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•USES OF OPTIMIZATION

•FORMULATION OF OPTIMIZATION PROBLEMS

•OVERVIEW OF COURSE

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OPTIMIZATION OF CHEMICAL PROCESSEST.F. EDGAR, D.M. HIMMELBLAU, and L.S. LASDONUNIVERSITY OF TEXASMCGRAW-HILL – 2001 (2nd ed.)

PART I – PROBLEM FORMULATION II – OPTIMIZATION THEORY AND METHODS III – APPLICATIONS OF OPTIMIZATION APPENDICES (MATRIX OPERATIONS)

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PHILOSOPHY OF BOOK

•Most undergraduates learn by seeing how a method is applied

•Practicing professionals need to be able to recognizewhen optimization should be applied (Problem formulation)

•Optimization algorithms for reasonably-sized problemsare now fairly mature

•Focus on a few good techniques rather than encyclopediccoverage of algorithms

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Chapter 1

The Nature and Organization of Optimization Problems

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WHY OPTIMIZE?

1. Improved yields, reduced pollutants

2. Reduced energy consumption

3. Higher processing rates

4. Reduced maintenance, fewer shutdowns

5. Better understanding of process (simulation)

But there are always positive and negative factors to beweighed

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OPTIMIZATION

• Interdisciplinary FieldMax ProfitMin CostMax Efficiency

• Requires

1. Critical analysis of process2. Definition of performance objective3. Prior experience (engr. judgment)

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Figure E1.4-3

Optimal Reflux for Different Fuel Costs

Floodingconstraint

Min reflux toachieve separation

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Material Balance Reconciliation

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Least squares solution:

2

1

)(minii B

P

iCA mmm

opt. mA is the “average” value

any constraints on mA?

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THREE INGREDIENTS IN OPTIMIZATION PROBLEM

1. Objective function economic model

2. Equality ConstraintsProcess model

3. Inequality Constraints

nx1

1

2

1. min f(x) x

2. subject to h( ) 0 (m )

3. g( ) 0 (m )

2(feasible region :)

3

dependent variables

independent variables

x

x

1 2relate to m and perhaps m

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TABLE 1THE SIX STEPS USED TO SOLVE OPTIMIZATION PROBLEMS

1. Analyze the process itself so that the process variables and specific characteristics of interest are defined, i.e., make a list of all of the variables.

2. Determine the criterion for optimization and specify the objective function in terms of the above variables together with coefficients. This step provides the performance model (sometimes called the economic model when appropriate).

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3. Develop via mathematical expressions a valid processor equipment model that relates the input-output variablesof the process and associated coefficients. Include both equality and inequality constraints. Use well-knownphysical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom).

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4. If the problem formulation is too large in scope:(A)Break it up into manageable parts and/or(B)Simplify the objective function

5. Apply a suitable optimization technique to the mathematical statement of the problem.

6. Check the answers and examine the sensitivity of theresult to changes in the coefficients in the problem andthe assumptions.

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EXAMPLES – SIX STEPS OF OPTIMIZATION

specialty chemical100,000 bbl/yr.

2 costs inventory (carrying) or storage, production cost >

how many bbl produced per run?

Step 1

define variables

Q = total # bbl produced/yr (100,000)D = # bbl produced per runn = # runs/yr

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Step 2

develop objective function

inventory, storage cost = k1D

production cost = k2 + k3 D per run (set up operating

cost) cost per unit

(could be nonlinear)

QkD

QkDkC

D

Qn

DkknDkC

321

321 )(

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Step 3

evaluate constraints

continuous

integern

D>0

Step 4

simplification – none necessary

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Step 5computation of the optimum

analytical vs. numerical solution

1

2

22

1 0

k

QkD

D

Qkk

dD

dC

opt

02

minimum? ifcheck

answer good

000,70000,30

optimumflat 622,31

10

0.4000,100.1

32

2

2

5

321

D

Qk

dD

Cd

D

D

Q

kkk

opt

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solution? analytical

02

run per cost suppose

2/34

22

1

2/142

D

Qk

D

Qkk

dD

dC

Dkk

Step 6Sensitivity of the optimumsubst Dopt into C

316.4

000,100

162.3

620,31

2

321

3

2

1

2

1

2

1

321

kQ

kk

Q

C

Qk

C

k

Qk

k

C

k

Qk

k

C

QkQkkC

opt

opt

opt

opt

opt

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000,1000.4000,100.1

158.02

1

00

581.12

1

810,152

1

321

1

2

3

21

2

2

11

2

1

1

2

Qkkk

Qk

Qk

Q

D

k

D

kk

Qk

k

D

kk

Qk

k

D

k

QkD

opt

opt

opt

opt

opt

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RELATIVE SENSITIVITY (Percentage change)

213

321

1

1

1

2

1

1

111

Con sens. abs.

Don sens. abs.

0683.0240,463

)0.1(31620

0932.0

5.0863.0

5.00683.0

5.00683.0

620,31

0.1240,463

ln

ln

/

/

1

3

3

22

11

1

kQkk

kQkk

C

k

k

CS

SS

SS

SS

SS

k

Qk

k

C

kC

k

C

kk

CCS

opt

optCk

Dk

CQ

DQ

Ck

Dk

Ck

Dk

Ck

opt

opt

optoptoptCk

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PIPELINE PROBLEM

variables parameters

V

p

f L

Re m

D pipe cost

electricity cost

#operating days/yr

pump efficiency

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Equality Constraints

2

2

0.2

4Re /

2

.046 Re

Dv m

Dv

Lp v f

D

f

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min (Coper + Cinv.)

subject to equality constraints

22 vD

Lfp

need analytical formula for f

tubessmoothf 2.0Re046.

pump power cost

o

pC m

2

mass flow rate 4

Dm v

substituting for ∆p,

)(5.11

0.28.22.08.4

annualizedDCC

mDCC

inv

ooper

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5.11

8.48.222.0 TC cost Total DCDmCo (constraint eliminated by substitution)

6.3 0.2 2 2.8

1

0.16

.32 .45 .03

1

opt

2

( )0 necessary condition for a minimum

solving,

( )

( )

opt velocity V

4(sensitivity analysis)

opt o

opt o

opt

d TC

dD

CD m

C

CD m

C

m

D

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optimum velocity

non-viscous liquids 3 to 6 ft/sec.gases (effect of ρ) 30 to 60 ft/sec.

at higher pressure, need to use different constraint (isothermal)

1

2 211 1

1 2

1 1

ln2

.32324

upstream velocity

or use Weymouth equation

ppp fL

p S Vp p D

S gV

for large L, ln ( ) can be neglectedexceptions: elevation changes, slurries (settling),

extremely viscous oils (laminar flow, f different)

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Heat Exchanger Variables

1. heat transfer area2. heat duty3. flow rates (shell, tube)4. no. passes (shell, tube)5. baffle spacing6. length7. diam. of shell, tubes8. approach temperature9. fluid A (shell or tube, co-current or countercurrent)10.tube pitch, no. tubes11.velocity (shell, tube)12.∆p (shell, tube)13.heat transfer coeffs (shell, tube)14.exchanger type (fins?)15.material of construction

(given flow rate of onefluid, inlet temperatures, one outlet temp., phys. props.)