Unified Heat Exchanger Network Synthesis via a ... - ACS Publications

Subscriber access provided by Iowa State University | Library

Process Systems Engineering

Unified Heat Exchanger Network Synthesis via a Stageless Superstructure Sajitha K Nair, and Iftekhar A Karimi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b04490 • Publication Date (Web): 01 Feb 2019 Downloaded from http://pubs.acs.org on February 5, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

Unified Heat Exchanger Network Synthesis via a Stageless Superstructure Sajitha K. Nair and Iftekhar A. Karimi* Department of Chemical & Biomolecular Engineering National University of Singapore, 4 Engineering Drive 4, Singapore 117585 Abstract The existing mathematical programming-based methods for simultaneous heat exchanger network synthesis (HENS) have used either match-centric or stage-centric superstructures. Most works over the past four decades have primarily used various modifications of essentially two superstructures whose configurational limitations are well established. We revisit a novel, promising but unsuccessful, exchanger-centric superstructure proposed by Huang and Karimi1. We modify the superstructure and the associated mixed-integer nonlinear programming formulation substantially, and develop an efficient algorithm for its solution. The superstructure simply assumes a pool of two-stream exchangers, to which hot and cold streams are assigned. Given a sufficiently large pool, it embeds all possible heat exchanger network (HEN) configurations (in contrast to previous superstructures) including repeated matches, cross flows, bypasses, etc. and allows multiple utilities. The novel HEN configurations obtained for three case studies and improvements in their total annual costs highlight the advantages of the proposed superstructure and model. Keywords: Heat exchanger network synthesis; mixed integer non-linear programming; optimization; superstructure; heat integration.

*

Corresponding author: Email [email protected], Phone +65 6516-6359, Fax +65 6779-1936.

ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.

Introduction

Heating and cooling needs form a major component of industrial energy consumption. A variety of utilities (steam, cooling water, chilled water, refrigerant, etc.) may be required for these, and the annual utility bill for an industrial facility is often quite large. Thus, efforts to reduce utility consumption have significant economic and environmental benefits. The chemical process industry has been a long-time user of heat integration. It examines and optimizes the use of heat in process streams to effect best inter-stream heat exchanges in a network of 2-stream heat exchangers to reduce external utilities. Comprehensive reviews exist in the literature on heat exchanger network synthesis (HENS) 2-8. Furman and Sahinidis9 proved that HENS is an NP-hard problem. Masso and Rudd10 were the first to formalize a problem statement for HENS. They proposed several heuristics for stage-by-stage structuring to synthesize a minimum-cost network. Hohmann11 and Linnhoff and Flower12 introduced tabular methods to obtain minimum utility requirements. Later, it evolved into graphical pinch analysis to obtain networks with maximum energy recovery, however, it left open the question of cost optimal HENS. Thus, several works developed mathematical programming models based on the concept of pinch methods to obtain and optimize heat exchanger networks (HENs). Cerda13 automated the pinch-based minimum utility computations in terms of a transportation model. Papoulias and Grossmann14 introduced a linear programming (LP) transshipment model for minimum utilities and a mixed-integer linear programming (MILP) transshipment model for exchange matches that minimize the number of heat exchanger (HE) units. Floudas et al.15 proposed a superstructure and NLP model to minimize the investment cost based on the matches obtained from the MILP transshipment model of Papoulias and Grossmann14. The sequential method of HENS comprised of three models for obtaining a) utility targets, b) unit targets, and c) area targets, which many earlier studies sought to enhance. Gundersen et al.16 developed another sequential framework to get near-optimum networks


Page 2 of 50

Page 3 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


using LP, MILP, and nonlinear programing (NLP) models. Pettersson17 presented a sequential approach that aimed for reducing matches in large-scale HENs. Anantharaman et al.18 and Anantharaman et al.19 presented improvements and reformulations for the subproblem of minimizing the HE units. Later, Chen et al.20 studied strategies such as priority branching and model reformulation comprehensively to solve large-scale transshipment models related to HENS. Recently, Letsios et al.21 developed and compared the heuristics for minimizing the HEN units. Concurrently, simultaneous cost-based synthesis of HENs using superstructure optimization via mathematical programming was introduced to overcome the several limitations of the above sequential methods. This approach traded off the capital and operating costs explicitly during synthesis. Ptáčník and Klemeš22 and François and Irsia23 augmented mathematical optimization with effective heuristics. Floudas and Ciric24 and Ciric and Floudas25 generalized the hyperstructure of Floudas et al.15 to include all potential stream matches. However, they did not allow cyclic matching in which two process streams can exchange heat in multiple HEs without having to split each into two separate streams in advance. In certain cases, such cyclic matching can help in reducing the heat exchanger area 26. Yee and Grossmann27 proposed a mixed integer nonlinear programming (MINLP) model with a staged superstructure (SS), where they split each stream into substreams to enable exchange among all streams. However, they proposed a very useful simplification of isothermal mixing, which forced all substreams to have the same temperature change within a stage. As a result, the entire problem formulation except the objective function becomes linear. This assumption is known to limit HEN configurations, thus resulting in suboptimal HENs. Hence, Björk and Westerlund28 modified this staged superstructure to allow non-isothermal mixing. Huang et al.29 improved stream temperature bounds and added logical constraints to obtain the best HENs with non-isothermal mixing. They also studied and proposed various ways to handle logarithmic mean temperature difference (LMTD). Recently, Mistry and



Misener30 proved the convexity characteristics of the LMTD and its reciprocal. Hasan et al.31, Huang and Karimi1 and Na et al.32 allowed utility substages between process stages, which enable multiple utility exchangers in series. Pavão et al.33 enhanced SS by introducing utility exchangers in parallel in each stage. While Beck and Hofmann34,35 presented tightening strategies and a novel linearization for SS, Huang and Karimi1 found better solutions by combining the hyperstructure of Floudas et al.15 with the SS of Yee and Grossmann27. In another vein, Jongsuwat et al.36 introduced sub-stages at each stage to overcome the limitations of SS and used an ad-hoc solution strategy. Pavão et al.37 developed a similar SS allowing substages, sub-splits, and cross flows. The generalized superstructure of Ciric and Floudas25 necessitated a non-convex model. A generalized model may also give undesirable networks with many exchangers and splits. The motivation for simultaneous HENS with such generalized superstructures was also low initially due to solution difficulty. For instance, Huang and Karimi1 proposed a novel exchanger-centric and stageless superstructure for grassroot designs that could not be solved efficiently. However, our ability to solve MINLP problems keeps improving. Toimil and Gómez38 have reviewed the meta-heuristic algorithms for HENS. Furman and Sahinidis39 have reported a set of approximations for obtaining the network with minimum matches in HENS. Wu et al.40 have employed a nonlinear approximation for the binary variable detecting the existence of a heat exchanger. Kim and Bagajewicz41 have used their partition-based global optimization strategy to solve several superstructures in the literature 41-43. All the work discussed above except that of Huang and Karimi1 employed either matchcentric or stage-centric superstructures for HENS. However, very few works have attempted to develop new superstructures for grassroot designs that consider the networks disallowed by the most widely used superstructures. Yee and Grossmann44 had proposed a generalized stageless superstructure for retrofitting heat exchanger networks with additional exchangers and the


Page 4 of 50

Page 5 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


associated piping modifications. With the aforementioned algorithmic advances and newer MINLP solvers (ANTIGONE, SCIP etc.) it is worthwhile to explore a novel generalized superstructure for HENS. In this work, we revisit the stageless superstructure of Huang and Karimi1. It assumes a pool of 2-stream exchangers and assigns hot and cold streams to achieve the best HEN. It offers at least three advantages: a) A staged superstructure requires a postulation of maximum stages. Since the stages do not directly reflect the number of exchangers in a network, this choice is not straightforward, especially when a single stream needs many exchangers in series. In contrast, the proposed exchanger-centric superstructure presumes maximum number of exchangers in the network, which directly reflects the physical structure of a network. Usually, we have a better feel for the number of exchangers in a network. b) The staged superstructure embeds specific possibilities of network structure. This leads to the preclusion of some structures, as has been observed with most existing superstructures. A stageless superstructure has the potential to allow all possible complex structures. c) The treatment of a retrofit synthesis is not obvious with a staged superstructure. A stageless superstructure on the other hand provides a seamless approach to address both grassroots and retrofit syntheses. In this work, we not only modify the superstructure of Huang and Karimi1, but also propose a novel MINLP model along with an efficient solution algorithm. Our algorithm is motivated from the global optimization strategy developed by Castillo et al.45 for solving nonlinear scheduling problems. We present the problem statement followed by the superstructure, model formulation, and solution algorithm. Later, we illustrate our algorithm with two case studies. 2.

Problem Statement

A plant has 𝐻𝑃 process streams (hot streams) to be cooled and 𝐶𝑃 process streams (cold streams)



to be heated. It has 𝐶𝑈 cold and 𝐻𝑈 hot utilities for this purpose. The utility costs can be reduced by using a network of 2-stream heat exchangers that enable some hot streams to heat some cold streams. While such a network may save utilities (thus operating costs), it will need several exchangers (thus capital costs). Therefore, it is economically attractive to configure a heat exchanger network (HEN) that trades off these two costs optimally to achieve the minimum total annualized cost (TAC). This problem of heat exchanger network synthesis (HENS) can be stated as follows. Given: 1) 𝐻𝑃 hot and 𝐶𝑃 cold process streams with their flow rates, heat capacities, initial temperatures, and target temperatures. 2) 𝐻𝑈 hot and 𝐶𝑈 cold utility streams with their initial and target temperatures. 3) Heat transfer coefficients for all potential 2-stream exchanges of individual streams. 4) Data and/or correlations for the purchase costs of 2-stream exchangers, utility costs, and annualization factors for the exchanger costs. Obtain: 1) A network of 2-stream exchangers with their hot/cold streams, stream flows, and stream temperatures. 2) Heat duties and heat transfer areas for all exchangers. Aiming to: Minimize the total annualized cost. Assuming: 1) The thermophysical properties of all streams are constant or averaged over appropriate temperature ranges. 2) All exchangers are 2-stream countercurrent. 3) A hot utility cannot heat a cold utility.


Page 6 of 50

Page 7 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4) A process stream cannot use a particular utility more than once. 5) If an exchanger uses a utility, then the utility enters the exchanger at its initial temperature and leaves at its target temperature. 6) Heat transfer coefficients of individual streams in the exchangers are constant (independent of flow). 3.

HEN Superstructure There are total 𝐼 hot streams including hot process streams and hot utilities. Similarly, there

are 𝐽 cold streams, which include cold process streams and utilities. Let 1 ≤ 𝑖 ≤ 𝐻𝑃 denote the hot process streams, 𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼 = 𝐻𝑃 + 𝐻𝑈 denote the hot utilities, 1 ≤ 𝑗 ≤ 𝐶𝑃 denote the cold process streams, and 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽 = 𝐶𝑃 + 𝐶𝑈 denote the cold utilities. Let 𝑠 ( 1 ≤ 𝑠 ≤ 𝑆 ) denote a generic stream (hot/cold process/utility) such that 1 ≤ 𝑠 ≤ 𝐻𝑃 denote the hot process streams, 𝐻𝑃 + 1 ≤ 𝑠 ≤ 𝐼 denote the hot utilities, 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃 denote the cold process streams, and 𝐼 + 𝐶𝑃 + 1 ≤ 𝑠 ≤ 𝑆 = 𝐼 + 𝐽 denote the cold utilities. With this, the plant has 𝐼 hot streams (𝑖 = 1, 2, … , 𝐻𝑃, 𝐻𝑃 + 1, . . , 𝐼 = 𝐻𝑃 + 𝐻𝑈), and 𝐽 cold streams (𝑗 = 1, 2, … , 𝐶𝑃, 𝐶𝑃 + 1, . . , 𝐽 = 𝐶𝑃 + 𝐶𝑈). Let 𝐹5 be the heat content flow (flow times heat capacity or kW/K) of a process stream (1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃). In addition, let 𝑇𝐼𝑁5 be the initial and 𝑇𝑂𝑈𝑇5 be the final or target temperature of a stream 𝑠 (1 ≤ 𝑠 ≤ 𝑆). For the desired HEN, we define a stageless superstructure (Figure 1) with 𝐸 (1 ≤ 𝑒 ≤ 𝐸) 2stream exchangers (𝐻𝐸> , 1 ≤ 𝑒 ≤ 𝐸), 𝐸 + 1 multi-stream hot splitters (𝐻𝑆𝑃> , 0 ≤ 𝑒 ≤ 𝐸), 𝐸 + 1 multi-stream cold splitters (𝐶𝑆𝑃> , 0 ≤ 𝑒 ≤ 𝐸), 𝐸 + 1 multi-stream hot mixers (𝐻𝑀𝑋> , 1 ≤ 𝑒 ≤ 𝐸 + 1), and 𝐸 + 1 multi-stream cold mixers (𝐶𝑀𝑋> , 1 ≤ 𝑒 ≤ 𝐸 + 1). Each hot process stream 𝑖 (1 ≤ 𝑖 ≤ 𝐻𝑃) enters 𝐻𝑆𝑃B at 𝑇𝐼𝑁C and leaves 𝐻𝑀𝑋(DEF) at 𝑇𝑂𝑈𝑇C . Similarly, each cold process stream 𝑗 (1 ≤ 𝑗 ≤ 𝐶𝑃) enters 𝐶𝑆𝑃B at 𝑇𝐼𝑁G and exits 𝐶𝑀𝑋(DEF) at 𝑇𝑂𝑈𝑇G . Each 𝐻𝐸> (1 ≤ 𝑒 ≤ 𝐸)



has one mixer at each inlet, 𝐻𝑀𝑋> at the hot and 𝐶𝑀𝑋> at the cold inlet. Similarly, it has one splitter at each outlet, 𝐻𝑆𝑃> at the hot and 𝐶𝑆𝑃> at the cold outlet. The stream from 𝐻𝑀𝑋> (1 ≤ 𝑒 ≤ 𝐸) becomes the hot, and that from 𝐶𝑀𝑋> (1 ≤ 𝑒 ≤ 𝐸) becomes the cold stream for 𝐻𝐸> . The hot process stream leaving 𝐻𝐸> (1 ≤ 𝑒 ≤ 𝐸) enters 𝐻𝑆𝑃> and the cold process stream leaving 𝐻𝐸> enters 𝐶𝑆𝑃> . The hot (cold) utility stream enters an exchanger at 𝑇𝐼𝑁C through 𝐻𝑀𝑋> (𝐶𝑀𝑋> ) at the entry of exchanger and exits from 𝐻𝑆𝑃> (𝐶𝑆𝑃> ) at 𝑇𝑂𝑈𝑇C . The inlet stream splitter 𝐻𝑆𝑃B (𝐶𝑆𝑃B ) splits the stream into 𝐸 + 1 substreams; one for each mixer 𝐻𝑀𝑋> (𝐶𝑀𝑋> ) at the inlet of exchanger 𝐻𝐸> (1 ≤ 𝑒 ≤ 𝐸) and one for the bypass stream directly exiting the HEN through mixer 𝐻𝑀𝑋DEF (𝐶𝑀𝑋DEF ). The mixer 𝐻𝑀𝑋> (𝐶𝑀𝑋> ) at inlet of exchanger mixes streams from all other exchangers, process stream from inlet stream splitter 𝐻𝑆𝑃B (𝐶𝑆𝑃B ) and utility streams. Similarly, the splitter 𝐻𝑆𝑃> (𝐶𝑆𝑃> ) at exchanger outlet splits into substreams for each of the exchanger, the outlet mixer 𝐻𝑀𝑋DEF (𝐶𝑀𝑋DEF ) and a utility stream. Thus, each mixer 𝐻𝑀𝑋> (𝐶𝑀𝑋> ) mixes 𝐸 + 1 hot (cold) substreams and splitter 𝐻𝑀𝑋> (𝐶𝑀𝑋> ) splits 𝐸 + 1 hot (cold) substreams. As illustrated in Figure 1, there are three types of hot stream flow to any exchanger 𝐻𝐸> : utility flow (𝐹C> ), process stream flow (𝐹C 𝑥CB> ) from the inlet splitter, process stream flow from other exchangers (𝑓> J > ). Similarly, at the exit of an exchanger, the stream splits into utility flow, flow to other exchangers and flow to the process stream exit mixer. Based upon which substream has nonzero flow, the configuration of exchangers would be decided. Some possible configurations for a hot process stream 𝑖 are shown in Figure 2. The substreams depicted in the figures have non-zero flow. In actual HEN, these configurations or a combination of such configurations can exist for both hot and cold streams.


Page 8 of 50

Page 9 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4.

Model Formulation

We select 𝐸 a priori. Define 𝑄5L = 𝐹5 |𝑇𝑂𝑈𝑇5 − 𝑇𝐼𝑁5 | as the heat exchange capacity of a process stream 𝑠 (1 ≤ 𝑠 ≤ 𝐻𝑃 ; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃 ). Then, with no loss of generality, we revise the stream indices of both hot and cold streams as follows. We assign 𝑖 = 1 to the hot stream with the highest capacity, 𝑖 = 2 to the hot stream with the second highest capacity, and so on. We do the same for the cold streams. The cold stream with the highest capacity becomes 𝑗 = 1 and the one with the lowest capacity becomes 𝑗 = 𝐶𝑃. 4.1

Network Structure A process stream 𝑠 will need at least one exchanger, unless 𝑄5L = 0. Thus, the HEN must

have at least 𝐸OCP = max(𝐻𝑃, 𝐶𝑃) exchangers and it can have at most 𝐸 exchangers. We eliminate any extra exchangers by defining the following 0-1 continuous variable. 1 𝑖𝑓 𝐻𝐸> 𝑖𝑠 𝑢𝑠𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝐻𝐸𝑁 𝑧> = U 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

1≤𝑒≤𝐸

Some binary variables defined later will force 𝑧> to be binary. Since at least 𝐸OCP exchangers must exist, we set 𝑧> = 1 ( 1 ≤ 𝑒 ≤ 𝐸OCP ) with no loss of generality. Furthermore, to eliminate equivalent permutations of the remaining exchangers, we prioritize the exchangers with lower indices. Thus, if an 𝐻𝐸> is not used, then 𝐻𝐸(>EF) , 𝐻𝐸(>E^) , to 𝐻𝐸D cannot be used either. 𝑧> ≥ 𝑧(>EF)

𝐸OCP + 1 ≤ 𝑒 ≤ 𝐸 − 1

(1)

Now, we assign streams to 𝐻𝐸> by defining the following binary variables. 1 𝑖𝑓 𝑠𝑡𝑟𝑒𝑎𝑚 𝑠 𝑢𝑠𝑒𝑠 𝐻𝐸> 𝑥5> = U 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

1 ≤ 𝑠 ≤ 𝑆; 1 ≤ 𝑒 ≤ 𝐸

We ensure that each existing 𝐻𝐸> must have exactly one hot and one cold stream via the following. 𝑧> = ∑cCdF 𝑥C> = ∑eGdF 𝑥G>

1≤𝑒≤𝐸

(2)

Eq. 2 forces 𝑧> to be binary. Then, we define 𝑥CG> = 𝑥C> · 𝑥G> , (1 ≤ 𝑖 ≤ 𝐼; 1 ≤ 𝑗 ≤ 𝐽) as a 0-1



Page 10 of 50

continuous variable and linearize it as below. ∑cCdF 𝑥CG> = 𝑥G>

1 ≤ 𝑒 ≤ 𝐸; 1 ≤ 𝑗 ≤ 𝐶𝑃

(3a)

∑eGdF 𝑥CG> = 𝑥C>

1 ≤ 𝑒 ≤ 𝐸; 1 ≤ 𝑖 ≤ 𝐻𝑃

(3b)

∑gh CdF 𝑥CG> = 𝑥G>

1 ≤ 𝑒 ≤ 𝐸; 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽

(3c)

∑ih GdF 𝑥CG> = 𝑥C>

1 ≤ 𝑒 ≤ 𝐸; 𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼

(3d)

We set 𝑥CG> = 0 for 𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼 and 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽 to prevent a hot utility from heating a cold utility. We avoid the variables associated with the matching of the hot utilities with the cold utilities throughout our model. Since a process stream can use a particular utility only once, we write, ∑D>dF 𝑥CG> ≤ 1

1 ≤ 𝑖 ≤ 𝐻𝑃; 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽

(4a)

∑D>dF 𝑥CG> ≤ 1

1 ≤ 𝑗 ≤ 𝐶𝑃; 𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼

(4b)

We also set limits [𝐸5j , 𝐸5L ] on the number of exchangers a stream 𝑠 (1 ≤ 𝑠 ≤ 𝑆) may use. Assuming 𝑄5 > 0 for all process streams, 𝐸5j = 1 for them, and 𝐸5L can be estimated from 𝑄5L . For utilities, pinch analysis may tell us if a utility will be absolutely required. Then, we can set 𝐸5j = 1 for such a utility. Therefore, we can write, 𝐸5j ≤ ∑D>dF 𝑥5> ≤ 𝐸5L

1≤𝑠≤𝑆

(5a)

If the optimal HEN hits this limit for any 𝑠 , then we can optimize again with a higher 𝐸5L . Additionally, pinch analysis may also help us fix some limits on the total number of utility exchangers. j L 𝐸gL ≤ ∑cCdghEF ∑D>dF 𝑥C> ≤ 𝐸gL

(5b)

j L 𝐸iL ≤ ∑eGdihEF ∑D>dF 𝑥G> ≤ 𝐸iL

(5c)

Since we know that 𝐻𝐸F to 𝐻𝐸Dmno must exist, we can assign specific process streams to them. If 𝐸OCP = max[𝐻𝑃, 𝐶𝑃] = 𝐻𝑃, then we assign a hot stream to each of them. If 𝐸OCP = 𝐶𝑃,


Page 11 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


then we assign a cold stream to each of them. 𝑥C> = 𝛿C>

1 ≤ 𝑒 ≤ 𝐻𝑃; 1 ≤ 𝑖 ≤ 𝐻𝑃; if 𝐻𝑃 ≥ 𝐶𝑃

(6a)

𝑥G> = 𝛿G>

1 ≤ 𝑒 ≤ 𝐶𝑃; 1 ≤ 𝑗 ≤ 𝐶𝑃; if 𝐶𝑃 ≥ 𝐻𝑃

(6b)

For using 𝐻𝐸Dmno EF to 𝐻𝐸D , we prioritize streams as follows. Process streams have a higher priority than utilities. A process stream with a larger 𝑄5 (or 𝑠) has a higher priority to use an 𝐻𝐸> with a lower 𝑒. Similarly, a utility with a higher temperature has a higher priority to use an 𝐻𝐸> with a lower 𝑒. We enforce these priorities as follows for 𝐸OCP = 𝐻𝑃. 𝑥F> ≥ 𝑥F(>EF)

𝐸OCP + 1 ≤ 𝑒 ≤ 𝐸 − 1

(7a)

𝑥F> + 𝑥^> ≥ 𝑥F(>EF) + 𝑥^(>EF)

𝐸OCP + 1 ≤ 𝑒 ≤ 𝐸 − 1

(7b)

⁞ 𝑥F> + 𝑥^> + … + 𝑥c,> ≥ 𝑥F(>EF) + 𝑥^(>EF) + ⋯ + 𝑥c,(>EF) 𝐸OCP + 1 ≤ 𝑒 ≤ 𝐸 − 1

(7c)

The above can be combined into a single equation for the hot streams as follows. ∑CC J dF 𝑥C J > ≥ ∑CC J dF 𝑥C J (>EF)

1 ≤ 𝑖 ≤ 𝐼; 𝐸OCP + 1 ≤ 𝑒 ≤ 𝐸 − 1

(7d)

For the cold streams, we do the following. 𝑥CG> + 𝑥CG J (>EF) ≤ 1 𝑗 ≤ 𝐶𝑃); 𝐸OCP + 1 ≤ 𝑒 ≤ 𝐸

(1 ≤ 𝑖 ≤ 𝐻𝑃; 1 ≤ 𝑗 s < 𝑗 ≤ 𝐽) 𝑎𝑛𝑑 (𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼; 1 ≤ 𝑗 s < (7e)

Similarly, for 𝐸OCP = 𝐶𝑃, we write, ∑GG J dF 𝑥G J > ≥ ∑GG J dF 𝑥G J (>EF) 𝑥CG> + 𝑥C J G(>EF) ≤ 1 𝑖 ≤ 𝐻𝑃); 𝐸OCP + 1 ≤ 𝑒 ≤ 𝐸

1 ≤ 𝑗 ≤ 𝐽; 𝐸OCP + 1 ≤ 𝑒 ≤ 𝐸 − 1

(8a)

(1 ≤ 𝑗 ≤ 𝐶𝑃; 1 ≤ 𝑖 s < 𝑖 ≤ 𝐼) 𝑎𝑛𝑑 (𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽; 1 ≤ 𝑖 s < (8b)

Our above discussion assumed that 𝐸OCP = max[𝐻𝑃, 𝐶𝑃] is a good estimate for the minimum number of exchangers in the network. Often, we can do better. For instance, pinch



Page 12 of 50

analysis may suggest that some utility exchangers (utility heaters for 𝐸OCP = 𝐻𝑃, and vice versa) must exist. Then, we can preassign such exchangers by fixing 𝑧> = 1 for and one or more specific utility streams to them. 4.2

Stream Flows and Balances

Let 𝑓>> J (1 ≤ 𝑒 ≤ 𝐸; 1 ≤ 𝑒 s ≤ 𝐸; 𝑒 s ≠ 𝑒) denote the heat content flow of the hot stream from 𝐻𝑆𝑃> (or 𝐻𝐸> ) to 𝐻𝑀𝑋> J (or 𝐻𝐸> J ), and 𝑔>> J (1 ≤ 𝑒 ≤ 𝐸; 1 ≤ 𝑒 s ≤ 𝐸; 𝑒 s ≠ 𝑒) denote the same of the cold stream from 𝐶𝑆𝑃> (or 𝐻𝐸> ) to 𝐶𝑀𝑋> J (or 𝐻𝐸> J ). Then, we can avoid circular flows between any two exchangers by demanding, 𝑓>> J 𝑓> J > = 0

1 ≤ 𝑒 < 𝑒s ≤ 𝐸

(9a)

𝑔>> J 𝑔> J > = 0

1 ≤ 𝑒 < 𝑒s ≤ 𝐸

(9b)

Since the above terms are nonlinear and zero; we add them as penalties to our objective of total annualized cost. Clearly, the flows between any two exchangers can be nonzero, if and only if a process stream uses them both, as different process streams cannot mix. To ensure this, we define another 0-1 continuous variable, namely 𝑥5>> J = 𝑥5> 𝑥5> J (1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃; 1 ≤ 𝑒 < 𝑒 s ≤ 𝐸) as follows. 1 𝑖𝑓 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑠𝑡𝑟𝑒𝑎𝑚 𝑠 𝑢𝑠𝑒𝑠 𝑏𝑜𝑡ℎ 𝐻𝐸> 𝑎𝑛𝑑 𝐻𝐸> J 𝑥5>> J = U 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃; 1 ≤ 𝑒 < 𝑒 s ≤ 𝐸 Then, we linearize 𝑥5>> J as follows. 𝑥5>> J ≤ 𝑥5>

1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃; 1 ≤ 𝑒 < 𝑒 s ≤ 𝐸

(10a)

𝑥5>> J ≤ 𝑥5> J

1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃; 1 ≤ 𝑒 < 𝑒 s ≤ 𝐸

(10b)

𝑥5>> J ≥ 𝑥5> + 𝑥5> J − 1

1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃; 1 ≤ 𝑒 < 𝑒 s ≤ 𝐸

(10c)

J

L ∑D>J |> 𝑥5>> J + ∑>> J }> dF 𝑥5> J > ≤ 𝑥5> (𝐸5 − 1)

1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃; 1 ≤ 𝑒 ≤ 𝐸


(10d)

Page 13 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Using these, we write the following to prevent the mixing of process streams. 𝑓>> J + 𝑓> J > ≤ ∑gh CdF 𝐹C 𝑥C>> J

1 ≤ 𝑒 < 𝑒s ≤ 𝐸

(11a)

𝑔>> J + 𝑔> J > ≤ ∑ih GdF 𝐹G 𝑥G>> J

1 ≤ 𝑒 < 𝑒s ≤ 𝐸

(11b)

Now, let 𝑥5B> (1 ≤ 𝑒 ≤ 𝐸 + 1) be the fractional heat content flow of a process stream 𝑠 from 𝐻𝑆𝑃B to 𝐻𝑀𝑋> (if 𝑠 is a hot stream) or from 𝐶𝑆𝑃B to 𝐶𝑀𝑋> (if 𝑠 is a cold stream). Similarly, 𝑥5>(DEF) (0 ≤ 𝑒 ≤ 𝐸 ) be the fractional heat content flow of a process stream 𝑠 from 𝐻𝑆𝑃> to 𝐻𝑀𝑋(DEF) or from 𝐶𝑆𝑃> to 𝐶𝑀𝑋(DEF) . Then, we have, 𝑥5B> ≤ 𝑥5>

1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃; 1 ≤ 𝑒 ≤ 𝐸

(12a)

𝑥5>(DEF) ≤ 𝑥5>

1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃; 1 ≤ 𝑒 ≤ 𝐸

(12b)

∑DEF >dF 𝑥5B> = 1

1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃

(12c)

∑D>dB 𝑥5>(DEF) = 1

1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃

(12d)

Let 𝐹> denote the heat content flow of the hot stream through 𝐻𝐸> (1 ≤ 𝑒 ≤ 𝐸), and 𝐺> denote the same for the cold stream. Let 𝐹5> (𝐻𝑃 + 1 ≤ 𝑠 ≤ 𝐼; 𝐼 + 𝐶𝑃 + 1 ≤ 𝑠 ≤ 𝑆) denote the utility flow in 𝐻𝐸> . Then, the flow balances around 𝐻𝑀𝑋> , 𝐶𝑀𝑋> , 𝐻𝑆𝑃> , and 𝐶𝑆𝑃> give us, c 𝐹> = ∑D>J dF,> J •> 𝑓> J > + ∑gh CdF 𝐹C 𝑥CB> + ∑CdghEF 𝐹C>

1≤𝑒≤𝐸

(13a)

c 𝐹> = ∑D>J dF,> J •> 𝑓>> J + ∑gh CdF 𝐹C 𝑥C>(DEF) + ∑CdghEF 𝐹C>

1≤𝑒≤𝐸

(13b)

e 𝐺> = ∑D>J dF,> J •> 𝑔> J > + ∑ih GdF 𝐹G 𝑥GB> + ∑GdihEF 𝐹G>

1≤𝑒≤𝐸

(14a)

e 𝐺> = ∑D>J dF,> J •> 𝑔>> J + ∑ih GdF 𝐹G 𝑥G>(DEF) + ∑GdihEF 𝐹G>

1≤𝑒≤𝐸

(14b)

To avoid small exchangers, we define 𝐹5j (1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃) as the minimum flow of stream s in an exchanger. Furthermore, the flow must be zero in non-existent exchangers. ∑cCdF 𝐹Cj 𝑥C> ≤ 𝐹> ≤ 𝐹>L 𝑧>

1≤𝑒≤𝐸

(15a)

∑eGdF 𝐹Gj 𝑥G> ≤ 𝐺> ≤ 𝐺>L 𝑧>

1≤𝑒≤𝐸

(15b)



Page 14 of 50

If we do not want streams to split, then the flow through the exchanger must be 𝐹5 (inlet heat content flow of stream). Thus, we can enforce no-splitting by simply setting 𝐹5j = 𝐹5 . 4.3

Stream Temperatures and Exchanger Duties

Define: 𝑇Cj = Minimum allowable temperature for stream 𝑖 in the process 𝑇Cj = 𝑇𝑂𝑈𝑇C for 𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼 𝑡GL = Maximum allowable temperature for stream 𝑗 in the process 𝑡GL = 𝑇𝑂𝑈𝑇G for 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽 𝑇𝐼> = Temperature of the hot stream into 𝐻𝐸> 𝑡𝑖> = Temperature of the cold stream into 𝐻𝐸> 𝑇𝑂> = Temperature of the hot stream out of 𝐻𝐸> 𝑡𝑜> = Temperature of the cold stream out of 𝐻𝐸> 𝐷> = Temperature drop for the hot stream in 𝐻𝐸> 𝑅> = Temperature rise for the cold stream in 𝐻𝐸> Then, 𝑇𝐼> and 𝑡𝑖> (1 ≤ 𝑒 ≤ 𝐸) must satisfy, j c c L L ∑gh CdF 𝑇C 𝑥C> + ∑CdghEF 𝑇𝐼𝑁C 𝑥C> + 𝑇> (1 − 𝑧> ) ≤ 𝑇𝐼> ≤ ∑CdF 𝑇𝐼𝑁C 𝑥C> + 𝑇> (1 − 𝑧> )

1≤𝑒≤𝐸

(16a)

c L ∑cCdF 𝑇Cj 𝑥C> + (𝑇>L − 𝐷>j )(1 − 𝑧> ) ≤ 𝑇𝑂> ≤ ∑gh CdF 𝑇𝐼𝑁C 𝑥C> + ∑CdghEF 𝑇𝑂𝑈𝑇C 𝑥C> + (𝑇> −

𝐷>j )(1 − 𝑧> )

1≤𝑒≤𝐸

(16b)

e L j ∑eGdF 𝑇𝐼𝑁G 𝑥G> + 𝑡>j (1 − 𝑧> ) ≤ 𝑡𝑖> ≤ ∑ih GdF 𝑡G 𝑥G> + ∑GdihEF 𝑇𝐼𝑁G 𝑥G> + 𝑡> (1 − 𝑧> )

1≤𝑒≤𝐸

(17a)

e e L j j j ∑ih GdF 𝑇𝐼𝑁G 𝑥G> + ∑GdihEF 𝑇𝑂𝑈𝑇G 𝑥G> + (𝑡> + 𝑅> )(1 − 𝑧> ) ≤ 𝑡𝑒> ≤ ∑GdF 𝑡G 𝑥G> + (𝑡> +

𝑅>j )(1 − 𝑧> )

1≤𝑒≤𝐸


(17b)

Page 15 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


𝑇𝑂> = 𝑇𝐼> − 𝐷>

1≤𝑒≤𝐸

(17c)

𝑡𝑜> = 𝑡𝑖> + 𝑅>

1≤𝑒≤𝐸

(17d)

Then, the energy balances around 𝐻𝑀𝑋> and 𝐶𝑀𝑋> give us, D 𝐹> 𝑇𝐼> ≤ ∑cCdghEF 𝑇𝐼𝑁C 𝐹C> + ∑gh CdF 𝑇𝐼𝑁C 𝐹C 𝑥CB> + ∑> J dF,> J •> 𝑓> J > 𝑇𝑂> J

1≤𝑒≤𝐸

(18a)

D 𝐺> 𝑡𝑖> ≥ ∑eGdihEF 𝑇𝐼𝑁G 𝐹G> + ∑ih GdF 𝑇𝐼𝑁G 𝐹G 𝑥GB> + ∑> J dF,> J •> 𝑔> J > 𝑡𝑜> J

1≤𝑒≤𝐸

(18b)

𝐷> and 𝑅> will be limited by the minimum temperature approaches and minimum acceptable drops and rises. Hence, ih L gL ih L ∑cCdF 𝑥C> 𝐷Cj + 𝐷>j (1 − 𝑧> ) ≤ 𝐷> ≤ ∑gh CdF ∑GdF 𝑥CG> 𝐷CG + ∑CdghEF ∑GdF 𝑥CG> 𝐷CG + iL L j ∑gh CdF ∑GdihEF 𝑥CG> 𝐷CG + 𝐷> (1 − 𝑧> )

1≤𝑒≤𝐸

(19a)

ih L gL ih L ∑eGdF 𝑥G> 𝑅Gj + 𝑅>j (1 − 𝑧> ) ≤ 𝑅> ≤ ∑gh CdF ∑GdF 𝑥CG> 𝑅CG + ∑CdghEF ∑GdF 𝑥CG> 𝑅CG + iL L j ∑gh CdF ∑GdihEF 𝑥CG> 𝑅CG + 𝑅> (1 − 𝑧> )

1≤𝑒≤𝐸

(19b)

where, 𝐷Cj = Minimum desired temperature drop for stream 𝑖 in an exchanger 𝐷Cj = |𝑇𝑂𝑈𝑇C − 𝑇𝐼𝑁C | for 𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼 𝑅Gj = Minimum desired temperature rise for stream 𝑗 in an exchanger 𝑅Gj = ‚𝑇𝐼𝑁G − 𝑇𝑂𝑈𝑇G ‚ for 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽 𝑀𝑇𝐴CG = Minimum allowable temperature approach for an exchange between 𝑖 and 𝑗 (1 ≤ 𝑖 ≤ 𝐼, 1 ≤ 𝑗 ≤ 𝐶𝑃 and 1 ≤ 𝑖 ≤ 𝐻𝑃, 1 ≤ 𝑗 ≤ 𝐽) 𝐷CGL = max„𝐷Cj , 𝑇𝐼𝑁C − max(𝑇Cj , 𝑇𝐼𝑁G + 𝑀𝑇𝐴CG )… for 1 ≤ 𝑖 ≤ 𝐼, 1 ≤ 𝑗 ≤ 𝐶𝑃 and 1 ≤ 𝑖 ≤ 𝐻𝑃, 1 ≤ 𝑗 ≤ 𝐽 L 𝑅CG = max [𝑅Gj , min (𝑡GL , 𝑇𝐼𝑁C − 𝑀𝑇𝐴CG ) − 𝑇𝐼𝑁G ] for 1 ≤ 𝑖 ≤ 𝐼, 1 ≤ 𝑗 ≤ 𝐶𝑃 and

1 ≤ 𝑖 ≤ 𝐻𝑃, 1 ≤ 𝑗 ≤ 𝐽



Page 16 of 50

𝑀𝑇𝐴j = min(𝑚𝑖𝑛FˆCˆc,FˆGîh „𝑀𝑇𝐴CG …, 𝑚𝑖𝑛FˆCˆgh,FˆGê „𝑀𝑇𝐴CG …) We can now compute the temperature approaches at the hot (𝐻𝑇𝐴> ) and cold (𝐶𝑇𝐴> ) ends of 𝐻𝐸> and make them exceed 𝑀𝑇𝐴CG . ih gL ih 𝑀𝑇𝐴j (1 − 𝑧> ) + ∑gh CdF ∑GdF 𝑥CG> 𝑀𝑇𝐴CG + ∑CdghEF ∑GdF 𝑥CG> 𝑀𝑇𝐴CG + iL j L j j ∑gh CdF ∑GdihEF 𝑥CG> 𝑀𝑇𝐴CG ≤ 𝐻𝑇𝐴> ≤ 𝑇𝐼> − 𝑡𝑜> + (𝑀𝑇𝐴 − 𝑇> + 𝑡> + 𝑅> )(1 − 𝑧> )

1≤𝑒≤𝐸

(20a)

ih gL ih 𝑀𝑇𝐴j (1 − 𝑧> ) + ∑gh CdF ∑GdF 𝑥CG> 𝑀𝑇𝐴CG + ∑CdghEF ∑GdF 𝑥CG> 𝑀𝑇𝐴CG + iL j L j j ∑gh CdF ∑GdihEF 𝑥CG> 𝑀𝑇𝐴CG ≤ 𝐶𝑇𝐴> ≤ 𝑇𝑂> − 𝑡𝑖> + (𝑀𝑇𝐴 − 𝑇> + 𝑡> + 𝐷> )(1 − 𝑧> )

1≤𝑒≤𝐸

(20b)

Let 𝑄> (1 ≤ 𝑒 ≤ 𝐸) denote the duty of 𝐻𝐸> , and let 𝑄5> = 𝑄> 𝑥5> (1 ≤ 𝑒 ≤ 𝐸;1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃). We linearize 𝑄5> by, c 𝑄> = ∑gh CdF 𝑄C> + ∑CdghEF(𝑇𝐼𝑁C − 𝑇𝑂𝑈𝑇C )𝐹C>

1≤𝑒≤𝐸

(21a)

e 𝑄> = ∑ih GdF 𝑄G> + ∑GdihEF‰𝑇𝑂𝑈𝑇G − 𝑇𝐼𝑁G Š𝐹G>

1≤𝑒≤𝐸

(21b)

𝑄> ≤ 𝐹> 𝐷>

1≤𝑒≤𝐸

(22a)

𝑄> ≤ 𝐺> 𝑅>

1≤𝑒≤𝐸

(22b)

e L 𝑄C> ≤ ∑ih GdF 𝑄CG 𝑥CG> + ∑GdihEF‰𝑇𝑂𝑈𝑇G − 𝑇𝐼𝑁G Š𝐹G>

1 ≤ 𝑒 ≤ 𝐸; 1 ≤ 𝑖 ≤ 𝐻𝑃

(23a)

L c 𝑄G> ≤ ∑gh CdF 𝑄CG 𝑥CG> + ∑CdghEF(𝑇𝐼𝑁C − 𝑇𝑂𝑈𝑇C )𝐹C>

1 ≤ 𝑒 ≤ 𝐸; 1 ≤ 𝑗 ≤ 𝐶𝑃

(23b)

𝑄5> ≤ 𝑄5L 𝑥5>

1 ≤ 𝑒 ≤ 𝐸; 1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃

∑D>dF 𝑄5> = 𝑄5L

1 ≤ 𝑠 ≤ 𝐻𝑃; 𝐼 + 1 ≤ 𝑠 ≤ 𝐼 + 𝐶𝑃

(24) (25a)

ih j gL ih j gh iL j ∑gh CdF ∑GdF 𝑥CG> 𝑄CG + ∑CdghEF ∑GdF 𝑥CG> 𝑄CG + ∑CdF ∑GdihEF 𝑥CG> 𝑄CG ≤ 𝑄> ≤ ih L gL ih L gh iL L ∑gh CdF ∑GdF 𝑥CG> 𝑄CG + ∑CdghEF ∑GdF 𝑥CG> 𝑄CG + ∑CdF ∑GdihEF 𝑥CG> 𝑄CG

where,


1≤𝑒≤𝐸

(25b)

Page 17 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


L L 𝑄CG = min( 𝑄CL , 𝑄GL , 𝐹C 𝐷CGL , 𝐹G 𝑅CG )

1 ≤ 𝑖 ≤ 𝐻𝑃; 1 ≤ 𝑗 ≤ 𝐶𝑃

L 𝑄CG = min( 𝑄CL , 𝐹C 𝐷CGL )

1 ≤ 𝑖 ≤ 𝐻𝑃; 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽

L L 𝑄CG = min( 𝑄GL , 𝐹G 𝑅CG )

𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼; 1 ≤ 𝑗 ≤ 𝐶𝑃

j 𝑄CG = max( 𝐹Cj 𝐷Cj , 𝐹Gj 𝑅Gj )

1 ≤ 𝑖 ≤ 𝐻𝑃; 1 ≤ 𝑗 ≤ 𝐶𝑃

j 𝑄CG = 𝐹Cj 𝐷Cj

1 ≤ 𝑖 ≤ 𝐻𝑃; 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽

j 𝑄CG = 𝐹Gj 𝑅Gj

𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼; 1 ≤ 𝑗 ≤ 𝐶𝑃

Using the above bounds, we get the following for the utility flows. L 𝐹C> |𝑇𝑂𝑈𝑇C − 𝑇𝐼𝑁C | ≤ ∑ih GdF 𝑄CG 𝑥CG>

1 ≤ 𝑒 ≤ 𝐸; 𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼

(26a)

L 𝐹G> ‚𝑇𝑂𝑈𝑇G − 𝑇𝐼𝑁G ‚ ≤ ∑gh CdF 𝑄CG 𝑥CG>

1 ≤ 𝑒 ≤ 𝐸; 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽

(26b)

j 𝐹C> |𝑇𝑂𝑈𝑇C − 𝑇𝐼𝑁C | ≥ ∑ih GdF 𝑄CG 𝑥CG>

1 ≤ 𝑒 ≤ 𝐸; 𝐻𝑃 + 1 ≤ 𝑖 ≤ 𝐼

(26c)

j 𝐹G> ‚𝑇𝑂𝑈𝑇G − 𝑇𝐼𝑁G ‚ ≥ ∑gh CdF 𝑄CG 𝑥CG>

1 ≤ 𝑒 ≤ 𝐸; 𝐶𝑃 + 1 ≤ 𝑗 ≤ 𝐽

(26d)

j j Finally, we can use the minimum hot (𝑄gL ) and cold (𝑄iL ) utility requirements from pinch analysis

as follows.

4.4

j ∑cCdghEF 𝐹C> |𝑇𝑂𝑈𝑇C − 𝑇𝐼𝑁C | ≥ 𝑄gL

1≤𝑒≤𝐸

(27a)

j ∑eGdihEF 𝐹G> ‚𝑇𝑂𝑈𝑇G − 𝑇𝐼𝑁G ‚ ≥ 𝑄iL

1≤𝑒≤𝐸

(27b)

Exchanger Areas and LMTDs

Following Huang et al.29, we compute the log mean temperature difference 𝐿𝑀𝑇𝐷> (1 ≤ 𝑒 ≤ 𝐸) of 𝐻𝐸> as follows. 𝐻𝑇𝐴> = 𝐶𝑇𝐴> + 𝐿𝑀𝑇𝐷> [ln(𝐻𝑇𝐴> ) − ln(𝐶𝑇𝐴> )]

1≤𝑒≤𝐸

(28a)

2𝐿𝑀𝑇𝐷> ≤ 𝐻𝑇𝐴> + 𝐶𝑇𝐴>

1≤𝑒≤𝐸

(28b)

Let 𝐴> (1 ≤ 𝑒 ≤ 𝐸) denote the area of 𝐻𝐸> , and 𝐴j denotes its lower limit to avoid numerical errors and exchangers with small areas. Then, the cost 𝐶𝐻𝐸> (1 ≤ 𝑒 ≤ 𝐸) of 𝐻𝐸> is,



Page 18 of 50

ih gL ih gh iL j ∑gh CdF ∑GdF 𝑥CG> 𝑈CG + ∑CdghEF ∑GdF 𝑥CG> 𝑈CG + ∑CdF ∑GdihEF 𝑥CG> 𝑈CG + 𝑈 (1 − 𝑧> ) ≤ 𝑈> ≤ ih gL ih gh iL j ∑gh CdF ∑GdF 𝑥CG> 𝑈CG + ∑CdghEF ∑GdF 𝑥CG> 𝑈CG + ∑CdF ∑GdihEF 𝑥CG> 𝑈CG + 𝑈 (1 − 𝑧> )

1≤𝑒≤𝐸

(29)

𝑄> ≤ 𝑈> 𝐴> 𝐿𝑀𝑇𝐷>

1≤𝑒≤𝐸

(30)

𝐴> ≤ 𝐴L> 𝑧> + 𝐴j> (1 − 𝑧> )

𝐸OCP ≤ 𝑒 ≤ 𝐸

(31)

𝐶𝐻𝐸> ≥ 𝐹𝐶𝑧> + 𝑉𝐶[𝐴> ]Ž − 𝑉𝐶[𝐴j> ]Ž (1 − 𝑧> )

1≤𝑒≤𝐸

(32)

where, 𝑈CG = Overall heat transfer coefficient for an exchange between i and j (kW/m2-K) (1 ≤ 𝑖 ≤ 𝐼, 1 ≤ 𝑗 ≤ 𝐶𝑃 and 1 ≤ 𝑖 ≤ 𝐻𝑃, 1 ≤ 𝑗 ≤ 𝐽) 𝑈 j = min (𝑚𝑖𝑛FˆCˆc,FˆGîh „𝑈CG …, 𝑚𝑖𝑛FˆCˆgh,FˆGê „𝑈CG …) 𝐹𝐶 = Fixed cost of a heat exchanger ($) 𝑉𝐶= Variable cost of a heat exchanger ($) 𝛼 = Exponent in the cost correlation for a heat exchanger The total annualized cost (TAC) is given by, TAC = ∑D>dF ∑ghEFˆ5ˆc;cEihEFˆ5ˆ• 𝑈𝐶5 𝐹5> |𝑇𝐼𝑁5 − 𝑇𝑂𝑈𝑇5 | + 𝛾 ∑D>dF 𝐶𝐻𝐸> D 𝐶𝑂𝑆𝑇 = 𝑇𝐴𝐶 + 𝜏[∑D“F >dF ∑> J dF,> J |> (𝑓>> J 𝑓> J > + 𝑔>> J 𝑔> J > )]

(33) (34)

where, 𝜏 is a multiplier of the order of 𝑇𝐴𝐶, 𝑈𝐶5 is the unit cost of utility 𝑠 ($/kW), and 𝛾 is the capital annualization factor. This completes our MINLP formulation (Eqs. 1-8, 10-34) for minimizing 𝐶𝑂𝑆𝑇 with all variables being non-negative. 5.

Solution Strategy

Our proposed MINLP model for HENS has been difficult to solve. Most commercial solvers including BARON, ANTIGONE, and SCIP failed to converge to a feasible solution. Hence, it became necessary for us to develop a heuristic outer-approximation strategy46 to obtain better


Page 19 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


rather than global solutions. Our strategy takes inspiration from the global optimization strategy of Castillo et al.45 for solving nonlinear scheduling problems and Viswanathan and Grossmann46 outer-approximation strategy for non-convex MINLP problems. The basic idea is to create an expanding pool of potential HEN configurations by solving a series of MILP relaxations of our original model (P-MINLP). Then, solve an NLP for each configuration by fixing the binaries in PMINLP. This gives us an upper bound on TAC, and the various MILP relaxations give us an approximate lower bound. Close the gap as much as possible or terminate on maximum iterations/solution time, and report the best HEN obtained in the process. As the first step in developing our series of MILP relaxations, we simplify P-MINLP as much as possible by fixing some variables related to the minimum exchangers. 5.1

Variable Elimination Recall that insights from pinch analysis can be used to increase 𝐸OCP . In addition, problem-

s specific analyses may suggest that the HEN will need some more exchangers, say 𝐸OCP > 𝐸OCP . s For such exchangers, we set 𝑧> = 1 (𝐸OCP + 1 ≤ 𝑒 ≤ 𝐸OCP ), but do not assign any streams to them, s as we cannot guarantee that they will exist. 𝐸OCP = 𝐻𝑃 + 𝐶𝑃 is often a good guess.

For the 𝐸OCP exchangers, we can fix some variables to eliminate several nonlinear terms and constraints. Consider a stream 𝑠 assigned to 𝐻𝐸> based on Eq. 6. Since 𝑠 must enter the HEN via at least one exchanger, we make 𝐻𝐸> the entry exchanger for 𝑠 with no loss of generality. Now, no flow from another exchanger can mix with this stream, because that would mean a recycle or back flow that would reduce the “temperature quality” of 𝑠. Therefore, we fix 𝑓> J > 𝑜𝑟 𝑔> J > = 0 and 𝑇𝐼> 𝑜𝑟 𝑡𝑖> = 𝑇𝐼𝑁5 . This makes Eq. 18 redundant. Further, we fix the utility flow 𝐹C> or 𝐹G> = 0 for 𝐸OCP exchangers with process streams. After simplifying P-MINLP, we develop a series of MILP relaxations. 5.2

P-MILP



Page 20 of 50

To obtain our first lower-bounding MILP relaxation (P-MILP), we drop a nonlinear constraint (Eq. 28a) related to LMTD and linearize/relax the various bilinear and trilinear terms in P-MINLP. For the six bilinear ( 𝐹> 𝑇𝐼> , 𝑓> J > 𝑇𝑂> J , 𝐺> 𝑡𝑖> , 𝑔> J > 𝑡𝑜> J , 𝐹> 𝐷> , 𝐺> 𝑅> ) and one trilinear (𝑈> 𝐴> 𝐿𝑀𝑇𝐷> ) terms, we use piecewise MILP relaxation 47-49. For this, we partition 𝐹> , 𝐺> , 𝐷> , 𝑅> , 𝐴> , 𝑇𝑂> J , 𝑡𝑜> J , and 𝐴> 𝐿𝑀𝑇𝐷> into 𝑛 equal intervals each, and employ the convex hull formulation for each partition. The constraints comprising these relaxations are listed in Supporting Information. Next, we linearize the bilinear flow terms in Eq. 34. For this, we define the following binary variables to force each flow between exchangers to be unidirectional, thus eliminating the bilinear penalty terms from Eq. 34. 𝑥ℎ>> J = ”

1 𝑖𝑓 𝑎 𝑓𝑙𝑜𝑤 𝑓𝑟𝑜𝑚 𝐻𝐸> 𝑡𝑜 𝐻𝐸> J 𝑖𝑠 𝑎𝑙𝑙𝑜𝑤𝑒𝑑 𝑓𝑜𝑟 𝑡ℎ𝑒 ℎ𝑜𝑡 𝑠𝑡𝑟𝑒𝑎𝑚 0 𝑖𝑓 𝑎 𝑓𝑙𝑜𝑤 𝑓𝑟𝑜𝑚 𝐻𝐸> J 𝑡𝑜 𝐻𝐸> 𝑖𝑠 𝑎𝑙𝑙𝑜𝑤𝑒𝑑 𝑓𝑜𝑟 𝑡ℎ𝑒 ℎ𝑜𝑡 𝑠𝑡𝑟𝑒𝑎𝑚 1 ≤ 𝑒 < 𝑒s ≤ 𝐸

1 𝑖𝑓 𝑎 𝑓𝑙𝑜𝑤 𝑓𝑟𝑜𝑚 𝐻𝐸> 𝑡𝑜 𝐻𝐸> J 𝑖𝑠 𝑎𝑙𝑙𝑜𝑤𝑒𝑑 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑐𝑜𝑙𝑑 𝑠𝑡𝑟𝑒𝑎𝑚 𝑥𝑐>> J = ” 0 𝑖𝑓 𝑎 𝑓𝑙𝑜𝑤 𝑓𝑟𝑜𝑚 𝐻𝐸> J 𝑡𝑜 𝐻𝐸> 𝑖𝑠 𝑎𝑙𝑙𝑜𝑤𝑒𝑑 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑐𝑜𝑙𝑑 𝑠𝑡𝑟𝑒𝑎𝑚 1 ≤ 𝑒 < 𝑒s ≤ 𝐸 Then, the following constraints will eliminate any bidirectional flows between two exchangers. 𝑓>> J ≤ min‰𝐹>L , 𝐹>LJ Š 𝑥ℎ>> J

1 ≤ 𝑒 < 𝑒s ≤ 𝐸

(35a)

𝑓> J > ≤ min‰𝐹>L , 𝐹>LJ Š (1 − 𝑥ℎ> J > )

1 ≤ 𝑒 < 𝑒s ≤ 𝐸

(35b)

𝑔>> J ≤ min‰𝐺>L , 𝐺>LJ Š 𝑥𝑐>> J

1 ≤ 𝑒 < 𝑒s ≤ 𝐸

(36c)

𝑔> J > ≤ min‰𝐺>L , 𝐺>LJ Š (1 − 𝑥𝑐> J > )

1 ≤ 𝑒 < 𝑒s ≤ 𝐸

(36d)

However, such flows can also cause circular flows across three, four, or all the exchangers using a given process stream. For instance, consider the flows from 𝑒 to 𝑒 s and 𝑒 s to 𝑒′′. Any flow from 𝑒′′ to 𝑒 will cause a circular flow, which we must prevent. In other words, 𝑓>> J . 𝑓> J > JJ . 𝑓> JJ > = 0, and we can add the following constraints.


Page 21 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


𝑥ℎ>> J + 𝑥ℎ> J > JJ + 1 − 𝑥ℎ>> JJ ≤ 2

1 ≤ 𝑒 < 𝑒 s < 𝑒 ss ≤ 𝐸

(37)

𝑥𝑐>> J + 𝑥𝑐> J > JJ + 1 − 𝑥𝑐>> JJ ≤ 2

1 ≤ 𝑒 < 𝑒 s < 𝑒 ss ≤ 𝐸

(38)

Note that special care in indices was required in using 𝑥ℎ>> JJ and 𝑥𝑐>> JJ in the above constraints. While circular flows can exist across four and more exchangers, the required constraints grow exponentially, so we do not impose them. Next, if the HE cost expression is not linear, then we underestimate 𝐶𝐻𝐸> by a linear cost. 𝐶𝐻𝐸> ≥ 𝐹𝐶𝐿𝑧> + 𝑉𝐶𝐿[𝐴> ] − 𝑉𝐶𝐿[𝐴j> ](1 − 𝑧> )

1≤𝑒≤𝐸

(39)

Finally, we impose one integer cut for each HEN configuration that has been evaluated previously to prevent it from being considered again. ∑> ∑C ∑G 𝑥CG> ≤ (∑D>dF 𝑧> ) − 1

(40)

where, the left side includes only those 𝑖, 𝑗, and 𝑒 for which 𝑥CG> = 1 in the HEN configuration. The above steps give us P-MILP, whose solution is a lower bound to the original P-MINLP. It involves minimizing TAC subject to Eqs. 1-8, 10-17, 19-21, 23-27, 28b, 29, 31, 35-40, and S1S14. Now, we tighten P-MILP further by contracting the bounds of the eight partitioned variables. Let P-MILPT denote the tightened P-MILP. 5.3

P-MILPT

Our partitioned variables are 𝐹> , 𝐺> , 𝐷> , 𝑅> , 𝐴> , 𝑇𝑂> , 𝑡𝑜> , and 𝐿𝑀𝑇𝐷> . We solve a series of simplified P-MILPs to estimate their tightest feasible bounds as follows. We set a single partition for all these variables in P-MILP. Then, we solve two P-MILPs for each variable; one to maximize the variable to update the upper bound and the other to minimize the variable to update the lower bound. We now add some more linearizations to obtain an outer-approximation for P-MINLP, which would give us a relaxed lower bound for TAC. We call the resulting relaxation P-MILPO.



5.4

P-MILPO

For every feasible NLP we solve, we get a point that can give us an outer-approximation50 based on the linearization of various nonlinear terms. Since our NLPs are nonconvex, we use slack variables that are penalized in the objective function through augmented penalty terms 46. These slack variables allow violations of the linearization, minimizing the possibility of cutting off parts of the feasible region. Specifically, we add the linearizations of Eqs. 18a, 18b, 22a, 22b, 28a, and 30 to P-MILP to obtain P-MILPO. The linearized equations with penalty terms are presented in Supporting Information. P-MILPO minimizes 𝐶𝑂𝑆𝑇 + ∑D>dF ∑™ ˜dF(𝜔˜F> 𝑠˜F> + 𝜔˜^> 𝑠˜^> + 𝜔˜š> 𝑠˜š> + 𝜔˜›> 𝑠˜›> + 𝜔˜œ> 𝑠˜œ> + 𝜔˜•> 𝑠˜•> ) where 𝑠˜F , 𝑠˜^ , 𝑠˜š , 𝑠˜› , 𝑠˜œ , and 𝑠˜• are the infeasibilities in the linearizations of Eqs. 18a, 18b, 22a, 22b, 28a, and 30 respectively at a given NLP solution 𝑘. Following Viswanathan and Grossmann46, we use 𝜔˜F> = 1000|𝜆˜F> |, 𝜔˜^> = 1000|𝜆˜^> | , 𝜔˜š> = 1000|𝜆˜š> | , 𝜔˜›> = 1000|𝜆˜›> | , 𝜔˜œ> = 1000|𝜆˜œ> | and 𝜔˜•> = 1000|𝜆˜•> |, where 𝜆˜ are the Lagrange multipliers corresponding to Eqs. 18, 22, 28a and 30 at the NLP solution 𝑘. The augmented penalty terms in the objective of P-MILPO may make its solution an invalid lower bound for P-MINLP. We now discuss the NLP that is required for developing P-MILPO. 5.5

P-NLP and P-NLPF

We obtain two NLPs by fixing the binaries in P-MINLP. In P-NLP, we minimize 𝐶𝑂𝑆𝑇 subject to Eqs. 1-8, 10-34. In P-NLPF, we minimize 𝐼𝑁𝐹 = 𝜇F + 𝜇^ + 𝜇š + 𝜇› subject to Eqs. 1-32, where 𝜇F , 𝜇^ , 𝜇š and 𝜇› are the infeasibilities in Eqs 18, 22, 28a and 30 respectively. The solution of PNLP for a feasible HEN configuration is a feasible solution to P-MINLP and gives us an upper bound. We now have all the constituents of our proposed HENS algorithm (Figure 3). 6.

HENS Algorithm


Page 22 of 50

Page 23 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1. Specify an iteration limit, overall optimality gap, tolerance for infeasibility, size of solution pool, and initial partitions for P-MILPO. Set iteration = 1. 2. Solve feasibility based bound tightening P-MILPT and update the bounds for the partitioned variables. 3. If iteration/solution time limit is exceeded, then go to step 6. Otherwise, solve P-MILPO with GAMS/CPLEX for a limited CPU time or until an optimality gap to generate a pool of integer solutions. We set the solution intensity parameter PopSolnPoolIntensity = 4 and SolnPoolPop = 2 to enumerate maximum solutions in the given time limit using the GAMS/CPLEX option solnpool. 4. If no integer solution is returned, then set the number of partitions 𝑛 = 1, and go to step 3. Otherwise, do the following. a. Take a solution from the pool. If no solution remains, then go to step 5. b. Check if the binary solution is new. If it is, then solve P-NLPF with IPOPT51 to minimize the infeasibility. If it is not new, then delete the solution from the pool, and go to step (4a). c. If the infeasibility is within the tolerance (1, 0.01 etc), then solve P-NLP using OQNLP52 (a global nonlinear solver) to obtain a feasible solution for P-MINLP. d. Update the upper bound for P-MINLP, delete this solution from pool and go to step (4a). 5. Add integer cuts for all the solutions in the pool and outer-approximation to the newly obtained feasible solutions to P-MILPO. If the upper bound does not improve, then set 𝑛 = min(2𝑛, 8). 6. Increment the iteration counter. If the iteration number or total solution time has not exceeded the limit, or the optimality gap is not reached, then go to step 3. Otherwise stop. Report the best solution. P-MILP is the lower bound and forms the basis for P-MILPT and P-MILPO. Since no integer cut



and NLP solution are available in the first iteration, P-MILPO reduces to P-MILP. P-MILPO with lower number of partitions is easier to solve, however, it is a weak relaxation of original problem. In contrast, P-MILPO with a greater number of partitions is difficult to solve but is a strong relaxation to P-MINLP. Thus, we vary the number of partitions in Step 3 and Step 5 so as to keep tightening the problem till it becomes intractable. Further, if we obtain two close P-NLP solutions with nearly equal TAC values, then we use only one solution for the outer-approximation in PMILPO to avoid clustering effect. 7.

Case Studies

We implemented the model and algorithm in GAMS version 25.153 to solve three case studies from Kim and Bagajewicz41 and Pavão et al.37 Kim and Bagajewicz41 had used a partition-based global optimization strategy to solve an extension of the superstructure proposed by Floudas et al.15 Pavão et al.33 had enhanced the stagewise superstructure of Yee and Grossmann27 by placing utility exchangers in parallel at each stage and used a meta-heuristic algorithm for its solution. Later, they used a superstructure with substages, splits, and cross flows37. The stream data for the case studies is listed in Table 1 and HEN results, comparison with the literature and model statistics is shown in Table 2, Table 3 and Table 4 respectively. The streams are arranged in the decreasing order of their capacity in the case studies. It should be noted that the model statistics vary in each iteration for P-MILPO depending on the interval size, number of P-NLP solutions and integer cuts. The composite curves for case studies are generated using Aspen Energy Analyzer and presented in Figure 4. Figure 5 present the various solution statistics for the algorithm. Figures 5(a), (c) and (e) depict the number of unique solutions generated in P-MILPO and P-NLP in each iteration for case studies. The dotted line shows the best TAC solution available at each iteration. The solid line is the solution of the lower bound model P-MILP. Figures 5(b), (d) and (f) depict the solution time for P-MILPO and total time for solving P-NLPF/P-NLP for the solution pool generated in each


Page 24 of 50

Page 25 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


iteration. 7.1

Case Study 1

The first example is from Faria et al.43 which was solved globally for the stagewise superstructure in Faria et al.43 and the generalized superstructure in Kim and Bagajewicz41. It has four hot and five cold streams with the stream data in Table 1. As all the streams have the same heat transfer coefficient, the overall heat transfer coefficient is the same for all exchangers. This simplifies the trilinear term 𝑈> 𝐴> 𝐿𝑀𝑇𝐷> to a bilinear term 𝐴> 𝐿𝑀𝑇𝐷> . Faria et al.43 reported a TAC of $99.6 million with 11 exchangers for the stage-wise superstructure. Later, Kim and Bagajewicz41 reported a similar TAC of $99.63 million with 12 exchangers for generalized superstructure. Our best HEN with TAC of $32.4 million and 13 exchangers is shown in Figure 6. Our superstructure allows various kinds of simple and novel configurations. First, the final HEN consists of series exchangers in substreams such as exchanger 7, and 9 for the cold stream C3. Secondly, it also has parallel exchangers such as exchanger 4 and 11 for cold stream C4. Further, there are bypasses for streams in an exchanger such as the bypass for hot stream H3 in exchanger 12 and for cold stream C3 in exchanger 10. The advantage of bypass for hot stream H3 in exchanger 12 is to control the inlet temperature of stream to E12 so as to not violate the MAT constraints. Finally, the utility heater 13 is in parallel with the process stream exchanger 12 for cold stream C4. Thus, our HEN has some interesting features that have not appeared in most previous works, even though they may be found in the industry. The HEN from Kim and Bagajewicz41 uses end coolers for hot streams H1 and H2 and endheater for cold stream C5. In contrast, our HEN suggests one end cooler for stream H2 and one utility heater for stream C4. Further, all the heaters/coolers in Kim and Bagajewicz41 are at the exit of the streams in series with other exchangers, whereas the utility heater 13 for C4 is in parallel with exchanger 12 in our HEN. The total area of the exchangers is 5008 m2 in our HEN compared



to 3323 m2 41. However, the total hot utility consumption is 71.8 % lower with 67.8 % lower utility costs ($ 32 million/a versus $99.3 million/a). This reduces our TAC to $32.4 million compared to $99.6 million, in spite of using one more exchanger (13 vs 12) than Kim and Bagajewicz41. This is a major improvement in TAC (67.5% compared to both Faria et al.43 and Kim and Bagajewicz41). We plotted the composite curves for the system in Figure 4(a). The hot and cold composite curves are close to each other over a wide range of temperature. Even for such tight curves, our network could reduce utilities owing to the novel configurations that our superstructure allows. These configurations ensured almost parallel temperature profile in a counter-current exchanger by maintaining almost the same flows of hot and cold stream for same temperature changes of hot and cold stream. As this case study involves unusually high utility costs, the TAC reduction is drastic. Considering that TAC solutions of Faria et al.43 and Kim and Bagajewicz41 were claimed to be globally minimum for their superstructures, our lower TAC is a clear proof of our superior superstructure. 7.2

Case Study 2

The second example is from an aromatics plant considered first by Khorasany and Fesanghary54 and then by Pavão et al.33 It has 16 process streams (6 hot and 10 cold), two hot utilities (steam and flue gas), and one cold utility (cooling water) as shown in Table 1. Khorasany and Fesanghary54 synthesized their HEN via a hybrid method involving a harmony search combined with sequential quadratic programming (SQP). They reported a TAC of $ 7.4 million with 18 heat exchangers. Pavão et al.33 solved the same case study using an enhanced stage-wise superstructure and a hybrid metaheuristic strategy involving simulated annealing (SA) combined with Rocket Fireworks Optimization (RFO). They reported a TAC of $ 6.8 million with 19 exchangers. In a subsequent paper37, they obtained a lower TAC of $ 6.71 million with 18 exchangers using a superstructure with substages, splits, and cross flows. With our superstructure, we obtain a slightly lower TAC of


Page 26 of 50

Page 27 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


$ 6.69 million with 19 exchangers (Figure 7 and Tables 2-3). Our HEN uses six coolers and one flue gas utility exchanger. In contrast, Pavão et al.37 used five coolers and two flue gas utility exchangers. The HEN in Pavão et al.37 have no splits for the cold streams, but we have bypass for cold stream C5 in exchanger E13 and E14. This bypass is mainly to maintain the minimum flow within exchangers without violating any temperature constraints. Since H1 has a large heat capacity and heat content compared to other hot streams, it has eight exchangers with several branches of stream splits. We have an extra match of stream H1 with C9 which is not present in Pavão et al.37 Although the other seven matches (C1, C2, C4, C5, C6, C7, CU) for H1 is same as Pavão et al.37, the topology of these matches is different in our work. The HEN in Pavão et al.37 has two major sub-streams for H1, which further splits into two and three substreams respectively. Our HEN has two major substreams for H1 with only one splitting further into three substreams. Although our HEN uses an extra cooler for stream H4, the total cold utility consumption is lower (412.25 kW vs 413.07 kW). Further, HEN in Pavão et al.37 has a flue gas heater for cold stream C5, which is not present in our HEN. However, our utility costs are almost the same. Furthermore, our total heat exchanger area is also 1.1 % lower (30667 m2 vs 31012 m2), and our TAC is 0.2 % lower. Thus, we obtain a HEN with both lower area and utility costs for one extra exchanger compared to Pavão et al.37 While the superstructure in Pavão et al.37 allows novel configurations such as cross-flows and exchangers in series in stages, it requires many pre-defined parameters such as maximum stages, sub-stages, and stream splits. In contrast, our superstructure has an advantage of only one predefined parameter, which is 𝐸, and we set it as 19 in this case study. 7.3

Case Study 3

We use crude preheating case study presented in Kim and Bagajewicz41 and Pavão et al.37. It has 11 hot and 2 cold process streams with one hot and cold utility. We assume a maximum of 18



exchangers and assign hot streams for the first 12 exchangers. Our best HEN shown in Figure 8 has slightly higher utility consumption, but lower total annualized costs (TAC), compared with Pavão et al.37 Since Kim and Bagajewicz41 and Kim et al.42 had used an MTA of 10 K, the final network had lower area with higher utility and total annualized costs. The network in Pavão et al.37 uses five cold utility exchangers and one hot utility exchanger. In contrast, our network uses only three coolers and one heater. Further, C1 in our network uses 9 exchangers compared to 10 in Pavão et al.37 Further, crude stream C2 uses six process exchangers in our network compared to three in Pavão et al.37. C2 stream has two major substreams in Pavão et al.37 whereas it has three major substreams in our network. This may be due to additional prefixed parameters in Pavão et al.37 Our network mostly has parallel configurations for cold streams with some cross-flows. Stream H2 and H4 split into two substreams, whereas there is no split for these streams in Pavão et al.37 Exchangers 1, 5, 6 and 7 are in parallel for stream C1; exchangers 6 and 17 are in series for hot stream H6. Further, it also has novel configurations, such as multiple exchangers in substreams for the process stream C2, cross-flows for stream C1 such as from exchanger 1 to exchanger 10. The network is simple with lower area compared to Pavão et al.37 (13432 m2 vs 13835 m2). As a result, our network is lower in capital costs, leading to 0.6 % lower TAC. Since we terminate on iteration/solution time limit, our algorithm may not converge to a globally optimal solution. In this case, another HEN configuration consisting only of 17 units can be easily derived from current configuration by combining exchanger 15 and 16 in to one big exchanger. It doesn’t affect the utility consumption or total area. However, the fixed costs of exchanger are reduced by eliminating one exchanger, thus lowering TAC to 3,344,318 (1.4% lower compared to literature best). 8. Conclusion


Page 28 of 50

Page 29 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


We developed a novel but conceptually simple exchanger-centric superstructure with only one prefixed parameter (maximum number of exchangers). In contrast to existing HEN superstructures, it allows all possible 2-stream exchange configurations such as bypasses, cross-flows, exchangers in sub-streams, etc. We have also developed an improved MINLP model and an effective outerapproximation algorithm to solve this difficult non-convex problem and obtained better solutions on three literature case studies. Our model and algorithm produce lower TACs for all the case studies. For the first case study with tight composite curves, our superstructure and model give 67.8 % lower hot utility consumption compared to global solutions for stagewise superstructure in Faria et al.43 and for match based superstructure in Kim and Bagajewicz41 with novel HEN features such as exchanger bypasses and utility heaters/coolers in parallel. For the second case study with multiple hot utilities and sixteen process streams and the third case study with thirteen process streams, we obtain a slightly lower TAC compared to the best available solution. The results for these case studies affirm the versatility and usefulness of our proposed superstructure and model. Acknowledgement Sajitha K Nair acknowledges the financial support under the President’s Graduate Fellowship from the National University of Singapore. The work was also funded in part by the National University of Singapore through a seed grant (R-261-508-001-646/733) for CENGas (Center of Excellence for Natural Gas). Supporting Information The constraints for piecewise MILP relaxations and outer-approximations are listed in Supporting Information. Table 1: Stream data for case studies Stream

𝑻𝑰𝑵 𝑲

H1

500.15

𝑻𝑶𝑼𝑻 𝒉 𝑲 𝒌𝑾/(𝒎𝟐 − 𝑲) Case Study 1 339.15 0.06


𝑭 𝒌𝑾/𝑲

𝑸 𝒌𝑾

4.431

713.39


Page 30 of 50

H2 472.15 339.15 0.06 5.319 707.43 H3 522.15 411.15 0.06 3.162 350.98 H4 433.15 366.15 0.06 2.634 176.48 C1 355.15 450.15 0.06 5.184 492.48 C2 366.15 478.15 0.06 4.17 467.04 C3 311.15 494.15 0.06 2.532 463.36 C4 333.15 433.15 0.06 2.286 228.60 C5 389.15 495.15 0.06 1.824 193.34 CU 311.15 355.15 0.06 HU 544.15 422.15 0.06 𝑀𝑇𝐴 = 10 𝐾; 𝐶𝑜𝑠𝑡 ($/𝑎) = 5291.9 + 77.79 𝐴; 𝐶iL = 53349 $/(𝑘𝑊 − 𝑎); 𝐶gL = 566167 $/(𝑘𝑊 − 𝑎) Case Study 2 H1 789.15 316.15 0.546 1198.96 567108.08 H2 922.15 316.15 1 116.00 70296.00 H3 490.15 316.15 1 186.22 32401.58 H4 658.15 432.15 1.238 131.51 29721.26 H5 405.15 355.15 0.771 378.52 18926.00 H6 364.15 333.15 0.859 589.55 18275.90 C1 372.15 744.15 1.129 191.05 71070.60 C2 351.15 691.75 1 160.43 54642.46 C3 436.15 922.15 1.81 95.98 46646.28 C4 303.15 658.15 1.85 119.10 42280.50 C5 710.15 794.15 0.815 377.91 31744.44 C6 529.15 539.15 2.085 2753.00 27530.00 C7 490.15 507.15 0.443 1297.70 22060.90 C8 322.15 422.15 1 197.39 19739.00 C9 332.15 436.55 1.063 123.16 12857.49 C10 492.15 494.45 1.377 1997.50 4594.25 HU1 (Fuel Gas) 2073.15 1073.2 1.2 HU2 (Steam) 509.15 508.65 1 CU (Water) 311.15 355.15 1 𝑀𝑇𝐴 = 1 𝐾; 𝐶𝑜𝑠𝑡 ($/𝑎) = 26600 + 4147.5 𝐴 0.6; 𝐶iL = 2.1 $/(𝑘𝑊 − 𝑎); 𝐶gLF = 35 $/(𝑘𝑊 − 𝑎); 𝐶gL^ = 27 $/(𝑘𝑊 − 𝑎)

Stream

𝑻𝑰𝑵 𝑲

H1 H2

523.75 413.35

𝑻𝑶𝑼𝑻 𝒉 𝑲 𝒌𝑾/(𝒎𝟐 − 𝑲) Case Study 3 363.15 0.26 312.65 0.26


𝑭 𝒌𝑾/𝑲

𝑸 𝒌𝑾

132.20 106.50

21231.32 10724.83

Page 31 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


H3 H4 H5 H6 H7 H8 H9 H10 H11 C1 C2 HU CU

576.75 543.35 0.41 234.98 7848.35 563.15 388.15 0.47 39.81 6966.46 483.15 436.15 0.33 115.76 5440.90 521.95 383.15 0.72 31.81 4414.61 550.15 395.05 0.57 24.58 3812.88 443.25 333.15 0.45 33.93 3735.14 451.75 382.05 0.6 47.85 3334.95 633.15 563.15 0.47 39.81 2786.58 632.75 553.15 0.47 24.53 1952.85 403.15 623.15 0.72 289.92 63781.67 303.15 403.15 0.26 202.48 20247.50 773.15 772.15 0.53 293.15 313.15 0.53 $ 𝑀𝑇𝐴 = 1 𝐾; 𝐶𝑜𝑠𝑡 ¹ º = 25,000 + 55 𝐴; 𝐶iL = 10 $/(𝑘𝑊 − 𝑎); 𝐶gL = 100 $/(𝑘𝑊 − 𝑎) 𝑎



Page 32 of 50

Table 2: Results for case studies 𝑬

𝑸 𝒌𝑾

𝑨𝒓𝒆𝒂 𝒎𝟐

1 2 3 4 5 6 7 8 9 10 11 12 13

492.5 467.0 93.2 11.1 193.3 146.1 68.8 79.8 45.0 176.5 153.2 21.7 42.7

1641.6 1022.9 163.2 11.1 423.5 263.0 229.4 170.0 42.9 429.6 510.6 71.6 28.4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

71070.6 54642.5 37957.6 42280.5 24338.4 27530.0 22060.9 19739.0 12857.5 4594.3 5388.0 8688.7 189.0 7217.1 312330.0 24932.3 32401.6 18926.0 18275.9

1893.0 1174.5 1095.8 612.6 2547.8 476.1 385.3 236.3 153.4 48.0 52.7 20.7 3.3 155.1 15485.4 1038.7 1642.9 926.2 2719.4

1 2

12495.9 7564.5

3280.7 879.6

𝑭𝑯 𝑭𝑪 𝒌𝑾/𝑲 𝒌𝑾/𝑲 Case Study 1 5.18 5.18 3.77 4.17 4.43 2.53 1.11 0.14 2.06 1.82 5.32 3.32 1.41 1.41 1.02 1.13 2.65 1.41 2.63 2.35 2.14 2.14 0.77 0.76 0.35 1.53 Case Study 2 318.90 191.05 291.15 160.43 116.00 95.98 197.20 119.10 880.06 377.91 391.71 2753.00 391.71 1297.70 103.31 197.39 391.71 123.16 131.51 1997.50 28.20 122.46 8.69 95.98 37.79 37.79 116.00 368.27 1198.96 7098.36 116.00 566.64 186.22 736.40 378.52 430.14 415.36 589.55 Case Study 3 118.17 132.20 378.23 75.12

𝑻𝑰 𝑲

𝑻𝑶 𝑲

𝒕𝒊 𝑲

𝒕𝒐 𝑲

460.15 500.15 360.18 421.15 505.15 366.63 472.15 513.69 522.15 433.15 414.58 443.15 544.15

365.15 376.15 339.15 411.15 411.15 339.15 423.26 435.48 505.15 366.15 343.15 414.78 422.15

355.15 366.15 311.15 333.15 389.15 311.15 413.26 423.15 462.15 347.95 333.15 404.58 405.19

450.15 478.15 347.95 411.15 495.15 355.15 462.15 494.15 494.15 423.15 404.58 433.15 433.15

789.15 566.29 761.50 573.82 858.31 531.08 761.50 547.09 789.15 761.50 705.18 634.89 761.50 705.18 623.22 432.15 634.89 602.07 658.15 623.22 623.22 432.15 2073.15 1073.15 922.15 917.15 920.52 858.31 576.65 316.15 531.08 316.15 490.15 316.15 405.15 355.15 364.15 333.15

372.15 351.15 436.15 303.15 710.15 529.15 490.15 322.15 332.15 492.15 311.15 831.62 789.15 774.55 311.15 311.15 311.15 311.15 311.15

744.15 691.75 831.62 658.15 774.55 539.15 507.15 422.15 436.55 494.45 355.15 922.15 794.15 794.15 355.15 355.15 355.15 355.15 355.15

523.75 413.35

403.15 293.15

508.90 313.15


429.23 312.65

Page 33 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

7848.4 5684.7 5440.9 3364.7 3313.7 3735.1 3334.9 2786.6 1952.8 20894.0 8735.4 3160.4 717.0 564.6 1049.9 499.1

742.1 1101.2 1495.6 725.6 678.4 651.2 425.3 177.2 125.0 373.5 1650.6 805.8 149.4 117.7 35.7 17.8

234.98 39.81 115.76 31.81 24.58 33.93 47.85 39.81 24.53 20894.01 132.20 31.38 22.27 17.54 31.81 24.58

155.35 40.36 73.78 31.73 25.89 37.35 56.18 59.70 34.51 289.92 87.35 77.77 12.08 9.51 52.50 24.96

576.75 563.15 483.15 521.95 550.15 443.25 451.75 633.15 632.75 773.15 429.23 413.35 420.35 420.35 416.16 415.36


543.35 420.35 436.15 416.16 415.36 333.15 382.05 563.15 553.15 772.15 363.15 312.65 388.15 388.15 383.15 395.05

493.70 403.15 403.15 403.15 403.15 303.15 343.79 518.71 508.90 551.08 303.15 303.15 343.79 343.79 293.15 293.15

544.22 544.00 476.90 509.20 531.15 403.15 403.15 565.39 565.49 623.15 403.15 343.79 403.15 403.15 313.15 313.15


References Faria et al.43 Kim and Bagajewicz4 1

This work

Table 3: Comparison for case studies Type of Hot utility TAC ($/a) Units superstructure (kW) Case Study 1 Stage-wise 99,606,280 11 151.39 Extension of Floudas et al.15 99,636,825 12 151.39 Exchangercentric

32,419,290

13

42.681

Page 34 of 50

Cold utility (kW)

Area (m2)

254.85

3071.5

254.85

3323

146.14

5008

Case Study 2 Khorasany and Fesanghary5

-

Pavão et al.33 Pavão et al.

Enhanced stagewise Stages, substages, sub-splits Exchangercentric

7,435,740

18

66.07 (MW) 469.62 (MW)

-

6,801,261

19

34.21 (MW) 437.78 (MW)

30897

6,712,551

18

9.50 (MW)

413.07 (MW)

31012

6,695,584

19

8.69 (MW)

412.25 (MW)

30667

Case Study 3 3,463,096 17*

23,566

11,783

9,794

3,586,052

17*

23,566

11,783

10,131

3,391,066

18

20,891

9,108

13,835

3,369,318

18

20,894

9,114

13432

3,344,318

17

20,894

9,114

13432

4

37

This work Kim and Bagajewicz4

Extension of Floudas et al.15

Kim et al.42

Stages and substages Stages, substages, sub-splits Exchangercentric Exchangercentric

1

Pavão et al. 37

This work This work

*𝑀𝑇𝐴 = 10 𝐾


Page 35 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 4: Model statistics for case studies Constraints Continuous Variables Case Study 1 P-MILPT 5325 2071 st P-MILPO (1 Iteration) 5844 2071 P-NLP/P-NLPF 582 403 Case Study 2 P-MILPT 14571 4673 st P-MILPO (1 Iteration) 15994 4673 P-NLP/P-NLPF 933 740 Case Study 3 P-MILPT 8759 2866 P-MILPO (1st Iteration) 9931 2866 P-NLP/P-NLPF 755 526 Model


Binary Variables 107 206 0 240 439 0 123 291 0


Figure 1. Stageless HEN superstructure


Page 36 of 50

Page 37 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2. Various Possible Configurations



Page 38 of 50

Start. Initialize algorithmic parameters.

Solve P-MILPT. Update bounds. Solve P-MILPO. Generate pool of binary solutions.

Yes

Non-empty solution pool?

No

Set n=1

Set n = min(2n, 8)

Binary solution unique? No

No Delete solution from pool Yes

Yes

Yes Solve P-NLPF

Upper bound improved?

No

Solution in pool?

No

INF ≤ tolerance value? Yes

Solve P-NLP. Update upper bound.

Add integer cuts and linearizations to P-MILPO

Obtain optimality gap, Iteration = Iteration +1

Check termination criteria Satisfied

Stop

Figure 3. Algorithm


Not Satisfied

Page 39 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Fig (4a) Case Study 1

Fig (4b) Case Study 2



Fig (4c) Case Study 3

Figure 4. Composite curves for case studies

Figures 5(a), 5(b) Case Study 1


Page 40 of 50

Page 41 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figures 5(c), 5(d) Case Study 2

Figures 5(e), 5(f) Case Study 3 Figure 5. Solution Pool Sizes, TAC and computational times Vs iteration for case studies



Figure 6: Best HEN for case study 1



Page 42 of 50

Page 43 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60





Nomenclature Subscripts 𝑒

Exchanger

𝑖

Hot stream

𝑗

Cold stream

𝑘

P-NLP solution

𝑠

Stream

Superscripts 𝐿

Lower limit

𝑈

Upper Limit

Parameters 𝐶𝑃

Number of cold process streams

𝐶𝑈

Number of cold utilities

𝐸

Number of exchangers

𝐹

Heat-content flow (mass flow x heat capacity or kW/K) of hot stream

𝐹𝐶

Fixed component of the cost of a heat exchanger ($)

𝐺

Heat-content flow (mass flow x heat capacity or kW/K) of cold stream

𝐻𝑃

Number of hot process streams

𝐻𝑈

Number of hot utilities

𝐼

Number of hot streams

𝐽

Number of cold streams

𝑀𝑇𝐴

Minimum allowable temperature approach in an exchanger (K)

𝑆

Number of streams


Page 44 of 50

Page 45 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


𝑇𝐼𝑁

Initial temperature of stream (K)

𝑇𝑂𝑈𝑇 Final (target) temperature of stream (K) 𝑈𝐶

Unit utility cost ($/kW-a)

𝑉𝐶

Coefficient in the variable component of the cost of a heat exchanger

𝛼

Exponent in the cost correlation for an exchanger

𝛾

Capital annualization factor

𝜆

Lagrange Multiplier

𝜏

Multiplier of the order of TAC

Continuous Variables 𝐴

Area of an exchanger (m2)

𝐶𝐻𝐸

Cost of heat exchanger ($)

𝐶𝑇𝐴

Temperature approach at the cold end of an exchanger (K)

𝐷

Temperature change for hot substream in an exchanger (K)

𝑓

Heat-content flow of hot substream (kW/K)

𝑔

Heat-content flow of cold substream (kW/K)

𝐻𝑇𝐴

Temperature approach at the hot end of an exchanger (K)

𝐿𝑀𝑇𝐷 Logarithmic Mean Temperature Difference (K) 𝑄

Heat duty (kW)

𝑅

Temperature change for cold substream in an exchanger (K)

𝑇𝐼

Entry Temperature of the hot stream in an exchanger (K)

𝑡𝑖

Entry Temperature of the cold stream in an exchanger (K)

𝑇𝑂

Exit Temperature of the hot stream in an exchanger (K)

𝑡𝑜

Exit Temperature of the cold stream in an exchanger (K)



𝑈

Overall Heat Transfer Coefficient (kW/m2-K)

Binary variables 𝑥

Existence of a stream in an exchanger

𝑧

Existence of an exchanger

Acronyms HEN

Heat Exchanger Network

HENS Heat Exchanger Network Synthesis HTC

Heat Transfer Coefficient

LP

Linear Programming

MILP

Mixed-Integer Linear Programming

MINLP Mixed-Integer Non-Linear Programming NLP

Non-Linear Programming

SS

Staged Superstructure

TAC

Total Annualized Costs


Page 46 of 50

Page 47 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

19.

Huang KF, Karimi IA. Simultaneous synthesis approaches for cost-effective heat exchanger networks. Chemical Engineering Science. 2013;98:231-245. Klemeš JJ, Kravanja Z. Forty years of Heat Integration: Pinch Analysis (PA) and Mathematical Programming (MP). Current Opinion in Chemical Engineering. 11// 2013;2(4):461-474. Chen Z, Wang J. Heat, mass, and work exchange networks. Front. Chem. Sci. Eng. 2012;6(4):484502. Morar M, Agachi PS. Review: Important contributions in development and improvement of the heat integration techniques. Computers and Chemical Engineering. 2010;34(8):1171-1179. Furman KC, Sahinidis NV. A Critical Review and Annotated Bibliography for Heat Exchanger Network Synthesis in the 20th Century. Industrial & Engineering Chemistry Research. 2002/05/01 2002;41(10):2335-2370. Jezowski J. Heat exchanger network grassroot and retrofit design. The review of the state-of-the art: part I. Heat exchanger network targeting and insight based methods of synthesis. Hungarian Journal of Industrial Chemistry. 1994;22(4):279-294. Jezowski J. Heat exchanger network grassroot and retrofit design. The review of the state-of-the art: part II. Heat exchanger network synthesis by mathematical methods and approaches for retrofit design. Hungarian Journal of Industrial Chemistry. 1994;22(4):295-308. Gundersen T, Naess L. The synthesis of cost optimal heat exchanger networks. An industrial review of the state of the art. Computers and Chemical Engineering. 1988;12(6):503-530. Furman KC, Sahinidis NV. Computational complexity of heat exchanger network synthesis. Computers and Chemical Engineering. 2001;25(9-10):1371-1390. Masso AH, Rudd DF. The synthesis of system designs. II. Heuristic structuring. AIChE Journal. 1969;15(1):10-17. Hohmann ECJ. Optimum networks for heat exchange, University of Southern California; 1971. Linnhoff B, Flower JR. Synthesis of heat exchanger networks: I. Systematic generation of energy optimal networks. AIChE Journal. 1978;24(4):633-642. Cerda J. Transportation models for the optimal synthesis of heat exchanger networks: Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh, PA.; 1980. Papoulias SA, Grossmann IE. A structural optimization approach in process synthesis—II: Heat recovery networks. Computers & Chemical Engineering. 1983;7(6):707-721. Floudas CA, Ciric AR, Grossmann IE. Automatic synthesis of optimum heat exchanger network configurations. AIChE Journal. 1986;32(2):276-290. Gundersen T, Traedal P, Hashemi-Ahmady A. Improved sequential strategy for the synthesis of near-optimal heat exchanger networks. Computers and Chemical Engineering. 1997;21(SUPPL.1):S59-S64. Pettersson F. Synthesis of large-scale heat exchanger networks using a sequential match reduction approach. Computers and Chemical Engineering. 2005;29(5):993-1007. Anantharaman R, Nastad I, Nygreen B, Gundersen T. The sequential framework for heat exchanger network synthesis - Minimum number of units subproblem. Paper presented at: CHISA 2008 - 18th International Congress of Chemical and Process Engineering2008; Prague; Czech Republic. Anantharaman R, Nastad I, Nygreen B, Gundersen T. The sequential framework for heat exchanger network synthesis-The minimum number of units sub-problem. Computers and Chemical Engineering. 2010;34(11):1822-1830.



20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

Chen Y, Grossmann IE, Miller DC. Computational strategies for large-scale MILP transshipment models for heat exchanger network synthesis. Computers and Chemical Engineering. 2015;82:6883. Letsios D, Kouyialis G, Misener R. Heuristics with performance guarantees for the minimum number of matches problem in heat recovery network design. Computers and Chemical Engineering. 2018;113:57-85. Ptáčník R, Klemeš J. An application of mathematical optimization methods in heat-exchange network synthesis. Computers and Chemical Engineering. 1988;12(2-3):231-235. François M, Irsia B. Synep1 : A methodology for energy integration and optimal heat exchanger network synthesis. Computers and Chemical Engineering. 1989;13(4-5):603-610. Floudas CA, Ciric AR. Strategies for overcoming uncertainties in heat exchanger network synthesis. Computers and Chemical Engineering. 1989;13(10):1133-1152. Ciric AR, Floudas CA. Heat exchanger network synthesis without decomposition. Computers and Chemical Engineering. 1991;15(6):385-396. Kemp IC. 4 - Heat exchanger network design. In: Kemp IC, ed. Pinch Analysis and Process Integration (Second Edition). Oxford: Butterworth-Heinemann; 2007:99-159. Yee TF, Grossmann IE. Simultaneous optimization models for heat integration--II. Heat exchanger network synthesis. Computers and Chemical Engineering. 1990;14(10):1165-1184. Björk K-M, Westerlund T. Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption. Computers & Chemical Engineering. 2002;26(11):1581-1593. Huang KF, Al-mutairi EM, Karimi IA. Heat exchanger network synthesis using a stagewise superstructure with non-isothermal mixing. Chemical Engineering Science. 2012;73:30-43. Mistry M, Misener R. Optimising heat exchanger network synthesis using convexity properties of the logarithmic mean temperature difference. Computers & Chemical Engineering. 2016;94:1-17. Hasan M, Jayaraman G, Karimi I, Alfadala H. Synthesis of heat exchanger networks with nonisothermal phase changes. AIChE journal. 2010;56(4):930-945. Na J, Jung J, Park C, Han C. Simultaneous synthesis of a heat exchanger network with multiple utilities using utility substages. Computers and Chemical Engineering. 2015;79:70-79. Pavão LV, Costa CBB, Ravagnani MASS. An Enhanced Stage-wise Superstructure for Heat Exchanger Networks Synthesis with New Options for Heaters and Coolers Placement. Industrial & Engineering Chemistry Research. 2018/02/21 2018;57(7):2560-2573. Beck A, Hofmann R. A Novel Approach for Linearization of a MINLP Stage-Wise Superstructure Formulation. Computers and Chemical Engineering. 2018;112:17-26. Beck A, Hofmann R. How to tighten a commonly used MINLP superstructure formulation for simultaneous heat exchanger network synthesis. Computers and Chemical Engineering. 2018;112:48-56. Jongsuwat P, Suriyapraphadilok U, Bagajewicz M. New Heat Exchanger Network Design Model. Paper presented at: 17th Conference on Process Integration, Modelling and Optimisation for Energy Saving and Pollution Reduction; August 2014, 2014. Pavão LV, Costa CBB, Ravagnani MASS. A new stage-wise superstructure for heat exchanger network synthesis considering substages, sub-splits and cross flows. Applied Thermal Engineering. 2018/10/01/ 2018;143:719-735. Toimil D, Gómez A. Review of metaheuristics applied to heat exchanger network design. International Transactions in Operational Research. 2017/01/01 2016;24(1-2):7-26. Furman KC, Sahinidis NV. Approximation algorithms for the minimum number of matches problem in heat exchanger network synthesis. Industrial and Engineering Chemistry Research. 2004;43(14):3554-3565.


Page 48 of 50

Page 49 of 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54.

Wu H, Yan F, Li W, Zhang J. Simultaneous Heat Exchanger Network Synthesis Involving Nonisothermal Mixing Streams with Temperature-Dependent Heat Capacity. Industrial and Engineering Chemistry Research. 2015;54(36):8979-8987. Kim SY, Bagajewicz M. Global optimization of heat exchanger networks using a new generalized superstructure. Chemical Engineering Science. 2016;147:30-46. Kim SY, Jongsuwat P, Suriyapraphadilok U, Bagajewicz M. Global Optimization of Heat Exchanger Networks. Part 1: Stages/Substages Superstructure. Ind. Eng. Chem. Res. 2017;56(20):5944. Faria DC, Kim SY, Bagajewicz MJ. Global Optimization of the Stage-wise Superstructure Model for Heat Exchanger Networks. Industrial & Engineering Chemistry Research. 2015/02/11 2015;54(5):1595-1604. Yee TF, Grossmann IE. A Screening and Optimization Approach for the Retrofit of Heat-Exchanger Networks. Industrial and Engineering Chemistry Research. 1991;30(1):146-162. Castillo PAC, Castro PM, Mahalec V. Global Optimization of Nonlinear Blend-Scheduling Problems. Engineering. 2017;3(2):188-201. Viswanathan J, Grossmann IE. A combined Penalty Function and Outer Approximation Method for MINLP Optimization. Comput. Chem. Eng. 1990;14:769. Bergamini ML, Aguirre P, Grossmann I. Logic-based outer approximation for globally optimal synthesis of process networks. Computers & Chemical Engineering. 2005/08/15/ 2005;29(9):1914-1933. Wicaksono DS, Karimi IA. Piecewise MILP under- and overestimators for global optimization of bilinear programs. AIChE Journal. 2008;54(4):991-1008. Karuppiah R, Grossmann IE. Global optimization for the synthesis of integrated water systems in chemical processes. Computers & Chemical Engineering. 2006/02/15/ 2006;30(4):650-673. Duran MA, Grossmann IE. An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathematical Programming. 1986;36(3):307-339. Wächter A, Biegler LT. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming. 2006// 2006;106(1):25-57. Ugray Z, Lasdon L, Plummer J, Glover F, Kelly J, Martí R. Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization. INFORMS Journal on Computing. 2007;19(3):328340. GAMS Development Corporation, General Algebraic Modeling System (GAMS) Release 25.1.2. Washington, DC, USA, 2018. Khorasany RM, Fesanghary M. A novel approach for synthesis of cost-optimal heat exchanger networks. Computers and Chemical Engineering. 2009;33(8):1363-1370.



Graphical abstract


Page 50 of 50

Unified Heat Exchanger Network Synthesis via a ... - ACS Publications

Recommend Documents