synthesis of wastewater treatment networks -...
TRANSCRIPT
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1
Juan M. Zamora
Department of Process EngineeringUNIVERSIDAD AUTONOMA METROPOLITANA
Iztapalapa, 09340 Mexico City, MEXICOEmail: [email protected]
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
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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
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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
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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
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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.
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• 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)
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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