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T he theoretical background of. The theoretical background of FactSage. The following slides give an abridged overview of the major underlying principles of the calculational modules of FactSage. Maxwell H, U, F. Phase Diagram. m i ,c p(i), H (i) ,S (i) ,a i ,v i. Gibbs-Duhem. - PowerPoint PPT Presentation
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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