Synthesis and Optimization of Separation Sequences

1
Synthesis and Optimization of Separation Sequences

• Libin Zhang
• and
• Andreas A. Linninger
• Laboratory for Product and Process Design,
• Department of Chemical Engineering, University of
  Illinois,
• Chicago, IL 60607, U.S.A.

2
Motivation
P1
PURITY SPECIFICATIONS XA 0.99, Xb 0.001,.
X2
P2
FEED Components A,B, C,D,E Known Compositions
X1
P3
Maximum profit and/or Minimum environmental
impact
X3
P4
X4
structure and operating conditions
P5
Mixture No. of Column No. of variables
3 2 (30 Trays) 560
5 4 (30 Trays) 1560

• Which SEPARATION configuration is optimal????

3
Outline

• Long-term objectives Computational synthesis of
  separation networks
• Algorithmic Approach to determine feasibility of
  separation specification
• Methodology - Minimum Bubble Point Distance
  Algorithm (MIDI)
• Assessing Feasibility of given Separation Tasks
• Feasible Range of Column Operations Minimum
  Maximum Refluxes
• Toward Computed Aided Distillation Synthesis
• Feasibility Test Genetic algorithm (Stochastic
  search)
• Temperature collocation on finite element
• Reduced search space, but rigorous models
  (non-ideal separations)
• Genetic algorithm and feasibility test
• Infeasible path algorithm

4
Quick and Reliable Feasibility Test
PRODUCT A PURITY SPECIFICATIONS XA 0.99, Xb
0.001,.
SIMPLE DISTILLATION COLUMN ?
FEED Components A,B, C Known Compositions
PRODUCT B
SEPARATION TASK FEASIBLE????
5
Solution Approaches
1. SIMULATION APPROACH
2. DESIGN APPROACH

• Performance Calculation
• Flowsheet Simulator
• Aspen Plus, HYSYS, Pro/II

• Design Calculation
• Given Feed 4 D.O.F
• SPECIFICATION FEASIBLE??

???
Given XA,D XB,D
DISTILLATE
F
Fixed Trays
??
Reflux
Given Feed
Given Feed
Given XA,B
???
BOTTOMS

• Trial and Error Approach
• Does not Assess Feasibility

• Direct Feasibility assessment
• Evaluate Column Profiles

6
Design Approach - Underwoods Equations
NUMERICAL PROBLEM ?? OR INFEASIBLE
SPECIFICATION??

• Underwoods Equations
• Highly Non-Linear
• Difficult to Converge

Rectifying Equations
(1)
Stripping Equations
(2)
Profile Intersection
(3)
7
A Robust Column Design Algorithm

• Main ideas
• 1. Model Column Profile
• Temperature (Not Trays)
• Continuous Profile Equations (Doherty, 1985)
• 2. Feasibility Check
• Minimum Bubble Point Distance between Rectifying
  Stripping Profiles
• 3. Solution Strategy
• Finite Element Collocation on Orthogonal
  Polynomials
• Supporting Concepts
• Pinch Point
• Attainable Temperature Window
• Bubble Point Distance

8
Reachable Temperatures - Pinch Points

• Fixed points in the composition profiles
• Pinch, saddle and unstable points
• Newton-Horner (Deflation),
• Continuation Method,
• Bounded Newton-Raphson algorithm

d distillation ? unstable point ? saddle
point ? pinch point
(OL)
(EQ)
At the pinch point Let
in equilibrium with xi,p
Pinch equation
?
?
?
Residual Error
d
(PL)
Temperature, ºF
9
Attainable Temperature Window

• Difference between boiling points and the stable
  points of rectifying and stripping section

10
Bubble Point Distance

• Bubble Point Distance
• The Euclidean difference of two points on the
  rectifying and stripping profile whose BP is
  equal to T
• Feasibility test ? p1, p2 p1?rectifying p2
  ?stripping ?(BPD(p1, p2))??

Feasible Min d 0
Infeasible d gt 0
T
d
11
2. Minimum Bubble Point Distance(MIDI)
Dimensionless Temperature
s.t.
12
Modeling Column ProfileContinuous Differential
Equation (Doherty, 1985)

• Continuous differential equation for evolution of
  column profile (Doherty,1985)
• Taylor expansion truncated after first term

Continuous column profile with independent
variable h
Stripping Section
Eq. (1)
Rectifying Section
Eq. (2)
13
Temperature Collocation of Column Profiles

• Temperature is monotonically increasing in
  distillation column (normally)
• Temperature is bounded from Distillate and
  Bottom temperature to the pinch points

Implicit Differentiation
Eq. (3)
Stripping profile
Eq. (4)
Rectifying profile
Eq. (5)
14
3. Temperature Finite Element Collocation

• Globally collocate entire profile between two
  temperatures
• Using 3-5 finite elements and 2-3 nodes

15
Element Placement

• Saddle temperature place an element boundary at
  the composition profile
• The saddle coincide with maximum curvature of the
  intermediate species

16
Column Profile Maps
Mark Peters, University of the Witwatersrand,
South Africa and Lei Huan
17
Derivation of CPMs
Column Section
Infinite reflux case (RD ? 8)
Difference Point Equation (DPE)
reflux ratio
difference point
net molar flow
Entire RCM
18
Derivation of CPMs
Finite reflux case
XD 0.9, 0.05, 0.05
RD 9
CPM is a simple transform of RCM
XD is in the MBT (Region 1)
19
What happens when XD is in the other Regions?
20
Pinch Point Loci
How do the nodes (pinch points) move as RD
changes for a set XD ?
21
Moving Triangles
A novel concept and design tool
22
Personalize Your Column...
23
Global Terrain Method
Global Terrain Method (example)

• Equations
• Feasible region
• Starting point
• (1.1, 2.0)

3D space of case 1
24
Global Terrain Method
Global Terrain Method

• Basic concept of Global Terrain Method (Lucia and
  Feng, 2002)
• A method to find all physically meaningful
  solutions and singular points for a given (non)
  linear system of equations (F0)
• Based on intelligent movement along the valleys
  and ridges of the least-squares function of the
  system (FTF)
• The task tracing out lines that connect the
  stationary points of FTF.

• Mathematical background
• Valleys and ridges in the terrain of FTF could be
  represented as the solutions (V) to

V opt gTg such that FTF L, for all L ? L
F a vector function, g 2JTF, J Jacobian
matrix, L the level-set of all contours
25
Global Terrain Method
Global Terrain Method

• Applying KKT conditions to this optimization
  problem we get the following
• Eigen value problem

Hi The Hessian for the i th function

• Thus solutions or stationary points are obtained
  as solutions to an eigen-value problem where the
  Eigen values are identical to the KKT multipliers

• Initial movement
• It can be calculated from M or H using Lanzcos or
  some other eigenvalue-eigenvector technique
  (Sridhar and Lucia, 2001)
• Direction
• Downhill Eigendirection of negative Eigenvalue
• Uphill Eigendirection of positive Eigenvalue

26
Application 1 Constant Alpha Mixtures

• Compute rectifying profile and stripping profile
  with pseudo-temperature respectively.
• Pseudo Temperature
• Then search the minimum distance between
  profiles.

ATW is not empty But Specification is
infeasible MIDI 0.295
B
Feasible MIDI ? 0
P1
Temperature
P2
D
Xd1 0.95xd20.049,xb10.05R 1.0
Xd1 0.95xd20.049,xb10.05R2.5
27
Application 2 Ideal Mixtures
Feasibility test works for both sloppy and sharp
split in ideal mixture
28
Application 3 Quaternary Mixtures
Constant alpha mixture
Ideal mixture
Non-ideal mixture
29
Application 4 Feasible Regions
Column Specification Search Space (10,000
possibilities)
Feasible region Constant alpha
39.4s
Feasible region Ideal Mixture
Feasible region Non-Ideal Mixture
x2
468.8s
1017.5s
x1

30
Synthesis of Separation Sequence
31
Challenges

• Problem Size of Column Sequences
• Large number of state variables (compositions,
  temperatures,)
• Highly non-linear relationships
• Vapor-liquid equilibrium model
• Local convergence
• Search for Structural and Parametric Design
  Variables
• Generate structural alternatives
• Find optimal parameters without getting trapped
  in local minima
• Converge to global solution in reasonable time
• Solution Approach
• Feasibility test and genetic algorithm
• Infeasible path algorithm
• Reduced search space and minimum design variable
  set
• Rigorous models

32
Temperature Collocation Algorithm(Zhang and
Linninger, IECR 2004)
Coordinate Transformation of Column Profiles into
DAE Orthogonal Collocation on Finite Element
Bubble points independent variable BPD (T)
Rigorous Feasibility Criterion min BPD 0
Infeasible Min bpd gt 0
Feasible Min bpd 0
TB
T
TA
x2
dB
dA
bpd
x1
33
Temperature Collocation Algorithm-Results(Zhang
and Linninger, 2004)

• Column Profiles DAE
• 10,000 specifications in 39.4 CPUs
• Robust Convergence for feasible and infeasible
  specification

10,000 random specifications
39.4 s for finding all feasible specs
Azeotropes/ non-ideal mix.
s.t.
34
Temperature Collocation-High Accuracy

• Size reduction 1 Order of Magnitude
• Hysys600 equation TC-OCFE50 equations
• High Fidelity Results
• Small differences attributable continuous vs,
  tray-by-tray (1st order appr.)
• V-L Equilibrium implementation

35
Reduced Search Space
Minimum set of design Variables
Each individual different designs (sequence
operating conditions)
MASTER GA
Genetic Algorithm
K individuals
D A BC
Feasibility Test
Population
State variables dramatically reduced by
temperature collocation
Feasibility test
(AB)(CD)
Feasibility test High performance Inverse problem
36
Problem Representation - Column Sequencing
Chromosome

• Only products ordered by relative volatility
• Mass flowrate in product of each specie
• Structure Integer string

Product I
Product II
Product III
Reflux
Structure Encoding
Operating Conditions
Product Specs
37
Crossover Example
2 Patents -gt Offspring with parameter and
structure variation
Mathematical Formulation
Operational Parameters Crossover
Structural Crossover
(2)
(4)
F1
After Crossover, mass balance is still valid
(1)
(3)
38
Structural and Parametrical Mutation
(2)
Operational parameters
(4)
F1
(1)
(3)
Integer parameters
P1
P2
F1
F2
P3
B1
F3
B2
B3
P4
39
Fitness and Convergence

12
10
Mutation crossover
8
Cost function

Op. Cost Cap. Cost Penalty
6
4
2
0
5
10
15
20
25
30
Generation

Crossover without mutation
Mutation without crossover
Cost function

Generation
40
Initial Population of Ternary Mixture

• Different initial population method
• Given the composition of all streams
• All candidate structure
• Only given the composition of products

Patterned initial guesses
All infeasible designs
Random initial guesses
41
Infeasible Path Approach

• Feasible Individual increase at beginning
  evolution
• Trend to stabilize at the end
• Optimal Sequences even from all infeasible
  initial population

No of Feasible sequences
No of Feasible sequences
Generation
Generation
Only infeasible designs initially
42
Case study I Ideal Ternary Mixture
Generation Evolution
Min. Cost 2.56 X103
Min. Cost 3.16 X103
43
Case study II Azeotropic Mixture
D1(Acetone)
1.0
D2(Chloroform)
D2
F1
0.8
F2
II
Azeotrope
0.6
B1
I
0.4
B2(Benzene)
0.2
B1
Generation evolution
D1
F1
B2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Penetrate the curved boundary
44
Case study III Entrainer Azeotropic Separation
D2
D1(Ethanol)
FUpper
D2(Water)
F
FLower
F2
F
B2
FLower
B3(EG)
B1
FUpper
D1
Flow sheet for separation of water and ethanol
with entrainer
Break azeotropic with entrainer
45
Case study III Entrainer Azeotropic Separation
Suboptimal solution
Best solution
Upper Feed Low Feed Distillate Bottom
Ethanol 1.0E-6 0.8564 0.99998 2.3E-6
Water 1.0E-5 0.1436 2.0E-5 0.9281
EG 0.99999 1.0E-6 1.0E-6 0.0719
46
Case study IV Quaternary Mixture
(2)
Ethanol
P2
Optimal Cost 5.98X103
P4
F1
P1
P3
Configuration Column sequence Min Cost
ABCD 5.98X103
ADBC 6.52X103
(AB)(CD) 6.61X103
DABC 7.72X103
DCAB 8.40X103
47
Case study V Five Component Mixture
48
Niche Evolutionary methods

• Niche methods work on the principles of genetics
  and natural selection.
• They work on a population of possible solutions,
  while other gradient based methods use a single
  solution in their iterations.
• They are probabilistic (stochastic), not
  deterministic.
• They are capable of detecting multiple optima

Flow chart of a continuous GA
49
Niche Evolutionary Methods

• Genetic algorithms for problem with multiple
  extrema (multi-modal)
• Use the concept of fitness sharing
• Restricts the number of individuals within a
  given niche by sharing their fitness, so as to
  allocate individuals to niches in proportion to
  the niche fitness

niche radius
No fitness sharing
Fitness sharing
50
Niche Evolutionary Methods (example)

• Spots 4 maxima and 1 saddle point

51
Conclusion

• Minimum bubble point distance algorithm to test
  feasibility or infeasibility of a design
  specification.
• Profile equation with temperature
• Finite element collocation method is used to
  instead the expensive tray-by-tray model.
• Applicability
• sloppy split and sharp split
• constant relative volatility, ideal, non-ideal
  mixtures
• multi-component mixtures
• Reduced hybrid Algorithm is robust and reliable
• Solution can be obtained from initial population
  without any feasible point
• Best sequence and operating conditions can be
  obtained at same time
• Rigorous distillation model -gt Necessary for
  non-ideal and azeotropic mixtures
• Reduced Space Genetic Algorithm
• Specialized chromosome mass consistency.
• Feasibility Test massive problem size reduction
  of state variables

52
Future Work

• Complex Columns

53
Future Work

• Heat Integration

54
Future Work

• Uncertainty

55

• Thank you!