1. Introduction

An efficiently designed heat-exchanger network (HEN) can be used to achieve significant reductions in energy consumption and pollutant emission into the environment by chemical plants. However, most design methods that have been presented for the synthesis of HENs

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have assumed fixed process-operating parameters. However, in reality, process parameters may fluctuate around some nominal operating points due to various factors such as changes in environmental conditions, plant start-ups/shutdowns, changes in product quality demand, and so on. In some other cases, the process parameters may deliberately be moved away from their set point/nominal conditions due to reasons such as planned transition from one product quality to another. For these cases, even though the set point tracking of the process parameters is required in ensuring a smooth transition to the new set of operating points, the network of heat exchangers still have to be flexible to handle these new set of operating conditions. Despite the fact that these new set of operating points lie within the possible range of variability of the process parameters, their length of duration needs to be taken into consideration while designing a flexible heat-exchanger network.

The methods that have been used for the synthesis of flexible heat-exchanger networks have been both sequential [1–3] and simultaneous in nature [4–16]. Some of the sequential methods are an automated multi-period version of the mixed integer linear programme (MILP) transshipment model [17] and the non-linear programme (NLP) minimum investment network cost model [18]. The simultaneous methods have mostly been based on a multi-period version of the simplified stage-wise superstructure (SWS) model [19]. Some of the existing design methods for flexible HENs may only be feasible to transfer heat for a finite set of processoperating parameters for which the network is designed in what is known as multi-period networks [4–8]. In the multi-period networks [4–8], each period of operation is distinct in that the process parameters, as well as the length of periods for each of the periods of operations, are known upfront. Since multi-period networks are capable of transferring heat within the specified finite set of operating periods, they can be termed flexible networks. However, their degree of flexibility may only be limited to the set of finite operating points for which the network is designed. The degree of flexibility of the SWS-based multi-period networks has been improved through the use of the timesharing mechanism [9], where heat exchangers may be shared by different stream pairs in more than one period of operation. This is unlike other SWS-based multi-period models where either the average area [4] or the maximum area [5–7] of the same pair of streams exchanging heat in the same stage of the superstructure, and in different periods of operations, is used as the representative heat-exchanger area in the objective function.

According to Jiang and Chang [9], a major shortcoming of the average area or maximum area approach, as presented in the literature, is that an exchanger may be overdesigned for some periods of operations such that when unforeseen changes in period durations occur, the multiperiod network may not be feasible to transfer heat any longer, or it may require a significantly higher utility flows. Even though the timesharing approach overcomes some of the aforementioned shortcomings of the other SWS-based methods [4–7], the complexities involved in having to thoroughly clean exchangers during the process of exchanger swapping can be enormous. Furthermore, additional costs and complex controllability issues will be incurred due to excessive piping and associated instrumentations. Hence, some other synthesis methods that result in networks that have a greater degree of flexibility have been presented in the literature [10–15]. One of the methods entails [10] a three-step approach where a network is designed based on a finite set of operating points in the first step. In the second step, the resulting network from the first step is tested for flexibility using the active set strategy [20]. The third step entails using integer cuts to exclude non-qualifying networks in the flexibility tests. Chen et al. [11] extended the aforementioned method by modifying the flexibility analysis step such that area restrictions are not considered during a first step of flexibility analysis but are considered in a later step.

have assumed fixed process-operating parameters. However, in reality, process parameters may fluctuate around some nominal operating points due to various factors such as changes in environmental conditions, plant start-ups/shutdowns, changes in product quality demand, and so on. In some other cases, the process parameters may deliberately be moved away from their set point/nominal conditions due to reasons such as planned transition from one product quality to another. For these cases, even though the set point tracking of the process parameters is required in ensuring a smooth transition to the new set of operating points, the network of heat exchangers still have to be flexible to handle these new set of operating conditions. Despite the fact that these new set of operating points lie within the possible range of variability of the process parameters, their length of duration needs to be taken into consideration

The methods that have been used for the synthesis of flexible heat-exchanger networks have been both sequential [1–3] and simultaneous in nature [4–16]. Some of the sequential methods are an automated multi-period version of the mixed integer linear programme (MILP) transshipment model [17] and the non-linear programme (NLP) minimum investment network cost model [18]. The simultaneous methods have mostly been based on a multi-period version of the simplified stage-wise superstructure (SWS) model [19]. Some of the existing design methods for flexible HENs may only be feasible to transfer heat for a finite set of processoperating parameters for which the network is designed in what is known as multi-period networks [4–8]. In the multi-period networks [4–8], each period of operation is distinct in that the process parameters, as well as the length of periods for each of the periods of operations, are known upfront. Since multi-period networks are capable of transferring heat within the specified finite set of operating periods, they can be termed flexible networks. However, their degree of flexibility may only be limited to the set of finite operating points for which the network is designed. The degree of flexibility of the SWS-based multi-period networks has been improved through the use of the timesharing mechanism [9], where heat exchangers may be shared by different stream pairs in more than one period of operation. This is unlike other SWS-based multi-period models where either the average area [4] or the maximum area [5–7] of the same pair of streams exchanging heat in the same stage of the superstructure, and in different periods of operations, is used as the representative heat-exchanger area in the objec-

According to Jiang and Chang [9], a major shortcoming of the average area or maximum area approach, as presented in the literature, is that an exchanger may be overdesigned for some periods of operations such that when unforeseen changes in period durations occur, the multiperiod network may not be feasible to transfer heat any longer, or it may require a significantly higher utility flows. Even though the timesharing approach overcomes some of the aforementioned shortcomings of the other SWS-based methods [4–7], the complexities involved in having to thoroughly clean exchangers during the process of exchanger swapping can be enormous. Furthermore, additional costs and complex controllability issues will be incurred due to excessive piping and associated instrumentations. Hence, some other synthesis methods that result in networks that have a greater degree of flexibility have been presented in the literature [10–15]. One of the methods entails [10] a three-step approach where a

while designing a flexible heat-exchanger network.

94 Heat Exchangers– Design, Experiment and Simulation

tive function.

Chen and Hung [12] extended the method of Chen and Hung [10] with some modifications to the synthesis of flexible heat and mass exchange networks. The flexibility test of this method is carried out on a large number of randomly generated parameters within the range of uncertainty. Another method [13] in the area of flexible HEN designs used a two-stage strategy which is based on the SWS model for the synthesis of flexible and controllable networks. Li et al. [14] developed a two-step approach for flexible HENS. The first step entails the synthesis of the network structure using the nominal set of operating conditions. The flexibility of the resulting network is further improved through a structural union with the topology of the critical operating points. In the second step, the areas of the heat exchangers obtained in the first step are further optimised considering flexibility and total annual cost (TAC). It should be known that in the method of Li et al. [14], the area optimisation is only done after obtaining structures that qualify from the flexibility step. This implies that a true simultaneous optimisation may somewhat be omitted. The method has the advantage that it can synthesise flexible networks with non-convex feasible regions. Li et al. [15] used a simulated annealing and decoupling strategy to determine the flexibility index of large-scale non-convex HEN optimisation problem.

It is worth stating at this point that, apart from the use of the multi-period version of the SWS model, a common feature of most of the flexible HENS methods is that they involve a first step where a candidate single period or a multi-period network is synthesised for a minimum total annual cost (TAC) scenario, followed by a flexibility analysis step. For the first step, the candidate multi-period network has mostly been made comprising few periods of operations which may include the nominal operating conditions and the critical operating points. The critical operating points are the periods of operations that require the maximum heat load. The authors of this chapter are of the view that if these sets of operating points that are used to generate the candidate multi-period network are carefully chosen, there may not be a need for complex and mathematically intensive flexibility analysis step, especially in small- to medium-scale HENS problems. This implies that as many as possible critical operating points that lie within the overall range of potential disturbance/fluctuation should be selected for participation in the candidate multi-period network synthesis of the first step. A further criterion that needs to be considered while designing the representative candidate multi-period network, which has erstwhile been ignored by existing flexible HENS methods, is the length of periods for each of the critical operating points used to generate the representative candidate multi-period network in the first step. The existing methods [5, 9–14] used the average costs of utility usage by each period of operation present in the first-step candidate multi-period network to determine its minimum TAC and associated network structure as shown in Eqs. (1) and (2)

$$\begin{aligned} \text{min}\,\text{TAC} &= A\boldsymbol{F} \cdot \left\{ \sum\_{i \neq j} \sum\_{j \in \text{kCl}} \text{CFz}\_{i,j,k} + \sum\_{i \neq j} \sum\_{i \in \text{I}} \text{CFz}\_{i,\text{cu}} + \sum\_{j \in \text{I}} \sum\_{i \in \text{I}} \text{CFz}\_{j,\text{hu}} \\ &+ \sum\_{i \neq j} \sum\_{j \in \text{I} \&\text{K}} A \mathbf{C}\_{i,j,k} A\_{i,j,k}^{AE\_{i,j}} + \sum\_{j \in \text{C}} A \mathbf{C}\_{j,\text{hu}} A\_{j,\text{hu}}^{AE\_{i,j}} + \sum\_{i \neq \text{I}} A \mathbf{C}\_{i,\text{cu}} A\_{i,\text{cu}}^{AE\_{i,\text{cu}}} \right\} \\ &+ \left[ \sum\_{p \in \text{P}} \frac{D \mathbf{OP}\_p}{N \text{OP}} \sum\_{j \in \text{C}} \text{CLH} q\_{j,\text{hu},p} + \sum\_{p \in \text{I}} \frac{D \mathbf{OP}\_p}{N \text{OP}} \sum\_{i \neq \text{I}} \text{CLCq}\_{i,\text{cu},p} \right] \\ &\quad \text{min}\,\text{TAC} = \frac{1}{N \text{OP} + 1} \left[ \sum\_{p \in \text{P} \text{H}} \text{CLCq}\_{i,\text{cu},p} + \sum\_{p \in \text{P} \in \text{C}} \text{CLH} q\_{j,\text{hu},p} \right] \\ &+ \sum\_{i \neq \text{I} \neq j \in \text{C} \&\text{K}} \text{AC}\_{i,j,k} A\_{i,j,k}^{AE\_{i,j}} + \sum\_{j \in \text{C$$

where AF is the annualisation factor, CF is the fixed charge for heat exchangers, AC is the area costs for heat exchangers, AE is the area cost exponent for heat exchangers, CUH and CUC are the costs of hot and cold utilities, respectively, DOP is the duration of period p, NOP is the number of periods/operating conditions, Ai,j,k is the area of heat exchanger for hot and coldprocess stream pairs i,j in interval k. Aj,hu and Ai,cu are the area of heat exchangers exchanging heat between hot utility and cold-process streams and cold utility and hot-process streams, respectively. H,C,HU,CU are the set of hot streams, cold streams, hot utilities and cold utilities, respectively. It should be known that the area Ai,j,k is the representative heat exchange area which, as explained previously, are used by the same pair of streams exchanging heat in the same interval of the superstructure at different periods of operations.

Eqs. (1) and (2) are the objective functions used in determining the TAC for the first-step initial candidate multi-period network that is later tested for flexibility using various kinds of approaches in some of the existing methods [5, 9–14]. It can be seen that the utility cost calculation component of these equations will result in allotting equal contributions, in terms of utility usage durations, for each of the periods of operations present in the first-step candidate multi-period network. This implies that the candidate multi-period network that is designed at the initial step, and later tested for flexibility, may be limited based on the fact that it is designed with the assumption that these initial candidate critical points have equal-period durations. Since TAC is being solved for at the first step, the objective functions in Eqs. (1) and (2) will aim to simultaneously minimise both utility consumption and investment costs. The investment cost is influenced by the size of heat-exchanger areas and the number of units. Allowing this limitation at the first step of the flexible network synthesis process means that the flexibility analysis step needs to be sophisticated so as to compensate for this limitation. This is because some candidate networks that lie in the uncertain process parameter range that are tested in the flexibility step may be disqualified from being included in the flexible network feasible space due to the fact that Eqs. (1) and (2) were used as the objective functions for generating the initial candidate multi-period network. Furthermore, even at the flexibility testing stage, the feasible solution space may further be limited or constrained based on the fact that equal-period duration scenario was assumed. The authors of this chapter are of the view that adequately incorporating period durations at the stage of generating the candidate multi-period network is vital so as to reduce the degree of complexity of the flexibility tests that would be carried out subsequently. Moreover, it can be said that a HEN is flexible not only when it is able to feasibly transfer heat for scenarios where each of the potential possible operating points that lie within the range of disturbance/fluctuation has equal lengths of periods, but also when their lengths of periods are significantly different from each other, as well as being uncertain. This implies that the total annual cost of a flexible HEN is not fixed but depends on which operating points (including period durations) within the uncertain range of variability are active.
