1

1

Juan M. Zamora

Department of Process EngineeringUNIVERSIDAD AUTONOMA METROPOLITANA

Iztapalapa, 09340 Mexico City, MEXICOEmail: jmzm@xanum.uam.mx

Synthesis of Wastewater Treatment Networks

Pan American Advanced Studies Institute Program on Process Systems EngineeringAugust 16-25, 2005, Iguazu Falls, Argentina.

2

Water is an Intensively Used Resource in Industry

FreshWater

Water usingprocesses

P1

P3

P2Wastewater

FreshWater

Water usingprocesses

P1

P3

P2Wastewater

Crude oil desaltingCrude oil distillationCoking operationsHydrocracking

FCCVisbreakingSweeteningHydrotreating

3

Steam generationCooling waterLiquid-liquid extractionWashing operationsCooling of blast furnacesEther SynthesisSteam strippingAlkylationMany other

4

Great Volumes of Contaminated Wastewaters are Generated

CONVENTIONALBiochemical oxygen demand (BOD)Total suspended solids (TSS)pHOil and Grease (O&G)

NON CONVENTIONALAmmoniaChemical oxygen demand (COD)ChlorineFluorides

TOXICAcrylonitrileBenzeneCarbon tetrachlorideChloroformPhenolLeadTolueneH2SMercaptansSulfidesCyanidesChromates

5

Effluents Discharge is Regulated by Environmental Standards

Biochemical Oxygen Demand (BOD)Total Suspended Solids (TSS)pHOil and Grease (O&G)Total Organic Carbon (TOC)Chemical Oxygen Demand (COD)Total Petroleum Hydrocarbons (TPH)TemperatureColorOdor

Limits are imposed on

6

Wastewater Treatment Costs

Type and concentration of contaminants present in effluent streams.

Amount of wastewater that is treated and discharged.

Discharge standards on the water quality of the receiving disposal site.

2

7

The Performance of a Processing Industry Depends Heavily on an Effective Effluent Treatment System

8

A Typical Centralized, Sequential Effluent Treatment System

Primary treatmentPhysical treatment processes

Secondary treatmentRemoval of soluble matter

Tertiary treatmentEffluent polishing to final discharge standards

FreshWater

Water usingprocesses

P1

P3

P2

Treatment

DisposalWastewaterT1 T2 T3

9

Main Features of a Sequential Effluent Treatment System

All effluent streams are mixed before treatment.

Large volumes of wastewater have to be processed.

Low concentration of contaminants in treatment units.

FreshWater

Water usingprocesses

P1

P3

P2

TreatmentDisposal

Wastewater

T1 T2 T3

10

A Distributed Effluent Treatment System

Subsets of effluent streams receive specialized treatment.

Reduced volumes of effluents are processed in treatment units.

Accounts for differences in concentrations of contaminants in the various effluent streams.

FreshWater

Water usingprocesses

P1

P3

P2

Treatment

DisposalWastewater

T1

T2

T3

11

A Decentralized, Distributed Effluent Treatment System

Targets specific types of contaminants at the source.Accounts for different costs between particular treatments.Costs of expensive treatment processes are reduced.

FreshWater

P1

P3

T1

DisposalT2

T3

P2

12

Synthesis of Integrated Water Systems

• All the water using processes and water treatment operations are integrated into a single system.

• The total cost of freshwater consumption and wastewater treatment is minimized.

FreshWater

P1

P3

T1

DisposalT2

T3

P2

Simultaneous synthesis of the water allocation, and the effluent treatment networks

3

13

Synthesis of Distributed Effluent Treatment Systems

Treatment System

DisposalEffluentStreams

T1

T2

T3

14

• A set of effluent streams, i I∈ iS i I∈

• A set of contaminants present in the effluent streams,

j J∈ , , i jC i I j J∈ ∈

• A set of treatment plants,

k K∈ , , j kR j J k K∈ ∈

• Treatment cost information,

( ) k k kCC CO t k Kβ+ ∈

U, , j e j ec c j J≤ ∈

Given

• The task of reducing the concentrations of all the contaminants in order to meet environmental limits before the final discharge.

Problem Statement

15

Determine

The topology and operating flowrates of the wastewater treatment network that will achieve the required removal of contaminants at minimum total cost.

Process plants

e.g. FCC

Utilities system

e.g.Boiler

Blowdown

Other Sources

e.g.Chem.Lab.

Was

tew

ater

dis

posa

l Si

te

Wastewater sources A wastewater treatment system

T1

T2

T

16

The enforcement of total, partial or no treatment at all for a subset of the wastewater streams.

The specification of minimum and maximum flowrates through treatment units.

The specification of maximum concentrations of contaminants at the inlet of treatment units.

The specification of target concentrations of contaminants at the outlet of the treatment units.

Additional design constraints

;L Uk k kt t t≤ ≤

, , , ,; ;U Uj k j k j k j kcin cin cout cout≤ ≤

, , ;Tj k j kcout cout=

17

State of the Art

18

Example 1A Design Problem Involving a Single Treatment Unit

900150700305

3003500124

350700200253

1500350382

400300930101

CBA

contaminant concentration (ppm)flowrate

(t/h)Stream number

Wastewater stream data for Example 1.

Environmental concentration limits (ppm)

70C

50B

60A

1

2

5

3

4

?

10

38

25

12

30

115

(Zamora, Hernández and Castellanos 2004)

4

19

Design constraints

• Treatment costs are proportional to total treatment flowrate.

• Concentration of A at the inlet of the treatment unit should be at most 430 ppm.

• Concentration of C at the outlet of treatment unit should be 45 ppm.

• At the inlet of treatment unit flowrate of stream 4 should be atleast one third of flowrate of stream 3.

808595TU

CBA

contaminantTreatment Unit

Removal ratios (%) for treatment unit

Treatment Unit

TUt

20

Single-Unit Basic Superstructure for Effluent Treatment(Zamora, Castellanos and Hernández, 1999; Zamora, Hernández and Castellanos 2004)

1 1, jS C,j ec

jR1f

2f

if

1,ef

2,ef

,i ef

jcin jcoutUT

1M 2M

2 2, jS C

,i i jS C

t

1 1, jS C,j ec

jR1f

2f

if

1,ef

2,ef

,i ef

jcin jcoutUTUT

1M 2M

2 2, jS C

,i i jS C

t

TU

21

No Loss of Wastewater Assumption

,

i i e if f S

i I

+ =

∈

ii I

f t∈

=∑ ,i e ei I

f t F∈

+ =∑

e ii I

F S∈

=∑

1 1, jS C,j ec

jR1f

2f

if

1,ef

2,ef

,i ef

jcin jcoutUT

1M 2M

2 2, jS C

,i i jS C

t

1 1, jS C,j ec

jR1f

2f

if

1,ef

2,ef

,i ef

jcin jcoutUTUT

1M 2M

2 2, jS C

,i i jS C

t

TU

22

( ), , , , 1 i e i j j i i j e j ei I i I

f C R f C F c j J∈ ∈

+ − = ∈∑ ∑

,

iI

ji ji

t cinf C

j J∈

=

∈

∑ , , ,

i e i j e j eI

ji

f C F c

j

t c

J

out∈

+ =

∈

∑

Material Balances for Contaminants

( )1

j jjt cout t

J

c n

j

i R= −

∈

1 1, jS C,j ec

jR1f

2f

if

1,ef

2,ef

,i ef

jcin jcoutUT

1M 2M

2 2, jS C

,i i jS C

t

1 1, jS C,j ec

jR1f

2f

if

1,ef

2,ef

,i ef

jcin jcoutUTUT

1M 2M

2 2, jS C

,i i jS C

t

TUU

, , j e j ec c j J≤ ∈

23

( )Minimize CC CO t+

LP Model for the Solution of Example 1

, i i e if f S i I+ = ∈

ii I

f t∈

=∑

,i e ei I

f t F∈

+ =∑

( ), , , , 1 i e i j j i i j e j ei I i I

f C R f C F c j J∈ ∈

+ − = ∈∑ ∑

,0 , i i e if f S i I≤ ≤ ∈

0 et F≤ ≤

, 0Ui i j j

i If C t cin

∈− ≤∑

3,10 0 L

j j i i ji I

m R f C j J∈

∆ − ≤ ∈∑

( ) ,1 0 Uj i i j j

i IR f C t cout j J

∈− − ≤ ∈∑

e ii I

F S∈

=∑

24

Optimal Solution of Example 1

38

UT

2

5

4

1

3

10

25

12

30

115

27.344

10.656

11.1310.869

28.6061.394

102.081

38

UT

2

5

4

1

3

10

25

12

30

115

27.344

10.656

11.1310.869

28.6061.394

102.081TU

5

25

Design of Distributed Wastewater Treatment Networks Involving Two or More Treatment Units

• Water pinch approach

Wang and Smith, 1994.Kuo and Smith, 1997.

• Approaches based on the optimization of a network superstructure.

Takama, Kuriyama, Shiroko and Umeda, 1980, 1981.Zamora and Grossmann, 1998.Galan and Grossmann, 1998.Alva-Argáez, Kokossis and Smith, 1998.Huang, Chang, Ling and Chang, 1999.Benko, Rév and Fonyó, 2000.Tsai and Chang, 2001.Lee and Grossmann, 2003.Hernández-Suárez, Castellanos-Fernández and Zamora, 2004.Gunaratnam, Alva-Argáez, Kokossis, Kim and Smith, 2005.

26

• TARGETING.Minimum effluent treatment flowrates through treatment plants are computed.

• SUBNETWORKS DESIGN.Alternative minimum treatment flowrate subnetworks are developedto accomplish the removal of particular contaminants.

• SUBNETWORK SELECTION.The subnetwork exhibiting the least exergy loss due to mixing isincluded in the overall network design.

• RETARGETTING.The set of streams that emerge from the selected subnetwork is considered in a re-targeting step for the removal of the remaining contaminants.

Water Pinch Approach(Wang and Smith 1994; Kuo and Smith, 1997)

27

Targeting Stage

Tm∆Mass of contaminant j to be removed

1S 2S

3S

4S

jC ,1

jC ,4

jC ,3

jC ,2

jsoutC ,

Composite Curve Tm∆

jC ,1

jC ,4

jC ,3

jC ,2

jsoutC ,

TREATMENT LINE

Composite curve and a treatment line

jC ,4

jinC ,

infjsoutC ,

joutC ,

Tm∆

PINCH POINT

jC ,4

jsoutC ,

joutC ,

jinC ,

Tm∆

inf

Treatment line for minimum treatment flowrate

28

Development of Treatment Subnetworks

TP

No Treatment

Total Treatment

Partial Bypass

Set o

f Str

eam

s for

a

Tar

getin

g St

age

Em

ergi

ng S

et o

f Str

eam

s for

a

Ret

arge

ting

Stag

e

• Streams above pinch are totally treated.• Streams at pinch are partially treated.• Streams below pinch are not treated at all.

Design Rules(Wang and Smith 1994; Kuo and Smith, 1997)

29

Stream Flowrate Contaminant

Number (t/h) A(ppm)

B(ppm)

C(ppm)

1 20 600 500 500

2 15 400 200 100

3 5 200 1000 200

Process Contaminant Removal Ratio (%)

Number A B C

I 90 0 0

II 0 99 0

III 0 0 80

Example 2Design of a Distributed Effluent Treatment System

(Kuo and Smith, 1997)

•Treatment costs are proportional to total treatment flowrate.

30

Minimum Treatment Flowrate Subnetworks (t/h)

Subnetwork for Contaminant A

I2

3

120

15

5

40

20

5

31.667

11.667

3.333

Subnetwork for Contaminant B

II2

3

120

15

5

40

155

23.283

1.717

18.283

Subnetwork for Contaminant C

III2

3

120

15

5

40

20

23.125

3.12515

1.875

6

31

Subnetwork for Contaminant A

I2

3

1

23.125

40

23.125

10.10415

4.896

1.8751.875

33.229

Subnetwork for Contaminant B

40II2

3

1

1.875

23.1251.513

21.612

23.487

1.87515

15

Retargeting

32

TOTAL TREATED FLOWRATE 80.78 t/h=

1 20

3.125

5 1.875

1.51

21.61240

23.487

9.158

5.842

3

2 15

III

II

I

SUBNETWORK III SUBNETWORK II SUBNETWORK I

Optimal Design for the Distributed Effluent Treatment System

33

Some Limitations of the Water Pinch Approach

• Due to graphical nature, It is not easy to manage design constraints.

• For multiple-unit, multicontaminant problems the minimum total treatment flowrate target is not rigorous.

• When two or more treatment units are capable of removing a givencontaminant, extra simplifying assumptions have to be made in order to resolve the issues of treatment unit arrangement, and mass load distribution.

• Suboptimal network designs might be obtained.

34

• A network superstructure is utilized.

• The optimization of the superstructure removes the redundant features of the design.

Superstructure Based Approaches for the Synthesis of Distributed Effluent Treatment Systems

T2

T1

35

Takama, Kuriyama, Shiroko and Umeda (1980)

First NLP approach reported for the simultaneous synthesis of the water allocation, and the effluent treatment networks.

36

P2

T2

T1

P1

The network superstructure by Takama et al. includes possibilities for water reuse, regeneration-reuse, recycling and treating.

7

37

An Obstacle for Superstructure Based Approaches

Simultaneous design techniques for the synthesis of distributed wastewater treatment systems (and integrated water management networks too!) rely on the solution of nonconvex mathematical models.

The development of globally optimal solutions for nonconvex mathematical models constitutes a challenging problem.

38

Zamora and Grossmann (1998)

The branch and contract algorithm is utilized for the global optimization of Problem Example 2.

T1

T2

T3

39

I

II

III

M1

M3

M2

M4

S6

S5

S4

40

5

S3

15S2

20S1

34.167 34.167

23.48723.487

23.125 23.12520.000

3.125

1.875

23.487

34.167

5.833

9.167

1.51321.612

0

5

10

15

20

25

30

0 200 400 600 800 1000 1200 1400 1600 1800 2000

CPU (s)

% G

AP

S5

S6

S7

0

5

10

15

20

25

30

0 200 400 600 800 1000 1200 1400 1600 1800 2000

CPU (s)

OPE

N N

OD

ES

S6

S7

40

A multi-start solution procedure based on LP relaxations to generate starting points for the NLP solver is utilized.

Galan and Grossmann (1998)

T2

T1

41

T3

T2

T1

T4

An MINLP model is also developed to account for alternative treatment technologies in the synthesis of distributed wastewater treatment networks.

42

Alva-Argáez, Kokossis, and Smith (1998)

MINLP approach for the simultaneous synthesis of the water allocation, and the effluent treatment networks.

The solution procedure is based on the decomposition of the original MINLP problem into a sequence of MILP problems.

Alva-Argáez (1999)

8

43

Huang, Chang, Ling and Chang (1999)

The superstructure by Takama et al. is extended by including multiple water sources and sinks, water losses and repeated water treatment units.

P2

T2

T1

P1

44

• NLP approach for the simultaneous synthesis of the water allocation, and effluent treatment networks.

• Initial feasible solutions for the NLP model are provided by water pinch or by fixing several key variables at “reasonable” levels.

• The idea of fixed outlet concentrations of contaminants at outlet of treatment units is introduced.

45

Benko, Rév and Fonyó (2000)

• Cover-and-eliminate NLP approach for the synthesis of integrated water management networks.

• All candidate system alternatives are covered by including them in a network superstructure.

• The recursive optimization of the network superstructure eliminatesunits and streams associated with inferior network designs.

46

Tsai and Chang (2001)

• The work by Huang, Chang, Ling and Chang (1999) is extended.

• The network superstructure is furnished with a set of fictitious mixer units that perform no stream transformation but expand the design search space by providing additional stream mixing and splittingnodes.

• The associated NLP model is solved by utilizing a genetic algorithm.

• Interesting results are presented for the retrofit of a water usage and treatment network in a refinery.

47

Rigorous global optimization algorithm for the solution of bilinear nonconvex generalized disjunctive programming problems.

Lee and Grossmann (2003)

48

EA v EB v EC

EG v EH v EI

ED v EE v EF

Discrete choices for process networks are expressed as disjunctions, which are relaxed by a convex hull formulation.

9

49

Heuristic solutionGalan and Grossmann (1998)

Optimal solutionLee and Grossmann (2003)

$1 697 253

$1 692 583(-0.27%)

50

Gunaratnam, Alva-Argáez, Kokossis, Kim and Smith (2005)

• MINLP model by Alva-Argáez (1999) for the synthesis of integrated water management networks is reformulated.

• Same network superstructure as in Alva-Argáez (1999), and Huang, Chang, Ling and Chang (1999). Multiple water sources and sinks, water losses and gains. Piping and sewer costs are included.

• Network complexity is controlled through constraints on flowrates, maximum number of streams at mixers and costs of piping.

• Solution approach based on the decomposition of MINLP model, and engineering insight to project bilinear terms in a recursive manner.

51

Comprehensive network

superstructure

Complex mathematical

model

Model Decomposition

Techniques

A Design Paradigm

Network Design

Optimal or suboptimal solution?

52

Superstructure Decomposition and Parametric Optimization Approach

Decomposition of Complex

Superstructure

Structured Mathematical

Model

ParametricOptimization

Network Design

How far can go with this approach?

Hernández-Suárez, Castellanos-Fernández and Zamora (2004)

53

2

1

Superstructure Decomposition

Network Superstructure

Design

Search Space

54

Basic Network Superstructures Partitioning the Design Search Space

(a) Arr. 1-2

1S

2S

nS

1 2,j ec

(b) Arr. 2-1

1S

2S

nS

2 1,j ec

(a) Arr. 1-2

1S

2S

nS

1 2,j ec

(a) Arr. 1-2

1S

2S

nS

1 2,j ec

1S

2S

nS

1 2,j ec

(b) Arr. 2-1

1S

2S

nS

2 1,j ec

(b) Arr. 2-1

1S

2S

nS

2 1,j ec

1S

2S

nS

2 1,j ec

Search Spacefor Arr. 1-2

Search Spacefor Arr. 2-1

10

55

1S

2S

nS

2 1,j ec

3

1

2

3

M1

M3

M2

M4

S4

S3

S2

S1

3S

2S

1S

S5

S6

1, jC

2, jC

3, jC

,1jR

,3jR

eF

1t

3t

2,ef

1,ef

3,ef

1,et

2,et

3,et

1,1t

1,2t1,3t

3,3t

3,1t3,2t

1,1f

1,2f1,3f

3,2f3,1f

3,3f

,1jcin

,3jcin

,1CUCout

,Uj eC

,2BUC in

,3ATCout

1S

2S

nS

3 2,j ec

1

1S

2S

nS

1 3,j ec

2

1S

2S

nS

1 2,j ec

3

1S

2S

nS

2 3,j ec

1

1S

2S

nS

3 1,j ec

2

56

Minimize ( ) k k kk K

t CC CO t∈

= +∑

NLP Model BNS-n

, , i k i e ik K

f f S i I∈

+ = ∈∑

, , ki k ki I K

k

tf t k Kα∈ ∈

<

+ = ∈∑ ∑l

l

l

l

, , 1 k k eKk

k Kα α∈>

+ = ∈∑ lll

,, k ee eI k

kii K

f Ftα∈ ∈

+ =∑ ∑

3,

,

3,

, , ,, ,

10 10(1 )

,

j ki k i j j

j j ki I Kk

jk

mf C R

R Rm

j J k K

α∈ ∈

<

∆+ −

∆=

∈ ∈

∑ ∑ ll

lll

l

( )3

, ,,

, ,,,,

10 1

i i j i k i j j k ej k

k e j ej ki I k K i I k K

S C f C R F cR

m

j J

α∈ ∈ ∈ ∈

− + − =∆

∈

∑ ∑ ∑ ∑

, Lj k j

k Km m j J

∈∆ ≥ ∆ ∈∑

LP Model BNS-nL

57

3, ,10 L U

j i i j i j ei I i I

m S C S c j J∈ ∈

∆ = − ∈∑ ∑

3,10 U

j i i ji I

m S C j J∈

= ∈∑

, ,31 1 (1 )

10Uj j k i i j

i Ik Km R S C j J

∈∈

⎡ ⎤∆ = − − ∈⎢ ⎥

⎢ ⎥⎣ ⎦∑∏

,

, ,(1 ) i i j

L i Ij e j k

ik Ki I

S Cc R j J

S∈

∈∈

⎡ ⎤= − ∈⎢ ⎥⎢ ⎥⎣ ⎦

∑∏ ∑

3, , ,10 0 ,U

j k j k j k km Cin R t j J k K∆ − ≤ ∈ ∈

3, , , ,10 (1 ) 0 ,T

j k j k j k j k kR m R Cout t j J k K− ∆ − ≤ ∈ ∈

Problem Parameters and Additional Constraints

58

Bounds

, , ,0 ULj e j e j ec c c j J≤ ≤ ≤ ∈

, ,0 , , i k i e if f S i I k K≤ ≤ ∈ ∈

0 L Uk k k et t t F k K≤ ≤ ≤ ≤ ∈

, , ,0 ,L U Uj k j k j k jm m m m j J k K≤ ∆ ≤ ∆ ≤ ∆ ≤ ∈ ∈

, ,0 , 1 , , k k e k K kα α≤ ≤ ∈ <l l l

59

6

5

4

3

2

01

1,2α

1,2 1,3 2,3, ,α α α

1,2 1,3 1,4 2,3 2,4 3,4, , , , ,α α α α α α

1,2 1,3 1,4 1,5 2,3 2,4 2,5 3,4 3,5 4,5, , , , , , , , ,α α α α α α α α α α

1,2 1,3 1,4 1,5 1,6 2,3 2,4 2,5 2,6 3,4 3,5 3,6 4,5 4,6 5,6, , , , , , , , , , , , , ,α α α α α α α α α α α α α α α

No. of Treatment

Units in Problem

Split Fraction Variables Present in Mathematical Model

LP Model BNS-nL

1,2 1α =

NLP Model BNS-n

1,2 1,2α α α= − ∆

60

Systematic Exploration of the Partitioned Design Region

1,2Loop 1: α

Infe

asib

le D

esig

n R

egio

n

BNS-2

BNS-3

BNS-4

BNS-3L

BNS-4L

BNS-2L

LP-Phase 1 NLP-Phase 2

1,3Loop 2: α

2K =

2,3Loop 3: α3K =

4K =

1,4Loop 3: α2,4Loop 3: α

3,4Loop 3: α

Feasible Designs

Loca

lly O

ptim

al

Des

igns

BNS-1L

11

61

Minimum Total Treatment Flowrate Surface

Min

imum

Tot

al T

reat

men

t Flo

wra

te (t

/h)

∗α 3,2

2,1α3,1α

62

Maximum no. of LP and NLP problems in exploration No. of treatment

units

No. of split fraction

variables 0.05∆α= 0.1∆α= 0.2∆α= 0.25∆α= 0.33∆α= 0.5∆α= 1∆α=

1 0 1 1 1 1 1 1 1 2 1 21 11 6 5 4 3 2 3 3 4,851 726 126 75 40 18 6 4 6 68.59 10x 3208 10x 7,056 2,625 800 180 24

5 10 991.3 10x 6209 10x 3889 10x 3184 10x 28,000 2700 120

6 15 154.9 10x 9627 10x 6224 10x 623.2 10x 61.57 10x 3680 10x 720

Maximum Costs for Systematic Exploration of the Design Region Associated with a Basic Network Superstructure

63

Wastewater stream data

Removal ratios (%) for treatment processes

507003

9870902

0099.91

SSOILH2Streatment processes

Environmental concentration limits

(ppm)

100SS

20OIL

5H2S

contaminants stream number

flowrate (t/h)

H2S (ppm)

OIL (ppm)

SS (ppm)

1 13.1 390 10 250 2 32.7 16,780 110 400 3 56.5 25 100 350

Example 3Design of a Distributed Effluent Treatment System

(Takama et al. 1980; Kuo and Smith, 1997)

•Treatment costs are proportional to total treatment flowrate.•Treatment unit 2 cannot precede treatment unit 3.

64

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1,21,3 = 0

1,2 = 1

3,2 = 1

= 176.57 t/htK

kk

1,3

(a) Arr. 1-3-2

Minimum Total Treatment Flowrate Maps and Network Designs

Arr. 1-3-2 Minimum Total Treatment Flowrate=176.57 t/h

32.7

13.1 8.1

56.5

5

37.7

36.520

102.3

1

2

3

1 2

3

65

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

3,1(b) Arr. 3-1-2

3,2

Fig. 20(c)

3,1 = 0

3,2 = 1

1,2 = 1

= 176.57 t/htK

kk

0.97 0.975 0.98 0.985 0.99 0.995 10

0.005

0.01

0.015

0.02

0.025

0.03

175

3,2

3,1(c) Arr. 3-1-2 cont.

3,1 = 1

3,2 = 0

1,2 = 1

= 173.83 t/htK

kk

32.7

13.1 8.1

56.5

5

33.3 38.3

102.3

0.6

55.9

1

2

3

3

2

132.7

13.1 8.1

56.5

5

33.3 38.3

102.3

0.6

55.9

1

2

3

3

2

132.7

13.1 8.1

56.5

5

33.3 38.3

102.3

0.6

55.9

1

2

3

3

2

1

Arr. 3-1-2Minimum Total Treatment Flowrate 173.83 t/h

66

0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

3,1

3,2

(d) Arr. 3-2-1

3,2 = 0.7241

3,1 = 0

2,1 = 1

= 216.67 t/htK

kk

Fig. 20e)

0.97 0.975 0.98 0.985 0.99 0.995 10

0.005

0.01

0.015

0.02

0.025

0.03

(e) Arr. 3-2-1 cont. 3,2

3,1 3,2 = 1

3,1 = 0

2,1 = 1

= 227.98 t/htK

kk

32.7

13.1

13

56.5

40.9

0.1

15.6

73.6 86.6 102.3

1

2

3 3

2 1

Arr. 3-2-1Minimum Total Treatment Flowrate 216.67 t/h

12

67

Effectiveness and Computational Costs

Actual number of LP and NLP problems solved in Example 3 Arr.

0.05∆α = 0.1∆α = 0.2∆α = 0.25∆α = 0.33∆α = 0.5∆α = 1∆α =

1-3-2 952 178 52 38 24 12 5 3-1-2 661 152 45 34 20 12 5 3-2-1 445 133 41 29 18 11 5

Minimum total treatment flowrate (t/h) Arr.

0.05∆α = 0.1∆α = 0.2∆α = 0.25∆α = 0.33∆α = 0.5∆α = 1∆α =

1-3-2 176.56 176.56 176.56 176.56 176.56 176.56 176.56

3-1-2 173.83 173.83 173.83 173.83 173.83 173.83 173.83

3-2-1 216.66 216.66 216.66 216.66 216.66 216.66 227.98

68

Example 4Analysis of the Treatment Section of an Integrated Water Network

(Tsai and Chang, 2001))

Water sources

Contaminants (ppm) Source

No. A B

Max Flowrate.

(t/h)

W1 10 20 ∞

W2 600 300 50

Unit No. Solute

Mass Load

(Kg/h)

Max Inlet Concentration

(ppm)

Max Outlet Concentration

(ppm)

U1 A 10 200 600

B 5 100 300

U2 A 2 40 120

B 8 120 360

Process Data for Water Using Units

Removal Ratio (%) Treatment

Unit A B

Max. Flowrate

(t/h)

T1 80 10 50

T2 20 70 50

Process Data for Treatment UnitsEnvironmental

concentration limits (ppm)

75B

75A

•Treatment costs are proportional to total treatment flowrate.

69

Flujo total a tratamiento = 87.90 t/h

39.7T1

T2

17.9

225.1

48.2

1.8

20

28.2

293

W2

W1 U1

U2

243

50

1S

2S

3S

*2,1 0.415α =

Flujo total a tratamiento = 87.90 t/h

39.7T1

T2

17.9

225.1

48.2

1.8

20

28.2

293

W2

W1 U1

U2

243

50

1S

2S

3S

*2,1 0.415α =

39.7T1T1

T2T2

17.9

225.1

48.2

1.8

20

28.2

293

W2W2

W1W1 U1U1

U2U2

243

50

1S

2S

3S

*2,1 0.415α = Genetic Algorithm

Tsai and Chang (2001)

Parametric Optimization

Hernández-Suárez (2004)

Total Treatment Flowrate = 87.90 t/h

T2

T1

17.9

225.138.8

48.2

11.20 19.1

19.7

293

W2

W1 U1

U2

243

50

*1,2 0.4923α =

1S

2S

3S

Arr. 1-2

Total Treatment Flowrate = 87.00 t/h

70

0.0 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

86

88

90

92

94

96

98

100

87.90

87.36

= 0.4038*α 2,1

arr. 2-1Fl

ujo

tota

l mín

imo

a tra

tam

ient

o (t

/h)

α

arr. 1-2

1,2 α 2,1o

= 0.4923*α 1,2

87.0 t/h

Tsai and Chang, 2001

Min

imum

Tot

al T

reat

men

t Flo

wra

te (t

/h)

Minimum Total Treatment Flowrate Curves

71

Allowing Repeated Treatment Units

T2-2

T1-117.9

28.2

41.8

T2-3

T1-2

T2-1

49.8

36.4

13.6

46.3

10.3 23.32.8

1.7

26.5

13.6

W250

W1

U2

46.1

96.1

1S

2S

3S

U1

Genetic AlgorithmTsai and Chang (2001)

Total Treatment Flowrate = 187.90 t/h

17.9 47.83 50.00

28.2

T1-2

29.8731.67

96.1

T1-1

T2-3

T2-1

28.216.23

11.60

20.07

11.97

9.80

38.03

18.33

T2-2

W2

W1 U1

U2

46.1

50

1S

2S

3S

Parametric Optimization

Hernández-Suárez (2004)

Total Treatment Flowrate = 187.57 t/h