GLOBAL CLIMATE AND ENERGY PROJECT | STANFORD UNIVERSITY
Energy Tutorial:
Exergy 101
Chris Edwards
Professor – Department of Mechanical Engineering
Stanford University
GLOBAL CHALLENGES – GLOBAL SOLUTIONS – GLOBAL OPPORTUNITIES
GCEP RESEARCH SYMPOSIUM 2012 | STANFORD, CA
Which would you choose?
1 kg Air
20 C
1 bar
1 kg Air
20 C
8 bar
Hint: Both have exactly the same amount of energy...
The ability to do work depends upon both the state of the
resource and the state of the surroundings.
Energy, Entropy, Exergy
• is the extensive, conserved quantity that
is inter-convertible with heat and work:
• is the extensive measure of the number
of microscopic rearrangements of energy:
ln
•
in out
B
dU Q W
Energy
Entropy
S k
is the potential of an energy resource to do work
in a given set of surroundings ( environment)a.k
Exerg
.
y
.a
• Resource in contact with environmental reservoir
- reservoir is
(fixed intensive state, not extensive)
- reservoir has
(boundary props fixed, irrev. in syste
large but finite
fast internal transport
0
any species
m)
:
:
ko
b k k
k k gen
ko
o o k o k k o gen
k
k
E dU Q W h N W
QS dS s N S
T
W dU P dV T dS h T s N T S
revW 0environmentalspecies i
Exergy
" "
• Reaction must reversibly transform all species ( )
to species that are naturally present in the environment ( ).
• Consider a reaction that does this transformation for species :
:j
resource j
i
j
Rxn A a A bB
with extent of reaction (extensive)
(signed coefs.)
• The balance for environmental species then becomes
:
while the balance for a non-environ
j
j j ij i
i
i i i ij j
j
cC dD
aA bB cC dD M M
i
N dN N
mental species is
:j j j j
j
N dN
i i ij j j
j
N dN dN
Exergy
Note: Any environmental species present in
the resource “reacts” to form itself.
Exergy Revisited
• RHS has exact differentials. The state of the environment enters
only through fixed, intensive parameters ( , , ).
• Since it is exact & with constant
rev o o io i io ij j j
i i j
o o io
W dU P dV T dS dN dN
T P
coef., integral is path independent.
• Integrate along a two-part path:
I: At to the thermo-mechanical ( )
dead state. (No diffusion or reaction per
fixed
mit
composition
fixed thermo
ted.)
II: At
restricted
, but with diffusion and
reaction to the environmenta
-mec
l (
ha
) dead state.
nical state
unrestricted
Exergy Revisited
max
T-M Dead State
Fixed Comp.maxResource State No Reaction
Dead State
T-M Dead State
• The fi
o
o
o o io i io ij j j
i i j
o o
ij
o o io i io j
i i j j T TP P
W dU P dV T dS dN dN
W dU P dV T dS
dU P dV T dS dN dN
rst integral is sometimes referred to as the
, . The second integral is the , .
• The of the resource is then:
• Adding th
-
e
TM C
int TM C
thermo mechanical
exergy chemical exergy
internal ex
X X
X Xergy
ext
X
erna
, gives the total exergy:
TM C
KE PE
X
l exergy
E
X X K PE
T-M Exergy
T-M Dead State
Fixed Comp.Resource State No Reaction
Resource Intensive State: , ,
Resource Extensive Composition:
Availability Fu
j
j
TM o o
TM TM o TM o TM
TM o o
T P x
N
X dU P dV T dS
X U U P V V T S S
or
X U PV T S
Thermo-mechanical Intensive State: , ,
Resource Extensive Composition:
= Gibbs Function in TM State, nction,
denotes any species present in the resou
o o j
j
TM o TM o TM
T P x
N
TMG
U PV T S
where j
A
rce.
TM TMX G A Original composition. Held fixed!
Chemical Exergy
Dead State
T-M Dead State
Gibbs Function in TM Dead State
o
o
TM
ij
C o o io i io j
i i j j T TP P
C TM o o TM o o TM o
io iTM io io ij j jTM jo
i i j
C TM o TM o TM
G
X dU P dV T dS dN dN
X U U P V V T S S
N N N N
X U PV T S
Environmental Intensive State: , ,
Unknown Extensive Composition:
Environ. UnknownIntensive Extensive
State Comp.
( )
o o io
io
o o o o o
T P x
N
io iTM io io ij j jTM jo
i i j
U PV T S
N N N N
The unknown composition in the system cancels out. (Whew!)
jN 00
Chemical Exergy
Chemical Potential
of Resource (at TM Dead State)
• We can interpret this on the basis of the chemical potential of the
initial resource and what it will become in the environment.
C TM io ij j
i
X G
Chemical Potential of Env. Species Formed from Species
Originally Present in Resource (at Env. Dead State)
• If we define the last term as
and
TM C
j
j
o
C TM o TM TM o
X X
N
X G X G G
G
G A G A oG
The chemical exergy is the difference between the chemical potential
(Gibbs function) of the resource before and after it has reacted and
diffused to become part of the environment (all at To and Po).
j j ij i
i
aA bB cC dD
aA bB cC dD
A A
C TM io ij j j
j i
X G N
Example:
2 2 2 2
4
4 2 2
, , , ,
4 2 2 2
F
• Environmental (dead) state: 25°C, 100 kPa
370 ppm, 3%, 20.
ind the chemic
3%, 76.66%
• 1 kmol, 2 2
1, 2,
al exergy of one kmol of meth e
an .
o o
CO o H O o O o N o
CH
CH O CO
T P
x x x x
N CH O CO H O
2
4 2 2 2
4 4 2 2 2
2 2 2 2 2 2
, , , ,
, ,o ,
,o , O,o ,
1, 2
• 2 2
, ln
ln , ln
• 130278 2 61090 3950 457232 19577
2 298091 8688 830 MJ (51.9
H O
C CH TM O o CO o H O o
o o
CH TM CH O O o O o
o o
CO CO o CO o H H O o H O o
C
X
RT x
RT x RT x
X
4MJ/kg CH )
C TM io ij j j
j i
X G N
Example:
4 3 8 2
4
, , ,
4 2 2 2 3 8 2 2 2
,
Find the chemical exergy of 1 kmol of methane mixed with
2 kmol propane and 1 kmol nitrogen. (Same
• 2 2 , 5 3 4
1 kmol, 2 kmol, 1 kmol
,
)
25%
CH C H N
CH
o i o
T
o
M
T P
CH O CO H O C H O CO H O
N N N
x
x
3 8 2
4 3 8 2 4 2 3 8
2 4 2 4 2 3 8 2 3 8 2
4 3 8 2 2
2 2 2 2 2
, ,
, ,
, , , ,
, , , ,
, , , , ,
50%, 25%
1, 1, 2, 5,
1, 2, 3, 4 0
• 2
2 1 2 2 5 2 3
C H TM N TM
CH C H O CH O C H
CO CH H O CH CO C H H O C H N
C CH TM C H TM N TM N o
O o CO o H O o O o CO o
x x
X
2 ,
, , ,o ,
2 4
ln , ln
• 5116 MJ (49.2 MJ/kg fuel)
H O o
o o
k TM k o k TM k k o k o
C
RT x RT x
X
C TM io ij j j
j i
X G N
4 2 2
4 2 2
4 2 2
4 2 2 2 2 2
, , ,
• 2 3.76 2 2 3.76
1 km
Find for 1 k
ol,
mol of methane mixed with s
2 kmol,
toichiometric a
7.52 kmol
9.51%, 19.01%, 71.48%
1, 2,
ir.
1,
CH O N
CH TM O TM N TM
CH O CO
C
CH O N CO H O N
N N N
x x x
X
2 2 2
4 2 2
2 2 2 2 2
, , ,
, , , , ,
, , ,o ,
4
2 0
• 2 7.52
2 7.52 2 2
ln , ln
• 822.5 MJ (51.4 MJ/kg CH or 2.83 MJ/kg mix)
H O N O
C CH TM O TM N TM
O o N o O o CO o H O o
o o
k TM k o k TM k k o k o
C
X
RT x RT x
X
Example:
C TM io ij j j
j i
X G N
2 2 2
2 2
4 2 2 2 2 2
, ,
Find for the of 1 kmol
of methane with a stoichiometric amount of a
• 2 3.76 2 2 3.76
1 kmol, 2 kmol, 7.52 kmol
9.51%, 19
ir.
CO H O N
CO TM H
C
O TM
CH O N CO H O N
N N N
x
X products of complete combustion
x
2
2 2 2 2 2 2
,
, , , , , ,
,
, ,o
,
4 4
.01%, 71.48%
• 2 7.52 2 7.52
ln
• 21.6 MJ (1.35 MJ/kg CH or 2.6% of for CH )
N TM
C CO TM H O TM N TM CO o H O o N o
k TM
k TM k o
k o
C C
x
X
xRT
x
X X
Example:
C TM io ij j j
j i
X G N
Example:
2
2
2
2
4 2
Find for 1 kmol of pure CO . (Inverse sequestration from air.)
Express the answer per unit mass of
1 kmol
All stoichiometric coefficients
CH that genera
are zero excep
te
t CO
d th
.
•
e CO .
CO
C CO
C
N
X
X
2
2 2
2
,
, , ,
,
2
4 4
ln ln
• 19.6 MJ 0.446 MJ/kg CO
1.23 MJ/kg CH (2.4% of for CH )
CO TM
TM CO o o o CO o
CO o
C
C
xRT RT x
x
X
X
There is sufficient exergy in the products of stoichiometric
methane-air combustion to drive the complete separation of all of
the CO2 produced by the reaction! (Lots of water!)
0 500 1000 1500 2000 2500 30000
200
400
600
800
1000
1200
T (K)
x (
MJ/k
mo
l) Methane
Carbon Dioxide
Water
Oxygen
Nitrogen
Single-Component Ideal Gases
0 500 1000 1500 2000 2500 30000
200
400
600
800
1000
1200
1400
1600
1800
T (K)
x (
MJ/k
mo
l-C
)
Methane-Air Reactants (Stoich.)
Methane-Air Products (Stoich.)
Carbon Dioxide
Methane Products & CO2
C TM io ij j j
j i
X G N
2 2 2
2 2
3 8 2 2 2 2 2
, ,
Find for the of
1 kmol of propane with
• 5 3.76 3 4 5 3.76
3 kmol, 4 kmol, 18.8 kmol
11.63%, 15.
stoichiometric air.
50%,
CO H O N
CO TM H
C
O TM N
C H O
X products of complete com
N CO H O N
N N N
x x
bustio
x
n
2
2 2 2 2 2 2
,
, , , , , ,
,
, ,o
,
2 2
72.87%
• 3 4 18.8
ln
• 56.7 MJ (18.9 MJ/kmol-CO . Less than pure CO .)
TM
C CO TM CO o H O TM H O o N TM N o
i TM
i TM i o
i o
C
X
xRT
x
X
Example:
0 500 1000 1500 2000 2500 30000
500
1000
1500
T (K)
x (
MJ/k
mo
l-C
)
Propane-Air Reactants (Stoich.)
Propane-Air Products (Stoich.)
Carbon Dioxide
Propane Products & CO2
Crosses at low T!
Separation possible!
Chemical Exergy of Common Fuels
Fuel Chemical Chem. Exergy† H° Reaction* G° Reaction* S° Reaction* Exergy
Species+ Formula MJ per fuel MJ per fuel MJ per fuel kJ/K per fuel to LHV
kmol kg kmol kg kmol kg kmol kg Ratio
Methane CH4 832 51.9 -803 -50.0 -801 -49.9 -5.2 -0.33 1.037
Methanol CH3OH 722 22.5 -676 -21.1 -691 -21.6 50.4 1.57 1.068 Carbon Monoxide CO 275 9.8 -283 -10.1 -254 -9.1 -98.2 -3.51 0.971 Acetylene C2H2 1267 48.7 -1257 -48.3 -1226 -47.1 -104.6 -4.02 1.008 Ethylene C2H4 1361 48.5 -1323 -47.2 -1316 -46.9 -25.2 -0.90 1.029 Ethane C2H6 1497 49.8 -1429 -47.5 -1447 -48.1 60.5 2.01 1.048 Ethanol C2H5OH 1363 29.6 -1278 -27.7 -1313 -28.5 117.7 2.56 1.067 Propylene C3H6 2001 47.6 -1926 -45.8 -1937 -46.0 36.6 0.87 1.039 Propane C3H8 2151 48.8 -2043 -46.3 -2082 -47.2 129.2 2.93 1.053 Butadiene C4H6 2500 46.2 -2410 -44.5 -2421 -44.7 36.9 0.68 1.038 i-Butene C4H8 2644 47.1 -2524 -45.0 -2560 -45.6 120.2 2.14 1.047 i-Butane C4H10 2800 48.2 -2648 -45.6 -2712 -46.7 214.4 3.69 1.058
n-Butane C4H10 2805 48.3 -2657 -45.7 -2717 -46.7 200.0 3.44 1.056 n-Pentane C5H12 3460 48.0 -3272 -45.3 -3353 -46.5 271.3 3.76 1.057 i-Pentane C5H12 3454 47.9 -3265 -45.2 -3347 -46.4 277.0 3.84 1.058 Benzene C6H6 3299 42.2 -3169 -40.6 -3190 -40.8 69.4 0.89 1.041 n-Heptane C7H16 4769 47.6 -4501 -44.9 -4625 -46.2 415.0 4.14 1.060 i-Octane C8H18 5422 47.5 -5100 -44.7 -5259 -46.0 531.4 4.65 1.063 n-Octane C8H18 5424 47.5 -5116 -44.8 -5261 -46.1 487.1 4.26 1.060 Jet-A C12H23 7670 45.8 -7253 -43.4 -7440 -44.5 626.4 3.74 1.057 Hydrogen H2 236 117.2 -242 -120.0 -225 -111.6 -56.2 -27.88 0.977
+All species taken as ideal gases. †Environment taken as: 25°C, 1 bar, 363 ppm CO2, 2% H2O, 20.48% O2, balance N2 .
*Reaction with stoichiometric air at 25°C, 1 bar. All products present as ideal gases, including water.
For simple fuels the exergy can be calculated directly.
For complex fuels (coal) it is not possible to calculate the exergy
(need entropy) and some form of correlation is required.
Standard-State Chemical Exergy
• may be expressed in terms of the standard state chemical
potential and chemical activity since ln .
• Defining the
; when is an environmen
-
tal sp
C
o
io i o i
o
ko
k
standard state chemical exer
X
R
g
k
y
T a
ecies
; when is a non-environmental species
and the
; when is an environmental species
; when is a non-environmental speciesik k
o
i ik k
i
k
ki
i
o
C TM k k o
k
effectiv
k
a k
a k
X G N R
e activit
T
y
lnk k
k
N
Standard-State Chemical Exergy
• Tabulated values of the standard-state chemical exergy can be
found in references such as:
J. Szargut, D.R. Morris, and F.R. Steward,
,
H
Exergy Analysis
of Thermal, Chemical, and Metallurgical Processes
emisphere, New York, 1988.
• Values found in the literature can differ from each other
according to the choice of species for the environmental dead
state (since the environment is not, itself, in equilibrium).
• For our purposes, tables of standard-state exergies are not
needed--we will calculate the chemical exergy directly.
Exergy Balances
= 0 0
(2nd L
2
aw) (Go
• Definition:
2
• Balance: :
o gen
TM C
o o o o o
system
External Internal
TransfersAccumulation T S
in out produced destroyed
X KE PE X X
mX V V mg z z U PV T S
X dX X X X X
G
uy-Stodola)
.
Carnot
• Transfers:
Other
Heat 1
1
Matter v
(m
fracti
olar
o
ex
n
Flow exergy
ergy)
comp./exp. o
rev
o
irrev o
o
W P P dV
W W
Q T T
W W T T
x P P N
where x X N
Example: LN2 Precooler
• Consider a precooler for hydrogen liquefaction
:
, steady
• Must consider 5 transfers as shown:
1 Heat 1
4 Matter v
in out dest
dest in out
o
o i i o ii
X X XdXX
dt dt dt dt
XX X X X
dt
Q T T
x P P N h T s
• If the device is configured such that the exergy of a stream cannot be
transferred, the exergy of that stream is necessarily destroyed.
(Recall extending the boundary to the environment in calcs
i
gen
m
S .?)
• How does the entropy generated in the ortho-para catalyst show up?
Comments on Efficiencies
I
,
Applies to heat engines only.• :
Is all heat equal in value?
Engine is modeled as heat engine.• :
Which heating value?
a.k.
Thermal
Fir
a
st
.:
-Law
outt
in
out
fuel in
W
Q
W
m HV
fuel conversion efficiency, ar
II
, Rev
Engine must still be modeled.• :
What are the prSecond-Law
Exerg
ocess constraints?
Model independent. (Neey
d dead state)• :
Applicable beyo
out
out
out outx
in in
bitrary overall efficiency
W
W
X W
X X
,
nd engines. (Any )
a.k.a.:
Assigns equal value to heat and work.• :
Which heating valuUti
e?
a.k.a.: - - - ( )
lization
out
out outu
fuel in
X
rational efficiency
W Q
m HV
combined heat and power CHP efficiency
Some Exergy Analysis Illustrations
• Exergy Resources
• GT/NGCC/STIG
• NG Reforming
• ASU
• SCATR
• SOFC/GT
• LHR Engines
• Environmental Impact
• Optimal Architectures
