A Mathematical Programming Method for Optimizing the Single-Contaminant Regeneration Heat-Integrated Water Networks
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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 programmingNomenclature
- 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})
- ξ_{1}
Constant which related to the freshwater consumption
- ξ_{2}
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
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.
Here, water unit j can be supplied by freshwater (F_{W, 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.
The maximum value of Q_{uti, 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:
- (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:
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- (2)
The mathematical model for the minimum regeneration contaminant load
- (3)
The mathematical model for the minimum total annual operating cost
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.
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.
- (1)
The mathematical model using the minimum amount of freshwater.
The restrictions are
- 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)
- 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)
- 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)
- d.The decomposition requirements of water unit$$ 0\le {a}_j\le 1 $$(37)
- 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)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)
Lr is the loss rate for regeneration unit and is taken as 0.9 in this paper according to the empirical data.
- 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)
- f.The connection among the water units:$$ {F}_{i,j}-{Uy}_{i,j}\le 0\left(i,j\in X\right) $$(56)
- g.The connection between regeneration unit and water unit:$$ {F}_{R,j}-{Uy}_{R,j}\le 0\left(j\in X\right) $$(57)
- h.The connection between water unit and regeneration unit:$$ {F}_{j,R}-{Uy}_{j,R}\le 0\left(j\in X\right) $$(58)
- i.Excluding the direct recycle in the system$$ {y}_{R,j}+{y}_{j,R}\le 1\left(j\in X\right) $$(59)
- 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
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 |
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 } | $150 kW^{−1} |
∆T_{min} | 10 °C |
Basic data of wastewater treatment process
The process of wastewater regeneration | The removal rate of contaminant (H_{2}S) (%) | Basic construction cost ($) | Operating cost ($ h^{−1}) |
---|---|---|---|
Stripping column | 99.9 | \( 16,800{F}_{RM}^{0.7} \) | F _{ RM } |
Biological treatment unit | 90 | \( 12,600{F}_{RM}^{0.7} \) | 0.0067F_{ RM } |
API separator | 0 | \( 4800{F}_{RM}^{0.7} \) | 0 |
Regeneration Recycling Optimization
Regeneration Reuse Optimization
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
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
Regeneration Reuse Optimization
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
Compliance with Ethical Standards
Conflict of Interest
The authors declare that they have no conflict of interest.
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