The document is a personal survey by Eric S Fraga of optimisation problems and methods used in chemical engineering over the past 20 years. It covers topics such as process synthesis, heat integration, carbon capture, and power generation. For each topic, it discusses mathematical models used to represent the systems, optimisation approaches applied, and examples of applications and results. The presentation aims to provide an overview of the wide range of problems addressed and methods developed and implemented, from off-the-shelf software to custom programs for specific applications.

1. #+TITLE: Optimisation problems and
methods in Chemical
Engineering: a personal survey
#+AUTHOR: Eric S Fraga
Centre for Process Systems Engineering
Department of Chemical Engineering
#+INSTITUTE: University College London (UCL)
#+DATE: 6 June 2014
One Day Workshop on Applied and Numerical
Mathematics
University of Greenwich
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2. Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
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3. What is Chemical Engineering?
Chemical Engineering about changing raw materials into
useful products.
Other engineering disciplines deal with things;
Chemical Engineering deals with stuff.
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4. Modelling
Mathematical models of
systems that work with stuff
are complex:
- nonlinear and nonsmooth,
e.g.
∆Hk = w
Qk
CHW
β
Lk
m
d −γ
m ykm
- combinatorial
- significant
uncertainties in
parameters and models
350
375
400
425
6 8 10
Annualizedcost(10
3
$/y)
Operating pressure, unit 4 (atm)
f(X)
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5. Outline
Presentation will be personal survey of problems and
optimisation methods used over the past 20 years,
ranging from off-the-shelf optimisation software through
to custom programs for specific applications.
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6. Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
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7. Process design
Objective is to determine
unit operations and their
interconnections so as to
achieve a specific task,
usually defined by a product
specification.
?
1
2
3
4
000000
000000000000
000000000
000
111111
111111111111
111111111
111
F
F
F
P
P
P
P
P
P
H SO
2 4
Reactor D1
Fluorspar
M
D2
Vapour effluent
HF
Excess
Makeup
M
HF +
A1
Makeup
Solid
Effluent
H SO
2
H SO
2
4
4
H SO
2 4
A
B
C
B
−−
C
−−
B
A
A
B
C
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8. Discrete Mathematical Approach
- Convert MINLP to discrete
problem: component flows,
stream enthalpies, unit
operations.
⇒ Search a large, but finite,
graph.
- Combine dynamic programming
with implicit enumeration:
f (s ) = min
u
cu (s ) +
np
i =1
f (pi )
ABD
AB/D A → C
AB
A→C
BC
BCD
BC/D
B/C
A B C D
ESF & K I M McKinnon (2004), Ind Eng Chem Res 43(1):144-160.
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9. Example model
A short-cut distillation column model:
Nmin =
log xd
1−xd
1−xb
xb
log αavg
Rmin =
i
αi xD ,i
αi − φ
− 1
where 1 − q =
i
αi zF ,i
αi − φ
N − Nmin
N + 1
= 0.75 1 −
R − Rmin
R + 1
0.5668
to estimate N stages and R reflux ratio necessary for
costing, and where α are thermophysical properties.
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10. N -best solutions
ESF (1996), in State of the Art in Global Optimisation, Kluwer, 627-651.
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11. Multiple objectives
-1.8
-1.5
-1.2
2e+07
4e+07
6e+07
0
2000
4000
Annual cost (M$)
SPI (m2/year)
CTWM(kgwater/year)
10 best flowsheets
according to cost
Minimum cost
flowsheet
#2 & #3
flowsheets
by cost
#1 SPI
B
A
#1 CTWM
M A Steffens, ESF & I D L Bogle (1999), Comp Chem Eng 23(10):1455-1467.
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12. Uncertainty
A R F O’Grady, ESF, I D L Bogle (2001), Chemical Papers 55:376-381.
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13. Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
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14. Process heat integration
Match
appropriate
heating and
cooling needs to
reduce costs and
environmental
impact.
P=8.1,R=11.0,S=79
P=5.6,R=0.9,S=24
P=7.8,R=5.9,S=62
P=¼.3,R=2.3,S=21
H1
H4
C4
C3C2
H2 H3
C1
Unit 3
Unit 1
Unit 2
Unit 4
n−Pentane
iso−Pentane
C3 C1 C2
Feed
Propaneiso−Butane
H4
H4
n−Butane
H4
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15. Simultaneous design
- Integrated process design is a nonlinear &
combinatorial problem:
minx ,y
f (x , y ) = g (x ) + h (x , y )
g (x ) base process design
h (x , y ) process heat integration
- Solve with an embedded hybrid method:
min
x
ˆf (x ) = g (x ) + min
y
h (x , y )
with gradient or direct search for x and GA for y .
ESF & A ˘Zilinskas (2003), Adv Eng Software 34:73-86.
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16. Model of heat exchanger
Cost model
C = α + βA γ
Area of exchanger
A =
Q
U ∆TLMTD
Driving force for heat exchange
∆TLMTD =
∆Tin − ∆Tout
log ∆Tin − log ∆Tout
and the various temperatures are the result of
thermophysical property predictions, functions of T and
P design variables.
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17. Outer methods for process structure
Key Code Method
PG gradproj project gradients
NP projbfgs quasi-Newton projected
ND method by Shor
NM fmins Nelder & Mead simplex
HJ hooke Hooke & Jeeves
IF imfil Implicit filtering
CS Coordinate search
using penalty functions for methods designed for
unconstrained optimisation (ND, NM, HJ, IF).
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18. Genetic algorithm for heat exchanges
Two implementations:
GA1 nl × nc ordered pairs (j , r ) where nl is number
of levels, nc number of cold streams,
j ∈ [0, nh ] index for hot stream and r fraction
of available heat to exchange.
GA2 vector of ne values i ∈ [0, nc × nh ] where ne
represents the number of exchanges to allow
and i the actual exchange to consider.
GA1 implements a larger and more comprehensive search
space; GA2 however is constant in length.
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19. Impact of embedded stochastic method
350
375
400
425
6 8 10
Annualizedcost(10
3
$/y)
Operating pressure, unit 4 (atm)
f(X,Y*
)
f(X)
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20. Zoomed view
342
344
7.6 8 8.4
Annualizedcost(10
3
$/y)
Operating pressure, unit 4 (atm)
f(X,Y*
)
f(X)
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21. Some performance results
Alg best ave std nf ninf time
(106
$) (106
$) (106
$) (s)
NM 8.40 9.35 0.813 906 13 1 472
HJ 8.39 8.39 0.001 810 78 1 594
IF 8.61 10.15 1.762 554 42 1 170
CS 8.81 10.06 1.584 170 44 244
GA-2L 8.53 8.75 0.192 814 58 1 703
GA-1L 8.39 8.51 0.122 107 399 2 239 1 239
- GA-1L solves combined problem, f (x , y ), directly for
benchmarking.
- GA-2L uses a GA for outer method.
- Hooke & Jeeves direct search algorithm is most
consistent and achieves best solution.
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22. Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
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23. Efficient carbon capture
Identification
of solvent or
nanoporous
material
Detailed
process modelling
Try
again
Process
integration
Molecular modelling
and/or experiments
Done
Yes
Yes
Yes
No
No
No
Evaluation
- reduce efficiency loss
due to carbon capture.
- combined materials and
process design.
- evaluation based on
experiments, detailed
modelling and process
simulation and
optimisation.
EPSRC EP/G062129/1
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24. Modelling I
Component mass balances (axial dispersed plug flow
model):
dc i
dt
+
1 − b
b
d¯Qi
dt
+
∂(uci )
∂z
+
∂Ji
∂z
= 0
d¯Qi
dt
= p
dc m
i
dt
+ (1 − p )
d¯qi
dt
= k p
i
Ap
Vp
(ci − c m
i )
Energy balance for the adsorbate in the gas phase:
b
dˆUf
dt
= −(1 − b )
∂ ˆUp
∂t
− b
∂(ˆHf u )
∂z
−
∂JT
∂z
−
Nc
i =1
∂(Ji ˆHi )
∂z
− hw
Ac
Vc
(Tf − Tw )
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25. Modelling II
Energy balance for the adsorbate in the solid phase:
∂ ˆUp
∂t
= p
dˆUp ,f
dt
+ (1 − p )
dˆUp ,s
dt
= hp
Ap
Vp
(Tf − Tp )
Energy balance in the bed wall:
ρw Cp ,w
∂Tw
∂t
= −hw
Ac
Vw
(Tw − Tf ) − U αwl (Tw − T∞)
and so on.
As simulation must reach cyclic steady state,
⇒ computational effort is significant.
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26. Behaviour of objective function
Objectivefunctionvalue
Along a line in design space
⇒ motivates use of surrogate modelling (response
surface modelling, meta-modelling, ...).
G Fiandaca, ESF & S Brandani (2009), Engineering Optimization 41(9):833-854.
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27. Surrogate model
- a fast approximation of model’s response
y (x ) : Rp
→ R where X ⊂ Rp
is the space with p
design variables.
- suitable for black box optimisation models as the
surrogate model is non-intrusive.
- based on training data: a set of known design
points.
Most surrogates have form
ˆy (x) =
q
k =1
βk hk (x) + (x)
with regressors hi (·) and a residual random process, (·).
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28. Kriging
A statistical interpolating approach used for
approximating deterministic models.
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
y(x)
x
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29. Optimisation
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30. Optimiser
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1
Purity(%)
λ
We use evolutionary stochastic methods to cater for
multi-modality of objective function.
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31. Case study: 6 step, 2 bed PSA
Bed
1
Bed
2
Vaccum
Tank
V1
V2
V4
V5
V7
V3 V6
Vent tank
Vent
Feed tank
Feed
BH
BH
BH
BH
- 6 design variables.
- 3 objective functions:
recovery, purity and power
(but will illustrate 2).
- computational effort large:
30-60 minutes per objective
function evaluation.
J Beck, D Friedrich, S Brandani & ESF (2012),
Proc 22nd ESCAPE, Elsevier, 1217-1221
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32. Pareto front: n = 64
0
20
40
60
80
100
0 20 40 60 80 100
Purity(%)
Recovery (%)
NSGA-II
SbNSGA-II
SbNSGA-II ALM
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33. Pareto front: n = 96
0
20
40
60
80
100
0 20 40 60 80 100
Purity(%)
Recovery (%)
NSGA-II
SbNSGA-II
SbNSGA-II ALM
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34. Pareto front: n = 176
0
20
40
60
80
100
0 20 40 60 80 100
Purity(%)
Recovery (%)
NSGA-II
SbNSGA-II
SbNSGA-II ALM
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35. Pareto front: n = 256
0
20
40
60
80
100
0 20 40 60 80 100
Purity(%)
Recovery (%)
NSGA-II
SbNSGA-II
SbNSGA-II ALM
–:–- talk.tex 83% 35/42 [Carbon capture] --------------------------
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36. Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
–:–- talk.tex 85% 36/42 [Power generation] ------------------------
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37. The problem
Determine power schedule
for minimum fuel cost for
set of online thermal
units subject to a number
of issues:
- prohibited operating
zones
- transmission losses
- valve point loadings
using mathematical
programming toolbox.
min
Pi
z =
i
Fi (Pi )
4000
6000
8000
10000
12000
150 200 250 300 350 400 450
F(P1)
P1 [MW]
Fi (Pi ) = ai P 2
i +bi Pi +ci +|ei sin (fi (Pi ,min − Pi ))|
L Yang, ESF & L G Papageorgiou (2013), Elec Power Sys Res 95:302-308.
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38. Results
Mean Cost Best Cost Method Year
CSOMA 121,415.05 121,414.70 Cultural algorithm 2010
FAPSO-VDE 121,412.61 121,412.56 Particle swarm 2011
DE 121,422.72 121,416.29 Differential Evolution 2008
SOMA 121,449.88 121,418.79 Self-org migration 2000
EP-SQP 122,379.63 122,323.97 Genetic Algorithm 2008
CDEMD 121,526.73 121,423.40 Differential Evolution 2009
BBO 121,512.06 121,418.27 Biogeography 2010
HGA 121,784.04 121,418.27 Genetic Algorithm 2008
HDE 122,304.30 121,698.51 Differential Evolution 2009
MTS 121,798.51 121,532.10 Tabu search 2011
UHGA 121,602.81 121,424.48 Genetic Algorithm 2008
MDE 121,418.44 121,414.79 Differential Evolution 2010
VLEMIQP 121,412.54 121,412.54 Mathematical Prog 2013
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39. Caveat: modelling
Chemical
Engineering
problems are
based on models
of the
transformation
of stuff. These
models are
difficult to
obtain and
sometimes the
results are not
correct.
0
500
1000
1500
2000
2500
3000
20 60 100
0
F(P7)
dF/dP
P7 [MW]
valve effects
no effects
dF/dP
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40. Restricted search
Problem Demand DVLMILP Excess DVLMILP Best Gap
output output cost known
(MW) (MW) (%) cost (%)
13 units 1800 1802 0.11 17964 17960 0.02
13 units 2520 2525 0.20 24174 24164 0.04
40 units 10500 10501 0.01 121986 121413 0.47
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41. Topic
* Introduction
* Process synthesis
* Heat integration
* Carbon capture
* Power generation
* Conclusions
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42. Summary
market
objective
process
structure
bespoke
software
optimised
operation
direct
search
process
scheduling
mathematical
programming
heat
integration
DS
+ GA
dynamic
operation
MOGA +
surrogate
Thanks to Dr Joakim Beck,
Professor David Bogle, Dr Rob
O’Grady, Professor Lazaros
Papageorgiou, Dr Mark Steffens
and Dr Lingjiang Yang at UCL;
Professor Stefano Brandani
(Edinburgh), Dr Daniel Friedrich
(Edinburgh), Professor Ken
McKinnon (Edinburgh) and
Professor Antanas ˘Zilinskas
(Lithuania).
http://www.ucl.ac.uk/~ucecesf/
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