he theoretcal backgroud of
-echologes he theoretcal backgroud of FactSage he followg sldes gve a abrdged overvew of the ajor uderlyg prcples of the calculatoal odules of FactSage.
-echologes he bbs Eergy ree Mawell H, U, F Phase Dagra,c p(), H (),S (),a,v bbs-duhe Legedre rasfor. Equlbra Partal Dervatves wth Respect to, or P Msato bbs Eergy Matheatcal ethods are used to derve ore forato fro the bbs eergy ( of phase(s) or whole systes ) Matheatcal Method Calculatoal result derved fro
-echologes herodyac potetals ad ther atural varables Varables bbs eergy: = (, p,,...) Ethalpy: H = H (S, P,,...) Free eergy: A = A (,V,,...) Iteral eergy: U = U (S, V,,...) Iterrelatoshps: A = U S H = U PV = H S = U PV S
-echologes P µ, V A, SP H, SV U, Mawell-relatos: herodyac potetals ad ther atural varables V P H S P P S U V H A S V ad
-echologes... S,V, cost. for 0, du U...,p, cost. for 0, d herodyac potetals ad ther atural Equlbru codto:... U,, cost. for 0, d A... S,p, cost. for 0, dh H... cost.u, V, for 0, ds S a
-echologes eperature Coposto p p p p p H c S H S, 2 2,,, Use of odel equatos perts to start at ether ed! bbs-duhe tegrato Partal Operator Itegral quatty:, H, S, c p Partal quatty: µ, h, s, c p() herodyac propertes fro the bbs-eergy
-echologes herodyac propertes fro the bbs-eergy J.W. bbs defed the checal potetal of a copoet as:,p Wth ( s a etesve property!) oe obtas
-echologes rasforato to ole fractos : 1 = partal operator herodyac propertes fro the bbs-eergy C p c p C p C p H H H h S s S S
-echologes bbs eergy fucto for a pure substace () (.e. eglectg pressure ters) s calculated fro the ethalpy H() ad the etropy S() usg the well-kow bbs-helholtz relato: H S I ths H() s H H 298 298 cp d ad S() s S S 298 298 c p d hus for a gve -depedece of the c p -polyoal (for eaple after Meyer ad Kelley) oe obtas for (): () A B C l D 2 E 3 F 2
-echologes bbs eergy fucto for a soluto As show above (,) for a soluto cossts of three cotrbutos: the referece ter, the deal ter ad the ecess ter. For a sple substtutoal soluto (oly oe lattce ste wth rado occupato) oe obtas usg the well-kow Redlch-Kster-Muggau polyoal for the ecess ters: (, ) o, R l j 0 j j L ( ) j ( )( j ) j k j k ( L jk ( ) j L jk j ( ) k L jk k ( )) /( j k )
-echologes Equlbru cosderatos a) Stochoetrc reactos Equlbru codto: Reacto : A A + B B +... = S S + +... eerally : or d 0 B 0 For costat ad p,.e. d = 0 ad dp = 0, ad o other work ters: d d
-echologes Equlbru cosderatos a) Stochoetrc reactos For a stochoetrc reacto the chages d are gve by the stochoetrc coeffcets ad the chage eted of reacto d. d d hus the proble becoes oe-desoal. Oe obtas: d d 0 [see the followg graph for a eaple of = () ]
bbs eergy -echologes Equlbru cosderatos a) Stochoetrc reactos = 400K = 500K = 550K 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Etet of Reacto bbs Eergy as a fucto of etet of the reacto 2NH 3 <=>N 2 + 3H 2 for varous teperatures. It s assued, that the chages of ethalpy ad etropy are costat.
-echologes Equlbru cosderatos a) Stochoetrc reactos Separato of varables results : d dξ µ 0 hus the equlbru codto for a stochoetrc reacto s: µ 0 Itroducto of stadard potetals ad actvtes a yelds: µ µ R l a Oe obtas: µ R l a 0
-echologes Equlbru cosderatos a) Stochoetrc reactos It follows the Law of Mass Acto: µ R l a where the product K a or s the well-kow Equlbru Costat. he REACION odule perts a ulttude of calculatos whch are based o the Law of Mass Acto. K ep R
-echologes Equlbru cosderatos b) Mult-copoet ult-phase approach Cople Equlbra May copoets, ay phases (soluto phases), costat ad p : wth o R l a or p
-echologes Equlbru cosderatos b) Mult-copoet ult-phase approach Massbalace costrat a j b j j = 1,..., of copoets b Lagragea Multplers M j tur out to be the checal potetals of the syste copoets at equlbru: j b j M j
-echologes Equlbru cosderatos b) Mult-copoet ult-phase approach a j j Phase as Slag Lq. Fe C o po et s Syst e C o po et s Fe N O C Ca S Mg Fe 1 0 0 0 0 0 0 N 2 0 2 0 0 0 0 0 O 2 0 0 2 0 0 0 0 C 0 0 0 1 0 0 0 CO 0 0 1 1 0 0 0 CO 2 0 0 2 1 0 0 0 Ca 0 0 0 1 0 0 0 C ao 0 0 1 0 1 0 0 S 0 0 0 0 0 1 0 SO 0 0 1 0 0 1 0 Mg 0 0 0 0 0 0 1 SO 2 0 0 2 0 0 1 0 Fe 2 O 3 2 0 3 0 0 0 0 C ao 0 0 1 0 1 0 0 FeO 1 0 1 0 0 0 0 M go 0 0 1 0 0 0 1 Fe 1 0 0 0 0 0 0 N 0 1 0 0 0 0 0 O 0 0 1 0 0 0 0 C 0 0 0 1 0 0 0 Ca 0 0 0 0 1 0 0 S 0 0 0 0 0 1 0 Mg 0 0 0 0 0 0 1 Eaple of a stochoetrc atr for the gas-etal-slag syste Fe-N-O-C-Ca-S-Mg
-echologes Equlbru cosderatos b) Mult-copoet ult-phase approach Modellg of bbs eergy of (soluto) phases,,p Pure Substace o, o, (, p) (stochoetrc) Soluto phase, ref, d, e S d Choose approprate referece state ad deal ter, the check for devatos fro dealty. See Page 11 for the sple substtutoal case.
-echologes Equlbru cosderatos Mult-copoet ult-phase approach Use the EQUILIB odule to eecute a ulttude of calculatos based o the cople equlbru approach outled above, e.g. for cobusto of carbo or gases, aqueous solutos, etal clusos, gas-etal-slag cases, ad ay others. NOE: he use of costrats such calculatos (such as fed heat balaces, or the occurrece of a predefed phase) akes ths odule eve ore versatle.
-echologes Phase dagras as projectos of bbs eergy plots Hllert has poted out, that what s called a phase dagra s dervable fro a projecto of a so-called property dagra. he bbs eergy as the property s plotted alog the z-as as a fucto of two other varables ad y. Fro the u codto for the equlbru the phase dagra ca be derved as a projecto oto the -y-plae. (See the followg graphs for llustratos of ths prcple.)
-echologes Phase dagras as projectos of bbs eergy plots g a b g b a b a b g a P P Uary syste: projecto fro --p dagra
-echologes Phase dagras as projectos of bbs eergy plots 1.0 0.5 0.0-0.5-1.0 300 400 1.0 N 0.8 0.6 0.4 0.2 N 0.0 Cu 700 600 500 Bary syste: projecto fro -- dagra, p = cost.
-echologes Phase dagras as projectos of bbs eergy plots erary syste: projecto fro - 1-2 dagra, = cost ad p = cost
-echologes Phase dagras geerated wth FactSage Use the PHASE DIARAM odule to geerate a ulttude of phase dagras for uary, bary, terary or eve hgher order systes. NOE: he PHASE DIARAM odule perts the choce of, P, (as R l a), a (as l a), ol () or weght (w) fracto as as varables. Mult-copoet phase dagras requre the use of a approprate uber of costats, e.g. a terary sopleth dagra vs oe olar rato has to be kept costat.
-echologes N-Copoet Syste (A-B-C- -N) Etesve varables q S V A B N U q -P µ A µ B µ N q j Correspodg potetals d U d S P d V d d q bbs-duhe: S d V d P d q d 0
-echologes Choce of Varables whch always gve a rue Phase Dagra N-copoet syste (1) Choose potetals: 1, 2,, (2) Fro the o-correspodg etesve varables (q +1, q +2, ), for (N+1-) depedet ratos (Q +1, Q +2,, Q N+1 ). N 1 Eaple: Q j q N2 J 1 q j 1 N 1 [ 1, 2,, ; Q +1, Q +2,, Q N+1 ] are the the (N+1) varables of whch 2 are chose as aes ad the reader are held costat.
-echologes MgO-CaO Bary Syste Etesve varables ad correspodg potetals S Chose aes varables ad costats 1 = for y-as V -P 2 = -P costat MgO CaO µ MgO µ CaO q q 3 4 MgO CaO Q 3 MgO CaO CaO for -as
-echologes Fe - Cr - S - O Syste S f 1 = (costat) V -P f 2 = -P (costat) O 2 O 2 3 O 2 -as S 2 S 2 4 S 2 -as Fe Cr Fe Cr q q 5 6 Cr Fe Q 5 Cr Fe (costat)
-echologes Fe - Cr - C Syste - proper choce of aes varables S f 1 = (costat) V -P f 2 = -P (costat) C Fe C Fe f 3 = C a C for -as ad Q 4 for y-as Q 4 Cr F e C r C (NO OK) Cr Cr Q 4 F e Cr C r (OK) dq j Requreet: 0 for 3 dq
Mole fracto of Cr -echologes Fe - Cr - C Syste - proper choce of aes varables 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 M 23 C 6 M 7 C 3 bcc fcc -3-2 -1 0 1 2 log(a c ) ceette hs s NO a true phase dagra. Reaso: C ust NO be used forula for ole fracto whe a C s a as varable. NOE: FactSage users are safe sce they are ot gve ths partcular choce of aes varables.