Process Integration using Block Superstructure

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Process Integration using Block Superstructure Jianping Li, Salih Emre Demirel, and M M Faruque Hasan Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b05180 • Publication Date (Web): 02 Mar 2018 Downloaded from http://pubs.acs.org on March 4, 2018

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Industrial & Engineering Chemistry Research

Process Integration using Block Superstructure Jianping Li, Salih Emre Demirel, and M. M. Faruque Hasan∗ Artie McFerrin Department of Chemical Engineering, Texas A&M University College Station, TX 77843-3122, USA

Abstract We propose a general mathematical model for various process integration problems, involving mass integration, heat integration, simultaneous mass and heat integration, and property integration.

The process units, including regenerators/interceptors,

mixers and splitters are represented using blocks.

The blocks are arranged in a

two-dimensional grid and the arrangement of these blocks gives rise to various process integration networks. The existence of connecting streams between adjacent blocks and jump flows among all blocks enables the necessary interaction between different blocks via material and energy flows. The size of block superstructure is determined by the number of layers with mixing operations, process units, product streams and heat integration stages. The general process integration model is formulated as a mixed-integer nonlinear optimization (MINLP) problem with the minimization of total annual cost as objective. We demonstrate our approach using four case studies from process integration literature.

∗

Correspondence concerning this article should be addressed to M.M. Faruque Hasan at [email protected], Tel.: 979-862-1449.

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Introduction

The conservation and sustainable use of energy, water and the environment is a grand challenge of the current time. 1 Specifically, among all the production sectors related with these challenges, the resource utilization in chemical industry towards energy conservation and waste reduction is receiving significant attention. While finding more economic and sustainable ways to produce desired products has been a major focus, 2–5 most efforts have been devoted to equipment design. 6–8 However, due to various interactions existing within a process plant involving many production routes, a system level targeting would be favored so as to account for trade-offs from a holistic perspective. This type of holistic approach is also referred as process integration which improves processes to reduce consumption of fresh resources or harmful emissions 9 and emphasizes the unity of the process. 10 The International Energy Agency adopted the definition of process integration as systematic methods for designing integrated production systems ranging from individual processes to total sites, with special focus on the efficient use of energy and reducing environmental effects. 11 Process integration can be mainly achieved through mass integration and energy integration. 12 Mass integration takes into account the production, separation,and routing of species and streams throughout the process. Energy integration, on the other hand, exploits energy through global allocation, generation, and exchange within the process. Property integration is also proposed to allocate and manipulate the streams and processing units by tracking, adjusting, assigning, and matching of their functionalities throughout the system. 13–15 The techniques for process integration in chemical industry can be broadly classified into pinch-based methods and optimization-based methods. 11 Pinch-based methods originate from efforts on achieving optimal design of heat exchanger networks. 16–18 Later on, pinch analysis in heat integration has been widely applied in chemical industry. Inspired by successful applications of pinch analysis on heat exchanger networks synthesis, researchers have extended the pinch-based methods to many other areas including mass exchange networks, 19 industrial water networks, 20,21 hydrogen pinch analysis, 22,23 oxygen pinch 2


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analysis, 24 and etc. Though it is easier to understand and implement, several limitations exist for pinch analysis. It is not always applicable for simple networks or cases where there are severe operating constraints. 25 Furthermore, it is difficult to apply the pinch-based methods for problems involving multiple components. 20 To overcome these limitations, optimization-based methods have gradually become the mainstream in process integration. Optimization-based methods often rely on the postulation of a problem-specific superstructure and the formulation of a corresponding mathematical programming problem. 3 These methods have been widely utilized in the area of process integration, especially for the design of heat exchanger networks (HENS) (an extensive review can be found in Furman and Sahinidis 26 ). In 1961, Westbrook first used optimization-based methods in heat integration problems. 27 Floudas et al. 28 constructed the superstructure involving the unit matches as predicted by the MILP transshipment model and developed a nonlinear programming formulation for automatically generating optimal heat exchanger network. Yee and Grossman 29,30 proposed the simultaneous HENS formulations, which considered utility cost, exchanger areas and selection of matches. In addition to the applications in heat integration, optimization-based methods are also used in other sub-systems of chemical plants including water networks, 31–35 simultaneous water and heat integration, 36–38 utility networks, 39 work exchange networks 40 and fuel gas networks. 41 Ahmetović and Grossmann 31 applied the optimization-based methods on water network design involving multiple water sources, water-using processes, wastewater treatment, and pre-treatment operations. Ahmetović and Kravanja 36 further extended the water network for both direct and indirect heat exchanges while allowing automatic identification of hot and cold streams. Elia et al. 39 developed an approach to generate heat exchange and power recovery networks (HEPN) to achieve simultaneous heat and work integration. Razib et al. 40 introduced a work exchanger network that accounts for the work exchange between process streams requiring compression and process streams requiring expansion. Nápoles-Rivera et al. 42 presented a mathematical programming model to achieve the optimal design of mass and property

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integration networks considering property interceptors for composition, toxicity, theoretical oxygen demand, pH, density and viscosity. It should be noted that the superstructures from which the mathematical formulations are derived directly influence the quality of solution for process integration problems. Several superstructure representation methods have been proposed for process synthesis and these methods are also widely applied in process integration.

State-Task-Network

representation and State-Equipment-Network representation are two most commonly used representation methods. These representation methods involve physicochemical properties of streams as states, processing steps as tasks and unit operations as equipment. 43 Bagajewic and Manousiouthakis 44,45 proposed a state-space conceptual framework for process synthesis, which involves the interaction of mass and heat exchange networks. Papalexandri and Pistikopoulos considered heat exchange network and mass exchange network as building blocks for representing process equipment and flowsheets. 46,47 A superstructure-free framework was developed via a two-level decomposition, in which the upper-level evolutionary algorithm generates structural alternatives and each alternative generated by the upper level is then optimized deterministically in the lower level. 48 Other superstructure representation methods include P-graph, 49,50 phenomena building blocks, 51–53 process-group contribution method, 54 graph theory-based network representation, 55 and unit-port-conditioning-stream (UPCS) approach. 56 A recurring feature in process integration superstructures is that they consist of several sinks with demands for certain flows and properties, and several sources to fulfill demands of sinks. For many cases, these superstructures also involve intermediate pools to achieve mixing, generation or separation.

The difference between the superstructure of mass

integration and energy integration arises from the type of properties exchanged within the system. These properties can be materials/contaminants as in the case of water networks, mass exchange networks, etc. The properties exchanged as energy can be in the form of heat as in heat exchanger networks, in the form of power as in work exchange networks, or in both

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heat and power form as in the utility networks. Though process integration of each subsystem has attracted significant attention, these general features are not addressed and represented in a systematic way. Each process integration problem requires a new superstructure to be postulated whenever a new problem is encountered. The existence of these common features in different process integration problems indicate that a unified superstructure representation for these different classes of process integration problems will be highly advantageous. Furthermore, there is a tight relation between process integration and process synthesis, both of which utilize available raw materials, identify unit operations, and interconnect different units within a single process flowsheet. 3,56 Process integration can be naturally incorporated into the framework of process synthesis through material recycle so as to reduce resource consumption. However, process integration typically improves a process by further minimizing the resource and energy consumptions after the optimal process flowsheet is identified using process synthesis methods. There are at least two reasons for this. In the case of mass integration, typical production processes are identified separately and assembled in an industrial park, which normally consists of multiple plants. A comprehensive evaluation of trade-offs among all production processes in an industrial park is challenging because of the complexity and the size of synthesis problems. As a result, holistic consideration on economic allocation and effective utilization of resources are lacking before all the process flowsheets are synthesized, which brings the opportunities for process integration. In the case of energy integration, an a priori characterization of process streams (i.e., cold or hot stream) is necessary for optimal design of heat exchanger networks. However, such characteristic is obtained after implementing process synthesis. To this end, methods also exist that investigates simultaneous process synthesis and integration. 57–59 This aims at depicting interactions among different sub-systems within a chemical industrial plant/park and provides an overall picture for plant design. In addition to the relation between process integration and process synthesis, process integration can be considered as a special case of process intensification. 60 It is also shown that process intensification is a limit case

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of tight integration via significant material recycling. 61 To achieve the grand unification of process synthesis, intensification and integration activities, the advancement of process integration techniques is also necessary in addition to that of process synthesis and process intensification. Recently, a systematic framework for process intensification based on building block superstructure has been proposed based on the idea of dissecting various unit operations into fundamental building blocks. 2 Later, this block superstructure is applied to address various process synthesis problems within a single framework, which does not require postulation of different superstructures for different synthesis problems. 62 The proposed approach can be used for automatic flowsheet generation and optimization. This block-based approach is also applied to investigate the synthesis of fuel gas network, 63 a special case of pooling problem. In this work, we extend the block-based approach for different integration problems towards a general framework for process synthesis, intensification and integration. We show that the framework reduces to an integration problem when each block allows the existence of feed inlet flow and product outlet flow, which serve as sources and sinks, respectively. Each stream connecting adjacent blocks are equipped with heaters/coolers so as to adjust the temperature for achieving the heating/cooling requirement. On top of these connecting streams between adjacent blocks, jump flows are introduced for the first time. The involvement of jump flows avoids the necessity of utilizing more intermediate blocks to connect blocks that are not adjacent while maintaining or increasing the number of process alternatives. These connecting streams and jump flows enable the interaction of different blocks via mass and energy flows. With these new features added to the block superstructure, different process integration problems can be represented using a single superstructure with I and J number of blocks in row and column respectively. We formulate the overall process integration problem using a mixed-integer nonlinear optimization (MINLP) model. The model includes mass and energy balances, property balances, product specifications, task assignments and etc. for mass integration, heat integration, simultaneous mass and heat

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integration, and property integration. The remaining of the article is structured as follows. First, we illustrate the representation of various integration networks using the block-based approach. Next, we formalize the problem statement for process integration and describe the corresponding MINLP model. Finally, we demonstrate the applicability of our approach with several case studies from literature.

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Process Integration using Block-Based Representation

Block-based representation of chemical processes were previously shown for process synthesis and intensification problems in which building blocks were used to represent different physicochemical phenomena, tasks, materials and/or equipment. 2,62 Here, we focus on the construction of block-based process integration networks which mainly relies on the superstructure connectivity. In this section, we first describe how the block-based approach can yield a generic representation method for process integration networks, and then we provide specific examples on how water and heat exchanger networks can be represented by the proposed method. The construction of block superstructure is realized via the interaction of individual blocks with each other.

Each block can serve as process sinks, process sources or

regenerators/interceptors for a process integration network. When these blocks are connected via material streams and positioned on a 2-D grid, block superstructure is obtained as illustrated in Figure 1a. Here, each block is identified via its position in the grid with its row and column indices, i.e. i and j respectively, and each block is designated as Bi,j . As shown in Figure 1b, superstructure connectivity is achieved either via interblock streams which build connections between the adjacent blocks, or jump flows which connect all the blocks in the superstructure to each other. In addition, each block allows external feeds and product streams. While the flowrate of component k in external feed stream f taken into block Bi,j is denoted as Mi,j,k,f , flowrate of component k in product stream s taken out from 7



block Bi,j is denoted as Hi,j,k,s . Note that while multiple external feed streams can be fed into the same block, only one external product stream can be taken out from the same block. b and block property yi,j,qu for quantifying the Each block also embeds block composition yi,j,k

component k and property qu specification of outlet stream from block Bi,j . [Figure 1 about here] Direct connectivity between adjacent blocks is achieved with interblock stream Fi,j,k,d , indicating the material flowrate of component k across the boundary of the block Bi,j in flow direction d. The flow direction d is equal to 1 when the stream is flowing in the horizontal direction, i.e. between Bi,j and Bi,j+1 and the flow direction d is equal to 2 when the stream is flowing in the vertical direction, i.e. between block Bi+1,j and Bi,j . Furthermore, these interblock streams are defined as bi-directional so as to increase the connectivity between adjacent blocks. This is indicated by the sign of the Fi,j,k,d which is positive when the flow is from block Bi,j to Bi,j+1 in horizontal direction (d = 1), and from block Bi,j to Bi+1,j in vertical direction (d = 2). Similarly, it is negative when the flow is from block Bi,j+1 to Bi,j in horizontal direction (d = 1), and from block Bi+1,j to Bi,j in vertical direction (d = 2). Unlike direct interblock streams between adjacent blocks, jump flows are used to build connection between all the blocks. The flowrate of a jump flow is denoted as Ji,j,i0 ,j 0 ,k and it designates the flowrate of component k from block Bi,j to block Bi0 ,j 0 . Note that this jump flow is an additional product stream taken out from block Bi,j , i.e. jump product for Bi,j , and an extra feed stream fed into the block Bi0 ,j 0 , i.e. jump feed for Bi,j . In overall, each block can have multiple jump feed streams coming from other blocks and can have multiple jump product streams that are taken out from this block. Each product stream leaving Bi,j , including the external product stream, have the same quality specifications, i.e. concentration, temperature, contamination, etc. These jump connectivities among all blocks and direct connectivities between adjacent blocks enable material and energy interactions among all blocks and the combination of these blocks in a 2-D grid gives rise to a highly connected superstructure. It should be noted that the connectivity can be solely 8


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realized via jump flows without the need for direct interblock streams between the adjacent blocks. Yet, these interblock streams can be used to represent the sharp/non-sharp splitters and separation related phenomena as it is discussed earlier (Please see the discussions on boundary assignments in Demirel et al. 2 ). Hence, direct streams are kept for the sake of generality and also utilized for heat integration network representation as it will be discussed below. With these connectivity features, stream mixers and splitters are represented via blocks with multiple inlets and multiple outlets, respectively. When multiple external feed streams enter into a block and get mixed, it serves as a source block. Here different mixing rules can be incorporated according to properties involved. Blocks with external product stream, on the other hand, serve as sinks. In addition to the representation of process sinks and sources, blocks can be also used to embed different process units, e.g. regenerators/interceptors, into the superstructure. In this case, each process unit is represented as a single block. These blocks representing process units can be used as intermediate pools which consume/generate materials or purify process streams for satisfying product requirements. With all these features, material flow within the process integration networks can be achieved via the block-based representation. These connecting streams and blocks representing process units constitute the mass and property integration. Energy flow in the superstructure is satisfied via the block and stream energy balances. Each direct interblock stream is assigned with cooler/heater so as to quantify the heat duty requirement for temperature manipulation operations. Although jump flows can be also embedded with heater/cooler units, just for the sake of simplicity, only the direct interblock streams are assigned with these units. Each outlet stream from block Bi,j has the same temperature: block temperature Ti,j . Additionally, each stream is assigned with a s stream temperature variable, Ti,j,d and this variable indicates the outlet temperature of the

heater/cooler unit on that stream. The inlet temperature for heaters/coolers, on the other hand, depends on the stream orientation and always equal to the source block temperature.

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The inlet temperature of heat heater/cooler unit on stream Fi,j,k,d is taken as Ti,j when the flow is in positive direction, as Ti,j+1 when the flow is in negative horizontal direction (d = 1) and as Ti+1,j when the flow direction is negative vertical direction (d = 2). With these variable assignments, energy flow in the block superstructure can be formulated for heat integration among all the stream heaters and coolers in the block superstructure. Utility consumption of each heater/cooler can be determined via an energy balance around each stream. If the utility requirement of a hot and cold stream at two different positions are the same, then they can be integrated to each other. If no such stream available, then the heating/cooling requirement can be satisfied via external utilities. Hence, energy integration is considered as a direct match between the heat duty of two different stream heat exchangers as it is illustrated in Figure 1c (heat exchange match is marked as red dash line). As an illustrative example, we use production of acetaldehyde via ethanol oxidation process to demonstrate the opportunities of process integration in an existing process and the construction of the corresponding block representations for the process integration superstructures.

The process flowsheet is adapted from the work of Al-Otaibi and

El-Halwagi 64 and shown in Figure 2. The ethanol feedstock is partially vaporized using flash operation before being fed to the reactor along with preheated air. In the reactor, acetaldehyde and water are generated and reaction products are sent to Scrubber 1 after being cooled down. Scrubber 1 is utilized to cool the reactor products further and scrub several species in the reactor outlet stream using dilute scrubbing solvent. The gaseous outlet flow from the top of the Scrubber 2 are scrubbed again with water to remove additional alcohol and acetaldehyde in Scrubber 2. The gaseous outlet flow of the Scrubber 2 is released into the atmosphere and the liquid product from the same unit is recycled back to utilize the remaining scrubbing agent. The liquid product from Scrubber 1 is directed to distillation column Dis 1 to recover acetaldehyde as the top product. The bottom product of Dis 1 is fed to Dis 2 to remove light organic wastes as the overhead product and the bottom product of Dis 2 is fed to Dis 3 to separate ethanol and other aqueous waste. The ethanol is obtained

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as the top product at Dis 3. [Figure 2 about here] In a typical chemical process, product/byproduct/waste streams can be further utilized to improve the process economics via process integration. In this case, these streams serve as material sources which can be recycled and reused in material sinks and they are typically comprised of the feed ports of the process units consuming the reusable material. Hence, in process integration networks, while feed/sources refer to product/byproduct/waste streams of the existing process, product/sinks refer to ports requiring feed streams of the existing process. In the process example above, there exists many integration opportunities for improving the resource and energy allocation within the process. For instance, the amount of fresh water sources utilized can be reduced via recycle, reuse and regeneration of waste water streams. As water feeds with less impurities are normally more expensive, effective utilization of water in the process waste can significantly reduce the cost and improve sustainability of the process. Several water sources and sinks exist in the process. There is one external fresh water source available, which is used as scrubbing agent in Scrubber 2 or cooling water in distillation condenser (Source 1). Furthermore, aqueous waste from Dist 3 (Source 2), bottom product from flash column (Source 3) and condenser outlet streams (Source 4) can be regarded as additional water sources that can be regenerated and recycled back into the process. There are two water sinks in the process: Scrubber 2 (Sink 1) and Distillation column condensers (Sink 2). Streams from Source 2, Source 3 and Source 4 can be sent to regenerators to remove the impurities and can be reused in water sinks of the process. There are two types of regenerators available in the process with different cost and performance. These water sinks, sources and regenerators are showed in Figure 2a. Now, to decide on the optimal water network with minimum cost of operation systematically, we can consider the classic source-pool-sink superstructure to include all connections discussed above (shown in Figure 2b). This block superstructure is equivalent to the classic superstructure representation (Figure 2c) in the sense that it contains the same process alternatives with 11



its classic counterpart. In this superstructure, two rows are utilized, in which the first row (i = 1) represents intermediate pools (regeneration operations), and the second row (i = 2) accommodates sink streams. The number of columns to be used is determined according to the maximum number of regenerators and process sinks. In this case, the column number is equal to two (J = 2) since two regenerators and two sink streams exist. In the first row, each block Bi,j allows to mix all the waste water source streams, Source 2, Source 3 and Source 4, and yields a jump product that can be fed into the both sinks. The jump products from different blocks can be recycled to other blocks except the block where the stream originates from. In the second row, the product streams represent the water sinks. Additionally, we can improve the energy allocation in the process via heat integration. In the process described above, there are three hot and cold process streams which can be integrated to decrease the utility consumption: H1 (Reactor outlet), H2 (Scrubber 2 outlet), H3 (Dist 1 top stream), C1 (Flash column top product), C2 (Air feed), and C3 (Dist 3 bottom stream). In HEN representation, we adopt a superstructure representation similar to the one proposed by Yee and Grossmann. 29 The block superstructure representation of HEN superstructure for this example is given in Figure 2d. In this representation, stream splitting is not allowed and each hot or cold stream occupies single row. As there are three hot and three cold streams, 3 stages are required. 29 In block superstructure, each stage refers to a column of interblock streams with heater/cooler units and columns (j = 2, 3) and (j = 4) are for these process stream matches. As shown in Figure 2d with dashed lines, only the streams that flow through the same column are allowed to match with each other. Similar to the original superstructure, external hot and cooling utility consumption are allowed at the two end points and hot and cold streams are only allowed to flow in reverse directions. As six process streams are present in the network, six rows are used: (I = 6). Also, six columns (J = 6) are used to allow for five interblock streams: three interblock streams are for process stream matches and two interblock streams are for the end heaters/coolers. This leaves one block boundary unused for hot and cold streams. These streams are not

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assigned with any heaters/coolers as there is no need to heat the hot streams and cool down the cold streams. This unused block is included just to establish similarities between the original superstructure representation given by Yee and Grossmann 29 and can be avoided. Note that, stream splitting can be achieved via increasing the number of rows occupied by a stream and allowing vertical flow between those rows. Furthermore, different configurations, such as stream bypasses, can be also incorporated via the use of jump flows. Although we show water network and heat integration networks separately, these two problems can also be considered together as heat-integrated water networks (HIWN) 37 and the proposed representation is general enough to consider these two networks simultaneously as well. This simultaneous case is demonstrated in Section 4 through a case study.

3

A General MINLP Model for Process Integration

In this section, we provide the problem description for the general process integration problem using the block superstructure. The proposed MINLP model consists of constraints on flow direction assignment, material balance, property balance, energy balance, product specification and task assignments. The objective is to minimize the total annual cost.

3.1

Problem Description

The problem description of general process integration problem using block superstructure can be stated as follows. Known a set K = {k|k = 1, ..., |K|} of chemical species and a set qu = {qu|qu = 1, ..., |QU |} of chemical qualities, for a set of fresh material sources f eed f eed F S = {f |f = 1, ..., |F S|} with composition specification yk,f , property specification yqu,f ,

h

min,prod a set P S = {s|s = 1, ..., |P S|} of process sinks with demand of Ds and purity range yk,s ,

i

h

i

max,prod in,min in,max yk,s for species k as well as other property specifications P Squ,s , P Squ,s , and a

set of P U = {p|p = 1, ..., |P U |} of process units which generate or remove certain amount of property qu, it is necessary to determine a process integration network with information on network interconnections, flowrates, contaminant concentrations and stream temperatures. 13



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The process units have different inlet and outlet property specification requirements as i

h

h

i

out,max out,min in,max in,min for property qu respectively. These process units , yqu,p and yqu,p , yqu,p yqu,p

have operating load of Li,j,qu,p as generation amount or consumption amount according to the unit information provided. The number of stages for heat integration is given by the set HS = {hs|hs = 1, ..., |HS|}. The flow alignment d is embedded in its corresponding set D = {d|d = 1, 2}. d = 1 when the stream is flowing in the horizontal direction; d = 2 when the stream is flowing in the vertical direction. The global temperature range and global h

i

h

i

flowrate range for the integration problem is set as T min , T max and F L, F U . The main assumption in this work includes constant properties (heat capacity, heat transfer coefficient, etc.), continuous steady-state operation, counter-current heat exchange and ideal mixing. The objective for the general process integration is to minimize the total annual cost. What follows is the MINLP formulation for process integration based on block superstructure.

3.2

Flow Directions

The adjacent blocks interact with each other through bidirectional connectivity Fi,j,k,d . The selection of flow direction is achieved through the following binary variable:     True    

P lus zi,j,d = 

     False

if Fi,j,k,d is from block Bi,j to Bi,j+1 (d = 1) or from block Bi,j to Bi+1,j (d = 2) otherwise

The positive and negative component of stream connectivity Fi,j,k,d are F Pi,j,k,d and F Ni,j,k,d respectively, while only one of them is active for each connectivity between adjacent blocks. Accordingly:

Fi,j,k,d = F Pi,j,k,d − F Ni,j,k,d

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i ∈ I, j ∈ J, k ∈ K, d ∈ D


(1)

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P lus F Pi,j,k,d ≤ F U zi,j,d ,

i ∈ I, j ∈ J, k ∈ K, d ∈ D

P lus F Ni,j,k,d ≤ F U (1 − zi,j,d ),

3.3

i ∈ I, j ∈ J, k ∈ K, d ∈ D

(2) (3)

Block Material Balance

Material balance for each block mainly involves horizontal stream (inlet and outlet), vertical stream (inlet and outlet), external feed stream, external product stream, jump feed stream and jump product stream. Hence, the general material balance for continuous steady-state operation is established as follows:

p f p f = 0, −Ji,j,k +Ji,j,k −Hi,j,k Fi,j−1,k,1 −Fi,j,k,1 +Fi−1,j,k,2 −Fi,j,k,2 +Mi,j,k

i ∈ I, j ∈ J, k ∈ K (4)

In the above material balance constraint, the last four terms indicate the flowrate of external feed f into block Bi,j , product stream p from block Bi,j , internal jump feed f into block Bi,j , jump product p withdrawn from block Bi,j . These mass flowrate can be obtained through the following relation.

f Mi,j,k =

X

Mi,j,k,f

i ∈ I, j ∈ J, k ∈ K

(5)

Hi,j,k,s

i ∈ I, j ∈ J, k ∈ K

(6)

f ∈F S

p Hi,j,k =

X s∈P S

f Ji,j,k =

X

Ji0 ,j 0 ,i,j,k

i ∈ I, j ∈ J, k ∈ K

(7)

Ji,j,i0 ,j 0 ,k

i ∈ I, j ∈ J, k ∈ K

(8)

(i0 ,j 0 )∈LN p Ji,j,k =

X (i0 ,j 0 )∈LN

Here the continuous variable Mi,j,k,f indicate the component flowrate k into block Bi,j through feed stream f . Hi,j,k,s gives the component flowrate k withdrawn from block Bi,j via product stream s. Jump flow Ji,j,i0 ,j 0 ,k refers to the flowrate of component k transporting from block Bi,j to Bi0 ,j 0 , where i 6= i0 and j 6= j 0 . LN (i, j, i0 , j 0 ) is a subset for denoting the

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connection between block Bi,j and Bi0 ,j 0 . We assume that the only interaction between the superstructure and the external environment are achieved through external feeds and product streams. Hence, we impose that Fi=I,j,k,1 = Fi,j=J,k,2 = 0 to prohibit the material flow through the superstructure boundary. Material availability for feed f is Fff eed .

The distribution of feed in the block

f eedf rac superstructure is achieved through the fraction zi,j,f ≥ 0 of external feed f in block Bi,j .

Hence, the amount of component k entering into block Bi,j via feed stream f is obtained as follows: f eed f eedf rac Mi,j,k,f = Fff eed yk,f zi,j,f ,

0≤

XX

i ∈ I, j ∈ J, k ∈ K, f ∈ F S

f eedf rac zi,j,f ≤ 1,

f ∈ FS

(9) (10)

i∈I j∈J

The total flowrate for all streams associated with the block Bi,j is obtained by summing over all components in the stream. Specifically, the total flowrate for F Pi,j,k,d , F Ni,j,k,d , T T T T T , Ji,j,i , F Ni,j,d Ji,j,i0 ,j 0 ,k , Mi,j,k,f , and Hi,j,k,s are F Pi,j,d 0 ,j 0 , Mi,j,f , and Hi,j,s respectively. They

are obtained through the following relations.

T F Pi,j,d =

X

F Pi,j,k,d ,

i ∈ I, j ∈ J, d ∈ D

(11)

F Ni,j,k,d ,

i ∈ I, j ∈ J, d ∈ D

(12)

(i, j, i0 , j 0 ) ∈ LN (i, j, i0 , j 0 )

(13)

k∈K

T F Ni,j,d =

X k∈K

T Ji,j,i 0 ,j 0 =

X

Ji,j,i0 ,j 0 ,k ,

k∈K T Mi,j,f =

X

Mi,j,k,f ,

i ∈ I, j ∈ J, f ∈ F S

(14)

Hi,j,k,s ,

i ∈ I, j ∈ J, s ∈ P S

(15)

k∈K T Hi,j,s =

X k∈K

To ensure that each flow leaving block Bi,j have the same composition for splitting operation, the following constraints are considered for all the direct connectivities, jump connectivities

16


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and product streams:

b T F Pi,j,k,d = yi,j,k F Pi,j,d

i ∈ I, j ∈ J, d ∈ D

(16)

b T F Ni,j−1,k,1 = yi,j,k F Ni,j−1,1

i ∈ I, j ∈ J

(17)

b T F Ni−1,j,k,2 = yi,j,k F Ni−1,j,2

i ∈ I, j ∈ J

(18)

b T Ji,j,i0 ,j 0 ,k = yi,j,k Ji,j,i 0 ,j 0 b T Hi,j,k,s = yi,j,k Hi,j,s

(i, j, i0 , j 0 ) ∈ LN (i, j, i0 , j 0 ), k ∈ K i ∈ I, j ∈ J, k ∈ K, s ∈ P S

(19) (20)

b Here the positive variable yi,j,k ≤ 1 indicates the block composition of component k, which

is also the composition of component k for all outlet streams associated with the block Bi,j . With these flowrate information, the inlet total flowrate of component k for block Bi,j , φi,j,k , can be determined as follows:

φi,j,k = F Pi,j−1,k,1 + F Ni,j,k,1 + F Pi−1,j,k,2 + F Ni,j,k,2 +

X

Mi,j,k,f

i ∈ I, j ∈ J, k ∈ K (21)

f

3.4

Block Property Balance

Since streams associated with each block normally involve other qualities such as contaminate (ions, heavy metals and etc.), heating value, we consider block property balance in addition to block material balance. This property balance also allows the involvement of property integration. With these total flowrate information, the stream property is determined by the multiplication of stream total flowrate and its property specification. The block property

17



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balance is established as follows: X

eed T ψ(ycfqu,f )Mi,j,f +

XX i0 ∈I

f ∈F S

T T ψ(yi0 ,j 0 ,qu )JiT0 ,j 0 ,i,j + ψ(yi,j−1,qu )F Pi,j−1,1 + ψ(yi,j+1,qu )F Ni,j,1

j 0 ∈J

T T + ψ(yi+1,j,qu )F Ni,j,2 + ψ(yi−1,j,qu )F Pi−1,j,2 = ψ(yi,j,qu )(

X

T Ji,j,i 0 ,j 0 +

(i0 ,j 0 )∈LN T T F Ni,j−1,1 + F Ni−1,j,2 +

X

T F Pi,j,d )+

X

X

T Hi,j,s +

(22)

s∈P S

i ∈ I, j ∈ J, qu ∈ QU

Sup Li,j,qu,p

p∈P U

d∈D

eed where the function ψ(y) is the mixing property operators. Parameter ycfqu,f quantify the

specification of property qu in feed stream f . Parameter Sup indicates the type of process units. Sup = 1 when the process unit p is utilized to remove certain amount of property and Sup = −1 when the process unit p is for generating property. Positive variable Li,j,qu,p indicates the generation or consumption amount of property qu in process unit p at block Bi,j . The block property specification yi,j,qu is also the property specification of outlet streams in for from the block Bi,j . Similarly, we can determine the inlet property specification yi,j,qu

the block Bi,j through the following constraint. X

eed T )Mi,j,f + ψ(ycfqu,f

XX

T T ψ(yi0 ,j 0 ,qu )JiT0 ,j 0 ,i,j + ψ(yi,j−1,qu )F Pi,j−1,1 + ψ(yi,j+1,qu )F Ni,j,1

i0 ∈I j 0 ∈J

f ∈F S

T T in + ψ(yi+1,j,qu )F Ni,j,2 + ψ(yi−1,j,qu )F Pi−1,j,2 = ψ(yi,j,qu )(

X

T Mi,j,f +

X

T F Ni,j,d )

JiT0 ,j 0 ,i,j

(23)

(i0 ,j 0 )∈LN

f ∈F S T T + F Pi,j−1,1 + F Pi−1,j,2 +

X

i ∈ I, j ∈ J, qu ∈ QU

d∈D

3.5

Product Specification

Purity constraints on the product streams are achieved through the following constraints:

min,prod yk,s

X k0 ∈K

max,prod Pi,j,k0 ,s ≤ Pi,j,k,s ≤ yk,s

X

Pi,j,k0 ,s ,

i ∈ I, j ∈ J, (k, s) ∈ kp

(24)

k0 ∈K

min,prod max,prod where, the parameters yk,s and yk,s denote the minimum and maximum allowed

concentration, respectively, for component k in product s.

The set kp designates the

key component k in product stream s which has component composition restrictions (e.g.

18


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contaminant concentration in a waste stream or minimum purity requirement for a product stream). If a product stream is not assigned to subset kp, then it does not have any concentration restrictions. In addition to purity requirement of key component k 0 in product stream s, possible requirement on ratio of different component k in product stream p is also considered.

Pi,j,k=k0 ,s ≥

X 00

πkprod 0 00 Pi,j,k 00 ,s ,k ,s

i ∈ I, j ∈ J, s ∈ P s

(25)

k ∈K

0

where πkprod is the minimum product ratio requirement between component k and 0 00 ,k ,s 00

component k for product s. In addition to the purity and ratio requirements, we impose the following demand constraint for product s:

DsL ≤

XX X

Pi,j,k,s ≤ DsU ,

s ∈ PS

(26)

i∈I j∈J k∈K

Here, DsL and DsU are lower bound and upper bound for product stream p respectively.

3.6

Block Energy Balance

The block energy balance ensures that the amount of enthalpy coming in is equal to the amount of enthalpy leaving the block. The inlet enthalpy amount for a block includes enthalpy via inlet material flow and enthalpy via feed streams. The outlet enthalpy amount involves enthalpy via outlet material streams and enthalpy carried out by product stream. Accordingly, we develop the continuous steady-state energy balance for block Bi,j as follows: out in in out − EPi,j − EJi,j = 0, EBi,j + EMi,j + EJi,j − EBi,j

i ∈ I, j ∈ J

(27)

in here, EBi,j is the stream enthalpy via direct connecting flows from adjacent blocks of block in Bi,j . EMi,j is the inlet enthalpy via feed streams entering block Bi,j . EJi,j is the inlet

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Page 20 of 92

out enthalpy coming into block Bi,j through jump flow Ji0 ,j 0 ,i,j,k . EBi,j represents the stream

enthalpy carried out by the outlet flow from block Bi,j to adjacent blocks. EPi,j is total out is the outlet enthalpy leaving block Bi,j enthalpy taken away by product stream. EJi,j

through jump flow Ji,j,i0 ,j 0 ,k . These energy flow variables and the system boundary for energy s balance are shown in Figure 3. The inlet temperature for the block Bi,j is Ti,j,d and the outlet

temperature is Ti,j . [Figure 3 about here] The enthalpy carried by a material stream is determined by its flowrate, component heat capacities and the temperature. Inlet streams coming from adjacent blocks to block Bi,j include F Pi,j−1,k,1 , F Ni,j,k,1 , F Pi−1,j,k,2 and F Ni,j,k,2 . Outlet streams from block Bi,j to adjacent blocks include F Pi,j,k,1 , F Ni,j−1,k,1 , F Pi,j,k,2 and F Ni−1,j,k,2 . Accordingly, inlet and outlet enthalpies for block Bi,j are calculated as: in EBi,j =

X

s s (F Pi,j−1,k,1 (Ti,j−1,1 − T ref ) + F Ni,j,k,1 (Ti,j,1 − T ref )+

k∈K

(28)

s s F Pi−1,j,k,2 (Ti−1,j,2 − T ref ) + F Ni,j,k,2 (Ti,j,2 − T ref ))Cpk

out EBi,j =

X

i ∈ I, j ∈ J

(F Pi,j,k,1 + F Ni,j−1,k,1 + F Pi,j,k,2 + F Ni−1,j,k,2 )(Ti,j − T ref )Cpk i ∈ I, j ∈ J (29)

k∈K

where Cpk is the heat capacity of component k. The inlet temperature information for each s s s s block, provided as Ti,j−1,1 , Ti,j,1 , Ti−1,j,2 , Ti,j,2 for streams F Pi,j−1,k.1 , F Ni,j,k,1 , F Pi−1,j,k,2 and

F Ni,j,k,2 respectively, is shown in Figure 3. The inlet and outlet enthalpy for block Bi,j through jump flows are determined as follows: in EJi,j =

X

X

(Ti0 ,j 0 − T ref )Cpk Ji0 ,j 0 ,i,j,k

i ∈ I, j ∈ J

(30)

(Ti,j − T ref )Cpk Ji,j,i0 ,j 0 ,k

i ∈ I, j ∈ J

(31)

k∈K (i0 ,j 0 )∈LN

out EJi,j =

X

X

k∈K (i0 ,j 0 )∈LN

20


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Similarly, the feed enthalpy and product enthalpy are calculated as follows:

EMi,j =

X X

Mi,j,k,f Cpk T f

i ∈ I, j ∈ J

(32)

Hi,j,k,s Cpk Ti,j

i ∈ I, j ∈ J

(33)

k∈K f ∈F

EPi,j =

X X k∈K s∈P S

Here T f is the inlet temperature of feed f .

3.7

Heat integration

Heat exchanger network synthesis formulation is based on the one proposed by Yee and Grossmann. 29 Yet, to keep the formulation general, it is extended to include unclassified streams where the identity of a stream as hot/cold is not known beforehand. The formulation, in general, contains heat duty determination, stream classification (i.e.

hot/cold) and

inlet and outlet temperature determination for each stream in the superstructure. Then, these variables are mapped into HENS part of the model where the matched streams and corresponding approach temperatures are calculated. Heat duty for each stream heater/cooler is calculated as follows:

P,h P,c QFi,j,d − QFi,j,d =

s F Pi,j,k,d Cpk (Ti,j,d − Ti,j )

X

i ∈ I, j ∈ J, d ∈ D

(34)

k∈K

Similarly, we derive the following constraint for negative flow F Ni,j,k,d from block Bi,j . The s outlet temperature for heat exchanger on each negative flow F Ni,j,k,d is Ti,j,d . The inlet

temperature for horizontal negative and vertical negative flow is Ti,j+1 and Ti+1,j respectively. N,h N,c QFi,j,1 − QFi,j,1 =

X

s F Ni,j,k,1 Cpk (Ti,j,1 − Ti,j+1 )

i ∈ I, j ∈ J

(35)

s F Ni,j,k,2 Cpk (Ti,j,2 − Ti+1,j )

i ∈ I, j ∈ J

(36)

k∈K

N,h N,c QFi,j,2 − QFi,j,2 =

X k∈K

P,h N,h If the stream is a cold stream, then QFi,j,d or QFi,j,d is activated. If the stream is a hot

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P,c N,c stream, then QFi,j,d or QFi,j,d is activated. As only one direction is allowed for a stream, heat

duty of each stream can be combined in a single variable depending on the classification of : or QF,cold the stream as QF,hot w w P,h N,h QFi,j,d + QFi,j,d = QF,hot w

i, j, d, w ∈ hipstreamsi,j,d,w

(37)

P,c N,c QFi,j,d + QFi,j,d = QF,cold w


(38)

Here, a subset hipstreamsi,j,d,w is defined to map superstructure indices (i, j, d) to a single index w in order to simplify the HENS formulation. Note that w ∈ W = 1, ..., |W | is an ordered set and it includes all the streams in the superstructure: |W | = (I − 1) × (J − 1) × 2. Accordingly, all the variables related with HENS are defined with index w. To account for the approach temperature in each heat exchanger, inlet and outlet temperature of each stream need to be determined. The inlet and outlet temperature of each stream through each boundary in the block superstructure assigned to a single stream inlet and outlet temperature as follow: pos neg T INw = T INi,j,d + T INi,j,d


pos max P lus ≤ Ti,j,d T INi,j,d zi,j,d pos T INi,j,d ≤ Ti,j

i ∈ I, j ∈ J, d ∈ D

i ∈ I, j ∈ J, d ∈ D

pos P lus T INi,j,d ≥ Ti,j − T max (1 − zi,j,d )

i ∈ I, j ∈ J, d ∈ D

(39) (40) (41) (42)

Similarly for the negative horizontal direction (i.e. d = 1):

neg max P lus T INi,j,1 ≤ Ti,j,1 (1 − zi,j,1 )

neg T INi,j,1 ≤ Ti,j+1

i ∈ I, j ∈ J, d ∈ D

i ∈ I, j ∈ J

neg P lus T INi,j,1 ≥ Ti,j+1 − T max zi,j,1

22

i ∈ I, j ∈ J


(43)

(44) (45)

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And negative vertical direction (i.e. d = 2):

neg max P lus T INi,j,2 ≤ Ti,j,2 (1 − zi,j,2 )

neg T INi,j,2 ≤ Ti+1,j

i ∈ I, j ∈ J

i ∈ I, j ∈ J

neg P lus T INi,j,2 ≥ Ti+1,j − T max zi,j,2

i ∈ I, j ∈ J

(46)

(47) (48)

s The outlet temperature of a stream is always equal to stream outlet temperature, i.e. Ti,j,d :

s T OU Tw = Ti,j,d


(49)

To classify each stream as hot/cold, two binary variables are defined:

zwhot =

zwcold =

    1

if stream Fw is a hot stream

   0

Otherwise

    1

if stream Fw is a cold stream

   0

Otherwise

Each stream can be either hot or cold stream:

zwhot + zwcold = 1

w∈W

(50)

And stream classification binary variables are related to the stream heat duty variables as follows: QF,hot ≤ q max zwcold w

w∈W

(51)

QF,cold ≤ q max zwhot w

w∈W

(52)

Note that, together with Eqs. 37 and 38, these relations also dictate that only one of the P,h P,c N,h stream heat duty variables for the positive (negative) direction, QFi,j,d or QFi,j,d (QFi,j,d or

23



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N,c QFi,j,d ), can be activated at one time.

For each match between two streams in the superstructure, heat duty is determined by hot cold hot qw,w 0 or qw,w 0 where qw,w 0 is the heat duty of the heat exchanger between hot stream Fw and cold cold stream Fw0 . Similarly, qw,w 0 is the heat duty of the heat exchanger between cold stream

Fw and hot stream Fw0 . QHU and QCU w w are the heat duty between the superstructure stream and hot external and cold external utility, respectively. Then, the overall energy balance around each stream in the superstructure becomes:

QF,hot = w

X

cold qw,w 0 +

w0 (w

Process Integration using Block Superstructure

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