fibonacci search for optimal feed location

8/19/2019 Fibonacci Search for Optimal Feed Location

1/14

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FIBONACCI SEARCH FOR OPTIMAL FEED LOCATIONJ.C. WANG

a

a Engineering Department , Union Carbide Chemicals and Plastics , P.O. Box 8361, South

Charleston, WV, 25303

Published online: 24 Apr 2007.

To cite this article: J.C. WANG (1980) FIBONACCI SEARCH FOR OPTIMAL FEED LOCATION, Chemical EngineeringCommunications, 4:6, 651-663, DOI: 10.1080/00986448008935937

To link to this article: http://dx.doi.org/10.1080/00986448008935937

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2/14

Chern.Eng. Commun. Vol.4, pp. 651-663

0098-6445/80/0406-0651S04.50/0

< lGordon and Breach, Science Publishers Inc., 1980

Printed in the U.S.A.

FIBONACCI SEARCH FOR OPTIMAL

FEED LOCATION

J e WANG*

Union Carbide Chemicals and Plastics,

Engineering Department,

P.o. ox 8361, South Charleston, WV 25303

Received

uly

23,1979; infinal

form

October 11,1979)

A Fibonacci search technique is used in conjunction with a rigorous multicomponent distillation computer

module to find the opt ima l feed location within a sect ion

o

a disti llat ion column. The funct ion to be

minimized can be one of the following:

keycomponentratiodifference

reflux

ratio or reboiler ratio

condenser

dutyor reboilerduty

his technique has been used successfully in the relocation

o

feed stages

o

many existing distillation

columns and thus saved energy consumption. We shall describe this technique with a sample problem.

INTRODUCTION

Feed location is an important factor in the design of a new disti llation column or in

evaluat ing the performance of an existing column. A mislocated feed not only can

seriously impair separation efficiency but also affect the energy consumption. The

subject of optimal feed location has been studied by many investigators.

McCabe and

Thiele, ' Gil li land,' and others define the optimum feed stage based on matching the

ratio of the light and heavy key components in the feed and on the feed stage. Scheibel

and Montross, Floyd and Hipkin;'

and

Maas,' have developed procedures to find

the

opt imum feed stage that produces the maximum separations. Shipman ' gave a

criterion of the optimal feed stage based on the minimization of the entropy

production. Although his approach is thermodynamica lly sound, the method is

difficult to apply and has not much practical value. Recently, Waller and Gustafsson

t reated the feed stage problem in their opt imizat ion of the steady state distiIJation

operation.

Distillation processes are the most energy intensive operation in the petroleum and

petrochemical industries. Prengle, et at, reported that up to 40 of the energy

consumed in a petrochemical plant is taken by the distillation columns. is therefore

more logical to define the optimal feed location based on the energy consumption.

Luyben has given such a cri ter ion by stating The minimum energy consumption is

achieved in a column with a fixed number of total t rays when not only

X

D

and

X

B

are

held at their specified values but also the feed is introduced onto the optimum feed

Dr.

Wang isnowwith Simulation Science, Inc., 1400 N. Harbor Blvd., Fullerton, CA 92635.


3/14

652

J.C.

WANG

tray. In this

paper

we shall show

that the

energy consumption of a distillation column

with a fixed

number of

stages for a specified

separation

is an unimodal function or the

feed

stage

location. This function can be

evaluated

by using a

steady

state

computer

simulation program.

The

problem of finding

the

optimal feed location is then to search

for

the min imum

point of this function. A Fibonacci search procedure is then

developed for solving this problem.

The

procedure is simple

and

can be readily

incorporated into any distillation simulation program.

THE

SIMULATION

MODULE

Ever since the invention of the digital computer, numerous methods have been

proposed for

the computer

solution of

the

distillation problem. is beyond

our

scope to

review

the

various methods in this paper. By selecting some

of the

methods we have

developed a sophisticated

computer

program not only for disti llat ion

bu t

also for

absorption (with or without reboiler)

and

liquid-liquid extraction.

These

operations

can be classif ied as the Multicomponent

Multistage

Separation

Processes and there

fore we have des ignated

our

program as

the MMSP

program. The program is

modular

and

it consists of various subprograms.

These

include

the thermodynamics

subprograms, the

mathematical

subprograms, the initiation subprograms,

and the

separation subprograms. It

can

also be readily incorporated to

any

process flowsheet

s imulator such as Union

Carbide s IPES (Integrated ro ess Engineering System)

program. The

input can

either

come from

the

standard

8

1

2

x

II data

sheets or from

a stored file.

The output can either

be printed or s tored in a file to be used for column

design

programs

such as

our APT ll

Purpose ray program. Figure I shows

the

information flow

of

such a program system.

The

methods used in

the MMSP program

are:

The Wang-Hcnkc

method

(DSTLWH)

for distillation;

the

Sujata s Sum

Rate '

method

(DSTLSR)

for absorb

ers or liquid-liquid extractors; and

the

Block Tridiagonal

matrix

method (DSTLBT)

developed by

Naphthali and

Sandholrn

and

modified by

Tang.

In

the

last method,

the

MESH (Material

Balance, Equilibrium,

Summation,

and Heat

balance)

equa-.

tions are solved simultaneously using the

Newton

Raphson technique. The MESH

equations

are arranged

in such a way

t ha t t he Jacobian matrix

is in a simple block

tridiagonal form. Using this method, var ious top

and

bottom specifications can be

made.

These

include condenser

and

reboiler

temperature,

condenser

and

reboiler

heat

duty, reflux

and

reboiler ratio,

and

most importantly,

the product

purities in

terms of

mole fraction, moles, or percent recovery

of

the key components.

For

the

design problems of

determining the

stage

and

reflux

requirements

in a

disti llat ion column for a specified key

component

separation, we employed

VanWin

kle s

shortcut

method

(SHTCUT). The number

of equil ibrium stages are first

computed from

SHTCUT

and

the

reflux

requirement and

energy consumption

are

computed by use of the Block Tridiagonal matrix method (DSTLBT).

The MMSP program

has been used within Union Carbide for the simulat ion,

design, control studies,

and

energy

audit

of

many

distillation columns, absorbers,

and

liquid-liquid extractors. The column

can

be a simple one with one feed

and

two

product s or a complex one with multiple feeds, multip le s idedraws,

and

multiple


4/14

Ul

8

~

«

z

~

o

0::

J:

I

z

«

J:

u

OPTIMAL

FEED

LOCATION

PHYSICAL

PROPERTY

rFEED AND COLUMN

INPUT DATA FORMS

INPUT DATA FORMS

•

•

HYSICAL

PROPERTY

~ D T

FEED AND

COLUMN

DATASET

PP-COMMON

BANK

INPUT DATA SET

~

-.

THERMODYNAMICS

-

COLUMN INITIATION

SUBPROGRAMS

-

SUBPROGRAMS

MATHEMATICAL

I

UBPROGRAMS

•

EPARATION SUBPROGRAMS

,

GFLASH

XFLASH

SHTCUT

DSTLBT

DSTLSR

DSTLWH

•

ROCESS FLOWSHEET

STAND ALONE PROGRAM

\ S

S

~

PROBLEMSL

OR PROP. \

FILES

•

•

t

P R I N T R

TRAY DESIGN

PROGRAM

O U T P U T

P T ~ L L PURPOSE TRAY

FIGURE

I I nfo rma ti on flow for MMSP simulation and design.

653

Ul

m

s

i

>

Ul

Ul

tl

o

0::

D.

z

J:

U

interstage heaters or coolers).

The

mi xtu re to be s ep ar at ed can be of any de gre e of

nonideality ranging from those obeying Raoult s law to highly nonideal systems such

as azeotropes. We have incorporated in

our

Thermodynamics Subprogram Library a

spectrum of equilibrium models. These include Chao Seader correlation, Robinson

Chao correlation, Poynting correction, and others for liquid phase fugacity; Redlich

Kister, Regulat Solution, NRTL Wilson equation, Margules, and Uniquac for liquid

activity coefficients; Redlich-Kwong, Second Virial, Prausnitz-Chueh modified


5/14

654 J.C. WANG

Redlich-Kwong and other equa tions of state to compute vapor phase fugacity

coefficient and vapor

enthalpy departure

We also

can

use the BWR Starling BWR

and

Soave modified Redlich-Kwong equation to

calculate

K-values

and

enthalpy

departures for hydrocarbon mixtures. The data required for all these models can be

retrieved from a comprehensive Data Bank in which the pure component physical

properties of more

than

1000 compounds

are

stored.

The

MMSP

program is used for the evaluat ion of the energy consumption of the

dist il la tion column as a function of the location of the main feed stream. The input

data

to the program include:

• total

number

of stages including condenser and reboiler

• feed conditions, rates and compositions

• sided raw locations and rates, if any

• purity specifications in terms of mole fraction or percent recovery of the

specified components in the distillate and bottom products

• the codes to indicate the models selected for calcu lat ing the K-values

and

the

enthalpies. The pure component data can be retr ieved from the Data Bank

bu t

binary interaction

parameters

must be supplied in the input sheets.

Most of the input data items listed above are provided with the item numbers in the

data sheets. Any of these items can be changed by supplying its item number and its

new value. At the end of each case, the computed resul ts are stored on the computer.

The

program will then proceed to read the

change

cards, change the values of the

specified items, and perform the computation for the new case. Since the new case

starts with a converged solution of the previous case, the new case converges very fast.

Finally, the resul ts of all the cases

are

printed in a single output.

This

change case

procedure can be overseen by a control subrou tine such as the one described in this

paper for the optimal feed location.

The

output

from the program includes the column profiles of stage temperatures

internal vapor

and

liquid rates , the stage vapor

and

liquid compositions

and

the

following items which are related to the energy consumption of the distillation

column:

• the reflux rat io or reflux rate,

• the reboi ler rat io or reboiler vapor rate

• the condenser

duty

• the reboiler duty.

addition the following two items related to

matching

the compositions in the feed

and

on the feed

stage can

also be computed:

• the absolute value of the difference between the key components

ratio

in the feed

and that on the feed stage, i.e.,


6/14

OPTIM L

FEED LO TION

655

• the sum of the absolute values of the difference between the feed composition and

the feed stage composition for all the components.

where Z - feed composition, m.f.;

feed stage composition, m.f. Subscripts i for the

ith component, l for the light key, for the heavy key.

EFFE T

OF

FEED LO TION

ON

DISTILL TION

To show the effect of the feed stage location on the performance of the distillation

column, let us consider the problem of benzene-toluene-xylene separation. As shown in

Fig. 2, 100 Kg molesjhr of the mixture is fed to a distillation column to be separated to

a disti llate containing 0.01 mole of toluene and a bottom product containing 0.01

mole

of benzene. A shortcut calculation indicated

that

40 stages are required at a

reflux ratio equal to 1.2 times the minimum. Therefore, 42 stages including condenser

and reboiler are used in the rigorous calculation. The stages are numbered from top to

bottom with the condenser as the first s tage and the reboiler as the 42nd stage. The

effect of the feed stage in the middle section of the column (Stage 16 to

Stage

29) is to

be studied by using the change case procedure.

The simulation was done by using the Block Tridiagonal matrix method with the

purity specifications for

the

distillate (0.0001 rn.f. of toluene) and the bottom product

(0.0001 m.f. of benzene). The Wang-Henke method was used for initiating the column

profiles to be used in the base case. The Chao-Seader correlation' was used to

calculate the K-values and the Yen-Alexander correlation was used for calculating

the vapor and liquid enthalpies. All the physical properties required were retrieved

from the Data Bank. The computed results are plotted in Figure 3. The smoothed

curves for the fourteen case studies with feed stage at 16 to 29 include the values of l

k

and l the reflux ratio, the reboiler ratio, the condenser duty, and the reboiler duty.

These results show clearly that all the quantities listed are unimodal functions of the

feed stage location with only one minimum point within

the

section of the column. It

was also noticed

that

the minimum points for all the quantit ies with the exception of

l

k

fall on the same point. This has been true for all the problems we have encountered.

Therefore, the criterion of optimal feed stage based on the matching of the key

component ratio is not correct. The criterion based on the energy consumption should

be used. Since the energy consumption is related to the reflux rate, the boil-up rate,

the condenser duty, arid the reboiler duty, the optimal feed location can also be defined

as the one that minimized one of these quantities (or the weighted sum of these

quantities). The optimal feed location problem can then be determined by applying a

unidimensional search technique.


7/14

656

p 101

KPA

F, 100 Kc;MoL.ES /HR

S TU RED L IQ U ID I

T 364

0

K

BENzENE, 0.6 M.F.]

TOL.UENE, 0.3

_XVL.ENE, 0. 1

e WANG

___

3 _

16

8

~ I t K G M o L E S

HR

0,0001

MIF.j

TOL.UENE,

0.7499

VL.ENE

1

0.2500

FIGURE 2 The sample problem.


8/14

1. 4

i.a

OPTIMAL

FEEO

LOCATION

5. 5 t \ \ 4 ~

4 4

. ~ I _ _ _ I _ l

0

5

4 t - - t - ~ - ; - -

...

0

0(

III

J:

5 •3 t l l

II:

III

5 • 2 r t t \ ~ I _ _

III

o

Z

~

5. 1 r - - ~ t - \ - - t _ _ - _ : ~ - I _ _ ~ r _ _ -

c

5.0

+-+----11--+-++

II:

III

: 4.

9 t I l 1 ~

o

lD

III

0::4.

8

r \ f \ ~

III

U

1. 2

4.6

II:

III

1.0

4. 5

c

0.8

4.4

0(

l:t:

0

i

0.4

z

o

E

0.2

rJl

o

L

0.0

29 28 27

NUMBER

FIGURE 3 Effect of feed stage on distillatiion (benzene-toluene-xylene separation).

THE SEARCH ALGORITHM

657

0

i=

0(

0::

3.7

II:

III

J

3. 6

0

lD

III

0::

3.5

1.8

1.7

~

0::

1. 6

s

i o

5

,

0::

x

::I

J

II

III

0::

There are various methods for the search of the minimum point of a function. One of

these is the graphical method that we just described in the previous section. Beveridge

and Schechter,' Wilde,19 and others have reviewed one dimensional search techniques.

Himmelblau' and Beveridge and Schechter ' compare the efficiency of the various

unidimensional search techniques. They show that the Golden Search and the

Fibonacci Search are the two most efficient methods. Since the Fibonacci search


9/14

658

J.e. WANG

t echn ique is based on the Fibonacci

numbers

it is especia lly convenient for integer

search problems such as our optimal feed stage problem. this case, the Fibonacci

number

F

is the

number

of s tages within a sect ion of columns in which

the

optimal

feed stage lies and n is the required number of function evaluat ions column

simulations .

Using the Fibonacci search technique, we have developed a computational proce

dure for locating the

optimum

feed stage with a section of the distillation column. The

function to be minimized can be one of the following:

1 = s, I Zlk

-

X

lk

· the key ratio difference

.

Zhk X

I

z, - X; iF I The

composition difference

R

/

D the reflux ratio

1 V

/

B the boilup ratio

1

Qc the condenser

heat duty

1 = QR, the reboiler heat

duty

1

2

3

4

5

6

where

J

F indicates the feed stage number.

The computational procedure for the

optimal

feed stage

search

algorithm includes

the following steps:

t p

I

Supply the input

data

for

the

column simulation except

the

feed

stage

location.

The

data requirements are outlined in the previous section. Also specify the function to be

minimized.

t p

2

Specify the section of the co lumn to be

searched

by giving the

stage

number J F near

the column top and the stage number

J

F

b

near the bottom.

The

difference between

J

F

b

and J F

must be a Fibonacci

number F• Then

find

the

number

of

function evaluations

n corresponding to F from the table of Fibonacci number Table I .

t p

3

Determine the first two stage numbers to be checked as follows:

JF - JF + F _

2


10/14

Step 4

OPTIM L

FEED LOC TION

T LE I

The First Fibonacci Numbers

n

0

I I

2 2

3

3

4 5

5 8

6 13

7

21

8

34

9

55

10 89

11

144

12

233

13 377

659

Eva lu at e the function to be minimiz ed as

I

and

12

c orr esp on di ng to the feed s ta ge

numbers

JF

and

JF

2

by performing the column simulation with the purity specifica-

tions for the top and bottom product streams. Also set the search number I 2.

t p 5

Checkj vs./2 If

II

< 2 go to Step 6 otherwise set:

and evaluate a new

12

corresponding to the new

JF

2

by p erf or min g the column

simulation then go to Step 7.

t p 6

Set


11/14

660

J.C. WANG

and

evaluate

a new corresponding to

the

new

I

by performing

t he column

simulation.

t p 7

Update

the search

counter

I as

I

=

I

I

t p

8

Cheek

I vs. n If I = n stop the i teration and go to

output.

Otherwise

repeat Step

5

through

Step

8.

This procedure has been programmed into

the

MMSP

program

as

the fBSRCH

subprogram. The

results from all the column simulations

are

stored and printed in

the

final

output

along with

the

column profiles corresponding to

the

optimal feed

stage

location.

These

results including

reflux ratio, reboiler

rate

condenser

duty

reboiler

duty number

of iterations used in

the

column simulation,

and the

convergence

of the column simulations are printed in

tabular

form.

APPLICATION

THE

SEARCH PROCEDURE

We shall i llustrate

the f ibonacci

search procedure by applying it to

the

sample

problem

of

benzene-toluene-xylene

separation

discussed previously.

The

section

of the

column between stage 16

and

29 was

searched

for

the

optimal feed

stage

location

that

produces the

minimum

reboiler heat duty.

The

total

number

of stages within

the

section is

the fibonacci number

13.

from

Table

I we found

that

n

-

6

and

therefore a

maximum

of

six 6 column simulations were required.

The search

procedure was

started

as follows.

figure

4 is

the

graphical representation of this procedure.

The

first two feed stages to be checked were

I

JF F

n

_

=

16

5 = 21

JF

2

= JF F

n

_

I 16

8 - 24

Column

simulations using

DSTLBT

were performed with these two feed stages.

The

corresponding reboiler duties were

obtained

as:

4.7702

GJ/hr

= 4.5997

GJ/hr

Since

>

the minimum

point

must

lie in

the

section between

stage

21 to

stage

29.

Therefore

we eliminated

the

section from

stage

16 to

stage

21 from

further

consideration.

We then

followed

Step

5 to set

the

top limit

of the

new section as

JF


12/14

OPTIMAL F EE D L OC AT IO N

661

1\

\

\

I

\

--_.

II

\

_. -

-- _

..-

i

..

-.-

_ _

--- -

f-- -

/

I

1\

i

z

J

Q

\

;

/

I

,

\

.J

{

lJJ

a

1/

t:

i:

I-

/

\

6

r

V

.

-

SECOND

,

5 ELIMINATION J

-

1 IRST

ELIMINATION

_

JF .

JFz JF.

JF ,

4

(MIN)

I

3

I

29

28

27

26 25

24

23

22 21

20

19 18

I

5. 4

4 .

4.

5. 5

5. 6

5. 7

5. 8

5. 9

4

(JF. i

It: 4 .

g 4. 7

lJJ

.It: 4.

e

5. 3

~ 5

FEED ST GE NUM ER

FIGURE 4 Graphical representation of the Fibonacci search procedure.

JF - 21. We also

s tf

-

f2

- 4.5997 GJ/hr, and JF - J

2

24. A new

J

2

was

obtained as

J

2

J

t

+

J

b

- JF

=

21 + 29 - 24 - 26

Another column simulation with feed stage at 26 was performed. The reboiler duty

was obtained as

f

4.7046 GJ

/hr

Since

f

<

f2

the optimal feed location must lie above stage 26. The column section

between stage 26 and stage 29 was eliminated from further consideration. Following

Step 6 we set J

b

- 26 /2 f - 4.5997

GJ/hr,

and J - JF 24. A new JF was

obtained as


13/14

662 J.e. WANG

J

I

= JF +

J

b

-

J

-

21 + 26 - 24 23

The

column simulation with

J

I

23 yielded

the

reboiler

duty

II

-

4.6242

GJ/hr

Since f

>

I the minimum point must lie in the section from stage 24 to

stage

26. The

section to be searched was finally reduced to three stages. The reboiler duties

corresponding to two of these stages as feed

stage

24 and 26 were already obtained.

The

final column simulation performed with feed

stage J

F

I

- 25 yielded a reboi ler

duty II = 4.6168 GJ /hr

This

was larger

than

the reboiler duty I 4.5997 obtained

for feed stage JF = 24. Therefore the opt imal feed location was found to be

stage

24

from the top.

Table

II lists the results obta ined from the MMSP program for the above search

procedure. The total

number

of column simulations used in this search procedure was

5 as compared to 13 used in the case study. In

Table

we compared the

computer

times used and the

computer

costs for these two procedures. An IBM

370/168

computer

was used for the simulations.

The advantage

of using the Fibonacci

Search

procedure becomes more obvious when larger numbers of stages are in the section to

be searched.

TABLE II

Results of Fibonacci Search Applied to the Sample Problem

Feed Reflux Boilup Condenser Duty Reboiler Duty

Stage No.

a,

a, Ratio tio GJjhr GJjhr

2\ 0 31\6 0 \024

1.4771

3.5292

4.6349 4.7702

26 0 6\80

0 \789 1.4417 3 48\4 4.5703 4.7056

23

0.0695 0.0610

1.3983

3.42\ 0 4.4887 4.6242

25

0.3422 0.0986 1.3942 3.4156

4.4818

4.6\68

24 0.\088

0.0589

\.3838 3 40\3

4.4625 4.5977

TABLE III

Comparison etween Fibonacci Search and Case Study

ibon cci

Search

Case Study

No of Column Simulations 5

\3

Total No. of Iterations

DSTLBT 20

50

Total CPU Time. Sec.

IBM 370/168

3.27 8.78

Total Cost.

Printing CPU

4.36 20.00


14/14

CONCLUSION

OPTIMAL FEED LOCATION

663

A computational procedure using the Fibonacci search technique has been developed

for the location of the opt imal feed

stage

within a section of the column such that the

min imum energy consumption of the distillation column can be achieved.

The

procedure is simple

and

can be readily incorpora ted into

any

rigorous distillation

simulat ion module. This procedure has been used in Union

Carbide

for relocation of

the feed stage in many existing columns. During 1978 over 25,000 column simulations

have been performed by use of the MMSP

computer

program. Eight percent of these

simulat ions used the Fibonacci search procedure for opt imal feed location. Most of

them were for feed

stage

relocation in existing disti llat ion columns to reduce energy

consumption in the plants. The saving of energy cost was estimated to be over a million

dollars per year.

LITERATURE CITED

Beveridge, G.S.G .. and Schechter, R.S., Optimization Theory and Practice. McGraw-Hili Book Co..

1970.

2. Chao, K.C.. and Seader, J.D., A General Correlation of Vapor-Liquid Equilibrium in Hydrocarbon

Mixtures, A./.Ch.E. Journal,

1,598

(1961).

3. Floyd, E.R., and Hipkin, H.G., Locating Feed Trays in Fractionators, Ind. Eng. Chem., 55 (6), 34

(1963). .

4. Gilliland, E.R.. Multicomponent

Rectification-Optimum

Feed Plate Composition, Ind. Eng.

Chern.. 32, 918 (1940).

5. Himmelblau. D.M., Applied Nonlinear Programming, McGraw-Hili BookCo., 1972.

6. Luyben, W.L.,

Steady-State

Energy Conservation Aspects of Distillation Control System Design,

Ind. Eng. Chem. Fundamentals, 14 (4), 321 (1975).

7. Maas, J .H., Optimum Feed Stage Location in Multicomponent Distillations, Chern. Eng., April 16,

1973, page 96.

8. McCabe, W.L., and Thiele, E.W., Ind. Eng. Chem.,

17,605

(1925).

9. Naphtali, L.M., and Sandholm, D.P. , Multicomponent Separation Calculation by Linearization,

A./.Ch.E. Journal, 17

1),148

(1971).

10. Prengle, H.W., Crump, J.R., Fang, e.S., Frupa, M., Henley, J. , and Worley, T., Potential for Energy

Conservation of Industrial Operation in Texas, Final Report on Project

SID-10,

Governor s Energy

Advisory Council, The State of Texas (1974).

II . Scheibel, E.G., and Montross, C.F.,

Optimum

Feed

Tray

in Multicomponent Distillation Calcula

tions, Ind. Eng. Chem.. 401 (8), 1398 (1948).

12. Shipman, e.W., On the Optimum Choice of Feed

Stage

in Staged Equilibrium Processes, A.I.Ch.E.

Journal, 18

6),1253

(1972).

13. Sujata, A.D., Absorber-Stripper Calculation Made Easier,

Hydrocarbon Processing and Pet. Ref.

40 12),137 (1961).

14. Tang, Y.P., Multicomponent Distillation Calculation by Newton s Method, Union Carbide Chemi

cals and rlastics. Engineering Department Report.

June 7,1972.

15. Union Carbide Chemicals and Plastics, User s Manual for the Integrated Process Engineering

System 1976.

16. Waller, K.V., and Gustafsson, T.K., On Optimal Steady-State Operation in Distillation, Ind. Eng.

Chern. Process Des. Dev., 17 (3), 313 (1978).

17. Wang,

J.e.

and Henke, G.E., Tridiagonal Matrix for Distillation, Hydrocarbon Processing, 45 (8),

155 (1966). .

18. Wang, J.e., User s Manual for Simulation of Multicomponent Multistage Separatian Processes,

Union Carbide Chemicals and Plastics, Engineering Department, 1977.

19. Wilde, DJ., Optimum Seeking Method, Prentice Hall, Inc., Englewood, N.J., 1964.

20. VanWinkle , M., and Todd, W.G.,

Optimum

Fractionator Design by Simple Graphical Methods,

Chern. Eng.. September

20,1971,

pp. 136-148.

21. Yen, L.e., and Alexander, R.E., Estimation of Vapor and Liquid Enthalpies, A.I.Ch.E. Journal. II

2),334 (1965).

fibonacci search for optimal feed location

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