Computational Study of System Dynamics (Chemical Kinetics)

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1 Computational Study of System Dynamics (Chemical Kinetics) 1 Review of Rate Laws J. Andraos, J. Chem. Ed., 76 (11), (1999). 2 Computational Methods PDynamics Methods < Chemical Kinetics Simlulator < A+BC: General Trajectory P Mathematics Methods < Integral < Differential P Graphical Interface Differential Equation Solvers 3

Dynamics Methods Chemical Kinetics Simulator http://www.almaden.ibm.

2 Dynamics Methods Chemical Kinetics Simulator PCKS does not integrate differential equations P CKS performs general, rigorously accurate stochastic algorithm to propagate reaction PSpeed < Comparable in efficiency to integration for simple < Significantly faster for stiff systems P Models complex reactions < Explosions < Changing volumes PHigh learning curve PResearch quality 4 5 Dynamics Methods A+BC: General Trajectory PLEPS diagram PMonte Carlo Collisions A b C θ B cm 6

integrate differential rate laws < Tables and Graphs 8 Review of Rate Laws Simple

3 7 Mathematics Methods General P Integral Methods < Use integrated rate laws < Tables and Graphs P Differential Methods < Use computational technique to integrate differential rate laws < Tables and Graphs 8 Review of Rate Laws Simple First Order Reactions k A B A = a, B = o o da = ka A = ae kt dt db dt ( 1 ) = ka B = a e kt 9

4 a = 1 Mathematics Methods--Integral k = 1 Spreadsheet t A B A B t 1 Mathematics Methods--Differential Spreadsheet (Based on Euler s Rule) a = 1 k = 1 da=-k A dt A = a - da B = a - A t da A B A B t 11 Mathematics Methods--Integral Mathcad a:= 1 k:= 1 At ():= ae k t Bt ():= a 1 e k t ( ) 1 At () Bt () t 12

5 Mathematics Methods--Differential k:= 1 Given Mathcad d dt At () k At () A ( ) 1 A := Odesolve( t, 1) Given d dt Bt () ka () t B ( ) B:= Odesolve( t, 1) 1 At () Bt () t 13 Mathematics Methods--Integral Mathematica 14 Mathematics Methods--Differential Mathematica 15

Mathematics Methods--Integral Matlab 16 db dt Review of Rate Laws Consecutive First Order Reactions k k A B C A = a, B = C = 1 2 o o o da = ka A = ae kt dt ak1 = ka 1 kb 2 B = e k k 2 1 kt 1 kt 2 ( e

6 Mathematics Methods--Integral Matlab 16 db dt Review of Rate Laws Consecutive First Order Reactions k k A B C A = a, B = C = 1 2 o o o da = ka A = ae kt dt ak1 = ka 1 kb 2 B = e k k 2 1 kt 1 kt 2 ( e ) dc dt = kb 2 C = a 1 17 kt ke 2 k k 2 1 ke 1 k k 1 kt Mathematics Methods--Integral k1 := 1 k2 :=.5 a := 1 At ():= ae k1 t ( ) Bt ():= ak1 e k1 t e k2 t k2 k1 k2 Ct () a 1 e k1 t k1 := e k2 t k2 k1 k1 k2 Note that k1 must be different than k2 in order for these integrated equations to work. Mathcad 1 1 At () Bt () Ct () t 1 18

7 Mathematics Methods--Differential k1:= 1 k2:= 1 a:= 1 b := c:= T:= 1 Mathcad Given d dt At () k1 At () A ( ) a 8 d dt Bt () k1 A() t k2 B() t B ( ) b NA() t 6 NB() t d dt Ct () k2 B() t C ( ) c NC() t 4 NA A NB := Odesolve B, t, T NC C t 1 19 Graphical Interface Differential Equation Solvers General P Model differential rate law P Software solves differential equations < Watch stiff P Tables and Graphs 2 Solving of Differential Equations Taylor Series Δt dy Δt d y Δt d y Δt d y yt ( m + Δt) = ym ! dt 2! dt 3! dt 4! dt 4 K t= t t= t t= t t= t m m m m P y m = value of reservoir at time t = t m P y(t m + t) = value of reservoir at time t = t m + t P Methods < Euler retains first two terms < Runga-Kutta 2 retains first three terms < RK4 retains first five terms 21

8 Commercial Products Graphical Interface Differential Equation Solvers (GIDES) PVensim PLE PBerkeley Madonna < Download demo version < Solves stiff differential equations PSTELLA < Java conversion available from Shodor PSimile P Model Maker PVisSim (Mathcad) PSimulink (Matlab) 22 Review of Rate Laws Simple First Order Reactions k A B A = a, B = o o da = dt ka db dt = ka 23 Graphical Interface Differential Equation Solvers Vensim PLE ( 1, 1, First Order A to B Time Reactant A : Current Product B : Current 24

9 Graphical Interface Differential Equation Solvers Berkeley Madonna 25 Graphical Interface Differential Equation Solvers Stella 26 Graphical Interface Differential Equation Solvers Simile 27

10 Graphical Interface Differential Equation Solvers VisSim (Mathcad) 28 Graphical Interface Differential Equation Solvers Simulink (Matlab) 29 Review of Rate Laws Dimerization of Cyclopentadiene 2nd Order Diels-Alder Reaction 2 dc 2 = kc dt 1 1 = + kt C C o 3

31 Review of Rate Laws A W B W C First Order and

Review of Rate Laws Michaelis-Menton After Len

11 31 Review of Rate Laws A W B W C First Order and Reversible First Order Consecutive First Order and Reversible First Order (Disequilibrium, Transient Equilibrium, Secular Equilibrium) 32 Review of Rate Laws Michaelis-Menton After Len Soltzberg (Simmons College) E + S º (ES) (ES) 6 P + E 33

12 34 Oscillating Reactions Belousov-Zhabotinskii BrO HBrO 2 + H 3 O BrO H 2 O 2 BrO Ce H 3 O HBrO Ce H 2 O P 18 reversible steps P 21 different chemical species 35 ~6 s intervals 36

13 Gentle Stirring 37 Oregonator after Len Soltzberg (Simmons College) A + Y 6 X + P X + Y 6 2P A + X 6 2X + 2Z 2X 6 A + P B + Z 6 (f/2) Y where A = BrO 3 - X = HBrO 2 Z = Ce 4+ P = HOBr B = organic Y = Br

14 4 41 Other Applications Debye Theory for Monatomic Crystals D( θ / T) = D 3 3 u 3 x dx x e 1 42

15 U U = 3RTD u Cv = 3R 4D 3 u e 1 u ln( 1 ) u [ 3ln( 1 ) ] 4 S = 3R D e 3 A U = RT e D

16 Vensim PLE Interface 46 Vensim PLE Interface Initial Window Move/Size words and arrows (select) 47 Vensim PLE Interface Variables Variable Auxiliary/Constant 48

17 Vensim PLE Interface Variables Box Variable Level (stock or reservoir) 49 Vensim PLE Interface Arrow Arrow (connector) 5 Vensim PLE Interface Rate Rate (flow) 51

18 Vensim PLE Interface Input Output Object Input Output Object 52 Vensim PLE Interface Delete Delete 53 Vensim PLE Interface Equations Equations 54

19 Vensim PLE Interface Formatting Model Set Surround Shape on Selected Variable (color, location of text, etc.) 55 Vensim PLE Interface Setup a Simulation Setup a Simulation 56 Vensim PLE Interface Run a Simulation Run a Simulation 57

20 Vensim PLE Interface Automatically Simulate on Change Automatically Simulate on Change 58 Vensim PLE Interface Control Panel Control Panel [setup graphs, etc.] 59

Computational Study of System Dynamics

Computational Study of System Dynamics Review of Rate Laws JCE Summary Computational Methods P Dynamics Methods < Chemical Kinetics Simulator < A+BC: General Trajectory P Mathematics Methods < Integral