Computational Study of System Dynamics

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1 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 < Differential P Graphical Interface Differential Equation Solvers

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

2 Dynamics Methods Chemical Kinetics Simulator P CKS does not integrate differential equations P CKS performs general, rigorously accurate stochastic algorithm to propagate reaction P Speed < Comparable in efficiency to integration for simple < Significantly faster for stiff systems P Models complex reactions < Explosions < Changing volumes P High learning curve P Research quality Dynamics Methods A+BC: General Trajectory P LEPS diagram P Monte Carlo Collisions A b C 2 B cm

3 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 Review of Rate Laws Simple First Order Reactions k A B A = a, B = 0 o o da = ka A = ae kt dt db dt ( 1 ) = ka B = a e kt

4 a = 1000 Mathematics Methods--Integral k = 1 Spreadsheet t A B A B t Mathematics Methods--Differential Spreadsheet a = 1000 k = 1 da=-k A dt A = a - da B = a - A t da A B A B t Mathematics Methods--Integral Mathcad a := 1000 k:= 1 At ():= ae k t Bt ():= a 1 e k t ( ) 1000 At () Bt () t

5 Mathematics Methods--Differential k:= 1 Given Mathcad d dt At () k At () A0 ( ) 1000 A := Odesolve( t, 10) Given d dt Bt () ka () t B0 ( ) 0 B:= Odesolve( t, 10) 1000 At () Bt () t Mathematics Methods--Integral Mathematica Mathematics Methods--Differential Mathematica

Mathematics Methods--Integral Matlab db dt k Review of Rate Laws Consecutive First Order Reactions k 1 2 A B C A = a, B = C = 0 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 db dt k Review of Rate Laws Consecutive First Order Reactions k 1 2 A B C A = a, B = C = 0 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 kt ke 2 k k 2 1 ke 1 k k 1 kt Mathematics Methods--Integral k1 := 1 k2 := 0.5 a := 1000 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 At () Bt () Ct () t 10

7 Mathematics Methods--Differential k1:= 1 k2:= 1 a := 1000 b := 0 c := 0 T:= 10 Mathcad Given d dt At () k1 At () A0 ( ) a 800 d dt Bt () k1 A() t k2 B() t B0 ( ) b NA() t 600 NB() t d dt Ct () k2 B() t C0 ( ) c NC() t 400 NA A NB := Odesolve B, t, T NC C t 10 Review of Rate Laws Several Consecutive First Order Reactions k k k k A B C D K Ao = a, Bo = CoK= 0 da = ka dt db = ka 1 kb 2 dt dc dt = kb 2 kc 3! Bateman solution: kt 1 kt 2 kt 3 N = c e + c e + c e + K+ c e c c c = = = n n kkklk a n n 1 ( k k )( k k ) L( k k ) kkklk a n n 2 ( k k )( k k ) L( k k ) kkklk a n n 3 ( k k )( k k )( k k ) L( k k )! kt n

8 Graphical Interface Differential Equation Solvers General P Model differential rate law P Software solves differential equations < Watch stiff P Tables and Graphs Graphical Interface Differential Equation Solvers Stella

9 Graphical Interface Differential Equation Solvers Berkeley Madonna Graphical Interface Differential Equation Solvers VisSim (Mathcad) Graphical Interface Differential Equation Solvers Simulink (Matlab)

= a, B = b, C = 0 o o o da db dc = = kab = dt dt dt kab

10 Graphical Interface Differential Equation Solvers Simile Review of Rate Laws Simple Second Order Reactions k A+ B C A = a, B = b, C = 0 o o o da db dc = = kab = dt dt dt kab Graphical Interface Differential Equation Solvers Change of Paradigm

11 Introduction to STELLA Structural Thinking Experiential Learning Laboratory with Animation Commerical Products Graphical Interface Differential Equation Solvers (GIDES) PStella < Free run-time version P Berkeley Madonna < Free run-time version < Solves stiff differential equations P VisSim (Mathcad) P Simulink (Matlab) PSimile P Model Maker Stella Interface Stella Layers Modeling P Construct Model using Building Blocks, Tools, Objects P Outputs Layer Navigation Map/Model Toggle Change to P 2 (or else!) Run Controller

Layers Equation P Initial Values of Stocks P Connector Information P

12 Stella Interface Stella Layers Mapping P Text P Output Tables and Graphs P Input using Slides and Dials P Pictures P Quick Time Movies Stella Interface Stella Layers Equation P Initial Values of Stocks P Connector Information P Differential Equations Representing the Time Dependence of Stocks Stella Interface Menu / Icons

13 Stella Interface Building Blocks Stocks Value Undergoing a Change Stella Interface Building Blocks Flow Change of Stock with Respect to Time Stella Interface Building Blocks Converter Constants Transformation Equations

14 Stella Interface Building Blocks Connectors Links Building Blocks Stella Interface Tools Hand General Purpose Editing Tool Cursor Pointer Stella Interface Tools Dynamite Delete (No Undo )

15 Stella Interface Objects Graph Pad Graphing Results Stella Interface Objects Table Pad Spreadsheet-like Table of Results

16 Solving of Differential Equations Taylor Series Solving of Differential Equations Euler Method First 2 terms

17 Solving of Differential Equations Runge-Kutta 2 First 3 Terms Solving of Differential Equations Runge-Kutta 4 First 5 Terms Simple Model Falling Calculator System y = height yn = velocity = (acceleration)(time) yo = acceleration = g = 9.8 m s -2

18 Simple Model Falling Calculator Stella Model

20 Simple Model Falling Calculator Stella Model Equation Layer

21 Simple Model Falling Calculator Stella Model Accuracy P Air Friction PBounce Chemical Kinetics Sure looked like a natural fit to me!! Kinetics Model Dimerization of Cyclopentadiene 2nd Order Diels-Alder Reaction 2 dc 2 = kc dt 1 1 = + kt C C o

22 Kinetics Model

25 Michaelis-Menton Len Soltzberg (Simmons College) E + S º (ES) (ES) 6 P + E

26 Oscillating Reactions Criteria P Two or more coupled reactions P Autocatalytic

27 Oscillating Reactions Briggs-Rauscher P H2O2 P KIO3 and H2SO4 P HOOCCH2COOH, MnSO4, starch ~15 s for each cycle Oscillating Reactions Belousov-Zhabotinskii BrO3- + HBrO2 + H3O+ 6 2 BrO2 + 2 H2O 2 BrO2 + 2 Ce H3O+ 6 2 HBrO2 + 2 Ce H2O P 18 reversible steps P 21 different chemical species

28 Oregonator by 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 -

30 Other Applications Rod Schluter (Formerly of CofC) Acid/Base Equilibrium H + + OH - 6 H 2 O

31 Other Applications Debye Theory for Monatomic Crystals D( θ / T) = D 3 3 u 3 x x e dx 1 U U = 3RTD 0 u Cv = 3R 4D 3 u e 1 4 S = 3R D e ln 3 u ( 1 ) u [ 3ln( 1 ) ] A U = RT e D 0

Computational Study of System Dynamics (Chemical Kinetics)

Computational Study of System Dynamics (Chemical Kinetics) 1 Review of Rate Laws J. Andraos, J. Chem. Ed., 76 (11), 1578-1583 (1999). 2 Computational Methods PDynamics Methods < Chemical Kinetics Simlulator