Differential Equations of Gas-Phase Chemical Kinetics

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1 Differetial Equatios of Gas-Phase Chemical Kietics Chemked A Program for Chemical Kietics of Gas-Phase Reactios M. Jeleziak ad I. Jeleziak Home page: ifo@chemked.com 02 Jue 2009, last updated 01 September 2017 Cotets 1. Itroductio 2. Gas Parameters ad Variables 3. Chemical Reactios 3.1 Reactio Rate 3.2 Heat of Reactio 4. Equatios of Chemical Kietics 4.1 Problem Formulatio 4.2 Costat Pressure Process 4.3 Costat Volume Process 5. Example of Calculatio 6. Coclusio Refereces Nomeclature 1

2 1. Itroductio Chemked [1] is a program desiged for creatig ad editig thermodyamic ad chemical kietics databases, for formatio of reactio mechaisms ad simulatio of problems of complex gas-phase chemistry. The program uses thermodyamic data ad chemical reactios that have the CHEMKIN-II descriptio. The iformatio is collected i databases that are stored i files of the Microsoft Access type. The data are displayed i the tables ad ca be easily hadled. Chemked has the multipledocumet iterface ad ca simultaeously process several reactio databases coected with a thermodyamic database. The program provides umerical ad graphical iformatio o thermodyamic fuctios ad reactio rate costats. Chemked icorporates a solver for itegratio of differetial equatios of gas-phase chemical kietics at costat pressure or costat volume (desity) without mass ad heat trasfer. The solver is a FORTRAN program based o the RADAU5 subrouties [2]. The ecessary iformatio o thermodyamics ad reactios is provided for the solver directly from the databases uder cosideratio. The solver output cotais umerical ad graphical iformatio o mixture-averaged gas parameters ad species cocetratios. The goal of the work preseted here is the accurate derivatio of equatios of gas phase chemical kietics that are used i the Chemked solver. These equatios are well-kow differetial equatios for temperature ad species cocetratios describig processes i a reactio vessel at costat pressure or costat volume. I additio, the defiitio of importat gas parameters ad reactio rates are give. For illustratio of the discussed items, a example of chemical evolutio of H 2 /O 2 mixture is preseted; the calculatio has bee made with Chemked. More iformatio about Chemked ca be foud at [3,4]. A brief list of useful literature related to topics discussed below is preseted i Referece Sectio [5-9]. 2. Gas Parameters ad Variables Here we itroduce gas parameters that will be used as variables i coservatio equatios or will be useful for aalysis of results. The choice of some of these parameters is obvious ad does ot require special commets. These parameters are the mixture-averaged gas parameters. Temperature T, [K], Pressure P, [atm], Total gas cocetratio C tot is a umber of molecules or miles per uit of volume, [1/cm 3 ] or [mol/cm 3 ]. For give P ad T, the C tot value is foud from the ideal-gas equatio of state. where R is the uiversal gas costat. P C tot =, (2.1) RT 2

3 Cocetratio of th species is deoted as [X ] ad has the uits of [1/cm 3 ] or [mol/cm 3 ]. The obvious formula relates the [X ] values to the total cocetratio C tot. C tot = [ X ], (2.2) where the summatio is over all species i the gas mixture. The species cocetratios ca be expressed also i terms of the mole fractios X. [ X ] X =. (2.3) Ctot Mea mass desity ρ (g/cm 3 ) ad mea molecular weight m (g/mol) are calculated as follows: where m is molecular weight of the th species. [ ] ρ =, (2.4) m = m X, (2.5) I the followig cosideratio, mass fractio Y of the th species will be employed. [ X ] Relatio of mole fractio to mass fractio is give by m Y =. (2.6) ρ X m Y = m. (2.7) We defie the Z as a umber molecules or moles per uit of total mass of the gas mixture, [1/g] or [mol/g]. Relatio of Z to the cocetratio per uit of volume is [ X ] ρz Substitutig Eq.(2.8) ito the equatio Eq.(2.6) we get =. (2.8) Y Z =. (2.9) m 3

4 3. Chemical Reactios I this sectio, we give oly a short descriptio of chemical reactios; more detailed iformatio o this subect ca be foud i literature, for example, i [5, 7-9]. 3.1 Reactio Rate The stoichiometric equatio of a chemical reactio ca be writte i the followig geeral form: ' v χ = v '' χ, (3.1). where the subscript is the reactio umber, χ is the chemical symbol of species, v is the stoichiometric coefficiet. The summatios are over the species ivolved i the reactio. The superscript of the stoichiometric coefficiets refers to the reactats, the superscript refers to the products. I this represetatio, the v coefficiets are iteger positive umbers or zeros. The forward ad reverse rates ad rate-of-progress of the rth reactio are q ' v v, for = k, for [ X ], q rev = k, rev [ X ] '',, q, et q, for q, rev =, (3.2). where k,for ad k,rev are the forward ad reverse rate costats of the th reactio. Remark I Eq. (3.2), the powers of cocetratios are reactio orders with respect to reactats. I more geeral expressios for reactio rates, the reactio orders are ot ecessarily equal to the stoichiometric coefficiets v ad v. Ofte, this situatio shows up at descriptio of o-elemetary reactios, for example, gas-surface reactios. Nevertheless, i gas reactio systems, the approach used here is usually valid The Arheius form is used most frequetly for calculatio of the reactio rate costat: a E =, for, for k A T, for, for exp, (3.3) RT where A,for, a,for ad E,for are the parameters that ate available i chemical kietics databases, for example [10]. The reverse rate costat is related to the forward rate costat by k k, for, rev =, (3.4) K ' '' where K is the equilibrium costat of the th reactio. The Arheius form is oe of the simplest expressios of the rate costat; i may cases the more sophisticated expressios are required. Descriptio of other reactio types ca be foud i [7]. 4

5 Remark The uits for the pre-expoetial factor A,for (A-uits) deped o the uits of species cocetratios. I Chemked, the followig uits are used. Species ccocetratio mol/cm^3 (moles/cm^3) 1/cm^3 (molecules/cm^3) Desigatio of A-uits mol-cm-sec cm-sec A-uits of 1 st order reactio 1/sec 1/sec A-uits of 2 d order reactio cm^3/mol sec cm^3/sec A-uits of 3 rd order reactio cm^6/mol^2 sec cm^6/sec Reactio rate mol/cm^3 sec 1/cm^3 sec (molecules /cm^3 sec). The uits for activatio eergy E,for (E-uits) are cal/mol, J/mol or K. Note that T is always i Kelvi. The A-uits ad E-uits, specified here, are most frequetly used i existig databases ad software [7] If we are iterested i productio of species the the productio rate of this species i the forward reactio is '' ' ( v v ) q, for This expressio reflects the fact that each evet of the th forward reactio yields the umber ( v ) '' v ' of species. Similarly, the productio rate of the species i the reverse reactio is ( v ' '' v ) q, rev The total productio rate is a sum of the productio rates i forward ad reverse reactios '' ' ( v v ) q et w, =. (3.5) The sig of w shows the actual situatio. If w > 0, the the species is produced; if w < 0, the the species is cosumed. The rate-of-progress q,et ad the species productio rate w have the uits of (1/cm 3 sec) or (mol/cm 3 sec). 3.2 Heat of Reactio The fial products of a chemical reactio are ot oly ew chemical compouds, but releasig (cosumig) of heat as well. The last process is a result of differece i eergies betwee products ad reactats. The heat of the th reactio Q is defied as a amout of heat that must be exchaged betwee reactio system ad its surroudig to keep a fixed system temperature. Below we will cosider two cases whe a reactio system is at costat pressure or at costat volume. Costat pressure, V#0. I this case, chage i the iteral eergy U must be equal to a sum of heat of reactio Q p ad work P V due to expasio (cotractio) of the gas volume (first low of the thermodyamics). The the heat of reactio is 5

6 Q p = U + P V, (3.6) whre the subscript p idicates costat pressure. It is coveiet to use the ethalpy istead of the iteral eergy. By defiitio, ethalpy of the th species is H = U + PV. (3.7) where H ad U are the ethalpy ad the iteral eergy of species. Takig ito accout Eq.(3.7), we may rewrite Eq.(3.6) i the form: '' ' Q = ( v v ) H p. (3.8) Costat volume, V=0. Uder these coditios the gas does ot make ay work (P V=0) ad the heat of reactio Q v is equal to chage i iteral eergy. '' ' Q = U, U = ( v v ) U v. (3.9) where the subscript v idicates costat volume process. Rate of heat productio. This value is give by the expressio s Q q, et =. (3.10) where q,et is rate-of-progress of the th reactio, Eq.(3.2). Eq (3.10) ca be rewritte i terms of productio rate of species ivolved i reactio (Eq. 3.5). s w H, (3.11a) p v = = s w U, (3.11b) The sig of s shows the actual situatio. If s < 0, the the heat is cosumed; if s > 0, the the heat is released. The heat of reactio Q (like H ad U ) has the uits of eergy per mole, (J/mol) or (cal/mol). Correspodigly, the rate-of-progress q,et, as well as productio rates of species w, must have the uits of molar productio, (mol/cm 3 sec). The the s values describe the rate of heat productio i the th reactio per uit volume, (J/cm 3 sec) or (cal/cm 3 sec). 6

7 4. Equatios of Chemical Kietics 4.1 Problem Formulatio We will cosider evolutio of parameters of a gas mixture i course of chemical reactios. Schematic diagram of a reactio vessel is give i Fig Whe describig the chemical processes, the followig assumptios will be used. There is perfect gas mixig i the vessel volume. The vessel walls are impermeable. There are o surface reactios. Adiabatic walls (the vessel walls are thermally isulated, i.e. there is o heat exchage betwee the reactio system ad surroudig). The vessel has a pisto that allows us to get two types of coditios. Whe the pisto ca move freely, chemical reactios take place at costat pressure, which is equal to surroudig gas pressure, Fig. 4.1a. Whe the pisto is fixed, chemical reactios take place at costat volume, Fig. 4.1b. We believe that chemical reactios occurrig i the gas mixture are kow ad their rate costats are specified. I additio, specific thermodyamic fuctios of species (ethalpy, iteral eergy ad heat capacity) are defied as well. Below we will obtai set of equatios that describe parameters of the reactio system i the course of reactios. a. Costat pressure V P T Y M Cheemical Reactios M V +dv P T+dT Y + dy b. Costat volume V P T Y M Cheemical Reactios M V P+dP T+dT Y + dy Fig Schematic diagram of chagig parameters i reactio system 4.2 Costat Pressure Process These coditios are met whe the pisto of the reactio vessel ca move freely, Fig. 4.1a. The total mass of gas ad volume are related by the followig equatio: M = ρ V = costat, (4.1) 7

8 I this case, the volume chages i curse of chemical reactios ad cosequetly the mea mass desity ρ chages as well. Coservatio of species. The coservatio of mass of species i the whole reactio volume ca be writte i the form: d ( MY ) = m w V, (4.2) w =. (4.3) w Here Y is mass fractio ad m is molecular weight of the th species, w is molar productio rate of species i the th reactio (Eq (3.5)), w is the total molar productio rate of the th species. Sum is over reactios that ivolve the th species. Thus, the left-had side of Eq (4.2) is time-derivative of the total mass of the th species i the whole volume V ad the right-had side is the mass productio rate of the same species i the same volume. Differetiatig the left-had side leads to Y d d +. (4.4) ( M ) M ( Y ) m w V = If we take ito accout Eq (4.1), the the differetial equatios for the mass fractios of species may be writte as dy m = w, (4.5) ρ It should be oted oce agai that here the mass desity ρ is a time-depedet parameter, Eq (4.1). Remark Whe aalyzig the calculatio results it is desirable to kow the itegral productio of species by each reactio. For this purpose, the equatio (4.5) ca be writte somewhat differetly. Dividig both sides of the equatio by molecular weight m gives dz = b, Y Z =, m w b =. (4.5a) ρ where Z is cotet of species per 1 gram of gas mixture Eq.(2.9), mol/g; b is productio rate of species i the th reactio i uits mol/g sec. The itegral productio of species o the time iterval t 1 -t 2 (mol/g) is ( t ) Z ( t ) = t 2 Z 2 1 B ; = t B b. The values B provide iformatio about the importace of the reactios i productio of species Coservatio of eergy. Uder costat pressure the ethalpy of a gas mixture i a closed vessel with isulated walls is held costat. 1 8

9 d Mh = 0, (4.6) where h is the mea specific ethalpy i mass uits (J/g or cal/g). h = Y h. (4.7) Here h is the specific ethalpy of the th species; the summatio is over all species of the gas mixture. Differetiatig Eq (4.6) gives dm dh h + M = 0. (4.8) The first term is equal to zero (see Eq (4.1)), the Eq (4.8) ca be writte i the form d Y h After differetiatig, Eq (4.9) ca be writte as dh = = 0. (4.9) dy Y h. (4.10) The ideal-gas ethalpy h is fuctio of gas temperature oly. I that case, the h derivative ca be evaluated by dh dt = c p, dh c = p dt, (4.11) where c p is the specific heat capacity of the th species at costat pressure i mass uits; (J/g K or cal/g K). Usig these relatios we get Eq (4.10) i the form: dt dy c p = h, (4.12) where the summatio is over all species of gas mixture. Mea specific heat capacity at costat pressure c is p c = Y c. (4.13) p Substitutig Eq (4.5) i Eq (4.12) gives us the fial equatio for the gas temperature. dt 1 = ρc p p H w, (4.14) 9

10 where H is the ethalpy of the th species i molar uits (J/mol or cal/mol). Remark H = m h. (4.15) Remark Eq (4.14) ca be also expressed i terms of the rates of heat productio i chemical reactios, Eq (3.11a). dt 1 = ρc where the summatio is over all chemical reactios p s p, (4.14a/) Iitial coditios. The differetial equatios (4.5) ad (4.14) must be accompaied by iitial coditios, that is, at a time istat t=0 we must specify the followig gas parameters i the reactio vessel: P 0 pressure T 0 temperature Y 0 species mass fractios Remark Ofte, species mole fractios X are specified as iitial coditios; coversio of X to Y is made with Eq (2.7) Hece, the equatios (4.5), (4.14) with iitial coditios for temperature T 0 ad species mass fractios Y 0 at a give costat pressure P completely determie parameters of the gas mixture i reactio vessel. Some additioal values (mass desity ρ, total gas volume V, etc) ca be foud from formulae of this sectio ad Sectio Costat Volume Process These coditios are met whe the pisto of the reactio vessel is fixed, Fig. 4.1b. The total mass of gas ad volume are related by Eq (4.1), as previously: M = ρ V = costat. I this case, the volume is costat i curse of chemical reactios ad cosequetly the mea mass desity ρ is costat as well. Coservatio of species. Whe derivig Eq (4.5) for species mass fractios we did ot explicitly use the costat pressure assumptio. So, Eq (4.5) has a geeral character ad ca be applied to the costat volume coditios as well. Uder these coditios some simplificatio is possible. Takig ito accout that the mass desity ρ is costat, we ca rewrite Eq (4.5) i the form. d [ X ] = w. (4.16) Here, the relatio (2.6) betwee molar cocetratio [X ] ad mass fractio Y was employed.

11 To obtai the itegral productio of species i reactios the method described i remark to Eq.(4.5a) ca be used. I costat volume process, the gas desity is costat. The the product ρb gives the itegral productio i uits mol/cm^ Coservatio of eergy. Iiteral eergy of a gas mixture i a closed vessel with isulated walls is. d Mu + dv P = 0, (4.17) where u is the mea specific iteral eergy i mass uits (J/g or cal/g). u = Y u. (4.18) Here, u is the specific iteral eergy of species ; the summatio is over all species of gas mixture. Here u is the specific iteral eergy of the th species; the summatio is over all species of gas mixture. The further coversio of Eq (4.17) is similar to the coversio of Eq (4.6) (Sectio 4.1); this gives us the familiar equatio for the gas temperature: dt = 1 ρc v U w (4.19) where c is the mea specific heat capacity at costat volume i mass uits (J/g K or cal/g K), U v is the iteral eergy of the th species i molar uits (J/mol or cal/mol), U = m u. (4.20) Iitial coditios. As previously metioed, the differetial equatios for species ad temperature ((4.16), (4.19)) must be accompaied by iitial coditios. I the reactio vessel, pressure P 0, temperature T 0 ad species mass fractios Y 0 must be specified at a time istat t=0. 5. Example of calculatio I order to illustrate the discussed above theory, we cosider a simple gas-phase system cosistig of H 2, O 2 ad Ar. The simulatio of chemical processes was carried out with a computer program Chemked [1]. The thermodyamic data ad detailed reactio mechaisms of hydroge combustio [10] was used. We will examie costat pressure ad costat volume processes that start at the idetical iitial coditios: P 0 = 0.3 atm T 0 = 900 K X H2 : X O2 : X Ar = 0.05 : : Results of the calculatio are preseted i Fig. 5.1 ad 5.2. Here we call attetio o some qualitative features of the results oly. As see from the figures, the gas temperature is otable greater i the costat volume process. I reactio system, the temperature elevatio is a result of trasformatio of the heat released i chemical reactios ito the iteral eergy of gas. At costat volume, the heat of 11

12 reactios goes totally ito the iteral eergy (Eq. (3.9)). At costat pressure, a part of the heat of reactios is used for the work of gas expasio ad rest of the heat goes ito the iteral eergy (Eq. (3.8)). For these reasos, there is a differece i the calculated temperatures i Fig. 5.1 ad 5.2. Oe must also pay attetio to the fact that the fial species cocetratios are greater at the costat volume. I costat pressure process, the reactio volume chages ad we ca determie the degree of the expasio (cotractio), Eq. (4.1). I particular, for coditios of Fig. 5.1 the volume icreases by the factor 1, Coclusio The equatios of gas phase chemical kietics are derived. These equatios are the differetial equatios for temperature ad species cocetratios describig processes i a reactio vessel at costat pressure ad costat volume, Eqs. (4.5, 4.14, 4.16, 4.19). Now we summarize the requiremets for use of these equatios. Let us cosider a gas-phase reactio system. We select a volume iside the system. The equatios will describe chemical evolutio of the selected gas volume if the followig coditios are satisfied. There are o mass fluxes through the boudary of the volume; the gas mass i the volume is costat. There are o heat fluxes through the boudary of the volume; well isulated boudaries. There are o temperature ad species gradiets withi the volume; the gas i the volume is always uiform. Course of chemical reactios takes place at a. costat volume b. costat pressure 12

13 Figure 5.1. Temperature, mass desity ad species profiles of H2/O2/Ar mixture, costat pressure process 13

14 H2 O2 Figure 5.2. Temperature, pressure ad species profiles of H2/O2/Ar mixture, costat volume process Refereces [1] M. Jeleziak, I. Jeleziak. Chemked A Program for Chemical Kietics of Gas-Phase Reactios. [2[] E. Hairer, G. Waer. Solvig Ordiary Differetial Equatios II: Stiff Differetial-Algebraic Problems. Spriger-Verlag, secod revised editio (1996). [3] M. Jeleziak, I. Jeleziak. Reactio Kietics Simulatio [4] M. Jeleziak, I.Jeleziak. Reduced Reactio Kietics [5] Gas-Phase Combustio Chemistry. W.C. Gardier, Jr. (Editor). Spriger-Verlag, New York (2000). [6] Geeral Chemistry, Fourth Editio by J. W. Hill, R. H. Petrucci, T. W. McCreary, S. S. Perry. Pretice Hall (2005). [7] R.J. Kee, F.M. Rupley, J.A. Miller, M.E. Coltri, J.F. Grcar, E. Meeks, H.K. Moffat, A.E. Lutz, G. Dixo-Lewis, M.D. Smooke, J. Waratz, G.H. Evas, R.S. Larso, R.E. Mitchell, L.R. Petzold, W.C. Reyolds, M. Caracotsios, W.E. Stewart, P. Glarborg, C. Wag, ad O. Adigu. CHEMKIN Collectio, Release 3.6, Reactio Desig, Ic., Sa Diego, CA (2000). [8] H. S. Fogler. Elemets of Chemical Reactio Egieerig (Pretice Hall Iteratioal Series i the Physical ad Chemical Egieerig Scieces). Pretice Hall PTR (2005). 14

15 [9] M. Jeleziak, I. Jeleziak. Gas-Phase Chemical Reactios [10] Chemical Kietics Database o the Web (NIST). A Compilatio of Kietics Data o Gas-Phase Reactios. [11] J. Li, Z. Zhao, A. Kazakov, F. L.Dryer. A Updated Comprehesive Kietics Model of H2 Combustio. Iteratioal Joural of Chemical Kietics,Vol. 36, p (2004). Nomeclature c p specific heat capacity at costat pressure of the th species (cal/g K), (J/g K) c p mea specific heat capacity at costat pressure (cal/g K), (J/g K) c v mea specific heat capacity at costat volume (cal/g K), (J/g K) C tot total gas cocetratio (1/cm 3 ), (mol/cm 3 ) h specific ethalpy of the th species (cal/g), (J/g) h mea specific ethalpy of a mixture (cal/g), (J/g) H molar ethalpy of the th species (cal/mol), (J/mol) k,for forward rate costat of the th reactio depeds o reactio k,rev reverse rate costats of the th reactio depeds o reactio K equilibrium costat for the th reactio depeds o reactio m molecular weight of the th species g/mol m mea molecular weight of a gas mixture g/mol M total mass of gas mixture i reactio vessel g P pressure atm q,for, forward rate of the th reactio (1/cm 3 sec), (mol/cm 3 sec) q rev, reverse rate of the th reactio (1/cm 3 sec), (mol/cm 3 sec) q rate-of-progress of the th reactio (1/cm 3 sec), (mol/cm 3 sec) Q heat of the th reactio ( cal/mol), (J/mol) R uiversal gas costat (cal/mol K), (J/mol K) s rate of heat productio of the th reactio (cal/cm 3 sec), (J/cm 3 sec) t time sec T temperature K u rate of heat productio i the th reactio (cal/cm 3 sec), (J/cm 3 sec) u specific iteral eergy of the th species (cal/g), (J/g) u mea specific iteral eergy (cal/g), (J/g) U iteral eergy of the th species (cal/mol), (J/mol) V volume of reactio vessel cm 3 15

16 w productio rate of species i the th reactio (1/cm 3 sec), (mol/cm 3 sec) X mole fractio of the th species [X ] cocetratio of the th species per uit volume (1/cm 3 ), (mol/cm 3 ) Y mass fractio of the th species Z cocetratio of the th species per uit ass (1/g), (mol/g) v stoichiometric coefficiet of the th species i the th reactio χ chemical symbol of species ρ mass desity g/cm 3 Subscripts p v costat pressure process costat volume process species umber reactio umber 16

Use of Chemked for Simulation of Gas-Phase Chemical Reactors

Use of Chemked for Simulation of Gas-Phase Chemical Reactors Use of Chemked for Simulatio of Gas-Phase Chemical Reactors Chemked A Program for Chemical Kietics of Gas-Phase Reactios M. Jeleziak ad I. Jeleziak Home page: http://www.chemked.com/, email: ifo@chemked.com