Mathematical programming synthesis of non-isothermal water networks by using a compact/reduced superstructure and an MINLP model

Abstract

The synthesis problems of non-isothermal water networks, combining heat exchanger network and water network (WN), usually consist of a significant number of constraints and variables, namely, flow rates, contaminant concentrations, temperatures and a large number of non-linear terms. In most cases, solving medium and large-scale synthesis problems is computationally too expensive and challenging. In order to circumvent that problem, we propose a compact superstructure and mixed-integer non-linear programming model for the simultaneous synthesis of non-isothermal WNs. The proposed superstructure includes heat integration stages enabling direct and indirect heat exchanges with a manageable number of hot and cold streams. This reduces the models size enabling easier solutions of the synthesis problems using local solvers. In addition, a superstructure reduction strategy is proposed making the superstructure flexible and adaptable for different types of problems, namely, pinched and threshold, and providing additional reduction of connections within the proposed superstructure. The proposed model is solved using a two-step solution strategy including initialisation and design steps. The model is applied to the examples of different complexities including single and multiple contaminant problems, and water-using and wastewater treatment units. Using the proposed iterative strategy, the improved locally optimal solutions are identified for most examples, minimising the total annual cost of the overall network.

Appendices

Appendix 1

Simultaneous optimisation and heat integration model (M2)

The minimum consumption of hot utility (qhu) is constrained by inequalities Eqs. (31), (32) according to the pinch location method (Duran and Grossmann 1986). Minimum consumption of cold utility (qcu) is defined by the global energy balance given by Eq. (33).

$$ {\text{qhu}} \ge \left( {\sum\limits_{{j \in {\text{CP}}}} {{{fc}}_{j} \cdot \left( {\hbox{max} \left( {0,\;{\text{tcout}}_{j} - \left( {{\text{thin}}_{{i^{'} }} - {\text{HRAT}}} \right)} \right) - \hbox{max} \left( {0,\;{\text{tcin}}_{j} - \left( {{\text{thin}}_{{i^{'} }} - {\text{HRAT}}} \right)} \right)} \right)} } \right) - \left( {\sum\limits_{{i \in {\text{HP}}}} {{{fh}}_{i} \cdot \left( {\hbox{max} \left( {0,\;{\text{thin}}_{i} - {\text{thin}}_{{i^{'} }} } \right) - \hbox{max} \left( {0,\;{\text{thout}}_{i} - {\text{thin}}_{{i^{'} }} } \right)} \right)} } \right)\quad \forall i^{'} \in {\text{HP}}, $$

(31)

$$ {\text{qhu}} \ge \left( {\sum\limits_{{j \in {\text{CP}}}} {{{fc}}_{j} } \cdot \left( {\hbox{max} \left( {0,\;{\text{tcout}}_{j} - {\text{tcin}}_{{j^{'} }} } \right)} \right) - \hbox{max} \left( {0,\;{\text{tcin}}_{j} - {\text{tcin}}_{{j^{'} }} } \right)} \right) - \left( {\sum\limits_{{i \in {\text{HP}}}} {{{fh}}_{i} \cdot \left( {\hbox{max} \left( {0,\;{\text{thin}}_{i} - \left( {{\text{tcin}}_{{j^{'} }} + {\text{HRAT}}} \right)} \right) - \hbox{max} \left( {0,\;{\text{thout}}_{i} - \left( {{\text{tcin}}_{{j^{'} }} + {\text{HRAT}}} \right)} \right)} \right)} } \right)\quad \forall j^{'} \in {\text{CP}}, $$

(32)

$$ {\text{qhu}} + \sum\limits_{{i \in {\text{HP}}}} {{\text{ech}}_{i} = {\text{qcu}} + \sum\limits_{{j \in {\text{CP}}}} {{\text{ecc}}_{j} } } . $$

(33)

Equations (34) and (35) give heat content of hot and cold streams.

$$ {\text{ech}}_{i} = {{fh}}_{i} \cdot \left( {{\text{thin}}_{i} - {\text{thout}}_{i} } \right)\quad \forall i \in {\text{HP,}} $$

(34)

$$ {\text{ecc}}_{j} = {{fc}}_{j} \cdot \left( {{\text{tcout}}_{j} - {\text{tcin}}_{j} } \right)\quad \forall j \in {\text{HC}} . $$

(35)

Within the model M1, flow rates of the process water streams are unknown as well as the inlet/outlet temperatures of the hot/cold streams. Accordingly, discontinuous derivatives appear in Eqs. (31) and (32) making the NLP/MINLP models (M1–2) difficult to solve. In order to circumvent this problem, a smooth approximation of the max operator was used (Biegler et al. 1997).

HEN synthesis model (M3)

The following Eqs. (36)–(56) were included in the modified HEN synthesis model proposed by Yee and Grossmann (1990):

Total energy exchanged by hot stream i and cold stream j

$$ {{fh}}_{i} \cdot \left( {{\text{thin}}_{i} - {\text{thout}}_{i} } \right) = \sum\limits_{{j \in {\text{CP}}}} {\sum\limits_{{k \in {\text{ST}}}} {q_{i,j,k} + {\text{qc}}_{i} } } \quad i \in {\text{HP,}} $$

(36)

$$ {{fc}}_{j} \cdot \left( {{\text{tcout}}_{j} - {\text{tcin}}_{j} } \right) = \sum\limits_{{i \in {\text{HP}}}} {\sum\limits_{{k \in {\text{ST}}}} {q_{i,j,k} + {\text{qh}}_{j} } } \quad j \in {\text{CP}} . $$

(37)

Energy exchanged by hot stream i and cold stream j in stage k

$$ {{fh}}_{i} \cdot \left( {{\text{th}}_{i,k} - {\text{th}}_{i,k + 1} } \right) = \sum\limits_{{j \in {\text{CP}}}} {q_{i,j,k} } \quad i \in {\text{HP}},\quad k \in {\text{ST,}} $$

(38)

$$ {{fc}}_{j} \cdot \left( {{\text{tc}}_{j,k} - {\text{tc}}_{j,k + 1} } \right) = \sum\limits_{{i \in {\text{HP}}}} {q_{i,j,k} } \quad j \in {\text{CP}},\quad k \in {\text{ST}} . $$

(39)

Energy exchanged by hot stream i with the cold utility and cold stream j with the hot utility

$$ {{fh}}_{i} \cdot \left( {{\text{th}}_{{{\text{i}},{\text{NOK}} + 1}} - {\text{thout}}_{i} } \right) = {\text{qc}}_{i} \quad i \in {\text{HP,}} $$

(40)

$$ {{fc}}_{j} \cdot \left( {{\text{tcout}}_{j} - t_{i,1} } \right) = {\text{qh}}_{j} \quad j \in {\text{CP}} . $$

(41)

Supply temperature of hot and cold streams i and j

$$ {\text{thin}}_{\text{i}} = {\text{th}}_{i,1} \quad i \in {\text{HP,}} $$

(42)

$$ {\text{tcin}}_{j} = {\text{tc}}_{{{{j}},{\text{NOK}} + 1}} \quad j \in {\text{CP}} . $$

(43)

Feasibilities of temperatures across temperature intervals

$$ {\text{th}}_{i,k} \ge {\text{th}}_{i,k + 1} \quad i \in {\text{HP}},\quad k \in {\text{ST,}} $$

(44)

$$ {\text{tc}}_{j,k} \ge {\text{tc}}_{j,k + 1} \quad j \in {\text{CP}}, \quad k \in {\text{ST,}} $$

(45)

$$ {\text{th}}_{{{{i}},{\text{NOK}} + 1}} \ge {\text{thout}}_{i} \quad i \in {\text{HP,}} $$

(46)

$$ {\text{tcout}}_{j} \ge {\text{tc}}_{j,1} \quad j \in {\text{CP}} . $$

(47)

Logical constraints for heat loads

$$ q_{i,j,k} - \hbox{min} \left( {{\text{ech}}_{i}^{\text{U}} ,\;{\text{ecc}}_{j}^{\text{U}} } \right) \cdot z_{i,j,k} \le 0\quad i \in {\text{HP}}, \quad j \in {\text{CP}},\quad k \in {\text{ST,}} $$

(48)

$$ {\text{qc}}_{i} - {\text{ech}}_{i}^{\text{U}} \cdot {\text{zcu}}_{i} \le 0\quad i \in {\text{HP,}} $$

(49)

$$ {\text{qh}}_{j} - {\text{ecc}}_{j}^{\text{U}} \cdot {\text{zhu}}_{j} \le 0\quad j \in {\text{CP}} . $$

(50)

Logical constraints for temperature differences

$$ \Delta t_{i,j,k} \le {\text{th}}_{i,k} - {\text{tc}}_{j,k} + \varGamma \cdot \left( {1 - z_{i,j,k} } \right)\quad i \in {\text{HP}},\quad j \in {\text{CP}}, \quad k \in {\text{ST,}} $$

(51)

$$ \Delta t_{i,j,k + 1} \le {\text{th}}_{i,k + 1} - {\text{tc}}_{j,k + 1} + \varGamma \cdot \left( {1 - z_{i,j,k} } \right)\quad i \in {\text{HP}},\quad j \in {\text{CP}},\quad k \in {\text{ST,}} $$

(52)

$$ \Delta {\text{thu}}_{j} \le {\text{thuout}} - {\text{tc}}_{j,1} \quad j \in {\text{CP,}} $$

(53)

$$ \Delta {\text{tcu}}_{i} \le {\text{th}}_{{{{i}},{\text{NOK}} + 1}} - {\text{tcuout}}\quad i \in {\text{HP}} . $$

(54)

Heat loads of hot stream i and cold stream j

$$ {\text{ech}}_{i} = {{fh}}_{i} \cdot \left( {{\text{thin}}_{i} - {\text{thout}}_{i} } \right)\quad \forall i \in {\text{HP,}} $$

(55)

$$ {\text{ecc}}_{j} = {{fc}}_{j} \cdot \left( {{\text{tcout}}_{j} - {\text{tcin}}_{j} } \right)\quad \forall j \in {\text{CP}} . $$

(56)

Connecting equations for the combined models

According to the proposed solutions strategy, as described in detail in section Solution approach, firstly a combined models M1–2 was solved providing initialisation and water/utility bounds, and afterwards a combined model M1–3 for the simultaneous synthesis of non-isothermal WNs is solved. In order to enable a solution of the combined models, appropriate connecting equations have to be defined. Heat capacity flow rates and inlet/outlet temperatures that appear in models M2 and M3 related to the flow rates and temperatures of the process water streams within the model M1 were given as follows.

Connecting equations for hot water streams

$$ {{fh}}_{i} = {\text{FHS}}_{{hs}} \cdot C_{p} \quad \forall {{hs}} \in {\text{HST}},\quad i \in {\text{HP}},\quad i = {{hs}}, $$

(57)

$$ {\text{thin}}_{i} = {\text{THS}}_{{hs}}^{{({\text{in}})}} \quad \forall {{hs}} \in {\text{HST}},\quad i \in {\text{HP}},\quad i = {{hs}}, $$

(58)

$$ {\text{thout}}_{i} = {\text{THS}}_{{hs}}^{{({\text{out}})}} \quad \forall {{hs}} \in {\text{HST}},\quad i \in {\text{HP}},\quad i = {{hs}}. $$

(59)

Connecting equations for cold water streams

$$ {{fc}}_{j} = {\text{FCS}}_{{cs}} \cdot C_{p} \quad \forall {{cs}} \in {\text{CST}},\quad {\kern 1pt} j \in {\text{CP}},\quad j = {{cs}}, $$

(60)

$$ {\text{tcin}}_{j} = {\text{TCS}}_{{cs}}^{{({\text{in}})}} \quad \forall {{cs}} \in {\text{CST}},\quad {\kern 1pt} j \in {\text{CP}},\quad j = {{cs,}} $$

(61)

$$ {\text{tcout}}_{j} = {\text{TCS}}_{{cs}}^{{({\text{out}})}} \quad \forall {{cs}} \in {\text{CST}},\quad j \in {\text{CP}},\quad j = {{cs}} . $$

(62)

Additional constraints

By solving the models (M1–2) an initialisation point can be provided for the models (M1–3), which is solved in the second synthesis step, as well as providing rigorous upper and lower bounds for freshwater and utilities consumption. The upper and lower bounds represent level values (LVs) of variables from the models M1–2. Equation (63) is applied in the case studies involving only one water source. Constraining freshwater consumption on the minimum values can cause significant increase in investment as addressed in Example 4 regarding the pinched problems with multiple water sources.

$$ \sum\limits_{{s \in {\text{SFW}}}} {{\text{FW}}_{s} \le \sum\limits_{{s \in {\text{SFW}}}} {{\text{FW}}_{s}^{\text{LV}} } } \quad |{\text{SFW|}} = 1 , $$

(63)

$$ \sum\limits_{{i \in {\text{HP}}}} {{\text{qc}}_{i} \ge {\text{qcu}}^{{{\text{LV,HRAT}} \le {\text{EMAT}}}} } , $$

(64)

$$ \sum\limits_{{j \in {\text{CP}}}} {{\text{qh}}_{j} \ge {\text{qhu}}^{{{\text{LV,HRAT}} \le {\text{EMAT}}}} } , $$

(65)

$$ \sum\limits_{{i \in {\text{HP}}}} {{\text{qc}}_{i} \le {\text{qcu}}^{{{\text{LV,HRAT}} \ge {\text{EMAT}}}} } , $$

(66)

$$ \sum\limits_{{j \in {\text{CP}}}} {{\text{qh}}_{j} \le {\text{qhu}}^{{{\text{LV}},{\text{HRAT}} \ge {\text{EMAT}}}} } , $$

(67)

$$ {\text{THS}}_{{hs}}^{{({\text{in}})}} \ge {\text{THS}}_{{hs}}^{{({\text{out)}}}} \quad {{hs}} \in {\text{HST,}} $$

(68)

$$ {\text{TCS}}_{{cs}}^{{({\text{out}})}} \ge {\text{TCS}}_{{cs}}^{{({\text{in}})}} \quad {{cs}} \in {\text{CST}} . $$

(69)

Heat exchanger related parameters

Heat exchange areas

$$ A_{i,j,k} = \frac{{q_{i,j,k} }}{{U_{i,j} \cdot {\text{LMTD}}_{i,j,k} }}\quad i \in {\text{HP}}, \quad j \in {\text{CP}},\quad k \in {\text{ST}}, $$

(70)

$$ A_{{{{i}},{\text{CU}}}} = \frac{{{\text{qc}}_{i} }}{{U_{{{{i}},{\text{CU}}}} \cdot {\text{LMTD}}_{{{{i}},{\text{CU}}}} }}\quad i \in {\text{HP,}} $$

(71)

$$ A_{{{{j}},{\text{HU}}}} = \frac{{{\text{qh}}_{j} }}{{U_{{{{j}},{\text{HU}}}} \cdot {\text{LMTD}}_{{{{j}},{\text{HU}}}} }}\quad j \in {\text{CP}} . $$

(72)

Temperature-driving forces

$$ {\text{LMTD}}_{i,j,k} = \left[ {\left( {\Delta t_{i,j,k} \cdot \Delta t_{i,j,k + 1} } \right) \cdot \frac{{(\Delta t_{i,j,k} + \Delta t_{i,j,k + 1} )}}{2}} \right]^{1/3} \quad i \in {\text{HP}}, \quad j \in {\text{CP}}, \quad k \in {\text{ST,}} $$

(73)

$$ {\text{LMTD}}_{{{{i}},{\text{CU}}}} = \left[ {\left( {\Delta {\text{tcu}}_{i} \cdot \left( {{\text{thout}}_{i} - {\text{tcuin}}} \right)} \right) \cdot \frac{{(\Delta t{\text{cu}}_{i} + {\text{thout}}_{i} - {\text{tcuin}})}}{2}} \right]^{1/3} \quad i \in {\text{HP,}} $$

(74)

$$ {\text{LMTD}}_{{{{j}},{\text{HU}}}} = \left[ {\left( {\Delta {\text{thu}}_{j} \cdot \left( {{\text{thuin}} - {\text{tcout}}_{j} } \right)} \right) \cdot \frac{{(\Delta {\text{thu}}_{j} + {\text{thuin}} - {\text{tcout}}_{j} )}}{2}} \right]^{1/3} \quad j \in {\text{CP}} . $$

(75)

Heat transfer coefficients

$$ \frac{1}{{U_{i,j} }} = \frac{1}{{h_{i} }} + \frac{1}{{h_{j} }}\quad i \in {\text{HP}}, \quad j \in {\text{CP}}, $$

(76)

$$ \frac{1}{{U_{{{{i}},{\text{CU}}}} }} = \frac{1}{{h_{i} }} + \frac{1}{{h_{\text{CU}} }}\quad i \in {\text{HP,}} $$

(77)

$$ \frac{1}{{U_{{{{j}},{\text{HU}}}} }} = \frac{1}{{h_{j} }} + \frac{1}{{h_{\text{HU}} }}\quad j \in {\text{CP}} . $$

(78)

Appendix 2

Variable bounds for the water network model (M1)

Based on the data given as parameters for each example (mass load of contaminants and maximum inlet and outlet contaminant concentration), the maximum and minimum water flowrate required for each process unit can be calculated as follows:

$$ {\text{FPU}}_{p,c}^{(\hbox{max} )} = \frac{{{\text{LPU}}_{p,c} }}{{({\text{xPU}}_{p,c}^{{({\text{out}},\hbox{max} )}} - {\text{xPU}}_{p,c}^{{({\text{in}},\hbox{max} )}} )}}\quad \forall p \in {\text{PU}}, \quad \forall c \in {\text{SC}}, $$

(79)

$$ {\text{FPU}}_{p,c}^{(\hbox{min} )} = \frac{{{\text{LPU}}_{p,c} }}{{({\text{xPU}}_{p,c}^{{({\text{out}},\hbox{max} )}} - 0)}}\quad \forall p \in {\text{PU}}, \quad \forall c \in {\text{SC}}, $$

(80)

$$ {\text{FPU}}_{p}^{(\hbox{max} )} = \max_{{c \in {\text{SC}}}} \left( {{\text{FPU}}_{p,c}^{(\hbox{max} )} } \right)\quad \forall p \in {\text{PU,}} $$

(81)

$$ {\text{FPU}}_{p}^{(\hbox{min} )} = \max_{{c \in {\text{SC}}}} \left( {{\text{FPU}}_{p,c}^{(\hbox{min} )} } \right)\quad \forall p \in {\text{PU}} . $$

(82)

Accordingly, if each process unit used freshwater (no water reuse), a process would consume a maximum amount of freshwater.

$$ {\text{FW}}^{{(\max )}} ~ = \sum\limits_{{p \in {\text{PU}}}} {{\text{FPU}}_{p}^{{(\max )}} } \quad \forall s \in {\text{SFW}}. $$

(83)

Equations (79)–(83) are used for defining upper bounds for flow rates for each water stream as follows:

$$ {\text{FW}}_{s}^{\text{U}} = {\text{FW}}^{{({ \hbox{max} })}} \quad \forall s \in {\text{SFW,}} $$

(84)

$$ {\text{FIHS}}_{{{{s}},{{hs}}}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall s \in {\text{SFW}},\quad \forall {{hs}} \in {\text{HST,}} $$

(85)

$$ {\text{FICS}}_{{{{s}},{{cs}}}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall s \in {\text{SFW}}, \quad \forall {{cs}} \in {\text{CST,}} $$

(86)

$$ {\text{FIP}}_{s,p}^{\text{U}} = {\text{FPU}}_{p}^{{ ( {\text{max)}}}} \quad \forall s \in {\text{SFW}}, \quad \forall p \in {\text{PU,}} $$

(87)

$$ {\text{FIE}}_{s}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall s \in {\text{SFW,}} $$

(88)

$$ {\text{FIT}}_{s,t}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall s \in {\text{SFW}}, \quad \forall t \in {\text{TU,}} $$

(89)

$$ {\text{FHS}}_{{hs}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{hs}} \in {\text{HST,}} $$

(90)

$$ {\text{FCS}}_{{cs}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{cs}} \in {\text{CST,}} $$

(91)

$$ {\text{FRHS}}_{{{{hs,hs}}^{'} }}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{hs}} \in {\text{HST}},\quad \forall {{hs}}^{'} \in {\text{HST,}} $$

(92)

$$ {\text{FRCS}}_{{{{cs}},{{cs}}^{'} }}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{cs}} \in {\text{CST}}, \quad \forall {{cs}}^{'} \in {\text{CST,}} $$

(93)

$$ {\text{FHSP}}_{{{{hs}},{{p}}}}^{\text{U}} = {\text{FPU}}_{p}^{(\hbox{max} )} \quad \forall {{hs}} \in {\text{HST}},\quad \forall p \in {\text{PU,}} $$

(94)

$$ {\text{FCSP}}_{{{{cs}},{{p}}}}^{\text{U}} = {\text{FPU}}_{p}^{(\hbox{max} )} \quad \forall {{cs}} \in {\text{CST}}, \quad \forall p \in {\text{PU,}} $$

(95)

$$ {\text{FHST}}_{{{{hs}},{{t}}}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{hs}} \in {\text{HST}},\quad \forall t \in {\text{TU,}} $$

(96)

$$ {\text{FCST}}_{{{{cs}},{{t}}}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{cs}} \in {\text{CST}}, \quad \forall t \in {\text{TU,}} $$

(97)

$$ {\text{FHSE}}_{{hs}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{hs}} \in {\text{HST,}} $$

(98)

$$ {\text{FCSE}}_{{cs}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{cs}} \in {\text{CST,}} $$

(99)

$$ {\text{FHSCS}}_{{{{hs}},{{cs}}}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{hs}} \in {\text{HST}}, \quad \forall {{cs}} \in {\text{CST,}} $$

(100)

$$ {\text{FCSHS}}_{{{{cs}},{{hs}}}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall {{cs}} \in {\text{CST}},\quad \forall {{hs}} \in {\text{HST,}} $$

(101)

$$ {\text{FPU}}_{p}^{{({\text{in}}),{\text{U}}}} = {\text{FPU}}_{p}^{{({ \hbox{max} })}} \quad \forall p \in {\text{PU,}} $$

(102)

$$ {\text{FPU}}_{p}^{{({\text{in}}),{\text{L}}}} = {\text{FPU}}_{p}^{{({ \hbox{min} })}} \quad \forall p \in {\text{PU,}} $$

(103)

$$ {\text{FPU}}_{p}^{{({\text{out}}),{\text{U}}}} = {\text{FPU}}_{p}^{{({ \hbox{max} })}} \quad \forall p \in {\text{PU,}} $$

(104)

$$ {\text{FPU}}_{p}^{{({\text{out}}),{\text{L}}}} = {\text{FPU}}_{p}^{{({ \hbox{min} })}} \quad \forall p \in {\text{PU,}} $$

(105)

$$ {\text{FP}}_{{p^{'} ,p}}^{\text{U}} = {\text{FPU}}_{{p^{'} }}^{{ ( {\text{max)}}}} \quad \forall p \in {\text{PU}},\quad \forall p^{'} \in {\text{PU}},\quad {\text{FPU}}_{{p^{'} }}^{(\hbox{max} )} \le {\text{FPU}}_{p}^{(\hbox{max} )} , $$

(106)

$$ {\text{FP}}_{{p^{'} ,p}}^{\text{U}} = {\text{FPU}}_{p}^{{ ( {\text{max)}}}} \quad \forall p \in {\text{PU}},\quad \forall p^{'} \in {\text{PU}},\quad {\text{FPU}}_{{p^{'} }}^{(\hbox{max} )} \ge {\text{FPU}}_{p}^{(\hbox{max} )} , $$

(107)

$$ {\text{FPHS}}_{{{{p}},{{hs}}}}^{\text{U}} = {\text{FPU}}_{p}^{(\hbox{max} )} \quad \forall p \in {\text{PU}},\quad \forall {{hs}} \in {\text{HST,}} $$

(108)

$$ {\text{FPCS}}_{{{{p}},{{cs}}}}^{\text{U}} = {\text{FPU}}_{p}^{(\hbox{max} )} \quad \forall p \in {\text{PU}},\quad \forall {{cs}} \in {\text{CST,}} $$

(109)

$$ {\text{FPE}}_{p}^{\text{U}} = {\text{FPU}}_{p}^{{ ( {\text{max)}}}} \quad \forall p \in {\text{PU,}} $$

(110)

$$ {\text{FPT}}_{p,t}^{\text{U}} = {\text{FPU}}_{p}^{{ ( {\text{max)}}}} \quad \forall p \in {\text{PU}}, \quad \forall t \in {\text{TU,}} $$

(111)

$$ {\text{FTU}}_{t}^{{({\text{in}}),{\text{U}}}} = {\text{FW}}^{(\hbox{max} )} \quad \forall t \in {\text{TU,}} $$

(112)

$$ {\text{FTU}}_{t}^{{({\text{out}}),{\text{U}}}} = {\text{FW}}^{(\hbox{max} )} \quad \forall t \in {\text{TU,}} $$

(113)

$$ {\text{FT}}_{{t,t^{'} }}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall t \in {\text{TU}}, \quad \forall t^{'} \in {\text{TU}}, $$

(114)

$$ {\text{FTE}}_{t}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall t \in {\text{TU,}} $$

(115)

$$ {\text{FTHS}}_{{{{t}},{{hs}}}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall t \in {\text{TU}}, \quad \forall {{hs}} \in {\text{HST,}} $$

(116)

$$ {\text{FTCS}}_{{{{t}},{{cs}}}}^{\text{U}} = {\text{FW}}^{(\hbox{max} )} \quad \forall t \in {\text{TU}}, \quad \forall {{cs}} \in {\text{CST,}} $$

(117)

$$ {\text{FTP}}_{t,p}^{\text{U}} = {\text{FPU}}_{p}^{{ ( {\text{max)}}}} \quad \forall t \in {\text{TU}},\quad \forall p \in {\text{PU,}} $$

(118)

$$ {\text{FWW}}^{{({\text{out}}),{\text{U}}}} = {\text{FW}}^{(\hbox{max} )} . $$

(119)

The upper and lower bounds on temperatures of hot and cold streams entering and leaving cooling and heating stages are based on the maximum and minimum temperature within the WN knowing the temperatures of process units, treatment units, freshwater and wastewater stream discharged into the environment.

$$ {\text{TFW}}^{(\hbox{min} )} = \mathop {\hbox{min} }\limits_{{s \in {\text{SFW}}}} \left( {{\text{TFW}}_{s} } \right), $$

(120)

$$ {\text{TPU}}_{\text{in}}^{(\hbox{min} )} = \mathop {\hbox{min} }\limits_{{p \in {\text{PU}}}} \left( {{\text{TPU}}_{p}^{(in)} } \right), $$

(121)

$$ {\text{TPU}}_{\text{in}}^{(\hbox{max} )} = \mathop {\hbox{max} }\limits_{{p \in {\text{PU}}}} \left( {{\text{TPU}}_{p}^{{({\text{in}})}} } \right), $$

(122)

$$ {\text{TPU}}_{\text{out}}^{(\hbox{min} )} = \mathop {\hbox{min} }\limits_{{p \in {\text{PU}}}} \left( {{\text{TPU}}_{p}^{{({\text{out}})}} } \right), $$

(123)

$$ {\text{TPU}}_{\text{out}}^{(\hbox{max} )} = \mathop {\hbox{max} }\limits_{{p \in {\text{PU}}}} \left( {{\text{TPU}}_{p}^{{({\text{out}})}} } \right), $$

(124)

$$ T_{\text{w}}^{(\hbox{min} )} = \hbox{min} \left( {{\text{TFW}}^{(\hbox{min} )} ,\; {\text{TPU}}_{\text{in}}^{(\hbox{min} )} ,\;{\text{TPU}}_{\text{out}}^{(\hbox{min} )} ,\; {\text{TWW}}^{{({\text{out}})}} } \right), $$

(125)

$$ T_{\text{w}}^{(\hbox{max} )} = \hbox{max} \left( {{\text{TPU}}_{\text{in}}^{(\hbox{max} )} ,\; {\text{TPU}}_{\text{out}}^{(\hbox{max} )} } \right). $$

(126)

The identified minimum and maximum temperatures of water streams within the network are used as lower and upper bounds for hot and cold water streams as follows:

$$ {\text{THS}}_{{hs}}^{{({\text{in}}),{\text{L}}}} = T_{\text{w}}^{(\hbox{min} )} \quad \forall {{hs}} \in {\text{HST,}} $$

(127)

$$ {\text{THS}}_{{hs}}^{{({\text{in}}),{\text{U}}}} = T_{\text{w}}^{(\hbox{max} )} \quad \forall {{hs}} \in {\text{HST,}} $$

(128)

$$ {\text{THS}}_{{hs}}^{{({\text{out}}),{\text{L}}}} = T_{\text{w}}^{(\hbox{min} )} \quad \forall {{hs}} \in {\text{HST,}} $$

(129)

$$ {\text{THS}}_{{hs}}^{{({\text{out}}),{\text{U}}}} = T_{\text{w}}^{(\hbox{max} )} \quad \forall {{hs}} \in {\text{HST,}} $$

(130)

$$ {\text{TCS}}_{{cs}}^{{({\text{in}}),{\text{L}}}} = T_{\text{w}}^{(\hbox{min} )} \quad \forall {{cs}} \in {\text{CST,}} $$

(131)

$$ {\text{TCS}}_{{cs}}^{{({\text{in}}),{\text{U}}}} = T_{\text{w}}^{(\hbox{max} )} \quad \forall {{cs}} \in {\text{CST,}} $$

(132)

$$ {\text{TCS}}_{{cs}}^{{({\text{out}}),{\text{L}}}} = T_{\text{w}}^{(\hbox{min} )} \quad \forall {{cs}} \in {\text{CST,}} $$

(133)

$$ {\text{TCS}}_{{cs}}^{{({\text{out}}),{\text{U}}}} = T_{\text{w}}^{(\hbox{max} )} \quad \forall {{cs}} \in {\text{CST}} . $$

(134)

For the given concentration of the contaminant within freshwater streams and maximum inlet/outlet contaminant concentrations within the process units, upper and lower bounds can be identified for any given process water stream:

$$ {\text{xPU}}_{p,c}^{{({\text{in}}),{\text{U}}}} = {\text{xPU}}_{p,c}^{{({\text{in}},\hbox{max} )}} \quad \forall p \in {\text{PU}},\quad \forall {\text{c}} \in {\text{SC,}} $$

(135)

$$ {\text{xPU}}_{p,c}^{{({\text{out}}),{\text{U}}}} = {\text{xPU}}_{p,c}^{{({\text{out,max}})}} \quad \forall p \in {\text{PU}}, \quad \forall c \in {\text{SC,}} $$

(136)

$$ {\text{xPU}}_{p,c}^{{({\text{out}}),{\text{L}}}} = \frac{{{\text{LPU}}_{p,j} }}{{{\text{FPU}}_{p}^{\hbox{max} } }}\quad \forall p \in {\text{PU}}, \quad \forall c \in {\text{SC,}} $$

(137)

$$ {\text{xTU}}_{t,c}^{{({\text{in}}),{\text{U}}}} = \mathop {\hbox{max} }\limits_{{p \in {\text{PU}}}} \left( {{\text{xPU}}_{p,c}^{{({\text{out,max}})}} } \right)\quad \forall t \in {\text{TU}}, \quad \forall c \in {\text{SC,}} $$

(138)

$$ {\text{xTU}}_{t,c}^{{({\text{out}}),{\text{U}}}} = {\text{xTU}}_{t,c}^{{({\text{in}}),{\text{U}}}} \cdot \left( {1 - {\text{RR}}_{t,c} } \right)\quad \forall t \in {\text{TU}},\quad \forall c \in {\text{SC,}} $$

(139)

$$ {\text{xFHS}}_{{{{hs}},{{c}}}}^{\text{U}} = \mathop {\hbox{max} }\limits_{{p \in {\text{PU}}}} \left( {{\text{xPU}}_{p,c}^{{({\text{out}},\hbox{max} )}} } \right)\quad \forall {{hs}} \in {\text{HST}},\quad \forall c \in {\text{SC,}} $$

(140)

$$ {\text{xFHS}}_{{{{hs}},{{c}}}}^{\text{L}} = \mathop {\hbox{min} }\limits_{{s \in {\text{SFW}}}} \left( {{\text{xFW}}_{s,c} } \right)\quad \forall {{hs}} \in {\text{HST}}, \quad \forall c \in {\text{SC,}} $$

(141)

$$ {\text{xFCS}}_{{{{cs}},{{c}}}}^{\text{U}} = \mathop {\hbox{max} }\limits_{{p \in {\text{PU}}}} \left( {{\text{xPU}}_{p,c}^{{({\text{out}},\hbox{max} )}} } \right)\quad \forall {{cs}} \in {\text{CST}},\quad \forall c \in {\text{SC,}} $$

(142)

$$ {\text{xFHS}}_{{{{cs}},{{c}}}}^{\text{L}} = \mathop {\hbox{min} }\limits_{{s \in {\text{SFW}}}} \left( {{\text{xFW}}_{s,c} } \right)\quad \forall {{cs}} \in {\text{CST}}, \quad \forall c \in {\text{SC,}} $$

(143)

$$ {\text{xWW}}_{c}^{{({\text{out}})}} = {\text{xWW}}_{c}^{{({\text{out}}),\hbox{max} }} \quad \forall {{c}} \in {\text{CS}},\quad |{\text{TU}}| > 0, $$

(144)

$$ {\text{xWW}}_{c}^{{({\text{out}})}} = \mathop {\hbox{max} }\limits_{{p \in {\text{PU}}}} \left( {{\text{xPU}}_{p,c}^{{({\text{out,max}})}} } \right)\quad \forall {{c}} \in {\text{CS}}, \quad |{\text{TU}}| = 0. $$

(145)

Variables bounds for the models M2 and M3

Upper and lower bounds for the heat integration (M2) and HEN synthesis (M3) models are defined in the following section. The upper bound for heat load required for cooling of hot stream i [Eq. (146)] and heating of cold stream j [Eq. (147)] can be calculated based on the upper bounds for the heat capacity flow rate and upper/lower bound for hot stream i and cold stream j temperatures as given by Eqs. (148)–(157)

$$ {\text{ech}}_{i}^{\text{U}} = {{fh}}_{i}^{\text{U}} \cdot \left( {{\text{thin}}_{i}^{\text{U}} - {\text{thout}}_{i}^{\text{L}} } \right)\quad \forall i \in {\text{HP,}} $$

(146)

$$ {\text{ecc}}_{j}^{\text{U}} = {{fc}}_{j}^{\text{U}} \cdot \left( {{\text{tcout}}_{j}^{\text{U}} - {\text{tcin}}_{j}^{\text{L}} } \right)\quad \forall j \in {\text{CP,}} $$

(147)

$$ {{fh}}_{i}^{\text{U}} = {\text{FHS}}_{{hs}}^{\text{U}} \quad i \in {\text{HP}}, \quad {{hs}} \in {\text{HST}},\quad i = {{hs,}} $$

(148)

$$ {{fc}}_{j}^{\text{U}} = {\text{FCS}}_{{cs}}^{\text{U}} \quad j \in {\text{CP}},\quad {{cs}} \in {\text{CST}},\quad j = {{cs,}} $$

(149)

$$ {\text{thin}}_{i}^{\text{U}} = {\text{THS}}_{{hs}}^{{ ( {\text{in),U}}}} \quad i \in {\text{HP}},\quad {{hs}} \in {\text{HST}},\quad i = {{hs}}, $$

(150)

$$ {\text{thin}}_{i}^{\text{L}} = {\text{THS}}_{{hs}}^{{ ( {\text{in),L}}}} \quad i \in {\text{HP}},\quad {{hs}} \in {\text{HST}},\quad i = {{hs,}} $$

(151)

$$ {\text{thout}}_{i}^{\text{U}} = {\text{THS}}_{{hs}}^{{ ( {\text{out),U}}}} \quad i \in {\text{HP}},\quad {{hs}} \in {\text{HST}},\quad i = {{hs,}} $$

(152)

$$ {\text{thout}}_{i}^{\text{L}} = {\text{THS}}_{{hs}}^{{ ( {\text{out),L}}}} \quad i \in {\text{HP}},\quad {{hs}} \in {\text{HST}},\quad i = {{hs,}} $$

(153)

$$ {\text{tcin}}_{j}^{\text{U}} = {\text{TCS}}_{{cs}}^{{ ( {\text{in),U}}}} \quad j \in {\text{CP}},\quad {{cs}} \in {\text{CST}},\quad j = {{cs,}} $$

(154)

$$ {\text{tcin}}_{j}^{\text{L}} = {\text{TCS}}_{{cs}}^{{ ( {\text{in),L}}}} \quad j \in {\text{CP}},\quad {{cs}} \in {\text{CST}},\quad j = {{cs,}} $$

(155)

$$ {\text{tcout}}_{j}^{\text{U}} = {\text{TCS}}_{{cs}}^{{ ( {\text{out),U}}}} \quad j \in {\text{CP}},\quad {{cs}} \in {\text{CST}},\quad j = {{cs,}} $$

(156)

$$ {\text{tcout}}_{j}^{\text{L}} = {\text{TCS}}_{{cs}}^{{ ( {\text{out),L}}}} \quad j \in {\text{CP}},\quad {{cs}} \in {\text{CST}},\quad j = {{cs}} . $$

(157)

The minimum approach temperatures and upper/lower bounds for temperatures of the streams across the temperature interval k are defined as follows:

$$ \Delta t_{i,j,k} \ge {\text{EMAT}}\quad \forall i \in {\text{HP}},\quad \forall j \in {\text{CP}},\quad \forall k \in {\text{ST}} $$

(158)

$$ \Delta {\text{thu}}_{j} \ge {\text{EMAT}}\quad \forall j \in {\text{CP,}} $$

(159)

$$ \Delta {\text{tcu}}_{i} \ge {\text{EMAT}}\quad \forall i \in {\text{HP,}} $$

(160)

$$ {\text{th}}_{i,k}^{\text{U}} = {\text{thin}}_{i}^{\text{U}} \quad i \in {\text{HP}},\quad k \in {\text{ST,}} $$

(161)

$$ {\text{th}}_{i,k}^{\text{U}} = {\text{thout}}_{i}^{\text{U}} \quad i \in {\text{HP}},\quad k \in {\text{ST,}} $$

(162)

$$ {\text{tc}}_{j,k}^{\text{U}} = {\text{tcout}}_{j}^{\text{U}} \quad j \in {\text{CP}},\quad k \in {\text{ST,}} $$

(163)

$$ {\text{tc}}_{j,k}^{\text{L}} = {\text{tcin}}_{j}^{\text{L}} \quad j \in {\text{CP}},\quad k \in {\text{ST}} . $$

(164)

The upper bound for driving forces is set to Γ = $T_{\text{w}}^{(\hbox{max} )} .$