5.68J/10.652J Spring 2003 Lecture Ntes Tuesday April 15, 2003 Kinetic Mdel Cmpleteness We say a chemical kinetic mdel is cmplete fr a particular reactin cnditin when it cntains all the species and reactins needed t describe the chemical prcesses at that reactin cnditin t sme specified level f accuracy. In ther wrds, the kinetic mdel is cmplete when all the reactins whse rate cnstants it sets equal t zer really are negligible fr the specified reactin cnditins and errr tlerance. Nte that this definitin f cmplete des NOT assume that the rate cnstants r any ther parameters in the mdel are crrect, just that the reactins excluded frm the mdel are indeed negligible. If yu believe yu can list all pssible reactins f the species in the mdel, a necessary cnditin fr a mdel t be cmplete is that the rates f these reactins t frm species nt included in the mdel must be smaller than sme errr tlerance; this apprach was intrduced by R.G. Susnw et al. (1997). Hwever, this cnditin is nt sufficient: e.g. ne f the minr neglected species culd be a catalyst r catalyst pisn with a very big influence n the kinetics at very lw cncentratins. If the kinetics are strictly linear yu can shw that this necessary cnditin is sufficient (Matheu et al. 2002, 2003); an imprtant example f strictly linear kinetics are the rdinary master equatins used t cmpute pressure-dependent reactin rates. Of curse, yu usually d nt knw all the pssible reactins. But fr purpses f kinetic mdel reductin ne usually assumes that the initial full mdel is cmplete; yu are satisfied if the reduced mdel reprduces the full mdel t within sme errr tlerances under sme reactin cnditins. Sensitivity Analyses A. Definitins f Sensitivities Suppse yu have a mdel yu think is cmplete at sme reactin cnditin, and yu have sme estimates fr k and Y s yu can numerically slve dy/dt = F(Y,k) Y(t ) = Y t get yur best predictin fr the trajectry Y(t). (Y is nrmally made up f many mass fractins y i (t) and a few ther time-varying state variables e.g. T(t).) Yu never knw the rate (and ther) parameters k and the initial cncentratins Y exactly, s yu are always interested in hw much the predicted trajectry wuld vary if these values were a little different than the values yu assumed, i.e. hw sensitive is yur predictin t the values f these parameters? A cmmn way t express this is t write Y(t) as a Taylr expansin in the parameters k and Y :
y i (t) = y i (t; k,y ) + Σ( y i (t)/ k n )(k n k n ) + Σ( y i (t)/ y m )( y m - y m) + The rate cnstants k usually depend n T and P, s if T and/r P vary with time the first term gets t be a mess. The mst ppular apprach (used in the latest versins f CHEMKIN, but nt cnsistently in earlier versins) is t imagine that all f the rate cnstants k n (T,P) are multiplied by scaling factrs D n. Our predicted trajectry Y(t) crrespnds t all D n =1. S the preferred expansin (fr T,P varying) is: y i (t) = y i (t; D=1,Y ) + Σ( y i (t)/ D n )(D n 1) + Σ( y i (t)/ y m )( y m - y m) + The derivatives in this expansin are smetimes called the sensitivities, thugh watch ut! - in practice what are usually reprted are the crrespnding unitless nrmalized sensitivities: Nrmalized sensitivity f species i t reactin n = (1/y i (t))( y i (t)/ D n )= (ln y i (t))/ (ln D n ) Nrmalized sensitivity f species i t initial cncentratin f species m = (y m /y i (t))( y i (t)/ y m ) = (ln y i (t))/ (ln y m) It must be emphasized that these sensitivities depend crucially n the assumed values f the rate cnstants k and all the ther parameters, and n (t-t ) and Y. If yu change the simulatin in any way yu will change all the cmputed sensitivities. The (first-rder) sensitivities are als very lcal in the sense that they are nly useful if the perturbed trajectry is very clse t the riginal trajectry Y(t), i.e. if the perturbatin in k r Y is very small. If yu care abut a big variatin (e.g. if ne f yur rate cnstants is extremely uncertain) yu may nt be able t get away with the first-rder Taylr expansin. If yu need t g t higher rders r therwise handle big variatins/uncertainties in the mdel parameters, see papers by H. Rabitz, G.J. McRae, and Nancy Brwn. One wuld ften like t knw the Green s functin sensitivity y i (t)/ y m (t ) i.e. hw much species i at time t wuld have changed if a little bit f species m was added at time t. This can be cmputed as an initial cncentratin sensitivity by setting t =t and Y =Y(t ). Hwever, it is usually impractical t d this fr many values f t. Finally, in additin t cncentratin sensitivities, ne can als cmpute rate sensitivities, which are hw much the instantaneus slpe dy i /dt wuld change if k r Y were perturbed. In additin t the rdinary integrated rate sensitivities (which depend n t-t, and capture the fact that the whle trajectry is perturbed starting at t ) ne can als
cmpute instantaneus rate sensitivities which are hw much the rate at time t wuld change if the cncentratins r rate cnstants were suddenly changed at time t. The instantaneus rate sensitivities are much simpler t cmpute than the ther sensitivities, just a little algebra is required: Instantaneus species i rate sensitivity t reactin n = F i (Y,k)/ k n Instantaneus species i rate sensitivity t species m = F i (Y,k)/ y m The instantaneus sensitivities d nt depend n the histry f the system, they will be the same numbers regardless f hw the system gt t its present state Y. Finally, sensitivities are partial derivatives, s it is imprtant t knw what ther variables are being held cnstant yu can get very different numerical values depending n what yu hld cnstant. In particular, it is imprtant t knw whether the thermchemistry is being held cnstant, r whether all the frward and reverse rates are individually being held cnstant. The frmer is generally preferred since yu are usually mre cnfident in the thermchemical parameter values than the kinetic parameters, and als there are usually fewer thermchemical parameters than reverse rate cnstants. (Thugh if yu are highly uncertain abut the thermchemistry f sme f yur species, yu will als be interested in the sensitivity t thse thermchemical parameters!). If the frward and reverse rates are treated separately, it is very imprtant t lk at sensitivities with respect t bth the frward and the crrespnding reverse rate simultaneusly: if they are ppsite in sign and nearly equal in magnitude, it means yu are really sensitive t the thermchemistry, nt the kinetics. (This usually happens when that reactin is fast and nearly equilibrated.) B. Cmputing Sensitivities The ther sensitivities are pretty hard t cmpute, they all require slving rather large systems f differential equatins. Fr example: d(dy i (t)/dd n )/dt=d(dy i (t)/dt)/dd n =d(f i (Y,D.*k)/dD n = Σ( F i / y m )(dy m /dd n ) + F i / D n defining S in (t) = dy i (t)/dd n evaluated n the reference trajectry Y(t) we can rewrite this: ds in /dt = Σ( F i / y m )S mn + F i / D n S in (t )=0 Fr the sensitivities t initial cncentratins Ζ il (t)= y i (t)/ y l ) the equatin is very similar: dz il /dt = Σ( F i / y m )Z ml Z il (t )=δ il In bth cases, the right hand side depends n the Jacbian F/ y which depends n Y (and s varies with time) s it is best t slve this simultaneusly with the riginal system f equatins dy/dt = F, t avid having t evaluate the Jacbian repeatedly. There are several tricks which take advantage f the special simple frm f the sensitivity equatins (e.g. linear in S), and gd prgrams which use these tricks fr cmputing the first-rder
sensitivities, including DASAC, DASPK, and DAEPACK. Fr details, see fr example recent papers by P.I. Bartn. Nte that there are a huge number f sensitivities O(Nspecies*Nreactins*Ntimesteps). T speed the cmputatins and reduce the utput, ne can nly ask fr the sensitivities t a few parameters (rather than all the rate cnstants), r fr the sensitivities f nly a few species. Exactly hw much this saves yu will depend n the details f the numerical algrithm. The algrithms in CHEMKIN are nt very efficient, and the pst-prcessr is wrse, s yu generally d much better if yu nly ask fr the utput yu really need. If yu need t slve large sensitivity prblems that CHEMKIN cannt handle, yu might want t ask Prf. Bartn fr advice. C. Sme Uses fr Sensitivities 1) Estimating effects f Varying Initial Cnditins Very ften yu are interested in the behavir f yur system as yu make (relatively small) variatins in the initial cnditins. Fr example, if yu are ding an ignitin prblem, what happens if the initial temperature is 10 K higher? In a chemical prcess, what happens if yu change the initial cncentratin f the ne f the reagents by a factr f tw? Are we sensitive t small variatins in any f the initial cncentratins? Y(t; Y + Y ) = Y(t; Y ) + Z(t)* Y Are any f the Z s fr minr species variatins greater than ε/y max, where Y max is the expected upper bund n cntaminants in the input stream, and ε is the acceptable upper bund n cntaminants in the utput stream? 2) Putting Errr Bars n the Mdel Predictins A first estimate f the uncertainty in the mdel predictin Y(t) is given by ln y i (t) Σ (ln y i (t))/ (ln D n ) * ln k n + Σ (ln y i (t))/ (ln y m) * ln y m The first term frequently dminates. Unfrtunately, the uncertainties estimated in this way are usually pretty large, accurately reflecting ur inability t precisely predict chemical kinetics. The errrs in the predicted selectivities (i.e. relative yields) are usually smaller than the errrs in the predicted abslute cncentratin prfiles. ln(y i (t)/y i (t)) Σ S in -S jn * ln k n 3) Identificatin f the Imprtant Uncertainties The sums abve that estimate the ttal uncertainties are usually dminated by a few terms. The uncertain parameters that crrespnd t these big terms are thse that matter the mst fr the simulatin at hand; if yu want a mre accurate simulatin yu need t tighten these errr bars. 4) Design f Kinetic Experiments In many kinetic experiments yu measure a signal S(t), and what yu want is fr S(t) t be very sensitive t the rate parameter k yu are trying t determine, and insensitive t the uncertainties in all the ther parameters. Then yu can have reasnable cnfidence
when yu say that yur bservatin f S(t) implies a certain value fr k. When yu design an experiment, yu are well-advised t immediately build a simulatin mdel, and check that yur bservable(s) really are sensitive t the rate cnstant yu are trying t determine. 5) Valid Range Analyses Each kinetic mdel is nly valid ver a finite range f reactin cnditins. Unfrtunately, this valid range is nt usually knwn. Fr ur mdel t be cmplete, we require that all the ignred side reactins have negligible rates. By determining the sensitivity f these side reactins t changes in the reactin cnditins we can estimate the valid range, i.e. the range ver which the mdel will be cmplete. (This presumes that the rate parameters in the mdel are nt t far ff.) See J. Sng et al. Chem. Eng. Sci. 2002. Flux Analyses It is ften very helpful t examine the rates at which each f the reactins in the system are prceeding. In particular, yu will want t knw which reactins have high net rates. By cnsidering the frward and reverse rates separately, yu can als identify the reactins which are in quasi-equilibrium. Often nly the reactins which have high net rates, high sensitivities, r which are quasi-equilibrated really matter fr the kinetics. Almst certainly yu cannt mdel the system crrectly if yu leave ut a reactin with a high net rate r a high sensitivity. Very ften it is interesting t knw which reactins cntribute the mst t the instantaneus frmatin r destructin f a key species. Often a species will be invlved in dzens f reactins, but nly three r fur have significant net rates. CHEMKIN will make these net rate plts fr yu. A prblem with net rates is that all the species in quasi-steady-state, and all the fast reactins which are nearly equilibrated, will have net rates near zer. It is therefre ften interesting t lk at the frmatin and destructin rates r the frward and reverse rates separately.