The “Gheorghe Asachi” Technical University of Iasi...
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The „Gheorghe Asachi” Technical University of Iasi – an excellent choice for highschool graduates, who decided to opt for a career in the provoking field ofengineering. All our eleven faculties, endowed with laboratories and the latestequipment, where worldwide recognized specialists perform their activity, areready to welcome and assist you in your endeavour. I am convinced that thediploma awarded by our university will become the key of success both in yourcareer and personal life.
We look forward to welcoming you at our university!
The “Gheorghe Asachi” Technical University of Iasi
Romania
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Gouy – Stodola Theorem
Prof. Dr. Eng. Gheorghe DUMITRASCU“Gheorghe ASACHI” Technical University of Iasi
Mechanical and Automotive Engineering Departmenthttp://www.tuiasi.ro
http://www.mec.tuiasi.ro/indextmtfc.html
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Assumptions
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1. They will be analyzed non steady-state enlarged basic open thermodynamic systems, including both the thermal system and, the external heat reservoirs controlling the heat transfers and, the environment allowing the mass transfers;
2. The working fluid is a mixture of different chemical species, the inlet and outlet compositions might be different because of chemical reactions that can appear during the flow through the thermal system;
3. The inner boundary of the flow path through the thermal system is deformable under the environmental pressure;
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Gouy – Stodola Theorem, EnginesFirst Law of Thermodynamics
Second Law of Thermodynamics
( )
∑∑
++−
+++
∂∂
−−−=∂∂
outlet
22
ee0
gZ2
VhmgZ2
Vhm
tVpWQQ
tU
inlet
0smsmTQ
TQ
tSS
outlet0
0irrevgen ≥⋅+⋅−
−−
∂∂
= ∑∑
inlet
M: Mass of the working fluid surrounded by the operating engine inner walls at a certain operational timeV: Working fluid volume defined by the operating engine inner walls at a certain operational timeU: Inner operating engine working fluid energy at a certain operational timeS: Entropy of the working fluid surrounded by the operating engine inner walls at a certain operational time
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inlet
outlet
Heat sinkT0
S
MU
Heat sourceT>T0
V
0Q
Q
eW
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Gouy – Stodola Theorem, Engines
Combined First and Second Laws of Thermodynamics
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( ) ( )
( ) irrevgene
outletinlet
irrevlost
revee
STSTVpU t
gZVsThmgZVsThm
TTQWWW
00
2
0
2
0
0
22
1
−−+∂∂
−
++−−
++−+
−=+=
∑∑
inlet
outlet
Heat sinkT0
S
MU
Heat sourceT>T0
V
0Q
Q
eW
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Gouy – Stodola Theorem, Engines
Reversible Engine Power
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( ) ( )
( )
methalpy the is gZVhh
where
STVpU t
sThmsThm TTQW
*
e
outletinlet
reve
++=
>−+∂∂
−
−−−+
−=
∑∑
2
0
1
2
0
0*
0*
0
inlet
outlet
Heat sinkT0
S
MU
Heat sourceT>T0
V
0Q
Q
eW
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Gouy – Stodola Theorem, Engines
Irreversible Lost Power
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irrevgen
irrevlost STW ⋅−= 0
inlet
outlet
Heat sinkT0
S
MU
Heat sourceT>T0
V
0Q
Q
eW
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Gouy – Stodola Theorem, Refrigeration Cycles
First Law of Thermodynamics
Second Law of Thermodynamics
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( )
∑∑
++−
+++
∂∂
−−−=∂∂
outlet
22
ee0
gZ2
VhmgZ2
Vhm
tVpWQQ
tU
inlet
0smsmTQ
TQ
tSS
outlet0
0irrevgen ≥⋅+⋅−
−−
∂∂
= ∑∑
inlet
Q
0Q
eW
inlet
outlet
Heat sourceT
Heat sinkT0
S
MU
V
M: Mass of the working fluid surrounded by the operating engine inner walls at a certain operational timeV: Working fluid volume defined by the operating engine inner walls at a certain operational timeU: Inner operating engine working fluid energy at a certain operational timeS: Entropy of the working fluid surrounded by the operating engine inner walls at a certain operational time
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Gouy – Stodola Theorem, Refrigeration Cycles
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Combined First and Second Laws of Thermodynamics
( ) ( )
( ) irrevgene
outletinlet
irrevlost
revee
STSTVpU t
gZVsThmgZVsThm
TTQWWW
00
2
0
2
0
0
22
1
−−+∂∂
−
++−−
++−+
−=+=
∑∑Q
0Q
eW
inlet
outlet
Heat sourceT
Heat sinkT0
S
MU
V
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Gouy – Stodola Theorem, Refrigeration Cycles
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Reversible Consumed Power
( ) ( )
( )
methalpy the is gZVhh
where
STVpU t
sThmsThm TTQW
*
e
outletinlet
reve
++=
<−+∂∂
−
−−−+
−=
∑∑
2
0
1
2
0
0*
0*
0
Q
0Q
eW
inlet
outlet
Heat sourceT
Heat sinkT0
S
MU
V
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Gouy – Stodola Theorem, Refrigeration Cycles
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Irreversible Lost Power
irrevgen
irrevlost STW ⋅−= 0
Q
0Q
eW
inlet
outlet
Heat sourceT
Heat sinkT0
S
MU
V
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Gouy – Stodola TheoremConclusions
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0→irrevgenS 00 →⋅−= irrev
genirrev
lost STW
revstoragee
revflowe
revQe
reve WWWW ,,,
++=
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Gouy – Stodola TheoremConclusions
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revstoragee
revflowe
revQe
reve WWWW ,,,
++=
•Relationship Q – W, engines, (heat exergy)
0TT1QW 0rev
Qe >
−=
,
•Relationship Q – W, refrigeration cycles, (heat exergy)
( ) 0TTT
Q1TTQW
0
0revQe <
−−=
−−=
,
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Gouy – Stodola TheoremConclusions
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revstoragee
revflowe
revQe
reve WWWW ,,,
++=
Relationship flow– W, (flow exergy)
( ) ( )∑∑ −−−=outlet
00revflowe sThmsThmW *
inlet
*,
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Gouy – Stodola TheoremConclusions
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revstoragee
revflowe
revQe
reve WWWW ,,,
++=
Relationship “energy storage” – W, non steady-state system operation,(storage exergy)
( )STVpU t
W 0erevstoragee −+
∂∂
−=,
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Introduction to Irreversible Cycles
Prof. Dr. Eng. Gheorghe DUMITRASCU“Gheorghe ASACHI” Technical University of Iasi
Mechanical and Automotive Engineering Departmenthttp://www.tuiasi.ro
http://www.mec.tuiasi.ro/indextmtfc.html
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Introduction to Irreversible Cycles
AssumptionsNo mass transfer:
and
Non deformable boundary walls:
Steady state operation:
0gZ2
VhmgZ2
Vhmoutlet
22
=
++−
++ ∑∑
inlet
0tVpe =∂∂
0tS and 0
tU
=∂∂
=∂∂
0smsmoutlet
=⋅+⋅− ∑∑
inlet
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Introduction to Irreversible Closed CyclesEndo-reversible Carnot Cycle
Entropy Generation by IrreversibilityEnlarged System – including both the thermal system and
the external heat reservoirsRelated to overall Irreversibility (external + internal)
T0
T
(T0+∆T0)
T
(T—∆T)
(T—∆T)
(T0+∆T0)
Q
Q
Q0
Q0
W
W 0TQ
TQ
S0
0overallgen ≥+−=
Entropy Generation by IrreversibilityThermal System – excluding the external heat reservoirs
Related to internal irreversibility
0TT
QTT
QS
00
0cyclegen ≥
∆−+
∆+−=
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or Nirrev is the overall irreversibility function related to T and T0,
is the number of internal irreversibility related to (T - ∆T) and (T0 - ∆T0),
Irr is the internal irreversibility function related to other reference temperatures on the cycle
and different from (T - ∆T) and (T0 - ∆T0)
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Introduction to Irreversible Closed CyclesEndo-reversible Carnot cycle
Entropy BalanceEnlarged System – including both the thermal system and the
external heat reservoirs
T0
T
(T0+∆T0)
T
(T—∆T)
(T—∆T)
(T0+∆T0)
Q
Q
Q0
Q0
W
W
0TQ
TQ
Irr0
0 =+−
Entropy BalanceThermal System – excluding the external heat reservoirs
Related to internal irreversibility
0TQIrr
TQ
TTQN
TTQ
0
0internalirrev
00
0 =+−=∆−
+∆+
− **
*0T
*T
IrrinternalirrevN
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Theat
source1
2rev
2irrev
s
equivalent to a throtling
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Cyclic Heat Input
Introduction to Irreversible non-Carnot Closed Cycles
( )1rev2rev21
mqrev21irrev12
qmqrevinputheat
rev2irrev2rev2irrev2
ssTmQQ
sTmQQTThh
−==
∆==
≅⇒=
−− ,
( )rev2irrev2irrev sss ,, −=∆
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Cyclic Heat Output
Introduction to Irreversible non-Carnot Closed Cycles
( )3rev4rev43
mqrev43irrev34
qmqrevinputheat
rev4irrev4rev4irrev4
ssTmQQ
sTmQQTThh
−==
∆==
≅⇒=
−− ,
3
4,rev4,irrev
T
s
heatsinkT0
equivalent to a throttling( )rev2irrev2irrev sss ,, −=∆
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Irreversible First Law Efficiency - Engines
Introduction to Irreversible non Carnot Closed Cycles
∆θ+−=
∆−−=
−=+
==η=
−qSLT
gen0
qt3r2
mq
gen00
rev
gen0
rev
reve
rev
irreve
revee
irrevirrevengines
smS
1TT1
sTmST
TT1
QST
QW
QWW
QWFLE
( )TTTT1N
TT1 0Carnot
01N
irrev0
irrev
irrev,η=−<−=η
>
Where T is the temperature of the heat source and, is a second law of thermodynamics correction
function linking the heat transfer to the involved mean thermodynamic temperatures
t3r2mq
SLT TT−=θ
Enlarged System Entropy Balance Equation
0TQN
T
Q
0
0irrev =+−
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Irreversible First Law Efficiency - Refrigeration Cycles
Introduction to Irreversible non Carnot Closed Cycles
rev
gen00
TT
rev
irrevloste
rev
reve
irrevloste
reve
rev
e
rev0W
eirrev
irrevionrefrigerat
QST
TT1
1
QW
QW
1
WWQ
WQ
WQCOPFLE
0
e
−−−=
+−=
+−=−===
<
<
,
,
∆θ+−
−=
∆θ−−
−=
−qSLT
gen0
qt5r4
mqSLT
gen00irrev
smS
1TT1
1
sTmST
TT1
1COP
( )0Carnot0
1N
irrev0
qSLT
gen0
irrev TTCOPTT
TTNT
T
Tsm
S1T
TCOPirrev
,=−
<−
=
−
∆θ+
=>
t5r4mq
SLT TT−=θWhere T is the temperature of the heat source and, is a second law of thermodynamics correction
function linking the heat transfer to the involved mean thermodynamic temperatures
Enlarged System Entropy Balance Equation
0TQN
T
Q
0
0irrev =+−
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Conclusions
1. ηirrev and COPirrev includes explicitly the overall irreversibility (internal and external one)
2. has the meaning of the overall number of irreversibility, it evaluates both
the external irreversibility due to the heat transfer at finite temperature differences, and the internal ones.
3. When Sgen → 0 then Nirrev → 1 ,
4. The correction function θSLT links the heat input at temperature T to the heat absorbed by the working
fluid at temperature Tmq < T
∆θ+=
qSLT
genirrev sm
S1N
Introduction to Irreversible non Carnot Closed Cycles
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Conclusions – Engines
T0 = 273,15 K
( )TTTT1N
TT1FLE 0Carnot
01N
irrev0
irrevirrevengines
irrev,η=−<−=η=
>
Introduction to Irreversible non Carnot Closed Cycles
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Conclusions – Refrigeration cycles
T0 + 293,15 K
Introduction to Irreversible non Carnot Closed Cycles
( )0Carnot0
1N
irrev0
qSLT
gen0
irrev TTCOPTT
TTNT
T
Tsm
S1T
TCOPirrev
,=−
<−
=
−
∆θ+
=>
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INFLUENCE OF WORKING FLUID PROPERTIES ON INTERNAL IRREVERSIBILITY OF POWER SYSTEMS
Gheorghe DUMITRASCU
Engineering Thermodynamics, Thermal Machines, and Refrigeration Dept.,
“Gh. Asachi” Technical University of Iasi,
ROMANIA,
[email protected], http://www.tuiasi.ro
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THE WAY TO OPTIMIZE THE IRREVERSIBLE CYCLES
The optimization of cycles: first and second law efficiencies (exergy analysis), entropy generation minimization, and sometimes Novikov–Curzon–Ahlborn maximum power issue.
The paper presents a concise method to evaluate directly the irreversibility, inside a unique criterion uniting first and second laws, called here the irreversible first law efficiency.
Key words: NTUS, second law effectiveness of external heat exchangesirreversible maximum power,number of internal irreversibility, number of external irreversibilityirreversible first law efficiency
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Engines Reversibility Principle- engineering realm -
The possible ideal engine cycles in temperature – entropy diagram
s1=s2 s3=s4
3
4
2 3
1 4
T
T
T0
dT
dT0
QR
2
s
c1-2=c3-4 ( )04312 ln TTcssss n=−=−
TT
rev01−=η
Any ideal cycle is characterized by no entropy generation
All Complete Reversible Engine Cycles
.0Sgen =
Non – CARNOT complete Reversible Engine Cycles
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Introduction“Optimization of Irreversible Cycles” , FTT or FST
The ideal cycles are characterized by no entropy generation,
The Finite Time Thermodynamics (FTT), or Finite Speed Thermodynamics (FST) is a featureof the originator works of:
CHAMBADAL (1957) and NOVIKOV (1958) – studies about nuclear cycles, CURZON et AHLBORN (1975) – the time-based thermodynamic analysis in view of the real
heat transfer made at finite temperature difference, YAN et al. (1989), GROSU et al. (2004), CHEN et al. (1997, 1999) – analyzed the
irreversible cycles with three external heat sinks
THE IDEAL REVERSIBLE CYCLES ARE CONSIDERED AS USELESS, THEY MIGHT SUPPLY THEMAXIMUM ENGINES WORK, BUT NO POWER. THE REQUIRED TIME BY THE REVERSIBLE
HEAT TRANSFER AT INFINITESIMAL TEMPERATURE DIFFERENCE REQUIRES AN INFINITEPROCESS TIME (I.E. THE POWER, WHICH IS RATIO WORK PER TIME IS IN FACT ZERO).
.0Sgen =
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The Irreversible First Law EfficiencyExternal Irreversibility
Endo-Reversible CARNOT Engine
The second law effectiveness of the heat exchange with the heat source
T
s
T
T0
∆T
∆T0
3
1 4
2
Heat exchanger at the hot reservoir
Heat exchanger at the cold sink
Endoreversible
CARNOT cycleqsm
UANTUS∆
=
Number of Transfer Units per Reversible Entropy Variationdue to reversible heat transfer
( ) revq QsTTmTUAQ
=∆∆−=∆=
Basic Equations
⇒+
=∆1NTUS
1TT ( )1NTUS
NTUSsTmsTTmQ qq +∆=∆∆−=
( ) ⇒<+
==ε∞→
11NTUS
NTUSQ
Q
NTUSII
The second law effectiveness at the heat exchange at the hot reservoir
( ) 0T ss
sTmQQQ
NTUSqq
qIIrev
IINTUSII
→∆∆=∆
⇒∆ε=ε=ε=
∞→
∞→
,max
revQQ max <
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The second law effectiveness of the heat exchange with the heat sinkT
s
T
T0
∆T
∆T0
3
1 4
2
Heat exchanger at the heat source
Heat exchanger at the heat sink
EndoreversibleCARNOT cycle
Basic Equations
The second law effectiveness of the heat exchange at the cold sink
0q
000 sm
AUNTUS,∆
=
( ) 0q000000 sTTmTAUQ ,∆∆+=∆=
⇒−
=∆1NTUS
1TT0
00 ( )1NTUS
NTUSsTmsTTmQ0
00q00q000 −
∆=∆∆−= ,,
( ) ⇒>−
==ε∞→
11NTUS
NTUSQ
Q
0
0
NTUS0
00II
0
, ( )0T ss
sTmQQQ
0NTUS0q0q
0q00II0IINTUS000
0
0
→∆∆=∆
⇒∆ε=ε=ε=
∞→
∞→
,,,
,,min,
minQQ0 >
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Number of Transfer Units per Reversible Entropy Variationdue to reversible heat transfer
The Irreversible First Law EfficiencyExternal Irreversibility
Endo-Reversible CARNOT Engine
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The dependence second law effectiveness – NTUS
0 0.5 1 1.5 2
∞→NTUS Decreasing
irreversibility: 0→∆T
sTmQQ ∆=→
max
0NTUS←∞ Decreasing
irreversibility: 00 T∆←
0min0 QQsTm
←=∆
ε → 1 The best way
1 ← ε0 The best way
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The Irreversible First Law EfficiencyExternal Irreversibility
Endo-Reversible CARNOT Engine
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The Power
εε
τ−ε∆=
εε
−ε=
−=−=
II
0IIIIq
II
0IIII
00
11sTmQ
Q1Q
Q
Q1QQQP ,,
max
minmax
The Irreversible First Law Efficiency
extirrII
0III N1111 ,
,
τ−=
εε
τ−=η
The Second Law Efficiency
11N11
N11IIextirr
extirr
II =η⇒→⇒
τ−
τ−
=η ,
,
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The Irreversible First Law EfficiencyExternal Irreversibility
Endo-Reversible CARNOT Engine
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Considerations on the Real Power Cycles
The real power cycles are also internally irreversible, respectively: the heat transfer processes, 2–3 and 4–1, are non-isothermal; the adiabatic processes, 2–3 and 4–1, are non-isentropic; the external heat reservoirs have finite heat capacities, respectively are non-
isothermal; the mean log temperature difference pertaining to the heat transfers has a value that it
is not equalizing the difference of the mean thermodynamic temperatures.
This dissimilarity can be solved step by step by introducing the necessary amendment coefficients in order to take into account also the internal irreversibility.
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The second law effectiveness of the heat exchange with the heat source
qsmUANTUS∆
=
Number of Transfer Units per Entropy Variation
Basic Equations
The second law effectiveness at the heat source
Tmq0
TmqΔTmq
ΔTmq0
T
s( ) qmqmqTmq sTTmCTUAQ ∆∆−=∆= ∆
⇒+⋅
=∆∆ 1CNTUS
1TTT
mqmq ( )1CNTUS
CNTUSsTmsTTmQT
Tqmqmqmq +⋅
⋅∆=∆∆−=
∆
∆
( ) ⇒<+⋅
⋅==ε
∆
∆∆
→∞→∞
11CNTUS
CNTUSCQ
Q
T
Ts
ANTUSII q
( )
( )→∞
→∞∆∆
→∞→∞
∆∆
=∆
∆=
⇒∆ε=ε=
ANTUSq
qs
mq
treansferheat
meanT
ANTUSqmqIIII
ss
C T
TC
sTmQQ
q,
max
maxQQ <
The Irreversible First Law EfficiencyExternal Irreversibility
Any Endo-Reversible Engine
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The second law effectiveness of the heat exchange with the heat sink
0q
000 sm
AUNTUS∆
=
Number of Transfer Units per Entropy Variation
Basic Equations
The second law effectiveness at the heat sink
Tmq0
TmqΔTmq
ΔTmq0
T
s( ) 0q0mq0mq0mq000 sTTmCTAUQ
0T∆∆+=∆=
∆
⇒−⋅
=∆∆ 1CNTUS
1TT0T0
0mq0mq ( )1CNTUS
CNTUSsTmsTTmQ0T0
0T00q0mq0q0mq0mq0 −⋅
⋅∆=∆∆−=
∆
∆
( ) ⇒>−⋅
⋅==ε
∆
∆∆
→∞→∞
11CNTUS
CNTUSCQ
Q
0T0
0T0s
ANTUS0
00II 0q
0
,
( )→∞
→∞∆∆
→∞→∞
∆∆
=∆
∆=
∆ε=ε=
ANTUS0q
0qs
mq
treansferheat
mean0T
ANTUS0q0mq0II0II0
00q s
sC
TTC
sTmQQ
,
,min,
minQQ0 >⇒
The Irreversible First Law EfficiencyExternal Irreversibility
Any Endo-Reversible Engine
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1 – 2r irreversible adiabatic compression, 2r – 3r irreversible heating 3r – 4r irreversible adiabatic expansion, 4r – 1 irreversible cooling
r2t3q sss −=∆
int,irrq
14irr
43irr
32irr
21irrq
t5r40q
Ns
sssss
sss
⋅∆=
=∆+∆+∆+∆+=
−=∆−−−−
( )1
ss
1Nq
irrirr >
∆
∆+= ∑ int
int,
4t
1=5r
2t 2r
3t 3r
4r
5t
21irrs −∆
32irrs −∆
43irrs −∆
14irrs −∆
s
T
The Irreversible First Law EfficiencyInternal Irreversibility
Any Basic Irreversible Engine
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engineintirrt3r2
mq
t5r4mq
qt3r2
mq
0qt5r4
mq
t3r2
t5r40
t3r2engine
NTT
1sTmsTm
1
Q
Q1
QP
,−
−
−
−
−
−
−
−=∆⋅⋅
∆⋅⋅−=
=−==η
4t
1=5r
2t 2r
3t 3r
4r
5t
21irrs −∆
32irrs −∆
43irrs −∆
14irrs −∆
s
T
The Irreversible First Law EfficiencyInternal Irreversibility
Any Basic Irreversible Engine
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4t
1=5r
2t 2r
3t 3r
4r
5t
21irrs −∆
32irrs −∆
43irrs −∆
14irrs −∆
s
T 1T
TC
sTmsTm
mq
t3r2mq
sqmq
qt3r2
mq32
t3r2
II q<=
∆
∆==ε
−
∆
−
−
−
max
1TT
CsTmsTm
0mq
t5r4mq
s0q0mq
0qt5r4
mq54
t5r4
0II 0q>=
∆
∆==ε
−
∆
−
−
−
min,
1C
CN
0q
q
s
s
II
0IIextirr >
εε
=∆
∆,,
int,,int,,
irrextirrirrs
s
II
0II
mq
0mqengine NN11N
C
C
TT
10q
q ⋅τ
−=εε
−=η∆
∆
The Irreversible First Law EfficiencyExternal and Internal Irreversibility
Any Basic Irreversible Engine
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1 – 2r irreversible adiabatic compression, 2r – 3r internal irreversible heating by heat regeneration, 3r – 4r irreversible heating, 4r – 5r irreversible adiabatic
expansion, 5r – 6r internal irreversible cooling by heat regeneration, 6r – 1 irreversible cooling.
4t
1=7r
2t 2r
3t3r
4r
6t
T
s
5r5t6r
7t
The Irreversible First Law EfficiencyInternal Irreversibility
Any Irreversible Engine with Internal Heat Transfer (Internal Regeneration of Heat)
r3t443q ssss −=∆=∆ −
( ) ( )r6t5t6r5
mqr2t3t3r2
mq ssTssT −=− −−
( )
76irr
65irr
54irr
43irr
32irr
21irr
t6r5mq
t3r2mq
r2t3qt7r60q
ssssss
TT
1ssssss
−−−−−−
−
−
∆+∆+∆+∆+∆+∆
+
−−+∆=−=∆
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The Irreversible First Law EfficiencyInternal Irreversibility
Any Irreversible Engine with Internal Heat Transfer (Internal Regeneration of Heat)
4t
1=7r
2t 2r
3t3r
4r
6t
T
s
5r5t6r
7t
( ) ( ) 1TT
1s
ssss
1N t6r5mq
t3r2mq
q
r2t3
q
irrirr >
−
∆−
+∆
∆+= −
−∑ intint,
int,irrt4r3mq
t7r6mq
qt4r3
mq
0qt7r6
mq0engine
NTT
1
sTmsTm
1Q
Q1
QW
−
−
−
−
−=
=∆⋅⋅
∆⋅⋅−=−==η
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The Irreversible First Law EfficiencyExternal and Internal Irreversibility
Any Irreversible Engine with Internal Heat Transfer (Internal Regeneration of Heat)
4t
1=7r
2t 2r
3t3r
4r
6t
T
s
5r5t
6r
7t
0
0 ,0,int
, ,int
1
11
εη
ε
τ
∆
∆
= − =
= − ⋅ ⋅
q
q
smq IIengine irr
mq II s
irr ext irr
CTN
T C
N N
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The Irreversible First Law EfficiencyInternal Irreversibility
Any Irreversible Co-generation Engine
r2t332q ssss −=∆=∆ −
( )
∆
∆+∆=∆
+∆=−=∆=∆
∑∑=
=
−
−
q
engineirr
q
1432y
4321x
yxirr
qt6r4140q
ss
1ss
sssss
int,,,
,,,
( )
∆
∆−∆+∆=∆−∆=−=∆=∆ ∑
−−q
cogengine
irrqcog14t6t515
cog0q s
ss1sssssss int
,
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The Irreversible First Law EfficiencyInternal Irreversibility
Any Irreversible Co-generation Engine
( )
cogirrt3r2
mq
t6r4mq
32t3r2
mq
54t5r4
mq14t6r4
mq
32
5414
32
54overall
NTT
1
sTmsTsTm
1
Q
QQ1
Q
QP
int,−
−
−−
−−
−−
−
−−
−
−
−=
=∆⋅⋅
∆⋅−∆⋅⋅−
=−
−=+
=η
( )
∆
∆−
∆
∆+=
−−
−
−
∑32
cogt6r4
mq
t5r4mq
32
engineirrcog
irr ss
TT
ss
1N intint, For endo-reversibility 0Ncog
irr →int,
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The Irreversible First Law EfficiencyExternal and Internal Irreversibility
Any Irreversible Co-generation Engine
For endo-reversibility and 0Ncogirr →int,
int,,
int,,
irrextirr
irrs
s
II
0II
mq
0mqoverall
NN11
NC
C
TT
10q
q
τ−=
=εε
−=η∆
∆
1overall →η
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Numerical Results
The computational procedure assumed the following restrictive conditions. Variable heat capacities of gases, fourth degree temperature polynomials;Adiabatic exponents of the reversible processes computed as the ratio of enthalpy variation to internal energy variation.In the case of engine heated by combustion, they were considered negligible dissociation during combustion, and the mass and energy balance equations of combustion gave the flue gases composition, the excess air value, and the fuel mass flow rate.
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Numerical Results
First Figures compare the oxy-combustion with 90% mass fraction of O2 in the oxygenated air, and the air combustion, for an open Joule – Brayton cogeneration cycle. The restrictive conditions were: maximum temperature on the cycle, 1200°C; isentropic efficiency of the compressor, 0.88; isentropic efficiency of the gas turbine, 0.94; pressure loss coefficient in the combustion chamber, 0.98; the fuel mass composition of 15% H2 and 85% C; higher heating value of the fuel ~46,000 kJ/kg.Next Figures compare six possible working fluids, air, O2, N2, CO2, H2 and He, in a closed Joule –Brayton engine cycle externally heated, e.g. by solar energy. The restrictive conditions were: maximum temperature on the cycle, 1000°C; the minimum temperature on the cycle, 20°C; isentropic efficiency of the compressor, 0.88; isentropic efficiency of the gas turbine, 0.94; pressure loss coefficient in the heat exchanger connecting the external hot source, 0.98; pressure loss coefficient in the heat exchanger connecting the external cold sink, 0.98; variable heat capacities; adiabatic exponent computed as the ratio of heat capacities at constant pressure and constant volume.
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Numerical Results
The dependence number of internal irreversibility – compression ratio for the basic engine cycle
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Numerical Results
The dependence overall number of internal irreversibility – compression ratio for the cogeneration cycle
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Numerical Results
The dependence specific power output on compression ratio for the basic engine cycle
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Numerical Results
The dependence irreversible first law efficiency (thermodynamic efficiency) on compression ratio for the basic engine cycle
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Numerical Results
Comparison of the dependence irreversible first law efficiency (thermodynamic efficiency) on compression ratio for both the basic engine cycle and the engine cycle with internal regeneration of heat, only for
CO2 as working fluid
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Numerical Results
The dependence number of internal irreversibility on compression ratio for the basic engine cycle
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Conclusions
This paper deals with a new optimization criteria, the irreversible first law efficiency applied to the real cycles. This new approach emphasizes the overall irreversibility by the means of the numbers of internal and external irreversibility, directly inside the first law efficiency.It yields that the maximum power is not unique, as it was assumed untill now, respectively depends on the nature of the working fluid and on the restrictive conditions.The irreversible first law efficiency conducts to the reversible limit, i.e. the reversible Carnot engine, respectively offers the simplest comparison.
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Conclusions
For complex cycles, we have only to deal carefully with the entropy variations and thermal interactions during the heat transfer processes.
Important remark: the second law effectiveness, defined in this paper, takes into account only the thermal interactions connecting the working fluids with external heat reservoirs, the internal heat exchanges of complex cycles, e.g. in the case of combined cycles, the heat transfer between the top cycle and the bottom one is included in the number of internal irreversibility; for instance, refer to the simple manner to manage the internal heat exchange of Power Cycles with Internal Heat Transfer.
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Conclusions
The direct method presented in this presentation is a concise one, equivalent to exergy analysis, since the lost exergy is proportional to the entropy irreversibly generated. The comparison of possible fluids used in a closed Joule – Brayton cycle, externally heated, shows that CO2 offers a substantial potential for internal regeneration of heat, see the below table, that includes the difference of temperatures of the fluid leaving the turbine and the compressor.
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Conclusions
Table 1. The difference between temperatures of the fluid leaving the turbine and the compressor, [°C]
Compression Ratio Air O2 N2 CO2 He H2
5,00 401 423 397 593 123 367
10,00 158 191 152 431 41 115
15,00 17 57 9 337
20,00 272
25,00 221
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Prof. Dr. Eng. Gheorghe DUMITRASCU“Gheorghe ASACHI” Technical University of Iasi
Mechanical and Automotive Engineering Department
Prof. PhD, Michel FEIDTINSTITUT NATIONAL POLYTECHNIQUE DE LORRAINE ET UNIVERSITE HENRI POINCARE
LABORATOIRE D'ENERGETIQUE ET DE MECANIQUE THEORIQUE ET APPLIQUEEC.N.R.S. UMR 7563
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The Influences of the Condenser and Evaporator Effectiveness on the Heat Pump Performances
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Overview
1. Introduction
2. Mathematical model
3. Results and discussion
4. Conclusions
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1. Introduction First and Second Laws Relationship – CV-HP
• First Law – Energy Balance
3
4t1
2t 2r
3t3r
4r1t
s
T
21irrs −∆
32irrs −∆
43irrs −∆
14irrs −∆
( )int∑∆+∆=
∆+∆+∆+∆+∆=−=∆
−=∆−−−−
irrevq
43irrev
32irrev
21irrev
14irrevqr2t30q
r4t1q
ss
ssssssss
sss
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( )( )
0QQQQW
0sTmssTmQQQ
0sTmssTmQQQ
00
0qt3r2
mqr2t3t3r2
mqt3r2c0
qt1r4
mqr4t1t1r4
mqt1r4e
<−=+=
<∆=−===
>∆=−===−−
−
−−−
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1. Introduction First and Second Laws Relationship – CV-HP
• First Law – Energy Balance
4
4t1
2t 2r
3t3r
4r1t
s
T
21irrs −∆
32irrs −∆
43irrs −∆
14irrs −∆
( )
1IrrTT
IrrTT
1NTT
NTT
1QQ
sTsTmsTm
QQQ
COP
e
c
e
cge
HP
irrt1r4mq
t3r2mq
irrt1r4mq
t3r2mq
0
0
qt1r4
mq0qt3r2
mq
0qt3r2
mq
0
0HP
−=
−=
−
=
=∆−∆
∆=
−=
−
−
−
−
−−
−
..
int,
int,
( )c
e32
ref
t3r2mq
irrq
irrirr T
TT
TNIrr or, 1
ss
1N −
−
=>∆
∆+= ∑
int,int
int,
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1. Introduction First and Second Laws Relationship – CV-HP
• Second Law – entropy balance• Internal irreversibility
5
4t1
2t 2r
3t3r
4r1t
s
T
21irrs −∆
32irrs −∆
43irrs −∆
14irrs −∆
0TQ
IrrTQ HP)-(CV e.g.or
0TQ
IrrTQor 0
TQ
NT
Q
c
c
e
e
32ref
014
reft3r2
mq
0irrt1r4
mq
=−
=−=− −−−−
int,
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1. Introduction First and Second Laws Relationship – CV-HP
• United First and Second Laws
6
4t1
2t 2r
3t3r
4r1t
s
T
21irrs −∆
32irrs −∆
43irrs −∆
14irrs −∆
1COP
COPCOPR where
TTCOPRIrr HP)-(CV e.g.or
TTCOPRIrror
TT
COPRN
HP
HP
c
e
32ref
14ref
t3r2mq
t1r4mq
irr
−=
=
== −
−
−
−
int,
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2. Mathematical model
7
Thermal Relationships of a Solar Assisted Compression
Vapor Heat Pump (CV-HP)
SC
E
K
CTV
Tsco Tsci
Tho Thi
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2. Mathematical model
8
• Solar collector
– Collector temperature under non-flow conditions
– Collector plate temperature
– Inlet temperature of the solar heat carrier
– Outlet temperature of the solar heat carrier
ηo is optical efficiency of solar collector, Ta is environmental temperature, UL is solar collector
heat-loss coefficient; εsc is solar collector effectiveness; I is total solar radiation.
L
oas U
ITT η+=
L
sssc U
ITT η−=
scsc
sscsci C
IATTε
η−=
sc
sscisco C
IATT
η+=
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2. Mathematical model
9
• Heating systemInlet/outlet temperatures of the useful heat carrier, were imposed by considering that the
heating system asks:
at Ta = TaN = 253.15K, Thi = ThiN = 323.15K and, Tho = ThoN = 333.15K;
at Ta = TaS = 288.15K, Thi = ThiS = 303.15K;
for 253.15 K < Ta < 288.15 K:
Troom = 295.15K.
807143.4675714286.0 +⋅−≅ aho TT
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( )( ) ( )ahi
aNroom
aroom
hoNhiN
hohi
aNheating
aheating TTTTTT
TTTT
TQTQ
⇒−−
≅−−
=
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2. Mathematical model
10
• Evaporator
evaporator heat rate
evaporator temperature
Where εe is evaporator effectiveness.
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( )sciscosce TTCQ −=
sce
escoe C
QTT
ε−=
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2. Mathematical model
11
• Heat pump irreversibility (entropy and energy balance)
cc
c
e
e Q 0TQ
TQIrr
⇒=−
( )c
e
c
e
HP
HP
TTCOPR
TT
COP1COPIrr =
+=
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2. Mathematical model
12
• CondenserCondensing temperature
Where εc is condenser effectiveness
hc
choc C
QTT
ε+=
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By CoolPack, and for adopted restrictive conditions inside the heatpump, it was interpolated, by square roots, the irreversibility, for303.15K < Tc < 338.15K, and, 258.15K < Te < 288.15K:
• refrigerant R22, errors: +1.1%, – 3.1%
• refrigerant R717, errors: +1.1%, – 1.8%
• refrigerant R410A, errors: +1.1%, – 6.6%
• refrigerant R407c, errors: +1.1%, – 4.9%
• refrigerant R290, errors: +1.1%, – 3.7%
• refrigerant R134a, errors: +1.1%, – 3.6%
2. Mathematical model
13
e
c
TT8506271909710030COPR .. +−=
e
c
TT5683381602290COPR .. +−=
e
c
TT5512542683264721COPR .. +−=
e
c
TT27284443123710456421COPR .. +−=
e
c
TT9518862751023481361COPR .. +−=
e
c
TT9103469052197969440COPR .. +−=
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2. Mathematical model
14
• Power balance
cece QQQQW −=+=
WQ
COP cHP
=
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3. Results and discussion
• The correlated compilation of previous equations allowed in MathLab to find out numerical results, depending only on εeand εc. Figures 2 to 5 show selected numerical results for:
15
75.0;75.0;1000;1000
;2;95.0;50;15.273 22
====
====
scshsc
Loa
KgKJC
KgKJC
KmWU
mWIKT
εη
η
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3. Results and discussion
16
Fig.2. R22
Fig.3. R717
( )ceW εε ,
( )ceW εε ,
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3. Results and discussion
17
Fig.4. R22
Fig.5. R717
( )ceHPCOP εε ,
( )ceHPCOP εε ,
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18
3. ConclusionsThe presentation presents a mathematical algorithmable in modeling the influences of evaporator andcondenser effectiveness upon the all operationalfeatures of a heat pump: power, heat rates, andtemperatures. The model start with the external heatsources imposed parameters and, combine these onesby the internal irreversibility of the heat pump (entropybalance).
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