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![Page 1: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/1.jpg)
EntropyProperty Relationships
Chapter 7b
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The T-ds relations
dUWQ outrevrev ,intint
Consider an internally reversible process occurring in a closed system
TdSQ rev int PdVW outrev ,int
dUPdVTdS
![Page 3: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/3.jpg)
PdvduTds Or…
T
Pdv
T
duds
To find s all you have to do is integrate!!!
First Gibbs equation – also calledFirst Tds relationship
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2nd Gibbs relationship
Recall that…
Pvuh
vdPPdvdudh
Find the derivative, dh
vdPPdvdhdu
Rearrange to find du
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PdvduTds vdPPdvdhdu
vdPdhTds Second Tds relationship, or
Gibbs equation
T
vdP
T
dhds
To find s all you have to do is integrate
First Tds relationship
![Page 6: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/6.jpg)
We have two equations for ds
T
vdP
T
dhds
T
Pdv
T
duds To find s, integrate
the equation that is the easiest, or for which you have the data
![Page 7: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/7.jpg)
First lets look at solids and liquids
T
Pdv
T
duds
T
vdP
T
dhds
Solids and liquids do not change specific volume appreciably with pressure
That means that dv=0, so the first equation is the easiest to use.
0
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T
duds
For solids and liquids…
![Page 9: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/9.jpg)
Recall that…
CdTdu For solids and liquids, so…
T
CdT
T
duds
1
2lnT
TCs
Integrate to give…
Only true for solids Only true for solids and liquids!!and liquids!!
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What if the process is isentropic? What happens to s?
0ln1
2
T
TCs
The only way this expression can equal 0 is if T2 = T1
For solids and liquids, isentropic processes are also isothermal, if they are truly incompressible
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Entropy change of ideal gases
RTPv
dTCdu v
dTCdh p
Some equations we know for ideal gases
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Let’s use these relationships with the Gibbs equations
T
Pdv
T
duds
T
vdP
T
dhds
v
Rdv
T
dTCv
P
RdP
T
dTC p
![Page 13: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/13.jpg)
We can integrate these equations if we assume constant Cp and constant Cv
2
1
2
1 v
Rdv
T
dTCs v
1
2
1
2 lnlnv
vR
T
TCs v
Only true for ideal gases, assuming constant heat capacities
First Gibbs equation
![Page 14: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/14.jpg)
We can integrate these equations if we assume constant Cp and constant Cv
2
1
2
1 P
RdP
T
dTCs p
1
2
1
2 lnlnP
PR
T
TCs p
Only true for ideal gases, assuming constant heat capacities
Second Gibbs equation
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Use which ever equation is easiest!!
Which should you use?
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Sometimes it is more convenient to calculate the change in entropy per mole, instead of per unit mass
2
1
2
112 P
dPR
T
dTCsss up
2
1
2
112 v
dvR
T
dTCsss uv
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1
2
1
212 lnln
v
vR
T
TCsss uv
1
2
1
212 lnln
P
PR
T
TCsss up
![Page 18: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/18.jpg)
What if it’s not appropriate to assume constant specific heats?
We could substitute in the equations for Cv and Cp, and perform the integrationsCp = a + bT + cT2 + dT3
That would be time consuming and error prone
There must be a better way!!
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What if it’s not appropriate to assume constant specific heats?
Someone already did the integrations and tabulated them for usThey assume absolute 0 as the starting
point
2
2 0
0 )(T
pT T
dTTCs
2
1
01
02 )(
T
T p T
dTTCss
1
1 0
0 )(T
pT T
dTTCs
See Table A-17, pg 910
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1
22
1lnP
PR
T
dTCs p
1
201
02 ln
P
PRsss
1
201
02 ln
P
PRsss u
So….
These two equations are good for ideal gases, and consider variable specific heats
Remember
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Entropy of an Ideal GasEntropy of an Ideal Gas6-12
The entropy of an ideal gas depends on both The entropy of an ideal gas depends on both T T and and PP. The function . The function ss° represents only the temperature-dependent part of entropy° represents only the temperature-dependent part of entropy
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Isentropic Processes of Ideal GasesMany real processes can be modeled
as isentropicIsentropic processes are the standard
against which we should measure efficiency
We need to develop isentropic relationships for ideal gases, just like we developed them for solids and liquids
![Page 23: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/23.jpg)
1
2
1
2 lnlnv
vR
T
TCs v For the
isentropic case, S=0
1
2
1
2 lnlnv
vR
T
TCv
Constant specific heats
vC
R
v v
v
v
v
C
R
T
T
2
1
1
2
1
2 lnlnln
![Page 24: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/24.jpg)
vC
R
v v
v
v
v
C
R
T
T
2
1
1
2
1
2 lnlnln
1
2
1
2
1
1
2
k
C
R
v
v
v
v
T
T v
First isentropic relation for ideal gases
vp CCR v
p
C
Ck and so 1k
C
R
v
Recall that…
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Similarly0lnln
1
2
1
2
P
PR
T
TCs p
pC
R
p P
P
P
P
C
R
T
T
1
2
1
2
1
2 lnlnln
k
k
P
P
T
T1
1
2
1
2
Second isentropic relationship
Only applies to ideal gases, with constant specific heats
![Page 26: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/26.jpg)
Since…
k
k
P
P
T
T1
1
2
1
2
1
2
1
1
2
k
v
v
T
T and
k
kk
P
P
v
v1
1
2
1
2
1
Which can be simplified to…
1
2
2
1
P
P
v
vk
Third isentropic relationship
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1
2
1
1
2
k
v
v
T
T
k
k
P
P
T
T1
1
2
1
2
1
2
2
1
P
P
v
vk
constant
constant
constant1
1
k
k
k
k
Pv
TP
Tv
Compact form
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That works if the heat capacities can be approximated as constant, but what if that’s not a good assumption?
1
201
02 ln
P
PRsss
We need to use the exact treatment
0
1
201
02 ln
P
PRss
This equation is a good way to evaluate property changes, but it can be tedious if you know the volume ratio instead of the pressure ratio
![Page 29: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/29.jpg)
Relative Pressure and Relative Specific Volume
1
201
02 ln
P
PRss
R
ss
P
P 01
02
1
2 exp
Rs
Rs
P
P01
02
1
2
exp
exp
s20 is a function
only of temperature!!!
1
201
02 ln
P
P
R
ss
![Page 30: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/30.jpg)
Rs
Rs
P
P01
02
1
2
exp
exp
1
2
1
2
r
r
P
P
P
P
Rename the exponential Pr , (relative pressure) which is only a function of temperature, and is tabulated on the ideal gas tables
You can use this equation or
1
201
02 ln
P
PRss
![Page 31: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/31.jpg)
What if you know the volume ratio?
2
22
1
11
T
vP
T
vP
Ideal gas law
2
1
1
2
1
2
P
P
T
T
v
v
2
1
1
2
r
r
P
P
T
T
1
1
2
2
T
P
P
T r
r
Rename this vr2
Rename this 1/vr1
1
2
r
r
v
v
Relative specific volumes are also tabulated in the ideal gas tables
Remember, these relationships only hold for ideal gases and isentropic processes
![Page 32: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/32.jpg)
Summary
We developed the first and second Gibbs relationships
PdvduTds
vdPdhTds
![Page 33: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/33.jpg)
Which can also be expressed as
T
Pdv
T
duds
T
vdP
T
dhds
![Page 34: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/34.jpg)
SummaryFor solids and liquids
T
Pdv
T
duds
Solids and liquids do not change specific volume appreciably with pressure, so dv=0
0
T
CdT
T
duds
1
2lnT
TCs
C can be approximated as a constant in solids and liquids
![Page 35: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/35.jpg)
SummaryFor ideal gases if we assume constant heat capacities…
T
Pdv
T
duds
T
vdP
T
dhds
v
Rdv
T
dTCv
P
RdP
T
dTC p
![Page 36: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/36.jpg)
Which can be integrated to give
1
2
1
2 lnlnv
vR
T
TCs v
1
2
1
2 lnlnP
PR
T
TCs p
True for ideal gas with constant heat capacities
![Page 37: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/37.jpg)
SummaryFor ideal gases with variable heat capacity
1
201
02 ln
P
PRsss
![Page 38: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/38.jpg)
SummaryWhat if its not an ideal gas?
You’ll need to use the tables
![Page 39: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/39.jpg)
SummaryIsentropic Processes – Ideal Gas and Constant Heat Capacity
1
2
1
1
2
k
v
v
T
T
k
k
P
P
T
T1
1
2
1
2
1
2
2
1
P
P
v
vk constant
constant
constant1
1
k
k
k
k
Pv
TP
Tv
![Page 40: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/40.jpg)
SummaryIsentropic processes for Ideal gases – Variable Heat Capacities
1
201
02 ln
P
PRss
1
2
1
2
r
r
P
P
P
P 2 2
1 1
r
r
v v
v v
![Page 41: Entropy Property Relationships Chapter 7b. The T-ds relations Consider an internally reversible process occurring in a closed system.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfed1a28abf838cb8f26/html5/thumbnails/41.jpg)
SummaryIsentropic processes if the gas is not ideal and the heat capacities are variable
Use the tables!!