Predmet:Nove fizičkohemijske metode Tema: Metode ispitivanja dinamike složenih reakcionih sistema
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1 Predmet:Nove fizičkohemijske metode Tema: Metode ispitivanja dinamike složenih reakcionih sistema Predavači: Ljiljana Kolar-Anić i Željko Čupić Sadržaj I čas 1. Složeni reakcioni sistemi 2. Dinamičke strukture složenih reakcionih sistema i samoorganizacija neravnotežnih sistema 3. Metode ispitivanja dinamike složenih reakcionih sistema Sadržaj II časa 1. Kvalitativna analiza dinamike složenih reakcionih sistema Kvalitativna analiza trendova Spektri snage i vejvletna transformacija Poenkareov presek Klasifikacija mehanizma oscilatora na osnovu bifurkacionih dijagrama (SNA) 2. Kvantitativna analiza dinamike složenih reakcionih sistema Parametrizacija modela Ljapunovljevi eksponenti Multifraktalna analiza
2 Kvalitativna analiza dinamike složenih reakcionih sistema Kvalitativna analiza trendova Spektri snage i vejvletna transformacija Poenkareov presek Klasifikacija mehanizma oscilatora na osnovu bifurkacionih dijagrama (SNA)
3 Kvalitativna analiza trendova Tabela 1. Opšti tipovi epizoda. Vreme Type A B C D E F G x(t) dx/dt d 2 x/dt Jedna varijanta metode dizajnirana za kontrolu kvaliteta proizvoda u automatizovanom upravljanju hemijskim procesima u industriji. S. Charbonnier, C. Garcia-Beltan, C. Cadet, S. Gentil, Engineering Applications of Artificial Intelligence 18 (2005) 21-36
4 Algoritam za automatsku identifikaciju matematičkog modela hemijske reakcije David Schaich, Ralf Becker, Rudibert King, Control Engineering Practice 9 (2001)
5 ANALIZA OSCILOGRAMA 7.344x10-2 Jedan segment oscilacije Koncentracije joda (a) Bray-Liebhafsky (BL) reaction IO 3, H H O 2H O O 2IO 2H 5H O I 5O 6H O I 5H O 2IO 2H 4H O Reakcije okidači: D R O / mol dm -3 IO 3 - R D O D R 7.342x x x Time / min I + H2O2 + H HIO + H2O (o) HIO + H O + I + H + O + H O (r) Željko Čupić, Davor Lončarević, Ana Ivanović, Predrag Banković, Srđan Petrović: Experimentally observable transitions between dynamical states in complex reaction systems, Computers and Chemical Engineering, 32 (2008) Korišćen je model iz referenci: Schmitz, G., J. Chim.Phys. 1987, 84, Kolar-Anić, Lj. et al, React.Kinet.Catal.Lett. 1995, 54, Kolar-Anić, Lj. et al, J.Serb.Chem.Soc. 1995, 60,
6 Spektri snage pogodnija metoda za oscilatorne procese Spektar snage je kvadrat modula furijeove transformacije signala. Prilikom udvajanja perioda dolazi do pojave subharmonika u spektru snage. Slika 1: Period-2 oscilacije kada je j 0 = min -1 ; (a) vremenska serija, i (c) spektar snage. Slika 2: Haos Kada je j 0 = min -1 ; (a) vremenska serija, i (c) spektar snage. Guy Schmitz, Ljiljana Kolar-Anić, Slobodan Anić, Tomislav Grozdić, Vladana Vukojević J. Phys. Chem. A, 110 (2006) Slika 1 Slika 2
7 Vejvletna transformacija Xue-Guang Shao, Alexander Kai-Man Leung, and Foo-Tim Chau, Wavelet: A New Trend in Chemistry, Acc. Chem. Res. 2003, 36, Signal se transformiše iz vremenske dimenzije u vremensko frekventnu, tako da se svakom vremenskom intervalu pripisuju udeli procesa na datoj frekvenciji skali. Za razliku od Furijeove analize ovde su talasni paketi vejvleti- lokalizovani u vremenu.
8 Poređenje Furijeove i vejvletne transformacije David Labat, Recent advances in wavelet analyses: Part 1. A review of concepts, Journal of Hydrology 314 (2005)
9 Fazni prostor i atraktor Periodične promene u vremenu su posledica kretanja dinamičkog sistema po zatvorenoj putanji u faznom prostoru. Haotičnoj dinamici odgovara otvorena putanja po ograničenom delu faznog prostora Atraktor je trajektorija dinamičkog sistema u faznom prostoru posle prolaska tranzijentnog perioda. 6.5 x H2O x I I x 10-8 (I-) I2 x 10-4
10 Rekonstrukcija atraktora Takens je dokazao da umesto 2n+1 generičkih signala, za prekrivanje n- dimenzionalnog atraktora može biti dovoljna konstrukcija sa vremenskim kašnjenjem izvedena iz samo jednog generičkog signala.
11 Matrica trajektorije Razlaganje po singularnim vrednostima postupak sličan razlaganju na svojstvene vrednosti, ali primenljiv i na nekvadratne (pravougaone) matrice. Singular value decomposition (SVD) A= Y Y. Y 1 2 = ν ν ( τ ) ν ( 2τ ). ν ( dτ ) ( 2τ ) ν ( 3τ ). ν (( d + a) τ ). ( Nτ ) ν (( N + 1) τ ). ν (( N + d ) τ ) N ν 1... A 11. AN1 V 11. VN S1 0 0 U 11. U1d... =... x... x S d U dxd d1. U dd A. A V. V 1d Nd Nxd 1d Nd Nxd Postupak SVD obezbeđuje dobijanje singularnih vrednosti u formi opadajućeg intenziteta. U idealnom slučaju samo nekoliko singularnih vektora odgovara singularnim vrednostima koje daju značajan doprinos, dok ostalima odgovaraju nule. Postupak se koristi i za eliminaciju šuma iz signala. T dxd
12 Primeri rekonstruisanih atraktora eksperimentalno snimljenih signala elektrodnog potencijala u oscilatornoj reakciji BL A. Z. Ivanović, Ž. D. Čupić, M. M. Janković Lj. Z. Kolar-Anić and S. R. Anić, The chaotic sequences in the Bray Liebhafsky reaction in an open Reactor, Phys. Chem. Chem. Phys., 2008, 10,
13 Poenkareovi preseci Dimenzionalnost dinamičkog sistema se smanjuje i Kontinualni dinamički sistem se diskretizuje Periodični sistemi imaju diskretan mali broj tačaka u Poenkareovom preseku Haotični sistemi imaju neograničen broj tačaka u Poenkareovom preseku
14 Mape Iteracione mape nam daju mogućnost da prikažemo Poenkareov presek u formi rekonstruisanog atraktora diskretizovanog dinamičkog sistema x x 10-4 [I2], n+1, mol x dm [I2]n+1, mol x dm x 10-4 [I2]n, mol x dm x 10-4 [I2]n, mol x dm-3 Periodika Haos
15 Upravljanje haosom H2 + O2 Davies,M.L.; Halford-Maw,P.A.; Hill, J.; Tinsley,M.R.; Johnson,B.R.; Scott,S.K.; Kiss,I.Z.; Gáspár,V.: Control of Chaos in Combustion Reactions, J. Phys. Chem. A, 2000, 104,
16 Klasifikacija mehanizma oscilatora na osnovu bifurkacionih dijagrama (SNA) Bifurkacija je kvalitativna promena dinamike do koje dolazi pri kontinualnoj promeni vrednosti kontrolnog parametra (npr. Brzina protoka kroz reaktor) Bifurkaciona tačka je vrednosti kontrolnog parametra pri kojoj dolazi do bifurkacije dinamičkog sistema Bifurkacioni dijagram je grafik bifurkacionih tačaka u parametarskom prostoru
17 Klasifikacija mehanizma oscilatora na osnovu bifurkacionih dijagrama (SNA) - Polazna ideja J. Phys. Chem. 1989, 93, Use of Bifurcation Diagrams as Fingerprints of Chemical Mechanisms Zoltan Noszticzius, William D. McCormick, and Harry L. Swinney
18 Matematički aparat SNA Stehiometrijska mrežna analiza Clarke, B. L. Adv. Chem. Phys. 1980, 43, 1. Clarke, B. Cell Biophysics 1988, 12, 237. Realizacija Tim Chevalier, Igor Schreiber, and John Ross, Toward a Systematic Determination of Complex Reaction Mechanisms, J. Phys. Chem. 1993,97, Metoda dopušta identifikaciju tipa mehanizma kome odgovara dati bifurkacioni dijagram oscilatorne reakcije
19 Kvantitativna analiza oscilograma Parametrizacija modela Ljapunovljevi eksponenti Multifraktalna analiza
20 Parametrizacija modela Metoda najmanjih kvadrata je najjednostavniji oblik parametrizacije i primenjuje se na linearne i nelinearne sisteme U slučaju složenijih nelinearnih sistema neophodna je primena računara za numeričku simulaciju
21 Neizotermalni sistemi Concentration (mol %) 0.2 CHL % (BE) % (PVP) % (A0) % (A4) CHN CHHP Temp. / 0 C k 1 O 2, cat. OOH Rezultate numericke simulacije je moguće direktno uporediti sa eksperimentalnim merenjima. Primenom nekog od postupaka optimizacije moguće je kroz niz iteracija doći do optimalnog seta kinetičkih parametara modela. k 2 k 3 OH O k 4 k 5 k 6 CO CO 2 D. Lončarević, Ž. Čupić, M. Odović: "Inhibition Effects in the Partial Oxidation of Cyclohexane on Polymer Supported Co(II) Catalysts". J.Serb.Chem.Soc., 70 (2005),
22 HETEROGENI SISTEMI Fitovanje eksperimentalnih podataka raspodelom B. Janković et al. / Thermochimica Acta 505 (2010)
23 Metoda Wei Prater za linearne sisteme k k k k C dc1 dt dc = k k k k C 2 dt dc3 Metoda omogućava određivanje svih konstanti brzina modela mreže reakcija pseudo-prvog reda. Zasniva se na jednostavnoj relaciji između konstanti brzina i svojstvenih vrednosti operatora dinamike. Atraktor linearnog sistemaje jedna tačka - ravnoteža k k k k C dt k31 k21 k12 k13 X1 X1 k k k k X =λ X k31 k32 k13 k 23 X 3 X 3 J. Wei, C. Prater, The structure and analysis of complex reaction systems, Advances in Catalysis, 13, 1962, 203.
24 Metoda prigušenja oscilacija Metoda prigušenja oscilacija omogućava određivanje konstanti brzina modela mehanizma oscilatorne reakcije. Kao i u metodi Wei Prater određuju se svojstvene vrednosti operatora dinamike, ali u ovom slučaju to je linearizovani operator u ustaljenom stanju Jakobijan. Hynne F, Sørensen PG. Quenching of Chemical Oscillations. (1987) J Phys Chem. 91, Vukojević V, Sørensen PG, Hynne F. Quenching Analysis of the Briggs-Rauscher Reaction. (1993) J Phys Chem. 97, Atraktor je granični krug
25 Parametrizacija haotičnih dinamičkih sistema Određivanje Ljapunovljevih eksponenata Ljapunovljev eksponent mera brzine razdvajanja dveju inicijalno veoma bliskih tačaka na atraktoru haotičnog dinamičkog sistema B 1old D( A 1 B 1 ) old DAB ( ) B A A 1 B 1new D( A 1 B 1 ) new Shematski prikaz procedure određivanja Ljapunovljevih eksponenata iz vremenskih serija Postoje procedure koje Ljapunovljeve eksponente određuju kao svojstvene ili singularne vrednosti operatora dinamike linearizovanog lokalno, u datoj tački na atraktoru. L. Dieci, C. Elia, Mathematics and Computers in Simulation 79 (2008)
26 Spektar Ljapunovljevih eksponenata vremenskih serija simulacije BL oscilatorne reakcije Osetljivost na početne uslove i na promene vrednosti parametara je još jedna važna osobina sistema sa determinističkim haosom. Ako je vrednost bar jednog najvećeg Ljapunovljevog eksponenta pozitivna, u vremenskoj evoluciji se javlja haos. A. Z. Ivanović, Ž. D. Čupić, M. M. Janković Lj. Z. Kolar-Anić and S. R. Anić, The chaotic sequences in the Bray Liebhafsky reaction in an open Reactor, Phys. Chem. Chem. Phys., 2008, 10,
27 Multifraktalna analiza Atraktori haotičnih dinamičkih sistema su često fraktali imaju razlomljenu - necelobrojnu dimenziju. Oni mogu biti mono ili multifraktali. Multifraktali imaju delove kojima odgovaraju različite fraktalne dimenzije. Dimenzija se određuje kao koeficijent skaliranja između gustine tačaka i veličine segmenta. Jedan od postupaka multifraktalne analize se zasniva na vejvletnoj transformaciji signala. wnload.html A. Z. Ivanović, Ž. D. Čupić, Lj. Z. Kolar-Anić, M. M. Janković and S. R. Anić, Russian Journal of Physical Chemistry A, 2009, Vol. 83, No. 9, pp L.D. Spectrum j 0 = j 0 = j 0 = j 0 = j 0 = Holder exponent [ I?], 10-8 mol dm [I 2 ], 10-3 mol dm -3
28 Hvala na pažnji.
Dr Željko Čupić, Principal Research Fellow
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