Subscriber access provided by Iowa State University | Library
Process Systems Engineering
Reverse Osmosis Network Rigorous Design Optimization Abdon Parra, Mario Andres Noriega, Lidia Yokoyama, and Miguel J. Bagajewicz Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 17 Jan 2019 Downloaded from http://pubs.acs.org on January 17, 2019
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
1
Reverse Osmosis Network Rigorous
2
Design Optimization
3
Abdon Parraa, Mario Noriegab, Lidia Yokohamaa and Miguel
4
Bagajewiczc, *
5 6
a.
Federal de Rio de Janeiro, Brasil.
7 8
b.
11 12
Departamento de Ingeniería Química y Ambiental, Universidad Nacional de Colombia, Colombia.
9 10
Departamento de Processos Inorgânicos, Escola de Química, Universidade
c.
School of Chemical Engineering and Material Science, University of Oklahoma, 100 East Boyd Street, T-335, Norman, OK 73019-0628, USA.
* Corresponding author:
[email protected] 13 14
KEYWORDS: Reverse osmosis; Desalinization; Metamodels; Optimization; Genetic
15
algorithms.
16 17
1 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 2 of 38
1
Abstract
2
In this work, we propose a methodology to solve a nonlinear mathematical model for the
3
optimal design of RO networks, which ameliorates the shortcomings of the computational
4
performance and sometimes convergence failures of commercial software to solve the
5
rigorous MINLP models. Our strategy consists of the use of a genetic algorithm to obtain
6
initial values for a full nonlinear MINLP model. In addition, because the genetic algorithm
7
based on the rigorous model equations is insurmountably slow, we use metamodels to
8
reduce the mathematical complexity and considerably speed up the run. We explore the
9
effect of the feed flow, seawater concentration, number of reverse osmosis stages, and
10
the maximum number of membrane modules in each pressure vessel on the total
11
annualized cost of the plant.
12
Highlights:
13 14
solution of rigorous MINLP RO network models.
15 16 17
Genetic algorithms based on metamodel are used to obtain good initial values for the
The effect of the feed flow, seawater concentration, number of reverse osmosis stages, and the number of membrane modules is depicted.
2 ACS Paragon Plus Environment
Page 3 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
Industrial & Engineering Chemistry Research
1.
INTRODUCTION
2
The increased demand for fresh water by a growing population and the per-capita
3
increase in the demand for fresh water due to industrialization and urbanization makes
4
desalination a competitive technology for the generation of pure water from seawater as
5
well as other low quality water containing salts and other dissolved solids1.
6
The available desalination technologies in the market can be classified as thermal-
7
based and membrane-based processes. Reverse osmosis (RO), multi-stage flash (MSF),
8
and multi-effect distillation (MED) are the main commercial desalination technologies,
9
with RO being the fastest growing2. This last technology (RO) is, in most cases, the
10
technology of choice for seawater desalination in places where inexpensive waste heat
11
is not available.
12
Desalination plants using RO have been traditionally designed by manufacturers
13
using empirical approaches and heuristics3. However, the cost performance of RO
14
desalination is sensitive to the design parameters and operating conditions4 and
15
therefore, attention needs to be placed in obtaining cost-optimal designs.
16
The problem of synthesizing a reverse osmosis network (RON) consists of
17
obtaining a cost-effective solution based on optimum values of the following: number of
18
stages, number of pressure vessels per stage, number of modules per pressure vessel,
19
number and type of auxiliary equipment, as well as the operational variables for all the
20
devices of the network.
21
After the early works of Evangelista5, El-Halwagi6 and Voros et al.7,8 many papers 9–15,
22
have followed
mainly using the solution-diffusion model proposed by Al-Bastaki et
23
al.16, a model that includes the effect of concentration polarization, which eliminates the 3 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 4 of 38
1
problem of significant overestimation of the total recovery17 and the economic model from
2
Malek et al.18.
3
Many authors proposed solving the problem of the design of a RON using Mixed
4
Integer Nonlinear Programming (MINLP) or Nonlinear Programming (NLP)3. For example,
5
Du et al.19 used a two stage superstructure representation and solved the resulting MINLP
6
using the solvers CPLEX/MINOS using several starting points to obtain the best solution.
7
They do not clarify how they generate these starting points.
8
proposed a MINLP model and solved the problem using an outer approximation algorithm
9
within gPROMS to evaluate the effects of temperature and salt concentration in the feed
10
current. They do that by generating “various structures and design alternatives that are
11
all candidates for a feasible and optimal solution”, without specifying how their initial
12
values are obtained. Alnouri and Linke21 explored different specific RON structures and
13
they optimized each using the ‘‘what’sBest’’ Mixed-Integer Global Solver for Microsoft
14
Excel by LINDO Systems Inc. The solver is global and does not require initial points. They
15
used “reduced super-structures resembling fundamentally distinct design classes”. Lu et
16
al.22 obtained an optimal RON using a two stages RON structure and used an MINLP
17
technique with several starting points obtained from an ad-hoc preliminary simulation.
18
Finally, Skyborowsky et al.23 proposed an optimization strategy with a special initialization
19
scheme where a feasible initial solution is obtained in two steps: first, all variables are
20
initialized with reasonable values (some obtained by heuristics) and then a solution is
21
obtained using SBB and SNOPT solvers. These local minima are reportedly obtained
22
within a few minutes of computation. They also reported an attempt to solve a RON using
Sassi and Mujtaba20
4 ACS Paragon Plus Environment
Page 5 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
1
the global solver Baron indicating that the solver finished after 240 hours with a relative
2
gap of 15.66%.
3
All the aforementioned works have a few things in common: they share the
4
complexity of the problem modeling and the difficulty of the solution procedure that stem
5
from the nonlinearities associated with the concentration polarization model. In some
6
cases, they do not indicate in detail what pre-processing was done and how they obtained
7
the initial data and/or the computing time. Regarding computing time, it varies: one to
8
few minutes23, 3 to 16 minutes10 or 5 to 28 hours19. Our experience indicates that without
9
the initial values, there is no convergence in several solvers. In addition, although
10
computational time is not critical in design procedures, we believe it ought to be
11
reasonable. In some cases, the computational time is unacceptable19, as it is in the order
12
of days. In this article, we intend to ameliorate these shortcomings providing these initial
13
data systematically, and reducing the computing time to the order of minutes.
14
To aid in our work, we also use surrogate models, often called Metamodels, which
15
have been proposed to address the issue of model complexity and the associated
16
difficulties of convergence when poor or no initial points are given. Such metamodels are
17
sets of equations of simple structure (low-order polynomial regression, and Kriging or
18
Gaussian process24) that facilitate an increased computational performance (mostly
19
time25). They are built with the information of the rigorous method and their functions
20
approximate well the image of the more complicated models26. Metamodels were
21
implemented in the optimization for heat exchanger network27,28, the optimization of water
22
infrastructure planning29,30, in stochastic structural optimization31 and building energy
23
performance32.
5 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 6 of 38
1
In this work, the RON optimization problem is formulated as an MINLP problem
2
that minimizes the total annualized cost (TAC). The model is first solved using a genetic
3
algorithm, which provides good initial values for the rigorous MINLP model. As we shall
4
observe, a genetic algorithm, using the full non-linear and rigorous equations is
5
computationally very expensive, while the MINLP is rather fast when good initial values
6
are provided. We will show that replacing the use of rigorous equations of the model by
7
the use of a metamodel in genetic algorithm speeds up the solution time orders of
8
magnitude and provides a similar solution.
9 10
2.
MATHEMATICAL MODEL
11
Figure 1 presents a reverse osmosis network involving two stages, pumps and
12
turbines. Each stage consists of a set of parallel pressure vessels. The figure also
13
presents the brine recycles for each stage. Figure 2, in turn, presents the structure of a
14
stage and the membrane modules in series.
15
The feed flow enters a high-pressure pump and is sent to the first RO stage where
16
it is separated into two streams: permeate and brine. The brine leaving the first RO stage
17
feeds the second RO stage, but a fraction could be recycled. Recycling increases the
18
velocity through the membrane module and thus reduces the concentration polarization.
19
The final brine from the second stage goes to a turbine to recover the residual energy.
20
The permeates from stages 1 and 2 are mixed to get the final permeate.
21
In this work, we consider spiral wound (SW) modules because of their high packing
22
density and relative low energy consumption. Thus, a diffusion model for spiral wound
23
(SW) modules with FilmTec™ SW30HR-380 membrane modules is used. 6 ACS Paragon Plus Environment
Page 7 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
1 2
3 4 5
Figure 1. Superstructure of a two-units reverse osmosis network.
6
7 8
Figure 2. Structure of a single stage.
7 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
We now show the equations of a rigorous MINLP model
2
Global Balances:
Page 8 of 38
FS FˆPO FBO
(1)
FS CS FˆPO CPO FBO CBO
(2)
6
F1IN FS FR1B
(3)
7
F1IN C1IN FS CS FR1B C1B
(4)
8
F2IN F1B FR1B FR2B
(5)
3 4 5
9 10 11
Recycle Balances:
F2IN C2IN F1B C1B FR1B C1B FR2B C2B
6)
Balances for the outlet permeate stream: N RO
FˆPO FmP
(7)
m 1
12
N RO
FˆPO CPO FmP CmP
(8)
CPO CPMax
(9)
m 1
13 14 15
16
17 18
Balances for the outlet brine stream: N RO
NT
m 1
t 1
FBO FmB FBO Ft T FBO N RO
NT
m 1
t 1
FBO CBO FmB FBO CmB Ft T FBO CtT
(10)
(11)
Reverse osmosis stages model: The feed of the RO stage is distributed equally to the Npv pressure vessels of each stage:
8 ACS Paragon Plus Environment
Page 9 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
Fminem Npvm FmIN
1
(12)
2
We decided to make this a continuous variable to reduce the computation
3
complexity, and the approximate result will be obtained by rounding the variable. Other
4
authors also use the same concept, except Du et al19, who used binaries to model the
5
number of pressure vessels.
6 7
The inlet flow of a membrane module its restricted to a specific range by the commercial provider: Fmmin in Npvm FmIN Fmmax in Npvm
8
(13)
9
For the remaining e 2,..., Ne modules within each pressure vessel the incoming
10
properties are equal to those of the brine of the previous module, so the properties of the
11
brine final current are obtained from the last series module
12
First module:
13
FmBe,1 Fminem FmPe,1
(14)
14
FmBe,1CmBe,1 FminemCmIN FmPe,1CmPe,1
(15)
15 16 17
Rest of the modules:
FmBe,e FmBe,e 1 FmPe,e ,
e 1
FmBe,eCmBe,e FmBe,e1CmBe,e1 FmPe,eCmPe,e , e 1
(16) (17)
18 19 20
The final brine current of the corresponding RO stage is obtained from:
FmB Npvm FmBe, Ne
(18) 9
ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 10 of 38
1
CmB CmBe, Ne
(19)
2
PmB PmBe, Ne
(20)
3 4
The flow of the permeate current is obtained from the mass and component balances of the hole stage as follows:
5
FmP FmIN FmB
(21)
6
FmP CmP FmIN CmIN FmB CmB
(22)
7
In practice, the modules inside the pressure vessel are all connected in series.
8
However, to be able to consider a variable number of pressure vessels, we consider the
9
O maximum number and add splits to remove some flowrate Fm ,e from each module (Figure
10
3) out of a total fixed number of pressure vessels. The optimization determines the optimal
11
number and will bypass the rest.
12
13 14
Figure 3. By-pass representation of a single pressure vessel.
15 16 17 18
To model these bypasses, we introduce binary variables ymO,e which are one if a O flow Fm ,e >0, and zero otherwise. Thus, we introduce the following equations:
FmO,e ymO,e 0
(23) 10
ACS Paragon Plus Environment
Page 11 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
Industrial & Engineering Chemistry Research
We force that only one of the flows is different from zero writing: Ne
y
2
e 1
3
6 7 8 9
( FmBe,e FmO,e ) (1 ymO,e ) 0
12
13
(25)
To obtain the number of modules Ne in a pressure vessel we use the next expression:
Nem 1* ymO,1 2* ymO,2 3* ymO,3 4* ymO,4 5* ymO,5 6* ymO,6 7 * ymO,7 8* ymO,8
(26)
Diffusion membrane model: The flux for water J mW,e for one membrane module is given by the following equations:
J mW,e aˆ Pmnd,e
10
11
(24)
1
Be O O In addition, to make sure that Fm ,e Fm ,e when ym ,e =1, we write:
4 5
O m ,e
(27)
where aˆ is the pure water permeability, with the net driving pressure difference given by:
P
nd m ,1
P
P
nd m ,e
IN m
P
PmB,1
Be m , e 1
2
PˆmPe,1 ( mW,1 mP,1 )
PmB,e 2
PˆmPe,e ( mW,e mP,e )
(28)
e 1
(29)
14
ˆ B , wall mW,e aˆ TC m ,e
(30)
15
ˆ Pe mP,e aˆ TC m ,e
(31)
16 17
bar . The brine pressure can be calculated as follows: K . ppm
6 where , aˆ 2.63 10
PmBe,1 PmIN PmB,1
(32)
11 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
PmBe,e PmBe,e1 PmB,e
1
e 1
Page 12 of 38
(33)
2
where, PmB,e is estimated from a SW RO membrane correlation given by the module
3
producer: 1.7
P
B m ,e
4
5
av m ,1
F
6 7
av m ,e
F
(34)
Fminem FmBe,1
(35)
2 FmBe,e 1 FmBe,e 2
e 1
(36)
In turn, the flux solute J mS ,e is given by:
J mS ,e bˆ CmB,,ewall CmPe,e
8 9
F av 9532.4 m ,e ˆ av
(37)
where, bˆ is the salt permeability, the membrane wall concentration is: Vw
10
CmB,,ewall
11
B , wall m ,e
CmIN CmBe,e ksmm,,ee Pe Pe Cm ,e Cm ,e e , e 1 2
(38)
Vw
C
C
Pe m ,e
CmBe,e 1 CmBe,e ksmm,,ee Pe Cm ,e e , e 1 2
(39)
12
where ksm ,e is the mass transfer coefficient, and Vmw,e the permeation velocity. These are
13
estimated using the following expressions:
14
15
ˆ ˆ 0.33 D ksm,e 0.04 Re0.75 m , e Sc dˆh w m ,e
V
J mW,e J mS ,e ˆ P
(40)
(41)
16
where Re m ,e , Sc , Dˆ and dˆh are the Reynold’s, Schmidt, salt diffusivity, and hydraulic
17
diameter, respectively. They are given by:
12 ACS Paragon Plus Environment
Page 13 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
ˆ ˆ Sc ˆ Dˆ
2
3 4
dˆhU ms ,e ˆ ˆ
Re m ,e
1
(42) (43)
s av The superficial velocity U m,e depends on the average flow rate Fm,e , the density and
the feed cross-section open area Sˆ fc .
U ms ,e
5
Fmav,e ˆ Sˆ
(44)
fc
6
Finally, the permeate concentration and flow rate are:
7
Pe m ,e
*1000
(45)
FmPe,e VmW,e Sˆmem ˆ P
(46)
C
8
J mS ,e VmW,e
9 10
Objective function
11
We minimize the Total Annualized Cost (TAC), given by:
TAC AOC ccf TCC
12 13 14 15 16 17 18 19
where ccf
(47)
(ir 1)t1 i . In turn, (ir 1)t1 1 TCC 1.25 (1.15CCequip )
(48)
The equipment cost is given by: CCequip CCswip CCHPP CCT CCmem CC pv
(49)
The salted water intake and pretreatment system cost:
CCswip 996(24(QSW IN ))0.8
(50)
where QSW IN is the feed flow rate to the system in m3/h.
13 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
Page 14 of 38
Cost of pump and turbines:
2
CCHPP 52 (PpHPP Q pHPP )
(51)
CCHPPR 52 (PmHPPR QRm )
(52)
p
3
m
CCT 52 (Pt T QtT )
4
(53)
t
5
where PpHPP , Q pHPP , PmHPPR , QRm and Pt T , QtT are the pressure drop in bar and flow rate
6
in m3/h for the high pressure pump, recycle pump and turbine respectively. We note that
7
this cost of capital is linear with power, a known shortcoming of previous models (all based
8
on the one proposed by Malek et al18) because it cannot capture the nonlinear behaviors
9
of costs33. We will discuss the impact of this assumption in the results’ section.
10
Membrane module cost: NRO
CCmem Npvm Nem cmem
11
(54)
m 1
12
Pressure vessels cost: NRO
CC pv Npvm cpv
13
(55)
m 1
14
Annual operational costs: AOC OClab OCchem OCm OCmemr OCins OC pow
15 16
Labor cost OClab clabQP ta
17 18 19 20 21
(56)
(57)
where the permeate production rate QP and the annual operational time ta are used. Cost of chemicals: OCchem 0.018QSW IN ta
(58)
Cost for replacement and maintenance: 14 ACS Paragon Plus Environment
Page 15 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
OCm 0.01TCC
1
(59)
Membrane replacement cost:
2 3
OCmemr 0.2CCmem
(60)
OCins 0.005TCC
(61)
Insurance costs:
4 5
Electric energy costs:
6
OC pow Cen ( PPSWIP PPRO )ta
7 8
Energy consumed for intake and pre-treatment system:
9
PPSWIP
1 QSW IN PˆSWIP ˆSWIP 36
(62)
(63)
Electric energy consumed by the reverse osmosis plant:
10
HPP HPP P HPPR QRm 1 Pp Q p PPRO m Pt T QtTˆT 36 p ˆHPP ˆHPPR m t
11
(64)
12 13 14
3.
METAMODELS
15
We use simple quadratic polynomials as metamodels: The equations of the
16
membrane diffusion model were solved numerically for different inlet conditions (values
17
selected according to the bounds of the problem) to obtain a mesh of input-output data
18
pairs. We identified two different regions for the permeate concentration: A Linear Region
19
and a Non-Linear Region, the expressions are presented next.
20
Linear Region: The concentration was adjusted to first order polynomial as follows:
21 22
C Pe K1 K 2 P IN K 3C IN K 4 F IN
(65)
Non-Linear Region: Permeate concentration was adjusted to a second order polynomial: 15 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 16 of 38
C Pe K1 K 2 P IN K 3 C IN K 4 F IN K 5 P IN K 6C IN 2
2
2
K 7 Fmin,e K8 P IN C IN K 9 P IN F IN K10 C IN F IN 2
1
K11 P
2
C F
IN 2
IN 2
IN 2
2
2
K12 P C K13 P F IN
IN
IN
2
IN
K14C F IN
2
(66)
IN
K15 P IN C IN F IN 2
The permeate flow was adjusted to a first order polynomial for all the operation region:
F Pe K1 K 2 P IN K 3C IN K 4 F IN K 5 P IN C IN
3
K 6 P IN F IN K 7C IN F IN K8 P IN C IN F IN
(67)
4
To obtain the coefficients of the metamodels we used a Diploid Genetic Algorithm
5
(DGA) programed in MATLAB®34. The code follows the description given by Fonteix et
6
al.35 The method tends to imitate principles of organic evolution processes as rules for
7
an optimization procedure, this is based in genetic concepts such as population,
8
recombination, and mutation as evolution rules to guide the optimum search. The
9
Simulations were developed in a laptop with processor Intel(R) Core(TM) i7 – 3610QM
10
CPU @ 2.3GHz (8 CPUs) with 8192MB of RAM. Each variable ( C Pe , F Pe ), was evaluated
11
using the equations (65-67) for different inlet conditions ( P IN , C IN and F IN ) using
12
estimated coefficients, to generate the metamodel solutions mesh. The same inlet
13
conditions were used with equations (27-46) to generate the rigorous solutions mesh. The
14
fitness function that was minimized to obtain the coefficients is the following: N
Fobj V Rig V Met
15
2
(68)
i
V
is the variable to be adjusted ( C Pe , F Pe ), N is the total number of points and
16
where
17
the superscripts Rig. and Met. represent the value obtained from the rigorous method and
16 ACS Paragon Plus Environment
Page 17 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
1
the metamodel prediction, respectively. Supplemental material shows additional details
2
of the metamodel.
3 4 5
4.
SOLUTION STRATEGY We tested different strategies:
6
a) A genetic algorithm (1000 individuals, 10 generations, 100 survivors and 100
7
mutants) using the rigorous equations and the metamodel equations as well.
8
The GA has 4 random variables per stage: Flow entering each pressure vessel,
9
pressure entering all pressure vessels, the number of modules in each stage
10
and the recycling ratio. Once these are fixed for each individual of the
11
population (1000), the respective model is solved calculating first the sequence
12
of membrane models using the transport phenomena equations or the
13
metamodel expressions according to the case, then the recycle is calculated
14
and the stage is recalculated based on the new inlet conditions of flow and
15
concentration until convergence. The rest is the classical set up of a diploid GA
16
algorithm. We considered the rigorous model and the linear and non-linear
17
metamodels depending on the regions where they are the most accurate.
18
Indeed, we used the linear model when the ratio between inlet concentration
19
and inlet pressure is lower than 950 and the nonlinear model when this ratio is
20
equal or greater than 950, this boundary ratio was obtained from the model
21
adjustment for different inlet flows.
22
b) A rigorous MINLP composed of the mass balances (Eqs. 1-11), the reverse
23
stages model (Eqs. 12-26), the membrane transport phenomena (diffusion) 17 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 18 of 38
1
model (Eqs. 27-46) and the economic model (Eqs. 48-64) solved with
2
DICOPT36
3
genetic algorithm as initial values. In particular, the MINLP is solved
4
considering the number of pressure vessels (Npv) as a continuous variable and
5
then is re-run fixing the Npv to the nearest integer value.
programmed in GAMS™37using the results obtained from the
6
c) Use of some global optimization solvers: Baron38, Antigone39 and Rysia40 for
7
the same rigorous MINLP model previously presented on b) . The first two are
8
commercial and the latter is not, but has proven to solve certain problems that
9
the aforementioned two cannot solve.
10 11
5.
RESULTS
12
The data for the modules is presented in the Table 1 and the economic parameters
13
are presented in Table 2. Table 3 shows the optimization results for the RO system for a
14
targeted permeate concentration of 500 ppm and a flow rate of 100kg/s, with an inlet
15
water concentration of 35,000 ppm. We also set a maximum of 87,000 ppm for the brine
16
concentration, a reasonable value before scaling onset. The necessary parameters to
17
describe the diploid GA35 used, were: 1000 individuals, 10 generations, 100 survivors,
18
100 mutants and a mutation rate equal to 0.01 In the table we report the final number of
19
vessels obtained considering it as a continuous value and in the case of the MINLP, we
20
report both, the continuous value obtained as well as the fixed value used for the final
21
MINLP run (in between parenthesis). The TAC reported for the MINLP is based on the
22
integer value of the number of vessels. The optimization renders the flow of the brine and
23
the inlet flow needed. 18 ACS Paragon Plus Environment
Page 19 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
Industrial & Engineering Chemistry Research
Table 1. Reverse osmosis module parameters. Parameter Effective area - Smem (m2) Water permeability - A (kg/m2∙s∙bar) Salt permeability - B (kg/m2∙s) Diffusivity coefficient - D (m2/s) Hydraulic diameter - dh (m) Feed cross-section open area - Sfc (m2) Maximum inlet flow rate – Fmaxin (kg/s) Minimum inlet flow rate – Fminin (kg/s)
Value 35.33 2.4 x 10-4 2 x 10-5 1.35 x 10-9 9.35 x 10-4 0.0147 5 1
2 3
Table 2. Economic parameters Parameter
4 5 6 7
Capital charge factor - ccf Membrane unitary cost - Cmem (USD) Pressure vessel unitary cost - Cpvm (USD) Labor cost factor – Clab (USD) Annual operational time - ta (h/y) Pressure difference for intake - ∆Pswip (bar) Energy price - Cen ($/kwh) Intake pump efficiency - ᶯswip High pressure pump efficiency - ᶯswip Turbine efficiency - ᶯswip
0.088 750 1,000 0.05 8,000 5 0.05 0.75 0.75 0.75
Table 3. Optimization results. Genetic Algorithm using the Metamodel $1,796,169 S1 S2 73 80 167.65 3.5 3.5
8
Value
Genetic Algorithm using the Rigorous Model $1,806,500 S1 S1 67.6 79.9 167.65 2.14 3.47
Rigorous MINLP
TAC $ 1,781,499 Stage S1 S2 Inlet Pressure (bar) 63 79.4 Salted water flow (kg/s) 166.57 Each Pressure vessel Inlet Flow 2.6 2.59 (kg/s) Number of modules 8 8 8 8 8 8 Number of pressure vessels As 47.9 25.4 78.3 24.5 63.6 (64) 37.2 (37) continuous variable- (As integer) Flow Recycle 1 (kg/s) 0 0 0 Flow Recycle 2 (kg/s) 0 0 0 Computing time ~20 min 280 hours 5.5 sec (Permeate concentration: 500 ppm; permeate flow rate: 100kg/s; Inlet water concentration: 35000 ppm)
9
19 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 20 of 38
1
The rigorous MINLP models using the results from the GA’s as a starting point,
2
give the same result with slightly different solution times (5.7 seconds for using the initial
3
values from the GA run using metamodel vs 6.8 seconds using the initial values from the
4
GA run with the rigorous model). We note that the genetic algorithm using the rigorous
5
model was disproportionally time consuming (280 hours vs. 20 minutes). The time-
6
consuming step is the iterative solution of the nonlinear set of equations. The fitness
7
(TAC) of each individual of the population at each GA iteration is evaluated by solving the
8
membrane module equations (Eqs 27-46). This is done as follows: The total inlet flow and
9
the inlet concentration are known from the problem specification then Eqs 27-46 are
10
solved to obtain permeate flow and concentration, the brine flow and concentration are
11
calculated from the mass balance. We remind the reader that the Matlab “solve” feature
12
is used in this step, of which we have little detailed information. These values are then
13
used as the inlet conditions for the next module. Once the last module is solved, the
14
concentration of the recycle is known and one can start at the first module again upon
15
convergence. Each solution of Eqs 27-46 takes for a single membrane module 5 to 60
16
seconds to be solved, and a single stage takes about 10 iterations to converge. We thus
17
explain the large computing time when we realize that we use 1000 individuals and 10
18
generations. Conversely, the metamodel does not need to iterate to solve the same
19
module equations.
20
Table 3 presented the TAC for the rigorous MINLP fixing the number of pressure
21
vessels Npv to the nearest integers values obtained from the rigorous MINLP (considering
22
Npv as continuous), the TAC values differs only in a 0.00033% indicating that the
23
approximation does not introduce a significant error. 20 ACS Paragon Plus Environment
Page 21 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
1
Because we made several runs using the GA with the metamodel followed by the
2
rigorous MINLP (as detailed below), finding similar times (less than 30 minutes), we are
3
confident that the pattern will repeat for other cases and/or with other parameter data.
4
The rigorous MINLP was also run without initial values in Dicopt36 and Antigone39
5
and we did not obtain a feasible solution. When trying Baron38, after a predefined time
6
limit of 250 hours the solver finished without attaining convergence, with the relative gap
7
at 18.72%. We also run Baron38 fixing the binary variables for the number of membrane
8
modules and we did not obtain a feasible solution.
9
In an attempt to implement Rysia40, we developed a relaxed version (linear lower
10
bound) of the rigorous MINLP.
Although the rational terms of the model can be
11
reformulated to render a bilinear model, the presence of exponential terms made us select
12
a different approach, which is the use of images of monotone functions in each domain
13
variable’s partition, as it was performed in several papers41–44 that follow the one
14
introducing Rysia40. When running Rysia, the upper bound is the rigorous MINLP and is
15
run using the results from the lower bound as initial values. Before trying bound
16
contraction, we attempted to increase the number of partitions in the lower bound to see
17
if the gap at the root node can be reduced. The result is that we reached a region where
18
there is no improvement in the objective value when the number of intervals for the
19
partitioned variables was increased. For example, when we run with 2,4,8 and 10 intervals
20
to partition the decision variables, we obtain a sequence of slowly increasing lower bound
21
values ($1,293,594, $1,318,176, $1,528,365, and $1,540,456) at an elevated
22
computational cost of 0.1,1.3,6.0, and 10 hours, respectively. In the meantime, the upper
23
bound rendered and optimum with an objective of $1,814,528, that is only 1.8% larger 21 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 22 of 38
1
than the optimum identified in Table 3. Clearly, an increase of the number of intervals
2
leads to unacceptable computing times. When bound contraction was attempted using
3
two intervals, none of the bounds for the partitioned variables could be contracted, we did
4
not try the bound contraction procedure using three intervals, since the computational
5
cost is already high, because the procedure is iterative and needs to be implemented for
6
each one of the 84 partitioned variables each time, and each run take 30 minutes, so it
7
will take about 42 hours to complete the first iteration, and we are not sure that it will take
8
just one pass.
9
All the above results show that the nonlinearity of the RON-MINLP model presents
10
important convergence difficulties, and is computationally expensive, especially if good
11
initial values are unknown for local solvers. While we cannot explain the reasons why
12
Baron and Antigone fail, in the case of Rysia, we can only say that it should be able to
13
solve the problem with enough number of partitions, if it was not for the computational
14
effort involved. Thus, for the rest of this paper, we will use a genetic algorithm using
15
metamodels to initialize a rigorous MINLP model.
16
The results for different inlet water concentrations and different targets permeate
17
flows and a permeate concentration of 500 ppm are shown, for two stages, in Table 4.
18
We note that in this case, we no longer fix the permeate flow, instead the inlet flow is fixed
19
maintaining a maximum brine concentration of 87,000 ppm.
20
Results presented in Table 3 and 4 do not show a brine recycle flow, despite that
21
recycling helps reducing concentration polarization (increasing the velocity through the
22
membrane module). Optimal solutions avoid it because the recycle pumps (HPPR)
22 ACS Paragon Plus Environment
Page 23 of 38
Industrial & Engineering Chemistry Research
1 2 3 1 needed to compensate the pressure drop of the membrane module, thus increasing the 4 5 2 total capital cost and power consumption since they are high-pressure pumps. 6 7 3 Incidentally, brine recycle also increases the system salt passage leading to unacceptable 8 9 10 4 salt permeate concentrations in some cases45. 11 12 5 13 14 6 15 16 17 7 Table 4. Optimal solutions for different scenarios 18 19 Feed seawater 30000 40000 50000 20 concentration (ppm) 21 GA with Rigorous MINLP GA with Rigorous GA with Rigorous 22 metamodel metamodel MINLP metamodel MINLP 23 24 TAC $4,723,835 $4,700,784 $2,582,401 $2,629,873 $532,703 $551,140 25 Stage S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 26 Inlet Pressure (bar) 78.4 80 55.8 73.5 73.4 78.8 65.9 75.3 75.4 77.6 78.1 76.8 27 Salted water flow 450 450 250 250 50 50 28 (kg/s) 29 Permeate flow (kg/s) 296.5 296.5 135.8 135.8 21.38 21.38 30 Pressure vessel Inlet 4.0 4.0 1.8 1.5 4.0 4.0 2.4 2.1 4.0 4.0 2.5 2.7 31 Flow (kg/s) 32 8 8 8 8 7 8 7 8 5 7 5 7 33 Number of modules 112.5 52.9 245.6 155.6 62.6 38.2 103.6 73.9 12.5 9.0 19.6 12.8 34 Number of pressure (246) (156) (104) (74) (20) (13) 35 vessels-As continuous 36 variable (As integer) 0 0 0 0 0 0 37 Flow Recycle 1 (kg/s) Flow Recycle 2 (kg/s) 0 0 0 0 0 0 38 Computing time ~20 min 4.2 sec ~22 min 5.1 sec ~24 min 4.7 sec 39 8 (Permeate maximum concentration: 500 ppm) 40 41 9 42 43 44 10 The main differences in the operation conditions (decision variables) between the genetic 45 46 47 11 algorithm using the metamodel and the rigorous MINLP (Table 3 and 4) are associated 48 49 12 to the precision of the metamodel, which underestimates some values of the operating 50 51 13 variables in the region closer to the bounds. Thus, although the genetic metamodel is not 52 53 54 55 56 57 58 23 59 ACS Paragon Plus Environment 60
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 24 of 38
1
completely suitable to determine operating conditions, it produces good initial estimates
2
as starting points for the rigorous MINLP.
3 4
We now turn to an analysis of the effect of the feed flow and the salt concentration
5
in the feed on the total annualized cost. Figure 4 presents the rigorous solution obtained
6
for a feed flow between 50 and 450 kg/s and a feed salt concentration between 20,000
7
and 50,000 ppm and running the GA with the metamodel, followed by the rigorous MINLP
8
solved using Dicopt with initial values obtained from the GA.
9
The results show that the TAC presents two regions with an apparent maximum at 30,000
10
ppm of feed concentration. We cannot say for sure that the maximum is exactly at 30,000
11
ppm, given the discrete nature of the number of points investigated. For a fixed flow the
12
differences in the extremes are -3,6% (cost for 20,000 ppm vs cost of 30,000 ppm) and -
13
2.6% (cost for 50,000 ppm vs cost of 30,000 ppp). For a feed concentration of 30,000
14
ppm and larger the TAC does not change significantly because the requirements for a
15
larger pump power are compensated by less membrane area due to higher
16
concentrations. On the other side, for feed concentrations of 20,000 and 25,000 ppm the
17
reduction is due to the use of membrane module with high retention parameters for
18
seawater, while these concentrations are more related to brackish water concentrations.
19
The reduction in TAC here is due to smaller pumping needs driven by lower
20
concentrations.
21 22
With such small differences in TAC for a large range of feed concentrations leads us to conclude that an average TAC for each flow is a good representation of all.
24 ACS Paragon Plus Environment
Page 25 of 38
6
x 10 5
4
TAC($)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
3
2
1
0 500
1 2
400
300
200
Feed Flow (kg/s)
100
0
20,000
25,000
30,000
35,000
40,000
45,000
50,000
Feed Concentration (ppm)
Figure 4. TAC for different inlet flows and seawater concentrations (two stages)
3 4
We now explore the influence of the number of RO stages. The results are
5
presented in Figure 5: the maximum TAC difference between 3 stages and 2 stages is
6
about 1.6% (Figure 5a); although it is not noticeable in the figure the surfaces have a
7
transition point from constant TAC to monotone TAC at 30,000 ppm, as explained above.
8
The TAC values for two stages are lower than for three stages for feed concentrations of
9
40,000 ppm and larger. For feed concentrations of 35,000 ppm and lower the design for
10
three stages is lower in cost than for two stages, again barely noticeable in the figure.
11
This behavior is obtained using linear cost for pumps as a function of power (Equations
12
51-53). One would argue that the use of a power law for cost, with other exponents might
13
change the results. Indeed, Lu et al.22 and Kim et al.14 used a power law with an exponent
14
of 0.96 and the general literature on costs suggests values as low as 0.5. We tested the
15
model for various points using 0.96, 0.7 and 0.5. The results are always the same: for
16
every flowrate, the region where 3 stages are of lower cost than 2 stages is for every flow 25 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research
1
below 35,000 ppm. For 40,000 ppm and above two stages are cheaper than three. This
2
is consistent with the fact that at higher concentrations the power is constant because the
3
pressure has reached its maximum. Thus, two pumps makes more sense than three
4
pumps in that region. Below 40,000 ppm, more pumps is compensated by a smaller
5
number of membrane modules (lower membrane area). The extreme differences (cost of
6
3 stages vs cost of two stages) are -1.59% and +5.97 % for n=0.96, -3.38% and +18.30
7
% for n=0.7, and -3.51% and +19.25 % for n=0.5. These are big differences and are a
8
warning about the cost functions that need to be used.
9 10
In turn, the optimal TAC results for one stage are larger than those for two stages with a maximum difference close to 7% (Figure 5b). 2 stages
6
x 10 5
4
TAC($)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 26 of 38
3
2
1
0 500
11 12
3 stages
400
300
200
Feed Flow (kg/s)
100
0
20,000
25,000
30,000
35,000
40,000
45,000
50,000
Feed Concentration (ppm)
(a)
26 ACS Paragon Plus Environment
Page 27 of 38
1 stage
6
x 10 5
4
TAC($)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
3
2
2 stages
1
0 500
1 2 3
400
300
200
100
0
20,000
Feed Flow (kg/s)
25,000
30,000
35,000
40,000
45,000
50,000
Feed Concentration (ppm)
(b) Figure 5. Effect of the number of stages. (a) 2 vs 3 stages, (b) 1 vs 2 stages
4 5
When analyzing the results, there were solutions that used the maximum
6
membrane modules allowed in a pressure vessel according to the fabricant, especially
7
those corresponding to 35,000 ppm or lower. So, a new set of solutions was obtained
8
allowing the variable to use up 16 membrane modules. In this case, we found solutions
9
using up to 14 membrane modules. These results are presented in Figure 6. The
10
maximum difference between the use of a maximum of 8 modules and 16 modules was
11
lower than 0.7% (indistinguishable in the figure), so there is no improvement in using a
12
larger number of membrane modules per vessel. This behavior is explained because an
13
increase in the maximum membrane modules is compensated with variations on the inlet
14
stage conditions (Pressure and pressure vessel inlet flow Fminem ), but these variations have
15
not significant effect on the TAC. We can conclude that the model is rather insensitive to
16
the number of modules and that also perhaps explains the convergence difficulties. 27 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research
1
6
x 10 5
4
TAC ($)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 28 of 38
3
2
1
0 500
2
400
300
200
100
Feed Flow (kg/s)
0
20,000
25,000
30,000
35,000
40,000
45,000
50,000
Feed Concentration (ppm)
3
Figure 6. TAC for different inlet flows and seawater concentrations max number of
4
membrane modules (Nmaxe) equal to 16
5 6
All the above results represent optimal values obtained minimizing the TAC for a
7
fixed concentration and flowrate of the permeate (Table 3), or flowrate of the feed (rest of
8
results). We now investigate the minimization of the cost per unit of freshwater produced
9
with a fixed targeted concentration. Results are shown in Figure 7. We observed a
10
sensibility to the feed concentration and the feed flow, different to the observed in figure
11
4. In the case of this Figure 4, for a fixed permeate concentration of 500 ppm, the TAC is
12
not significantly affected by the change in the feed concentration when the feed is
13
maintained constant. However, when the inlet concentration varies, the amount of
14
permeate flow decreases with feed concentration (shown in Figure 8). Since the TAC
15
values for both figures (Figure 4 and Figure 7) are the same, but the amount of obtained
28 ACS Paragon Plus Environment
Page 29 of 38
1
freshwater varies with the inlet concentration the cost per unit of produce fresh waters
2
increases with the feed concentration.
3
4
Cost per Unit permeate ($/kg)
x 10 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 500
5
400
300
200
100
0
20,000
25,000
Feed Flow (kg/s)
4
30,000
35,000
40,000
45,000
50,000
Feed Concentration (ppm)
Figure 7. Optimal TAC per unit permeate flow. Concentration of permeate: 500 ppm.
350
Permeate Flow (kg/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
300 250 200 150 100 50 0 500
400 300
200
100 0
6 7
Feed Flow (kg/s)
20,000
25,000
30,000
35,000
40,000
45,000
50,000
Feed Concentration (ppm)
Figure 8. Permeate flow (2 stages, 500 ppm permeate).
8 29 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
6.
Page 30 of 38
CONCLUSIONS
2
A new methodology to solve a nonlinear mathematical model for the optimal design
3
of RO is proposed. Metamodels were used to reduce the mathematical complexity and
4
get accurate solutions using a genetic algorithm. Then, the results were used as initial
5
values to solve the full nonlinear model using GAMS/DICOPT. This allows to get optimal
6
solutions for a complex MINLP problem with less computational effort. One of the major
7
advances of our approach is that initial values are not needed (always a problem for
8
practitioners using MINLP codes), as the GA provides them.
9
The total annualized cost increases with an increase in the feed flow and presents
10
little variations for different feed salt concentrations at a fixed inlet flow, indicating that a
11
reverse osmosis plant could have adaptation capability for variations in the inlet
12
concentration without major effects on the TAC.
13
The effect of the number of stages was studied for different feed flows and
14
seawater concentrations, finding that one stage has the largest TAC and the differences
15
between two and three stages are small, but dependent on the costing of pumps used.
16
The effect of the number of membrane modules in a pressure vessel was also
17
investigated, finding that increasing the maximum number of membranes allowed in a
18
commercial pressure vessel does not have any advantage over the TAC values obtained.
19 20
Nomenclature
21 22
AOC
Annual operational cost [$]
23
C
salt concentration [ppm] 30 ACS Paragon Plus Environment
Page 31 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
1
C B , wall
membrane wall concentration [ppm]
2
CPMax
maximum permeate concentration [ppm]
3
CCequip
total equipment cost [$]
4
ccf
capital charge factor
5
CCHPP
high pressure pump cost [$]
6
CCmem
membrane module cost [$]
7
clab
labor cost factor [$]
8
Cen
electricity cost [$/(kWh)]
9
cmem
membrane module unitary cost [$]
10
CC pv
total pressure vessel cost [$]
11
cpv
unitary pressure vessel cost [$]
12
CCswip
Seawater intake and pretreatment system cost [$]
13
CCT
turbine cost [$]
14
F
flow rate [kg/s]
15
F av
average flow rate [kg/s]
16
FR
recycle flow rate [kg/s]
17
Js
solute flux [kg/(s.m2)]
18
Jw
water flux [kg/(s.m2)]
19
ir
annual interest rate
31 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
ks
mass transfer coefficient [m/s]
2
Ne
number of membrane modules per pressure vessel
3
Npv
number of pressure vessels of the stage m
4
NRO
number of reverse osmosis stages
5
OCchem
cost of chemicals [$]
6
OCins
Insurance costs [$]
7
OClab
labors cost [$]
8
OC pow
Electric energy costs [$]
9
OCm
cost for replacement and maintenance [$]
10
OCmemr
Membrane replacement cost [$]
11
P
pressure [bar]
12
PPSWIP
Energy consumed for intake and pre-treatment system [kWh]
13
PPRO
Electric energy consumed by the reverse osmosis plant [kWh]
14
Q
flow rate [m3/h]
15
QR
recycle flow rate [m3/h]
16
QSW IN
feed flow rate to the system [m3/h]
17
Re
Reynold’s number
18
Sc
Schmidt number,
19
TAC
total annualized cost [$]
20
TCC
total capital cost [$]
Page 32 of 38
32 ACS Paragon Plus Environment
Page 33 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
1
Us
superficial velocity [m/s]
2
Vw
permeation velocity [m/s]
3
y
binary variable
4
ta
annual operation time [hours]
5
t1
lifetime of the plant [years]
6
P B
brine side pressure difference [bar]
7
P nd
net driving pressure difference [bar]
8
P
pressure difference [bar]
9
Parameters
10
aˆ
pure water permeability [kg/(s.m2.bar)]
11
aˆ
Van’t Hoff ´s constant [bar/(K.ppm)]
12
bˆ
salt permeability [kg/(s.m2)]
13
Dˆ
salt diffusivity [m2/s]
14
dˆh
hydraulic diameter [m]
15
Pˆ Pe
permeate outlet pressure [bar]
16
Tˆ
inlet temperature [K]
17
Sˆ fc
feed cross-section open area [m2]
18
Sˆmem
active membrane area [m2]
19
PˆSWIP
seawater intake pressure difference [bar] 33 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
Superscripts
2
av
average
3
B
brine
4
Be
brine current of module e from the stage m
5
B FBO
interconnection between brine and brine final discharge
6
HPP
high pressure pump
7
HPPR
recycle high pressure pump
8
IN
inlet
9
inem
inlet of firsts modules of the stage m
10
O
outlet
11
P
permeate
12
Pe
permeate current of module e from the stage m
13
RO
reverse osmosis
14
T
turbine
15
T FBO
interconnection between turbine and brine final discharge
16
W
membrane wall
17
Subscripts
18
BO
brine final discharge
19
e
membrane module e
20
m
RO stage m
Page 34 of 38
34 ACS Paragon Plus Environment
Page 35 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Industrial & Engineering Chemistry Research
1
p
pump p
2
pv
pressure vessel
3
PO
permeate final current
4
S
feed current
5
SWIP
seawater intake and pretreatment
6
t
turbine t
7
Symbols:
8
ˆ
density [kg/m3]
9
ˆ
dynamic viscosity [kg/m.s]
10
efficiency
11
osmotic pressure [bar]
12
7. References
13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
(1)
Kucera, J. Desalination: Water from Water, 1st ed.; John Wiley and Sons, 2014.
(2)
Ghaffour, N.; Bundschuh, J.; Mahmoudi, H.; Goosen, M. F. A. Renewable Energy-Driven Desalination Technologies: A Comprehensive Review on Challenges and Potential Applications of Integrated Systems. Desalination 2015, 356, 94–114.
(3)
Ghobeity, A.; Mitsos, A. Optimal Design and Operation of Desalination Systems : New Challenges and Recent Advances. Curr. Opin. Chem. Eng. 2014, 6, 61–68.
(4)
Choi, J. S.; Kim, J. T. Modeling of Full-Scale Reverse Osmosis Desalination System: Influence of Operational Parameters. J. Ind. Eng. Chem. 2015, 21, 261–268.
(5)
Evangelista, F. A Short Cut Method for the Design of Reverse Osmosis Desalination Plants. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 211–223.
(6)
El-Halwagi, M.; Al-Ahmad, M.; Zhu, M. Optimal Design and Scheduling of Flexible Reverse Osmosis Networks. J. Memb. Sci. 1997, 129, 161–174.
(7)
Voros, N.; Maroulis, Z. B.; Marinos-Kouris, D. Optimization of Reverse Osmosis Networks for Seawater Desalination. Comput. Chem. Eng. 1996, 20, S345–S350.
(8)
Voros, N. G.; Maroulis, Z. B. Short-Cut Structural Design of Reverse Osmosis Desalination Plants. 1997, 127, 47–68.
35 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
Page 36 of 38
(9)
Maskan, F.; Wiley, D. E.; Johnston, L. P. M.; Clements, D. J. Optimal Design of Reverse Osmosis Module Networks. AICHE J. 2000, 46 (5).
(10)
Marcovecchio, M. G.; Aguirre, P. a.; Scenna, N. J. Global Optimal Design of Reverse Osmosis Networks for Seawater Desalination: Modeling and Algorithm. Desalination 2005, 184 (1-3), 259– 271.
(11)
Geraldes, V.; Pereira, N. E.; Norberta de Pinho, M. Simulation and Optimization of Medium-Sized Seawater Reverse Osmosis Processes with Spiral-Wound Modules. Ind. Eng. Chem. Res. 2005, 44, 1897–1905.
(12)
Lu, Y. Y.; Hu, Y. D.; Zhang, X. L.; Wu, L. Y.; Liu, Q. Z. Optimum Design of Reverse Osmosis System under Different Feed Concentration and Product Specification. J. Memb. Sci. 2007, 287 (2), 219–229.
(13)
Vince, F.; Marechal, F.; Aoustin, E.; Bréant, P. Multi-Objective Optimization of RO Desalination Plants. Desalination 2008, 222 (1-3), 96–118.
(14)
Kim, Y. S. Y. M.; Kim, S. J.; Kim, Y. S. Y. M.; Lee, S.; Kim, I. S.; Kim, J. H. Overview of Systems Engineering Approaches for a Large-Scale Seawater Desalination Plant with a Reverse Osmosis Network. Desalination 2009, 238 (1-3), 312–332.
(15)
Oh, H. J.; Hwang, T. M.; Lee, S. A Simplified Simulation Model of RO Systems for Seawater Desalination. Desalination 2009, 238 (1-3), 128–139.
(16)
Al-Bastaki, N. M.; Abbas, A. Predicting the Performance of RO Membranes. Desalination 2000, 132 (1-3), 181–187.
(17)
Wang, J.; Dlamini, D. S.; Mishra, A. K.; Theresa, M.; Pendergast, M.; Wong, M. C. Y.; Mamba, B. B.; Freger, V.; Verliefde, A. R. D.; Hoek, E. M. V. A Critical Review of Transport through Osmotic Membranes. J. Memb. Sci. 2014, 454, 516–537.
(18)
Malek, A.; Hawlader, M. N. A.; Ho, J. C. Design and Economics of RO Seawater Desalination. Desalination 1996, 105 (3), 245–261.
(19)
Du, Y.; Xie, L.; Wang, Y.; Xu, Y.; Wang, S. Optimization of Reverse Osmosis Networks with SpiralWound Modules. Ind. Eng. Chem. Res. 2012, 51, 11764–11777.
(20)
Sassi, K. M.; Mujtaba, I. M. Effective Design of Reverse Osmosis Based Desalination Process Considering Wide Range of Salinity and Seawater Temperature. Desalination 2012, 306, 8–16.
(21)
Alnouri, S. Y.; Linke, P. A Systematic Approach to Optimal Membrane Network Synthesis for Seawater Desalination. J. Memb. Sci. 2012, 417-418, 96–112.
(22)
Lu, Y.; Liao, A.; Hu, Y. Design of Reverse Osmosis Networks for Multiple Freshwater Production. Korean J. Chem. Eng. 2013, 30 (5), 988–996.
(23)
Skiborowski, M.; Mhamdi, A.; Kraemer, K.; Marquardt, W. Model-Based Structural Optimization of Seawater Desalination Plants. Desalination 2012, 292, 30–44.
(24)
Kleijnen, J. P. C. Regression and Kriging Metamodels with Their Experimental Designs in Simulation : A Review. 2017, 256, 1–16.
(25)
Mahmoudi, S.; Trivaudey, F.; Bouhaddi, N. Bene Fi Ts of Metamodel-Reduction for Nonlinear Dynamic Response Analysis of Damaged Composite Structures. Finite Elem. Anal. Des. 2016, 119, 1–14.
(26)
Practice, S. M. Simulation Modelling Practice and Theory Interval Metamodels for the Analysis of Simulation Input – Output Relations. 2015, 54, 86–100.
(27)
Wen, J.; Yang, H.; Jian, G.; Tong, X.; Li, K.; Wang, S. International Journal of Heat and Mass Transfer Energy and Cost Optimization of Shell and Tube Heat Exchanger with Helical Baffles Using Kriging Metamodel Based on MOGA. 2016, 98, 29–39.
(28)
Psaltis, A.; Sinoquet, D.; Pagot, A. Systematic Optimization Methodology for Heat Exchanger Network and Simultaneous Process Design. Comput. Chem. Eng. 2016, 95, 146–160.
36 ACS Paragon Plus Environment
Page 37 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
Industrial & Engineering Chemistry Research
(29)
Broad, D. R.; Dandy, G. C.; Maier, H. R. Environmental Modelling & Software A Systematic Approach to Determining Metamodel Scope for Risk- Based Optimization and Its Application to Water Distribution System. Environ. Model. Softw. 2015, 69, 382–395.
(30)
Beh, E. H. Y.; Zheng, F.; Dandy, G. C.; Maier, H. R.; Kapelan, Z. Environmental Modelling & Software Robust Optimization of Water Infrastructure Planning under Deep Uncertainty Using Metamodels. Environ. Model. Softw. 2017, 93, 92–105.
(31)
Bucher, C. Metamodels of Optimal Quality for Stochastic Structural Optimization. Probabilistic Eng. Mech. 2018, 54, 131–137.
(32)
Jaffal, I.; Inard, C. A Metamodel for Building Energy Performance. Energy Build. 2017, 151, 501– 510.
(33)
Guthrie, K. M. Data and Techniques for Preilinary Capital Cost Estimation. Chem. Eng. 1969, No. March 24, 114–142.
(34)
The Mathworks Inc. MATLAB - MathWorks http://www.mathworks.com/products/matlab/.
(35)
Fonteix, C.; Bicking, F.; Perrin, E.; Marc, I. Haploid and Diploid Algorithms, a New Approach for Global Optimization: Compared Performances. Int. J. Syst. Sci. 1995, 26 (10), 1919–1933.
(36)
Grossmann, I.; Viswanathan, J.; Vecchietti, A.; Raman, R.; Kalvelagen, E. DICOPT https://www.gams.com/.
(37)
Rosenthal, R. E. GAMS — A User ’ S Guide. 2015, No. October.
(38)
Sahinidis, N. V. BARON: A General Purpose Global Optimization Software Package. J. Glob. Optim. 1996, 8 (2), 201–205.
(39)
Misener, R.; Floudas, C. A. ANTIGONE: Algorithms for Continuous / Integer Global Optimization of Nonlinear Equations. J. Glob. Optim. 2014, 59 (2-3), 503–526.
(40)
Faria, C.; Bagajewicz, M. J. A New Approach for Global Optimization of a Class of MINLP Problems with Applications to Water Management and Pooling Problems. AICHE J. 2012, 58 (8).
(41)
Carvalho, M.; Secchi, A. R.; Bagajewicz, M. Model Reformulation and Global Optimization of Oil Production Using Gas Lift. Ind. Eng. Chem. Res. 2016, 55 (38).
(42)
Kim, S. Y.; Bagajewicz, M. Global Optimization of Heat Exchanger Networks Using a New Generalized Superstructure. Chem. Eng. Sci. 2016, 147, 30–46.
(43)
Kim, S. Y.; Jongsuwat, P.; Suriyapraphadilok, U.; Bagajewicz, M. Global Optimization of Heat Exchanger Networks. Part 1: Stages/Substages Superstructure. Ind. Eng. Chem. Res. 2017, 56 (20), 5944–5957.
(44)
Kim, S. Y.; Bagajewicz, M. Global Optimization of Heat Exchanger Networks. Part 2: Stages/Substages Superstructure with Variable Cp. Ind. Eng. Chem. Res. 2017, 56 (20), 5958– 5969.
(45)
Dow Water & Process Solutions. FILMTEC TM Reverse Osmosis Membranes Technical Manual, Formno.609-00071-1009.; 2011.
38 39 40 41
37 ACS Paragon Plus Environment
Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 38 of 38
1
Supporting Information:
2
Metamodel adjustment, operational region, set of constants for the linear and nonlinear-
3
regions, influence of decision variables. This information is available free of charge via
4
the Internet at http://pubs.acs.org/.
5 6
Abstract Graphic
7
38 ACS Paragon Plus Environment