Article pubs.acs.org/IECR
Effective Synthesis and Optimization Framework for Integrated Water and Membrane Networks: A Focus on Reverse Osmosis Membranes Esther Buabeng-Baidoo and Thokozani Majozi* School of Chemical and Metallurgical Engineering, University of the Witswatersrand, 1 Jan Smuts Avenue, Johannesburg 2001, South Africa ABSTRACT: Strict environmental regulations and social pressures have created the need for water and energy minimization in the process industries. Therefore, this work looks at the incorporation of a detailed reverse osmosis network (RON) superstructure within a water network superstructure in order to simultaneously minimize water, energy, operation, and capital costs. The water network consists of water sources, water sinks, and RO units for the partial treatment of the contaminated water. An overall mixed-integer nonlinear programming framework is developed that simultaneously evaluates both water recycle−reuse and regeneration reuse−recycle opportunities. The solution obtained from optimization provides the optimal connections between various units in the network arrangement, size and types of RO units, booster pumps, as well as energy recovery turbines. The paper looks at four cases to highlight the importance of including a detailed regeneration network within the water network instead of the traditional “black-box” model. The importance of using a variable removal ratio in the model is also highlighted by applying the work to a literature case study which leads to a 28% reduction in freshwater consumption and 80% reduction in wastewater generation.
1. INTRODUCTION The scarcity of water and strict environmental regulations have made sustainable engineering a prime concern in the process and manufacturing industries.1 Water minimization involves the reduction of freshwater use and effluent discharge in chemical plants. This is achieved through water reuse, water recycle, and water regeneration. Water reuse involves the use of wastewater in other operations except the process where it was originally used. Water recycle, however, allows the effluent to be used in any process including the process in which it was produced. In water regeneration, the effluent is partially treated before it is recycled or reused in other processes. Partial treatment can be achieved by using water purification units often classified as membrane and nonmembrane processes, e.g., reverse osmosis (RO) membranes and steam stripping, respectively.2 The purification of water through membrane systems is an energy intensive process. The minimization of energy within the water networks is also needed for sustainable development. Energy usage within the water network is largely associated with the regeneration units (membrane units). In most published work, however, membrane systems have been represented using the “black-box” approach3−5 which uses a simplified linear model to represent the membrane systems.5 This approach does not give an accurate representation of the energy consumption and associated costs of the membrane systems. A more rigorous representation of the regeneration unit is therefore needed.6 The regenerator unit considered in this paper is the reverse osmosis membrane. RO membranes are highly favored among other separation units because of their low energy consumption, ease of operation, and high product recovery and quality.7 RO membranes separate a water stream into a lean stream of low contaminant concentration known as permeate and a highly contaminated stream known as the retentate © 2015 American Chemical Society
stream. The process is achieved by applying an external pressure to the feed solution in order to reverse the osmotic phenomenon. As a result of this process, retentate streams exit the membrane at a high pressure. Energy recovery turbines are often placed in the retentate streams to offset the energy requirements of the entire unit. Figure 1 depicts the principle of RO membranes.
Figure 1. Schematic representation of a hollow fiber reverse osmosis membrane (HFRO).
The rigorous design of RO units has been extensively studied in the literature. Following from Evangelista,8 El-Halwagi7 introduced the “state space approach” for the design of RO networks (RONs). A superstructure representation of the RON was introduced with the objective of synthesizing a network Received: Revised: Accepted: Published: 9394
May 15, 2015 August 16, 2015 August 31, 2015 August 31, 2015 DOI: 10.1021/acs.iecr.5b01803 Ind. Eng. Chem. Res. 2015, 54, 9394−9406
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optimal WNS. A fixed flow rate model that considers the concept of sources and sinks is adopted. The model takes into account streams with multiple contaminants. The idea of using a variable removal ratio to describe the performance of the regenerators is also explored in this work.
with the optimal number of RO units, booster pumps, and energy recovery turbines that resulted in minimal freshwater consumption. Saif et al.9 and Sassi and Mujtaba10 have modified the model used by El-Halwagi7 by including scheduling, adding further design rules, simplifying the modeling of the RON, and including fouling effects in the modeling of the network. Saif et al.11 extended the superstructure of Saif et al.9 by applying an efficient branch-and-bound algorithm to obtain global optimality for the RON. This was achieved by the introduction of additional constraints to tighten the mathematical programming. Saif et al.11 obtained a total annualized cost (TAC) that was 14.8% lower than that obtained by El-Halwagi.7 There has, however, been few works that consider a detailed RON superstructure within a water network synthesis.6 There are two major approaches adopted in addressing water network synthesis, namely, insight-based techniques and mathematical model-based optimization methods. Insightbased techniques involve the water pinch analysis, which is a graphical method based on the concept of a limiting water profile which is the most contaminated water that can be fed into a particular operation. This method was first proposed by Wang and Smith.12 Hallale13 then proposed a graphical method that was based on nonmass transfer operations with single contaminants. Recent studies have extended water pinch analysis to algebraic methods, primarily water cascade analysis.14,15 The water pinch method proves unsuccessful for complex problems involving multiple contaminants16 and various topological constraints.4 The computation burden of this method is, however, lower than that experienced by mathematical modelbased optimization methods. The mathematical optimization approach employs a superstructure which identifies an optimal configuration for the process from a number of alternatives. This idea was first proposed in the work of Takama et al.17 They proposed a nonlinear model that incorporates both water-using and wastewater-treating units for multiple contaminant systems. Significant developments in the area have been achieved which includes the work of Galan and Grossmann,18 Karuppiah and Grossmann,19 and Tan et al.5 who explored different techniques for modeling regenerators and developing strategies for the complex mixed-integer nonlinear programming (MINLP) models. However, this method is computational expensive. Khor et al.2 presented a detailed membrane regenerator model which was incorporated into a water network superstructure (WNS). The model they proposed consisted of continuous variables for the contaminants and the flow rates as well as binary variables for the piping interconnections in conjunction with a nonlinear RON model. The resultant WNS was a MINLP model in structure. The multicontaminant model enabled direct water reuse−recycle, regeneration reuse, or regeneration recycling. The work of Khor et al.2 assumed a single regenerator with a fixed design, which implies that the number of regenerators needed, number of pumps, and number of energy recovery turbines were specified a priori. This limited the flexibility of the model, which could result in a suboptimal solution. The current work proposes a superstructure optimization for the synthesis of a detailed RON within a WNS. The overall mathematical formulation is used to minimize both water and energy simultaneously. A rigorous nonlinear RON superstructure model, which is based on the state space approach by El-Halwagi,7 is included in the WNS to determine the optimum number of RO units, pumps, and turbines required for an
2. PROBLEM STATEMENT The problem addressed in this work can be stated as follows: Given: (i) A set of water sources, I, with known flow rates and known contaminant concentration (ii) A set of water sinks, J, with known flow rates and known maximum allowable contaminant concentration (iii) A network of RO regenerators, Q, with known liquid recovery and design parameters (iv) A freshwater source, FW, with known contaminant concentration and variable flow rate (v) A wastewater sink, WW, with known maximum allowable contaminant concentration and variable flow rate Determine: (i) The minimal freshwater intake, wastewater generation, the energy consumed in the RON and the TAC (ii) The optimal configuration of the water network (iii) The optimal number of RO units, pumps, and energy recovery turbines (iv) The optimal operation and design conditions of the RON such as feed pressure, number of hollow fiber modules per regenerator, stream distributions, separation levels, etc. 3. MATHEMATICAL MODEL The overall MINLP model is based on the superstructure represented in Figure 2, which is adapted from the work of El-Halwagi7 and Khor et al.2 In the state space approach, the RO networks are split into four distribution boxes as shown in Figure 2: a pressurization−depressurization stream distribution box (PDSDB), pressurization−depressurisation matching box (PDMB), a RO stream-distribution box (ROSDB), and a RO matching box (ROMB). The purpose of the distribution boxes is to allow all possible combinations of stream mixing, splitting, recycle, and bypass. In the PDSDB, the sources, permeate, and retentate streams are fed into the box and are distributed to the final permeate and retentate stream or recycled to a regenerator. The streams then proceed to the PDMB where they can be sent to either a pump, energy recovery turbine, or directly to a regenerator without undergoing any pressure change. Thereafter, streams are fed to the ROSDB where they are allocated to the appropriate unit in the ROMB. In the ROMB, the streams are fed to the RO membranes where they are separated into a permeate stream and a retentate stream. The permeate and retentate streams are then fed to the PDSDB, and the cycle continues until an optimal network is obtained. The RON superstructure proposed by El-Halwagi7 must, however, be modified in order to incorporate it within the WNS. This is achieved by modifying the PDSDB and the PDMB sections of the superstructure. The properties of the updated PDSBD and PDMB are detailed below. (i) Water sources are fed directly to node n for regeneration and are not mixed with retentate or permeate streams. This was incorporated to ensure that each reteantate and 9395
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Figure 2. Superstructure representation of the RON superstructure within the WNS.
RON, wastewater sink, or to the water sinks. The flow rate balance is shown in constraint 1.
permeate stream leaves its respective regenerator without further contamination. (ii) Permeate and retentate streams are not allowed to mix in order for each stream to be fed directly from regenerator to the sinks. It is also assumed that each permeate stream will leave the regenerator at atmospheric pressure. Retentate streams, however, leave the RO at high pressures and are therefore passed through an energy recovery turbine for reduction in pressure to atmospheric pressure before distribution to the sinks. (iii) Different retentate streams or permeate streams are also not allowed to mix in order to feed each stream directly to the water sinks. Mixing of the streams within the water sinks is decided by water quality requirements of the sink. (iv) Each retentate stream or permeate stream can, therefore, go directly to a retentate node or can be recycled back to node n for further cleaning by the regenerators. (v) A stream that does not require a pressure change can be fed directly to the ROSDB where it is then fed to the ROMB. (vi) Inlet streams to box PDMB can either go to a pump or to an energy recovery turbine. The illustration of this idea is modified in order to clearly explain the original idea proposed by El-Halwagi.7 These modifications are illustrated in Figure 3a,b. Figure 3a shows the original PDSDB proposed by El-Halwagi,7 and Figure 3b shows the modified PDSDB and PDMB which will be incorporated with the WNS. 3.1. Water Balances for the Sources. Figure 4 shows a schematic representation of the water sources. From the diagram it can be seen that a water source can be fed to the
J
Fi =
N
∑ Fis,j + ∑ Fid,n j=1
∀i∈I (1)
n=1
It should also be noted that the freshwater source is included in the model as the last source within the model formulation. It can also be sent to the regenerators for further cleaning as its contaminant concentration is not zero. 3.2. Water Balances for the Sinks. Figure 5 shows a schematic representation of the water sinks. From the diagram it can be seen that the water sinks receive water from the water sources, permeate and retenetate of the regeneration units, as well as the freshwater source. This flow rate balance is shown in constraint 2. I
Fjw =
Q
Q
∑ Fis,j + ∑ Fqr,j + ∑ Fqp,j i=1
q=1
∀j∈J (2)
q=1
Each sink can, however, handle a certain concentration limit. Constraint 3 implies that the load to each sink must not exceed the maximum allowable load to that particular sink. I
CjU, m
≥
Q
Q
∑i = 1 Fis, jCis, m + ∑q = 1 Fqr, jCqr, m + ∑q = 1 Fqp, jCqp, m Fjw
∀j∈J ∀m∈M 9396
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Figure 3. (a) Original PDSDB and PDMB by El-Halwagi7 and (b) new modification to PDSDB and PDMB.
It should be noted that the wastewater sink is considered as the last sink. The maximum allowable load to this sink is also given in order to comply with the standard effluent discharge limits imposed by environmental regulations. In order to forbid the mixing of permeate and retentate streams from one regenerator in the same sink, constraint 4 is added to the model as follows: yqp, j + yqr, j ≤ 1
∀j∈J ∀q∈Q
(4)
3.3. Regeneration Unit (RON Superstructure). Figure 2 shows the schematic representation of the updated RON superstructure within the WNS superstructure. Figure 6 shows the interaction of the PDSBD with the sources and sinks of the WN. 3.3.1. Performance Equations. The performances of the RO regenerators are represented by means of the liquid recovery αq and removal ratio RRq,m. The liquid recovery is the amount of the feed flow rate into the regenerator that exits in the permeate stream. The removal ratio RRq,m refers to the fraction
Figure 4. Schematic representation of the water sources. 9397
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Figure 5. Schematic representation of the water sinks. I
of the inlet mass load that exits in the retentate stream of the regenerators.2 Constraints 5 and 6 represent the αq and RRq,m, respectively. αq =
Fqp Fqf
RR q , m =
FnaCna, m =
i=1
(5)
FL ≤
∀q∈Q ∀m∈M
Pqf ≤ Pmax
≤ FU
(6)
N
Fqp =
∀q∈Q
J
∑ Fqp,nCqp,m + ∑ Fqp,jCqp,m
N
Fqr =
∑ Fqr,n
i=1
q=1
q=1
j=1
∀q∈Q ∀m∈M
(12)
J
∑ Fqr,n + ∑ Fqr,j n=1
(13) J
∑ Fqr,nCqr,m + ∑ Fqr,jCqr,m n=1
∀n∈N
∀q∈Q
j=1 N
FqrCqr, m =
Q
∑ Fqp,n +
(11)
The balances for the flow rate and concentration of the retentate stream entering the PDSDB to the sinks are shown in constraints 13 and 14, respectively.
(8)
∑ Fid,n +
∀q∈Q
j=1
n=1
∀q∈Q
Q
J
N
FqpCqp, m =
(7)
I
q=1
(10)
∑ Fqp,n + ∑ Fqp,j n=1
3.3.2. RON Superstructure Equations. 3.3.2.1. Constraints for PDSDB. Constraint 9 shows the flow rate balance for the outlet junction of the PDSDB as can be seen in Figure 6. The node n represents a mixing junction at the outlet of the PDSBD. Fna =
q=1
The balances for the flow rate and concentration of the permeate stream entering the PDSDB to the sinks are shown in constraints 11 and 12, respectively.
The recommended operating flow rate for RO modules is given in constraint 7 and is determined by the manufacturers. Constraint 8 gives the upper bound for the feed pressure into the RO membranes. Fqf Nqs
Q
∀n∈N ∀m∈M
∀q∈Q Cqr, mFqr Cqf , mFqf
Q
∑ Fid, nCid, m + ∑ Fqp,nCqp,m + ∑ Fqr,nCqr,m
j=1
∀q∈Q ∀m∈M
(14)
Because the permeate and retentate streams from the regenerator are at different pressures, constraints have to be given to ensure that streams are at the same pressures before they mix. This is shown in constraints 15, 16, and 17 for the
(9)
Constraint 10 shows the corresponding concentration balance for the outlet junction of the PDSDB. 9398
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Figure 6. Schematic representation of the modified PDSDB.
feed, permeate, and retentate streams, respectively. Constraint 18 shows the isobaric mixing of streams within the ROSDB. (Pna − Piw )Fid, n = 0
∀n∈N ∀i∈I
(15)
(Pna − Pqp)Fqp, n = 0
∀n∈N ∀q∈Q
(16)
(Pna − Pqr)Fqr, n = 0
∀n∈N ∀q∈Q
(17)
(Pna − Pno)Fna, q = 0
∀n∈N ∀q∈Q
(18)
3.3.2.2. Constraints for PDMB and ROSDB. In the PDMB, the turbine is used to reduce the pressure of a stream while the pump is used to increase the pressure. Constraints 19 and 20 represent the principles of an energy recovery turbine and a pump, respectively. Figure 7 shows the schematic representation of the PDMB and RODB. (Pni − Pna) ≥ 0
∀n∈N
(19)
(Pni
∀n∈N
(20)
−
Pno)
≥0
Figure 7. Schematic representation of the PDMB and ROSDB.
The flow rate balance for the inlet of the ROSDB is given in constraint 21.
The outlet flow rate and concentration balance for the ROSDB are given in constraints 22 and 23, respectively.
Q
Fna =
∑ Fna,q q=1
N
Fqf
∀n∈N (21)
=
∑ Fna,q n=1
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Fqf Cqf , m =
∑ Fna,qCna,m q=1
⎛ ΔPqm ⎞ ΔPq = Pqf − ⎜⎜ + Pqp⎟⎟ ⎝ 2 ⎠
∀q∈Q ∀m∈M
(23)
CqU, m ≥
M
∑q = 1 Fna, qCna, m
∀q∈Q
Fqf
∀m∈M
Δπq = OS (24)
P Lbn ≤ Pni − Pna ≤ PUbn
∀n∈N
(25)
P Ltn ≤ Pni − Pno ≤ PUtn
∀n∈N
(26)
∀n∈N
Fqp Nqs
(30)
= ASm(ΔPq − Δπq)
∀q∈Q (31)
The average concentration, constraint 32. Cqav, m
=
Cav q,m,
on the feed side is given by
Cqf , m + Cqr, m
∀q∈Q
2
∀m∈M
(32)
The concentration of contaminants on the feed side must also be described in terms of the pressure drop and the osmotic pressure. This is described in constraint 33.
(27)
∀q∈Q
∀q∈Q
The permeate flow rate per module is given in constraint 31.
Cqp, m
Constraint 28 indicates the existence of a RO unit which is defined by the flow rate of the permeate stream from the regenerator q. FlLrq ≤ Fqp ≤ FlUrq
∑ Cqav,m m=1
3.3.2.3. Binary Variables for the Existence of Units. Constraint 25 shows that a booster pump exists in the RON if the Pin is larger than the pressure of the stream entering the PDMB, and this forces the binary variable bn to become 1. A similar concept is used to represent the existence of an energy recovery turbine and is given in constraint 26. However, It is illogical to pressurize and depressurize a stream simultaneously. Constraint 27 is therefore needed to prevent a turbine and pump from occurring in series.
bn + tn ≤ 1
(29)
The osmotic pressure, Δπq, is defined as a function of the contaminant concentration on the feed side9 and is shown in constraint 30.
The maximum inlet concentration limit to the regenerators must also be specified because not all of the waste streams can be fed to the RO membrane, and this is shown in constraint 24. Q
∀q∈Q
=
kmCqav, m
∀q∈Q
A(ΔPq − Δπq)γ
∀m∈M
(33)
A mass and concentration balance around the regenerator is also needed and is described in constraints 34 and 35, respectively.
(28)
Fqf = Fqp + Fqr
3.3.2.4. Constraints for ROMB. The characteristic of the RO membrane needs to be described in order to relate flow rate to pressure. The pressure drop across the membrane, ΔPq, is given in constraint 29.2 The equation was simplified by assuming a linear-shell side concentration and pressure profiles.20 The schematic representation of the ROMB is given in Figure 8.
∀q∈Q
(34)
Fqf Cqf , m = FqpCqp, m + FqrCqr, m
∀q∈Q ∀m∈M
(35)
3.4. Big-M Constraints. In order to determine the existence of piping interconnections, logical constraints and discrete variables will be adopted. This formulation makes use of the big-M parameters adopted by Khor et al.2 In the big-M parameters, M is a valid upper or lower bound denoted by U and L, respectively. Constraints 36−39 represent the big-M parameters for the piping interconnections between the different units.
Figure 8. Schematic representation of the ROMB.
MiL, jyi , j ≤ Fis, j ≤ MiU, j yi , j
(36)
MqL, jyqp, j ≤ Fqp, j ≤ MqU, jyqp, j
(37)
MqL, jyqr, j ≤ Fqr, j ≤ MqU, jyqr, j
(38)
Table 1. Limiting Data for Water Network sources, i
sinks, j contaminant concentration (kg/m3)
max contaminant concentration (kg/m3)
i
unit
flow rate (kg/s)
TDS
COD
J
unit
flow rate (kg/s)
TDS
COD
1 2 3 4
amine sweeting distillation
7.3 10.65 3.5
3.5 4 1 2
3.5 4 3 1
1 2 3 4
caustic treating menox-I sweeting desalting wastewater
0.83 40 5.56
2.5 2 2.5 25
2.5 2 2.5 25
hydrotreating freshwater
9400
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Table 2. Manhattan Distance for the Case Study
(39)
sinks
3.5. Objective Function. The objective function of the combined RON superstructure and WNS is used to minimize the overall cost of the regeneration network on an annualized basis, which consists of the following: (i) TAC of the RON (ii) Cost of freshwater (FW) (iii) Treatment cost of wastewater (WW) (iv) Capital and operation costs of the piping interconnection The total annualized cost of the RON consists of the capital cost of RO modules, pump, and energy recovery turbines; operating cost of pumps and turbines; as well as pretreatment of chemicals. The operating revenue of the energy recovery turbine is also considered in the determination of the TAC and is shown in constraint 40. N
j=1
⎛ ∑N (P i − P a)F a ⎞ n n n⎟ + C elec AOT⎜⎜ n = 1 ⎟ ηpump ⎝ ⎠
J
J
− C elec AOT(∑ (Pqr − P jr)Fqr, j)ηtur − C elec AOT(∑ (Pni − Pno)Fna)ηtur j=1
j=1
Q
I
+ C mod ∑ Nqs + C chem AOT ∑ Fid, n q=1
∀q∈Q
(40)
i=1
3
4
1
2
1 2 3 4 regenerator unit 1 2
50 60 50 60
50 50 50 50
50 60 50 60
60 70 60 70
50 40 65 100
50 40 50 50
80 60
70 10
60 40
70 20
The piping cost of components will be formulated by assuming a linear fixed-charge model. In their formulation, a particular cost of a pipe is incurred if the particular flow rate through the pipe falls below the threshold value. This is achieved by using 0−1 variables. Constraint 41 represents the objective function of the total regeneration network. It is also assumed that all the pipes share the same properties of pc and qc and a 1-norm distance, D. ⎛Q N ⎞ ⎜ ∑ ∑ TACq , n + AOTCCwater FW + AOTCwaste WW⎟ ⎜ q=1 n=1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ I J ⎞⎞ ⎛ p Fis, j ⎜+ AA⎜ ⎟ c ⎟ ⎟ ⎜ + D q y c i , j ⎟⎟ ⎜∑ ∑ i , j⎜⎝ 3600v ⎜ ⎟ ⎠ ⎠ ⎝ i=1 j=1 ⎜ ⎟ ⎜ ⎟ p ⎛ ⎞ q J ⎞ ⎛ p Fq , j ⎜ ⎟ c p p ⎟ ⎜ + qcyq , j ⎟⎟⎟ ⎟ min⎜+ AA⎜∑ ∑ Dq , j⎜⎜ ⎝ 3600v ⎠⎠ ⎜ ⎟ ⎝q=1 j=1 ⎜ ⎟ ⎜ ⎟ ⎛ q J ⎛ p Fqr, j ⎞⎞ r ⎟ ⎜+ AA⎜∑ ∑ Dr ⎜ c ⎟ ⎟ + q y q , j⎜ c q , j ⎟⎟ ⎜ ⎜ ⎟ ⎝ 3600v ⎠⎠ ⎝ q=1 j=1 ⎜ ⎟ ⎜ ⎟ d ⎛ ⎞ I N ⎛ ⎞ pF ⎜ ⎟ d ⎜ c i,n d ⎟⎟ ⎜ + qcyi , n )⎟⎟ ⎜⎜+ AA⎜∑ ∑ Di , n⎜ ⎟⎟ ⎝ 3600v ⎠⎠ ⎝ i=1 n=1 ⎝ ⎠
1.82 × 10−8 m/s 0.75 m 0.075 m 42 × 10−6 m 21 × 10−6 m 180 m 0.001 kg/(m s) 0.69 101 325 Pa 0.7 0.7 0.7 4.14 × 10−7 Pa 0.11 $/kg 0.06 $/(kW h) 6.5 $/(yearW0.65) 18.4 $/(yearW0.43) 2300 $/(year module) 0.27 kg/s 0.21 kg/s
parameter
value
annual operating time, AOT unit cost of freshwater, Cwater unit cost of wastewater, Cwaste interest rate per year, m number of years, n parameter p for carbon steel piping parameter q for carbon steel piping velocity, v
8760 h 1 $/t 1 $/t 5% 5 year 7200 250 1 m/s
Table 5. Summary of Results for Cases 1−3
where AA = m(1 + m) /[(1 + m) − 1 ] The overall model results in a nonconvex MINLP model due to the bilinear terms as well as the power function in the constraints. n
5.50 × 10−13 m/(s Pa) 4.05 × 104 Pa
Table 4. Economic Data and the Model Parameters for the WNS
(41) n
value
pure water permeability, A shell side pressure drop per module per regenerator, Pm solute permeability coefficient, km fiber length, L seal length, Ls outside radius of fiber, ro inner radius of fiber, ri membrane area, Sm water viscosity, μ dimensionless constant, ϒ permeate pressure per regenerator, Pp(q) pump efficiency, ηpump turbine efficiency, ηtur liquid recovery for all regenerators, α(q) osmotic constant, OS cost parameter for chemicals, Cchemical cost of electricity, Celec cost coefficient for pump, Cpump cost coefficient for turbine, Ctur cost per module of HFRO membrane, Cmod maximum flow rate per hollow fiber module, FU minimum flow rate per hollow fiber module, FL
n=1
+ C tur(∑ (Pqr − P jr)Fqr, j)0.43
2
parameter
N
n=1
1
Table 3. Process and Economic Data for the Detailed RON
TAC(q , n) = C pump(∑ (Pni − Pna)Fna)0.65 + C tur(∑ (Pni − Pno)Fna)0.43 J
regenerator unit
sources
n
freshwater flow rate (kg/s) wastewater flow rate (kg/s) cost of regeneration (million $/year) total cost (million $/year) CPU time (h)
4. ILLUSTRATIVE EXAMPLE The above model was applied to a literature-based refinery case study.2 The model was implemented in GAMS 24.2 using the 9401
no regeneration
single regenerator
two regenerators
(case 1)
fixed RR (case 2)
fixed RR (case 3)
38.40 13.40
32.54 7.59 0.068
28.87 3.91 0.23
1.70 0
1.40 0.13
1.32 6
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Figure 9. Network obtained for case 1 (no regeneration).
Figure 10. Network for case 2 based on the distribution boxes.
general purpose global optimization solver BARON, which obtains a solution by using a branch-and-reduce algorithm. The network consists of four sources and four sinks. The limiting water data for the sources and sinks is given in Table 1. Table 2 shows the Manhattan distances between different units. The distances between the regenerators and the sinks for both permeate and retentate streams is the same. Table 3 presents the process and economic data for the detailed RON. The economic data and the model parameters are given in Table 4.
Four scenarios will be compared to highlight the importance of incorporating a detailed RON superstructure within the water network. (i) First, the case in which no regeneration is considered within the water network is modeled to provide a basis (base case) for comparison (case 1). (ii) In the second case, a single regenerator is incorporated within the WNS with fixed removal ratio (case 2). 9402
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Figure 11. Network obtained for case 2 (single regenerator with fixed removal ratio).
Figure 12. Network obtained for case 3 (multiple regenerators with fixed removal ratio).
The results obtained from the optimization are given Table 5 for cases 1−3. In cases 2 and 3, the regenerators had a fixed removal ratio of 0.95. In the first scenario, the water network with no regeneration had a higher total cost due to the high
(iii) The third case considers multiple regenerators within the WNS with fixed removal ratio (case 3). (iv) Case 4 considers multiple regenerators with variable removal ratio. 9403
DOI: 10.1021/acs.iecr.5b01803 Ind. Eng. Chem. Res. 2015, 54, 9394−9406
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Industrial & Engineering Chemistry Research consumption of freshwater, which can be seen in Table 5, and the network is shown in Figure 9. The second scenario in which a single regenerator was used led to a 15.26% reduction in freshwater usage and a 43.36% in reduction in wastewater generation in comparison with the base case. The overall cost of the network was minimized by 17.6% because of the incorporation of the RO regenerator. The use of the energy recovery turbines in the RON led to a reduction in the
regeneration cost of the network. Figure 10 shows the complete water network and RON obtained for case 2. This diagram includes the distribution boxes as shown in Figure 2. Figure 10 can be translated into a simplified schematic diagram showing only the relevant physical units, i.e., RO membranes, pumps, turbines, mixes, and splitters. Figure 10 shows the water network for case 2. In Figure 11 it can be seen that one pump and turbine are needed for the regeneration as well as 20 HFRO modules. For simplicity, in cases 3 and 4 only the simplified water network is presented. Case 3 led to a 24.82% reduction in freshwater consumption and 70.82% reduction in wastewater generation in comparison with case 1. The total cost of the network was also reduced by 22.35%. The low cost of the water network is due to the low freshwater consumption and wastewater generation. The introduction of a second regenerator, case 3, leads to further reduction in the total cost. This is due to the lower consumption in freshwater and wastewater generation. Figure 12 shows the water network for case 3. In Figure 12 it can be seen that two pumps and turbines are needed for the regeneration as well as 37 HFRO modules per regenerator. A parallel configuration was chosen by the model.
Table 6. Summary of Results for Cases 3 and 4 multiple regenerators fixed RR (case 3) freshwater flow rate (kg/s) wastewater flow rate (kg/s) cost of regeneration (million $/year) total cost (million $/year) network configuration number of HFRO modules CPU time (h)
variable RR (case 4)
28.87 3.91 0.23
27.68 2.72 0.096
1.32 parallel 37 for each regenerator 6
1.11 parallel 15 for each regenerator 54
Figure 13. Network obtained for case 4 (multiple regenerators with variable removal ratio). 9404
DOI: 10.1021/acs.iecr.5b01803 Ind. Eng. Chem. Res. 2015, 54, 9394−9406
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Table 6 shows the comparison between cases 3 and 4. The removal ratio chosen by the model in case 4 was 0.97 for all contaminants. Case 4 led to 3.12% reduction in freshwater and 30.43% reduction in wastewater generation in comparison with case 3. A 15.91% reduction in the total network cost was also achieved. The large decrease in the total cost of the network in case 4 can be attributed to the high removal ratio which was selected by the model in comparison to the value that was initially predicted. In comparison with the case in which no regeneration was considered, case 4 leads to a 28% reduction in freshwater consumption and 80% reduction in wastewater generation. However, the modeling of case 4 is computationally expensive, as can be seen in Table 6. The best case used 15 HFRO modules per regenerator. The model selected two regenerators, two pumps, and two energy recovery turbines, as can be seen in Figure 13. It can also be seen that a parallel configuration of the network was chosen by the model. Flow rates obtained for the different streams are indicated in Figures 9−13. The high computational time for solving the model in case 3 was due to the complexity of the problem as well as the large number of 0−1 variables. The model solves more quickly when tighter bounds are imposed on the feed and retentate pressure. The use of the energy recovery turbines in the RON led to a reduction in the regeneration cost of the network and, as a result, a reduction in energy usage by the system was achieved. Table 7 contains the statistics of the model for all four cases.
number of equations number of continuous variables number of discrete variables optimality gap
single regenerator
(case 1)
fixed RR (case 2)
fixed RR (case 3)
variable RR (case 4)
60
168
282
282
46
134
208
212
16
32
48
48
0.1
0.1
0.1
0.1
AUTHOR INFORMATION
Corresponding Author
*Tel.: (+27) 11 7177567/7384. E-mail: thokozani.majozi@ wits.ac.za. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank the National Research Foundation (NRF) for funding this work under the NRF/DST Chair in Sustainable Process Engineering at the University of the Witwatersrand, Johannesburg, South Africa.
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NOMENCLATURE
Sets
I = {i|i = water source} J = {j|j = water sink} M = {m|m = contaminants} Q = {q|q = regeneration units} Parameters
αq = liquid recovery RRq,m = removal ratio FU = maximum flow rate per hollow fiber module FL = minimum flow rate per hollow fiber module ΔPmq = shell side pressure drop per module MU = upper bound of big-M constant for interconnections between streams ML = lower bound of big-M constant for interconnections between streams AOT = annual operating time pc = parameter for carbon steel piping based on CEPCI value of 318.3 qc = parameter for carbon steel piping based on CEPCI value of 318 v = velocity A = water permeability coefficient Pmax = maximum allowable pressure for the regenerators km = solute permeability constant L = fiber length Ls = seal length ro = outside radius of fiber ri = inner radius of fiber Sm = membrane area per module PU = an arbitrary big value for pressure PL = an arbitrary small value for pressure ϒ = a dimensionless constant ηpump = pump efficiency ηtur = turbine efficiency OS = proportionality constant between the osmotic pressure and average salt mass fraction on the feed side CUj,m = maximum allowable contaminant concentration m in sink j CUq,m = maximum allowable contaminant concentration m into a regenerator q Dai,j = Manhattan distance between water source i and sink j Dpq,j = Manhattan distance between regenerator q and sink j Drq,j = Manhattan distance between regenerator q and sink j Ddi,n = Manhattan distance between source i and node n Ci,m = mass fraction of contaminant m within water source i Cchem = cost parameter for chemicals Celec = cost of electricity
Table 7. Model Statistics for Cases 1−4 no regeneration
Article
multiple regenerators
5. CONCLUSION This paper has addressed the synthesis of a water regeneration network that incorporates the detailed synthesis of a RON. The proposed model was applied to a literature case study and was then solved using GAMS/BARON to highlight its practicality. The results show that the use of multiple regenerators in the water network can lead to a reduction in the total cost of the network due to the significant reduction in freshwater consumption and wastewater generation. It can also be concluded that there is a significant benefit in allowing the removal ratio in the model to be a variable as this has great significance on the cost and structure of the network. The implications of this study show that detailed optimization of regenerators within water networks can significantly improve wastewater management within process plants. However, large computational times were incurred because of the complex nature and structure of the model. It is also noteworthy that the proposed model was limited to one membrane technology. However, multiple membrane technologies such as ultrafiltration can be incorporated in the membrane network, thus offering a scope for future work. 9405
DOI: 10.1021/acs.iecr.5b01803 Ind. Eng. Chem. Res. 2015, 54, 9394−9406
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Industrial & Engineering Chemistry Research Cmod = cost per module of HFRO membrane Cpump = cost coefficient for pump Ctur = cost coefficient for turbine μ = water viscosity Ppq = pressure of a permeate stream from regenerator q Pwi = pressure of source i Prj = pressure of the retentate stream in sink j FlL = lower bound on flow rate FlU = upper bound on flow rate PL = lower bound on pressure PU = upper bound on pressure
ydi,n = 1 if piping exists between source i and node n; 0 otherwise Integer Variables
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N sq = the number of hollow fiber modules of regenerator q
REFERENCES
(1) Bandyopadhyay, S.; Cormos, C. Water management in process industries incorporating regeneration and recycle through a single treatment unit. Ind. Eng. Chem. Res. 2008, 47, 1111−1119. (2) Khor, C.; Foo, D.; El-Halwagi, M.; Tan, R.; Shah, N. A superstructure optimization approach for membrane separation-based water regeneration networks synthesis with detailed nonlinear mechanistic reverse osmosis model. Ind. Eng. Chem. Res. 2011, 50, 13444−13456. (3) Alva-Argáez, A.; Kokossis, A.; Smith, R. Wastewater minimisation of industrial systems using an integrated approach. Comput. Chem. Eng. 1998, 22, S741−S744. (4) Khor, C.; Chachuat, B.; Shah, N. A superstructure optimization approach for water network synthesis with membrane separationbased regenerators. Comput. Chem. Eng. 2012, 42, 48−63. (5) Tan, R.; Ng, D.; Foo, D.; Aviso, K. A superstructure model for the sybthesis of single-contaminant water network with partitioning regenerators. Process Saf. Environ. Prot. 2009, 87 (3), 197−205. (6) Khor, C.; Chachuat, B.; Shah, N. Optimization of water network synthesis for single-site and continous processes: milstones, challenges, and future directions. Ind. Eng. Chem. Res. 2014, 53, 10257−10275. (7) El-Halwagi, M. Synthesis of reverse osmosis networks for waste reduction. AIChE J. 1992, 38, 1185−1198. (8) Evangelista, F. A short-cut method for the design of reverseosmosis desalination plants. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 211. (9) Saif, Y.; Elkamel, A.; Pritzker, M. Optimal design of reverse osmosis networks for wastewater treatment. Chem. Eng. Process. 2008, 47, 2163−2174. (10) Sassi, K.; Mujtaba, I. OPtimization of design and operation of reverse osmosis based desalination process using MINLP approach incorporating fouling effects. Comput.-Aided Chem. Eng. 2011, 29, 206−210. (11) Saif, Y.; Elkamel, A.; Pritzker, M. Global optimization of reverse osmosis network for wastewater treatment and minimization. Ind. Eng. Chem. Res. 2008, 47 (1), 3060−3070. (12) Wang, Y. P.; Smith, R. Wastewater Minimisation. Chem. Eng. Sci. 1994, 49 (7), 981−1006. (13) Hallale, N. A new graphical targeting method for water minimisation. Adv. Environ. Res. 2002, 6, 377−390. (14) Ng, D.; Foo, D.; Tan, R. Targeting for total water networks.1. waste stream identification. Ind. Eng. Chem. Res. 2007, 46, 9107−9113. (15) Manan, Z.; Tan, Y.; Foo, D. Targeting the minimum water flow rate using water cascade analysis technique. AIChE J. 2004, 50 (12), 3169−3183. (16) Faria, D.; Bagajewicz, M. On the appropriate modelling of process plant water systems. AIChE J. 2009, 56, 668−689. (17) Takama, N.; Kuriyama, T.; Shiroko, K.; Umeda, T. Optimal water allocation in a petroleum refinery. Comput. Chem. Eng. 1980, 4, 251−258. (18) Galan, B.; Grossmann, I. Optimal design of distributed wastewater treatment networks. Ind. Eng. Chem. Res. 1998, 37 (10), 4036−4048. (19) Karuppiah, R.; Grossmann, I. Global optimization for the synthesis of integrated water systems in chemical processes. Computers and Chemical Engineering 2006, 30 (4), 650−673. Comput. Chem. Eng. 2006, 30 (4), 650−673. (20) El-Halwagi, M. Pollution Prevention through Process Integration; Academic Press: San Diego, CA, 1997.
Continuous Variables
Fsi,j = allocated flow rate between source i and sink j Fdi,n = allocated flow rate between source i and node n Fi = flow rate of sources i FPq,j = flow rate of the permeate stream from regenerators q to sink j Frq,j = flow rate of the retentate stream from regenerators q to sink j Fan,q = flow rate of streams from node n to regenerator q Ffq = flow rate leaving the outlet junction of ROSDB Fpq = flow rate of permeate stream leaving the regenerator q Fpq,n = flow rate of permeate stream regenerator q to node n Frq = flow rate of retentate stream leaving the regenerator q Frq,n = flow rate of retentate stream from regenerator q to node n Fan = flow rate of streams from node n Fwj = flow rate of sink j Can,m = concentration of contaminant m in stream leaving node n Cfq,m = concentration of contaminant m in the feed to the regenerator q Cpq,m = concentration of contaminant m in permeate stream leaving regenerator q Crq,m = concentration of contaminant m in retentate stream leaving regenerator q Cav q,m = average concentration of contaminant m in the highpressure side of regenerator q Pan = pressure of streams leaving node n Pin = pressure of an inlet stream to an energy recovery turbine from node n Pon = pressure of an outlet stream from an energy recovery turbine from node n ΔPq = pressure drop over regenerator q Pfq = feed pressure into regenerator q Prq = pressure of a retentate stream from regenerator q Ppq = pressure of a permeate stream from regenerator q Δπq = osmotic pressure on the retentate side of regenerator q FW = freshwater flow rate WW = wastewater flow rate Binary Variables
bn = 1 if a pump exists; 0 otherwise tn = 1 if a turbine exists; 0 otherwise rq = 1 if regenerator q exists; 0 otherwise ypq,j = 1 if piping exists between the permeate streams and sink j; 0 otherwise yrq,j = 1 if piping exists between the retentate streams and sink j; 0 otherwise yi,j = 1 if piping exists between source i and sink j; 0 otherwise 9406
DOI: 10.1021/acs.iecr.5b01803 Ind. Eng. Chem. Res. 2015, 54, 9394−9406
