T. Gundersen

T. Gundersen MP 01

Process Integration Methods

HierarchicalAnalysis

HeuristicMethods

KnowledgeBased Systems

OptimizationMethods

ThermodynamicMethods

Pinch AnalysisExergy Analysis

Stochastic MethodsMathematical Programming

Rules of ThumbExpert Systems qualitative

quantitative

interactiveautomatic

Process, Energy and System

Optimization MethodsForward

T. Gundersen MP 02

Limitations in Pinch Analysis & PDM

• A lot of “heuristics”, not very rigorous (N – 1) rule for minimum number of units Bath formula for minimum total area

• Composite Curves cannot handle Forbidden matches between streams Limitations in for example distillation

• Pinch Design Method is Sequential Targeting before Design before Optimization One match at a time, one loop at a time, etc.

• Time consuming but gives “good” designs


Optimization Methods

T. Gundersen MP 03

What is Mathematical Programming?

• Numerical Optimization Techniques• Can handle various Design Problems

Discrete Decisions related to Equipment Continuous Decisions related to Operation

• Process Constraints can easily be included Material and Energy Balances, Specifications Equality and Inequality Constraints

• Can handle multivariable Trade-offs• Framework for Automatic Design

“wouldn’t it be nice to have?”



T. Gundersen MP 04

A small Linear Programming (LP) Problem



1 2

2 1

1 2

1

2

min ( ) 2subject to: 2 (a) 8 (b) 2 (c) 1 (d)

f x x

x xx x

xx

x Solve the Objective Function andConstraints (a) and (b) as Equations

with respect to variable x2

2 1

2 1

2 1

Objective Function: 2Constraint (a): 2Constraint (b): 8

x x fx xx x

The LP Problem can be solved by the well-known and heavilyapplied Simplex Method, but it can also be solved graphically

f=0

T. Gundersen MP 05

Graphical Solution for small LP Problem



0 1 2 3 4 5 6 7 80

7

6

5

4

3

2

1

8

x2

x1

1 2 8x x

2 1 2x x f=12f=4 f=8

1 2x

2 1x

Optimum:at Vertex

Algorithm:Simplex

Solution:x1=2 , x2=4

Objective:f = 0

2 12x x f

T. Gundersen MP 06

Mathematical Programming & Superstructure

Ref.: Papoulias & GrossmannComput. Chem. Engng, 1983



T. Gundersen MP 07

Mathematical Programming

min f(x,y)s.t. g(x,y) ≤ 0 h(x) = 0

x ε Rn y ε <0,1>m

General MINLP:

f, g, h linear => MILP (or LP)

dim(y) = 0 => NLP (or LP)

Start

MILPmaster

NLPsub-problem

End

LB > UB

Branch& Bound

ReducedGradient



T. Gundersen MP 08

Problems with Mathematical Programming

Non-Linear Part

Local Optima

y1

y3

y2

1 0

1 0 1 0

Binary Part

Combinatorial Explosion



T. Gundersen MP 09

Stream Ts Tt mCp ΔH°C °C kW/°C kW

H1 180 80 1.0 100H2 130 40 2.0 180C1 30 120 1.8 162C2 60 100 4.0 160

ST 280 280 (var)CW 15 20 (var)

WS-4Forbidden

Matches

Specification:

ΔTmin = 10°C

Q: What is the effect if H2 and C1 are not allowed to exchange heat?Find QH,min , QC,min and the Heat Exchanger Network with andwithout this forbidden match. Discuss the Degrees of Freedom.



T. Gundersen MP 10

Pinch70°

C2100° 60°

C1120° 30°

H2130° 40°

H1180°

60°

3

3

Hb

1

1

2

2

90°

100 kW

40 kW 120 kW

Cb

6 kW

54 kW

60°

70°

80°

43°

mCp(kW/°C)

1.0

2.0

1.8

4.0

MER Design without Constraints

Ha

8 kW

115.6°

U = 6



T. Gundersen MP 11

“Extended”Heat Cascade



40°C 30°C

ST

C2

C1

H2

H1QH1,1=50

180°C 170°C

130°C 120°C

70°C 60°CQH2,2=120

QC1,3=54

QH

RST,1

QC

QC1,2=108

CW

RH1,1

RH2,2

QH1,2=50

RH1,2

RST,2QH2,3=60

QC2,2=160

2

3

1

T. Gundersen MP 12


QP = QPH = 54 kW



40°C 30°C

ST

C2

C1

H2

H150

50

180°C 170°C

130°C 120°C

70°C 60°C120

40

54

102

102

QC

60

48

CW

50

54

120 60

60

T. Gundersen MP 13

Pinch70°

C2100° 60°

C1120° 30°

H2130° 40°

H1180°

60°

3

3

Hb

1

1

2

2

90°

60 kW 54 kW

120 kW

Cb

60 kW

40 kW

60°

70°

80°

mCp(kW/°C)

1.0

2.0

1.8

4.0

Design with Constraints

Ha

48 kW

140°

93.3°

QP = QPH = 54 kW

U = 6



T. Gundersen MP 14

40°C 30°C

ST

C2

C1

H2

H150

50

180°C 170°C

130°C 120°C

70°C 60°C120

40

54

102

102

QC

60

48+x

CW

50

54-x

120 60-x

60

0+x


QP = QPP = 54 kW

Choice: x = 54 kW



T. Gundersen MP 15

Pinch70°

C2100° 60°

C1120° 30°

H2130° 40°

H1180°

60°

3

3

1

1

2

2

90°

6+54 kW

120 kW

Cb

60 kW

40 kW

70°

80°

mCp(kW/°C)

1.0

2.0

1.8

4.0


Ha

102 kW

140°

63.3°

QP = QPP = 54 kW

U = 5



QP = QPP + QPH

= 40 + 14 kW

T. Gundersen MP 16

40°C 30°C

ST

C2

C1

H2

H150

50

180°C 170°C

130°C 120°C

70°C 60°C120

40-y

54

102

102

QC

60

48

CW

50

54-y

120 60

60

0+y

0+y


Choice: y = 40 kW



T. Gundersen MP 17

Pinch70°

C2100° 60°

C1120° 30°

H2130° 40°

H1180°

60°

Hc

1

1

2

2

90°

60+40 kW 14 kW

120 kW

Cb

60 kW

70°

80°

mCp(kW/°C)

1.0

2.0

1.8

4.0


Ha

48 kW

93.3°

QP = QPH +QPP = 54 kW

Hb

40 kW

37.8°

U = 6



T. Gundersen MP 18



' '

'

', , 1

,

', , 1

,

,0 ,

min

subject to:

0

0

k k

k k

k

k k

k

i ik j jkk TI i HU j CU

i k i k ijk ik kj C CU

i k i k ijk ik kj C

ijk jk ki H HU

ijk jk ki H

i i K

c Q c Q

R R Q Q i H

R R Q Q i HU

Q Q j C

Q Q j CU

R R

,0 0 0

0 ( , )i k ijk

ijk

R Q

Q i j P

LP Model forForbidden

Matches

Easily solved bythe Simplex

Algorithm

T. Gundersen MP 19



' '

'

, ,

', , 1

,

', , 1

,

,0 , ,

min

subject to:

0 0 0

k k

k

k k

k

iji H HU j C CU

i k i k ijk ik kj C CU

i k i k ijk ik kj C

ijk jk ki H HU

ijk jk ki H

i i K i k ijk

ijk

y

R R Q Q i H

R R Q Q i HU

Q Q j C

Q Q j CU

R R R Q

Q

0ij ijk TI

U y

MILP Model forfewest Number

of Units

Logical Constraintsrelating Discrete

& ContinuousVariables

T. Gundersen MP 20

Status for Mathematical Programming?• Considerable Research in the 1980’s/90’s

CMU, Princeton, Caltech, Imperial College• One “Road” towards Automatic Design

Math Programming provides the Framework Has the Potential to identify Superior Solutions

• Obstacles against Industrial Use Lack of Knowledge about the Methods Lack of user friendly Software Applications require Expertise Considerable Numerical Problems

• The Advantages are many Can handle Multiple Trade-offs, Discrete

Decisions and Constraints in the Design



T. Gundersen MP 21



The Sequential Framework − SeqHENS

Surprisingly few Iterations are neededto identify the Global Optimum

Reason: SeqHENS is strongly based on Insight from PA

T. Gundersen MP 22

UMIST Comments after Sabbatical

Promoting Mathematical Programmingwas quite challenging in those Days !



Documents

T. Gundersen