computational study of system dynamics · computational study of system dynamics review of rate...
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Computational Study ofSystem Dynamics
Review of Rate LawsJCE Summary
Computational Methods
PDynamics Methods< Chemical Kinetics Simulator< A+BC: General Trajectory
PMathematics Methods< Integral< Differential
PGraphical Interface Differential EquationSolvers
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Dynamics MethodsChemical Kinetics Simulator
http://www.almaden.ibm.com/st/msimPCKS does not integrate differential equationsPCKS performs general, rigorously accurate
stochastic algorithm to propagate reactionPSpeed< Comparable in efficiency to integration for simple< Significantly faster for “stiff” systems
PModels complex reactions< Explosions< Changing volumes
PHigh learning curvePResearch quality
Dynamics MethodsA+BC: General Trajectory
http:/qcpe.chem.indiana.edu/cgi-bin/catalog/view.pl
PLEPS diagramPMonte Carlo Collisions
A
B
C
cm
2b
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A B A a Bko o⎯ →⎯⎯ = =, 0
− = = −dAdt
kA A ae kt
( )dBdt
kA B a e kt= = − −1
Mathematics MethodsGeneral
P Integral Methods< Use integrated rate laws< Tables and Graphs
PDifferential Methods< Use computational technique to integrate
differential rate laws< Tables and Graphs
Review of Rate LawsSimple First Order Reactions
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a = 1000k = 1
t A B0 1000 0
0.1 904.837418 95.1625820.2 818.730753 181.2692470.3 740.818221 259.1817790.4 670.320046 329.6799540.5 606.53066 393.469340.6 548.811636 451.1883640.7 496.585304 503.4146960.8 449.328964 550.6710360.9 406.56966 593.43034
1 367.879441 632.1205591.1 332.871084 667.1289161.2 301.194212 698.8057881.3 272.531793 727.4682071.4 246.596964 753.4030361.5 223.13016 776.869841.6 201.896518 798.1034821.7 182.683524 817.3164761.8 165.298888 834.7011121.9 149.568619 850.431381
2 135.335283 864.6647172.1 122.456428 877.5435722.2 110.803158 889.1968422.3 100.258844 899.7411562.4 90.7179533 909.2820472.5 82.0849986 917.9150012.6 74.2735782 925.7264222.7 67.2055127 932.794487
0 2 4 6 8 10 12t
0
200
400
600
800
1000
1200
AB
a = 1000k = 1
dA=-k A dt A = a - dA B = a - At dA A B
0 0 1000 00.1 -100 900 1000.2 -90 810 1900.3 -81 729 2710.4 -72.9 656.1 343.90.5 -65.61 590.49 409.510.6 -59.049 531.441 468.5590.7 -53.1441 478.2969 521.70310.8 -47.82969 430.46721 569.532790.9 -43.046721 387.420489 612.579511
1 -38.742049 348.67844 651.321561.1 -34.867844 313.810596 686.1894041.2 -31.38106 282.429536 717.5704641.3 -28.242954 254.186583 745.8134171.4 -25.418658 228.767925 771.2320751.5 -22.876792 205.891132 794.1088681.6 -20.589113 185.302019 814.6979811.7 -18.530202 166.771817 833.2281831.8 -16.677182 150.094635 849.9053651.9 -15.009464 135.085172 864.914828
2 -13.508517 121.576655 878.4233452.1 -12.157665 109.418989 890.5810112.2 -10.941899 98.4770902 901.522912.3 -9.847709 88.6293812 911.3706192.4 -8.8629381 79.7664431 920.2335572.5 -7.9766443 71.7897988 928.210201
0 2 4 6 8 10 12t
0
200
400
600
800
1000
1200
AB
a 1000:=
k 1:=
A t( ) a e k− t⋅⋅:= B t( ) a 1 e k− t⋅
−( )⋅:=
0 2 4 6 8 100
500
1000
A t( )
B t( )
t
Mathematics Methods--IntegralSpreadsheet
Mathematics Methods--DifferentialSpreadsheet
Mathematics Methods--IntegralMathcad
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k 1:=
Given
tA t( )d
dk− A t( )⋅ A 0( ) 1000
A Odesolve t 10,( ):=
Given
tB t( )d
dk A t( )⋅ B 0( ) 0
B Odesolve t 10,( ):=
0 5 100
500
1000
A t( )
B t( )
t
Mathematics Methods--DifferentialMathcad
Mathematics Methods--IntegralMathematica
Mathematics Methods--DifferentialMathematica
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A B C A a B Ck ko o o
1 2 0⎯ →⎯⎯ ⎯ →⎯⎯ = = =,
− = = −dAdt
kA A ae kt
( )dBdt
k A k B Bak
k ke ek t k t= − =
−−− −
1 21
2 1
1 2
dCdt
k B C ak ek k
k ek k
k t k t
= = −−
−−
⎛
⎝⎜
⎞
⎠⎟
− −
22
2 1
1
1 21
1 2
k1 1:= k2 0.5:=
a 1000:=
A t( ) a e k1− t⋅⋅:=
B t( )a k1⋅
k2 k1−e k1− t⋅ e k2− t⋅
−( )⋅:=
C t( ) a 1k2
k2 k1−e k1− t⋅⋅−
k1k1 k2−
e k2− t⋅⋅−⎛⎜
⎝⎞⎠
⋅:=
Note that k1 must be different than k2 in orderfor these integrated equations to work.
0 2 4 6 8 100
200
400
600
800
10001000
0
A t( )
B t( )
C t( )
100 t
Mathematics Methods--IntegralMatlab
Review of Rate LawsConsecutive First Order Reactions
Mathematics Methods--IntegralMathcad
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k1 1:= k2 1:=
a 1000:= b 0:= c 0:=
T 10:=
Given
tA t( )d
dk1− A t( )⋅ A 0( ) a
tB t( )d
dk1 A t( )⋅ k2 B t( )⋅− B 0( ) b
tC t( )d
dk2 B t( )⋅ C 0( ) c
NA
NB
NC
⎛⎜⎜⎝
⎞
⎠Odesolve
A
B
C
⎛⎜⎜⎝
⎞
⎠t, T,
⎡⎢⎢⎣
⎤⎥⎥⎦
:=
0 2 4 6 8 100
200
400
600
800
10001 103×
0
NA t( )
NB t( )
NC t( )
100 t
A B C D A a B Ck k k ko o o
1 2 3 4 0⎯ →⎯⎯ ⎯ →⎯⎯ ⎯ →⎯⎯ ⎯ →⎯⎯ = = =K K,
− =dAdt
kA
dBdt
k A k B= −1 2
dCdt
k B k C= −2 3
N c e c e c e c enk t k t k t
nk tn= + + + +− − − −
1 2 31 2 3 K
( )( ) ( )ck k k k a
k k k k k kn
n1
1 2 3 1
2 1 3 1 1
=− − −
−L
L
( )( ) ( )ck k k k a
k k k k k kn
n2
1 2 3 1
1 2 3 2 2
=− − −
−L
L
( )( )( ) ( )ck k k k a
k k k k k k k kn
n3
1 2 3 1
1 3 2 3 4 3 3
=− − − −
−L
L
Mathematics Methods--DifferentialMathcad
Review of Rate LawsSeveral Consecutive First Order Reactions
!
Bateman solution:
!
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Graphical Interface DifferentialEquation Solvers
General
PModel differential rate lawPSoftware solves differential equations< Watch “stiff”
PTables and Graphs
Graphical Interface DifferentialEquation Solvers
Stella
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Graphical Interface DifferentialEquation Solvers
Berkeley Madonna
Graphical Interface DifferentialEquation Solvers
VisSim (Mathcad)
Graphical Interface DifferentialEquation Solvers
Simulink (Matlab)
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A B C A a B b Cko o o+ ⎯ →⎯⎯ = = =, , 0
− = − = =dAdt
dBdt
kABdCdt
kAB
Graphical Interface DifferentialEquation Solvers
Simile
Review of Rate LawsSimple Second Order Reactions
Graphical Interface DifferentialEquation Solvers
Change of Paradigm
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Introduction toSTELLA
Structural Thinking Experiential LearningLaboratory with Animation
Commerical ProductsGraphical Interface Differential Equation Solvers (GIDES)
PStella< Free run-time version
PBerkeley Madonna< Free run-time version< Solves “stiff” differential equations
PVisSim (Mathcad)PSimulink (Matlab)PSimilePModel Maker
Stella InterfaceStella “Layers” – Modeling
PConstructModel usingBuilding Blocks,Tools, Objects
POutputs
Layer Navigation
Map/Model ToggleChange to P2 (or else!)
Run Controller
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Stella InterfaceStella “Layers” – Mapping
PTextPOutput Tables
and GraphsP Input using
Slides andDials
PPicturesPQuick Time
Movies
Stella InterfaceStella “Layers” – Equation
P Initial Values ofStocks
PConnectorInformation
PDifferentialEquationsRepresentingthe TimeDependence ofStocks
Stella InterfaceMenu / Icons
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Stella InterfaceBuilding Blocks – Stocks
Value Undergoing a Change
Stella InterfaceBuilding Blocks – Flow
Change of Stock with Respect to Time
Stella InterfaceBuilding Blocks – Converter
ConstantsTransformation Equations
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Stella InterfaceBuilding Blocks – Connectors
Links Building Blocks
Stella InterfaceTools – Hand
General Purpose Editing ToolCursor Pointer
Stella InterfaceTools – Dynamite
Delete (No “Undo”)
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Stella InterfaceObjects – Graph Pad
Graphing Results
Stella InterfaceObjects – Table Pad
Spreadsheet-like Table of Results
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Solving of Differential EquationsTaylor Series
Solving of Differential EquationsEuler Method
First 2 terms
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Solving of Differential EquationsRunge-Kutta 2
First 3 Terms
Solving of Differential EquationsRunge-Kutta 4
First 5 Terms
Simple Model – Falling CalculatorSystem
y = heightyN = velocity = (acceleration)(time)yO = acceleration = g = 9.8 m s-2
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Simple Model – Falling CalculatorStella Model
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Simple Model – Falling CalculatorStella Model – Equation Layer
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dCdt
kC
C Ckt
o
= −
= +
2
1 1
Simple Model – Falling CalculatorStella Model – Accuracy
PAir FrictionPBounce
Chemical KineticsSure looked like a natural fit to me!!
Kinetics ModelDimerization of Cyclopentadiene2nd Order Diels-Alder Reaction
2
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Kinetics Model
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Michaelis-MentonLen Soltzberg (Simmons College)
E + S º (ES)
(ES) 6 P + E
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Oscillating ReactionsCriteria
PTwo or more coupled reactionsPAutocatalytic
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Oscillating ReactionsBriggs-Rauscher
PH2O2
PKIO3 and H2SO4
PHOOCCH2COOH, MnSO4, starch
~15 s for each cycle
Oscillating ReactionsBelousov-Zhabotinskii
P 18 reversible stepsP 21 different chemical species
BrO3- + HBrO2 + H3O+ 6 2 BrO2 + 2 H2O
2 BrO2 + 2 Ce3+ + 2 H3O+ 6 2 HBrO2 + 2 Ce4+ + 2 H2O
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A + Y 6 X + PX + Y 6 2PA + X 6 2X + 2Z2X 6 A + PB + Z 6 (f/2) Y
whereA = BrO3
-
X = HBrO2Z = Ce4+
P = HOBrB = organicY = Br -
Oregonator by Len Soltzberg (Simmons College)
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Other ApplicationsRod Schluter (Formerly of CofC)
Acid/Base Equilibrium
H+ + OH- 6 H2O
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D Tu
xe
dxD x( / )θ =−∫
313
3
( )
( )[ ]
U U RTD
C R Du
e
S R D e
A U RT e D
v u
u
u
− =
= −−
⎛⎝⎜
⎞⎠⎟
= − −⎡⎣⎢
⎤⎦⎥
− = − −
−
−
0
0
3
3 4 31
343
1
3 1
ln
ln
Other ApplicationsDebye Theory for Monatomic Crystals