*5.1. Model for HEN Synthesis*

The model of the HEN is formulated to determine the existences of heat exchangers, operating parameters of heat exchangers, required heat transfer area, capital cost and utility cost. The model formulation presented below consists mainly of mass and energy balance equations. Temperature feasibility constraints are also included. Several sets are defined to assist the expression of involved parameters and variables: *i*∈*I*, *j*∈*J* denote hot and cold process streams, *k*∈*K* denotes temperature interval of inner- and inter-stage, *m*∈*M* is the order of temperature location which is corresponding to stage *k*, *n*∈*N* denotes the steam levels in different pressures.

Heat balances of hot and cold process streams are defined by Equations (1) and (2), wherein *FHi* and *FCj* denote the total heat capacity flowrates of a hot process stream *i* and a cold process stream *j*; *thini*, *thouti*, *tcinj*, *tcoutj* denote supply and target temperatures of streams; *qi,j,k*, represents the heat load of a heat exchanger between hot and cold streams; *qhuj,k,n* and *qhusj,k,n* are the heat satisfied by steam *n* within inter-stage and inter-stage, respectively. *qhuj,0,n* is the heat satisfied by steam *n* at the end of cold stream *j*. Equation (1) is used to ensure each hot stream is cooled from its supply temperature to target by performing adequate heat exchanges with cold streams and cold utility. Analogously, Equation (2) is involved to guarantee that the required heat of each cold stream is satisfied by inner-stage stream-stream exchange, inter-stage stream-steam exchange (including the matching at stream end) and inner-stage stream-steam exchange:

$$\sum\_{j} \sum\_{k} q\_{i,j,k} + qc\_i = FH\_i (thin\_i - that\_i) \tag{1}$$

$$\sum\_{i} \sum\_{k} q\_{i,j,k} + \sum\_{k} \sum\_{n} qh\mu\_{j,k,n} + \sum\_{k} \sum\_{n} qh\text{us}\_{j,k,n} + \sum\_{n} qh\mu\_{j0,n} = \text{FC}\_{j}(\text{tout}\_{j} - \text{tin}\_{j}) \tag{2}$$

Equations (3)–(6) present the energy balances of hot and cold process streams within innerand inter-stage. *tii,m* and *tjj,m* are temperature locations of process streams corresponding to stage *k*. Equation (3) shows that, within inner-stage, hot streams only exchange heat with cold streams. Equation (4) defines that cold process streams can be heated up by hot process stream and steam *n* in parallel form within inner-stage *k*, and more than this, the inter-stage heating from steam is also available, as indicated in Equation (5). The relationship of temperature location *m* and stage *k* is worth

mentioning here. In view of the existence of inner- and inter-stage, the presentation of temperature location becomes complicated. In order to enhance the universality of the model, constraints are employed to determine the relationship of *m* and *k*. For inner-stage *k*, the left temperature location *m* of process stream corresponds to 2*k*−1, right temperature location *m* of process stream corresponds to 2*k*. While for inter-stage *k*, its left temperature location is the same as right temperature of inner-stage *k*, so its left temperature location is 2*k*, and the right temperature location corresponds to 2*k*+1:

$$FH\_i(t i\_{i,m-1} - t i\_{i,m}) = \sum\_j q\_{i,j,k} \quad m = 2k \tag{3}$$

$$FC\_j(tj\_{j,m-1} - tj\_{j,m}) = \sum\_{i} q\_{i,j,k} + \sum\_{n} qhus\_{j,k,n} \quad m = 2k \tag{4}$$

$$FC\_j(tj\_{j,m} - tj\_{j,m+1}) = \sum\_{n} qh\mu\_{j,k,n} \quad m = 2k \tag{5}$$

$$FC\_j(tcount\_j - t j\_{j,m}) = \sum\_{n} qln\_{j,0,n} \quad \text{ } m = 1 \tag{6}$$

Equations (7)–(9) express the mass balances of hot and cold process streams within the innerand inter-stages. A hot stream must be split into branches to perform the potential heat exchanges with cold streams within each inner-stage (Equation (7)), but for a cold stream, there are two kinds of matches within inner-stage *k*, with hot process streams and with steam (Equation (8)). For inter-stages, split branches can only match with steam (Equation (9)):

$$FH\_i = \sum\_j fhs\_{i,j,k} \tag{7}$$

$$FC\_j = \sum\_i fcs\_{i,j,k} + \sum\_n fcsp\_{j,k,n} \tag{8}$$

$$FC\_{\bar{\jmath}} = \sum\_{n} f \text{csk}\_{\bar{\jmath},k\jmath} \tag{9}$$

The characteristics of an exchanger include its location, inlet and outlet temperatures, hot and cold stream (branch) flowrates, heat load and heat transfer area. Due to the variety of the exchangers in the presented superstructure, the models for exchangers are presented in five sets: inner-stage stream-stream exchangers, inter-stage stream-steam exchangers, inner-stage stream-steam exchangers, heaters at end of cold process streams, and coolers at the end of hot streams. All these exchangers are expressed with similar heat balances and area equations in the modeling.

Equations (10)–(16) are used to calculate the heat load *qi,j,k* and heat transfer area *Ai,j,k* of heat exchangers between hot and cold streams within inner-stage *k*. Heat balance Equations (10) and (11) are used to constraint the inlet and outlet temperatures of the unit that, the released heat from hot stream *i* equals to that obtained by the pairing cold stream *j*. *dtij*1*i,j,k* and *dtij*2*i,j,k* in Equations (12) and (13) denote the temperature difference at the two sides of the heat exchanger. In order to circumvent the presence of logarithm and the resultant difficulty in model solving, the logarithmic mean temperature difference used for transfer area calculation (Equation (15)) is approximated by Equation (14) [23]. In the model, binary variable *zi,j,k* is employed to indicate the existence of heat exchangers, that is, *zi,j,k* = 1 means the heat exchanger exists, otherwise the exchanger is not involved. Heat loads are highly related to the locations of exchangers, so Equation (16) is used by defining sufficiently large constant *Q*max to assist with the constraint:

$$q\_{i,j,k} = fh s\_{i,j,k} (t i\_{i,m-1} - t l\_{i,j,k}) \quad m = 2k \tag{10}$$

$$q\_{i,j,k} = f \text{cs}\_{i,j,k}(tc\_{i,j,k} - tj\_{j,m}) \quad m = 2k \tag{11}$$

$$
\dot{d}tij1\_{i,j,k} \le t i\_{i,m-1} - t c\_{i,j,k} + T\_{\max} (1 - z\_{i,j,k}) \quad m = 2k \tag{12}
$$

$$\text{dtif}j2\_{i,j,k} \le t h\_{i,j,k} - t j\_{j,m} + T\_{\text{max}} (1 - z\_{i,j,k}) \quad m = 2k \tag{13}$$

$$
\Delta T i j\_{i,j,k} = \left( d t i j \mathbf{1}\_{i,j,k} \times d t i j \mathbf{2}\_{i,j,k} \times \left( d t i j \mathbf{1}\_{i,j,k} + d t i j \mathbf{2}\_{i,j,k} \right) \times 0.5 \right)^{1/3} \tag{14}
$$

$$A\_{i,j,k} = q\_{i,j,k} / \left( h \times \Delta T i j\_{i,j,k} \right) \tag{15}$$

$$q\_{i,j,k} \le z\_{i,j,k} \times Q\_{\text{max}} \tag{16}$$

In similar way, Equations (17)–(23) are included to describe the steam heaters within each inner-stage. It is stipulated that steam, at most one type, is allowed for each possible stream-steam match, so Equation (22) is introduced into the model, by using *zhusj,k,n* to denote the existences of the heaters:

$$
gamma\_{j,k,n} = f \text{csp}\_{j,k,n}(t \text{csp}\_{j,k,n} - t \text{j}\_{j,n}) \quad m = 2k \tag{17}
$$

$$\text{s.dths1}\_{j,k,n} \le \text{thus}\_{j,k,n} - \text{tcsp}\_{j,k,n} + T\_{\text{max}} (1 - z \text{lns}\_{j,k,n}) \quad m = 2k \tag{18}$$

$$\text{splits2}\_{j,k,n} \le \text{thus}\_{j,k,n} - tj\_{j,n} + T\_{\text{max}}(1 - z\text{hus}\_{j,k,n}) \quad m = 2k \tag{19}$$

$$
\Delta \text{Thus}\_{j,k,n} = \left( d \text{ths} \mathbf{1}\_{j,k,n} \times d \text{ths} \mathbf{2}\_{j,k,n} \times \left( d \text{ths} \mathbf{1}\_{j,k,n} + d \text{ths} \mathbf{2}\_{j,k,n} \right) \times 0.5 \right)^{1/3} \tag{20}
$$

$$\text{Alus}\_{j,k,n} = \text{qhus}\_{j,k,n} / (\text{h1} \times \Delta \text{Thus}\_{j,k,n}) \tag{21}$$

$$\sum\_{n} zhws\_{j,k,n} \le 1\tag{22}$$

$$
gamma\_{j,k,n} \le z \text{lus}\_{j,k,n} \times Q\_{\text{max}} \tag{23}
$$

Equations (24)–(30) are formulated to determine the steam heater within each inter-stage. *zhuj,k,n* denotes the existence of the heater having steam *n* involved. Equation (29) is employed to ensure that a cold process stream can only be heated up by one steam at most within each inter-stage:

$$f \csc\_{j,k,n}(tj\_{j,m} - tj\_{j,m+1}) = qh\mu\_{j,k,n} \quad m = 2k \tag{24}$$

$$\text{s.tth1}\_{\text{j},\text{k},\text{u}} \le t \text{h} \mu\_{\text{j},\text{k},\text{u}} - t \dot{\eta}\_{\text{j},\text{m}} + T\_{\text{max}} (1 - z \text{h} \mu\_{\text{j},\text{k},\text{u}}) \quad m = 2k \tag{25}$$

$$\text{lth2}\_{j,k,n} \le \text{th}\mu\_{j,k,n} - \text{tf}\_{j,m+1} + T\_{\text{max}}(1 - zh\mu\_{j,k,n}) \quad m = 2k \tag{26}$$

$$
\Delta \text{Thu}\_{j,k,n} = \left( dth \mathbf{1}\_{j,k,n} \times dth \mathbf{2}\_{j,k,n} \times \left( dth \mathbf{1}\_{j,k,n} + dth \mathbf{2}\_{j,k,n} \right) \times 0.5 \right)^{1/3} \tag{27}
$$

$$A \text{llu}\_{j,k,n} = \eta \text{llu}\_{j,k,n} / \left(\text{l1} \times \Delta T \text{llu}\_{j,k,n}\right) \tag{28}$$

$$\sum\_{n} zh\iota\_{j,k,n} \le 1\tag{29}$$

$$
gamma\_{j,k,n} \le zh\mu\_{j,k,n} \times Q\_{\text{max}}\tag{30}
$$

Equations (31)–(37) are employed to determine the size of the heaters at the end of cold process streams. As mentioned in the superstructure section, the number of inter stages is one more than that of inner stages, so the exchange for this stage is expressed with the subscript '0':

$$f \text{cs}k\_{j,0,n}(t \text{count}\_j - t \dot{t}\_{j,m}) = q \text{ln}\_{j,0,n} \quad m = 1 \tag{31}$$

$$
\varepsilon dt \mathbf{1}\_{j,0,n} \le t \mathbf{h} u\_{j,0,n} - \mathbf{t} \mathbf{out}\_j + T\_{\max} (1 - z h u\_{j,0,n}) \tag{32}
$$

$$\text{s.tth2}\_{j,0,n} \le \text{thu}\_{j,0,n} - \text{tf}\_{j,n} + T\_{\text{max}}(1 - zh\mu\_{j,0,n}) \quad m = 1\tag{33}$$

$$
\Delta T h \mu\_{j,0,n} = \left( d th \mathbf{1}\_{j,0,n} \times d th \mathbf{2}\_{j,0,n} \times \left( d th \mathbf{1}\_{j,0,n} + d th \mathbf{2}\_{j,0,n} \right) \times 0.5 \right)^{1/3} \tag{34}
$$

$$\text{Alu}\_{j,0,n} = \text{qhu}\_{j,0,n} / (h1 \times \Delta Thu\_{j,0,n}) \tag{35}$$

$$\sum\_{n} zh\mu\_{j,0,n} \le 1\tag{36}$$

$$
gamma\_{j,0,n} \le zh\mu\_{j,0,n} \times Q\_{\text{max}}\tag{37}$$

Equations (38)–(44) present the calculation of heat load *qci* and heat transfer area *Acui* of the cooler at *i*th stream end. *tcuin* and *tcuout* are the specified inlet and outlet temperatures of cold utility. *zcui* denotes the existence of a cooler:

$$FH\_i(ti\_{i,m} - thout\_i) = qc\_i \quad m = M \tag{38}$$

$$\mathbf{Acu}\_{i} = q\mathbf{c}\_{i}/(\mathbf{h} \times \Delta T \mathbf{c}u\_{i})\tag{39}$$

$$\text{l.} \,\text{tcx} \mathbf{1}\_i \le \text{ti}\_{i,m} - \text{tcx} \,\text{out} + T\_{\text{max}} \left(1 - \text{zcx}\_i\right) \quad m = M \tag{40}$$

$$\text{dtcu2}\_{i} \le \text{thout}\_{i} - \text{tcuin} + T\_{\text{max}}(\mathbf{1} - z\mathbf{c}u\_{i}) \tag{41}$$

$$
\Delta Tcu\_i = \left(dtcu1\_i \times dtcu2\_i \times \left(dtcu1\_i + dtcu2\_i\right) \times 0.5\right)^{1/3} \tag{42}
$$

$$
\mathfrak{qc}\_i \le zc u\_i \rtimes \mathbb{Q}\_{\max} \tag{43}
$$
