Transcript The theoretical background of FactSage

The theoretical background of
GTT-Technologies
The theoretical background of
FactSage
The following slides give an
abridged overview
of the major
underlying principles
of the
calculational modules
of
FactSage.
GTT-Technologies
The Gibbs Energy Tree
Maxwell
H, U, F
Phase Diagram
mi,cp(i),H(i),S(i),ai,vi
Gibbs-Duhem
Legendre Transform.
Equilibria
Partial Derivatives
with Respect to
x, T or P
Minimisation
Gibbs
Energy
Mathematical methods are used to derive more
information from the Gibbs energy ( of phase(s)
or whole systems )
Mathematical
Method
Calculational
result derived
from G
GTT-Technologies
Thermodynamic potentials and
their natural variables
Variables
Gibbs energy:
Enthalpy:
Free energy:
Internal energy:
G
H
A
U
=
=
=
=
G
H
A
U
(T, p, ni ,...)
(S,P, ni ,...)
(T,V, ni ,...)
(S,V, ni ,...)
Interrelationships:
A = U  TS
H = U  PV
G = H  TS = U  PV  TS
GTT-Technologies
Thermodynamic potentials and
their natural variables
Maxwell-relations:
H
V
P
S
S
U
H
PP
VV
A
G
TT
G
 S
T
and
 U 
 G 
 H 
 A 
  
  
  

µi  
 ni T,P  ni S,P  ni S,V  ni  T,V
GTT-Technologies
Thermodynamic potentials and
their natural
Equilibrium condition:
G  min 
for const.T, p,ni, ...

dG  0 
H  min 
for const.S,p,ni, ...

dH  0 
A  min 
 for const.T,U, ni, ...
dT  0 
U  min 
for const.S,V,ni, ...

dU  0 
S  max 
 for const.U, V,ni, ...
dS 0 
GTT-Technologies
Thermodynamic properties
from the Gibbs-energy
Temperature
 G 
S  


T

 p,ni
 G 
H  G  TS  G  T


T

 p,ni
  2G 
 H 
cp  
   T  2 
 T  p,ni
 T  p,n
i
Composition
Integral quantity: G, H, S, cp
Partial Operator
Gibbs-Duhem integration
Partial quantity: µi, hi, si, cp(i)
Use of model equations permits to start at either end!
GTT-Technologies
Thermodynamic properties
from the Gibbs-energy
J.W. Gibbs defined the chemical potential
of a component as:
With
G   ni Gm
one obtains

mi 
ni
(G is an extensive property!)
 n G
i
 G 

m 
i  n 
 i T,p
m

 Gm   ni  Gm
ni
GTT-Technologies
Thermodynamic properties
from the Gibbs-energy
Transformation to mole fractions : ni  xi


mi  Gm 
Gm   xi
Gm
xi
xi
hi  Hm 
si
1


  xi
xi
xi
= partial operator


Hm   xi
Hm
xi
xi
Sm
cpi
Sm
Cpm
Sm
Cpm
Cpm
GTT-Technologies
Gibbs energy function
for a pure substance
• G(T) (i.e. neglecting pressure terms) is calculated from the
enthalpy H(T) and the entropy S(T) using the well-known
Gibbs-Helmholtz relation: G  H  TS
• In this H(T) is
• and S(T) is
H  H298
T


298
cp  dT
T
S  S298   cp T  dT
298
• Thus for a given T-dependence of the cp-polynomial (for
example after Meyer and Kelley) one obtains for G(T):
G(T) A B T  C T  lnT  D T2  E T3  F T2
GTT-Technologies
Gibbs energy function
for a solution
• As shown above Gm(T,x) for a solution  consists
of three contributions: the reference term, the
ideal term and the excess term.
• For a simple substitutional solution (only one
lattice site with random occupation) one obtains
using the well-known Redlich-Kister-Muggianu
polynomial for the excess terms:
nij
Gm (T, xi )   xi Gio,  RT xi ln xi   xi x j  L(ij ) (T)( xi  x j )
i
i
i
j
 0
  xi x j xk ( xi Lijki (T)  x j Lijkj (T)  xk Lijkk (T)) /(xi  x j  xk )
i
 j k
GTT-Technologies
Equilibrium considerations
a) Stoichiometric reactions
Equilibrium condition:
G  min
or
dG  0
Reaction : nAA + nBB + ... = nSS + nTT + ...
Generally :
 iBi  0
i
For constant T and p, i.e. dT = 0 and dp = 0,
and no other work terms:
dG   mi dni
i
GTT-Technologies
Equilibrium considerations
a) Stoichiometric reactions
For a stoichiometric reaction the changes dni are
given by the stoichiometric coefficients ni and the
change in extend of reaction dx.
dni   idx
Thus the problem becomes one-dimensional.
One obtains:
dG   mi idx  0
i
[see the following graph for an example of G = G(x) ]
GTT-Technologies
Equilibrium considerations
a) Stoichiometric reactions
Gibbs energy G
T = 400K
T = 500K
T = 550K
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Extent of Reaction x
0.8
0.9
1.0
Gibbs Energy as a function of extent of the reaction
2NH3<=>N2 + 3H2 for various temperatures. It is assumed,
that the changes of enthalpy and entropy are constant.
GTT-Technologies
Equilibrium considerations
a) Stoichiometric reactions
dG
  i µi  0
Separation of variables results in :
dξ
i
Thus the equilibrium condition
for a stoichiometric reaction is:
G   i µi  0
i
Introduction of standard potentials mi° and activities ai
yields:
µi  µi  RTln ai
One obtains:


µ
 i i  RT  i ln ai   0
i
i
GTT-Technologies
Equilibrium considerations
a) Stoichiometric reactions
It follows the Law of Mass Action:
G   i µi  RTln  ai i
i
i
where the product
K   ai
i
i
or
 G 

K  exp 
 RT 
is the well-known Equilibrium Constant.
The REACTION module permits a multitude of
calculations which are based on the Law of Mass Action.
GTT-Technologies
Equilibrium considerations
b) Multi-component multi-phase approach
Complex Equilibria
Many components, many phases (solution phases),
constant T and p :
G  min
with

G   ni mi   ni mio  RTln ai
i
or
i
 mp   
G     nm  Gm
  i


GTT-Technologies
Equilibrium considerations
b) Multi-component multi-phase approach
Massbalance constraint
a n  b
ij i
j
j = 1, ... , n of components b
i
Lagrangeian Multipliers Mj turn out to be the
chemical potentials of the system components at
equilibrium:
G   bj Mj
j
GTT-Technologies
Equilibrium considerations
b) Multi-component multi-phase approach
Phase
aij
j
Gas
i
Slag
Liq. Fe
Components
Fe
N2
O2
C
CO
CO 2
Ca
CaO
Si
SiO
Mg
SiO 2
Fe2O 3
CaO
FeO
MgO
Fe
N
O
C
Ca
Si
Mg
Fe
1
0
0
0
0
0
0
0
0
0
0
0
2
0
1
0
1
0
0
0
0
0
0
N
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
System Components
O
C
Ca
0
0
0
0
0
0
2
0
0
0
1
0
1
1
0
2
1
0
0
1
0
1
0
1
0
0
0
1
0
0
0
0
0
2
0
0
3
0
0
1
0
1
1
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
Si
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
0
0
0
1
0
Mg
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
Example of a stoichiometric matrix for the gas-metal-slag system Fe-N-O-C-Ca-Si-Mg
GTT-Technologies
Equilibrium considerations
b) Multi-component multi-phase approach
Modelling of Gibbs energy of (solution) phases


Gm  Gm T,ni ,p
Pure Substance Gm  m o,  Go, (T,p) (stoichiometric)
Solution phase Gm  Gm,ref
 Gm,id
 Gm,ex
 TS 
id
m
Choose appropriate reference state and ideal term, then check for deviations from ideality.
See Page 11 for the simple substitutional case.
GTT-Technologies
Equilibrium considerations
Multi-component multi-phase approach
Use the EQUILIB module to execute a
multitude of calculations based on the
complex equilibrium approach outlined
above, e.g. for combustion of carbon or
gases, aqueous solutions, metal inclusions,
gas-metal-slag cases, and many others .
NOTE: The use of constraints in such calculations (such
as fixed heat balances, or the occurrence of a
predefined phase) makes this module even
more versatile.
GTT-Technologies
Phase diagrams as projections
of Gibbs energy plots
Hillert has pointed out, that what is called a
phase diagram is derivable from a projection of a
so-called property diagram. The Gibbs energy as
the property is plotted along the z-axis as a
function of two other variables x and y.
From the minimum condition for the equilibrium
the phase diagram can be derived as a projection
onto the x-y-plane.
(See the following graphs for illustrations of this principle.)
GTT-Technologies
Phase diagrams
as projections of Gibbs energy plots
g
a
b
m
b
g
b
g
T
a
b
a
a
T
P
Unary system: projection from m-T-p diagram
P
GTT-Technologies
Phase diagrams
as projections of Gibbs energy plots
1.0
0.5
0.0
G
-0.5
-1.0
300
400
500
600
1.0
Ni
0.8
0.6
0.4
xNi
0.2
T
700
0.0
Cu
Binary system: projection from G-T-x diagram, p = const.
GTT-Technologies
Phase diagrams
as projections of Gibbs energy plots
Ternary system: projection from G-x1-x2 diagram,
T = const and p = const
GTT-Technologies
Phase diagrams generated with
FactSage
Use the PHASE DIAGRAM module to generate a
multitude of phase diagrams for unary, binary, ternary
or even higher order systems.
NOTE: The PHASE DIAGRAM module permits the choice of
T, P, m (as RT ln a), a (as ln a), mol (x) or weight (w)
fraction as axis variables. Multi-component phase diagrams
require the use of an appropriate number of constants, e.g.
in a ternary isopleth diagram T vs x one molar ratio has to
be kept constant.
GTT-Technologies
N-Component System (A-B-C-…-N)
Extensive
variables
qi
 U 
i   
 qi qj
S
V
nA
nB
T
-P
µA
µB
nN
µN



Corresponding
potentials



dU  TdS  PdV   mi dni  i dqi
Gibbs-Duhem:
SdT  VdP   ni d mi   qi di  0
GTT-Technologies
Choice of Variables which always
give a True Phase Diagram
N-component system
(1) Choose n potentials: 1,  2, … ,  n
n  N  1
(2) From the non-corresponding extensive variables
(qn+1, qn+2, … ), form (N+1-n) independent ratios
(Qn+1, Qn+2, …, QN+1).
Example:
Qj 
qi
n  1  i  N  1
N 2
q
J n1
j
[ 1,  2, … ,  n; Qn+1, Qn+2, …, QN+1] are then the (N+1) variables
of which 2 are chosen as axes
and the remainder are held constant.
GTT-Technologies
MgO-CaO Binary System
Extensive variables
and corresponding
potentials
Chosen axes variables
and constants
S
T
1 = T
for y-axis
V
-P
2 = -P
constant
nMgO
µMgO
nCaO
µCaO
q3  nMgO
nCaO

Q3 
nMgO  nCaO
q4  nCaO 
for x-axis
GTT-Technologies
Fe - Cr - S - O System
S
V
T
-P
nO2
mO
nS2
mS
nFe
mFe
nCr
mCr
2
2
f1 = T
(constant)
f2 = -P
(constant)
3  mO
x-axis
4  m S
x-axis
2
2
q5  nCr 
nCr

 Q5 
nFe
q6  nFe
(constant)
GTT-Technologies
Fe - Cr - C System - improper
choice of axes variables
S
T
V -P
f1 = T
(constant)
f2 = -P
(constant)
nC
mC
f3 = mC
nFe
mFe
Q4 
nCr
 nFe  nCr  nC  (NOT OK)
mCr
Q4 
nCr
 nFe  nCr 
(OK)
dQ j
0
i 3
nCr
Requirement:
dqi

for
aC for x-axis and
Q4 for y-axis
GTT-Technologies
Fe - Cr - C System - improper
choice of axes variables
1.0
0.9
This is NOT a true phase
diagram.
M23C6
Mole fraction of Cr
0.8
0.7
Reason: nC must NOT be
used in formula for mole
fraction when aC is an
axis variable.
M7C3
0.6
0.5
0.4
0.3
bcc
0.2
0.1
fcc
0
-3
-2
-1
0
log(ac)
1
cementite
2
NOTE: FactSage users
are safe since they are
not given this particular
choice of axes variables.