t he theoretical background of
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The theoretical background of
GTT-Technologies
The theoretical background of FactSage
The following slides give an abridged overviewof the major underlying principles of the calculational modules of FactSage.
GTT-Technologies
The Gibbs Energy Tree
Mathematical methods are used to derive more information from the Gibbs energy ( of phase(s)or whole systems )
GibbsEnergy
Minimisation
Gibbs-Duhem
Legendre Transform.Partial Derivativeswith Respect tox, T or P
Equilibria
Phase DiagramMaxwellH, U, F mi,cp(i),H(i),S(i),ai,vi
Mathematical Method
Calculational result derived
from G
GTT-Technologies
Thermodynamic potentials and their natural variablesVariables
Gibbs energy: G = G(T, p, ni ,...) Enthalpy: H = H
(S, P, ni ,...) Free energy: A= A (T,V, ni ,...)
Internal energy: U = U(S, V, ni ,...)
Interrelationships:A = U TSH = U PVG = H TS =
U PV TS
GTT-Technologies
PTii n
Gµ,
VTin
A
,
PSin
H
,
VSin
U
,
Maxwell-relations:
Thermodynamic potentials and their natural variables
VPH
STG
PP TT
S U V
H A
G
S V
and
GTT-Technologies
...nV,S,const.for0 i,
dUU min
...np,T,const.for0 i,
dGG min
Thermodynamic potentials and their naturalEquilibrium condition:
...nU,T,const.for0 i,
dTA min
...np,S,const.for0 i,
dHH min
...nV,U,const.for0 i,
dSS max
GTT-Technologies
Temperature
Composition
ii
i
i
npnpp
np
np
TGT
THc
TGTGSTGH
TGS
,2
2
,
,
,
Use of model equations permits to start at either end!
Gibbs-Duhem integrationPartial Operator
Integral quantity: G, H, S, cp
Partial quantity: µi, hi, si, cp(i)
Thermodynamic propertiesfrom the Gibbs-energy
GTT-Technologies
With (G is an extensive property!)
one obtains
T,pinG
i
mJ.W. Gibbs defined the chemical potential of a component as:
mi GnG
Thermodynamic propertiesfrom the Gibbs-energy
mi
im
mii
i
Gn
nG
Gnn
m
GTT-Technologies
Transformation to mole fractions :
mi
imi
mi Gx
xGx
G
mi
ii x
xx
1 = partial operator
ii xn
Thermodynamic propertiesfrom the Gibbs-energy
mpCipc mpC
mpC
mi
imi
mi Hx
xHx
Hh
mSis mS mS
GTT-Technologies
Gibbs energy functionfor 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-knownGibbs-Helmholtz relation:
• In this H(T) is
• and S(T) is
• Thus for a given T-dependence of the cp-polynomial (for example after Meyer and Kelley) one obtains for G(T):
TSHG
T
p dTcHH298298
T
p dTTcSS298298
232ln TFTETDTTCTBAG(T)
GTT-Technologies
Gibbs energy functionfor 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:
)/())()()((
))((ln),( )(,
kjii j k
ijkkk
ijkjj
ijkiikji
i j
n
jiijjii
iii
oiiim
xxxTLxTLxTLxxxx
xxTLxxxxRTGxxTGij
0
GTT-Technologies
Equilibrium condition: or
Reaction : nAA + nBB + ... = nSS + nTT + ...Generally :
For constant T and p, i.e. dT = 0 and dp = 0,and no other work terms:
min G 0 dG
i
iiB 0
i
iidndG m
Equilibrium considerationsa) Stoichiometric reactions
GTT-Technologies
For a stoichiometric reaction the changes dni are given by the stoichiometric coefficients ni and the change in extend of reaction dx.
Thus the problem becomes one-dimensional.One obtains:
[see the following graph for an example of G = G(x) ]
x d dn ii
0i
id dG xm i
Equilibrium considerationsa) Stoichiometric reactions
GTT-Technologies
Gibbs Energy as a function of extent of the reaction2NH3<=>N2 + 3H2 for various temperatures. It is assumed,that the changes of enthalpy and entropy are constant.
Extent of Reaction x
Gib
bs e
nerg
y G
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T = 400K
T = 500K
T = 550K
Equilibrium considerationsa) Stoichiometric reactions
GTT-Technologies
Separation of variables results in :
Thus the equilibrium condition for a stoichiometric reaction is:
Introduction of standard potentials mi° and activities ai yields:
One obtains:
0i
ii µdξdG
0 i
ii µG
iii aRTµµ ln
0 i
iii
ii aRTµ ln
Equilibrium considerationsa) Stoichiometric reactions
GTT-Technologies
It follows the Law of Mass Action:
where the product
or
is the well-known Equilibrium Constant.
i i
iiiiaRTµG ln
i
iiaK
Equilibrium considerationsa) Stoichiometric reactions
RTGK
exp
The REACTION module permits a multitude of calculations which are based on the Law of Mass Action.
GTT-Technologies
Complex EquilibriaMany components, many phases (solution phases), constant T and p :
with
or
i
ioi
iiii aRTnnG lnmm
m
m
im GnG
p
minG
Equilibrium considerationsb) Multi-component multi-phase approach
GTT-Technologies
Massbalance constraint
j = 1, ... , n of components b
Lagrangeian Multipliers Mj turn out to be the chemical potentials of the system components at equilibrium:
i
jiij bna
j
jjMbG
Equilibrium considerationsb) Multi-component multi-phase approach
GTT-Technologies
System ComponentsPhase ComponentsFe N O C Ca Si Mg
Fe 1 0 0 0 0 0 0N2 0 2 0 0 0 0 0O2 0 0 2 0 0 0 0C 0 0 0 1 0 0 0CO 0 0 1 1 0 0 0CO2 0 0 2 1 0 0 0Ca 0 0 0 1 0 0 0CaO 0 0 1 0 1 0 0Si 0 0 0 0 0 1 0SiO 0 0 1 0 0 1 0
Gas
Mg 0 0 0 0 0 0 1SiO2 0 0 2 0 0 1 0Fe2O3 2 0 3 0 0 0 0CaO 0 0 1 0 1 0 0FeO 1 0 1 0 0 0 0
Slag
MgO 0 0 1 0 0 0 1Fe 1 0 0 0 0 0 0N 0 1 0 0 0 0 0O 0 0 1 0 0 0 0C 0 0 0 1 0 0 0Ca 0 0 0 0 1 0 0Si 0 0 0 0 0 1 0
Liq. Fe
Mg 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
aij j
i
Equilibrium considerationsb) Multi-component multi-phase approach
GTT-Technologies
Modelling of Gibbs energy of (solution) phases
Pure Substance (stoichiometric)
Solution phase
,pT,nGG imm
),(,, pTGG oom
m
ex
m
idm
idm
refmm
GSTG
GG
,
,
,
Equilibrium considerationsb) Multi-component multi-phase approach
Choose appropriate reference state and ideal term, then check for deviations from ideality.See Page 11 for the simple substitutional case.
GTT-Technologies
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.
Equilibrium considerationsMulti-component multi-phase approach
GTT-Technologies
Phase diagrams as projections of Gibbs energy plotsHillert 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
a
b g
P
Tab
bg
ag
a
b
g
ab
g
m
PT
Unary system: projection from m-T-p diagram
Phase diagrams as projections of Gibbs energy plots
GTT-Technologies
Binary system: projection from G-T-x diagram, p = const.
300
400
500
600
700
1.0
0.5
0.0
-0.5
-1.0
1.0 0.8 0.6 0.4 0.2 0.0
T
CuxNiNi
G
Phase diagrams as projections of Gibbs energy plots
GTT-Technologies
Ternary system: projection from G-x1-x2 diagram, T = const and p = const
Phase diagrams as projections of Gibbs energy plots
GTT-Technologies
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.
Phase diagrams generated with FactSage
GTT-Technologies
0i i i iSdT VdP n d q dm Gibbs-Duhem:
i i i idU TdS PdV dn dqm
N-Component System (A-B-C-…-N)
SVnAnB nN
T-P µAµB µN
Extensive variables
Corresponding potentials
jqii q
U
iq
GTT-Technologies
N-component system(1) Choose n potentials: 1, 2, … , n (2) From the non-corresponding extensive variables
(qn+1, qn+2, … ), form (N+1-n) independent ratios(Qn+1, Qn+2, …, QN+1).
Example:
Choice of Variables which always give a True Phase Diagram
1Nn
11 Nin
2
1
N
nJj
ij
q
[ 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
1 = T for y-axis
2 = -P constant
for x-axis
S T
V -P
nMgO µMgO
nCaO µCaO
Extensive variables and corresponding potentials
Chosen axes variables and constants
CaOMgO
CaO
CaO
MgO
nnnQ
nq
nq
3
4
3
GTT-Technologies
S T
V -P
nFe mFe
nCr mCr
f1 = T (constant)
f2 = -P (constant)
x-axis
x-axis
(constant)
Fe - Cr - S - O System
Fe
Cr
Fe
Cr
S
O
nnQ
nq
nq
5
6
5
4
3
2
2
m
m
2
2
S
O
m
m
2
2
S
O
n
n
GTT-Technologies
Fe - Cr - C System - improper choice of axes variablesS T
V -P
nC mC
nFe mFe
nCr mCr
f1 = T (constant)
f2 = -P (constant)
f3 = mC aC for x-
axis andQ4 for y-axis
(NOT OK)
(OK)
4
4
Cr
Fe C
Cr
e
r
F Cr C
nQn n n
nQn n
Requirement: 0 3j
i
dQfor i
dq
GTT-Technologies
This is NOT a true phase diagram.
Reason: nC must NOT be used in formula for mole fraction when aC is an axis variable.
NOTE: FactSage users are safe since they are not given this particular choice of axes variables.
M23C6
M7C3
bcc
fcc
cementitelog(ac)
Mol
e fr
actio
n of
Cr
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-3 -2 -1 0 1 2
Fe - Cr - C System - improper choice of axes variables
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