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