# A Mathematical Programming Method for Optimizing the Single-Contaminant Regeneration Heat-Integrated Water Networks

• Renjie Shen
• Yu Zhang
• Jian Ma
Review Article

## Abstract

A superstructure-based mathematical model is established for single-contaminant heat-integrated water networks (HIWN), and the water loss of both wastewater regeneration recycling and regeneration reuse is considered. Furthermore, a sequential optimization procedure is established to achieve multi-objective optimization. The loss rate of water is taken into the mathematical model as there must have water loss in regeneration process, so that the optimization would be more realistic. Two cases are optimized using proposed model with regeneration recycling and regeneration reuse considered, respectively; the results show the differences between these two modes. Compared with the optimization method in the literature, the proposed method is more realistic.

## Keywords

Heat-integrated water networks Regeneration recycling Regeneration reuse Mathematical programming

## Nomenclature

a

The partition coefficient of contaminant mass load between two subunits

B

The cost coefficient of regenerated water

c

The contaminant concentration of streams

Cp

Specific heat at constant pressure (kJ (kg °C)−1)

F

Flow rate of stream

Lr

Loss rate of water in regeneration process

M

The contaminant load of water using unit (g h−1)

N

The annual running time (h)

P

The set of water units before decomposition

PR

Proportion of annual running time

Q

Heat (kW)

r

The removal rate of the contaminant in the regeneration unit R (g h−1)

R

Regeneration unit

T

Temperature (°C)

u

The unit cost ($) U Constant related to the number of connections X The set of water units after decomposition y Binary variables for network connection Z The annual total operating cost of system ($ a−1)

ξ1

Constant which related to the freshwater consumption

ξ2

Constant which related to the amount of regenerated water

## Superscript

com

Parameters related to decomposition process

D

Discharge of water using unit

in

Inlet of water using unit

min

The minimum value of parameters

max

The maximum value of parameters

out

Outlet of water using unit

s

Definite value

W

Freshwater consumption

## Subscript

C

Set of cold streams

CU

Cooling utility

F

Flow rate of streams

H

Set of hot streams

HU

Heating utility

k

Temperature degrees

K

Temperature intervals

n

Temperature degrees

m

Hot streams in the set H

q

Cold streams in the set C

R

Regeneration unit

uti

Utility consumption

## Indices

b

Water using unit after decomposition

d

Water using unit

i

Water using unit

j

Water using unit

## Introduction

Rapid development of industry and dramatic growth of population lead to increasing environmental burdens, including severe water and energy shortages. As chemical processes involving extensive use of water and energy, it is urgent to save energy and reduce pollution to achieve sustainable development.

In industrial processes, water networks (WN) and heat exchange networks (HEN) co-exist as water is a carrier for both contaminants and energy. To save energy and water as much as possible, the optimization for HEN should also be considered together with that of water networks. As a result, optimization methods for heat-integrated water networks (HIWN) were generated, including both pinch-based conceptual and mathematical programming methods. Pinch-based conceptual methods for HIWN are mainly graphical methods, including source-demand energy composite curves (Savulescu et al. 2002), graphical thermodynamic rule that instruct the mix and spilt on the temperature-heat capacity diagram (Sorin and Savulescu 2004), water energy balance diagram (Leewongtanawit and Kim 2009) and so on. Bagajewicz et al. (1998) proposed a state and space approach to minimize the total annualized cost (TAC) of heat/mass exchange networks. Savulescu et al. (2005a, b) studied the optimization of HIWN using two-dimensional grid method where maximum wastewater reuse was analyzed. Mao et al. (2010) proposed a design method considering the non-isothermal mixing of a HIWN and the concept of segmenting temperature. In addition, they put forward the mixing rules for non-isothermal mixing heat transfer after studying the effects of homogeneous mixing and heterogeneous mixing approach on the utility consumption of a system. Martínez et al. (2011) proposed the temperature-concentration graphical method for designing HIWN. Conceptual design methods are generally divided into two steps: firstly, designing the water network with minimum amount of freshwater and identifying the hot and cold streams in the water network successively and secondly, developing the heat exchange networks and HIWN can be developed.

Graphical method is concise, clear, and easy to be applied for simple networks, while mathematical programming method is rigorous and has wide range of applications. As for the system that is in large scale and multi-contaminant, mathematical programming method performs better. Liao et al. (2007) constructed direct and indirect heat transfer in one transshipment model for the water utilization. Dong et al. (2008) modified state-space method and constructed a mixed integer non-linear programming (MINLP) model for HIWN aiming to minimize total annual operating cost considering all the direct and indirect heat transfer. Bogataj and Bagajewicz (2008) performed simultaneous synthesis and optimization of HIWN through a new superstructure of heat exchange network (HEN) synthesis based on MINLP model. For HIWN design, Liao et al. (2011) introduced a stage-wise HEN superstructure and then a mixed integer linear programming (MILP) model which has less flexibility compared with non-isothermal mixing and freely splitting nature. Boix et al. (2011) proposed a MILP model for solving HIWN and identifying numerous optimal solutions for a fixed number of allocations in network. Hu et al. (2011) proposed three sequential mathematical models and related optimization procedure to optimize single/multiple-contaminant regeneration reuse water networks with process decomposition. Boix et al. (2012) presented a mathematical programming formulation for the design of water networks and HEN based on the two-step methodology. The first step is using a MILP model to solve the water and energy problem, and the second step is improving the best results of the first step with energy integration into the water network. Sahu and Bandyopadhyay (2012) used linear programming (LP) model complemented by concept-based pinch analysis results to target the minimum energy requirements in a heat-integrated fixed flow rate water allocation networks. Ahmetović and Kravanja (2014) extended their formal superstructure and simultaneous optimization model of HIWN involving process-to-process streams, and other streams within the overall network, for heat integration, whose objective was to minimize the total annual network cost. Liu et al. (2014) combined a new water network model with the LP transshipment model where non-isothermal mixing is introduced to improve the energy performance of water network and reduce the complexity. Ahmetović et al. (2014) proposed a new superstructure with WN (water usage, wastewater treatment, and recycling) and HEN (direct and indirect heat exchanges) combined into an overall network and the optimization performed through a MINLP model. Chen et al. (2014) established a mathematical model to illustrate the modified state-space method, in which the water exchange networks (WEN), HEN, and the interactions among them were combined together. Li et al. (2015) considered the temperature limit of non-isothermal mixing rules and established a LP model for water network with wastewater reused directly. Zhao et al. (2015) combined the concentration potentials and LP approach for the minimization of freshwater and regenerated stream flow rates considering the wastewater regeneration recycling. Yan et al. (2016) revised a superstructure by changing the position of heaters and coolers and used a reformed approximated equation for the logarithmic mean temperature difference (LMTD), which simplified non-linear programming (NLP) model. Hong et al. (2016) carried out a MINLP model for simultaneous optimization of HIWN featuring parallel HEN structure, and the model is suitable to both uniform and separate wastewater treatment cases. Liu et al. (2017) presented a methodology for synthesizing inter-plant HIWN in industrial parks considering the coupling of water allocation and heat exchange. Both sequential and simultaneous designs were used to solve this NLP problem, and the sequential design featured more targeting flexibility and lower requirement on solution process.

In previous studies of both conceptual and mathematical programming methods for HIWN, the regeneration reuse of wastewater is generally neglected. In addition, the loss of water in regeneration process has not been considered, as well as its impact on the usage of freshwater and regenerated water consumption. This paper introduces the regeneration of wastewater into mathematical programming model for HIWN, which includes regeneration recycling and regeneration reuse. The loss rate of water in regeneration process is added into the model to make the optimization more realistic. A new superstructure and a new heat exchange model are established based on features of regeneration system, and the freshwater consumption, water quality, water temperature, regenerated water loss, and heat load are considered. Furthermore, five objective functions are considered sequentially to reduce complexity. Two cases are optimized to show the application of the proposed.

## Problem Statement

The problem addressed in this paper can be stated as follows:

A set of water using units that has a certain water quality and temperature is given; the water loss of regeneration process and contaminant concentration after regeneration are specified. Sequential optimization is used as it can achieve the same number of objectives while has reduced complexity than simultaneous optimization, and can also adjust the match based on real situation. The aim is to obtain a HIWN with minimum annual operating cost and least number of network connections.

For the water using unit, the limiting inlet and outlet contaminant concentrations, contaminant load, and operating temperature are assumed to be constant. For regeneration unit, the limiting contaminant concentration should be satisfied, no matter regeneration recycling or regeneration reuse approaches are applied. In HEN, the non-isothermal mixing and fliting rules for streams are adopted. In addition, to concentrate on the interactions between water network and HEN, a single-contaminant process is assumed. The optimal HIWN should include (1) the flow rate of every water using unit, (2) the regeneration units and their parameters, (3) the number of heat exchangers and their duties, and (4) the consumption of freshwater, the hot and cold utilities.

## Methods

Generally, optimization and integration for HIWN that uses the mathematical programming method mainly have two steps: the first one is establishing a physical model and a mathematical model, and the second is carrying out the sequential optimization. The procedure for systems with regeneration recycling is similar to that with the regeneration reuse. The specific step is shown in Fig. 1.

## Regeneration Recycling Model

Based on above algorithms, the design of the regeneration recycling HIWN can be carried out within the superstructure, heat exchange model, and mathematical model established in turn.

First, it is necessary to sort all the water units in the order of descending temperature. The water unit with the highest temperature is in the first place, following with that has lower temperature until the end. Figure 2 shows the superstructure of wastewater regeneration recycling HIWN. The new superstructure has one more regeneration unit R. Different from other units, unit R can only receive water from other water using units and regeneration unit whose contaminant concentration can satisfy its limiting inlet concentration. The regenerated water can be sent to other water using unit, regeneration unit, or discharged. The inlet and outlet water of the regeneration unit formed a cycle in the regeneration recycling HIWN. F R,j denotes the rate of water flow from regeneration unit to water using unit j; C R,out is the outlet contaminant concentration of regeneration unit, which is used when deciding whether regenerated water can be reused to a certain water using unit or not. Other symbols in the manuscript are named in the same way.

Here, water unit j can be supplied by freshwater (FW, j), water sources from other units (F i,j ), and regeneration unit (F R,j ), while the outlet water of unit j can supply for other water units, the regeneration unit, or be discharged. The regeneration unit can be supplied by each unit, while the outlet water can supply to all the water units whose contaminant concentration is qualified.

Water network can be designed based on the superstructure for single-contaminant regeneration recycling HIWN. There are many match patterns that can satisfy the requirements of water networks. However, the minimum utility consumption cannot be guaranteed.

With the minimum heat transfer temperature difference taken ∆T, the match between hot and cold streams can be identified with the temperature of hot/cold streams decrease/increase by 0.5∆T. With all temperatures changed, n + 1 temperature degrees will be generated; the highest temperature degree 0, and the lowest temperature degree n + 1, creating n temperature interval. Temperature degree 0 to temperature degree 1 correspond to temperature interval 1, and so on. Each temperature interval has heat savings and losses. Suppose that in temperature interval K, after the heat exchange between the hot and cold streams, the remaining heat is Q K , whose value can be negative, zero, or positive. A positive Q K means that there is heat that can be released to temperature interval K + 1. A negative Q K means that heat needs to be added (Quti, K) from an external utility to temperature interval K − 1 to ensure that all the requirements of temperature intervals can be satisfied. To ensure that the remaining cumulative heat of each temperature interval is 0, the sum of the remaining heat (Quti, K) and the heat from previous temperature interval should be zero. The heat transfer model is shown in Fig. 3.

The maximum value of Quti, K is the minimum heating utility. Generally, the heat should be input from the highest temperature interval, and the value should be non-negative.

Based on the above process model and mass balance calculation of water quality and contaminant concentration for each water unit, the mathematical model for wastewater regeneration recycling HIWN can be set up to optimize utilities and connection numbers. The optimization is a multi-target optimization problem, which involves the usage of freshwater and regenerated water, the regenerated contaminant load, the total annual operating cost, and the number of allocations. According to the research results of relative importance of each objective (Feng et al. 2008), the optimization order for these objectives can be set as follows:

The entire process for establishing mathematical models can be divided into the following five steps:
1. (1)
The mathematical model using the minimum amount of freshwater:
$$\mathit{\min}\kern1.5em \sum \limits_{j\in P}{F}_j^W\kern0.2em$$
(1)

Restrictions:

1. a.
The water balance between the inlet and outlet of water supply unit j:
$${F}_j^W+\sum \limits_{\begin{array}{c}i\in P\\ {}i\ne j\end{array}}{F}_{i,j}+{F}_{R,j}={F}_j^D+\sum \limits_{\begin{array}{c}d\in P\\ {}d\ne j\end{array}}{F}_{j,d}+{F}_{j,R}$$
(2)

2. b.
Contaminant balance at the mixed node of water unit j:
$$\sum \limits_{\begin{array}{c}i\in P\\ {}i\ne j\end{array}}\left({F}_{i,j}{c}_i^{out}\right)+{F}_{R,j}{c}_R^{out}=\left({F}_j^W+\sum \limits_{\begin{array}{c}i\in P\\ {}i\ne j\end{array}}{F}_{i,j}+{F}_{R,j}\right){c}_j^{in}$$
(3)

3. c.
Contaminant balance for the inlet and outlet of water unit j:
$$\left({F}_j^W+\sum \limits_{\begin{array}{c}i\in P\\ {}i\ne j\end{array}}{F}_{i,j}+{F}_{R,j}\right){c}_j^{in}+{M}_j=\left({F}_j^W+\sum \limits_{\begin{array}{c}i\in P\\ {}i\ne j\end{array}}{F}_{i,j}+{F}_{R,j}\right){c}_j^{out}$$
(4)

4. d.
The limit of contaminant concentration at the inlet and outlet of unit j:
$${c}_j^{in}\le {c}_j^{in,\mathit{\max}}$$
(5)
$${c}_j^{out}\le {c}_j^{out,\mathit{\max}}$$
(6)

5. e.
The water balance at the inlet and outlet of regeneration unit R:
$$\sum \limits_{j\in P}{F}_{j,R}\cdot Lr=\sum \limits_{j\in P}{F}_{R,j}$$
(7)

Lr is the loss rate for regeneration unit, and it is taken as 0.9 in this paper according to the empirical data.
1. f.
The contaminant balance at the inlet mixing node of regeneration unit R:
$$\sum \limits_{j\in P}\left({F}_{j,R}{c}_j^{out}\right)=\sum \limits_{j\in P}{F}_{R,j}{c}_R^{in}$$
(8)

2. g.
Auxiliary restrictions (the limited post-regeneration concentration):
$${c}_R^{out}={c}_R^{out,s}$$
(9)
$${r}_s=\left({c}_R^{in}-{c}_R^{out}\right)/{c}_R^{in}$$
(10)
$${c}_R^{in}\le {c}_R^{in,\mathit{\max}}$$
(11)
$${c}_R^{out}\le {c}_R^{out,\mathit{\max}}$$
(12)

Based on the minimum freshwater consumption obtained in formal calculation, the amount of regenerated water can be optimized. The smaller amount of regenerated water leads to smaller running costs and lower operating costs. The aim to minimize regenerated water consumption is shown as follows:
$$\mathit{\min}\kern1.5em \sum \limits_{j\in P}{F}_{R,j}\kern0.2em$$
(13)
The restrictions from Eqs. (2) to (13) and the following one should be satisfied.
$$\sum \limits_{j\in P}{F}_j^W\le {F}_W^{min}\left(1+{\xi}_1\right)$$
(14)
In Eq. (14), constant ξ1 is used to control the amount of freshwater. Its value satisfies $$0\le {\xi}_1\le {F}_W^{\mathit{\min},0}/{F}_W^{min}-1$$, and $${F}_W^{\mathit{\min},0}$$ means the minimum amount of freshwater usage when the wastewater is reused directly. If ξ1 = 0, the maximum amount of freshwater used in the system is $${F}_W^{min}$$, which is obtained from the Eq. (1); if $${\xi}_1={F}_W^{\mathit{\min},0}/{F}_W^{min}-1$$, regeneration recycling is not necessary.
1. (2)

The mathematical model for the minimum regeneration contaminant load

For a single-contaminant water network with only one regeneration unit, minimizing the regeneration contaminant load is equal to minimizing the value $$\sum \limits_{j\in P}{F}_{R,j}\left({c}_R^{in}-{c}_R^{out}\right)$$, where $$\sum \limits_{j\in P}{F}_{R,j}$$ has been optimized and $${c}_R^{out}$$ is given. The optimization for regeneration contaminant load is changed into optimizing regeneration concentration. Therefore, the minimum regeneration concentration is chosen as the objective function.
$$\mathit{\min}\left({c}_R^{in}\right)$$
(15)
This model also satisfies the restrictions for Eq. (2) to (13) and (15), and
$$\sum \limits_{j\in P}{F}_{j,R}\le {F}_R^{min}\left(1+{\xi}_2\right)$$
(16)
where ξ2 is a constant that controls the consumption of regenerated water. If ξ2 = 0, the minimum usage of regenerated water is $${F}_R^{min}$$, which is calculated by Eq. (2).
1. (3)

The mathematical model for the minimum total annual operating cost

Z is the annual total operating cost and objective function, the mathematical model is as follows:
$$\min Z$$
(17)
$$Z= PR\left(N{u}_F\sum \limits_{j\in P}{F}_j^W+{NBF}_R+{u}_H{Q}_{HU,\mathit{\min}}+{u}_C{Q}_{CU,\mathit{\min}}\right)$$
(18)
This model also satisfies the restrictions for Eqs. (2) to (13), (15), and (17), and
$${T}_{n+1}\le {T}_R\le {T}_0$$
(19)
where T0 is the temperature of the water unit, with the highest temperature of all, and Tn + 1 is the temperature of the water unit with the lowest temperature.
$${c}_R^{in}\le {c}_R^{in,\mathit{\min}}$$
(20)
$${Q}_K={C}_p\left(\sum \limits_{m\in {H}_{k,k+1}}{F}_m-\sum \limits_{q\in {C}_{k,k+1}}{F}_q\right)\left({T}_{k+1}-{T}_k\right)$$
(21)
Q K is the cumulated heat in temperature interval K, and for each temperature interval, the heat utility is
$${Q}_{uti,K}+{Q}_K=0\left(K=1,L,n\right)$$
(22)
$${Q}_{HU,\mathit{\min}}=\max {Q}_{uti,K}$$
(23)
$${Q}_{CU,\min }=\min {Q}_{uti,K}$$
(24)
The mathematical model for the minimum number of allocations:
$$\mathit{\min}\kern0.5em \left[\sum \limits_{j\in P}{y}_j^W+\sum \limits_{i\in P}\sum \limits_{\begin{array}{c}j\in P\\ {}i\ne j\end{array}}{y}_{i,j}+\sum \limits_{j\in P}{y}_{j,R}+\sum \limits_{j\in P}{y}_{R,j}+\sum \limits_{j\in P}{y}_j^D\right]$$
(25)
This model, as well, satisfies the restrictions for Eq. (2) to (13), (15), (17), (20) to (25), and
$${F}_R={F}_R^{\mathrm{min}}$$
(26)
$${F}_j^W-{Uy}_j^W\le 0\left(j\in P\right)$$
(27)
$${F}_{i,j}-{Uy}_{i,j}\le 0\left(i,j\in P\right)$$
(28)
$${F}_{R,j}-{Uy}_{R,j}\le 0\left(j\in P\right)$$
(29)
$${F}_{j,R}-{Uy}_{j,R}\le 0\left(j\in P\right)$$
(30)
$${F}_j^D-{Uy}_j^D\le 0\left(j\in P\right)$$
(31)
where U is the upper bound of all flow rate F. When binary variable y is 0, the flow rate F should also be 0 and invalid. Otherwise, when y is 1, the flow rate F can be above 0 and thus valid.

### Regeneration Reuse Model

The design process for the HIWN with regeneration reuse is similar to that of the system with regeneration recycling. The superstructure, heat exchange model, and the mathematical model can be established.

Assumed that there was only one regeneration unit, R, in a given series of water units P. If the wastewater is regenerated and reused, some water units may need to be decomposed in series or in parallel. Assuming that all the water units are decomposed only once, the set of water units that after the decomposition is represented by X. Figure 4 shows the superstructure of the HIWN with regeneration reuse. In this superstructure, unit j cannot receive regenerated water from the regeneration unit R while supplying wastewater to it. Pj in Fig. 4 is a set which contains all water using units except for the one that discharge wastewater to regeneration unit, R. In addition, the indirect cycle such as unit j supply water to unit R, unit R discharge regenerated water to unit i, later unit i finally supply water to unit j should also be excluded. Restrictions (59, 60) can ensure that there is no cycle between unit R and unit j. To be more realistic, water loss during the regeneration process into the superstructure and the outlet flow that symbolized water loss are added. This superstructure can help obtaining the connection patterns of whole network, from which we can chose the one that has the minimum annual cost.

When designing the HEN, the same HEN model for the single-contaminant HIWN with regeneration recycling can be used to the system with regeneration reuse.

The mathematical model for a single-contaminant HIWN with regeneration reuse is similar to that for the system with regeneration recycling. But the decomposition should be considered when optimizing sequentially. The optimization process for the system with regeneration reuse is similar to that of the system with regeneration recycling.
1. (1)

The mathematical model using the minimum amount of freshwater.

Set the minimum freshwater consumption as objective function, and the water balance, contaminant balance, and limiting inlet and outlet concentration of all the water units and regeneration units as restrictions.
$$\mathit{\min}\sum \limits_{j\in X}{F}_j^W\kern0.2em$$
(32)

The restrictions are

1. a.
The water balance between the inlet and outlet of water supply unit j
$${F}_j^W+\sum \limits_{\begin{array}{c}i\in X\\ {}i\ne j\end{array}}{F}_{i,j}+{F}_{R,j}={F}_j^D+\sum \limits_{\begin{array}{c}d\in X\\ {}d\ne j\end{array}}{F}_{j,d}+{F}_{j,R}\left(j\in X\right)$$
(33)

2. b.
The contaminant balance at the mixed node of water unit j
$$\sum \limits_{\begin{array}{c}i\in X\\ {}i\ne j\end{array}}\left({F}_{i,j}{c}_i^{out}\right)+{F}_{R,j}{c}_R^{out}=\left({F}_j^W+\sum \limits_{\begin{array}{c}i\in X\\ {}i\ne j\end{array}}{F}_{i,j}+{F}_{R,j}\right){c}_j^{in}\left(j\in X\right)$$
(34)

3. c.
The contaminant balance between inlet and outlet of water unit j:
$$\left({F}_j^W+\sum \limits_{\begin{array}{c}i\in X\\ {}i\ne j\end{array}}{F}_{i,j}+{F}_{R,j}\right){c}_j^{in}+{M}_j{a}_j=\left({F}_j^W+\sum \limits_{\begin{array}{c}i\in X\\ {}i\ne j\end{array}}{F}_{i,j}+{F}_{R,j}\right){c}_j^{out}\left(j\in P\right)$$
(35)
$$\left({F}_{b+j}^W+\sum \limits_{\begin{array}{c}i\in X\\ {}i\ne b+j\end{array}}{F}_{i,b+j}+{F}_{R,b+j}\right){c}_{b+j}^{in}+{M}_j\left(1-{a}_j\right)=\left({F}_{b+j}^W+\sum \limits_{\begin{array}{c}i\in X\\ {}i\ne b+j\end{array}}{F}_{i,b+j}+{F}_{R,b+j}\right){c}_{b+j}^{out}\left(j\in P\right)$$
(36)

4. d.
The decomposition requirements of water unit
$$0\le {a}_j\le 1$$
(37)

1. 1)
The restriction of parallel decomposition
$${c}_j^{in}\le {c}_j^{in,\mathit{\max}}\left(j\in P\right)$$
(38)
$${c}_j^{out}\le {c}_j^{out,\mathit{\max}}\left(j\in P\right)$$
(39)
$${c}_{b+j}^{in}\le {c}_j^{in,\mathit{\max}}\left(j\in P\right)$$
(40)
$${c}_{b+j}^{out}\le {c}_j^{out,\mathit{\max}}\left(j\in P\right)$$
(41)

2. 2)
The restriction of serial decomposition
$${c}_j^{in}\le {c}_j^{in,\mathit{\max}}\left(j\in P\right)$$
(42)
$${c}_j^{out}\le {c}_j^{com}\left(j\in P\right)$$
(43)
$${c}_{b+j}^{in}\le {c}_j^{com}\left(j\in P\right)$$
(44)
$${c}_{b+j}^{out}\le {c}_j^{out,\mathit{\max}}\left(j\in P\right)$$
(45)
$${c}_j^{com}={c}_j^{in,\mathit{\max}}+{a}_j\left({c}_j^{out,\mathit{\max}}-{c}_j^{in,\mathit{\max}}\right)\left(j\in P\right)$$
(46)

The water balance between the import and export of regeneration unit R:
$$\sum \limits_{j\in X}{F}_{j,R}\cdot Lr=\sum \limits_{j\in X}{F}_{R,j}$$
(47)

Lr is the loss rate for regeneration unit and is taken as 0.9 in this paper according to the empirical data.

The contaminant balance at inlet mixing node of regeneration unit R:
$$\sum \limits_{j\in X}\left({F}_{j,R}{c}_j^{out}\right)=\sum \limits_{j\in X}{F}_{R,j}{c}_R^{in}$$
(48)
Secondary conditions (limited post-regeneration concentration):
$${c}_R^{out}={c}_R^{out,0}$$
(49)
$${r}_s=\left({c}_R^{in}-{c}_R^{out}\right)/{c}_R^{in}$$
(50)
$${c}_R^{in}\le {c}_R^{in,\mathit{\max}}$$
(51)
$${c}_R^{out}\le {c}_R^{out,\mathit{\max}}$$
(52)
1. e.
The features of regeneration reuse process:
$$\sum \limits_{j\in X}{F}_j^W\ge \sum \limits_{j\in X}{F}_{j,R}\kern0.2em$$
(53)

The inlet restriction of water using subunit:
$$\left({F}_j^W+\sum \limits_{\begin{array}{c}i\in X\\ {}i\ne j\end{array}}{F}_{i,j}+{F}_{R,j}\right)-{a}_jU\le 0\left(j\in P\right)$$
(54)
$$\left({F}_{b+j}^W+\sum \limits_{\begin{array}{c}i\in X\\ {}i\ne b+j\end{array}}{F}_{i,b+j}+{F}_{R,b+j}\right)-\left(1-{a}_j\right)U\le 0\left(j\in P\right)$$
(55)
1. f.
The connection among the water units:
$${F}_{i,j}-{Uy}_{i,j}\le 0\left(i,j\in X\right)$$
(56)

2. g.
The connection between regeneration unit and water unit:
$${F}_{R,j}-{Uy}_{R,j}\le 0\left(j\in X\right)$$
(57)

3. h.
The connection between water unit and regeneration unit:
$${F}_{j,R}-{Uy}_{j,R}\le 0\left(j\in X\right)$$
(58)

4. i.
Excluding the direct recycle in the system
$${y}_{R,j}+{y}_{j,R}\le 1\left(j\in X\right)$$
(59)

5. j.
Excluding the indirect recycle in the system
$${y}_{R,j}\ge {y}_{i,j}+{y}_{j,R}-1\left(j\in X\right)$$
(60)

GAMS is used to solve the MINLP problem to find the optimal global solution for the five objective functions described above. The results of the previous formulation are the restrictions of next model. For example, the optimal solutions of objective functions (1) and (2) should be used as restrictions when solving objective function (3). Most importantly, the complexity of model increases along the number of units, and the solution would have difficulty in convergence; it would be better to set a proper initial value when solving.

## Case Study

### Example 1

This case was selected from the literature Savulescu and Smith (1998). Table 1 shows the limiting data for process. The proposed method is applied to analyze this case, and the improved HIWN can be obtained, which has different water and energy consumption compared with that in the literature. From the comparison, the advantages of new model can be seen.
Table 1

Example data from Savulescu and Smith (1998)

Water unit

C L (g s−1)

$${C}_{in}^{\mathrm{lim}}$$ (ppm)

$${C}_{out}^{\mathrm{lim}}$$ (ppm)

T o (°C)

1

5

50

100

100

2

30

50

800

75

3

4

400

800

50

4

2

0

100

40

Tables 2 and 3 show the relevant parameters that were selected from the literature (Liao et al. 2011). As for this model, only one contaminant was considered in the optimization. All the basic data is same as that in the literature Leewongtanawit and Kim (2009), so that the optimization results have comparability. The cost models of these regeneration units are adopted from Kuo and Smith (1998). Suppose that the investment payoff period of regeneration equipment is 10 years; the annual investment rate of basic construction cost, λ, can be set as 0.1.
Table 2

Operating parameters and values

Operating parameters

The value of the parameters

PR

0.95

N

8000 h

u F

$2.5 t−1 u H$260 kW−1

u c

$150 kW−1 Tmin 10 °C Table 3 Basic data of wastewater treatment process The process of wastewater regeneration The removal rate of contaminant (H2S) (%) Basic construction cost ($)

### Regeneration Reuse Optimization

The optimization for the regeneration reuse system is similar to that for the regeneration recycling system. Figure 6 shows the final HIWN created by the sequential optimization method for the single-contaminant regeneration reuse system. The minimum amount of freshwater for this system is 20 kg s−1; the minimum amount of regenerated water is 120.6 kg s−1, which is same as that of regeneration recycling approach for this system. But the minimum heating and cooling utility are more than those of regeneration recycling approach, which are 6215.3 and 5373.4 kW, respectively, using three heaters and two coolers. And the number of network connections is 10.
Table 4 shows the comparison of optimization results for example 1 in literature and that in this paper using different regeneration approaches, respectively, where it shows the specific difference of energy and water savings among them clearly. After regeneration, the freshwater consumption is reduced from 90 to 20 kg s−1; 120.6 kg s−1 of regenerated water is reused, which means the same amount of wastewater emission is reduced as well. Utility consumption is increasing, as well as the number of heat exchangers. But overall, the total annual operating cost for the regeneration recycling is less than that for regeneration reuse approach, both much fewer than that in literature, which reach up to about 7.7 M$annually. Furthermore, since the water loss in the regeneration process is considered in the optimization, the freshwater consumption will increase while results of this case are more realistic. For cases that contaminant accumulation has no influence on the system operation, the regeneration recycling model for HIWN can be used. Table 4 Comparison of results for example 1 Results Regeneration recycling Regeneration reuse Liao et al. (2011) Minimum freshwater (kg s−1) 20.0 20.0 90 Minimum regenerated water (kg s−1) 120.6 120.6 Minimum heating utility (kW) 4911.2 6215.3 3780 Minimum cooling utility (kW) 4094.4 5373.4 0 The number of the heat exchanger 8 8 4 The number of unit connections 11 10 Total annual operating cost (M$ a−1)

2.02

2.52

7.67

### Example 2

This case was selected from literature Bagajewicz et al. (2002) (Table 5). Table 4 shows the limiting data for process. The other operating parameters and regeneration data are same as example 1, as shown in Tables 2 and 3. This case is optimized using regeneration recycling and regeneration reuse model, respectively.
Table 5

Example data from Bagajewicz et al. (2002)

Water unit

C L (g s−1)

$${C}_{in}^{\mathrm{lim}}$$ (ppm)

$${C}_{out}^{\mathrm{lim}}$$ (ppm)

T o (°C)

1

5

50

100

100

2

30

50

800

75

3

50

800

1100

100

### Regeneration Recycling Optimization

With the regeneration recycling model used to optimize example 2, the optimal network is shown in Fig. 7. The minimum freshwater consumption is 11.5 kg s−1; the minimum regenerated water is 114.7 kg s−1; the minimum heating utility is 1465.6 kW; the minimum cooling utility is 991.8 kW, and the number of heat exchanger is 7. The number of network connections is 11. Only one heaters and one cooler are used in the HIWN after heat integration. The heat load of exchangers in network is listed below the network. Wastewater of all units can discharge to the unit R, while regenerated water can supply to any water using unit that is in need as regeneration recycling approach is applied. When regeneration process is added in this system, there is no wastewater discharge as it can be regenerated and reused by other water using units. Furthermore, freshwater consumed here is mainly used to balance the water loss in the regeneration process for system.

### Regeneration Reuse Optimization

With the regeneration reuse model used to optimize example 2, the optimal network is shown in Fig. 8. The minimum freshwater consumption is 20.3 kg s−1, more than that of regeneration recycling approach. While the minimum regenerated water is 100 kg s−1, less than that of regeneration approach. The minimum heating utility is 2423.3 kW; the minimum cooling utility is 1569.1 kW, and the number of heat exchanger is 7. Three heaters and three coolers are used in this approach as more utility is needed. The number of network connections is 11. Heat load of every heat exchangers and utility is listed below the network, as shown in Fig. 8. Freshwater is needed for unit 1 and unit 2; wastewater from unit 3 is discharged, while that of other unit is reused to save water and energy. The freshwater consumption is more than wastewater emission as water loss was considered in the system.
The results of this paper are compared with that in literature Dong et al. (2008), which optimized the same example and considered the wastewater treatment. The comparison is shown in Table 6, which contains comparisons from many aspects that value the results scientifically. The optimization in this paper consumes more freshwater as water loss during regeneration process is considered and fewer utilities as energy contained in the regeneration process is reused than that in literature. Wastewater regenerations are considered here, as well as that in literature; differently, freshwater is needed here, while optimization in literature needs no freshwater at all, as optimization here considered water loss during the regeneration process which is more realistic and practicable. Especially, regeneration recycling approach here consumes almost the same amount of annual cost, 0.67 M$, while needs less utility and is more close to real operation situation. Table 6 Comparison of results for example 2 Results Regeneration recycling Regeneration reuse Dong et al. (2008) Minimum freshwater (kg s−1) 11.5 20.3 0.0 Minimum regenerated water (kg s−1) 114.7 100.0 114.7 Minimum heating utility (kW) 1465.6 2423.3 4818.6 Minimum cooling utility (kW) 991.8 1569.1 4818.6 The number of the heat exchanger 7 7 6 The number of unit connections 11 11 _ Total annual operating cost (M$ a−1)

0.67

1.02

0.66

## Conclusions

A new superstructure is established for wastewater regeneration water network considering the loss of water. According to the characteristic of regeneration recycling and regeneration reuse, the superstructure is modified to be more suitable. Based on the superstructure and the idea of sequential optimization, a mathematical programming model was established and used to design single-contaminant wastewater regeneration recycling and regeneration reuse HIWN, to achieve the multi-target for minimum freshwater consumption, minimum regenerated water consumption, minimum regeneration contaminant load, minimum annual total operating cost, and least number of network connections. When these two models were applied to a specific case, the optimization approach should be chosen based on the specific situation. Regeneration recycling and regeneration reuse are significant approaches for water saving, while optimization for HEN is of great significance to energy saving. Regeneration for wastewater not only reduces the freshwater consumption but also saves heating and cooling utilities when the wastewater was regenerated, and at the same time minimizes the number of the network connections and total annual operating cost as much as possible. The consideration of water loss during regeneration process makes this model more realistic, and the optimization results have more practicality. The optimization for HIWN is of great value for both water saving and energy saving. This is much more important for the sustainable development of the society.

## Notes

### Funding information

The financial support for this research is provided by the National Natural Science Foundation of China under grant 21736008.

### Conflict of Interest

The authors declare that they have no conflict of interest.

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© Springer Nature Singapore Pte Ltd. 2018

## Authors and Affiliations

• Renjie Shen
• 1
• Yu Zhang
• 1
• Jian Ma
• 1
1. 1.Department of Chemical EngineeringXi’an Jiaotong UniversityXi’anChina