Analyss of the chemcal equlbum of combuston at constant volume Maus BEBENEL* *Coesondng autho LIEHNICA Unvesty of Buchaest Faculty of Aeosace Engneeng h. olzu Steet -5 6 Buchaest omana mausbeb@yahoo.com DI:.3/66-8.4.6.S.3 Abstact: Detemnng the comoston of a mxtue of combuston gases at a gven temeatue s based on chemcal equlbum when the equlbum constants ae calculated on the assumton of constant essue and temeatue. In ths ae an analyss of changes occung when combuston takes lace at constant volume s esented devng a secfc fomula of the equlbum constant. he smle eacton of cabon combuston n ue oxygen n both cases (constant essue and constant volume) s next consdeed as examle of alcaton obsevng the changes occung n the comoston of the combuston gases deendng on temeatue. ey Wods: mathematcal modelng combuston chemcal equlbum themodynamc oetes ntenal enegy. INDUCIN he mathematcal modelng of the chemcal eactons usually egads the behavo n eactos whee ocesses ae steady so that the temeatue and essue ae constant. he comoston of the eacton oducts at equlbum s detemned by the law of mass acton usng the equlbum constants whch take nto account a constant essue. Howeve thee ae cases when the eacton s caed out at constant volume (fo examle combuston n a ston engne cycle). We wll show that at constant volume the exessons of equlbum constant dffe fom those whch ae secfc to eactons at constant essue. he nfluence of the mathematcal model changes on the comoston of the combuston oducts deendng on the combuston temeatue wll be analyzed n ths ae.. LAW F MASS ACIN We wll efe to the law concenng the chemcal equlbum. Fo a chemcal eacton of the fom ' A ' A... ' ' A ' ' A... () the law of the mass acton can be exessed by the equaton: '' C C C ' INCAS BULLEIN Volume 6 Secal Issue / 4. 5 34 ISSN 66 8 ()
Maus BEBENEL 6 whee C = equlbum constant n tems of concentatons C = concentatons of seces [ ]. = stochometc coeffcents A smla exesson of equlbum constant can be deved n tems of essues o mola factons. he last shows a secal nteest snce t can be exessed as a functon of themodynamc oetes of the eacton atcants: INCAS BULLEIN Volume 6 Secal Issue / 4 whee = equlbum constant n tems of mola factons (used fo eactons n gas hase). = mola facton of seces [ ]. It s well known that fo the eactons occung at constant essue and temeatue the chemcal equlbum s eached then when the bbs fee enegy of the gven system gets mnm: '' ' equlbu m syst mn dsyst (4) whch s equvalent to the condton: whee (3) (5) bbs fee eneges of seces [ ] of eactants and of eacton oducts esectvely the themodynamc functon beng defned be the exesson U V S (6) wheen U = ntenal enegy S = entoy = essue = absolute temeatue V = volume. By ocessng equaton (5) an exlct fomula of the equlbum constant n tems of mola factons s deved whch s vey useful fo the eactons n gas hase occung at constant essue and temeatue [] []: ex whee = standad efeence essue =.3 ba then (8) (9) and = unvesal gas constant = 8.34 kj/kmol. notaton [ ] efeng to the mola quanttes. (7)
7 Analyss of the chemcal equlbum of combuston at constant volume 3. EQUILIBIUM NSAN A NSAN VLUME In what follows a smla fomula of the equlbum constant wll be deved but fo a eacton takng lace at constant volume. Accodngly unde the st law of hemodynamcs the themal effect of the eacton s elated to the change n ntenal enegy of the system: Q du Q U U U () n the othe hand accodng to the nd law the total (unvesal) entoy system+suoundngs must ncease: that: S S S () unv Usng the notaton S syst S one deduces fom () the equaton: syst su (whee [ ] means eacton ) and takng nto account Q S () su Q S Q S (3) Snce the eacton s consdeed to take lace at constant volume and constant temeatue equal to that of the suoundngs by alyng () the above nequalty becomes: S Accodng to the defnton of Helmholtz fee enegy: the condton (4) educes to U (4) F U S df du ds ( const.) (5) df (6) At equlbum the Helmholtz fee enegy eaches ts mnmum value: F F eq mn (7) Unde these ccumstances smlaly to equaton (7) the equlbum constant fo the eactons takng lace at constant volume and temeatue wll have the exesson: V eq ex F he above equaton can be ocessed as below consdeng the elaton between the fee eneges: (8) INCAS BULLEIN Volume 6 Secal Issue / 4
Maus BEBENEL 8 F F F F (9) () whee F s gven by (9). () F () By elacng n (8) the fomula of the equlbum constant fo the eactons at constant volume s found: V ex (3) o by comang to (7): V ex (4) whee the essue n (3) s the fnal essue obtaned when a chemcal eacton at constant volume eaches ts equlbum. It can be obseved that fo a eacton showng a dffeence between the total numbe of moles of eactants and the eacton oducts the equlbum constant at constant volume s dffeent fom that at constant essue. hus: - f the total numbe of moles of eacton oducts exceeds the total numbe of moles of eactants then the eacton constant at constant volume s hghe than n the case of constant essue hence the equlbum shfts to oducts fomng; - f the total numbe of the eacton oducts moles s less than the total numbe of the eactants moles then the eacton constant at constant volume s lowe than n the case of constant essue hence the equlbum shfts to the eactants. 4. EEMLIFICAIN F HE CASE F MBUSIN F CABN IN YEN Let s consde the chemcal eacton of combuston of ue cabon (gahte) n oxygen wheen oducts of ncomlete combuston ae esent as well. lobally ths can be descbed by the equaton: C C (5) s s INCAS BULLEIN Volume 6 Secal Issue / 4
9 Analyss of the chemcal equlbum of combuston at constant volume he comoston of combuston oducts at vaous temeatues wll be analyzed by comason consdeng the chemcal equlbum seaately at constant essue and constant volume. he equatons of smle eactons leadng to oveall eacton (5) ae: Cs (6) (7) Fo the above eactons the equlbum constants n mola factons wll be consdeed gnong the sold hase: (8) / / / / By addng the nomaton equaton: (9) (3) the system of equatons whch detemnes the mola factons of combuston oducts n gas hase wll be found. By solvng the system one yelds: 4 (3) 4 (3) 4 (33) Fomulas (3) (3) and (33) detemne the comoston of the combuston oducts of (5) at equlbum at a gven temeatue and essue deendng on the equlbum constants. hese equlbum constants wll be comuted n the followng at both eacton condtons: constant essue and constant volume. Calculaton of the equlbum constants fo the eacton at constant essue = he equlbum constants wll be detemned by usng fomula (7) that s: INCAS BULLEIN Volume 6 Secal Issue / 4
Maus BEBENEL 3 INCAS BULLEIN Volume 6 Secal Issue / 4 ex ex (34) ex ex (35) he comoston of the eacton oducts wll be analyzed at 3 temeatues: and 3. he themodynamc data ae taken fom [3] [4] and [5] and the values of the comuted equlbum constants ae esented n tables and 3. he total numbe of the combuston oducts moles n the gas hase whch occus n equaton (5) s: g (36) so that the mola factons of coesondng comonents can be exessed as: (37) ne yelds fo and : (38) he followng notatons wll be used n the next: (39) S S H H - (4) S H (4) d c H (4) d c S (43)
3 Analyss of the chemcal equlbum of combuston at constant volume ab. Standad enthaly and entoy of fomaton and combuston of [4] [5] hemochemcal data H [kj/kmol] S [kj/kmol.] Fomaton -53-6.9 Combuston -83-86.3 ab. Enthaly and entoy vaaton fo seces esent n the eactons of fomaton and combuston of wthn temeatue ange ( ) (data taken fom [3]) emeatue = = 3 = 3 H [kj/kmol] 7 597 98 S [kj/kmol.] 38.5 63.6 79.4 H [kj/kmol] 69 5674 935 S [kj/kmol.] 36.8 6. 75.9 H [kj/kmol] 334 944 58 S [kj/kmol.] 55.5 95.5.4 ab.3 Calculaton of equlbum constants fo eactons of fomaton and combuston of at constant essue emeatue = = 3 = 3 [kj/kmol] -4763 57 787 [kj/kmol] -967-4 -4 [kj/kmol] -5484-5975 475 [kj/kmol] -9579-685 -85 73..64.569.69 777.9 3.99 he esults fom table 3 wll be ntoduced n (3) (3) (33) and (38) and the values esented n table 4 wll be obtaned. INCAS BULLEIN Volume 6 Secal Issue / 4
Maus BEBENEL 3 ab.4 Comoston of gases esulted fom cabon combuston at 3 temeatues and constant essue emeatue = = 3 = 3 8.8-4 4.64-4.57.8-4.56.89.9998.943.454..94.4695 -. -.85.33 Calculaton of the equlbum constants fo the eacton at constant volume and fnal essue = Fo the alcaton of fomula (4) the vaaton of the moles numbe (gas hase) wll be comuted n the two smle eactons (fomaton and combuston of ): C ; s (44) ; (45) By alyng fomula (4) to the equlbum constants fom table 3 the values esented n table 5 wll be found: ab.5 Comoston of gases esulted fom cabon combuston at 3 temeatues and constant volume emeatue = = 3 = 3 7. 4.39.938. 47.8.88 8.9-4 4.47-4.7 3.43-4.9.47.9997.98.366.7-4.48.535 -.7-4 -.473 -.78 In fgues and 3 the vaaton of the mola factons of and wth temeatue s lotted by comason at constant essue and at constant volume. INCAS BULLEIN Volume 6 Secal Issue / 4
33 Analyss of the chemcal equlbum of combuston at constant volume ln -6 _ - -8-4 -3 3 [] Fg. Vaaton of ecentage n combuston gas mxtue wth temeatue by comason at constant essue and constant volume ln - V = const. _ -4-6 = const. -8-3 [] Fg. Vaaton of ecentage n combuston gas mxtue wth temeatue by comason at constant essue and constant volume ln -. -.4 _ = const. -.6 -.8 V = const. - -. 3 [] Fg. 3 Vaaton of ecentage n combuston gas mxtue wth temeatue by comason at constant essue and constant volume INCAS BULLEIN Volume 6 Secal Issue / 4
Maus BEBENEL 34 5. NCLUSINS AND MMENS N HE ESULS By analyzng the data shown n the esented tables and dagams the followng obsevatons come out: ) he ecentage of unconsumed oxygen n the combuston gases s almost the same n both models (constant essue and constant volume esectvely). Howeve the amount of unconsumed oxygen nceases along wth the combuston temeatue. ) he ecentage of n the combuston gases s hghe n the case of constant volume and nceases along wth the combuston temeatue. In othe wods the combuston at constant volume s moe ollutng than at constant essue. 3) he ecentage of obtaned by combuston at constant volume s lowe than at constant essue the dffeences sng along wth the combuston temeatue. 4) In tables 4 and 5 small negatve values of coeffcent can be obseved whch means that n fact moe than mole of cabon gets n the eacton. Howeve at constant essue (table 4) one obseves that at 3 =.33 whch shows taces of smog n the combuston oducts. Accodng to ths model at constant volume no smog s esent n the comoston of the combuston oducts. 5) Inceasng of the ecentage of ncomlete combuston oducts along wth the temeatue se s n ageement wth Le Chatele ncle whch states that n the case of exothemc eactons a temeatue se shfts the equlbum towads the eactants. Based on the above analyss we can conclude that n the case of the cabon combuston n ue oxygen at low temeatues no sgnfcant dffeences between the esults gven by the models of equlbum at constant essue and constant volume occu whle at hghe temeatues (as usual combuston ocesses wth enegetc effects haen) combuston at constant volume changes the weght of ncomlete eacton oducts. In the case of the cabon combuston n oxygen at constant volume when comang to the constant essue condton the equlbum moves accodng to the vaaton of the moles numbe n the smle eactons. hus: - at fomaton of the vaaton of the moles numbes s ostve equlbum shfts to the ght; - at combuston of the vaaton of the moles numbes s negatve equlbum shfts to the left. It s also exected that othe featues of combuston (heat eleased enegetc effcency flammablty lmts vesus a/fuel ato and othes) n the case of the combuston at constant volume wll be affected by adotng the model of constant volume equlbum dffeent fom that of constant essue. EFEENCES [].. Balme Moden Engneeng hemodynamcs Elseve Inc. USA (66-63). []. Ahuja Chemcal Engneeng hemodynamcs HI Leanng vate Ltd. 9 New Delh (59-593). [3] *** NIS Chemsty Webbook Standad efeence Data www.webbook.nst.gov [4] *** DAA ey Values fo hemodynamcs Hemshee ublshng Co. New Yok 989. [5] Hütte - Manualul Ingneulu - Fundamente ed. ehncă Bucueşt 995. INCAS BULLEIN Volume 6 Secal Issue / 4