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Reverse Osmosis Network Rigorous Design Optimization Abdon Parra, Mario Andres Noriega, Lidia Yokoyama, and Miguel J. Bagajewicz Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02639 • Publication Date (Web): 17 Jan 2019 Downloaded from http://pubs.acs.org on January 26, 2019

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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

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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.


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1


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


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


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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.



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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

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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.



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:


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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,e1CmBe,e1  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

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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

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1


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)



PmBe,e  PmBe,e1  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:


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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.



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

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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 QtTˆT   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


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


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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


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

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1


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



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

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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


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)


Page 23 of 38


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


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.


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


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


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


(a)


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


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


(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


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


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


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


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


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


Figure 8. Permeate flow (2 stages, 500 ppm permeate).

8 29 ACS Paragon Plus Environment


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

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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



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


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


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


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


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


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

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38 39 40 41



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


Reverse Osmosis Network Rigorous Design Optimization | Industrial

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