5.68J/10.652J Feb 13, 2003 Numerical Solution to Kinetic Equations Lecture 1

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1 5.68J/10.652J Feb 13, 2003 Numercal Soluton to Knetc Equatons Lecture 1 The Problem We want to solve 1 dn l H = ν r ( T l V dt n, ) Transport of ' ' out of V Eq.(1) Where n s the number of moles of speces, ν s the matrx of stochometrc coeffcents, and = Rateof Step 'l' (unts: moles/volume-second) For a typcal bmolecular elementary step reacton A+B = C+D: = k forward ( T )[ A][ B ] k ( T )[ C][ D] For a typcal gas-phase dssocaton reacton A= C + D: = k forward ( T, P )[ A ] k ( T, P )[ C][ D] For a typcal gas-phase somerzaton reacton A = C: = k forward ( T, P )[ A ] k ( T, P )[ C] The rate constants also usually depend (weakly) on the chemcal envronment (e.g. delectrc constant), so they strctly are not really constant but (very weak) functons of all the n s. Note that the forward and rates are normally related by the equlbrum constant, related to the free energy of reacton (thermochemstry). It s mportant to apprecate that except n the smplest cases, there s no analytcal soluton to ths system of dfferental equatons, so we MUST use the computer to solve them numercally. Approxmatons lke quas-steady-state extend the range of systems where analytcal (approxmate) solutons are possble, but you don t have to get very complcated before even the steady-state-approxmaton solutons are ntractable.

2 More About the Fundamental Equaton Chemstry textbooks usually gve nstead of Eq. (1) the equaton: d [ ] l H = ν r ( l, dt n T ) Eq.(2) But Eq.(2) s true only f dv/dt = 0 ( sochorc ) and there s no transport (.e. perfectly homogeneous, no convectve flow). These textbook approxmatons are good for dlute reactons n soluton or for constant volume apparatus (e.g. bomb calormeter, shock tube). In many mportant cases the system s sobarc,.e. dp/dt = 0, so f the temperature ncreases or there s a change n the number of moles of gas V wll change. It s convenent to express the equaton n terms of ntensve quanttes rather than the extensve varables n and V. The ntensve varables that work best are mass fractons yι: n W = [] W y = M total ρ dy W 1dn = Transport of ' ' out of dt ρ V dt V You can plug n the same expresson as above for the term n parentheses. Note that mole fractons x are not such a good choce for knetcs as mass fractons, snce the number of moles can change n chemcal reactons, but the mass does not. However one wrtes the chemcal reacton equatons, one must be explct about the model for dt/dt, snce the knetcs nvarably depends strongly on T. The common textbook assumpton s sothermal (dt/dt = 0) whch s okay f the reacton s very slow and the reagents dlute. In many cases of nterest the system s closer to beng adabatc, so dt/dt s gven by energy conservaton. (For the energy conservaton equaton around a small control volume wth unform T, P, and composton, and an elementary dscusson of how ODE s are solved numercally, see: ). Note that to accurately compute dt/dt one must have a way of computng the enthalpes and heat capactes of all the speces at any T. Thermochemcal Data Because one needs the thermochemcal data both to compute dt/dt and also to compute the reacton rate constants, t s usual to buld up a lbrary fle contanng all ths data for all the speces of nterest. For (stupd) hstorcal reasons the data are usually represented as the coeffcents n a pecewse fttng form known as the NASA polynomals. For some of the arcane detals about NASA polynomals, see the CHEMKIN manual and Beware that there s more than one format for the NASA polynomals! The NIST Webbook gves thermochemcal data n a dfferent Shomate format whch s better

3 behaved than the NASA form. The other popular form s the dscrete values of Benson (used also by THERM and the NIST S&P program) dscussed later n the course. All forms take advantage of the fact that the T-dependence of the enthalpy and the entropy s easly computed (often analytcally) from the heat capacty. So the man ssues are whether one knows H and S exactly at any T for the speces of nterest and then how well the T-dependence of the heat capacty (usually Cp, measured at constant pressure) can be determned and represented. Very often S and Cp are calculated usng statstcal mechancs rather than measured, you need to read carefully to determne whether the numbers reported n a table are measured or just computed, and f they are computed what assumptons were made. The reported S and H are usually for an deal gas state at 1 atm at 298 K, even f the molecules are actually lquds or solds at 298 K and 1 atm. So the reported H(298) s often calculated, perhaps from an expermentally measured H at some dfferent temperature and the computed heat capactes. In many real-world applcatons of knetcs, the estmated barrer heghts of the slow steps are strongly correlated wth the estmates of ther endothermctes, and many of the fast reactons are nearly equlbrated. Hence one s more senstve to uncertantes n the thermochemcal data than to the (usually much larger) uncertantes n the ndvdual rate constants. So a knetcst MUST be concerned about the qualty of the thermochemcal data fle employed n the smulatons. There s plenty to worry about. For example, n 2001 t was dscovered that the heat of formaton of OH was computed ncorrectly many years ago, and that therefore the many speces whose thermochemstry was nferred relatve to OH were also ncorrect. Presumably many smlar errors exst n the databases, watng for some brght graduate student to dentfy them. Note that you wll get observably dfferent results f you use the default therm.dat fle shpped wth CHEMKIN or the GRI-MECH thermo fle. Beware: Inhomogenety can develop Spontaneously If dt/dt s nonzero, often gradents wll develop n the system even f t was ntally homogeneous due to heat losses at the boundares. In some cases these temperature gradents can become extreme (e.g. n a flame). Pressure equlbrates at about the speed of sound, much faster than speces concentraton or temperature gradents can equlbrate, so very often t s a good approxmaton that P s unform n a system. However, n some combuston systems the reacton s so fast that sgnfcant pressure gradents develop; ths can cause loud noses or n an extreme cases detonatons and other types of explosons. These nhomogenetes are reduced by usng a smaller control volume; for reactng flow smulatons people often use control volumes less than a mllmeter across so that ther approxmaton that the system has a sngle temperature and composton s accurate. Requred Inputs What nputs do we need for a Knetc Smulaton? 1) A Thermo Fle (wth data for all the speces n the smulaton) 2) Reacton Mechansm: A lst of the reactons and forward rate constants 3) Reacton condtons (e.g. ntal concentratons, T, P, tmescale, adabatc or sothermal, flow stuaton, etc.) to fully specfy physcal stuaton.

4 4) Smulaton detals: Specfy outputs requred, tolerances, numercal methods, ntal guesses, etc. In CHEMKIN, nput 1 s called somethng lke therm.dat and usually resdes n the /data/ drectory, nput 2 s called somethng lke chem.np (and f there s surface chemstry an addtonal fle surf.np), and nputs 3 and 4 are combned n a sngle fle, e.g. aurora.np. How are ODE s solved numercally? (the short short verson) If we defne Y={y, T} and there s no transport, then Eq.(1) s of the form: dy/dt = F(Y) and we usually know the ntal condtons: Y(t o )=Y o The general procedure s to step forward n tme wth some formula Y(t+ t) = Y(t) + G(Y) t Eq.(3) where G s our best estmate of the average of F(Y(t )) over the trajectory from Y(t =t) to Y(t =t+ t), usng lots of lttle tmesteps t untl we reach t fnal. In the smplest approxmaton called Forward (or Explct) Euler G=F(Y(t)). Ths turns out to be pretty naccurate (just lke the rectangle rule s not a very accurate way to compute numercal ntegrals) and also numercally unstable unless t s very small. The fundamental problem s that we don t know the trajectory Y(t ) all we can do s extrapolate to estmate Y(t ) at future tmes. The extrapolaton at each tmestep ntroduces a lttle bt of error, and then the extrapolaton at the next tmestep s bult on an ncorrect startng pont and an ncorrect estmate of the slope G(Y). If one s not careful, the errors can cumulate n a very unfavorable way, makng the whole procedure numercally unstable (even f the ODEs and the physcal stuaton they descrbe s perfectly stable). There are two general ways of copng wth ths. One s to use relatvely smple estmates of G and just use tny tmesteps t much smaller than any of the physcal tmescales n the system. Ths general approach s called explct, the most famous algorthms that work ths way are called Runge-Kutta methods. The alternatve approach s to use very complcated methods for estmatng G that are guaranteed to be numercally stable, these are called mplct methods because n these methods G s a functon of Y(t+ t), so Eq.(3) becomes an mplct (usually) nonlnear system of equatons, very dffcult to solve. The hope s that by mprovng the numercal stablty, one could use large t s and so reduce the number of tmesteps requred to reach t fnal. The dffcult aspect of chemcal knetcs s that one often has reactve ntermedates wth lfetmes tmes shorter than the relatvely nert startng materals. The short lfetmes of the reactve speces force t n the explct methods to be extremely small, so then a huge number of tmesteps (and a correspondngly huge amount of CPU tme) are requred before the startng materals are consumed sgnfcantly. When the range of tmescales s very large, the ODE system s called stff. Stffness ntroduces many numercal problems, even for mplct ODE solvers; several algorthmc trcks must

5 be used smultaneously n order to solve stff systems wthout ntroducng huge numercal errors. The frst algorthm that could correctly handle stff ODE systems was dscovered by Wllam Gear n the 1970 s. The best programs now avalable for solvng large stff ODE systems are VODE, DASSL/DASPK, and DAEPACK. I beleve the current verson of CHEMKIN calls VODE and DASSL.