From Macroscopic World to Microscopic World through Mazes of Process Graphs and from
Microscopic World to Mesoscopic World through Drunkards’ Paths
Computing in Chemical Engineering AwardLecture
ByL. T. Fan
Department of Chemical EngineeringKansas State University
Computing and Systems Technology (CAST) BanquetNovember 18, 2003
AIChE Annual MeetingSan Francisco, CA, November 16 – 21, 2003
G1
“From Macroscopic World to Microscopic World through Mazes of Process Graphs”
P1
Pre- Process-graph days: early 1950’s ~ late 1980’s• Heuristics• Dynamic programming• Structural parameter• Super-structure with structural parameters:
conventional search techniques; random search; adaptive random search
• Conventional graphs Successes: only with small problems Computational complication
• Local optima: nonlinearity• Combinatorial complexity: exponential
(2n – 1) or (3n – 1)
P2
Meeting of minds: 1989 ~ 90 • Fan & Friedler at the North American-German
Workshop on Chemical Engineering Mathematics and Computation, Göttingen, West Germany, July 18 – 23, 1989
• Fan’s struggle to overcome combinatorial complexity in process synthesis without approximation
• Friedler’s struggle to overcome the difficulty in establishing a formal framework for combinatorial process synthesis on the basis of process graphs
• 1-year stint by Friedler and Tarjan with Fan in 1990
P3
First Milestone: early 1990’s• Presentations of a series of papers at major
technical conferences• Review of the 1st major manuscript for
journal publication by late Professor Rippin and his daughter, a mathematician
• Publications of the first 2 refereed journal papers:
Friedler, F., K. Tarjan, Y. W. Huang, and L. T. Fan, Combinatorial Algorithms for Process Synthesis, Computers Chem. Eng., 16, S313-320 (1992).
Friedler, F., K. Tarjan, Y. W. Huang, and L. T. Fan, Graph- Theoretic Approach to Process Synthesis: Axioms and Theorems, Chem. Eng. Sci., 47, 1973-1988 (1992).
P4
Graphs• Natural Language or logical tool for
describing and representing networks• Examples of networks: gas or oil pipelines;
waterways or irrigation channels; process flowsheets; highways; railroads; telephone lines; family trees; social relationships; structures of any organizations, etc., etc., etc., …; some visible and some invisible
• Structures of graphs: nodes ;arcs , • Types of graphs: monopartite; bipartite
P5
Process graphs (P-graphs)• What?: unique bipartite graphs
P6
P-graphs of some operating units and their concomitant material streams:
(a) Materials A, B, and C, and operating unit ({A, B}, {C})(b) Material C, D, and E, and operating unit ({C}, {D, E})
A B
C
C
ED
(a) (b)
• Why?a. Syntactically and semantically richb. Unique representation possiblec. Special classes of graphs, e.g., call
graphs, social graphs and highway graphs, needed for different fields; according to Hayes (Graph Theory in Practice: Part I, American Scientist, January- February, 2000), “The next step is to develop a mathematical model of the structure, which typically takes the form of an algorithm for generating graphs with the same statistical properties. Such models of very large graphs will be the subject of…”
P7
• How?a. Recognition of the nature of the networks
involved: materials transfer and transformation networks
b. Identification of statistical properties of the graphs to represent the networks: perfect statistics representing mass conservation and stoichiometry
c. Axioms based on perfect statisticsd. Algorithms derived from the axioms:
MSG for maximal structure generation; SSG for solution structure generation; and ABB for accelerated branch-and-bound for generating the optimal and near-optimal solutions
P8
• Illustration
P9Reduction in the search space of (2n – 1)
P o ss ib le n e tw o r k s3 4 3 6 1 0 7
C o m b in a to r ia llyfe a s ib le n e tw o r k s
3 4 6 5
O p tim a ln e tw o r k
1
N e a r-o p tim a ln e tw o r k s
5
In p u tn = 3 5
M S G
S S G
A B B
9 9 .99 99 9%red u ction
in th e search sp ace
Second Milestone: mid 1990’s
• Favorable comments by Prof. Sargent in a report of Center for Process Systems Engineering of Imperial College
• Citation by Prof. Sargent in the Rippin memorial issue of Computers Chem. Eng. (Vol. 22, No. 1–2, 1998), “The task performance models also set conditions (both qualitative and quantitative)......, as considered by Friedler et al. (1992)...... . Armed with these rules and conditions we can devize an algorithm which sistematically generates all feasible state-task-networks”
P10
Third Milestone: mid 1990’s ~ late 1990’s• Realization for potential adaptation of P-graphs
for mesoscopic-level process synthesis and identification of molecular-level networks
• Publication of a series of papers on azeotropic distillation; for exampleFeng, G., L. T. Fan, F. Friedler, and P. A. Seib, Identifying
Operating Units for the Design and Synthesis of Azeotropic- Distillation Systems, Ind. Eng. Chem. Res., 39, 175-184 (2000).
• Publication of a series of papers on catalytic-reaction and metabolic pathways: combinatorial complexity, (3n – 1); for exampleFan, L. T., B. Bertok, and F. Friedler, Combinatorial Framework for
the Systematic Generation of Reaction Pathways, presented at the
AIChE Annual Meeting, Dallas, TX, U.S.A., October 31 – November 5, 1999, ISBN 0-8169-0805-2P11
Fourth Milestone: late 1990’s ~ early 2000
• Endorsement by Dr. Keller: Institute Lecture at the 1999 AIChE annual meeting; CEP Vol. 96, No. 1, “... the P-graph may be the fastest computationally, as well as the method most likely to find a truly optimal solution.”
• Accelerated pace of adoption of P-graphs by other researchers and citation in their presentations and publications including books, monographs and journal articles; none reviewed by us
P12
Fifth Milestone: early 2000• Selection by Prof. Timmerhaus as the method
of choice for algorithmic flowsheeting
• Inclusion of a 30-page section entitled ALGORITHMIC FLOWSHEET GENERATION in the 5-th edition of “Plant Design and Economics for Chemical Engineers” by Peters, Timmerhaus, and West
• Publication of a landmark paper by Brendel, Friedler and Fan, Computers Chem. Eng., Vol. 24, 2000: an additional proof for the rigorousness and superiority of the P-graph-based method for process synthesis
P13
Current status and future prospect of P-graphs• Downstream processing in biochemical
production• Identification of catalytic pathways• Metabolic flux analysis• Molecular design (conventional bipartite
graph)• Azeotropic-distillation system synthesis• Design of alternative synthetic routes; profit-
potential estimation• Further generalization, automation,
computational acceleration through parallel and grid computing
P14
Algorithmic Identification of Stoichiometrically Exact, Plausible Mechanisms of the Catalytic Ethylene Hydrogenation
Reactionby Fan, Shafie, Khaitan, More, Bertok, and Friedler
(AIChE Annual Meeting, Nov. 18, 2003) A graph- theoretic algorithmic
method Process graphs (P-graphs)
Axioms Feasible reaction pathways Combinatorially feasible
reaction networks Algorithms
RPIMSG PBT
Input Overall reaction: 1 Elementary reactions: 7 Combinatorial complexity:
(3n – 1) = 37 – 1 = 2,186
Overall reaction Σ:
C2H4 + H2 C2H6
Elementary reactions:proposed mechanism for two active sites
H2 + 21 2H1
H2 + 22 2H2
C2H4 + 21 1C2H41
1C2H41 + H1 1C2H5 + 21
1C2H41 + H2 1C2H5 + 1 + 2
1C2H5 + H1 C2H6 + 21
1C2H5 + H2 C2H6 + 1 + 2
P15
Results Feasible pathways:
independent; 8 combined acyclic; 17
Conventionally accepted mechanism for one active site:
Pathway 3 Computational
efficiency: less than a second with a PC (Intel Pentium III, 533 MHz, 128 MB RAM)
P16
P-graph representation of independent pathway 7
H2 + 21 2H1
H2 + 22 2H2
2C2H4 + 41 21C2H41
21C2H41 + 2H1 21C2H5 + 41
21C2H5 + 2H2 2C2H6 + 21 + 22
P17
Feasible and Optimal Flowsheets for Downstream Processing in Biochemical Production of Butanol, Ethanol and Acetone: Inclusion
of Pervaporation by Liu, Fan, Seib, Friedler, and Bertok(AIChE Annual Meeting, Nov. 20, 2003)
Comprehensive flowsheet with inclusion of pervaporation: P-graph
A graph- theoretic approach Process graphs (P-graphs)
Axioms Feasible reaction pathways Combinatorially feasible
reaction networks
Algorithms MSG SSG ABB
Input: 25 operating units
P18
S 5 5S 5 6
U 1
S 5 7
P 1
E 1
S 11
S 0 8
S 1
S 1 6
S 1 3
D 1
S 0 5
G 1
S 0 6 S 0 7
D 2
S 0 0
S 5 1 S 5 2
C 1
S 5 3
B 1 B 2
S 5 4
B 3B 4
A1 D3 A2
S06
D7
D8
S15
D5
D6
S31
D9
D10
S32
D11
D12
D13 D14
S34
S33
D15
D16
S36
S35
D17
D18
S38S37
D19
D20
S39 S40
D21
D22
S46S43
S44 S45
D25
D26
S50S47
S48 S49
D27
D28 D29
S01
S02
S03
S09
S19
S20
Comparison of the total costs of the 10-best flowsheets generated
4000
4500
5000
5500
6000
6500
7000
7500
8000
1 2 3 4 5 6 7 8 9 10
Rank
Cos
t (10
00 U
S$/
Yea
r)
4000
4500
5000
5500
6000
6500
7000
7500
8000
1 2 3 4 5 6 7 8 9 10
Rank
Cos
t (10
00 U
S$/
Yea
r)
4000
4500
5000
5500
6000
6500
7000
7500
8000
1 2 3 4 5 6 7 8 9 10
Rank
Cos
t (10
00 U
S$/
Yea
r)
4000
4500
5000
5500
6000
6500
7000
7500
8000
1 2 3 4 5 6 7 8 9 10
Rank
Cos
t (10
00 U
S$/
Yea
r)
Combinatorial complexity: (2n – 1) = 225 – 1
= 33.554 106
Results Optimal and
near-optimal flowsheets: 4 sets of 10 each for parametric study with respect to the cost of pervaporation
Conclusion Profound computational
efficiency: less than 5 s for each set with PC (Pentium 266 Mhz; 65 MB RAM; W95)
Novel paradigm for process design and development
Retrofitting vs new design
(a) Conventional operating units only
(b) Current best estimate
(c) 84% reduction (d) 97% reduction
P19
The optimal flowsheet with pervaporating and ultrafiltering units included
P -g rap h rep re sen tatio n
S 20 S 19
S 39S 09 S 40
20 -1 (D 2 1)
20 -2 (D 2 2)
S 00
S 56
26 (U 1 )
S 08
S 55
S 57
27 (P 1 )
C o nv en tion al rep resen ta tion
S 09
S 20
S 19S 40
S 39
A ce tone (A )A 7
B utano l(B )E th an o l(E )
B u tano l(B )E th an o l(E )
E th an o l(E )E 2
B u tano l(B )B 26
S 00A ceto ne (A )
B u tan o l(B )S o lid s(D )
E tha no l(E )W ate r(W )
A 11
B 27S 87
E 4W 17 73
S 56E tha no l(E )W ate r(W )B u tan o l(B )S o lid s (D )
E 1W 24 9B 1S 87
S 55A ceto ne (A )
B u tan o l(B )
E tha no l(E )W ate r(W )
A 11E 3
B 26W 15 24
U 1
S 57A ceto ne (A )E tha no l(E )W ate r(W )
A 4E 1W 89 9
P 1
S 08A ceto ne (A )
B u tan o l(B )E tha no l(E )
A 7
B 26E 2
D 21
D 22
P20
Synergistic identification of multiple flux distributions and multiple metabolic pathways: Application to the E. coli model
by Lee, Fan, Park, Lee, Shafie, Bertok, and Friedler(AIChE Annual meeting, Nov. 21, 2003)
• Model– Metabolites: 52– Reactions: 48
P21
• Algorithmic Method
– P-graph Representation of the E. coli Model: Input
• Glycolytic pathway• PPP• TCA
– Algorithm RPIMSG
– Algorithm PBT
P22
• Results– Normalized Multiple Flux Distributions for the Maximum
Ethanol Production• Four different flux distributions leading to the same external
state: the net reaction balance, GLCxt 2 ETHxt + 2 CO2xt• Pathway redundancy: cell robustness which is a unique feature
of complex systems• Computational efficacy: less than 1 second with a PC (Intel
Pentium IV, 1.8 GHz, 768 MB RAM); extension to the large-scale models (300 & 700 metabolic reactions)
P23
“From Microscopic World to Mesoscopic World through Drunkards’ Paths”
S1
Drunkards Paths, Random Walks and Stochastic Processes
• Stochastic processes: a rigorous branch of mathematics or mathematical statistics
Statistics Stochastic Processes
Statistics > Stochastic Processes; random behavior evolving with time or operation according to a certain mathematical property defined by a distribution of the random variable
S2
Awakening at Dawn: 1950’s• Residence-time distribution age
distribution of molecules Microscopic & discrete; Continuum, statistical or stochastic?
• Dancing particles, droplets and bubbles everywhere
Mesoscopic, finite & discrete; Continuum, statistical or stochastic?
• Stochastic processes: population of discrete entities evolving with time
Natural & logical language for the mean behavior of microscopic entities Natural & logical language for the mean & fluctuating behavior of mesoscopic entities
S3
Encounter with Markov: mid 1960’s ~ early 1970’s • Feller, Cox & Miller, Chiang, etc.• Frequent consultations & extensive
collaboration with statisticians• Markovian Property
S4
• Classification Markov chain (time discrete) Markov process (time continuous) Diffusion process (state and time continuous): Fokker-Planck
• Strictly linear processes: analytical solutions for moments of the evolving distribution of a random variable
n n 1 1 2 2 n 1 n 1
n n n 1 n 1
Pr[X(t ) x X(t ) x ,X(t ) x ,...,X(t ) x ]
Pr[X(t ) x X(t ) x ]
Encounter with Prigogine: early 1980’s • Nicolis, G., & I. Prigogine “Self-organization
in Non-Equilibrium Systems,” 1977• Non-linearity• Stochastic mechanics
Encounter with van Kampen: mid 1980’s • van Kampen, N. G., “Stochastic Processes in
Physics and Chemistry,” 1982• Master equation: probability balance
gain-loss equation• Non-linearity: system-size expansion
S5
Publications of series of papers, each ranging two to more than a dozen, on various subjects such as chemically reacting systems, solids mixing, grinding, fluidization, crystallization, filtration, biochemical processes, mass transport and residence-time distribution
Noteworthy articles• Fan, L. T., B. C. Shen, and S. T. Chou, “The
Surface‑Renewal Theory of Interphase Transport: A Stochastic Treatment,” Chem. Eng. Sci., 48, 3971‑3982 (1993).
• Fan, L. T., B. C. Shen, and S. T. Chou, “Stochastic Modeling of Transient Residence‑Time Distributions during Start‑Up,” Chem. Eng. Sci., 50, 211‑221 (1995).
S6
Current status and future prospect of stochastic analysis and modeling• Stochastic Modeling of Thermal Death
Kinetics of a Cell Population: Revisited• Stochastic Modeling and Simulation of the
Formation of Carbon Molecular Sieves by Carbon Deposition
• Nanoparticles formation• Tumor growth• Human neuron cells decay• Monographs on stochastic modeling of
particulate systems; chemical reacting systems; biochemical systems
S7
Stochastic Modeling of Thermal Death Kinetics of a Cell Population: Revisited by Fan, Argoti-Caicedo,
Chou, and Chen(Chemical Engineering Education, 37, 228-235)
• Microorganisms: Discrete and randomly behaving
• Stochastic modeling: Pure-death process
• Mean and higher moments: Variance, skewness, and kurtosis.
• Comparison with experimental data: Mean in accord with experimentally measured data
n n 1 n 1 n n
dp (t) p (t) p (t)
dt
S8
Electron microscopic image of S. aureus (From: http://www2.uol.com.br/cienciahoje/chdia/n468.htm)
Normalized mean, m, and normalized standard deviation, σ, as functions of the dimensionless time, , for the low-range of the number concentration of live cells
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0 2 4 6 8 10 12 14 16 18 20 44 46 48 50
Dimensionless time,
0.040
0.060
0.080
0.100
0.120
0.140
0.160
0.180
0.200
Experim ental data
T = 52 Co
T = 54 Co
T = 56 Co
n = 100
n = 1000
n = 100,0000
M ean
m ( ) ( ) ± /n 0
m(
) o
r
m
()
)/n
± (
0
S9
Stochastic Modeling and Simulation of the Formation of Carbon Molecular Sieves by Carbon Deposition by
Fan, Argoti, Walawender, and Chou(AIChE Annual Meeting, Nov. 20, 2003)
• CMS Formation: Complex and random
• Stochastic modeling: Pure-birth process
• Mean and higher moments: Variance, skewness, kurtosis, etc
• Comparison with experimental data: Mean in accord with experimentally measured data
Side view of the progression of CMS formation:
Carbon source ; Fine carbon particle ; Carbon packet
n n 1 n 1 n n
dp (t) p (t) p (t)
dt
S10
Dimensionless mean, m, and dimensionless standard deviation, σ, for
the pore-narrowing as functions of the dimensionless time, S11
Concluding Remarks
Obviously borrowing heavily from General Douglas MacArthur, “Old professors never die;
they just (asymptotically) fade away.”
G2
Acknowledgement
Many thanks to:
All my current and former students, assistants, associates, collaborators, and teachers
All my current and former colleagues and staff in the
department
All organizations and agencies in and out of the University which supported my research
All attendants who suffered through my “random”
talks
All my family members, especially my wife, Eva, who has accompanied me for 45 years in the journey
through the “Mazes of Process Graphs” and “Drunkards’ Paths”
G3
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