts
Helmholtz free
energy free
:
Gibb’s
energy g
:  h  Ts
da  du  Tds  sdT
 da  sdT  Pdv
dg  dh  Tds  sdT
 dg  sdT  vdP
 M   N 

 
 y  x  x  y
Since u, h, a, and g are total derivative ;dz  Mdx  Ndy  
 T 
 P 
du  Tds  Pdv  
  

 v s
 s  v
 T   v 
dh  Tds  vdP  
  
 P s  s  P
 s   P 
da  sdT  Pdv     

 v T  T  v
 s 
 v 
dg  sdT  vdP  
  

 P T
 T  P
http://procom.chonbuk.ac.
kr
3
Propulsion and
Combustion Lab.
Clapeyron Equation
Clapeyron Equation : P, v, T를 통해 증발엔탈피(h fg )와 같은 상변화와 관계있는 엔탈피 변화를 구하는 관계식
Let’s consider the 3rd Maxwell relation ;
 s   P 
  

 v T  T  v
상변화가 일어나는 동안 압력은 온도에
만 의존하고 비체적에는 무관한 포화압
력을 유지함. 즉,
Psat  f Tsat 
 P   dP 

 

 T  v  dT sat
등온 액체-증기 상변화과정에 대해서
세번째 Maxwell 관계식을 적분하면 ;
s fg
 dP 
 dP 
sg  s f  
 vg  v f  
 
 dT sat
 dT sat v fg


이 과정 동안 압력도 일정하게 유지되므로,
dh  Tds  vdP 
따라서,

g
f
dh 
g
 Tds  h
f
fg
 Ts fg
h fg
 dP 



 dT sat Tv fg
http://procom.chonbuk.ac.
kr
4
Propulsion and
Combustion Lab.
Clapeyron-Clausius Equation
Clapeyron-Clausius Equation : Clapeyron 방정식에 약간의 근사를 사용하여 액체-증기와 고체-증기의 상변
화에 적용함.
저압상태일 때 ;
vg
v f  v fg  v g
증기를 이상기체로 가정하면 ;
vg 
RT
P
따라서,
h fg
Ph fg
h fg  dT 
 dP 
 dP 
 dP 











 2
2
 dT sat Tv fg
 dT sat RT
 P sat R  T 
작은온도구간에 대하여 h fg 는 어떤 평균값으로 일정하므로,
h fg  1 1 
P 
 ln  2  
  
P
R
 1 sat
 T1 T2 sat
hig(승화엔탈피)로 대치함으로서 고체-증기
윗 식에서 h fg를
영역에서도 사용함.
http://procom.chonbuk.ac.
kr
5
Propulsion and
Combustion Lab.
Relations between du, dh, ds, Cv and Cp
(1/6)
• Change of Internal Energy
 u 
 u 
 u 
 dT    dv  C v dT    dv
 T  v
 v T
 v T
If, u  u T , v   du  
 s 
 s 
 dT    dv
 T  v
 v T
If, s  s T , v   ds  
since
du  Tds  Pdv
 s 

  s 

 s 
 s 
du  T 
 dT    dv   Pdv  T 
 dT  T    P  dv
 v T 
 T  v
  v T

 T  v
 s  C v
Therefore,

 
 T  v T
 u 
 s 
 P 
  T  P T
 P
 v T
 v T
 T  v
Finally,
and
  P 

du  C v dT  T 

P

 dv

T
v
 

u2  u1 
http://procom.chonbuk.ac.
kr

T2
T1
C v dT 
6

v2
v1
  P 

T

P


 T
 dv
v
 

Propulsion and
Combustion Lab.
Relations between du, dh, ds, Cv and Cp
(2/6)
• Change of Internal Energy - Example
http://procom.chonbuk.ac.
kr
7
Propulsion and
Combustion Lab.
Relations between du, dh, ds, Cv and Cp
(3/6)
• Change of enthalpy
 h 
 h 
 h 
 dT    dP  C p dT    dP
 T  P
 P T
 P T
If, h  h T , P   dh  
s 
 s 
 dT    dP
 T  P
 P T
If, s  s T , P   ds  
since
dh  Tds  vdP
 s 


 s 
 s 
 s  
dh  T 
 dT    dP   vdP  T 
 dT   v  T    dP
 P T 
 T  P
 P T 
 T  P

Therefore,
Cp
 s 



 T  P T
 h 
 s 
 v 

  v T 
  v T 

 P T
 P T
 T  P
Finally,
and

 v  
dh  C p dT   v  T 
  dP

T

P 

h2  h1 
http://procom.chonbuk.ac.
kr

T2
T1
C p dT 

8
P2
P1

 v  
v

T

  dP


T

P 

Propulsion and
Combustion Lab.
Relations between du, dh, ds, Cv and Cp
(4/6)
• Change of Entropy
Cv
 s 
 s 
 P 
dT  
 dT    dv 
 dv

T

v
T

T

v
 T

v
If, s  s T , v   ds  
Therefore,
s2  s1 

T2
T1
Cv
dT 
T

v2
v1
 P 

 dv
 T  v
Cp
 s 
 s 
 v 
s

s
T
,
P

ds

dT

dP

dT  


If,




 dP
T
 T  P
 P T
 T  P
Therefore,
s 2  s1 
http://procom.chonbuk.ac.
kr

T2
T1
Cp
T
dT 

P2
P1
9
 v 

 dP

T

P
Propulsion and
Combustion Lab.
Relations between du, dh, ds, Cv and Cp
(5/6)
• Specific Heat (1/2)
 2 P 
C v
  Cv 
  P 
T 2 
 1       
v  T T T  T  v
v
 T  v
C
 P 
ds  v dT  
 dv
T
 T  v
C p
 2 v 
  Cp 
  v 
 T  2 
2        
P  T T
T  T  P
P
 T  P
Cp
 v 
ds 
dT  
 dP
T
 T  P
 v 
 P 
Take T  2    1  C p  C v dT  T 
 dP  T 
 dv

T

T

P

v

P

P




P  P T , v   dP  
 dT  
 dv  3 
 T  v
 v T

Substitute (3’) into (3) ;

C
p
 3

 v   P 
 P 
 P 
 C v dT  T 
 
 dT  
 dv   T 
 dv

T

T

v

T

 P 
v

T 

v

 v   P   P  
 v   P 
T
dT

T
 

 T   v    T   dv
 T  P  T  v
P 
T 
v 

 v   P 
T
 
 dT

T

T

P 
v
 v   P 
C p  C v  T 
 

 T  P  T  v
http://procom.chonbuk.ac.
kr
10
Propulsion and
Combustion Lab.
Relations between du, dh, ds, Cv and Cp (6/6)
• Specific Heat (2/2)
 P   T   v 
 P 
 v   P 
 
 
  1  
  
 

 T  v  v  P  P T
 T  v
 T  P  v T
2
2
 v   P 
 v   P   T 
C p  C v  T 
 
  
   

 T  P  T  v
 T  P  v T
where,
1 v
   
: volume expansivity
  T 
We know the cyclic relations as : 
P
1 v
    
  P T
: isothermal
compressibility
Comments :
 1 C p  C v   0
 2  C p  C v as T  0
 3  for incompressible liquid and solid , v  constant  C p  C v
http://procom.chonbuk.ac.
kr
11
Propulsion and
Combustion Lab.
Joule-Thomson Coefficient (1/2)
• Joule-Thomson Coefficient : 교축(h=constant) 과정 중의 유체의 온도 변화
 0 temperature increases

T

 JT      JT  0 temperature remains constant
 P  h
 0 temperature decreases

http://procom.chonbuk.ac.
kr
12
Propulsion and
Combustion Lab.
Joule-Thomson Coefficient (2/2)

1 
 v  
 T 
 v  
dh  0  C p dT   v  T 
dP





v

T
 



 
JT
C p 
 T  P 
 P h
 T  P 

http://procom.chonbuk.ac.
kr
13
Propulsion and
Combustion Lab.
No Homework !
http://procom.chonbuk.ac.
kr
14
Propulsion and
Combustion Lab.