General Method for Calculating Chemical Equilibrium Composition
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- Laurence Howard
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1 AE 6766/Setzma Sprg 004 Geeral Metod for Calculatg Cemcal Equlbrum Composto For gve tal codtos (e.g., for gve reactats, coose te speces to be cluded te products. As a example, for combusto of ydroge wt ar we mgt cose te followg products:,,,,,,, ad. I terms of a coverso reacto, we would wrte: φ + ( were φ represets a arbtrary umber of moles of te fuel (ad also te way ts equato was cose to be wrtte t s also called te equvalece rato, were a value of φ represets just te rgt umber of moles of fuel to react wt te oxyge to form oly te most stable combusto products, ts case. For a mxture of M speces, tere are geerally M+ ukows: M cocetratos,, ad two tesve propertes, e.g., T ad. For a mxture cosstg of R atoms, e.g. R4 for te C/// system, we ca wrte R atom coservato equatos. You ca tk of eac of te coservato equatos as tal codtos or costrats. (ote, tese R equatos may ot be depedet, wc case we ca oly use as may as are depedet. If we furter specfy two termodyamc propertes, we typcally ave M-R ukows. e could specfy te T ad of te products, or for example adabatc flame temperature calculatos, you would specfy ad of te products. We solve for te M-R ukows usg stocometrc reacto relatosps to gve us eoug depedet p. To come up wt te M-R reactos, oe metod s to wrte formato reactos for eac speces preset, except for te elemet speces. (ote: ts metod s ot so elpful f oe of te elemet speces ot part of te mxture. I our example (M8,R3, we eed 5 formato reactos: ext, wrte equlbrum relatosps for eac formato reactos usg te pf, for eac. For our ydroge/ar example, we ave:
2 f, f, f, f, f, 4 ow, we clude te (depedet atom coservato equatos. Aga for our example, we get: atoms atoms atoms φ ( ( 3.76 ( were s te al umber of product moles per φ moles of ad s ukow at ts pot. 5 To remove te depedece, we use atom balace ratos (pyscally, t s tese ratos, ot te al umber of moles, wc are most mportat, ad we add te costrat tat te mole fractos must sum to uty, e.g., atoms atoms atoms atoms φ ( 3.76 ( ( ( 6 ow gve two termodyamc propertes, we ave eoug formato to solve for te ukow. If T ad of te products are kow, te soluto smply cossts of determg te pf, from a source suc as te JAAF tables. If te fal temperature s ukow, for example a adabatc flame temperature calculato, te te soluto s teratve: guessg T, fdg te product composto, te calculatg ts assocated T ad usg t to mprove your guess at T. As a alteratve for calculatg te fal temperature, oe ca realze tat most of te eergy s assocated wt te presece of te major (largest speces. Teore you ca gore all te oter/ mor speces o your frst terato ad get a very close estmate of T usg te major speces oly. Te go back ad reterate, ow cludg te mor speces. For roug estmates of product compostos, you ca smply take te temperature ad te speces mole fractos foud from te major speces product calculatos ad use tem, alog wt stocometrc reactos tat form te mor speces from te major speces (.e., approprate p, to calculate te mor speces cocetratos. Ts approac s kow as te major-mor model or major-mor speces approxmato. I some cases, lke te - -
3 ydroge/ar example descrbed above, oe ca get a smple algebrac soluto for te mole fractos of te major products usg te major speces model (see below. f course, te easest way to solve te problem s to use a cemcal equlbrum computer code/tool. You stll ave to determe te products to be cluded te calculato, ad te tal codtos, e.g., tal atom ratos, but te te computer ca perform te termodyamc property evaluatos ad te teratos!! 7 Major-Mor Model: To llustrate te use of te major-mor model, let s estmate te flame temperature for te ydroge/ar combusto example. Frst, we coose te major speces; we let te products be, ad eter (for lea mxtures or (for rc mxtures. Wrtg te reactos for te two cases ad deotg te stocometrc coeffcets for te products terms of φ from smple atom balaces, we ave φ for φ< : φ + ( φ for φ> : φ + ( ( φ Major Speces Mole Fractos: From te above reacto equatos ad wt algebra, we get: φ< (lea φ> (rc φ φ φ φ. ( φ+ 88. φ ( φ+ 88. ( φ ( φ Tus smply gve φ (te : rato, we kow te product composto ad ca calculate te fal temperature. For example wt φ.3 (rc combusto, we get, Adabatc Flame Temperature: Assumg a tal temperature of -55 C (8 for te reactats, we fd te adabatc flame temperature usg 0 ad R0,.e., roducts ( T ( ad Reactats [( + ] [( + ] Tad T roducts 8 Reactats 8 T - 3 -
4 mol Wrtg out te summato for eac product ad reactat, ad usg te umber of moles of eac for our φ.3 flame we ave,.3mol [( T + ] +.88mol [( ] [( ] T mol T + ad, ad ad f T [( ] [( ] [( ] mol mol 8 + Sce te etalpy of formato for elemets s zero, we get mol.3 [( T + ] +.88mol [( ] [( ] T + 0.3mol T ad ad ad T moles [( ] [( ] [( ] moles moles 8 Data for f of water ad te sesble etalpy cages for eac speces * ca be foud a umber of sources, e.g., te JAAF tables. Usg ts data, oe fds Tad 300. Mor Speces Mole Fractos: We ca wrte te followg stocometrc relatosps betwee te mor speces ad te major speces of our rc ydroge/ar flame (for empass, te mor speces are wrtte bold letters. Eac reacto represets a metod for producg te mor speces usg oly te major products of our ydroge/ar flame ow we ca wrte te followg expressos for te mole fractos of te mor speces terms of te major speces : f, f, f, f, f, f, f, f, * T It s ot reasoable to assume tat c p s costat c ( T T T p dt for our large temperature rage T
5 were we ave used te fact tat p for eac of te stocometrc reactos s smply a fucto of te formato equlbrum costats of te speces te reacto. For example for te reacto + te equlbrum costat s gve by p f, f, f, f, sce te formato equlbrum costat pf of a elemet s uty (by defto. Usg te p f from te JAAF tables at 300, te estmated for te major speces, ad assumg a pressure of bar, we get te values for te mor speces lsted te table below. As a comparso, te table below also cludes results from a complete soluto obtaed wt te STAJA cemcal equlbrum code. Wle ot completely accurate, te major-mor model does a good job of predctg te flame temperature (+5 or ~% relatve error ad te major speces mole fractos (<% relatve error, ad gves a reasoable estmate of te mor speces mole fractos (wt a 40% relatve error, ad certaly better ta oe order-of-magtude. Tus for roug approxmatos, te major-mor model s smple ad reasoably accurate. Speces/Tad Major-Mor Model Full Calculato Error(% Tad % 59.0% % 3.% % 9.4% % (3000 ppm 0.6% (600 ppm 5 0.6% (600 ppm 0.3% (300 ppm 3 0 ppm 80 ppm 49 ppm 36 ppm ppb.5 ppb 36 ppmparts per mllo ( 0-6, ppbparts per bllo (
We have already referred to a certain reaction, which takes place at high temperature after rich combustion.
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