ChemE 260 - Thermodynamics
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Transcript ChemE 260 - Thermodynamics
ChemE 260
Entropy Generation
Fundamental Property Relationships
Dr. William Baratuci
Senior Lecturer
Chemical Engineering Department
University of Washington
TCD 7: C & D
CB 6: 2, 7 & 9
May 6, 2005
Principle of Increasing Entropy
• Clausius:
Q
0
T
• Apply to Cycle 1-A-2-B
Q
Q
Q
0
T 1 T A 2 T B
2
• Introduce
Entropy:
1
Q
S1 S 2
T
int rev
2
1
Q
1 T A S1 S 2 0
2
Q
S 2 S1
T
A
1
2
• Substitute into Clausius:
• Rearrange:
• Differential Form:
Q
dS
T
Baratuci
ChemE 260
May 6, 2005
Entropy Generation
• Newest Statement of the 2nd Law : dS
• Make Clausius into an equality:
Q
T
Q
dS
dSgen
T
• Entropy Generation, Sgen
Baratuci
ChemE 260
May 6, 2005
– Internally reversible processes:
S gen 0
– Irreversible processes:
S gen 0
– Impossible processes:
S gen 0
TS Diagram
• Area under
the curve:
2
Area T dSˆ
1
• But:
ˆ
ˆ
Q
Int Re v TdS
2
ˆ
ˆ
Area Q
Int Re v Q Int Re v
1
• Irreversible
ˆ
Q
Processes: dSˆ Irrev dSˆ gen
T
ˆ
ˆ
ˆ
Q
Irrev TdS TdSgen
2
ˆ
ˆ
Q
Irrev Area TdS gen
ˆ
Q
Irrev Area
1
• Sgen is a path variable ! We must use Sˆ gen:
Baratuci
ChemE 260
May 6, 2005
dS
Q
Sgen
T
Isolated Systems
• Entropy Statement of the
2nd
Law:
Q
S
Sgen
T
• Isolated Systems, Q = 0, therefore:
Sisolated Sgen 0
• The universe is an
isolated system:
Suniv Sgen 0
The Principle of Increasing Entropy
(a consequence of the 2nd Law)
• Divide the universe into two regions
– The system and the surroundings:
Suniv Ssys Ssurr Sgen 0
Baratuci
ChemE 260
May 6, 2005
The Gibbs Equations
• Definition of entropy:
QInt Re v TdS
• 1st Law, closed system, Wb only:
Q W dU
• Boundary work for an
internally reversible process:
WInt Re v P dV
• 1st Gibbs Equation:
dU TdS P dV
• Definition of enthalpy:
dH dU d(PV)
dU PdV VdP
• Substitute in the 1st Gibbs Eqn:
dH TdS P dV P dV V dP
• 2nd Gibbs Eqn:
dH TdS V dP
Baratuci
ChemE 260
May 6, 2005
Incompressible Liquids
• Assumptions:
V0
• Gibbs Equations:
dV 0
CP CV C
dU TdS PdV
dH TdS V dP
• Enthalpy, internal energy
and heat capacity:
dU CV dT C dT
dH CP dT C dT
• Result:
Baratuci
ChemE 260
May 6, 2005
dH dU
C
dS
dT
T
T
T
C
dS dT
T
Ideal Gases
•
Gibbs Equations:
dU TdS PdV
dH TdS VdP
•
Ideal Gas Heat
Capacities and EOS:
2
2
2
2
2
2
dU
P
S dS
dV
T 1T
1
1
dH
V
S dS
dP
T 1T
1
1
2
2
2
2
CoV
CoV
P
R
S
dT dV
dT dV
T
T
T
1
1
1
1 V
2
2
2
2
2
CoP
dH
V
R
S dS
dP
dT dP
T 1T
T
P
1
1
1
1
•
Integration yields:
V2
CoV
S
dT R Ln
T
V1
1
2
P2
CoP
S
dT R Ln
T
P1
1
2
Baratuci
ChemE 260
May 6, 2005
Using the Shomate Equation
• This method is accurate for ideal gases, but it is tedious.
T
2
CoP
1
B
C
D
2
3
2 E
dT
A
T
T
T
1000
dT
3
2
1 T
T T 1000 10002
1000
T
1
2
A Ln
1
T2
B
C/ 2
D/ 3
1
2
2
3
3
2 E
T
T
T
T
T
T
1000
2 1
2 1 10003 2 1
T1 1000
2 T22 T12
10002
2
2
T2
CoV
CoP R
CoP
T T dT T T dT T T dT R Ln T1
1
1
1
T2
Baratuci
ChemE 260
May 6, 2005
T
T
Ideal Gas Entropy Function
• Definition:
T
S
o
T
Tref
• Relationship with
Shomate Eqn :
• 1st Gibbs Eqns
for ideal gases :
CoP
dT
T
Where Tref is the reference
temperature at which S = 0.
T2
CoP
o
o
dT
S
S
T2
T1
T T
1
V
CoV
S
dT R Ln 2
T
V1
1
2
V2
T2
CoP
dT R Ln R Ln
T
T1
V1
1
2
• Substitute the Ideal Gas
Entropy Function :
Baratuci
ChemE 260
May 6, 2005
V2
T2
S S S R Ln R Ln
T1
V1
o
T2
o
T1
Ideal Gas Entropy Function
• Definition:
T
S
o
T
Tref
• Relationship with
Shomate Eqn :
• 2nd Gibbs Eqns
for ideal gases :
CoP
dT
T
Where Tref is the reference
temperature at which S = 0.
T2
CoP
o
o
dT
S
S
T
T1
T T
2
1
P2
CoP
S
dT R Ln
T
P1
1
2
• Substitute the Ideal Gas
P
Entropy Function :
S SoT SoT R Ln 2
2
Baratuci
ChemE 260
May 6, 2005
1
P1
Next Class …
• Polytopic Processes: P V = C
– More to this type of process than meets the eye
– Determination of Boundary Work
– PV Diagrams Revisited !
• Isentropic Processes
– A special case of the polytropic process
– The Relative Property Method of analysis
Baratuci
ChemE 260
May 6, 2005