non-statistical thermodynamic optimization: an extravagance or a useful tool?
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Non-statistical thermodynamic optimization: an extravagance or a useful tool? . Cong (Leo) Dai. Outline. Thermodynamic modeling CALPHAD method Principles and examples Advantages and deficiencies Kantorovich idea Conclusions and future efforts. Thermodynamic modeling. B. C. A. - PowerPoint PPT PresentationTRANSCRIPT
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Non-statistical thermodynamic optimization: an extravagance or a useful tool?
Cong (Leo) Dai
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2Outlin
e
• Thermodynamic modeling
• CALPHAD method
• Principles and examples
• Advantages and deficiencies
• Kantorovich idea
• Conclusions and future efforts
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3Thermodynamic modeling
PhasSage, www.factsage.com PhasSage; http://honghua66.en.ecplaza.net; Solutionizing Heat Treatment Furnace (27982050)
A
B C
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4
What is needed to use thermodynamic modeling?
• Analytical descriptions of the Gibbs energies of stoichiometric compounds and solutions• Publications including
tables of thermochemical data
• Databases
• A program minimizing the Gibbs energy of a system• Thermo-Calc• FactSage• Pandat• MALT2
Reliability of the analytical representations of the Gibbs energies?
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5Phase diagrams of the Fe–Nb system
TCFE2 database TCFE6 database
BA
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6Constructing the Gibbs energies
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7CALPHAD technique is not unique
• Step 1: a modelThe coin is a disc• Step 2: identify unknown model’s
parametersRadius r, thickness h, density ; they all
must be positive• Step 3: collect all available experimental
datar, h, d, l, a, V, m, • Step 4: solve an optimization problem
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8Finding r, h and
2 2 2 2
1 1 1 1
2 2 2
1 1 1
2
1
2 2 2
2 2
m
0,
i
0
n
, 0
, ,
h d lr
a V m
n n nni i i i
i i i ii i i i
n n ni i i
i i ii i i
ni
i i
r r h r
r r h h r d r lr h d l
a V ma V m
h
h
h
r
r
Non-linear least squares problem with linear constraints
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9Almost the same happens in CALPHAD
• Experimental data1. Enthalpies of mixing at 1150°C2. (Ag)+L / L liquidus temperatures3. L / L+(Cu) liquidus temperatures4. Chemical potential of Cu in (Ag)5. Chemical potential of Ag in (Cu)
u
Ag
L
C
Ag AgA
Cu CuCuex Cu C
L LLex L L
gex Ag
u Cu0
A Ag
L0
g0
1
,
1
,
1
,
n
i ii
i
n
i
i
ii
n
i
G x x
G x x C x T
C T
G x x C x T
x
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10Building the Gibbs energies of the phases
2exp calc L
9
exp1
2 2exp calc L exp calc L
2 5
exp exp1 1
2exp calc Ag expCu Cu Ag A
expCu
L
L Ag
Ag
CuL
, ,
, , , , , ,
, ,
i i i
i i
i i i i i i
i ii i
H H x T
H
T T x T T T x T
T T
x T
C
C C CC
C
2
calc Cug
expAg
Cu
CuAgL, ,
min , ,x T
C CC
C
A traditional CALPHAD method rests on a statistical foundation
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Statistical description of data
1
2
1
2
1
Mean: ,
( ) Variance: ,
1
( ) Standard error of the mean:
( 1)
n
ii
n
ii
n
ii
xx
n
x x
n
x x
n n
• Measurement is non-repeated
• Measurement is insufficient
• Measurement errors are affected by random errors and systematical errors
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Kantorovich idea(1962)
Instead of minimizing the sum of squared derivations, simultaneous inequalities should be used, which can be solved by linear programming methods.
Kantorovich, L.V. Sib. Mat. Zh., 1962, vol.3, No.5, p. 701.
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Interval data
• All possible outcomes of an experiment belong to a finite interval!
• Any value in the same interval data is equally acceptable!
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Non-statistical approach14
1. Error analysis
2. Postulate a model
3. Solve inequalities
i iy a b x
i i iy y y
i i iy a b x y
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Data from Kawakami
Kawakami, M.: Sci. Rep. Res. Inst. Tohoku Univ. 7 (1930) 351.
Reach the required temperature
Remove the porcelain tube
Measure the temperature variation
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Mixing Enthalpy of liquid Mg-Ag alloys
( )E
E EP
GH G TT
Kawakami, M.: Sci. Rep. Res. Inst. Tohoku Univ. 7 (1930) 351.
A B
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Mixing Enthalpy of liquid Mg-Ag alloys
No feasible region.
)21()1(
)21()1(
MgMgMgE
MgMgMgE
xbaxxH
xbaxxG
axxH
axxG
MgMgE
MgMgE
)1(
)1(
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Data from Gran
J. Gran et al./ CALPHAD 36 (2012 89-93)
Heat the furnace
Hold at a predetermined temperature
Quenched in an Argon-stream
Analyzed by ICP-AES analysis(Inductively Coupled Plasma-
Atomic Emission Spectrometry)
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Activity of Mg in liquid Mg-Ag alloys19
ln( ) ln( )Mg MgE
MgMgMg
a aRT G RT
xx
J. Gran et al./ CALPHAD 36 (2012 89-93)
(1 )E
E EMg Mg
Mg
GG G xx
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20
Activity of Mg in liquid Mg-Ag alloys
axG
axxG
MgEMg
MgMgE
2)1(
)1(
bxxaxG
xbaxxG
MgMgMgEMg
MgMgMgE
)41()1()1(
)21()1(
22
No feasible region.
Comparison Future work!
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Conclusions21
• CALPHAD technique is successful but not
always satisfactory.
• A non-statistical approach is established and
applied to thermodynamic optimization.
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Future efforts22
• Comparisons in liquid Mg-Ag alloys.
• Rules of assigning interval data.
• Non-statistical approach applies to other
interesting thermodynamic assessments.
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23
Acknowledgements
Supervisor:
- Dr. Dmitri Malakhov
Thanks for your attention!