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

Qq

=

AAx H= A

Ay L= AAyx1 0=−−

0p

LL Tcm

Qq

=

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