Transcript ChemE 260 - Thermodynamics

ChemE 260
Lost Work &
2nd Law Efficiency
Dr. William Baratuci
Senior Lecturer
Chemical Engineering Department
University of Washington
TCD 8: D
CB 7: Supplement
April 16, 2005
Lost Work
• Definition:
WS,lost  WS,rev  WS,act
– Applies to any device, so watch the signs carefully.
– WS,rev
• Rate at which work is done by or on a completely reversible process
operating between the same initial state AND the same final state as
the actual process.
• Assume for now that heat exchange only occurs with the
surroundings and not with a thermal reservoir.
Baratuci
ChemE 260
April 16, 2005
nd
2
Law Efficiency
• 2nd Law Efficiency:
ii ,comp 
W Sh,rev
ii,turb 
W Sh,act
W Sh,act
W Sh,rev
– More fair than S because initial and final states are the same.
– Best measure of performance for processes that produce or
consume work.
• In terms of Lost Work:
ii,comp 
WSh,act  WSh,lost
WSh,act
Baratuci
ChemE 260
April 16, 2005
ii,turb 
W Sh,act
W Sh,act  W Sh,lost
Apply 1st Law & Definition of S
•
1st Law:
ˆ
Q  WSh  m H
•
Solve for WS:
ˆ
WSh  Q  m H
•
From the definition
of entropy:
Q int rev
m
2
  T dSˆ  T Sˆ  Tsurr Sˆ
1
(Tsys = Tsurr, completely reversible)


•
For a reversible process:
ˆ
WSh,rev  m Tsurr Sˆ  H
•
For the actual process:
ˆ
WSh,act  Qact  m H
•
Lost Work:

ˆ    Qact  m H
ˆ 
WSh,lost   mTsurr Sˆ  m H

 

•
Canceling terms gives:
WSh,lost  mTsurr Sˆ  Qact
Baratuci
ChemE 260
April 16, 2005
2nd Law & External Sgen
•
2nd Law:
Q
m Sˆ  act  Sgen,int
Tact
•
Algebraic slight of hand:
 1
Qact
1
1 
 Qact 



Tact
T
T
T
act
surr 
 surr
•
More algebra:
 1
Qact Qact
1 

 Qact 


Tact Tsurr
T
T
surr 
 act
•
Substitute back into
2nd
Law:
•
External Entropy Generation:
•
A more robust statement
of the 2nd Law:
Baratuci
ChemE 260
April 16, 2005
 1
Qact
1 
ˆ
m S 
 Qact 

  Sgen,int
Tsurr
 Tact Tsurr 
S gen,ext
 1
1 
 Qact 


T
T
surr 
 act
Q
m Sˆ  act  Sgen,ext  Sgen,int
Tsurr
Lost Work
• 2nd Law:
Q
Q
m Sˆ  act  Sgen,ext  Sgen,int  act  Sgen,tot
Tsurr
Tsurr
• Total Entropy
Generation:
Sgen,tot
• Lost Work
revisited:
WS,lost  mTsurr Sˆ  Qact
• Algebra:
Tsurr Sgen,tot  mTsurr Sˆ  Qact
• BIG result:
WS,lost  Tsurr Sgen,tot
Baratuci
ChemE 260
April 16, 2005
Qact
ˆ
 m S 
Tsurr
Lost Work for a Cycle
•
Total entropy generation
for a cycle:
Sgen,tot
 m Sˆ 
Pr ocesses

i
•
Lost Work
for a cycle:
WSh,lost  Tsurr
Pr ocesses

i
•
BIG Result:
WSh,lost  Tsurr
Pr ocesses

i
•
Typical HE:
Baratuci
ChemE 260
April 16, 2005
WSh,lost
Qact,i
Tres,i


S
gen,tot


i
 Qact,i
Tres,i

QC,act QH,act

 Tsurr

 TC,res TH,res





Example #1
QH
•
Consider the HE shown here:
– QH = 800 J
QC = 300 J
WSh,act =
500 J
– TH = 1000 K
TC = 300 K
Tsurr = 300
K
• Determine
both the reversible and the lost
work.
•
Hot Reservoir
Method #1:
rev 
WSh,rev
QH
 1
HER
QC,rev
Wrev
Cold Reservoir
TC
300
 1
 0.7
TH
1000
WSh,rev  0.7 QH  0.7  800J   560J
WSh,lost  WS,rev  WS,act  560 J  500 J  60 J
•
Method #2:
Baratuci
ChemE 260
April 16, 2005


Q
Q
C,act
H,act 
WSh,lost  Tsurr 

 TC,res TH,res 


 300 J 800 J 
W Sh,lost  300 K 

  300  1  0.8  J  60 J
 300 K 100 K 
Example 8D - #2
• Why is there lost work for irreversible heat transfer ?
• How can we calculate the lost work for irreversible heat transfer
?
T1
T1
Qwaste
T2
Wlost
Qwaste
WHE
T1 > T2 > Tsurr
Qwaste
•
WHP
QC,HE
T2
•
HER
HPR
QC,HP
Tsurr
By allowing the heat to flow spontaneously from T1 to T2, we lost
the opportunity to recover work.
We call this “Lost Work”.
WHE > WHP
Wlost > 0
Baratuci
ChemE 260
April 16, 2005
Lost Work
•
1st Law,
Reversible HE:
•
1st Law,
Reversible HP:
•
Lost Work:
WHE  Qwaste  QC,HE
QC,HE

 Qwaste  1 
Qwaste



Tsurr 
  Qwaste  1 

T

1 

WHP  Qwaste  QC,HP
QC,HP

 Qwaste  1 
Qwaste



Tsurr 
  Qwaste  1 

T

2 

Wlost  WHE  WHP
1
1
Wlost  Qwaste Tsurr   
 T2 T1 
T1
Qwaste
Wlost
Qwaste
WHE
HER
QC,HE
Baratuci
ChemE 260
April 16, 2005
T2
HPR
WHP
T1 > T2 > Tsurr
Tsurr
QC,HP
Sgen & Suniv
•
Definition:
•
Reservoirs are isothermal and internally
reversible:
Sgen  Suniv  Sres,1  Sres,2
S res,1
S res,2
 Q waste

T1
Q waste

T2
 Q waste Q waste


T1
T2
•
Algebra:
S gen  S univ
•
Big result:
1
1
S gen  Suniv  Q waste   
 T2 T1 
Baratuci
ChemE 260
April 16, 2005
T1
T1 > T2 > Tsurr
Qwaste
T2
Conclusion
•
Lost Work:
•
Entropy
Generation:
•
Baratuci
ChemE 260
April 16, 2005
Conclusion:
1
1
Wlost  Qwaste Tsurr   
 T2 T1 
S gen  Suniv
1
1
 Q waste   
 T2 T1 
Wlost  Tsurr Sgen
Next Class …
• Ch 9 – Power Systems
– Review vapor & gas power systems
• With particular attention to the 2nd Law
– The Rankine Cycle
• A practical version of the Carnot Cycle
• Back-Work Ratio
• Improving Efficiency
– Increase boiler pressure
– Decrease condenser pressure
• Problem Session
Baratuci
ChemE 260
April 16, 2005