Synthesis of Heat Integrated Resource Conservation Networks with

Article pubs.acs.org/IECR

Synthesis of Heat Integrated Resource Conservation Networks with Varying Operating Parameters Yin Ling Tan,†,* Denny K. S. Ng,‡ Mahmoud M. El-Halwagi,§,|| Dominic C. Y. Foo,‡ and Yudi Samyudia† †

Chemical Engineering Department, Curtin University, Sarawak Campus, CDT 250, 98009 Miri, Sarawak, Malaysia Department of Chemical and Environmental Engineering/Centre Excellence for Green Technologies, University of Nottingham Malaysia, Broga Road, 43500 Semenyih, Selangor, Malaysia § Chemical Engineering Department, Texas A&M University, College Station, Texas 77843, United States ‡

ABSTRACT: This paper presents the synthesis of heat integrated resource conservation networks (HIRCNs), covering both concentration- and property-based direct reuse/recycle systems. This newly proposed method adopts the targeting concept of the insight-based pinch approach where the minimum consumption of fresh resources and energy utilities is targeted prior to the detailed HIRCN design. Furthermore, this method is capable of handling HIRCN problems with varying operating range of process parameters (i.e., flow rates, temperatures, and properties). The proposed method is formulated as a mixed integer nonlinear program (MINLP). As the temperature of stream is uncertain, the f loating pinch concept is adopted to identify hot and cold utilities. Besides, a recently developed discretization approach is also used to solve the MINLP problem. Three literature case studies are solved to illustrate the proposed method.

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INTRODUCTION Traditionally, process industries have focused on conventional end-of-pipe waste treatment. Over the past decades, the center of attention has shifted toward sustainable operations with resource conservation activities. Among the few reasons that have resulted in this change are environmental sustainability and stringent emission legislation as well as the increase of fresh resources and waste treatment costs. Process integration has been commonly used as an effective tool for resource conservation and waste reduction. El-Halwagi1 defines process integration as a holistic approach to process design, retrof itting, and operation which emphasizes the unity of the process. Process integration can be generally categorized as heat and mass integration. Heat integration is a systematic methodology that provides a global understanding of heat utilization within the process and employs this understanding in identifying the utility targets and optimizing heat recovery as well as energyutility systems.1 On the other hand, mass integration is a systematic approach that provides a fundamental understanding of the global flow of mass within the process and employs the understanding in the identification of performance targets and optimization of the generation and routine of species throughout the process.1 Extensive reviews on energy and mass integration can be found in refs 1−8. One of the most established area for mass integration is resource conservation networks (RCNs). A particular application with extensive works reported is the synthesis of water network. Takama et al.9,10 initiated mathematical optimization approaches for water network synthesis. Later, various mathematical optimization works on synthesis of water network were presented, which may be categorized as direct reuse/recycle11−20 and regeneration systems.15,21,22 On the other hand, insight-based approaches for water network synthesis were first reported by Smith and co-workers for reuse and regeneration systems.23−25 Other works on regeneration systems were also reported.26,27 © 2012 American Chemical Society

However, these works were limited to f ixed load problem. Various works for a more generalized f ixed f low rate problem were also reported for direct reuse/recycle28−39 and for regeneration systems.40−42 Furthermore, the use of combined conceptualbased and mathematical optimization techniques was also reported for a water network synthesis with reuse/recycle, regeneration, and total water network.11,43−45 The significance of mass integration techniques for material reuse and recycle is well established. However, the major drawback of the previous works is that they are limited to “chemo-centric”.1,46 Note that many design problems are not limited by the nature and quantity of chemical of the streams. Other properties or functionalities of the streams (e.g., pH, density, color, etc.) are usually involved.1,8 Furthermore, the effluent regulation is commonly defined not only in term of pollutant concentration but also based on stream properties, such as pH, color, etc. To overcome this limitation, El-Halwagi and co-workers46,47 introduced the notion of property integration, which is defined as a f unctionality-based, holistic approach to the allocation and manipulation of streams and processing units, which is based on the tracking, adjustment, assignment, and matching f unctionalities throughout the process. Various methodologies have also been established for the design of property-based RCN. These include graphical,47−49 algebraic,49,50 and optimization43−45,51 techniques. Furthermore, some works have also been extended to batch RCN,52−54 as well as simultaneous mass and property integration.55,56 Special Issue: PSE-2012 Received: Revised: Accepted: Published: 7196

September 14, 2012 November 28, 2012 November 29, 2012 November 29, 2012 dx.doi.org/10.1021/ie302485y | Ind. Eng. Chem. Res. 2013, 52, 7196−7210

Industrial & Engineering Chemistry Research

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PROBLEM STATEMENT The Problem to Be Addressed Is Given As Follows. Given NSOURCES number of process sources and NSINKS number of process sinks. Process sources may be allocated for reuse/ recycle to the process sinks, or be discharge as waste. Each process source i, has a fixed flow rate (Wi), property operator (ψi,p) and temperature (Ti). Process sinks are units that can receive process sources. Each sink, j, has an acceptable range of flow rate (Gj), property operator (ψj,p), and temperature (Tj), given as in eqs 1−3.

Most of the previous works do not consider temperature as part of the process constraints. However, in many cases, the temperature is an important design parameter. For example, when solvent is utilized in an industry for extraction purposes, heating or cooling is needed to achieve the desired operating conditions before the solvent is fed into the extraction column. Therefore, resource conservation and heat management need to be considered simultaneously as both the quality and temperature of solvent are equally important. In addition, in some cases, the property of the solvent may be sensitive with temperature, therefore, simultaneous consideration of property and heat recovery should be addressed. Over the past decades, simultaneous energy and water minimization have received particular attention and many studies have been conducted. Salveski and Bagajewicz57 considered water and energy consumptions separately and solving the problem in a sequential manner. Later, both insight-based and mathematical optimization techniques were established. Note that the insightbased techniques are developed based on a sequential approach. Meanwhile, sequential and simultaneous approaches can be considered in mathematical optimization techniques. As reported in the literature, the currently available insightbased techniques include two-dimensional grid diagram with separate systems,58−60 source-demand energy composite curves,61−63 stream merging principles,64 graphical thermodynamic rules,65,66 energy recovery algorithm,67 modified problem table algorithm,68 and temperature versus concentration diagram.69 However, these methods are limited to single property measurement problems. To address multiple property measurements, mathematical programming techniques have been proposed. Sequential linear programming models have been developed to first determine the minimum fresh water consumption followed by minimum energy requirement.70−73 A detailed heat integrated water network is then identified via mixed integer linear programming (MINLP) models.70−73 Various discontinuous nonlinear programming (DNLP),73 MINLP,74−79 and mixed integer linear programming (MILP)80 models have been presented to address nonisothermal mixing issue in HIRCNs. Recently, Kheireddine et al.81 further extended the area of mass and property integration with thermal constraints. In the previous work, a nonlinear programming (NLP) model that minimizes the total annualized cost of direct reuse/recycle network to satisfy a set process and environmental constraints is presented. Furthermore, the model also accounts for the heat of mixing and the interdependence of properties. However, the main drawback of the proposed model is that it involves direct mixing without heat integration. In some cases, when the thermal constraints are unable to be satisfied through direct mixing, external heating and cooling utilities are needed, which is the subject of this work. In this work, the main objective is to develop a generic superstructure for the synthesis of concentration- and property-based HIRCNs. The model is capable in solving cases with varying process parameters such as flow rates, temperatures, and properties. The f loating pinch concept is adopted for the targeting of energy utilities; while a discretization approach is used to solve the MINLP problem. Note that on the basis of the proposed approach, the sink−source matching of the HIRCN is determined based on the superstructure. Meanwhile, the design of the heat exchanger network (HEN) of the HIRCN is performed using the well-established pinch design method.2 Three literature case studies are used to illustrate the proposed method.

Gjmin ≤ Gj ≤ Gjmax

j ∈ NSINKS

ψ jmin ≤ ψj , p ≤ ψ jmax ,p ,p

(1)

j ∈ NSINKS

p ∈ NPROP (2)

T jmin

≤ Tj ≤

T jmax

j ∈ NSINKS

(3)

max min max min max where Gmin are the respective lower j , Gj , ψj,p , ψj,p , Tj , and Tj

and upper bounds of the admissible flow rate, property operator p, and temperature for sink j. In addition, NFRESH number of external fresh resources are available at property operator p (ψr,p) and temperature (Tr) which may be supplement to the sinks. The flow rate of fresh resource r is to be determined as part of the solution model. A general linearized mixing rule is needed to define all possible mixing patterns among the individual properties, which can take the form of concentration, pH, density, etc. This can be given by eq 4 that follows:46

ψ (p ̅ ) =

∑ xmψm,p

(4)

m

where ψm,p and ψ(p̅) are the property operator p for stream m and mixture property p;̅ while xm is the fractional distribution of stream m of the total mixture flow rate. The problem can be represented by a source−sink−HEN superstructure given as in Figure 1. Each source is segregated and supplied to all sinks directly. Besides, each source is also split and sent to all mixing points before the heat exchanger network (HEN). These mixed streams are classified into hot and cold streams. Hot stream is defined as a stream with supply temperature higher (Tinh ) than the target temperature (Tout h ); while cold stream has supply temperature (Tinc ) lower than the target temperature (Tout c ). In HEN, heat exchangers are used to recover the heat from the hot streams to cold streams. Heaters and coolers may be needed to provide external hot (Qh) and cold (Qc) utilities. Note that after passing through the HEN, the temperature of those mixed streams may increase or decrease, but their compositions and flow rates remain unchanged. These streams are then segregated at the splitting points to be sent to all sinks. The objective of this work is to synthesize a HIRCN with minimum cost, which may take the form of minimum annual operating cost (AOC) or total annualized cost (TAC).

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MODEL FORMULATION In the following section, both models for concentration- and property-based RCN as well as HEN of the HIRCN are presented. Concentration- and Property-Based RCN. For each node of the HIRCN in Figure 1, its mass and energy balances can be defined as follows: Splitting of fresh source r: Fr =

∑

fr , j +

j ∈ NSINKS

∑ h ∈ NHOT

r ∈ NFRESH

fr ,h +

∑

fr ,c

c ∈ NCOLD

(5)

where f r,j, f r,h, and f r,c are the flow rates of fresh source r to sink j as well as to hot stream h and cold stream c in HEN, respectively. 7197

dx.doi.org/10.1021/ie302485y | Ind. Eng. Chem. Res. 2013, 52, 7196−7210


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Figure 1. Source−sink representation for a HIRCN.

Splitting of process source i: Wi =

∑

∑

wi , j +

j ∈ NSINKS

mcCPc(Tcin − To) =

∑

wi ,h +

h ∈ NHOT

wi ,c + Bi, waste (6)

∑

wi ,h +

i ∈ NSOURCES

∑

mc =

∑

fr ,h

wi ,c +

i ∈ NSOURCES

∑

fr ,c

(7)

∑

[ψi , pwi ,h] +

h ∈ NHOT

[ψi , pwi ,c] +

i ∈ NSOURCES

c ∈ NCOLD

∑

∑ r ∈ NFRESH

fr ,h CPr(Tr − To)

c ∈ NCOLD

wi , j +

i ∈ NSOURCES

+ (10)

(13)

(14)

∑

gc, j

∑

∑

fr , j +

r ∈ NFRESH

gh, j

h ∈ NHOT

j ∈ NSINKS (15)

c ∈ NCOLD

where Gj is the total flow rate of sink j. Mass balance of waste:

∑

Bwaste =

Bi ,waste +

i ∈ NSOURCES

wi ,hCPi (Ti − To)

+

i ∈ NSOURCES

+

gc, j + Bc,waste

∑

Gj =

[ψr , pfr ,c ]

r ∈ NFRESH

∑

h ∈ NHOT

where gh,j and gc,j are the flow rates of hot stream h and cold stream c in HEN to sink j, while Bh,waste and Bc,waste are the flow rates of hot stream h and cold stream c in HEN to be discharged as waste, respectively. Mass balances at the mixing point before sink j:

[ψr , pfr ,h ]

p ∈ NPROP

− To) =

gh, j + B h,waste

j ∈ NSINKS

(9)

∑

∑

mc =

where ψi,p, ψr,p, ψh,p and ψc,p are the property operator p for source i, fresh resource r, hot stream h, and cold stream c in HEN, respectively. Energy balances of hot stream h and cold stream c at the mixing point before HEN are mh CPh(Thin

(12)

j ∈ NSINKS

r ∈ NFRESH

p ∈ NPROP

∑

ψc, pmc =

c ∈ NCOLD

(8)

r ∈ NFRESH

i ∈ NSOURCES

∑

mh = c ∈ NCOLD

where mh and mc are the flow rates of hot stream h and cold stream c. Based on eqs 7 and 8, the operator balances for property p of hot stream h and cold stream c at the mixing point before HEN are ψh, pmh =

fr ,c CPr(Tr − To)

where CPi, CPr, CPh, and CPc are heat capacities of source i, fresh resource r, hot stream h, and cold stream c, respectively; To, Ti and Tr are the reference temperature and the temperatures of sources i and fresh resource r while Tinh and Tinc are the supply temperatures of hot stream h and cold stream c, respectively. Mass balances of hot stream h and cold stream c at the splitting point after HEN are

h ∈ NHOT

r ∈ NFRESH

wi ,cCPi (Ti − To)

r ∈ NFRESH

where wi,j, wi,h, wi,c, and Bi,waste are the flow rates of source i recovered to sink j, to hot stream h, and cold stream c in HEN and to be discharged as waste, respectively. Mass balances of hot stream h and cold stream c at the mixing point before HEN are mh =

∑

+

c ∈ NCOLD

i ∈ NSOURCES

∑ i ∈ NSOURCES

∑

∑

Bc,waste

c ∈ NCOLD

h ∈ NHOT

B h,waste

h ∈ NHOT

(16)

where Bwaste is the flow rate of waste.

(11) 7198



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Figure 2. (a) Hot and cold composite curves. (b) Shifted hot and cold composite curves at true pinch point. (c) Infeasible hot and cold composite curves.

Operator balance for property p at the mixing point before sink j:

∑

ψj , pGj =

∑

+

∑

[ψi , pwi , j] +

i ∈ NSOURCES

merged by individual streams via linear superposition, and plotted on a temperature versus enthalpy diagram. These curves are then brought together via horizontal displacement until they reached a minimum temperature difference (ΔTmin). The vertical overlapping region between the two composite curves shows the energy recovery from the hot streams to the cold streams.2,7 If both composite curves are shifted vertically (such that hot composite curve is ΔTmin/2 cooler; and cold composite is ΔTmin/ 2 hotter) until they touch at the true pinch point, as shown in Figure 2b, one can observe the following: (1) For constant heat capacities, the potential pinch point candidate(s) are those corner points on the composite curves, that correspond to the inlet temperatures of any hot and cold streams.85 (2) The total energy balance for the HEN must always be achieved (total heat removal of the hot streams and that supplied by the external hot utility should equal to the total heat gained by the cold streams and that of external cold utility). To minimize the external hot and cold utilities, no energy should be transferred across the pinch.2,7 Hence, the true pinch point divides the composite curves into two regions, a heat source and a heat sink. The energy balance for both regions must also be satisfied. (3) To ensure feasible heat transfer, the hot composite curve must always stay above the cold composite curve, with both composite curves touching only at the true pinch point. (4) If the composite curves touch at any potential pinch point candidates other than the true pinch point (Figure 2c), some portions of the hot composite curve may lie below the cold composite curve in the same temperature range. For such a situation, energy transfer between the composite curves is thermodynamically infeasible. The basic idea of floating pinch is to postulate a set of pinch point candidates based on the inlet temperatures of the hot and cold streams. Constraints are then developed for each of the postulated pinch point candidates, which will then identify the true pinch point and also the minimum hot and cold utilities. The following section presents the HEN model of the HIRCN in detail, with the adoption of the floating pinch concept. The supply and target temperatures of the hot and cold streams from fresh resource r and source i to sink j are first shifted by subtracting ΔTmin/2 for hot streams; while adding ΔTmin/2 for cold streams. These temperatures are given by the equations as follow,

[ψr , pfr , j ]

r ∈ NFRESH

∑

[ψh, pgh, j] +

c ∈ NHOT

[ψc, pgc, j]

c ∈ NCOLD

j ∈ NSINKS

p ∈ NPROP

(17)

Note that eqs 9, 10, and 17 are used to determine the mean property value of each hot stream h, cold stream c, and sink j, and should be carried out for all properties in concern for each hot stream, cold stream, and process sink. Energy balance at the mixing point before the sink j:

∑

GjCPj(Tj − To) =

wi , jCPi (T − To)i

i ∈ NSOURCES

∑

+

fr , j CPr(Tr − To) +

r ∈ NFRESH

∑

+

∑

gh , jCPh(Thout − To)

h ∈ NHOT

gc, jCPc(Tcout − To)

j ∈ NSINKS

c ∈ NCOLD

(18)

out where CPj is the heat capacity of sink j while Tj, Tout h and Tc are the temperature of sink j and the target temperatures of hot stream h and cold stream c in HEN, respectively. Energy balance at the mixing point before waste:

Bwaste CPwaste(Twaste − To)

∑

=

Bi ,waste CPi (Ti − To)

i ∈ NSOURCES

+

∑

B h,waste CPh(Thout − To)

h ∈ NHOT

+

∑ c ∈ NCOLD

Bc,waste CPc(Tcout − To)

(19)

where CPwaste and Twaste are the heat capacity and temperature of waste, respectively. HEN Model. Owing to varying flow rates and temperatures in the problem formulation, the established heat integration models with fixed stream conditions (e.g., problem table,82 transportation problem,83 trans-shipment model84) cannot be applied in this case. Therefore, the f loating pinch method established by Duran and Grossmann85 and El-Halwagi and Manousiouthakis86 has been adopted in this work. To understand the concept of f loating pinch method, an understanding of the heat transfer composite curves is needed. As shown in Figure 2a, the hot and cold composite curves are

ΔTmin 2

Ths = Thin − Tht = Thout − Tcs = Tcin + 7199

ΔTmin 2 ΔTmin 2

h ∈ NHOT h ∈ NHOT c ∈ NCOLD

(20)

(21)

(22)


Industrial & Engineering Chemistry Research Tct = Tcout +

Tsh

ΔTmin 2

c ∈ NCOLD

Article

The total energy balance is expressed as (23)

∑

Tth

where and are the shifted supply and target temperatures of the hot streams, while Tsc and Ttc are the shifted supply and target temperatures of the cold streams in HEN. The potential pinch point candidate, Tq corresponds to the inlet temperatures of the hot and cold streams, ⎧ s h ∈ NHOT ⎪ Th Tq = ⎨ s c ∈ NCOLD ⎪Tc ⎩ q ∈ NPINCH

h ∈ NHOT

⎧1 if T t < T h q ⎪ =⎨ ⎪ 0 if Tht ≥ Tq ⎩ s ⎧ ⎪1 if Th < Tq =⎨ s ⎪ ⎩ 0 if Th ≥ Tq

λh,s q

ηc,t q

ηc,s q

⎧1 if T t < T q c ⎪ =⎨ t ⎪ 0 if Tc ≥ Tq ⎩ s ⎧ ⎪1 if Tc < Tq =⎨ s ⎪ ⎩ 0 if Tc ≥ Tq t λ h,q ,

s λ h,q ,

t η c,q

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SOLUTION STRATEGY As shown in eqs 29 and 30 the supply and target temperatures as well as the property operator p for each hot stream h and cold in out stream c (Tinh , Tout h , Tc , Tc , ψh,p and ψc,p) are unknown variables; this leads to a MINLP model. As the proposed model results in a large and highly complex model in both nonlinearity and integer aspects, it is computationally extensive. To overcome this issue, a discretization approach presented by Pham et al.87 is adopted in this work. According to Pham et al.,87 the unknown variables are discretized into several known values. As a result of this, it transforms the formulation into a MILP model which enables a global optimum solution to be obtained. With sufficient discretization, the global solution of the discretized problem can approximate (or coincide with) the true global optimal of the original problem.87 The following section explains the proposed strategy in detail. Step 1: Discretization of Property Operators for Each Hot Stream h and Cold Stream c. In this work, we proposed the use of property operators ψh,p and ψc,p as the discretized variables. The property operators of hot stream h and cold stream c are bounded by the property operators of the fresh resources, process sources and process sinks. Using the following notation: max ψmin HEN, p = min {ψr,p, ψi,p, ψj,p} and ψHEN, p = max {ψr,p, ψi,p, ψj,p}, then the domain of ψr,p and ψc,p are given as follow,

(25)

(26)

c ∈ NCOLD (27)

c ∈ NCOLD q ∈ NPINCH

(28)

s η c,q

where, and are the binary integer variables. Equation 29 shows the energy balance below the pinch point candidates. Qc ≥

∑

mh CPh{λh,t q(Tq − Tht) − λh,s q(Tq − Ths)}

h ∈ NHOT

−

∑

mcCPc{ηc,s q(Tq − Tcs) − ηc,t q(Tq − Tct)}

c ∈ NCOLD

q ∈ NPINCH

(30)

(3) Case 3: hot stream h lies across a potential pinch point q, which means that Tsh is above Tq while Tth is below Tq. Equation 25 and 26 determine that λsh,q = 0 and λth,q = 1. Therefore, heat lost by hot stream h below the potential pinch point is given as mhCPh (Tq − Tth).

h ∈ NHOT

q ∈ NPINCH

c ∈ NCOLD

mh CPh{(Tq − Tht) − (Tq − Ths)} = mh CPh(Ths − Tht)

h ∈ NHOT

q ∈ NPINCH

mcCPc(Tct − Tcs)

To demonstrate the usefulness of eq 29, let us consider the following situations where all possible locations of hot stream (with respect to the pinch point) are considered. (1) Case 1: hot stream h lies completely above the potential pinch point q. Based on eqs 25 and 26, integer λsh,q and λth,q are set to zero. Thus, heat lost by hot stream h below the potential pinch point is zero. (2) Case 2: hot stream h appears completely below the potential pinch point q. Equations 25 and 26 next determine that λsh,q = λth,q = 1. As a result, heat lost by hot stream h below the potential pinch point as given by the second term of eq 29 takes the following form.

(24)

q ∈ NPINCH

∑

+ Qh − Qc = 0

To identify the true pinch point and to ensure thermodynamic feasibility, the total energy balance is to be used together with energy balance above or below the pinch point candidate. In this work, energy balance below the pinch point candidate is used. Stream location parametrization is needed in order to determine the energy balance below the pinch point candidates. In the floating pinch approach for HEN presented by Duran and Grossmann,85 the authors used maximum operators to parametrize the stream locations. However, this approach causes nondifferentiability in the mathematical program and thus a special nonsmooth optimization algorithm is required to solve it. In this work, binary variables are used to parametrize the stream locations, following the floating pinch method for mass exchange network,86 given by the constraints in eqs 25−28. λh,t q

mh CPh(Ths − Tht) −

(29)

min max ψHEN, ≤ ψh, p ≤ ψHEN, p p

(31)

min max ψHEN, ≤ ψc, p ≤ ψHEN, p p

(32)

ψmin HEN, p

ψmax HEN, p

where and are the minimum and maximum property operator p in HEN respectively. The discretization approach limits the search space of property operator p of hot stream h and cold stream c whose values are max bounded between ψmin HEN, p and ψHEN, p. Discretization of the property operator p of hot stream h and cold stream c may be conducted in many ways. In this work, it is discretized based on

where mh, mc, Tth, and Ttc are flow rates and shifted target temperatures for hot stream h and cold stream c in HEN, respectively. Note that the second term of the equation shows the heat lost by hot stream h below a potential pinch candidate q; while the third term shows the heat gained by cold stream c below the potential pinch point. 7200



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property operator p of fresh resources, process sources, and process sinks (ψr,p, ψi, p and ψj, p) using the following notation: ψh, p = {ψr , p , ψi , p , ψj , p}

(33)

ψc, p = {ψr , p , ψi , p , ψj , p}

(34)

To reduce the size of the search space of the supply temperature of hot stream h and cold stream c (Tinh and Tinc ), convex hull approach proposed by Pham et al.87 is used. The main idea of this approach is that the supply temperature of any hot stream h and cold stream c, which is a result of mixing of various process sources, will be enclosed in the convex hull constructed by the convex combination of the properties of the individual process sources. To illustrate this concept, let us consider the same case study in Table 1. Without convex hull approach, the original search space of the supply temperature of hot stream h is bounded by 0 ppm ≤ ψh ≤ 1100 ppm and 20 °C ≤ Tinh ≤ 100 °C (Figure 3a). On the basis of the data given in Table 1, all four process sources (SR1, SR2, SR3, and FW1) are allocated as dots on the search space. Connecting these dots will then produce the convex hull (shaded area in Figure 3b) which is also referred to as the attainable region for the hot stream h. For given flow rates of SR1, SR2, SR3, and FW1, any possible mixture will fall within the attainable region. Therefore, values outside the convex hull are not needed. The application of a convex hull approach will significantly reduce the size of the search space. For the case in Figure 3, a 52% reduction in search space was achieved. This value is achieved by comparing the original search space region (Figure 3a) with the attainable region after implementing the convex hull approach (Figure 3b). The construction of a convex hull is relatively simple for a single property problem. However, it gets more challenging when more property operators are involved and may require a convex hull algorithm from the field of geometrical mathematics such as the Graham Scan Algorithm.88 Table 2 shows the Tinh and Tout h range values for data given in Table 1. Tinh corresponding to each ψh is defined based in Figure 3b,

For example, a single property operator case study is shown in Table 1, ψr,1 = {0} ppm ψi,1 = {100 ppm, 800 ppm, 1100 ppm} Table 1. Limiting Data for Case Study 1

Sink SK1 SK2 SK3 wastewater (WW) Source SR1 SR2 SR3 fresh water (FW1)

flow rate, F (kg/s)

temperature, T (°C)

concentration, C (ppm)

100 40 166.67

100 75 100 30

50 50 800

100 40 166.67

100 75 100 20

100 800 1100 0

and ψj,1 ={50 ppm, 800 ppm}. The property operator of both hot stream h and cold stream c is discretized as 0 ppm, 50 ppm, 100 ppm, 800 ppm, and 1100 ppm. The intention is to reduce the number of discretization values. Furthermore, it also reduces the piping needed for mixing and splitting of streams as compared to the equal distribution method proposed by Pham et al.87 Step 2: Discretization of Temperatures for Each Hot Stream h and Cold Stream c. Next, discretization of supply and target temperatures of hot stream h and cold stream c (Tinh , in out Tout h , Tc , Tc ) is conducted. The supply temperature is bounded by the source temperatures while the target temperature is bounded by the sinks temperatures. This results in Tinmin = out min{Ti}, Tinmax = max{Ti}, Tout min = min{Tj} and Tmax = max{Tj}. in out in out Then the boundaries for Th , Th , Tc , Tc are as follows, in in Tmin ≤ Thin ≤ Tmax

(35)

in Tmin

(36)

≤

Tcin

≤

in Tmax

out out Tmin ≤ Thout ≤ Tmax

(37)

out out Tmin ≤ Tcout ≤ Tmax

(38)

Table 2. Convex Hull for Case Study 1 ψh,1 (ppm)

Tinh,min (°C)

Tinh,max (°C)

Tout h,min (°C)

Tout h,max (°C)

0 50 100 800 1100

20 23.4 26.9 75 100

20 60 100 100 100

30 30 30 30 30

100 100 100 100 100

while Tout h is bounded based on the process sink temperatures. For example, for ψh of 100 ppm, the corresponding Tinh,min and Tinh,max are obtained from the vertical dotted line at 100 ppm in Figure 3b, and give 26.9 and 100 °C, respectively. On the other out hand, the corresponding Tout h,min and Th,max is based on the temperature of WW and SK1 or SK3 in Table 1. With these ranges

Figure 3. (a) Search space without convex hull approach. (b) Attainable region with convex hull approach. 7201



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Figure 4. HIRCN for Case Study 1 with m = 4.

and is supplied at 20 °C. On the other hand, wastewater has to be discharged at a temperature of 30 °C. The objective of this case study is to minimize the AOC, which consists of the operating costs for fresh resources as well as hot and cold utilities in the HIRCN. The optimization objective is given as follows:

defined, these variables are then discretized on the basis of the equal distribution method proposed by Pham et al.87 The same approach is applied for cold streams. Step 3: Application of Integer Cuts for Hot and Cold Streams. It is computationally extensive to have a huge number of hot streams h and cold streams c. Therefore, the actual number of hot streams h and cold streams c should be limited to m. To resolve this, an integer cut proposed by Pham et al.87 is used to select the desired number of hot streams h and cold streams c. Two binary variables f h and fc are introduced for hot stream h and cold stream c, respectively. If a hot stream h or cold stream c is used in the solution, then its binary variable is assigned a value of 1 or else 0. The number of hot streams h and cold streams c is restricted by using the following linear constraints:

∑

wi ,h ≤ Uhfh

h ∈ NHOT

wi ,c ≤ Ucfc

c ∈ NCOLD

∑

fh +

∑ c ∈ NCOLD

fc ≤ m

∑

Cost rFr + CosthQ h + CostcQ c}

r=1

(42)

where Costr is the cost of fresh resources. In this case study, heat capacities of all streams (CPc, CPh, CPi, CPr, and CPj) are assumed to take a constant value of 4.2 kJ/kg·K as the process streams are mainly water. In this case study, the actual number of hot and cold streams, m is assumed as 4. Solving the optimization objective in eq 42, subject to the constraints in eqs 1−30 and eqs 39−41, yield the minimum AOC of $1,383,822, with the optimal HIRCN shown in Figure 4. The values of Fr, Qh, and Qc for the HIRCN are determined as 77.27 kg/h, 4472.8 kW, and 1227 kW, respectively. To further verify the targeted results of Qh and Qc, the HEN for this case study is synthesized using the classical pinch design method7 and is presented in Figure 5. Note that identical energy targets are obtained as per that obtained via the proposed model. Note that the AOC depends on the actual number of hot and cold streams. There is an implied trade-off as the AOC may reduce if the actual number of hot and cold streams increases as more streams are allocated for HEN, which may reduce the external hot and cold utilities needed; however, more heat exchangers may be needed, which leads to higher capital cost. Sensitivity analysis is conducted to observe how the actual number of hot and cold streams affects the AOC. Figure 6 shows the plot of AOC versus actual number of hot and cold streams for Case Study 1. As shown in the figure, as the actual number of hot and cold streams increases, the AOC decreases proportionally and eventually levels off when the actual number of hot and cold streams reached 6. Figures 7 and 8 show the optimal HIRCN and HEN with an actual number of hot and cold streams of 6, which are also the same results for actual numbers of hot and cold streams of 7 and 8. The values of Fr, Qh and Qc for this HIRCN

(40)

i ∈ NSOURCES

h ∈ NHOT

Fr , Q h , Q c

(39)

i ∈ NSOURCES

∑

NFRESH

min AOC = k*{

(41)

where Uh and Uc are the maximum flow rates of the hot stream h or cold stream c and are set as the summation of total sources. The presented solution strategy linearizes the original proposed MINLP, which leads to an MILP model. Three case studies are solved in the following section, to illustrate the proposed approach.

■

CASE STUDIES All case studies are solved using Extended LINGO v11.0 with Global Solver. In all case studies, the costs of hot and cold utilities (Costh and Costc) are given as $3.758/kW·h and 0.005$/kW·h, respectively. In addition, total operating hour (k) and ΔTmin are taken as 8000 h/y and 10 °C respectively. Case Study 1. Case Study 1 reports a single property-based water network,72 consisting of three process sinks and three process sources, with limiting data given in Table 1. The fresh resource is pure fresh water (0 ppm), with a unit cost of $0.45/ton 7202



Article

Figure 5. HEN for HIRCN in Case Study 1 with m = 4.

Figure 6. Sensitivity analysis for Case Study 1: Effect of actual number of hot and cold streams on AOC.


utilities as well as the AOC. On the other hand, five additional heat exchangers are needed when the actual number of hot and cold streams increases from 4 to 6. However, when the actual number of hot and cold streams reaches 6, the heat recovery between hot

are determined as 81.65 kg/h, 3445.6 kW, and 13.2 kW, respectively. Comparing Figure 6 with Figure 8, it is noted that as more actual number of hot and cold streams is assigned, more heat recovery can take place, which further reduce the hot and cold 7203



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Figure 9. VAM process flow diagram.

stream then enters the second absorber where water is used to remove AA. On the other hand, the bottom product of the first absorber is sent to the primary distillation column. The top product of the primary tower includes VAM along with some water and AA while the bottom product of the primary tower is mixed with the bottom product of the second absorber. This waste stream is sent to a neutralization system followed by biotreament. Tables 3 and 4 show the limiting data for process sinks and sources, respectively. Note that both process sinks operate within a range of temperature and mass fraction. To include the operating ranges, eqs 2 and 3 are modified as follows:

and cold streams (which is constraint by the process sinks temperature) reaches its maximum. Therefore, a further increase of actual number of hot and cold streams can no longer reduce the AOC, as no additional hot and cold streams are needed. Case Study 2. The case study of vinyl acetate monomer (VAM) plant1 is adopted for use here. The process flow diagram for this process is shown in Figure 9. VAM and water are produced when acetic acid reacts with oxygen and ethylene. In this process, a fresh feed of acetic acid (AA) along with water is vaporized in the acid tower. The vapor is fed with oxygen and ethylene into a reactor where VAM is formed. The reactor off-gas is first cooled before entering the first absorber where AA is used as solvent. The exit at the top of the absorber consists of most of the gases and AA. This

0 ≤ ψ1,1 ≤ 0.05 7204

(43)



Article

Table 3. Limiting Data for Case Study 2 sink SK1 SK2

flow rate (kg/h)

minimum temperature (K)

maximum temperature (K)

minimum concentration (mass fraction)

450 300

500 350

0 0

5100 10200 source

maximum concentration (mass fraction)

flow rate (kg/h)

temperature (K)

0.05 0.10 concentration (mass fraction)

1400 9100

370 470

0.14 0.25

SR1 SR2

Table 4. Limiting Data for Case Study 3 sink SK1 SK2 SK3

flow rate (kg/h)

minimum temperature (°C)

2718 1993 1127 source

maximum temperature (°C)

70 30 25 flow rate (kg/h)

SR1 SR2 SR3 FR1 FR2

85 50 65

3661 1766 1485

minimum vapor pressure (kPa)

maximum vapor pressure (kPa)

minimum concentration (mass fraction)

15 10 13 temperature (°C)

35 25 40 vapor pressure (kPa)

75 65 40 25 35

38 25 7 3 6

0 0 0

maximum concentration (mass fraction)

0.013 0.013 0.1 concentration (mass fraction) 0.016 0.024 0.22 0 0.012


0 ≤ ψ2,1 ≤ 0.10

(44)

450 K ≤ T1 ≤ 500 K

(45)

300 K ≤ T2 ≤ 350 K

(46)

are shown in Figures 10 and 11. The Qh and Qc results of the HEN design match with those that were obtained from the optimization model. Case Study 3. Case Study 3 is adopted from Kheireddine et al.,81 where multiple properties (concentration, temperature, and vapor pressure, VP) and heat of mixing are considered. Note that heat of mixing is temperature-dependent; while other properties are interdependent of temperature effect. Table 4 shows the limiting data for process sinks and sources as well as fresh resources for this case study. Note that two fresh resources are available to fulfill the process sinks requirement, i.e. pure fresh water (FR1) and fresh water with 0.012 mass fraction of impurity (FR2). The unit cost of these sources correspond to $0.00132/kg (FR1) and $0.00088/kg (FR2), respectively. As heat of mixing is taken into consideration, the energy balances of hot stream h and cold stream c at the mixing point before HEN (eqs 11 and 12) as well as energy balance

The objective of this case study is to synthesize a HIRCN by minimizing AOC, which is contributed by fresh acetic acid, as well as hot and cold utilities. In this case study, the fresh acetic acid cost (Costr) is given as $0.625/kg. Meanwhile, heat capacity values of all streams (CPc, CPh, CPi, CPr, and CPj) are assumed to be constant at 2.5 kJ/(kg·K), as most streams consist of acetic acid. In this case study, the actual number of hot and cold streams, m is selected as 5 as it gives the lowest AOC. Solving eq 42 subject to constraints in eqs 1−18, eqs 20−30, eqs 39−41, and eqs 43−46 with this m value, determine the minimum AOC as $62,482,140, with the values of Fr, Qh, and Qc reported as 9587.12 kg/h, 0 kW, and 101.02 kW, respectively. The optimal HIRCN and HEN for this case study 7205



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at the mixing point before the sink j (eq 18) are replaced as follow,

∑

mh CPh(Thin − To) =

∑

fr ,h CPr(Tr − To) +

+ FjΔHjmix

h ∈ NHOT

∑

fr , j CPr(Tr − To)

r ∈ NFRESH

∑

gc, jCPc(Tcout − To)

c ∈ NCOLD

j ∈ NSINKS

(49)

where ΔHmix is the heat of mixing of sink j. i Note that an additional term of heat of mixing is included in the above equations. All the heat capacities in this case study can be determined via equation below:81

(47)

∑

gh, jCPh(Thout − To) +

h ∈ NHOT

r ∈ NFRESH

mcCPc(Tcin − To) =

∑

+

i ∈ NSOURCES

+

wi , jCPi (Ti − To) +

i ∈ NSOURCES

wi ,hCPi (Ti − To) mh ΔHhmix

∑

GjCPj(Tj − To) =

wi ,cCPi (Ti − To)

i ∈ NSOURCES

+

∑

fr ,c CPr(Tr − To) + mcΔHcmix

CP =

c ∈ NCOLD

∑ xkCPk

k ∈ NCOMP

(50)

r ∈ NFRESH

(48)

ΔHmix h

ΔHmix c

where and streams, respectively.

where xk is the mole fraction of component k and CP for each component can be determined using eq 51.81 Note that the heat capacity values are temperature-dependent.

are heat of mixing of hot and cold 7206


Industrial & Engineering Chemistry Research CPk = ak + bk T

k ∈ NCOMP

Article

h = index for hot streams i = index for the internal sources j = index for the sinks k = index for the components p = index for the properties q = index for the pinch point candidates r = index for the fresh sources waste = index for waste

(51)

where ak and bk are parameters in linearized temperaturedependent expression for heat capacity of each component. Based on Kheireddine et al.,81 a and b parameters for phenol are given as 0.4685 J/(g·K) and 0.0044 J/(g·K) while a and b parameters for water are 1.3724 J/(g·K) and 0.0083 J/(g·K) respectively. Besides, the heat of mixing for binary systems can be determined as81 ΔH

mix

⎛ Λ12b12 Λ 21b21 ⎞ = −Rx1x 2⎜ + ⎟ x 2 + x1Λ 21 ⎠ ⎝ x1 + x 2 Λ12

Sets

NCOLD = {c∣c is one of the cold streams} NCOMP = {k∣k is one of the components} NFRESH = {r∣r is a fresh source} NHOT = {h∣h is one of the hot streams} NPROP = {p∣p is one of the properties} NPINCH = {q∣q is one of the pinch point candidates} NSINKS = {j∣j is a sink} NSOURCES = {i∣i is an internal source}

(52)

with ln Λ12 = a12 +

b12 T

(53)

b21 (54) T where x is the mole fraction, T is the absolute temperature, and R is the ideal gas constant; (8.314J/(K mol)). Furthermore, a12, a21, b12, and b21 are the binary parameters in the Wilson equation for phenol and water solutions; these values are taken as 2.4395, −3.2239, −2229.9297, and 1046.1246, respectively. The optimization objective of this case study is to minimize AOC, which consists of operating costs for fresh resources, waste as well as hot and cold utilities in the HIRCN, as shown in eq 56.

Parameters

ln Λ 21 = a 21 +

ac = parameter in linearized temperature-dependent expression for heat capacity of the pure component; ac = 1.3724 J/(g K) for water, 0.4685 J/(g K) for phenol bc = parameter in linearized temperature-dependent expression for heat capacity of the pure component; bc = 0.0083 J/(g K) for water, 0.0044 J/(g K) for phenol a12 = binary parameter in the Wilson equation for phenol and water solution, a12 = 2.4395 a21 = binary parameter in the Wilson equation for phenol and water solution, a21 = −3.2239 b12 = binary parameter in the Wilson equation for phenol and water solution, b12 = −2229.9297 K b21 = binary parameter in the Wilson equation for phenol and water solution; b21 = 1046.1246 K CPc = heat capacity of a cold stream c CPh = heat capacity of a hot stream CPi = heat capacity of process source i CPk = heat capacity of the pure component CPr = heat capacity of fresh source r Costc = unit cost of cold utility Costh = unit cost of hot utility Costr = unit cost of fresh source r Costwaste = unit cost of waste Wi = total flow rate from process source i Gj = total flow rate inlet process sink j k = total operating hours per year R = ideal gas constant, is taken as 8.314 J/(Kmol) To = reference temperature, assumed to be 0°C Ti = temperature of process source i Tmax = upper bound temperature of process sink j j Tmin = lower bound temperature of process sink j j Tr = temperature of fresh source r Uc = maximum flow rate of cold stream c Uh = maximum flow rate of hot stream h ψi,p = property operator p of process source i ψmax j,p = upper bound property operator p of process sink j ψmin j,p = lower bound property operator p of process sink j ψr,p = property operator p of fresh source r

NFRESH

min

Fr , Fw , Q h , Q c

AOC = k*{

∑

Cost rFr + CostBBB

r=1

+ CosthQ h + CostcQ c}

(55)

where CostB is the cost of waste and is taken as $0.002/kg. Equation 56 is solved subject to the constraints in eqs 1−10, 13−17, 20−34, 39−41, and 47−55. It was determined that the actual number of hot and cold streams of 2 gave the lowest AOC of $49,959, with the optimized HIRCN presented in Figure 12. As shown, the synthesized HIRCN only consumes 1234.1 kg/h fresh resource 1 (FR1), without any utility targets needed.

■

CONCLUSION In this work, a novel approach has been presented for the synthesis and optimization of HIRCNs. An MINLP formulation has been developed to identify the minimum cost of a HIRCN with varying process parameters (e.g., flow rates, temperatures, and properties). Since the model is nonlinear and nonconvex, a discretization approach has been proposed in this work to guarantee solution quality and efficiency. Three case studies are solved to illustrate the proposed approach. On the basis of the results, it is shown that the proposed model is efficient and can be used to solve various types of problems. Future work will be directed to extend this methodology by including the capital cost of heat exchangers as well as piping cost for the HIRCNs.

■ ■

ACKNOWLEDGMENTS Financial support provided by Shell Malaysia Grant is deeply appreciated.

Variables

CPj = heat capacity of process sink j CPk = heat capacity of component k CPwaste = heat capacity of waste Bwaste = flow rate of waste Bc,waste = flow rate of cold stream c to waste

NOMENCLATURE

Indices

c = index for cold streams 7207



Article

Bh,waste = flow rate of hot stream h to waste Bi,waste = flow rate of process source i to waste Fr = flow rate of fresh resources r f r,c = flow rate of fresh resources r to cold stream c f r,h = flow rate of fresh resources r to hot stream h f r,j = flow rate of fresh resources r to process sink j fc = binary variable for cold stream c f h = binary variable for hot stream h gc,j = flow rate of cold stream c to process sink j gh,j = flow rate of hot stream h to process sink j mc = flow rate of cold stream c mh = flow rate of hot stream h Qh = external hot utility Qc = external cold utility Tinc = supply temperature of cold stream c Tout c = target temperature of cold stream c Tsc = shifted supply temperature of cold stream c Ttc = shifted target temperature of cold stream c Tinh = supply temperature of hot stream h Tout h = target temperature of hot stream h Tsh = shifted supply temperature of hot stream h Tth = shifted target temperature of hot stream h Twaste = temperature of waste Tq = potential pinch point candidate q wi,c = flow rate of process source i to cold stream c wi,h = flow rate of process source i to hot stream h wi,j = flow rate of process source i to process sink j xk = mole fraction of component k ψc,p = property operator p of cold stream c ψh,p = property operator p of hot stream h ψmin HEN,p = minimum property operator p in HEN ψmax HEN,p = maximum property operator p in HEN

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Greek Symbols

ΔHmix c = heat of mixing of cold stream c ΔHmix h = heat of mixing of hot stream h = heat of mixing of sink j ΔHmix j ΔTmin = minimum temperature difference Λ12, Λ21 = binary variables in the Wilson equation λth,q, λsh,q, ηtc,q, ηsc,q = binary variables in the energy balance equation

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +60-85-443833. Fax: +60-85-443837. E-mail: tan.yin. [email protected] (Y.L.T.); [email protected] (D.K.S.N.); [email protected] (M.M.E.-H.); Dominic.Foo@ nottingham.edu.my (D.C.Y.F.); [email protected] (Y.S.). || Adjunct Professor at King Abdul-Aziz University, Jeddah, Saudi Arabia Notes

The authors declare no competing financial interest.

■

REFERENCES

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