J.C. Ordóñez
Juan C. Ordonez
Department of Mechanical Engineering Energy and Sustainability Center
Center for Advanced Power SystemsFlorida State University
A Thermodynamic Perspective on Energy Efficiency
J.C. Ordóñez
Nicolas Léonard Sadi Carnot. 1796-1832
J.C. Ordóñez
J.C. Ordóñez
Thermal engine
J.C. Ordóñez
Drawn after Bejan –AET and Cardwell “From Watt to Clausius”
Engine Efficiencies in times of Carnot
J.C. Ordóñez
J.C. Ordóñez
Sources: Bejan (AET); and H.B. Callen (T)
J.C. Ordóñez
J.C. Ordóñez
Observed efficiency
J.C. Ordóñez
J.C. Ordóñez
Chambadal (1957)Novikov (1958)Curzon & Alhborn (1975)
endoreversible
External irreversibilityLinear heat transfer model
External irreversibilityLinear heat transfer model
Endoreversible model
J.C. Ordóñez
J.C. Ordóñez
Taylor expansion
Recent studies suggest that at maximum power
J.C. Ordóñez
What is the maximum power that can be extracted from a hot stream? What are realistic limits for this power?
J.C. Ordóñez
reversible power plant
TH, P0
m.T0, P0
Q0
.
atmosphere: T0
Wrev
.If the stream interacts only with the atmospheric temperature reservoir (T0), the maximum power that can be extracted is the flow exergy:
−−=
0
H
0
H0prev T
Tln1TTTcmW
(ideal gas)
The actual power will always be lower than rev because of the irreversibilities of the heat transfer between the hot stream and the rest of the power plant.
W
1. Power extraction:
J.C. Ordóñez
(i) temperature difference between the stream (T) and the heat exchanger surface (Ts) and
reversible power plant
TH, P0
m. T0, P0
Q0
.
atmosphere: T0
Tout
Wrev>W. .
The heat transfer irreversibility is due to:
(ii) thermal mixing of the discharged stream.
J.C. Ordóñez
The case where the collecting stream is single phase was studied by Bejan and Errera, 1998.
We use a second stream to collect the power from the hot stream. This second stream is the working fluid of the power producing device
power plant
Q0
.
atmosphere: T0
W
Hot stream
collecting stream
J.C. Ordóñez
Power extraction from a hot stream
J.C. Ordóñez
Entropy generation analysis
( ) e00H QQhhmW −−−=
( ) 0ssmT
QQS H0
0
e0gen ≥−+
+=
gen0H,x STemW −=
( ) ( )000H0HH,x sThsThe −−−=
Now for the heat exchanger, and external cooling alone:
( ) ( ) 0Qhhmhhm e12w0H =−−−−
( ) ( ) 0TQ
ssmssmS0
e12wH0gen ≥+−+−=
( )1,x2,xwH,xgen eememSTo −−= ( )1,x2,x eemW −=
W.
T0
TH m cp.
A
water mw.
reversible compartment
Q0
.This image cannot currently be displayed.
Qe
.
Tout T0
J.C. Ordóñez
We can maximize the power output using:
or directly using
( )1,x2,x eemW −=
In dimensionless form:
gen0H,x STemW −=minimize
( )H,x
1,x2,xw
H,xII em
eememW
−==η
J.C. Ordóñez
Maximization of the second law efficiency by selecting the mass flow rate of the water stream
mmw
=
[IJHMT Vargas, Ordonez and Bejan, 1999]
J.C. Ordóñez
Effect of the mass flow rate on the allocation of area among the sections of the heat exchanger
mmw
=
Optimal system structure
J.C. Ordóñez
m cp.
TH
A
watermw.
Tout
0
0.5
1
0.3 0.35 0.4 0.45 0.5
1-x-y=Aw
/A
y = Ab /A
x = AS /A
N=10τ
Η = 4
τb =1.98
τ1 =1.8
M
steam
boiling
liquid
Effect of the mass flow rate on the allocation of area among the sections of the heat exchanger
J.C. Ordóñez
Concluding remarks
• Sadi Carnot, laid out the foundations of thermodynamics exploring limits of operation of thermal engines that lead to maximum power.
• An interesting question that can be asked is why do we observe a pattern in efficiency at maximum power?
• This maybe linked to symmetry in physical laws
J.C. Ordóñez
Concluding remarks
• Chambadal, Novikov, Curzon and Ahlborn derived efficiency expression that predicts well performance of thermal plants.
• The Novikov–Chambadal-Curzon-Ahlborn expression has been derived in different context:• Classical thermodynamics• Endoreversible thermodynamics (finite times,
finite sizes)• Linear Irreversible Thermodynamics
J.C. Ordóñez
Concluding remarks:
• The extraction of power and refrigeration from a hot stream can be maximized by properly matching the stream with a receiving stream of cold fluid, across a finite-size heat transfer area
counterflow
+
0.66
0.68
0.7
0.72
0.32 0.34 0.36 0.38 0.4 0.42 0.44
M
ηΙΙ
N=10τ
Η=4
τb=1.98
τ1=1.8
ideal-gas model for steam
tabulated steam properties
TH m cp.
A
watermw.
Tout
0
0.5
1
0.3 0.35 0.4 0.45 0.5
1-x-y=Aw
/A
y = Ab /A
x = AS /A
N=10τ
Η = 4
τb =1.98
τ1 =1.8
M
steam
boiling
liquid
Optimal mass flow rate ratio
There is an associated optimal allocation of heat exchanger inventory:
J.C. Ordóñez
Concluding remarks
• System structure appears as a result of optimization -> maximization of flow access
• Constructal Design: “Generation of architecture under global constraints”
J.C. Ordóñez
Thank you!
J.C. Ordóñez
• Prof. S.A. Sherif• Professors A. Bejan, J.V. C. Vargas and C. Harman • The comments of the reviewers of this paper are greatly
appreciated.• Center for Advanced Power Systems at Florida State
University.
Acknowledgements:
J.C. Ordóñez
Constructal Design: “Generation of architecture under global contraints”
System structure appears as a result of optimization
J.C. Ordóñez
J.C. Ordóñez
Concluding remarks:
How does the optimal area allocation appears in practice?
0
0.5
1
0.3 0.35 0.4 0.45 0.5
1-x-y=Aw /A
y = Ab /A
x = AS /A
N=10τ
Η = 4
τb =1.98
τ1 =1.8
M
steam
boiling
liquid
a) One counterflow heat exchanger, three sections:The three sections rearrange themselves
b) Three sections, boiling in contact with hottest gases:
Here a ‘morphing’heat exchanger is needed
System structure appears as a result of optimization (Constructal theory)
J.C. Ordóñez
Concluding remarks: Optimal HT area allocation
TH m cp.
A
watermw.
Tout
0.66
0.68
0.7
0.72
0.32 0.34 0.36 0.38 0.4 0.42 0.44
M
ηΙΙ
N=10τ
Η=4
τb=1.98
τ1=1.8
ideal-gas model for steam
tabulated steam properties
0.15
0.2
1 1.5 2 2.5 3 3.5 4 4.5 5
τH = 4
τL = 0.9
τ1 = 1.1
r
qL
5U~ H LO =
x = 0.2y = 0.2
0
0.5
1
0.3 0.35 0.4 0.45 0.5
1-x-y=Aw
/A
y = Ab /A
x = AS /A
N=10τ
Η = 4
τb =1.98
τ1 =1.8
M
J.C. Ordóñez
Maximization of the second law efficiency using Toluene as working fluid
Duke UniversityMechanical Engineering and Material Science Department
J.C. Ordóñez
J.C. Ordóñez
Maximum power from a hot stream
• In engineering thermodynamics it is usually assumed that the heat that drives a power plant is already available from a hot temperature reservoir.
•In most applications a fuel is burn, and a hot stream becomes the input to the power plant.
Wpowerplant
hot
cold
Wpowerplant
cold
combustion
hot stream
- What is the maximum power that can be extracted from a hot stream?
J.C. Ordóñez
• Power and refrigeration systems are assemblies of streams and hardware (components).
• The size of the hardware is constrained.
Thermodynamic optimization methodology:
Rankine powercycle
Department of Mechanical EngineeringJ.C. Ordóñez
J.C. Ordóñez
Each stream carries exergy (useful work content), which is the life blood of the power system. Exergy is destroyed (or entropy is generated) whenever streams interact with each other and with components. Our objective is to optimize the streams and components, so that they generate minimum entropy subject to the constraints.
fuel
heaterturbine
pump condenser
Department of Mechanical EngineeringJ.C. Ordóñez
J.C. Ordóñez
THE METHOD OF THERMODYNAMIC OPTIMIZATION
SYSTEM
ENVIRONMENT
-Thermodynamics providesthe basic equations
-Flows, flow resistances, losses (irreversibility, “dissipation”) and interactions are integrated from related disciplinesInteractions
Starts with:
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
Department of Mechanical EngineeringJ.C. Ordóñez
J.C. Ordóñez
∑ ∑ ∑=
−+−=n
0i in out
00i hmhmWQ
dtdE
∑ ∑∑ ≥+−−== in out
n
0i i
igen 0smsm
TQ
dtdSS
We start from the 1st and 2nd laws of thermodynamics:
out
in
T1
T1
Q1
.T2
T2
Q2
.Tn
Tn
Qn
.
T0
T0Q0
.
Environment (T0P0)
System (M, E, S)
W.
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
Department of Mechanical EngineeringJ.C. Ordóñez
J.C. Ordóñez
( ) ( ) ( ) gen0out
00
in0
0i
n
1i i
00 STsThmsThmQ
TT
1STEdtdW −−−−+
−+−−= ∑∑∑
=
( ) ( ) ( )∑∑∑ −−−+
−+−−=
= out0
0
in0
0i
n
1i i
00rev sThmsThmQ
TT
1STEdtdW
Non-flow exergy
Exergy, heat transfer interactions
Flow-exergy associated to mass flows
The two laws combined (eliminating Q0):.
In the reversible limit ( ),0Sgen =
(E1)
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
Actual work
Work in the reversible limitrevWW <
destroyed exergy
``Exergy Analysis”
Department of Mechanical EngineeringJ.C. Ordóñez
J.C. Ordóñez
gen0revlost STWWW =−=
Sustracting them we get the Gouy-Stodola theorem: The destroyed power is proportional to the rate of entropy generation.
(E.3)
EGM starts from (E.3). We want to be as close as possible to the reversible limit ( ), then we should work in the minimization of the entropy generation ( ).
revW
genS
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
Department of Mechanical EngineeringJ.C. Ordóñez
J.C. Ordóñez
CONSTRAINED OPTIMIZATIONfunction, constraints and degrees of freedom
The FUNCTION to be optimized, is related to PURPOSE e.g:
-Max. power extraction-Min. power requirement-Max. of exergy collection-Min. ratio destroyedexergy/ supplied exergy
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
CONSTRAINTS. e.g:Total volume, area, material amount, operationtemperatures.
DEGREES OF FREEDOM-Operation temperatures-Charging/discharging times-Dimensions, thickness-Spacing among components-Material properties
Department of Mechanical EngineeringJ.C. Ordóñez
J.C. Ordóñez
The total surface constraint (c1), can be written as,
ww
sb
b
ss N
UU
NUU
NN µ′++µ=p
s
cmAU
N
= constant
AA
x s=
AA
y b=
AA
yx1 w=−−
In the numerical computations, we defined the following area fractions
superheater (steam)
boiling
Preheater (liq. water)
The equations we have until now allow us to compute the temperature distribution. We need the work output.
Department of Mechanical EngineeringJ.C. Ordóñez
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
J.C. Ordóñez
Constrained optimization:
ww
sb
b
ss N
UU
NUU
NN µ′++µ=p
s
cmAU
N
= constant
AA
x s=
AA
y b=
AA
yx1 w=−−
In the numerical computations, we defined the following area fractions
superheater (steam)
boiling
Preheater (liq. water)
The equations we have until now allow us to compute the temperature distribution. We need the work output.
Department of Mechanical EngineeringJ.C. Ordóñez
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
The total area constraint, can be written as,
J.C. Ordóñez
Concluding remarks (1/3):
• The extraction of power from a hot stream can be maximized by properly matching the stream with a receiving stream of cold fluid, across a finite-size heat transfer area
counterflow
+
0.66
0.68
0.7
0.72
0.32 0.34 0.36 0.38 0.4 0.42 0.44
M
ηΙΙ
N=10τ
Η=4
τb=1.98
τ1=1.8
ideal-gas model for steam
tabulated steam properties
TH m cp.
A
watermw.
Tout
0
0.5
1
0.3 0.35 0.4 0.45 0.5
1-x-y=Aw
/A
y = Ab /A
x = AS /A
N=10τ
Η = 4
τb =1.98
τ1 =1.8
M
steam
boiling
liquid
Optimal mass flow rate ratio
There is an associated optimal allocation of heat exchanger inventory:
J.C. Ordóñez
We want to search for an optimal matching between the streams.
Thermodynamic optimization
SYSTEM
ENVIRONMENT
-Thermodynamics providesthe basic equations
-Flows, flow resistances, losses (irreversibility, “dissipation”) and interactions are integrated from related disciplinesInteractions
Starts with:
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
J.C. Ordóñez
A
m cp.
TH
T2T3
T4
Tout
T1
TbTb
cs
cw
A s A b A w
steam superheating boiling liquid water
preheating
heat transfer surface
mw.
0
Temperature distribution along the threesections of the heat exchanger
The case where the collecting stream experience a phase changewas studied by Vargas, Ordonez and Bejan (IJHMT, 1999).
Florida State University, J.C. Ordóñez, S. Chen
J.C. Ordóñez
Heat transfer analysis classical effectiveness Ntu analysis
Superheater Boiling Preheating
( )[ ]( )[ ]µ−−µ−
µ−−−=ε
1Nexp11Nexp1
s
ss
( )bH
3Hs TT
TT−µ
−=ε
b3
b2s TT
TT−−
=ε
p
sw
cmcm
=µ
sw
sss cm
AUN
=
)Nexp(1 bb −−=ε
bH
3Hb TT
TT−−
=ε
p
bbb cm
AUN
=
( )[ ]( )[ ]'
w
'w
w 1Nexp11Nexp1
µ−−µ−µ−−−
=ε
14
1bw TT
TT−−
=ε
( )41
4outw TT
TT−µ′−
=ε
p
ww
cmcm
=µ′
ww
www cm
AUN
=
1 ,1 <µ′<µ
s
fg3 c
hTTH µ=−
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
J.C. Ordóñez
The total area constraint
can be written as,
ww
sb
b
ss N
UU
NUU
NN µ′++µ=
p
s
cmAU
N
=
TH m c p.
A
Tout
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
Constrained optimization:
In the numerical computations, we defined the following area fractions
AA
x s=
AA
y b=
AA
yx1 w=−−
superheater (steam)
boiling
Preheater (liq. water)
Maximize power extraction (efficiency)
For fixed total area, A
Degree of freedom, wm
J.C. Ordóñez
OPTIMIZED SYSTEMMODEL
CONSTRAINEDOPTIMIZATION
J.C. Ordóñez
Heat transfer analysis classical effectiveness Ntu analysis
Superheater Boiling Preheating
( )[ ]( )[ ]µ−−µ−
µ−−−=ε
1Nexp11Nexp1
s
ss
( )bH
3Hs TT
TT−µ
−=ε
( )bH
b2s TT
TT−µ
−=ε
p
sw
cmcm
=µ
sw
sss cm
AUN
=
)Nexp(1 bb −−=ε
b3
34b TT
TT−−
=ε
p
bbb cm
AUN
=
( )b3pfg TTcmhm −=
( )[ ]( )[ ]'
w
'w
w 1Nexp11Nexp1
µ−−µ−µ−−−
=ε
14
1bw TT
TT−−
=ε
( )41
4outw TT
TT−µ′−
=ε
p
w
cmcwm
=µ
ww
www cm
AUN
=
1 ,1 <µ′<µ
J.C. Ordóñez
Maximization of the second law efficiency by selecting the mass flow rate of the water stream
Duke UniversityMechanical Engineering and Material Science Department
J.C. Ordóñez
mmw
=
J.C. Ordóñez
Effect of the mass flow rate on the allocation of area among the sections of the heat exchanger
Duke UniversityMechanical Engineering and Material Science Department
J.C. Ordóñez
mmw
=
J.C. Ordóñez
Effect of the heat transfer area size on the “match” between the temperature distributions of the two streams
J.C. Ordóñez
Effect of heat exchanger size on the second law efficiency and on the allocation of heat transfer area
Duke UniversityMechanical Engineering and Material Science Department
J.C. Ordóñez
J.C. Ordóñez
Effects of varying the working-fluid inlet temperature and boiling temperature
Duke UniversityMechanical Engineering and Material Science Department
J.C. Ordóñez
J.C. Ordóñez
J.C. Ordóñez
0
0.5
1
6 9 12 15
ηII,max
M opt
xopt
yopt
τH = 4 , τ
b = 1.98 , τ
1 = 1.8
N
Effect of heat exchanger size on the second law efficiency and on the allocation of heat transfer area
Notice “robustness” of the optimal ‘match’
J.C. Ordóñez
Effect of the heat transfer area size on the “match” between the temperature distributions of the two streams
0
1
2
3
4
5
0 1A
τ
Ab A
wA
S
N = 10τ
H = 4
τb =1.98
τ1 =1.8
Mopt
= 0.405
τΗ
τ3
τ4 τ
out
τ1
τ2
τb
gas
water
boilingsection liquid water sectionsteam
section
τb τ
b
0
1
2
3
4
5
0 1A
τ
Ab
Aw
AS
N = 6τ
H = 4
τb =1.98
τ1 =1.8
Mopt
= 0.48
τΗ
τ3
τ4
τout
τ1
τ2
τb
gas
water
boilingsection
liquid water sectionsteamsection
τb τ
b
J.C. Ordóñez
0
0.5
1
1 1.3 1.6 1.9
ηΙΙ, max
Mopt
xopt
yopt
N = 10, τ Η
= 4, τb = 1.98
τ1
Effect of varying the working-fluid inlet temperature
Effect of varying the boiling temperature
J.C. Ordóñez
0.66
0.68
0.7
0.72
0.32 0.34 0.36 0.38 0.4
M
ηΙΙ
N=10τ
Η=4
τb=1.98
τ1=1.8
Ub/U
s = 100 and U
w/U
s = 10
Ub/U
s = 1 = U
w/U
s
Effect of overall heat transfer coefficients on the second law efficiency
mmw
=
J.C. Ordóñez
Concluding remarks :
Optimal matching among the hot and collecting stream
(counterflow configuration, optimal mass flow rate ratio, optimal allocation of heat exchanger inventory)
Collecting stream is single phase
Collecting stream experience phase change
Collecting stream experience phase change and boiling section is in contact with hottest gases
IJHMT Bejan and Errera, 1998
IJHMT Vargas, Ordonez and Bejan, 1999
ASME HT Charlotte Ordonez and Chen, 2004
J.C. Ordóñez
J.C. Ordóñez
Refrigeration system driven by a hot stream through a counterflow heat exchanger
3. Optimal Matching for Refrigeration:
J.C. Ordóñez
( ) ( ) 0QQTTcm 0L12rp =−+−
( ) 0TQ
TQ
TTlncm
C0
0
LC
L
1
2rp =−+
)TT()UA(Q 0C000 −=
)TT()UA(Q LCLLL −=
( )[ ]( )[ ]r1Nexpr1
r1Nexp1
H
H
−−−−−−
=ε
p
HHH cm
AUN
=
r <1
( )[ ]( )[ ]1
H1
1H
r1Nexpr1r1Nexp1
−−
−
−−−−−−
=ε
r >1
( )1H12 TTrTT −ε=−
( )rp
p
cmcm
r
=
( )1H12 TTTT −ε=−L0H AAAA ++=
Thermodynamics Heat Transfer:
Constraint:
J.C. Ordóñez
0TT
=τ0p
00 Tcm
=
AAx H= A
Ay L= AAyx1 0=−−
0p
LL Tcm
=
p
00 cm
AUU~
=p
HH cm
AUU~
=p
LL cm
AUU~
=
Dimensionless groups:
J.C. Ordóñez
0.15
0.2
1 1.5 2 2.5 3 3.5 4 4.5 5
τH = 4
τL = 0.9
τ1 = 1.1
r
qL
5U~ H LO =
x = 0.2y = 0.2
Illustration of the existence of an optimal capacity rate ratio
An optimal matching for refrigeration
( )rp
p
cmcm
r
=
J.C. Ordóñez
THE EFFECT OF THE HOT-STREAM INLET TEMPERATURE ON THE OPTIMAL ALLOCATION OF HEAT EXCHANGER INVENTORY
J.C. Ordóñez
Maximized refrigeration rate and optimal capacity rate ratio of the countreflow system
Here the heat exchanger area allocation has been optimized
J.C. Ordóñez
Effects of refrigeration temperature
J.C. Ordóñez
Effects of the matching stream inlet temperature
J.C. Ordóñez
EFFECT OF HOT SIDE OVERALL HEAT TRANSFER COEFFICIENT ON THE REFRIGERATION RATE AND THE EXISTENCE OF AN OPTIMAL CAPACITY RATE RATIO.
J.C. Ordóñez
Placing the boiling section in contact with the hottest gases will prevent pipe overheating (materials constraints).
An alternative configuration:
2. Phase change under limiting collecting temperatures
J.C. Ordóñez
Temperature distribution along the three sections of the heat exchanger
TH m cp.
A
watermw.
Tout
J.C. Ordóñez
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
Entropy generation analysis
( ) e00H QQhhmW −−−=
( ) 0ssmT
QQS H0
0
e0gen ≥−+
+=
gen0H,x STemW −=
( ) ( )000H0HH,x sThsThe −−−=
Now for the heat exchanger, and external cooling alone:
( ) ( ) 0Qhhmhhm e12w0H =−−−−
( ) ( ) 0TQ
ssmssmS0
e12wH0gen ≥+−+−=
( )1,x2,xwH,xgen0 eememST −−= ( )1,x2,x eemW −=
W.
T0
TH m cp.
A
watermw.
reversible compartment
Q0
.This image cannot currently be displayed.
Qe
.
Tout T0
J.C. Ordóñez
We can maximize the power output using:
or directly using
( )1,x2,x eemW −=
In dimensionless form:
gen0H,x STemW −=minimize
( )H,x
1,x2,xw
H,xII em
eememW
−==η
OPTIMIZED SYSTEMMODELCONSTRAINEDOPTIMIZATION
J.C. Ordóñez
0.66
0.68
0.7
0.72
0.32 0.34 0.36 0.38 0.4 0.42 0.44
M
ηΙΙ
N=10τ
Η=4
τb=1.98
τ1=1.8
ideal-gas model for steam
tabulated steam properties
mmw
=
Maximization of the second law efficiency by selecting the mass flow rate of the water stream
Optimal matching
J.C. Ordóñez
Concluding remarks:
0
1
2
3
4
5
0 1A
τ
Ab A
wA
S
N = 10τ
H = 4
τb =1.98
τ1 =1.8
Mopt
= 0.405
τΗ
τ3
τ4 τ
out
τ1
τ2
τb
gas
water
boilingsection liquid water sectionsteam
section
τb τ
b
0
0.5
1
6 9 12 15
ηII,max
M opt
xopt
yopt
τH = 4 , τ
b = 1.98 , τ
1 = 1.8
N
Optimal ratio is robust with respect to total surface area
J.C. Ordóñez
Optimal area allocation is robust with respect to refrigeration temperature