Simultaneous Optimization of Property-Based Water-Allocation and

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Simultaneous optimization of property-based water-allocation and heat-exchange networks with state-space superstructure Ruijie Zhou, and Lijuan Li Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b01486 • Publication Date (Web): 29 Sep 2015 Downloaded from http://pubs.acs.org on October 3, 2015

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Industrial & Engineering Chemistry Research

Simultaneous optimization of property-based water-allocation and heat-exchange networks with state-space superstructure Rui-Jie Zhou* Department of Electrical Engineering, Stanford University, Stanford, 94305, USA Li-Juan Li Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, 12180, USA

Abstract A novel mathematical model for simultaneous optimization of property-based water-allocation and heat-exchange network (WAHEN) is presented in this work, where both linear and nonlinear dependent properties are taken into account. Specifically, state-space representation is modified to capture the structural characteristics of the property-based WAHEN, and a mixed-integer nonlinear programming (MINLP) model is formulated correspondingly to minimize the total annualized cost (TAC) of the network. In the proposed model, not only property and energy integrations are carried out simultaneously, but all possible waste regeneration opportunities are considered as well. To enhance the solution quality, sampling and spectral clustering techniques are introduced to capture the common characteristics of classes of good networks and guide the refined search toward a potential global optimum. Three examples are presented in this paper to demonstrate the validity and advantages of the proposed approach.

*Corresponding author. Email address: [email protected] (R.-J. Zhou), [email protected] (L.-J. Li)

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1. Introduction Water and energy are two of the most important resources for process industries. A large amount of water is needed not only as an effective mass-separating agent in separation operations but also as heat carriers in the utility systems to produce steam and/or cooling water. Recently, escalating price of energy, shortage of fresh water as well as strict environmental regulations have created increasing incentives to use an integrated approach for water-allocation and heat-exchange network (WAHEN) designs. This gives rise to the development of WAHEN design methodologies, which can facilitate simultaneously the optimal distribution of water and energy resources to satisfy process demands as well as environmental regulations at minimum total cost. The importance of simultaneous minimization of water and energy in heat integrated water system was brought to attention by Savelski and Bagajewicz1 as well as by Savulescu and Smith.2 Since then, the WAHEN design has long been the research focus of process systems engineering community. The techniques to synthesize WAHEN can be classified into three general categories: the insight-based pinch techniques,3-12 mathematical-based optimization approaches13-26 and the hybrid approaches.27-29 In the past few years, many conceptual tools for WAHEN design, such as separate system,3,4 source-demand energy composite curves,5,6 heat surplus diagram,7 superimposed mass and energy curves,8 temperature and concentration order composite curves,9 have been presented. These works based on pinch techniques have also considered a wide range of practical WAHEN design issues such as network complexity,10 sub-system analysis,11 homogeneous and heterogeneous mixing12 and multi-component systems.9 On the other hand, superstructure-based mathematical optimization methods have also been widely studied and relevant works can be classified into two general categories: sequential13-17 and simultaneous approaches.18-26 Bagajewicz et al.13 presented a sequential optimization method for energy efficient water network design. In


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their proposed method, a linear programming model is first solved to find the minimum water usage and minimum utility target values and then detailed allocation network can be found by solving a mixed-integer linear programming (MILP) model. Feng et al.14 proposed two methods to reduce the number of temperature fluctuations in water networks, which is helpful for the energy performance of the entire system. Liao et al.15 proposed a two-step optimization model for WAHEN design, where a transshipment model is first applied to target the number of heat exchange matches and detailed network design can be then obtained with a modified a stage-wise model.30 Biox et al.16 introduced a two-step mathematical optimization model for WAHEN design, where an MILP model is first solved to determine the water and energy allocation under several objectives and the best result obtained is improved through a mixed-integer nonlinear programming (MINLP) model. In a recent study, a two-step solution procedure was developed by Ibric et al.17 In their work, a nonlinear programming targeting model is first solved to minimize the operating cost of the network with a fixed heat recovery approach temperature and an MINLP synthesis model is then solved based on the initial values and bounds obtained in the first step. Although the abovementioned methods are relatively easy to implement, yet the sequential approaches cannot explore the interactions between different subsystems of WAHEN and as a result trade-offs between the capital and operation costs cannot be optimally balanced. To overcome this problem, many simultaneous WAHEN design methods have been proposed by explicitly considering the complex trade-offs between freshwater usage, utility consumption and investment for heat exchangers. Bogataj and Bagajewicz18 modified the stage-wise superstructure for heat-exchange network (HEN) by allowing cross stream mixing within stages and presented an MINLP model for the simultaneous synthesis of integrated WAHEN with regeneration units. In a later study by Zhou et al.,19 both MINLP model and equilibrium constraints model were developed for the WAHEN



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design and complementary formulations were also applied in modeling discrete decisions instead of binary variables. Although stream splitting and mixing as well as non-isothermal mixing were included in the above two works, yet mixing across stages cannot be achieved in the stage-wise framework and a large class of potential optimal network structure may be eliminated from the model formulation. On the other hand, Dong et al.20 proposed a modified state-space framework to capture all possible network structures of the integrated WAHEN, and an MINLP model with a hybrid solution strategy was proposed to minimize the TAC. In their model, not only all possible water reuse and treatment options were incorporated, but all potential direct and indirect heat-exchange opportunities were considered as well. Zhou et al.21,22 then extended the state-space framework and proposed a unified MINLP model for interplant WAHEN design with both direct and indirect integration schemes. Interplant WAHEN integrations for fixed contaminant-load processes, fixed flow rate processes as well as interplant WAHEN integrations between the above two processes were all addressed in their series of works. Later, Yang and Grossmann23 developed a linear programming water targeting model and incorporated this model with the available heat targeting models into an simultaneous optimization framework, where better operating conditions can be obtained. In a recent study, Ahmetovic and Kravanja24 combined water using network and HEN superstructure with interconnecting hot and cold streams and developed an MINLP model for WAHEN design. In their latest studies,25,26 the authors extended their previous study to include process-to-process stream for heat integration and wastewater treatment networks in WAHEN design. For a comprehensive review on methodologies for the synthesis of WAHENs, the reader is referred to the work by Ahmetovic et al.31 It is worth noting that in all above-mentioned works only the mass flow rate, compositions, and temperature of streams have been explicitly considered in constraints. However, as observed by


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El-Halwagi et al.,32 processes often impose constraints based on the functionalities or properties of streams that are difficult or even impossible to quantify from the compositions and temperature. For optimization of property-based water-allocation network (WAN), several methodologies have been suggested by Foo et al.,33 Ponce-Ortega et al.,34,35 Napoles-Rivera et al.,36 and Rojas-Torres et al.37 These works have considered various aspects of property-based WAN designs including property cascade analysis,33 life cycle assessment,34 global optimization algorithms,35 recycle and reuse schemes,36 thermal effects and temperature-based property operators.37 Recently, Kheireddine et al.38 considered the heat of mixing and the interdependence of properties in mass and property integration and proposed a nonlinear programming model that minimizes the TAC of direct reuse/recycle network with process and environmental constraints. However, only direct mixing was considered and the thermal constraints might not be satisfied only through direct mixing. To overcome this problem, Tan and coworkers39 developed an MINLP model for direct reuse/recycle systems based on the source-sink-HEN superstructure where each source is segregated and supplied to all sinks and all mixing points before the HEN. Minimum consumption of utilities was targeted prior to the detailed network design and floating pinch concept was adopted to identify hot and cold utilities. Later, Jimenez-Gutierrez and coworkers40 proposed an MINLP model to integrate energy, mass and properties in water network design. The mathematical model was developed based on a staged superstructure, where an energy integration was first considered, followed by a mass and property integration network, and ending in an additional stage for energy integration. Distributed property interceptors and stage-wise superstructure were used respectively for property and energy integrations and methods for identifying hot or cold stream were also proposed. Despite significant contribution accomplished by Jimenez-Gutierrez et al.,40 there are several drawbacks that inevitability lead to suboptimal designs of the resulting networks. One limitation is



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that the staged superstructure has reduced the design space and leads to the preclusion of a class of optimal network structures, where the cost-optimal scheme may actually lie. For instance, stream mixings are only allowed at the mixers and there is a lack of enough opportunities for both direct heat transfers and property blending in the network. In addition, indirect heat exchanges between streams in the two separate HENs are precluded. The second important problem is that the property interceptors were used in an end-of-pipe manner and no regeneration reuse and/or recycling option was considered. The design of property integration system with regeneration opportunity can not only improve the efficiency of the network structures, but also bring considerable economic and environmental savings. Finally, only linearly dependent properties (i.e. the property which is the same as the property operator) were considered in their work. Since many properties, such as pH value, in WAHEN design have nonlinear property operators, it is necessary and critical to study those properties in the context of WAHEN optimization. In this paper, a rigorous and novel mathematical programming model based on state-space framework is presented for the property-based WAHEN design problem under a recycle and reuse configuration. First, a state-space superstructure with both HEN and property interceptor operators is proposed to capture all possible structural characteristics of the WAHEN. Splitting and merging of streams are allowed throughout the network and property and energy integrations are performed in a simultaneous manner in the superstructure. Then, waste streams from the property interceptors are treated as part of the integration network and all potential regeneration reuse and/or recycling opportunities are considered. In order to address the nonlinearities introduced by property mixings (especially mixings for nonlinear dependent properties) as well as the combinatorial nature of the HEN design, a novel sampling and clustering based algorithm is developed and implemented in this work. The rationale for a clustering-based approach is that a class of good solution exhibits certain


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structural similarities. Specifically, spectral clustering is used to identify the most suitable region as well as initial guesses that facilitate the deterministic optimization solver to find the true optimal solutions. With all aforementioned novel features and improvements, it is expected that the TAC of a property-based WAHEN can be reduced significantly. To illustrate the overall design methods developed in this work, the rest of this paper is organized as follows. Assumptions and descriptions of the problem are formally stated in Section 2. The modified state-space representation and the model formulation are given in Section 3. Section 4 gives the outline and detailed procedures of the proposed solution strategy. Three demonstrative examples are then presented in Section 5, and the conclusions of this research are provided in the last section.

2. Problem Statement With the assumption that all properties in the system are independent, the WAHEN design problem can be stated as follow. Given a set of process and utility streams, a set of property interceptors for wastewater treatment, it is desired to synthesis a cost-optimal WAHEN satisfying the mass, property, temperature and flow rate requirement imposed at all sinks in the network. More specifically, the model inputs include: (1) the specifications of every source (i.e., its flow rate, temperature, heat capacity and each property value), (2) the specifications of every sink (i.e., its flow rate, temperature, upper and lower bound values of every property), (3) the specifications of each interceptor (i.e., its unit cost and conversion factor), (4) the costs freshwater as well as hot and cold utilities, (5) the cost of heat exchange units and (6) the mixing operators of all properties. The resulting property-based WAHEN design should include: (1) the throughput of every interceptor, (2) the number of heat exchangers and their duties, (3) the consumption rates of freshwater as well as the hot and cold utilities, (4) the values of all properties at the inlet of all sinks, and (5) the network configuration and the flow rate, temperature and value of each property in every branch stream.



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3. Superstructure and Mathematical Model 3.1 Superstructure Similar to any other optimization-based approaches in WAHEN design, it is necessary to build a superstructure in which all possible network configurations are embedded. In order to overcome the above-mentioned problems in the staged superstructure, we build our MINLP model based on a modified state-space framework. The state-space superstructure was first proposed by Bagajewicz and Manousiouthakis41 as the superstructure for the mass and heat exchange network design. This original structure is modified in the present work to incorporate the property and energy integration options. Figure 1 shows a schematic representation of the modified state-space superstructure. The overall framework can be viewed as a system of two interconnected blocks. One is referred to as the distribution network (DN), where all splitters, mixers, and the connections between them are embedded. The other is the so-called process operator (PO), which can be further divided into two sub-blocks, i.e., property interceptor operators and heat exchange operators. In the superstructure, every input to DN is split into several branches at the splitting node and each of them is connected to a mixing node leading to PO blocks or to the sinks. While property treatment and indirect heat exchange are carried out respectively through interceptor operator and HEN operator, direct heat exchange and property blending can be realized via direct stream mixing at all mixers attached on DN. The state-space approach has the following noticeable advantages over the staged framework for WAHEN design as it doesn't contain any sequence assumption or structural simplification of the mass and energy integration processes. Specifically, stream splitting and mixing can be realized at every splitter and mixer attached on DN and all regeneration reuse and/or recycle opportunities can be included and modeled via the interactions between DN and OP. For brevity, detailed descriptions on this state-space framework as well as the relations between the number of process streams and


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the number of mixing and splitting nodes attached to DN are omitted. The readers can refer to the work by Li et al.42 for further details. The mathematical models for both DN and PO are described in the following subsections. 3.2 Distribution network All operation flows in the systems are characterized by the splitting and/or mixing nodes (i.e., the splitters and/or mixers) attached to DN. For simplicity, set SP is introduced to represent all splitting nodes on DN, while set MX is used to denote all mixing nodes attached to DN. Furthermore, superscript "in" and "out" are used to represent respectively the physical quantities at the inlet and outlet of DN. Notice that, at every splitter and mixer, the property and energy balances must all be satisfied, i.e., f spin = ∑ fssp ,mx

∀sp ∈ SP (1)

mx

f mxout = ∑ fssp ,mx

∀mx ∈ MX (2)

sp

out in f mxout ⋅ψ mx , p = ∑ fssp , mx ⋅ψ sp , p

∀mx ∈ MX , p ∈ P (3)

sp

out f mxout ⋅ tmx = ∑ fssp ,mx ⋅ tspin

∀mx ∈ MX , (4)

sp

Fmxout,min ≤ f mxout ≤ Fmxout,max out max Ψ min mx , p ≤ ψ mx , p ≤ Ψ mx , p

out Tmxmin ≤ t mx ≤ Tmxmax

(5)

∀mx ∈ SINK

∀mx ∈ SINK , p ∈ P ∀mx ∈ SINK

(6)

(7)

where set P represents the set of all properties considered in the system; set SINK denotes the set of all sinks; f spin denotes the total mass flow rate at splitting node sp; f mxout stands for the total mass flow rate at mixing node mx; fssp ,mx denotes the mass flow rate from nodes sp to mx; ψ spin , p and out ψ mx , p represent, respectively, the values of property operator for property p at nodes sp and mx; out denote respectively the temperature of stream at nodes sp and mx; Fmxout,min and tspin and t mx

Fmxout,max are the lower and upper bounds of mass flow rate at node mx; Tmxmin and Tmxmax represent



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max respectively the lower and upper bounds of temperature at mixer mx; Ψ min mx , p and Ψ mx , p are,

respectively, the lower and upper bounds for property operator p at mixing node mx. Table 1 shows some property mixing operators used in this work and it can be found that some of them are linear (i.e., concentration, toxicity and chemical oxygen demand) and others are nonlinear (i.e., pH value). Property balances with above property operators at each mixing node are first proposed by Ponce-Ortega et al.35 and are shown in equation 3. Since the mass flow rate at mixer mx (i.e., f mxout ) is not fixed a priori, a set of bilinear terms appears in equation 3 as the product of mass flow rate and property operators. Solution to model with those constraints using local optimization solvers may produce suboptimal solutions and a hybrid algorithm is introduced in section 4 to mitigate the computational problem. Environmental and temperature restrictions for all process and waste sinks are enforced by equations 6 and 7. Finally, if not all streams in the system are allowed to mix, the following constraints should be enforced: fssp , mx = 0

∀sp ∈ SP , mx ∈ N sp

(8)

where set N sp denotes all forbidden mixing nodes of stream from node sp. 3.3 HEN Operators It is imperative to place enough number of heat exchangers in the state-space framework to match the hot and cold streams in the HEN operator. This appropriate number can be determined by the heuristic rules in section 3.2 of the work by Li et al.42 The mass and property balances around each heat exchange unit can be written as the following equalities: f mxout = f spin

∀he ∈ HE , mx ∈ H heout , sp ∈ H hein

(9)

f mxout = f spin

∀he ∈ HE , mx ∈ Cheout , sp ∈ Chein

(10)

out in ψ mx , p = ψ sp , p

∀he ∈ HE , mx ∈ H heout , sp ∈ H hein , p ∈ P

(11)

out in ψ mx , p = ψ sp , p

∀he ∈ HE , mx ∈ C heout , sp ∈ Chein , p ∈ P

(12)


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where set HE denotes the set of heat exchangers embedded in the superstructure; H heout and Cheout , the subset of MX, denote respectively the mixing node of the hot and cold streams at the inlet of unit he; H hein and Chein , the subset of SP, denote respectively the splitting node of hot and cold stream at the outlet of unit he. Constraints 9-12 are obvious since the all mass flow rates and property values should be identical around each heat exchange unit. To determine the existence and the heat load of each heat exchange unit, the following equations are needed: out qhe = f mxout ⋅ Cp ⋅ ( tmx − t spin ) out qhe = f mxout ⋅ Cp ⋅ ( t spin − tmx )


(13)


(14)

Q min ⋅ whe + ε ≤ qhe ≤ Q max ⋅ whe + ε

∀he ∈ HE

(15)

where Q min and Q max correspond to the lower and upper limits of allowed heat load; Cp stands for the heat capacity of streams in the system; binary variable whe denotes whether heat exchanger he is selected or not in the final network. Bilinear constraints 13 and 14 enforce that an equivalent amount of heat qhe is removed from the hot stream and then transferred to the cold stream. Constraint 15 enforces binary variable whe equals 1 when there is a heat load qhe between Q min and Q max . Finally, the temperature of hot and cold streams around every heat exchanger should satisfy the following thermodynamic constraints: 1 out ∆the = tmx − tspin

∀he ∈ HE , mx ∈ H heout , sp ∈ Chein

(16)

out ∆the2 = t spin − tmx

∀he ∈ HE , mx ∈ Cheout , sp ∈ H hein

(17)

out tmx − tspin ≥ 0 out t spin − tmx ≥0


(18)


(19)

1 ∆the + Γ ⋅ [1 − whe ] ≥ ∆Themin

∀he ∈ HE

(20)

∆the2 + Γ ⋅ [1 − whe ] ≥ ∆Themin

∀he ∈ HE

(21)

1 where ∆ t he and ∆ t he2 denote the temperature driving forces at both ends of the heat exchanger



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he . By specifying the minimum temperature difference ∆Themin , constraints 20 and 21 ensure that exchangers with infinite sizes do not occur in the network. Furthermore, if a match does not occur, the associated binary variable equals zero and the large positive upper bound Γ renders these two equations redundant.

3.4 Property Operators In this work, both treatment for discharge to environment and treatment for regeneration reuse and/or recycle are considered. Specifically, the outlet stream from each property interceptor can be sent to the process sinks, waste sinks, heat exchange units, and property interceptors (including the interceptors where the stream has been treated). The mathematical model for property operators are expressed as follows:

f mxout = f spin out tmx = tspin out pspin = α int p pmx

∀int ∈ INT , mx ∈ int out , sp ∈ int in ∀int ∈ INT , mx ∈ int out , sp ∈ int in

(22) (23)

∀int ∈ INT , mx ∈ int out , sp ∈ int in , p ∈ P

(24)

where set INT represents the set of property interceptors; sets int out and int in represent the mixing out and splitting nodes around interceptor int; pmx and pspin are, respectively, the values of property

p at node mx and sp; α int is the conversion factor of property p at interceptor int. While the mass p flow rate and temperature are the same around each interceptor, the inlet and outlet property values at each interceptor are related through the conversion factors given by equation 24.

3.5 Objective Function The objective function in this model is to minimize the TAC, taking into account, (1) the cost of freshwater, (2) the cost of hot and cold utilities, (3) the installation cost of heat exchange units and (4) the cost of property interceptors. The objective function can be written as follows:


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Objective function = Cost of freshwaters + Cost of hot and cold utilities + Cost of property interceptors + Cost of heat exchangers Cost of freshwater= Cost of hot and cold utilities =

∑

∑

fw ∈FW

h∈H

cost fw ⋅ f fwin

(26)

cost h ⋅ qh + ∑ c∈C cost c ⋅ qc

Cost of property interceptors= ∑ int∈INT cost int ⋅ fintout Cost of heat exchangers = ∑ he∈HE cost fix ⋅ whe

(25)

(27)

(28) (29)

where cost fw , cost h , cost c , cost int and cost fix are, respectively, the annualized cost coefficients for freshwater fw, hot utility h, cold utility c, property interceptor int and heat exchange units. Notice that heat transfer area and variable cost of heat exchangers are not considered due to the computational complexity introduced by the extra nonlinear terms in the constraints and objective function. In addition, the cost of pipelines are left out of consideration in the objective function since (1) the pipeline cost often accounts for a very small percentage of the TAC and (2) networks with multistage mixing and splitting can often be found based on state-space framework and it's practically impossible to characterize the identity or origin of each stream and quantify the unit cost between any pair of nodes in the system.

4. Solution strategies Given the large design space generated by state-space framework in one-step optimization, the present model still calls for the development of a more powerful solution strategy. In previous works, several hybrid algorithms20,43,44 involving both stochastic and deterministic components have been developed to avoid getting trapped into poor suboptimal solutions. In this work, a two step sampling and clustering based approach is proposed and the solution procedures are given in the flowchart shown in Figure 2. In the proposed algorithm, step I is used to find the distribution of the local optimal solutions of the MINLP problem and step II is aimed to locate the initial starting points that will guide the deterministic solver to the true optimum with clustering and perturbation ACS Paragon Plus Environment


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techniques. The DICOPT, CONOPT and CPLEX solvers in GAMS environment are used in this procedure to solve the MINLP model and GAMS is interfaced with MATLAB for executing the clustering analysis. The first step in step I of the proposed procedure is to produce a set of initial guesses of the decision variables with a random number generator and supply these guesses to the deterministic solvers. In the MINLP model, the selected variables for initial guesses are the binary variable whe , out in out and continuous variables fssp ,mx , ψ spin , p , ψ mx , p , t sp and t mx . The primary objective at this point is

to obtain the local optimal solutions of the original MINLP problem directly by solving the MINLP model with the randomly generated initial guesses. If the original MINLP model is infeasible, a relaxed MINLP (RMINLP) is formulated and then solved with the same initial values. If this attempt is successful, the solution of RMINLP is slightly modified to create an additional set of initial guesses for the original MINLP problem. In particular, if a relaxed binary variable is close to 1 (say, larger than 0.75), the corresponding initial value is set to one. A similar practice can be used to set the initial value of a close-to-zero (say, less than 0.25) binary variable. All other binary variables should be left unspecified in the new MINLP problem. However, if the RMINLP model is infeasible, the search procedure should be restarted by generating another set of initial guesses randomly. The above procedures are repeated until a reasonable number of feasible solutions are obtained. It should be noted that the aforementioned steps are almost identical to the procedure proposed by Dong et al.20 As observed in previous work,42 when the solution space is of moderate size, good local optimal solution might be found with step I. Effective clustering algorithm can capture the common structural and economic features that a class of optimal solutions share, such as source and interceptors connections, waste regeneration schemes and cost distributions. To enhance the solution quality, the spectral clustering technique is


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adopted to identify the candidate regions for further refined search in step II. Spectral clustering techniques make use of the spectrum of the similarity matrix of the data sets to perform dimension reduction before clustering in lower dimensions using K-means. In this work, the features selected for clustering analysis include the objective value, the cost for each utility and property interceptor as well as the mass flow rates between sources and interceptors. The normalized spectral clustering algorithm proposed by Ng et al.45 is used in this work and the detailed procedures are provided in Appendix A. To make the clustering algorithm stable, the number of clusters is chosen so as to maximizes the eigengap (i.e., the difference between consecutive eigenvalues) of the normalized graph Laplacians matrix. The benefit of using spectral clustering algorithm is that compared with the traditional K-means, spectral clustering can identify clusters with non-convex boundaries.45 After finding the candidate regions with spectral clustering, random perturbations are introduced into the immediate neighborhood of the centroids of each candidate region and new initial guesses are then created. More specifically, the flow rates in the new guesses are adjusted as follows:

fsspnew,mx = fsspold,mx (1 + ε sp , mx )

(30)

where ε sp ,mx is a random perturbation chosen in interval [-0.05, 0.05]. In addition, the binary variables in the new guesses are adjusted one-at-a-time. Each time one of whe is selected randomly and its value is then changed from 0 to 1 or vice versa. The original MINLP model is then solved with the perturbed initial guesses to search for improved solutions. The entire algorithm procedures of step II are illustrated in Figure 2(b), where K and P represent respectively the number of clusters and perturbations performed for each cluster.

5. Application Examples Three examples are presented to illustrate the advantages and effectiveness of the proposed MINLP model and solution strategy for property-based WAHEN design. The first two examples



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with single linear dependent property (i.e., toxicity) were originally solved with an MINLP model based on the staged superstructure by Jimenez-Gutierrez et al.40 The last example is adapted from the case by Napoles-Rivera et al.36 and is used to demonstrate the capabilities of the proposed approach for a more complex WAHEN design problem with multiple nonlinear dependent properties. All case studies are solved with a 2.9 GHz Intel i7-3520 processor in GAMS 23.7 environment.

5.1 Example 1 The first example is a single-contaminant WAHEN design problem adopted from Savalescu et al.4 The toxicity property was later introduced into this system by Jimenez-Gutierrez et al.40 The process data, which include the flow rates and operating conditions, for all sources and sinks are reported in Tables 2 and 3 respectively. To be able to compare different design strategies on the same basis, other model parameters are all taken from the work by Jimenez-Gutierrez et al.40 Specifically, the heat capacity for all streams are fixed at 4.3 kJ/kg K. Unit costs for concentration and toxicity interceptors are set to $25.74 and $38.88 h(kg y)-1 respectively and the corresponding conversion factors are chosen to be 0.02 and 0. The costs of cold and hot utilities are taken to be $10 and $110 (kW y)-1 respectively. Minimum temperature difference is assumed to be 10℃ and annualized fixed cost for heat exchangers is $15,000. Jimenez-Gutierrez et al.40 have solved this problem based on the staged superstructure and obtained the network with a TAC of approximately $18,839,670 without the pipeline cost. In this work, the same problem is solved based on the state-space MINLP model, which includes a set of 287 equations, 656 continuous and 24 binary variables. It can be found that the size of the proposed model is significantly smaller than the one proposed by Jimenez-Gutierrez and coworkers. This is due to the facts that (1) one single HEN operator, rather than two stage-wise HEN superstructure, is


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sufficient to consider all indirect heat exchange opportunities and that (2) classification of hot or cold stream is not needed in the state-space superstructure when mixing between streams is allowed throughout the network. A total of 142 different feasible WAHEN designs have been obtained in step I based on 1500 random guesses. Then four clusters are identified from spectral clustering in step II and for each cluster 25 perturbed initial guesses are passed to the MINLP solver for refined search. Table 4 shows the computation time and best solution found in each step. From Table 4, it can be found that step II is computationally cheaper than step I and a 1.6% reduction on the TAC is achieved with the aid of the refined search in step II. The optimal network found in step II, which has two heat exchangers and one heater, is shown in Figure 3. The corresponding TAC is found to be $11,642,570, which represents a 38.2% improvement. Table 5 provides some details about our solution and a comparison with the solution reported by Jimenez-Gutierrez et al.40 Specifically, although the consumptions of hot utility and fresh waters are approximately five times higher than those in previous work, the cost of interceptors is drastically reduced from $18,839,670 to $11,642,570, leading to a significant lower TAC. The overall cost reduction results from the better trade-offs among all cost terms by removing the structural assumption and allowing stream mixing in the state-space framework. In particular, since (1) there are more available design options for the interceptor systems and (2) the costs of freshwater are relative cheaper compared with those of interceptors, better trade-offs lead to a reduction in interceptor cost and an increase in freshwater consumption. The consumption of hot utility also increases considerably due to the large amount of cold freshwater (i.e., 100kg/s) used in the network. Finally, without the fixed sequence of mass and energy integrations in the staged superstructure, indirect heat exchange can be carried out between any pair of streams in the system. This is evident from the network in Figure 3, as heat exchanger 1 is used for indirect heat exchanges between the stream leading to toxicity interceptor and sink 3 and



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one split of stream from concentration interceptor. It should be noted that such structure can never be obtained with the original staged framework.

5.2 Example 2 The second study is based on the process of phenol production from cumene peroxide which was originally proposed by Ponce-Ortega et al.35 and Kheireddine et al.38 For brevity, we omit the detailed flowsheet of this process and the reader should refer to the above works for detail descriptions. This problem can be abstracted into a property-based WAHEN design problem with three waste sources and three sinks. Available for uses are two different fresh water sources as well as hot and cold utilities. Tables 6 and 7 give the process data and operation condition for each source and sink. All other parameters, including heat capacity, cost of utilities, annualized cost of heat exchangers and minimum temperature difference, are the same as those in example 1. The resulting MINLP model has 268 equations and 642 variables, out of which 9 are binary variables. 1500 MINLP problems based on random initial guesses have been solved in step I with 2590s and 132 different feasible solutions are found. Three clusters are then found from clustering analysis but better solution is not found in further refined search in step II. This is due to the fact that when the problem size or the solution space is not large, solving the MINLP repeatedly based on random guesses in step I may be sufficient to find a satisfied local optimal solution. The optimal network structure is shown in Figure 4 and the corresponding TAC is found to be $122,636. Table 8 compares the solution in Figure 4 with the one found by Jimenez-Gutierrez et al.40 While the costs of freshwater and heat exchangers increased from $0 to $1107 and $30,000 to $45,000 respectively in this work, savings in property treatment as well as hot and cold utilities amount to 62.6% and 20.1% respectively. Since property treatment cost is the main contributor, the entire trade-offs translate into a 46.2% reduction in the TAC. Like previous case, it can be found that a


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significant reduction in property treatment expenditure is accompanied by a slight increase in freshwater consumption rate. It is worth mentioning that the resulting network has one less heat exchanger than the network reported in reference 40. Indirect heat exchange between streams might be preferred when properties are involved as direct stream mixing will average the property values in merged stream and this may further increase the cost of property interceptors. In addition, while the proposed model can be extended to consider the distributed property interceptor system, it is highly desirable to have less property interceptors since this can save additional cost and avoid unfavorable environmental impact that are not taken into account in the objective function. Finally, both concentration and toxicity interceptors are used in an end-of-pipe manner and regeneration scheme is neither necessary nor cost-optimal when only two properties (i.e. composition and toxicity) are considered in the system.

5.3 Example 3 The last example in this work is adapted from the work by Napoles-Rivera et al.36 This case is studied to demonstrate the capabilities of the proposed model and algorithm for solving a more complicated problem with both linear and nonlinear mixing operators, a problem that has never been properly addressed in any of the previous studies. In this example, the performance of refined search in step II based on clusters obtained from K-means and spectral clustering is compared. For brevity, the process data for sources, sinks and interceptors is given in supporting information associated with this work. It is further assumed that the plant operates continuously for 8000h per year. Heat capacity values for all streams, minimum temperature difference and fixed cost of heat exchangers are the same as previous examples. The unit cost of fresh water is $0.03/kg and the annualized costs for hot and cold utilities are $388 (kW y)-1 and $189 (kW y)-1 respectively. In this example, it can be first observed that the density and viscosity of all process and utility



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streams are within the maximum lower bounds and minimum upper bounds of these values at all sinks. Since it has been assumed that all properties are independent, the above two properties can be eliminated from consideration in the proposed mathematical model. Although density is needed in the formulation of a rigorous chemical oxygen demand balance, yet the densities of all streams are approximately identical and the effect of mixing resulting from density difference can be safely ignored. This simplified MINLP model has 451 equations and 1008 variables, out of which 36 are binary variables. 2000 MINLP problems based on random initial guesses have been solved in step I with 4455s and 107 different feasible solutions are found. Based on the distribution of the 107 feasible solutions, four clusters can be found with both K-means and spectral clustering. 30 perturbations are performed for each cluster and the TAC can be further improved by 5.4% and 11.7% based on the clusters obtained from K-means and spectral clustering respectively. While the computational time using spectral clustering (i.e., 1168 s) is slightly higher than that of K-means (i.e., 1106 s), a further 6.3% cost reduction can be obtained via spectral clustering. The computation time and the optimal solutions found in each step are summarized in Table 9. Like the results in example 1, it can be seen that the computational time in step II is relatively small compared with the time spent on sampling in step I. The optimal network with a minimum TAC of $529,300 is shown in Figure 5 and the costs of hot utility, cold utility and interceptors are, respectively, $52,787, $18,997 and $427,516. Table 10 provides the values of the properties at all sinks in the resulting network. It can be observed that the compositions and the pH values at all sinks are driven respectively to the allowed upper and lower limits. The features and capabilities of the state-space based model can be clearly observed from the resulting network designs. First, since all process streams are mixable, indirect heat exchanges between streams are not needed and there is only one heater and one cooler in this network. Moreover, the network in Figure 5 includes an in-plant property treatment system


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as opposed to an end-of-pipe treatment facility and the waste streams are treated as part of the property integration network. Process sources 1, 2, 3, 4 and 6 are treated for composition; 149 kg/h of the output from the composition interceptor is sent directly to the COD interceptor; the output of COD interceptor together with one split from source 2 are mixed with the outputs from composition and pH interceptors at the inlet of the toxicity interceptor, which is used in an end-of-pipe manner. While the total amount of source 5 bypasses the use of interceptors and is directly sent to process sinks 2, 3 and 4, some branches from the outlets of pH and concentration interceptors are mixed multiple times and sent to the pH interceptor for further treatment. It is worth mentioning that such regeneration recycle structure has never been reported in previous literature. Finally, freshwater is not consumed in the system and as a result no additional wastewater is discharged to the waste sink.

6.Conclusions A state-space MINLP model has been presented in this work for one-step optimization of property-based WAHEN with both linear and nonlinear dependent properties. The advantage of this approach is that all possible design options, e.g., the integration of property and energy network, the regeneration of waste streams, the property blending and indirect heat exchanges with stream mixing, can be easily incorporated in the model formulation. A hybrid optimization algorithm has also been developed in this work to guarantee the solution quality and efficiency. Three examples were presented to demonstrate the applicability of the proposed model and algorithm, and results obtained so far showed that better overall network designs with less TAC can be obtained with the proposed approach.

Acknowledgements The authors would like to thank the anonymous reviewers for valuable suggestions.



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Supporting Information The process data for sources, sinks and interceptors in example 3. This information is available free of charge via the Internet at http://pubs.acs.org/.


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Notations Superscripts in=inlet to distribution network out=outlet from distribution network

Sets and Indices C= set of coolers FW = set of freshwaters H = set of heaters HE = set of heat exchangers, including heaters and coolers H heout , Cheout = the mixing nodes of the hot and cold streams at the inlet of unit he

H hein , Chein = the splitting nodes of the hot and cold streams at the outlet of unit he INT = set of property interceptors

int out , int in = the mixing and splitting nodes around interceptor int N sp =set of all forbidden mixing nodes of stream from sp P=set of all properties considered in the system SINK=set of all process and waste sinks SOURCE=set of all process sources SP=set of all splitting nodes in distribution network, SP = H hein ∪ Chein ∪ FW ∪ SOURCE ∪ INT in MX=set of all mixing nodes in distribution network, MX = H heout ∪ Cheout ∪ SINK ∪ INT out

Parameters Fmxout,min , Fmxout,max = the lower and upper bounds of the mass flow rate at mixing node mx max Ψ min mx , p , Ψ mx , p = the lower and upper bounds of the property operator p at mixing node mx



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Tmxmin , Tmxmax = the lower and upper bounds of the temperature at mixing node mx

Q min , Q max = the lower and upper limits of allowed heat load of each heat exchanger

∆Themin = minimum temperature difference for heat exchanger he

Cp = heat capacity of all streams in the system

α int = the conversion factor of property p at interceptor int p cost fw = annualized cost coefficient for freshwaters cost h , cost c = annualized cost coefficient for hot and cold utilities cost int = annualized cost coefficient for interceptor int cost fix = annualized fixed cost coefficient for heat exchangers

Γ = a sufficient large positive number

Continuous variables

f mxout = total mass flow rate at mixing node mx f spin = total mass flow rate at splitting node sp

fssp , mx = mass flow rate from splitting node sp to mixing node mx out ψ spin , p ,ψ mx , p = the value of property operator for property p at nodes sp and mx out = the temperature of stream at nodes sp and mx tspin , t mx

qhe = heat exchange rate of heat exchanger he 1 ∆ t he , ∆ t he2 = the temperature driving forces at both ends of the heat exchanger he out , pspin = the values of property p at nodes mx and sp pmx

Binary variables whe = binary variables denoting the existence/nonexistence heat exchanger he in the network


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(29) Sahu, G. C.; Bandyopadhyay, S. Energy optimization in heat integrated water allocation networks. Chem. Eng. Sci. 2012, 69, 352-364. (30) Yee, T. F.; Grossmann, I. E. Simultaneous optimization models for heat integration. II. Heat exchanger network synthesis. Comput. Chem. Eng. 1990, 14, 1165-1184. (31) Ahmetovic, E.; Ibric, N.; Kravanja, Z.; Grossmann, I. E. Water and energy integration: A comprehensive literature review of non-isothermal water network synthesis. Comput. Chem. Eng. 2015, 82, 144-171. (32) El-Halwagi, M. M.; Glasgow, I. M.; Qin, X. Y.; Eden, M. R. Property integration: componentless design techniques and visualization tools. AIChE J. 2004, 50, 1854-1869. (33) Foo, D. C. Y.; Kazantzi, V.; El-Halwagi, M. M.; Manan, Z. A. Surplus diagram and cascade analysis technique for targeting property-based material reuse network. Chem. Eng. Sci. 2006, 61, 2626-2642. (34) Ponce-Ortega, J. M.; Mosqueda-Jimenez, F. W.; Serna-Gonzalez, M.; Jimenez-Gutierrez, A.; El-Halwagi, M. M. A property-based approach to the synthesis of material conservation networks with economic and environmental objectives. AIChE J. 2011, 57, 2369-2387. (35) Ponce-Ortega, J. M.; El-Halwagi, M. M.; Jimenez-Gutierrez, A. Global optimization of property-based recycle and reuse networks including environmental constraints. Comput. Chem. Eng. 2010, 34, 318-330. (36) Napoles-Rivera, F.; Ponce-Ortega, J. M.; El-Halwagi, M. M.; Jimenez-Gutierrez, A. Global optimization of mass and property integration networks with in-plant property interceptors. Chem. Eng. Sci. 2010, 65, 4363-4377. (37) Rojas-Torres, M. G.; Ponce-Ortega, J. M.; Serna-Gonzalez, M.; Napoles-Rivera, F.; El-Halwagi, M. M. Synthesis of water networks involving temperature-based property operators and thermal integration. Ind. Eng. Chem. Res. 2013, 52, 442-461. (38) Kheireddine, H.; Dadmohammadi, Y.; Deng, C.; Feng, X.; El-Halwagi, M. M. Optimization of direct recycle networks with the simultaneous consideration of property, mass, and thermal effects. Ind. Eng. Chem. Res. 2011, 50, 3754-3762. (39) Tan, Y. L.; Ng, D. K. S.; El-Halwagi, M. M.; Foo, D. Y. C.; Samyudia, Y. Synthesis of Heat Integrated Resource Conservation Networks with Varying Operating Parameters. Ind. Eng. Chem. Res. 2013, 52, 7196-7210. (40) Jimenez-Gutierrez, A.; Lona-Ramirez, J.; Ponce-Ortega, J. M.; El-Halwagi, M. M. An MINLP model for the simultaneous integration of energy, mass and properties in water networks. Comput. Chem. Eng. 2014, 71, 52-66. (41) Bagajewicz, M.; Manousiouthakis, V. Mass/heat-exchange network representation of ACS Paragon Plus Environment


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Appendix A. Normalized Spectral Clustering Algorithm

Input: Given a set of points S = {s1 ,..., sn } in

•

l

, number of k cluster to construct.

Construct a fully connected similarity graph using Gaussian similarity function. Let W be its weighted adjacency matrix.

•

Compute the normalized Laplacian Lsys .

•

Compute the first k eigenvectors u1 ,..., uk of Lsys .

•

Let U ∈

•

Construct the matrix T ∈

n ×k

be the matrix containing vectors u1 ,..., uk as columns. n ×k

from U by normalizing the rows to norm 1, that is set

ti , j = ui , j / (∑ k ui2,k )1/ 2 . •

For i = 1,..., n , let yi ∈ R k be the vector corresponding to the i-th row of T .

•

Cluster the points ( yi )i =1,...,n with k-means algorithm into clusters C1 ,..., Ck .

Output: Cluster A1 ,..., Ak with Ai = { j | y j ∈ Ci } .



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List of tables Table 1. Property mixing operators Table 2. Source data for example 1 Table 3. Sink data for example 1 Table 4. Computation time and best solution in each step for example 1 Table 5. Comparisons of cost in example 1 Table 6. Source data for example 2 Table 7. Sink data for example 2 Table 8. Comparisons of cost in example 2 Table 9. Computation time and best solution in each step for example 3 Table 10. Properties at the sinks for example 3


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Table 1. Property mixing operators Property

Mixing operator

Concentration

ψ (z) = z

Toxicity

ψ (Tox ) = Tox

Chemical oxygen demand

ψ (COD ) = COD

pH

ψ ( pH ) = 10 pH



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Table 2. Source data for example 1 Sources

Flow rate (kg/s)

Temperature (ºC)

Source 1 Source 2 Source 3 Source 4 Fresh water 1 Fresh water 2

20 100 40 10

40 100 75 50 20 15

Phenol concentration (ppm) 100 100 800 800 0 10


Toxicity

Cost ($ h)/ (kg y)

0.3 0.5 1.5 0.75 0 0

NA NA NA NA 2.376 2.538

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Table 3. Sink data for example 1 Sinks

Flow rate (kg/s)

Temperature ( ºC)

Sink 1 Sink 2 Sink 3 Sink 4 Waste sink

20 100 40 10

40 100 75 50 30

Minimum phenol concentration (ppm) 0 40 40 50 0


Maximum phenol concentration (ppm) 0 50 50 80 20

Maximum Toxicity 0 0.9 1.6 0.85 0


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Table 4. Computation time and best solution in each step for example 1 Best solution found in each step Time (s) Utility Interceptors Heat exchanger ($/year) ($/year) ($/year) Step I 2445.2 1336532 10450811 45000 Step II 632.7 1328360 10269210 45000


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TAC ($/year) 11832343 11642570

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Table 5. Comparisons of cost in example 1 Hot utility ($/year) 40 Jimenez-Gutierrez et al. 92,400 This paper 473,000


Fresh waters ($/year)

Interceptors ($/year)

171,070 855,360

18,531,200 10,269,210


Heat exchangers ($/year) 45,000 45,000

TAC ($/year) 18,839,670 11,642,570


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Table 6. Source data for example 2 Sources

Flow rate (kg/h)

Temperature (ºC)

Washer 101 Decanter 101 Washer 102 Fresh water 1 Fresh water 2

3661 1766 1485

75 65 40 25 35

Phenol concentration (mass fraction) 0.016 0.024 0.22 0 0.012


Toxicity

Cost ($ h)/ (kg y)

0.3 0.5 1.5 0 0

NA NA NA 2.376 2.538

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Table 7. Sink data for example 2 Sinks

Flow rate (kg/h)

Temperature (ºC)

Wash 101 Wash 102 Neutralizer R104 Waste

2718 1993 1127

80 75 65 25

Maximum phenol concentration (mass fraction) 0.013 0.011 0.1 0.15


Maximum Toxicity 2 2 2 0


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Table 8. Comparisons of cost in example 2 Hot utility Cold utility Fresh waters ($/year) ($/year) ($/year) 40 Jimenez-Gutierrez et al. 5,641 300 0 196 This paper 4,549 1,107 a This TAC is calculated by subtracting the pipeline cost from the original TAC


Interceptors ($/year) 192,027 71,784

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Heat exchangers ($/year) 30,000 45,000

TAC ($/year) 227,968a 122,636

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Table 9. Computation time and best solution in each step for example 3 Best solution found in each step Time (s) Utility Interceptors Heat exchanger ($/year) ($/year) ($/year) Step I 4455 88735 465506 45000 Step II with K-means 1106 84323 452549 30000 1168 71784 427516 30000 Step II with spectral clustering


TAC ($/year) 599241 566872 529300


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

Table 10. Properties at the sinks for example 3 Composition Sinks Toxicity (%) (ppm) Sink 1 0.1 0.807 Sink 2 0.01 0.864 Sink 3 0.04 0.662 Sink 4 0.02 0.851 Sink 5 0.01 0.826 Sink 6 0.01 0.826 Waste sink 0.005 0

THOD (mg O2/kg) 86.08 96.039 97.831 96.048 95.612 95.612 50


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pH 5.8 5.5 5.4 5.5 5.5 5.5 5.4

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List of Figures Figure 1. The state-space superstructure Figure 2. Solution Strategies Figure 3. Optimal network structure for example 1 Figure 4. Optimal network structure for example 2 Figure 5. Optimal network structure for example 3



M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

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

Figure 1. The state-space superstructure


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Start

Start Identify K centroids based on spectral clustering

Generate random initial guesses for selected continuous and binary variables

k=1

N

p=1 Y Solve MINLP

Feasible ?

N Feasible ?

Perturb the split ratio and binary variable of jth solution

Solve RMINLP

Solve MINLP

Y Store the solution

Store the solution

Y p=p+1 p

Simultaneous Optimization of Property-Based Water-Allocation and

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