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A shortcut model for energy efficient water network synthesis
Accepted Manuscript Title: A shortcut model for energy efficient water network synthesis. Author: Yingzong Liang, Chi-Wai Hui PII: DOI: Reference:
S1359-4311(15)01256-9 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.020 ATE 7301
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
30-3-2015 9-11-2015
Please cite this article as: Yingzong Liang, Chi-Wai Hui, A shortcut model for energy efficient water network synthesis., Applied Thermal Engineering (2015), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.020. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Title: A Shortcut Model for Energy Efficient Water Network Synthesis. Authors: Yingzong Liang, Chi-Wai Hui * Affiliation: Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email: [email protected] (Y. Liang), [email protected] (C. W. Hui). * Corresponding author Tel: +852-2358-7137 Fax: +852-2358-0054
Page 1 of 13
Highlights
Shortcut model is formulated for energy efficient water network synthesis. Energy consumption is minimized by minimizing temperature fluctuations. The model is formulated as MILP to achieve global optimum. Example shows the model generates water network with minimum energy consumption.
Graphical Abstract
Temperature fluctuation Temperature ‘Valley’
ABSTRACT This paper presents a shortcut model for energy efficient water network synthesis with single contaminant. The proposed model is based on the idea of reducing repeated heating and cooling proposed by Feng et al. [1]. To avoid sub-optimum that can be generated from Feng’s model, the proposed model only minimizes the number of temperature ‘valleys’ instead of the total number of ‘peaks and valleys’ of the water network. With the new formulation, the proposed model not only guarantees global optimum but also becomes much easier to be solved. Keywords: Water Allocation Network, Heat Integration, Mixed-integer Linear Programming.
1. Introduction Process industry is water and energy consuming. For example, oil refinery requires sheer bulk of water for washing, striping, and extraction. In certain cases, large amount of water needs to be heated up or cooled down to meet the temperature requirement of the operations. As a result, considerable amount of utility is necessary for water heating and cooling.
Page 2 of 13
In order to reduce water and energy consumption, researchers developed different methods to address the water and energy usage problem based on conceptual design or mathematical programming. El-Halwagi and Manousiouthakis proposed a procedure for mass exchange network [2]. Wang and Smith introduced an approach for wastewater minimization with the underlying concept of pinch analysis [3]. The approach is able to determine minimum water using target without mathematical programming. Bagajewicz and Savelski presented a linear programming model for water allocation network design [4], and discussed the necessary conditions of optimal water networks [5]. Savulescu and Smith introduced a study on simultaneous energy and water minimization [6]. Savulescu et al. provided a systematic method for heat exchanger network and water network design [7,8]. Bagajewicz et al. proposed a mathematical programming method to generate water and heat exchanger network achieving minimum water and energy target [9]. Feng et al. introduced a mixed integer non-linear programming (MINLP) model to construct energy efficient network by minimizing number of temperature ‘peaks’ and ‘valleys’ of the water network [1]. The method is based on the observation that fewer temperature fluctuations in a water network results in smaller energy consumption. Since unnecessary repeated heating and cooling of water leads to utility increase of the network, by minimizing temperature fluctuation, the resulting water network may feature minimum energy consumption if there is no repeated heating or cooling. However, since the required water using temperature is often higher than fresh water temperature and wastewater discharged temperature, temperature ‘peaks’ do not necessarily indicate repeated heating or cooling, minimizing the total number of temperature ‘peaks’ and ‘valleys’ may lead to a sub-optimal solution. In this paper, a shortcut model that only minimizes the number of temperature “valleys’ is proposed. The paper outline is as follows: the problem definition is stated in Section 2, a mixed-integer linear programming (MILP) model is developed in Section 3, and an illustrative example, results, and discussion are presented in Section 4. 2. Problem Definition In process operation, fresh water or reused water is applied to remove contaminant from the processes. And due to the temperature difference between processes and fresh water, heat exchange is necessary to heat up or cooled down the water streams to target temperature. The water network problem can be defined as follows.
Page 3 of 13
a. Single contaminant is assumed in the water network. b. Water using processes have specific requirement of maximum inlet and outlet contaminant concentrations of the water. c. The processes’ contaminant loads are fixed. Since the contaminant levels in the water streams are very low, the contaminant loads will not be added up to the water flowrate. d. Water streams need to be heated up or cooled down to the specific temperature before entering the processes and it is assumed that the water has no heat gain or loss in the processes. e. It is assumed that the water streams can only exchange heat with other water streams or utilities. f. Fresh water temperature and wastewater discharged temperature are given. The purpose of the following formulation is to synthesize a water network with good energy performance. The synthesis of heat exchanger network is not included in this work. 3. Model Formulation In this section, a MILP model is formulated according to the problem definition. The model adopts some constraints from Bagajewicz [9] and Feng [1]. 3.1 Mass Balance Constraints Mass balance is necessary to generate a feasible water network. Eq. (1)-(4) can be found in Bagajewicz’s work [9]. (1) (2) (3) (4) Eq. (1) is the water balance of the process. Eq. (2) is contaminant balance of the process, and it is assumed the contaminant in the water reaches maximum concentration at each process outlet. Eq. (3)
Page 4 of 13
guarantees inlet contaminant concentration is lower than required maximum value. Eq. (4) ensures minimum water usage of water network, and
is minimum water flow rate of water network
obtained by wastewater minimization approach [3]. 3.2 Logic Constraints The concepts of temperature ‘peak’ and ‘valley’ proposed Feng [1] is adopted. However, in this work, only temperature ‘valleys’ are considered. The constraints used for identifying ‘valleys’ are linearized in the proposed model to make it easier to be solved. Existence of Water Reuse. Eq. (5) determines the existence of water reuse between units. If there is water reusing from process i to process j, the water flow variable,
must be larger than zero. The binary
, in Eq. (5) must be equal to 1 to hold the constraint. (5)
Existence of Temperature ‘Valley’. In Eq. (6), if and from process
to
exists, and
,
, that water reuse from process to , then
, which denotes the existence of
temperature valley at process , equals to 1. Since the objective is to minimize the total number of ‘valleys’, if
or
is not equal to 1,
is forced to be 0. Eq. (7) defines the set of possible
temperature ‘valleys’. (6) (7) 3.3 Objective Function The objective of the proposed model is to minimize the number of ‘valleys’. With constraints Eq. (1)-(7), the proposed MILP model generates a feasible energy efficient water network. (Obj) 4. Example In this section, the proposed shortcut model is applied to a water allocation network synthesis example taken from Bagajewicz [9] and Feng [1] work. The example involves 8 water using
Page 5 of 13
processes. Fresh water temperature is 20°C, wastewater discharged temperature is 30°C, and minimum heat recovery approach temperature is 10○C, heat capacity of water is 4.2 kJ/kg○C. Water using data are listed in Table 1. In this example, the proposed model is implemented on GAMS 23.7.0 and solved by CPLEX 12.3 on an Intel(R) Core(TM) i5 3.10 GHz CPU and 8 GB of RAM Computer. Only one core is used for the model solving. 0% optimality tolerance is set. Feng’s model [1] is also implemented and solved by on GAMS for comparison. The models’ performance is listed in Table . Regarding solution efficiency, the proposed model is solved in 0.148 s with 18 iterations, faster than Feng’s 6.32 s with 34614 iterations. The relaxation gaps [10] shown in Table 2 indicate the tightness of the models. In general, the smaller the relaxation gap is, the easier the model is solved. The relaxation gap of the proposed model is 0%, which is much smaller than 3185% of Feng’s model. In terms of the models’ sizes, the proposed model has 94 variables and 61 constraints in this example; Feng’s [1] model is reformulated in GAMS resulted in 143 variables and 126 constraints, slightly different from the reported 140 variables and 103 constraints, and both are larger than the proposed model. The result water network generated by the proposed model is shown in Fig. 1. The network structure is different from Bagajewicz and Feng’s network shown in Fig. and Fig. 2, respectively. Table is the sub-stream data calculated by the proposed model. The fresh water feed is 125.943 kg/s, same as Bagajewicz [9] and Feng’s [1] networks. On the other hand, the energy consumption of the result network is 5289.61 kW, which is equal to Bagajewicz’s result, and is equal to the minimum possible hot utility target calculated by following equation [8,9]. (8) Noted that the energy consumption is greater than Feng’s 5135.76 kW [1], however, it should be pointed out that in Feng’s network, a 10 kg/s fresh water inlet to process 8 was missed in their utility calculation, leading to infeasible energy consumption lower than the minimum target. If the fresh water inlet to process 8 is taken into consideration, the minimum energy consumption of Feng’s network should be 5555.76 kW, a sub-optimal solution, higher than optimal solution of 5289.61kW.
Page 6 of 13
It is interesting to note that the resulted water network has 10 ‘peaks’ and no ‘valley’, indicating that there is no repeat heating or cooling. While according to Fig. 2, Feng’s network has 10 ‘peaks’ and 1 ‘valley’, suggesting that some repeating heating and cooling occurred. As it is mentioned in section 1 that minimizing repeating heating and cooling may result in water network with minimum energy consumption, the generated network achieves minimum energy target. Moreover, the proposed method is formulated as MILP model, thus, it guarantees the solution to be global optimum. It should also be noted that the proposed model is a shortcut model minimizing the number of temperature ‘valleys’. In case the minimum number of ‘valleys’ is not 0, that there is unavoidable repeating heating and cooling in the water network with the minimum water using constraint, the resulted energy consumption may not be the minimum. 5. Conclusion A shortcut model has been proposed to synthesize energy efficient water network. Instead of minimizing energy consumption, the proposed model minimizes the number of temperature ‘valleys’ and results in a water network with minimal energy requirement. The model is formulated as a MILP allowing it to be solved efficiently. Acknowledgement The authors would acknowledge the financial support from the Hong Kong RGC-GRF grant (613513), the UGC-Infra-Structure Grant (FSGRF13EG03), the Studentship from the Energy Concentration program of the School of Engineering at HKUST. Nomenclature Indices , ,
= water using process.
Sets ={
is a water using process}
={ |
}
={ |
}
Page 7 of 13
= temperature ‘valley’ candidates Parameters = maximum contaminant concentration of water inlet to process (ppm), = maximum contaminant concentration of water outlet from process (ppm), = contaminant load of process (mg/s), = minimum water flow rate of water network (kg/s), = heat capacity of water (kJ/kg°C), = fresh water temperature (°C), = wastewater discharged temperature (°C), = temperature of process (°C), = heat recovery approach temperature (°C), = minimum hot utility (kW). Positive variables = fresh water flow rate to process (kg/s), = water flow rate from process to (kg/s), = discharged wastewater flow rate from process (kg/s). Binary variables = existence of reusing water from process to process , = existence of temperature valley at process . Reference
Page 8 of 13
[1] X. Feng, Y. Li, R. Shen, A new approach to design energy efficient water allocation networks, Applied Thermal Engineering. 29 (11–12) (2009) 2302-2307. [2] M.M. El-Halwagi, V. Manousiouthakis, Synthesis of mass exchange networks, AIChE Journal. 35 (8) (1989) 1233-1244. [3] Y.P. Wang, R. Smith, Wastewater minimisation, Chemical Engineering Science. 49 (7) (1994) 981-1006. [4] M. Bagajewicz, M. Savelski, On the Use of Linear Models for the Design of Water Utilization Systems in Process Plants with a Single Contaminant, Chemical Engineering Research and Design. 79 (5) (2001) 600-610. [5] M.J. Savelski, M.J. Bagajewicz, On the optimality conditions of water utilization systems in process plants with single contaminants, Chemical Engineering Science. 55 (21) (2000) 5035-5048. [6] L. Savulescu, R. Smith, Simultaneous energy and water minimisation, (1998). [7] L. Savulescu, J. Kim, R. Smith, Studies on simultaneous energy and water minimisation—Part I: Systems with no water re-use, Chemical Engineering Science. 60 (12) (2005) 3279-3290. [8] L. Savulescu, J. Kim, R. Smith, Studies on simultaneous energy and water minimisation—Part II: Systems with maximum re-use of water, Chemical Engineering Science. 60 (12) (2005) 3291-3308. [9] M. Bagajewicz, H. Rodera, M. Savelski, Energy efficient water utilization systems in process plants, Computers & Chemical Engineering. 26 (1) (2002) 59-79. [10] J.M. Zamora, I.E. Grossmann, Continuous global optimization of structured process systems models, Computers & Chemical Engineering. 22 (12) (1998) 1749-1770.
Page 9 of 13
Fig. 1 Water Network of Example (kg/s)
4 7
1 3 2 5
6 8 Fig. 2 Bagajewicz’s Network (kg/s)
Page 10 of 13
1
5 7
2
3
6 4
8 Fig. 2 Feng’s Network (kg/s)
Page 11 of 13
Table 1 Water Using Data of Example Process 1 2 3 4 5 6 7 8
(g/s) 2 2.88 4 3 30 5 2 1
(ppm) 80 90 200 100 800 800 600 100
(ppm) 25 25 25 50 50 400 400 0
(°C) 40 100 80 60 50 90 70 50
Table 2 Performance of the Proposed Model and Feng’s Model
Proposed Model Feng’s Model
Continuous Variables
Binary Variables
Constraints
CPU time (s)
Iterations
Relaxation Gap (%)
43
51
61
0.148
18
0
43
100
126
6.325
34614
3185
Page 12 of 13
Table 3 Results of Example Sub-streams Flow rate/(kg/s) Path/°C Minimum energy consumption/kW 24.34 20-60-30 1022.28 1( 25 20-40-60-30 1050 2( 11.825 20-80-30 496.65 3( 5 20-80-70-30 210 4( 6.603 20-100-60-30 277.326 5( 3.175 20-100-80-30 133.35 6( 22.222 20-100-50-30 933.324 7( 17.778 20-50-30 746.676 8( 2.857 20-50-80-30 119.994 9( 7.143 20-50-90-30 300.006 10( Sum 125.943 5289.61 Number of ‘valley’ 0
Page 13 of 13
S1359-4311(15)01256-9 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.020 ATE 7301
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
30-3-2015 9-11-2015
Please cite this article as: Yingzong Liang, Chi-Wai Hui, A shortcut model for energy efficient water network synthesis., Applied Thermal Engineering (2015), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.11.020. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Title: A Shortcut Model for Energy Efficient Water Network Synthesis. Authors: Yingzong Liang, Chi-Wai Hui * Affiliation: Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email: [email protected] (Y. Liang), [email protected] (C. W. Hui). * Corresponding author Tel: +852-2358-7137 Fax: +852-2358-0054
Page 1 of 13
Highlights
Shortcut model is formulated for energy efficient water network synthesis. Energy consumption is minimized by minimizing temperature fluctuations. The model is formulated as MILP to achieve global optimum. Example shows the model generates water network with minimum energy consumption.
Graphical Abstract
Temperature fluctuation Temperature ‘Valley’
ABSTRACT This paper presents a shortcut model for energy efficient water network synthesis with single contaminant. The proposed model is based on the idea of reducing repeated heating and cooling proposed by Feng et al. [1]. To avoid sub-optimum that can be generated from Feng’s model, the proposed model only minimizes the number of temperature ‘valleys’ instead of the total number of ‘peaks and valleys’ of the water network. With the new formulation, the proposed model not only guarantees global optimum but also becomes much easier to be solved. Keywords: Water Allocation Network, Heat Integration, Mixed-integer Linear Programming.
1. Introduction Process industry is water and energy consuming. For example, oil refinery requires sheer bulk of water for washing, striping, and extraction. In certain cases, large amount of water needs to be heated up or cooled down to meet the temperature requirement of the operations. As a result, considerable amount of utility is necessary for water heating and cooling.
Page 2 of 13
In order to reduce water and energy consumption, researchers developed different methods to address the water and energy usage problem based on conceptual design or mathematical programming. El-Halwagi and Manousiouthakis proposed a procedure for mass exchange network [2]. Wang and Smith introduced an approach for wastewater minimization with the underlying concept of pinch analysis [3]. The approach is able to determine minimum water using target without mathematical programming. Bagajewicz and Savelski presented a linear programming model for water allocation network design [4], and discussed the necessary conditions of optimal water networks [5]. Savulescu and Smith introduced a study on simultaneous energy and water minimization [6]. Savulescu et al. provided a systematic method for heat exchanger network and water network design [7,8]. Bagajewicz et al. proposed a mathematical programming method to generate water and heat exchanger network achieving minimum water and energy target [9]. Feng et al. introduced a mixed integer non-linear programming (MINLP) model to construct energy efficient network by minimizing number of temperature ‘peaks’ and ‘valleys’ of the water network [1]. The method is based on the observation that fewer temperature fluctuations in a water network results in smaller energy consumption. Since unnecessary repeated heating and cooling of water leads to utility increase of the network, by minimizing temperature fluctuation, the resulting water network may feature minimum energy consumption if there is no repeated heating or cooling. However, since the required water using temperature is often higher than fresh water temperature and wastewater discharged temperature, temperature ‘peaks’ do not necessarily indicate repeated heating or cooling, minimizing the total number of temperature ‘peaks’ and ‘valleys’ may lead to a sub-optimal solution. In this paper, a shortcut model that only minimizes the number of temperature “valleys’ is proposed. The paper outline is as follows: the problem definition is stated in Section 2, a mixed-integer linear programming (MILP) model is developed in Section 3, and an illustrative example, results, and discussion are presented in Section 4. 2. Problem Definition In process operation, fresh water or reused water is applied to remove contaminant from the processes. And due to the temperature difference between processes and fresh water, heat exchange is necessary to heat up or cooled down the water streams to target temperature. The water network problem can be defined as follows.
Page 3 of 13
a. Single contaminant is assumed in the water network. b. Water using processes have specific requirement of maximum inlet and outlet contaminant concentrations of the water. c. The processes’ contaminant loads are fixed. Since the contaminant levels in the water streams are very low, the contaminant loads will not be added up to the water flowrate. d. Water streams need to be heated up or cooled down to the specific temperature before entering the processes and it is assumed that the water has no heat gain or loss in the processes. e. It is assumed that the water streams can only exchange heat with other water streams or utilities. f. Fresh water temperature and wastewater discharged temperature are given. The purpose of the following formulation is to synthesize a water network with good energy performance. The synthesis of heat exchanger network is not included in this work. 3. Model Formulation In this section, a MILP model is formulated according to the problem definition. The model adopts some constraints from Bagajewicz [9] and Feng [1]. 3.1 Mass Balance Constraints Mass balance is necessary to generate a feasible water network. Eq. (1)-(4) can be found in Bagajewicz’s work [9]. (1) (2) (3) (4) Eq. (1) is the water balance of the process. Eq. (2) is contaminant balance of the process, and it is assumed the contaminant in the water reaches maximum concentration at each process outlet. Eq. (3)
Page 4 of 13
guarantees inlet contaminant concentration is lower than required maximum value. Eq. (4) ensures minimum water usage of water network, and
is minimum water flow rate of water network
obtained by wastewater minimization approach [3]. 3.2 Logic Constraints The concepts of temperature ‘peak’ and ‘valley’ proposed Feng [1] is adopted. However, in this work, only temperature ‘valleys’ are considered. The constraints used for identifying ‘valleys’ are linearized in the proposed model to make it easier to be solved. Existence of Water Reuse. Eq. (5) determines the existence of water reuse between units. If there is water reusing from process i to process j, the water flow variable,
must be larger than zero. The binary
, in Eq. (5) must be equal to 1 to hold the constraint. (5)
Existence of Temperature ‘Valley’. In Eq. (6), if and from process
to
exists, and
,
, that water reuse from process to , then
, which denotes the existence of
temperature valley at process , equals to 1. Since the objective is to minimize the total number of ‘valleys’, if
or
is not equal to 1,
is forced to be 0. Eq. (7) defines the set of possible
temperature ‘valleys’. (6) (7) 3.3 Objective Function The objective of the proposed model is to minimize the number of ‘valleys’. With constraints Eq. (1)-(7), the proposed MILP model generates a feasible energy efficient water network. (Obj) 4. Example In this section, the proposed shortcut model is applied to a water allocation network synthesis example taken from Bagajewicz [9] and Feng [1] work. The example involves 8 water using
Page 5 of 13
processes. Fresh water temperature is 20°C, wastewater discharged temperature is 30°C, and minimum heat recovery approach temperature is 10○C, heat capacity of water is 4.2 kJ/kg○C. Water using data are listed in Table 1. In this example, the proposed model is implemented on GAMS 23.7.0 and solved by CPLEX 12.3 on an Intel(R) Core(TM) i5 3.10 GHz CPU and 8 GB of RAM Computer. Only one core is used for the model solving. 0% optimality tolerance is set. Feng’s model [1] is also implemented and solved by on GAMS for comparison. The models’ performance is listed in Table . Regarding solution efficiency, the proposed model is solved in 0.148 s with 18 iterations, faster than Feng’s 6.32 s with 34614 iterations. The relaxation gaps [10] shown in Table 2 indicate the tightness of the models. In general, the smaller the relaxation gap is, the easier the model is solved. The relaxation gap of the proposed model is 0%, which is much smaller than 3185% of Feng’s model. In terms of the models’ sizes, the proposed model has 94 variables and 61 constraints in this example; Feng’s [1] model is reformulated in GAMS resulted in 143 variables and 126 constraints, slightly different from the reported 140 variables and 103 constraints, and both are larger than the proposed model. The result water network generated by the proposed model is shown in Fig. 1. The network structure is different from Bagajewicz and Feng’s network shown in Fig. and Fig. 2, respectively. Table is the sub-stream data calculated by the proposed model. The fresh water feed is 125.943 kg/s, same as Bagajewicz [9] and Feng’s [1] networks. On the other hand, the energy consumption of the result network is 5289.61 kW, which is equal to Bagajewicz’s result, and is equal to the minimum possible hot utility target calculated by following equation [8,9]. (8) Noted that the energy consumption is greater than Feng’s 5135.76 kW [1], however, it should be pointed out that in Feng’s network, a 10 kg/s fresh water inlet to process 8 was missed in their utility calculation, leading to infeasible energy consumption lower than the minimum target. If the fresh water inlet to process 8 is taken into consideration, the minimum energy consumption of Feng’s network should be 5555.76 kW, a sub-optimal solution, higher than optimal solution of 5289.61kW.
Page 6 of 13
It is interesting to note that the resulted water network has 10 ‘peaks’ and no ‘valley’, indicating that there is no repeat heating or cooling. While according to Fig. 2, Feng’s network has 10 ‘peaks’ and 1 ‘valley’, suggesting that some repeating heating and cooling occurred. As it is mentioned in section 1 that minimizing repeating heating and cooling may result in water network with minimum energy consumption, the generated network achieves minimum energy target. Moreover, the proposed method is formulated as MILP model, thus, it guarantees the solution to be global optimum. It should also be noted that the proposed model is a shortcut model minimizing the number of temperature ‘valleys’. In case the minimum number of ‘valleys’ is not 0, that there is unavoidable repeating heating and cooling in the water network with the minimum water using constraint, the resulted energy consumption may not be the minimum. 5. Conclusion A shortcut model has been proposed to synthesize energy efficient water network. Instead of minimizing energy consumption, the proposed model minimizes the number of temperature ‘valleys’ and results in a water network with minimal energy requirement. The model is formulated as a MILP allowing it to be solved efficiently. Acknowledgement The authors would acknowledge the financial support from the Hong Kong RGC-GRF grant (613513), the UGC-Infra-Structure Grant (FSGRF13EG03), the Studentship from the Energy Concentration program of the School of Engineering at HKUST. Nomenclature Indices , ,
= water using process.
Sets ={
is a water using process}
={ |
}
={ |
}
Page 7 of 13
= temperature ‘valley’ candidates Parameters = maximum contaminant concentration of water inlet to process (ppm), = maximum contaminant concentration of water outlet from process (ppm), = contaminant load of process (mg/s), = minimum water flow rate of water network (kg/s), = heat capacity of water (kJ/kg°C), = fresh water temperature (°C), = wastewater discharged temperature (°C), = temperature of process (°C), = heat recovery approach temperature (°C), = minimum hot utility (kW). Positive variables = fresh water flow rate to process (kg/s), = water flow rate from process to (kg/s), = discharged wastewater flow rate from process (kg/s). Binary variables = existence of reusing water from process to process , = existence of temperature valley at process . Reference
Page 8 of 13
[1] X. Feng, Y. Li, R. Shen, A new approach to design energy efficient water allocation networks, Applied Thermal Engineering. 29 (11–12) (2009) 2302-2307. [2] M.M. El-Halwagi, V. Manousiouthakis, Synthesis of mass exchange networks, AIChE Journal. 35 (8) (1989) 1233-1244. [3] Y.P. Wang, R. Smith, Wastewater minimisation, Chemical Engineering Science. 49 (7) (1994) 981-1006. [4] M. Bagajewicz, M. Savelski, On the Use of Linear Models for the Design of Water Utilization Systems in Process Plants with a Single Contaminant, Chemical Engineering Research and Design. 79 (5) (2001) 600-610. [5] M.J. Savelski, M.J. Bagajewicz, On the optimality conditions of water utilization systems in process plants with single contaminants, Chemical Engineering Science. 55 (21) (2000) 5035-5048. [6] L. Savulescu, R. Smith, Simultaneous energy and water minimisation, (1998). [7] L. Savulescu, J. Kim, R. Smith, Studies on simultaneous energy and water minimisation—Part I: Systems with no water re-use, Chemical Engineering Science. 60 (12) (2005) 3279-3290. [8] L. Savulescu, J. Kim, R. Smith, Studies on simultaneous energy and water minimisation—Part II: Systems with maximum re-use of water, Chemical Engineering Science. 60 (12) (2005) 3291-3308. [9] M. Bagajewicz, H. Rodera, M. Savelski, Energy efficient water utilization systems in process plants, Computers & Chemical Engineering. 26 (1) (2002) 59-79. [10] J.M. Zamora, I.E. Grossmann, Continuous global optimization of structured process systems models, Computers & Chemical Engineering. 22 (12) (1998) 1749-1770.
Page 9 of 13
Fig. 1 Water Network of Example (kg/s)
4 7
1 3 2 5
6 8 Fig. 2 Bagajewicz’s Network (kg/s)
Page 10 of 13
1
5 7
2
3
6 4
8 Fig. 2 Feng’s Network (kg/s)
Page 11 of 13
Table 1 Water Using Data of Example Process 1 2 3 4 5 6 7 8
(g/s) 2 2.88 4 3 30 5 2 1
(ppm) 80 90 200 100 800 800 600 100
(ppm) 25 25 25 50 50 400 400 0
(°C) 40 100 80 60 50 90 70 50
Table 2 Performance of the Proposed Model and Feng’s Model
Proposed Model Feng’s Model
Continuous Variables
Binary Variables
Constraints
CPU time (s)
Iterations
Relaxation Gap (%)
43
51
61
0.148
18
0
43
100
126
6.325
34614
3185
Page 12 of 13
Table 3 Results of Example Sub-streams Flow rate/(kg/s) Path/°C Minimum energy consumption/kW 24.34 20-60-30 1022.28 1( 25 20-40-60-30 1050 2( 11.825 20-80-30 496.65 3( 5 20-80-70-30 210 4( 6.603 20-100-60-30 277.326 5( 3.175 20-100-80-30 133.35 6( 22.222 20-100-50-30 933.324 7( 17.778 20-50-30 746.676 8( 2.857 20-50-80-30 119.994 9( 7.143 20-50-90-30 300.006 10( Sum 125.943 5289.61 Number of ‘valley’ 0
Page 13 of 13