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Synthesis of Interplant Water-Allocation and Heat-Exchange Networks. Part 2: Integrations between Fixed Flow Rate and Fixed Contaminant-Load Processes Rui-Jie Zhou,† Li-Juan Li,‡ Hong-Guang Dong,‡,* and Ignacio E. Grossmann§ †

Center for Process Systems Engineering, Imperial College London, London, SW7 2AZ, U.K. School of Chemical Engineering, Dalian University of Technology, Dalian, 116012, PRC § Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, 15213, United States ‡

ABSTRACT: Part 1 of the series proposes a multiscale state-space superstructure for interplant water-allocation and heatexchange networks (IWAHENs) design with direct and indirect integration schemes in fixed flow rate (FF) processes (Zhou, R. J.; Li. L. J.; Dong, H. G.; Grossmann, I. E. Synthesis of Interplant Water-Allocation and Heat-Exchange Networks. Part 1: Fixed Flowrate Processes. Ind. Eng. Chem. Res. 2012, 51, 4299). Based on the same superstructure, part 2 of this series extends the IWAHEN integration methods to fixed contaminant-load (FC) processes as well as integration of FF and FC processes. The integrations are performed by solving the corresponding mixed-integer nonlinear programming (MINLP) models based on the multiscale state-space superstructure. Several relevant examples, including both direct and indirect integrations for IWAHENs, are presented to illustrate various aspects of the proposed approach.

1. INTRODUCTION

2. PROBLEM STATEMENT The IWAHEN design problem with both FF and FC processes addressed in this paper is stated as follows: Given is an industrial complex with a set of process plants. Each plant has either a set of water using units, where a set of transferable contaminants are to be removed, or a set of water sources and sinks with fixed flow rates and operating temperatures. Available for services are utilities (freshwater, cold and hot utilities), a set of interplant junctions as well as wastewatertreatment units. The aim of IWAHEN design is to integrate the individual WAHEN with the two proposed integration schemes, in which the optimum performance, such as TAC, can be achieved under all process and environmental constraints. Two kinds of integrations, namely direct and indirect integrations, are considered in this work. In direct IWAHEN design, the overall complex will be treated as a single plant and all process streams are allowed to be directly mixed at any points in the network. On the other hand, no direct stream connection between stand-alone plants is considered in indirect IWAHEN. Individual plants are interconnected via central junctions or treatment units.

In part 1 of this series, a systematic superstructure-based optimization framework is presented for the interplant water-allocation and heat-exchange network (IWAHEN) design for fixed flow rate (FF) processes. Specifically, the state-space superstructure for a stand-alone plant is first introduced and a mixed-integer nonlinear programming (MINLP) model was solved to obtain the resulting individual water-allocation and heat-exchange networks (WAHENs). Then, by embedding such individual state-space framework into the interplant state-space representation, all intra- and interplant connections for both water-allocation networks (WANs) and heat-exchange networks (HENs) can be captured. For all cases shown in part 1, better network structures with lower total annualized cost (TAC) can always be found for both direct and indirect integrations by the proposed methods. In part 2 of this series, we focus on the development of a mathematical model for the IWAHEN designs with (1) fixed contaminant-load (FC) processes and (2) integrations of FC and FF processes. To the best knowledge of the authors, this is the first time that hybrid IWAHEN designs with both types of processes have been addressed. Like in part 1 of our work, the multiscale state-space superstructure will be used to integrate these two different kinds of processes. The most important benefit of using such representation is that IWAHEN design can be addressed in its entirety, since the design alternatives and interactions between (1) WANs and HENs, (2) FF and FC processes, and (3) intra- and interplant structures are all considered simultaneously in the framework. The rest of this paper is structured as follows. First, a brief statement of the IWAHEN design problem is given in the next section. Next, the superstructures as well as the mathematical model are presented. Finally, three examples with both direct and indirect integrations are solved to demonstrate our approach in IWAHEN designs. © 2012 American Chemical Society

3. MULTISCALE STATE−SPACE SUPERSTRUCTURE In part 1 of this series, the state-space superstructure is used to develop the framework for both individual WAHEN and IWAHEN. In addition to the capability of including all possible network structures, the state-space superstructure can also effectively handle the synthesis of networks with multiple physical properties. As stated in part 1, all flows in the system are identified Received: Revised: Accepted: Published: 14793

July 24, 2012 September 19, 2012 October 14, 2012 October 15, 2012 dx.doi.org/10.1021/ie3019752 | Ind. Eng. Chem. Res. 2012, 51, 14793−14805

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by both the splitting node where the streams emanate and the mixing node where the streams merge. One major benefit of such an approach is that it provides more opportunities for stream mixing1 (i.e., water reusing/recycling and heat recovery opportunities). Also, since the network flows are featured with nodes, we only need to embed the FC-based water user in the IWAHEN superstructure and develop the corresponding models for the mass removal processes. No additional work is needed to distinguish the input/output for a FC water-using unit and the sinks/sources for the FF system in the mass and energy balances. In this work, the state-space superstructure proposed by part 1 of this work is adopted as the WAHEN superstructure for a single plant with both FF and FC water-using operations. Detailed description of this framework can be found in section 3.1 in our previous work.2 The final superstructure for IWAHEN is obtained by embedding the superstructure for individual WAHEN (see section 3.1 in part 1 of the series) into the PO_PLANT operators in the IWAHEN framework (see section 3.2 in part 1 of the series). For simplicity, the descriptions of these two superstructures are omitted and the reader can refer to section 3 of our previous work.2

CIN, COUT = the cold stream entering to and released from heat exchange unit CPS = the cold process streams CUS = the cold utility streams H = heaters HE = heat exchangers (not including the heaters and coolers) HIN, HOUT = the hot stream entering to and released from heat exchange unit HPS = the hot process streams HUS = the hot utility streams IP = the splitting nodes for source i in DN p JP = the mixing nodes for sink j in DN p JUN = junctions K = components NIP = all forbidden mixing nodes of stream from node IP TR = wastewater treatment units W = fixed contaminant-load water-using units With these definitions, the mathematical model is described in the next sections. 4.1. Model for DNs. At every splitter and mixer, the mass and energy balances must all be satisfied; that is,

4. MATHEMATICAL MODEL Within our methodology, superstructures are built from three basic element types: process units, nodes, and streams. Process units include all water-users and heat-exchangers. Nodes are used to connect between process units and the streams. Each node is characterized by the following three sets of variables: flow rate, temperature, and concentrations. Although different nodes are used to denote the different families of sources and sinks for the FF and FC-based processes, to ensure the general expressions of our model we make no distinction of the identities of splitting/ mixing nodes in formulating the mass and energy balances. Furthermore, different process models are developed for different types of unit operations and all units in each type are grouped together in the same process operator (PO). Therefore, the entire model is made up of one set of general mass and energy balances for distribution network (DN) and different sets of process models for POs. Minimum effort is needed when we incorporate other types of process units, such as FC-based water users in the flowsheet. For completeness and clarity, the mathematical model for this study is as follows:

f iout = p

∑ fsi ,j

f jin = p

∀ jp ∈ JP

p ′ , jp

ip ′

p

(2)

∑ fsi

p ′ , jp

t iout p′

∀ jp ∈ JP

ip ′

f jin c inj , k = p

(1)

∑ fsi

f jin t inj = p

∀ ip ∈ IP

p p′

jp ′

p

(3)

∑ fsi

p ′ , jp

ciout p′, k

∀ jp ∈ JP , k ∈ K

ip ′

(4)

in where f out ip and f jp represent respectively the total flow rates at in each splitting and mixing node; tout ip′ and tjp′ stand for respectively the temperatures at each splitting and mixing points; cjinp,k and ciout p′,k denote respectively the concentration of pollutant k at each mixing and splitting point; fsip,jp stands for the flow rate from ip to jp. Since the inlet and outlet flow rates and concentrations of internal sources and sinks for each individual plant are given, we have

1 Indices c = cooler cin, cout = the cold stream entering to and released from heat exchange unit e = heat exchange unit (including heat exchanger, heater and cooler) h = heater he = heat exchanger hin, hout = the hot stream entering to and released from heat exchange unit i, i′ = index of splitters j, j′ = index of mixers k = index of components p,p′ = index of all DNs (including DN for individual plant and interplant representation) p1 = index of DNs for individual plant w = fixed contaminant-load water-using units 2. Sets C = coolers

f iout = Fiout p1

∀ ip1 ∈ IP1

p1

f jin = F jin p1

∀ jp1 ∈ JP1

p1

t iout = Tiout p1 p1 t inj = Tinj p1

∀ ip1 ∈ IP1 ∀ jp1 ∈ JP1

p1

(5) (6) (7) (8)

ciout = Ciout p1 , k p1 , k

∀ ip1 ∈ IP1 , k ∈ K

(9)

c inj , k ≤ C jin , k

∀ jp1 ∈ JP1 , k ∈ K

(10)

p1

p1

in out in out in where Fout ip1 and Fjp1, Tip1 and Tjp1, Cipl,k and Cjpl,k are all given parameters and bounds regarding the flow rates, temperatures, and concentrations of process streams. Furthermore, as not all

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where f inhine and f out houte are the flow rates of hot streams entering to and exiting from unit e; f incine and f out coute stand for the flow rates of cold streams entering to and exiting from unit e; cinhine ,k and cout houte ,k denote the concentrations of k in hot streams entering to and exiting from unit e; cincine ,k and cout coute ,k represent the concentrations of k in cold streams entering to and exiting from unit e. In eqs 14−17, we assume that there is no mass loss and therefore all flow rates and concentrations are identical around each unit. b. Heat Load of Heat Exchanger Units. To determine the existence as well as the heat load of each heat exchanger, the following equations are needed:

streams are allowed to mix at certain mixing points, the following constraints are enforced: fsip , j = 0

∀ ip ∈ IP , jp ∈ NIP

p

(11)

where NIP is a subset of JP and is determined by the different mechanisms of integration. Finally, unreasonable amounts of flow not allowed in the optimal operating policy and such amount can be limited by the addition of the following constraint: fs min nfsip , j ≤ fsip , j ≤ fs max nfsip , j p′

p′

p′

∀ ip ∈ IP , jp ′ ∈ JP ′ (12)

where nfsip,jp are binary variables that stand for the existence/ nonexistence of the flow between nodes ip and jp. Here, it should be noted that the use of eq 11 and eq 12 depends on the following situations: (1) Direct integration with process streams. Constraint 11 is introduced to remove the possibilities of mixing between process streams and hot or cold utility streams in the state-space superstructure. Constraint 12 is then used specify binary variables (nfsip,jp) which are needed to calculate the fixed pipeline cost between plants. However, if fixed costs of pipelines are not included, binary variables are removed and constraint 12 is reformulated as follows to bound the flow rates between the two plants: 0 ≤ fsip , j ≤ fs max p′

∀ ip ∈ IP , jp ′ ∈ JP ′

in out in out in f hin Cphin (tin hinhe − thinhe) = f cin Cpcin (tcinhe − tcinhe) he

e

e

c

in out f cin = f cout e

e

in in f cin Cpcin (tout cinh − tcinh) = qh h

out c in hine, k = c houte, k

(13)

in c cin = c out coute, k e, k

h

(19)

∀ h ∈ H , cinh ∈ CPS

∀ he ∈ HE, hinhe ∈ HPS, cinhe ∈ CPS

(21)

Q minw(hinh, cinh) ≤ qh ≤ Q maxw(hinh, cinh) ∀ h ∈ H , hinh ∈ HUS, cinh ∈ CPS

(22)

Q minw(hinc , cinc) ≤ qc ≤ Q maxw(hinc , cinc) ∀ c ∈ C , hinc ∈ HPS, cinc ∈ CUS

(23)

In these equations, tinhinhe and tout hinhe denote, respectively, the temperature of hot stream at the inlet and outlet of heat exchanger he; tincinhe and tout cinhe represent, respectively, the temperature of cold stream at the inlet and outlet of heat exchanger he; the binary variables w(hinhe,cinhe), w(hinh,cinh) and w(hinc,cinc) represent, respectively, the existence/nonexistence of heat exchanger, heater, and cooler. They are set to one if there is a heat load between Qmin and Qmax, which correspond to the lower and upper bounds of the heat duties. While constraints 18−20 describe the energy balances around each heat exchanger, heater and cooler, constraints 21−23 are introduced to limit the heat load of the corresponding equipment. c. Temperature Differences. The temperature differences in PO_HEN and PO_IHEN are calculated as follows:

(14)

in out Δte1 = thin − tcout e e

(15)

∀ e ∈ HE ∪ H ∪ C , hine ∈ HINE , coute ∈ COUTE (24) (16) out in Δte2 = thout − tcin e e

∀ cine ∈ CINHE ∪ CINH ,

coute ∈ COUTHE ∪ COUTH , k ∈ K

c

Q min·w(hinhe, cinhe) ≤ qhe ≤ Q max ·w(hinhe, cinhe)

∀ hine ∈ HINHE ∪ HINC ,

houte ∈ HOUTHE ∪ HOUTC , k ∈ K

(18)

(20)

∀ cine ∈ CINHE ∪ CINH ,

coute ∈ COUTHE ∪ COUTH

c

c

∀ c ∈ C , hinc ∈ HPS, cinc ∈ CUS

∀ hine ∈ HINHE ∪ HINC ,

houte ∈ HOUTHE ∪ HOUTC

he

∀ he ∈ HE, hinhe ∈ HPS, cinhe ∈ CPS

in out in out in f hin Cphin (tin hinc − thinc) = f cin Cpcin (tcinc − tcinc) = qc

(2) Indirect integration with central junctions and treatment units. In this case, constraint 11 is used to exclude the direct water flow from one plant to another as well as the possibilities of mixing between process streams and utility streams. Then, constraint 12 is needed to specify the connections between each individual plant and junctions. In particular, Ip is used to represent the water sources in individual plants and Jp′ denotes the junctions between plants. All binary variables in 12 are used for the calculation of fixed cost of interplant pipelines. Like previous cases, if such cost is excluded from the objective function, then constraint 12 can also be replaced by 13. 4.2. Model for POs. Notice that for a given design problem the number of splitting and mixing nodes (also the number of heat exchangers) attached on DN must be selected before solving the corresponding model. The appropriate number of these nodes attached on DN and PO block can be determined with the heuristic rules in section 3.2 of our previous work.1 All mathematical constraints for the POs are written as follows. 4.2.1. Heat Exchange Units. a. Mass and Heat Balances for Heat Exchange Units. The mass balances around each heat exchange unit can be written as the following equations: in out f hin = f hout

he

he

= qhe

∀ e ∈ HE ∪ H ∪ C , houte ∈ HOUTE , cine ∈ CINE

(17)

(25) 14795


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controllability and operational flexibility. The relevant mass balance for all junctions can be written as follows:

∀ e ∈ HE ∪ H ∪ C

in out f jun = f jun

(26)

Δt1e

Δt2e

where and denote the temperature driving forces at both ends of the heat exchanger e; Δte stands for the logarithmic mean temperature difference for heat exchanger e and Chen’s approximation3 is adopted, as shown in eq 26. Finally, the temperatures of hot and cold streams around every heat exchange unit must satisfy the following thermodynamic constraints: Δte1

+ Γ[1 − w(hine , cine)] ≥ ΔT

in out c jun, k = c jun, k

in out t jun = t jun

(27)

Δte2 + Γ[1 − w(hine , cine)] ≥ ΔT min ∀ e ∈ HE ∪ H ∪ C , hine ∈ HINE , cine ∈ CINE (28)

out tcou te

−

in tcin e

≥0

∀ e ∈ HE ∪ H ∪ C , cine ∈ CINE ,

∀w∈W ∀ w ∈ W, k ∈ K

(33)

out c out w , k ≤ Cw , k

∀w∈W

(34)

twout = Twout

(41) (42)

∀w∈W

∀w∈W

∀ tr ∈ TR

(43)

in where f intr and f out tr denote the inlet and outlet flow rates of tr; ctr,k and cout represent the inlet and outlet concentrations of pollutant tr,k k of tr; tintr and tout tr are respectively the inlet and outlet temperatures of tr; rtr,k represents the removal ratio of component k in unit tr. Like FC-based water-using units, different inlet and outlet temperatures for treatment (Tintr and Tout tr ) can be assigned in eqs 42 and 43 to meet process specifications. 4.3. 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 costs of heat exchange units, (4) the fixed and variable costs of wastewater-treatment units and (5) the cost of crossplant pipelines and junctions. The objective function can be written as follows:

(32)

∀w∈W

=

∀ tr ∈ TR, k ∈ K

∀ tr ∈ TR

t trout = Ttrout

(31)

in c in w , k ≤ Cw , k

Twin

(40)

(30)

in f win (c out w , k − c w , k ) = Δmw , k

twin

∀ tr ∈ TR

t trin = Ttrin

By specifying the minimum temperature difference ΔTmin, eqs 27 and 28 also ensure that exchangers with large 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. 4.2.2. Water-Using Units, Junctions, and Treatment Units. For plants with FC-based operations, the corresponding waterusing units should be placed in the superstructure. The mass and heat balance for every FC-based unit can be written as follows: f win = f wout ≤ f wlim

(38) (39)

f trin ctrin, k(1 − rtr , k) = f trout ctrout, k

(29)

coute ∈ COUTE

∀ jun ∈ JUN, k ∈ K

∀ jun ∈ JUN

f trin = f trout

∀ e ∈ HE ∪ H ∪ C , hine ∈ HINE ,

houte ∈ HOUTE

(37)

out where f in jun and f jun denote the inlet and outlet flow rates of in out jun; cjun,k and cjun,k represent the inlet and outlet concentrations of pollutant k of jun; tinjun and tout jun are respectively the inlet and outlet temperatures of jun. It is worth to note that the centralized junction among plants can serve as the function of water regeneration units where the water source quality is improved before it is sent to other plants. The models for fixed removal ratio wastewater treatment units are formulated as follows:

min

∀ e ∈ HE ∪ H ∪ C , hine ∈ HINE , cine ∈ CINE

in thin − thout oute ≥ 0 e

∀ jun ∈ JUN

Objective function = cost of freshwater + cost of utilities

(35)

+ cost of treatment unit + cost of heat exchange units

(36)

+ cost of piplines and junctions

where f inw and f out w denote the inlet and outlet flow rates of waterusing unit w; f lim w the upper bound of the inlet flow to each unit; cinw,k and cout w,k represent the inlet and outlet concentrations of pollutant k of w; Δmw,k is the mass load of contaminant k in water user w. Cinw,k and Cout w,k stand for the upper bounds of the inlet and outlet concentrations of each unit; tinw and tout w are respectively the inlet and outlet temperatures of w. To account for the nonisothermal operations, different inlet and outlet temperatures (Tinw and Tout w ) can be assigned to meet the operating conditions. Junctions are introduced with the purpose of indirect integration where process streams are not allowed to directly mix with streams in other plants for further water and heat recovery. In such scenarios, junctions act as storage tanks that store and transfer cross-plant flows, and can significantly improve

cost of freshwater =

(44)

∑ cost fwf fwout

(45)

fw

cost of utilities = (∑ costhuqh + h

+ γ(∑ costhuq′h + h

∑ costcuqc) c

∑ costcuq′c )

(46)

c

var β α cost of treatment unit = AF ∑ cost fix tr f tr + ∑ cost tr f tr tr

tr

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used in examples 2 and 3, since the fixed cost of the pipelines is very small.

∑ costfixhew(hinhe, cinhe) he

⎛ qhe ⎞ var + ∑ costhe ⎜ ⎟ + ⎝ U Δthe ⎠ he

∑ costhfixw(hinh, cinh)

⎛ q ⎞ + ∑ costhvar⎜ h ⎟ + ⎝ U Δth ⎠ h

∑ costcfixw(hinc , cinc)

b

5. APPLICATION EXAMPLES Three examples are presented to demonstrate the IWAHEN design with both FF and FC processes. The first two examples show the two IWAHEN design schemes for FC-based waterusing operations. Then, IWAHEN designs involving both FF and FC processes are studied for both direct integration and indirect integrations in example 3. The MINLP problem is solved using the algorithm of Viswanathan and Grossmann,4 which is available in GAMS as the solver DICOPT. The MINLP problem is decomposed into a series of NLP and MILP subproblems, which are solved respectively by the CONOPT and CPLEX solvers. 5.1. Example 1. An example that consists of two plants with FC operations is first solved to illustrate the proposed approach. Table 1 shows the process data for all streams in both plants

h

b

c

⎛ qc ⎞b var + ∑ costc ⎜ ⎟ ⎝ U Δtc ⎠ c

(48)

where costfw, costhu, and costcu denote respectively the annualized unit costs of freshwater, hot and cold utilities; costfix tr and costvar tr represent the annualized fixed and variable cost var parameters for treatment; costfix e and coste (e ∈ HE ∪ H ∪C) for the annualized fixed and variable cost coefficients for heat exchange unit e; α, β, and b are the cost coefficients for treatment unit and heat exchange unit; γ is the cost parameter considering the additional cost incurred when hot and cold utilities are used across plants; AF is used to denote the annualized factor. In eq 46, qh′ and qc′ represent consumption rates of hot and cold utilities in plants where utilities from other plants are used. The expressions for q′h and q′c are the same as constraints (18) and (19). Finally, the capital cost for the cross-plant pipeline in eq 44 is expressed respectively with both direct and indirect integration schemes. Neglecting the piping cost within stand-alone plants, the fixed and variable costs of pipelines for direct integration with process streams can be expressed as follows: ⎡ ⎢ var ∑ cost of pipelines = AF⎢cost pipe i p ∈ IP ⎣ + cost fix pipe

⎤ ⎥ Dip , j nfsip , j ⎥ p′ p′ jp ′ ∈ JP ′ ⎦

∑ ∑ ip ∈ IP

∑

Table 1. Process Data for Example 1

p ≠ p′ (49)

var where costfix pipe and costpipe denote respectively the fixed and variable cost parameters of pipelines; Dip,jp′ stands for the distance of pipelines from sources ip to jp′; ρ and v stand for the density of the fluids and the stream velocity, respectively. For indirect integration, the interplant piping cost in eq 44 is reformulated as eq 50. In addition, the cost of central junctions, which is proportional to the total junction flow rate, is also included.

cost of pipelines and junctions ⎧⎡ Di , j fsi , j ⎪ var AF⎨⎢cost pipe + cost fix ∑ ∑ pipe ⎢ ρ v ⎪⎣ i ∈ I j ∈ ∪ JUN SINK P ⎩ ×

∑

∑

var Di , j nfsi , j + cost pipe

i ∈ IP j ∈ JUN ∪ SINK

∑ ∑

in ⎬ ∑ ∑ Di ,jnfsi ,j⎥⎥ + cost jun ∑ f jun

i ∈ JUN j ∈ JP ′

p ≠ p′

ρv ⎫ ⎪

⎤

+ cost fix pipe

Di , j fsi , j

i ∈ JUN j ∈ JP ′

⎦

jun

mass load (g/s)

max. inlet conc. (ppm)

max. outlet conc. (ppm)

plant 1

1 2 3 4 5 6 7

2 5 30 4 5 30 50

0 50 50 400 50 50 800

100 100 800 800 100 800 1100

max. flow rate (kg/s)

temp (°C)

100 40 166.7

40 100 75 50 100 75 100

(taken from Savulescu et al.5,6 and Bagajewicz et al.,7 respectively). In addition to the process data in the table, other parameters are set as follows. The unit price for the freshwater (20 °C) is 0.375 $/ton; the inlet and outlet temperatures of cooling water are set at 10 and 20 °C, respectively, and its cost is 189 $/(kW yr); the temperatures of low and medium-pressure steams are assumed to be 120 and 150 °C, respectively, and the corresponding costs are 377 and 388 $/(kW yr); the overall heattransfer coefficient and the minimum temperature approach are assumed to be 0.5 (kW m−2 K−1) and 10 °C; the annualized capital cost model for a conventional shell-and-tube heat exchanger is 8000 + 1200AREA0.6, where the area of heat exchanger is in m2; the temperature and maximum component concentration at the sinks are chosen to be 30 °C and 20 ppm, respectively; the annualized factor is set at 10% and the plants are assumed to be operated continuously for 8000 h a year. In addition, it is assumed that two classes of treatment units are available in each plant. The removal ratios for the treatment unit in plant 1 and plant 2 are 95% and 90%, respectively, and their operating temperatures are both required to be maintained at 30 °C. The capital costs ($/yr) are determined according to the formulas 16800F0.7 and 12600F0.7, while the corresponding operating costs ($/yr) are calculated with the formulas F and 0.76F. All above cost coefficients are adopted from Dong et al.8 Finally, to account for the cost incurred in interplant integration, γ is set to 1.1 throughout the following case studies and all cost parameters with the cross-plant pipelines and junctions are the same as the values in part 1 of this series. Two cases are introduced for the study of the direct integration of the single-component WAHENs. The first case provides two separate WAHENs which serve as the basic case for further comparison. Then, the direct integration of the aforementioned two plants is solved to illustrate the benefits of our approach.

p′

ρv

jp ′ ∈ JP ′

unit

plant 2

Dip , j fsip , j p′

plant

⎪ ⎭ (50)

However, we should clarify that the fixed cost of pipelines and the corresponding binary variables (nfsi,j) in eqs 49 and 50 are not 14797



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In case 1, by allowing that all process streams and fresh water in a single plant can be mixed, two WAHENs each with a waste sink are synthesized without considering any form of integration. The resulting networks are shown in Figures 1 and 2, where the blue

Table 2. A Comparison between Stand-Alone and Integrated Designs in Example 1 stand-alone plants cost terms

plant 1

plant 2

direct integration

fresh water hot utility (LP steam) cold utility heat exchangers treatment pipelines individual TAC overall TAC

$1,301,358 $1,907,935

$1,479,130 $2,168,569

$2,470,658 $3,622,216

$293,046 $2,685,763

$280,228 $2,445,316

$ 435,400 $5,285,839 $333,144

$6,188,102 $ 6,373,243 $12,491,345

$12,147,257

noting that compared to that in the network structure in a previous work,8 the TAC obtained in plant 1 has been cut down to $6,188,102, which represents a 5.6% improvement compared to the previous study.8 More specifically, the consumption of freshwater (120.5 kg/s) significantly reduces the cold utility cost ($0 in this case) and wastewater treatment costs. Finally, from Table 2, it is obvious that in both plants treatment and utility costs constitute the major part of TAC, while the capital investment of heat exchangers only account for less than 5% of the TAC. The direct IWAHEN problem is solved in case 2 to obtain the network structure with minimum TAC. In this case, we further assume that both freshwater source and waste sink are placed in plant 2. (In general, the placement of the fresh water sources and waste sinks will affect the results. We did not look further into this issue since our major concern is the effects of integration.)

Figure 1. Optimal network for plant 2 in example 1.

line (see the web version of this article) stands for the flow of freshwater. From the figures, we can see that both plants 1 and 2 contain two heat exchangers and one heater. All cost items on the two plants are given in Table 2. It can be found that only hot utility (LP steam) is used in both plants. Furthermore, it is worth

Figure 2. Optimal network for plant 1 in example 1. 14798



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Figure 3. Optimal network for direct integration in example 1.

5.2. Example 2. In this example, IWAHEN designs with indirect integration for multicomponent systems are studied. We consider the process data (see Table 3) used by Dong et al.8 and Bogataj and Bagajewicz.9 Specifically, in the first plant, the unit prices for primary (20 °C) and secondary (80 °C) water are set to 0.375 and 0.45 $/ton, respectively. All contaminant concentrations for both primary and secondary water are 0 ppm. While there is no concentration constraint on the waste sinks, the maximum wastewater temperature is set to 60 °C. Additionally, all other data are exactly the same as those introduced in example 1. For the second plant, the cost of freshwater (20 °C) is 2.5 $/ton and maximum wastewater temperature is 30 °C; there is no constraint on the concentrations at the sink; the costs of hot (126 °C steam) and cold (15−20 °C) utilities are set to 260 and 150 $/(kW yr); the models for heat

The resulting network with seven cross-plant pipelines is presented in Figure 3, where two heat exchangers and one heater are used in plant 2 and no heat exchanger is needed in plant 1. The corresponding minimum TAC of the direct integrated plants has been reduced to $12,147,257, which represents a 2.8% improvement compared to stand-alone WAHEN designs. A comparison of the key features of the two cases is summarized in Table 2. It can be found that there are more water reuse and/or recycle opportunities in direct integration and the optimal network favors a higher treatment cost and lower freshwater cost. Furthermore, more stream mixing opportunities also provide more direct heat exchange opportunities and this leads to a further decrease in the cost of hot utility. Finally, like case 1, while treatment and utility costs constitute the major part of TAC, costs of heat exchangers and pipelines accounts for only 6.3% of the TAC. 14799



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Table 3. Process Data for Example 2 mass load(g/s) plant a

plant 1

plant 2b a

max inlet concn (ppm)

max outlet concn (ppm)

temp (°C)

unit

k1

k2

k3

k1

k2

k3

k1

k2

k3

inlet

outlet

1 2 3 4 5

3 4 1.5 1.67 1.39

2.4 3 0.6 0.83 2.22

1.8 3.6 2 1.11 0.28

0 50 50 5 150

0 40 50 150 120

0 15 30 100 60

100 150 125 50 300

80 115 80 200 150

60 105 130 200 30

100 75 35 25 100

100 75 35 35 85

U = 0.5 (kW m−2 K−1). bU = 0.5 or 0.833 (kW m−2 K−1) depending on the presence of hot steam.

exchange unit and treatment are 10000 + 860AREA0.75 and + 0.95NhFtr, where Nh represents the number of 20000F0.78 tr operating hours each year and the flow rate is in ton/h; the removal ratios of the three components are 0.75, 0.9, and 0.9, respectively, and the inlet and outlet temperatures are 40 and 37 °C. To be able to compare different approaches and examples on the same basis, we assume (1) both plants operate 8000 h annually and (2) that TACs of these published network structures (see the figures in these two original works8,9) can be calculated according to our cost functions in eqs 45−48. The estimated TACs, capital and operating costs of these two plants are summarized in Table 4. From the results, it

TACs in the two plants. Detailed costs are listed in Table 4. From this table, it is apparent that both capital and operating costs have been reduced significantly. Such dramatic savings result from the optimal choice of utilities as well as the installation of heat exchangers. In particular, cost-effective supplies of freshwater in plant 1 and hot utility in plant 2 are chosen in this optimal structure. Similarly, capital investments tend to be cheaper in plant 1 and therefore the resulting structure favors the use of heat exchangers in plant 1. Since a large amount of fresh water is consumed, it is not cost-optimal to use the waste sink with temperature requirement of 60 °C. A final observation for this network is that with the use of cheaper freshwater (0.375 $/ton) in the integrated network, it is no longer cost-effective to use the treatment unit in plant 2. To demonstrate the capability of state-space superstructure for incorporating the central wastewater treatment unit in IWAHEN design, case 2 is performed by imposing an extra upper bound of 20 ppm on all contaminant concentrations at the sink. In addition, wastewater treatment units from both plants are allowed to be used for central treatment and all freshwater sources and waste sinks are placed between the two plants with the same distance as case 1. By including the treatment options in our superstructure and solving the resulting model, an optimal IWAHEN design can be obtained (see Figure 5). Notice that, other than water-using units, there is one wastewater treatment unit, one central junction, one heat exchanger and one heater. Notice also that there are nine cross-plant pipelines and cold utility is not consumed in the system. Its minimum TAC and the corresponding utility and treatment costs were found to be $6,709,719, $4,183,053, and $2,133,932, respectively. The freshwater (0.375 $/ton) flow rate in this case is 190.29 kg/s, while the consumption rate of hot utility (126 °C steam) is 7440 kW. Finally, we have to mention that due to the extra bounds of the contaminant concentrations at the waste sink, the TAC for this case is larger than the TAC of case 1 as well as the sum of TAC of two individual plants. 5.3. Example 3. In the last example, we consider the IWAHEN designs with both FF and FC processes. The data for operating conditions of water using processes is given in Tables 5 and 6. In these two plants, the overall heat exchange coefficients are set to 0.5 and 0.833 (kW m−2 K−1) depending on the use of hot steam. All other cost and operating parameters (cost of fresh water, utility, heat exchangers, and wastewater treatment units as well as operating conditions for treatment units) are the same as those adopted in example 1. Three cases are considered in the current example. Case 1 introduces two optimal individual WAHENs with potential opportunities for integrations. Then both direct and indirect integration of these two separate WAHENs are investigated in the following two cases respectively. By allowing all process streams to be mixable, the solutions for the two stand-alone plants are presented in Figures 6 and 7 and the capital and operating costs are summarized in Table 7. It can

Table 4. A Comparison between Stand-Alone and Integrated Designs in Example 2 stand-alone plants indirect integration without treatment

indirect integration with treatment

cost terms

plant 1

plant 2

freshwater hot utility cold utility heat exchangers treatment pipelines and junctions individual TAC overall TAC

$907,200 $475,020 $1,349,460

$1,600,000 $366,340 $240,300

$1,464,753 $1,451,174

$2,055,101 $2,127,952

$173,627

$340,948

$203,468

$194,229

$850,009

$2,905,307

$2,133,932 $228,476

$198,505

$3,347,871

$6,709,719

$3,397,597

$6,302,904

can be seen that the cost of cold utility and freshwater account for nearly half of the TAC in the two plants, respectively. Two cases with indirect integrations are provided in this example. In both cases, the MINLP problems are first solved by removing the binary cost term in eq 50. Then, we added the corresponding fixed pipeline cost to obtain the exact TAC. As the cost for pipelines carries few weights in the final objective function, good local optimal solutions can always be captured with this approach. The first one considers indirect integration with two central junctions and the second case is studied to illustrate the impacts of central treatment units on the entire network. In case 1, it is assumed that all freshwater sources and waste sinks are placed between the two plants with a distance of 180 m. Furthermore, we allow the use of all available freshwater sources and hot and cold utilities as well as sinks in both plants. On the basis of the same solution strategy proposed in part 1 of this series, we obtain the optimal network structure in Figure 4, where the entire network consists of one heat exchanger, one heater, and eight cross-plant pipelines. The TAC for the resulting network is $3,347,871, which is only 53.1% of the sum of the 14800



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Figure 4. Optimal network for indirect integration in example 2.

be found that cooling of 6972 kW and heating of 6443 kW are required, respectively, in these plants. In addition, wastewater treatment is only needed in plant 2 and such cost accounts for almost half of its TAC. The TAC of these two networks running separately is $12,940,250, of which approximately 62.7% corresponds to the costs of all utilities, 33.8% to the treatment cost, and only 3.5% to capital investment. Case 2 is studied to investigate the effects of direct integration with process streams between two plants. To explore the possible cross-plant structure of freshwater supply network, we assume that freshwater is available in both plants, and that the waste sink of the system is embedded in the second plant. Moreover, since the costs of pipelines and junctions are almost negligible compared with utility cost and capital investment, we exclude this cost in the IWAHEN integration in this and the following cases. Although one may need to calculate and add such costs after identifying the optimal network, the advantage of such approach is that it reduces the number of binary variables and corresponding inequality constraints. On the basis of these assumptions and such solution methods, we obtain the optimal network for direct integration, which is depicted in Figure 8. Under more strict concentration requirements for wastewater (20 ppm for all components), the minimum TAC ($10,244,215) is only 79.2% of the sum of TACs for individual plants. From the list of all costs in Table 7, we can find that the cost of utilities has been reduced significantly, and this is simply due to both the additional direct and indirect heat exchange and mass transfer opportunities provided by direct integration. One particular example for indirect heat exchange is the cold stream in plant 2, which is first directed to E1 in plant 1 and is then transferred back to plant 2 via cross-plant pipeline. The entire network is assembled with eight cross-plant pipelines with negligible cost.

Figure 5. Optimal network for indirect integration with treatment unit in example 2. 14801



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Table 5. Process Data for Plant 1a in Example 3 concn (ppm)

a

max. concn (ppm)

source

flow rate (kg/s)

temp (°C)

k1

k2

k3

sink

flow rate (kg/s)

temp (°C)

k1

k2

k3

1 2 3 4 fresh water

25 32 25 30

100 90 95 25 20

120 95 110 110 0

90 70 100 60 0

70 70 55 85 0

1 2 3 4 waste sink

20 30 25 25

30 25 25 80 30

50 40 40 40 30

40 40 50 35 30

50 40 35 35 30

Maximum allowable concentrations for all components at the sink are 20 ppm.

Table 6. Process Data for Plant 2a in Example 3 mass load (g/s)

a

max inlet concn (ppm)

max outlet concn (ppm)

unit

temp (°C)

k1

k2

k3

k1

k2

k3

k1

k2

k3

1 2 3

100 50 75

6 4 8

10 7 6

12 8 15

50 40 60

60 50 50

100 100 120

150 100 150

160 140 140

300 200 300

Maximum allowable concentrations for all components at the sink are 20 ppm.



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and wastewater regeneration unit. In this case, both freshwater source and sink (with maximum allowable concentrations of 20 ppm for all components) are placed between the two plants with a distance of 180 m. To further explore the desirable network structure for IWAHEN design, we allow (1) that heat exchangers, heaters and coolers can be used for indirect heat exchange for streams between plants, and (2) that all junctions, treatment units and sinks between plants can be connected. The optimal network with a minimum TAC of $11,733,889 is depicted in Figure 9. Notice that the network between two plants consists of one heat exchanger, one heater as well as 15 crossplant pipelines and two junctions. Even the unit costs of utilities and capital investment are exactly the same in both plants, compared with the stand-alone operation, considerable savings in TAC can still be obtained for indirect integration under more strict environmental regulations. In the final analysis, although the TAC for indirect integration is higher than that for direct integration, the penalty of additional costs might be negligible

Table 7. A Comparison between Stand-Alone and Integrated Designs in Example 3 stand-alone plants Indirect integration without cost of pipelines and junctions

cost terms

plant 1

plant 2

direct integration without cost of pipelines

freshwater hot utility cold utility heat exchangers treatment individual TAC overall TAC

$2,710,800

$1,656,754 $2,428,986

$5,060,475 $816,441

$5,537,660 $1,516,047

$372,231

$389,166

$365,341

$4,378,555

$3,978,133

$4,314,841

$10,244,215

$11,733,889

$1,317,708 $75,216

$ 4,103,724

$8,836,526

$12,940,250

The final case in this example (case 3) deals with the indirect integration of both FF and FC processes with central junctions

Figure 8. Optimal network for direct integration in example 3. 14803



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Figure 9. Optimal network for indirect integration with treatment in example 3.

Table 8. Problem Size and Computing Time for Example Studies problem

equations

constraints

binary variables

continuous variables

major iterations

CPU time (sec)

case 2 in example 1 case 1 in example 2 case 2 in example 2 case 2 in example 3 case 3 in example 3

1−12, 14−49 1−11, 13−48, 50 1−11, 13−48, 50 1−11, 13−49 1−11, 13−48, 50

570 484 505 592 631

332 12 12 16 16

640 591 640 923 1047

7 4 6 9 12

4.45 3.98 4.24 6.16 7.44

when other benefits, such as improved controllability and operational flexibility, are taken into account. 5.4. Computational Results. Table 8 shows the size of each example as well as the CPU time required to solve them with appropriate initial values in an Intel(R) Core(TM) i7 CPU 920 @ 2.67 GHz processor. In all cases, we have used several initial guesses and random perturbations to identify the best possible solutions. One can notice that computing times are relatively small in both direct and indirect IWAHEN designs. From the designs produced above, it can be concluded that the proposed methods are indeed suitable for obtaining optimal designs for IWAHENs in a computationally effective manner. However, we have to mention that the global optimal solutions of all cases

cannot be guaranteed, because of the nonlinearity and nonconvexity of the proposed mathematical model.

6. CONCLUSIONS A general MINLP model has been presented in this work for the optimization of IWAHENs with both FF and FC operations. The IWAHEN designs are based on the multiscale state-space superstructure which incorporates all network connections of all types of process units in all plants. Three examples have been presented, clearly showing that the proposed method can be used in the design of single contaminant and multicontaminant IWAHENs with both direct and indirect integrations. 14804



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Near global or at least good suboptimal solutions can be obtained in all cases presented.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China, under Grant No. 20876020. The authors in Imperial College London and Dalian University of Technology would also like to acknowledge the support from Center for Advanced Process Decision-making (CAPD) at Carnegie Mellon University.

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REFERENCES

(1) Li., L. J.; Zhou, R. J.; Dong, H. G.; Grossmann, I. E. Separation network design with mass and energy separating agents. Comput. Chem. Eng. 2010, 35, 2005. (2) Zhou, R. J.; Li., L. J.; Dong, H. G.; Grossmann, I. E. Synthesis of interplant water-allocation and heat-exchange networks. Part 1: Fixed flowrate processes. Ind. Eng. Chem. Res. 2012, 51, 4299. (3) Chen, J. J. J. Letter to the editor: Comments on improvement on a replacement for the logarithmic mean. Chem. Eng. Sci. 1987, 42, 2488. (4) Viswanathan, J.; Grossmann, I. E. A combined penalty function and outer approximation method for MINLP optimization. Comput. Chem. Eng. 1990, 14, 769. (5) Savulescu, L. E.; Kim, J. K.; Smith, R. Studies on simultaneous energy and water minimization-Part I: Systems with no water re-use. Chem. Eng. Sci. 2005, 60, 3279. (6) Savulescu, L. E.; Kim, J. K.; Smith, R. Studies on simultaneous energy and water minimization-Part II: Systems with maximum re-use of water. Chem. Eng. Sci. 2005, 60, 3291. (7) Bagajewicz, M.; Rodera, H.; Savelski, M. Energy efficient water utilization systems in process plants. Comput. Chem. Eng. 2002, 26, 59. (8) Dong, H. G.; Lin, C. Y.; Chang, C. T. Simultaneous optimization approach for integrated water-allocation and heat-exchange networks. Chem. Eng. Sci. 2008, 63, 3664. (9) Bogataj, M.; Bagajewicz, M. Synthesis of non-isothermal heat integrated water networks in chemical processes. Comput. Chem. Eng. 2008, 32, 3130.

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