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Pressure Drop Consideration in Cooling Water Systems with Multiple Cooling Towers
I.A. Karimi and Rajagopalan Srinivasan (Editors), Proceedings of the 11th International Symposium on Process Systems Engineering, 15-19 July 2012, Singapore. © 2012 Elsevier B.V. All rights reserved.
Pressure Drop Consideration in Cooling Water Systems with Multiple Cooling Towers Khunedi V. Gololoa,b, Thokozani Majozia a
Department of Chemical Engineering, University of Pretoria, Pretoria, South Africa Council for Scientific and Industrial Research, Advanced Modelling and Digital Science, Pretoria, South Africa b
Abstract Pressure drop consideration has shown to be an essential requirement for synthesis of cooling water network where reuse/recycle philosophy is employed. This is due to an increased network pressure drop associated with additional reuse/recycle streams. This paper presents a mathematical technique for pressure drop optimization in cooling water systems consisting of multiple cooling towers. The proposed technique is based on the Critical Path Algorithm (CPA) and the superstructural approach. The CPA is used to select the cooling water network with minimum pressure drop whilst the superstructure allows for cooling water reuse. This technique which was previously used in a cooling water network with single source is modified and applied in a cooling water network with multiple sources. The mathematical formulation exhibits a mixed integer nonlinear programming (MINLP) structure. The cooling tower model is used to predict the exit conditions of the cooling towers, given the inlet conditions from the cooling water network model. Keywords: Cooling water system, Pressure drop, Critical Path Algorithm, Optimization
1. Introduction Cooling water systems are used in many industries to remove waste heat from the process to the environment. Research in this area has focused mostly on optimization and synthesis of cooling water systems in which the technique of recycle and reuse is explored. In most cases the synthesized cooling water network is more complex thus resulting in a higher pressure drop. Kim and Smith (2001) used the graphical technique to debottleneck a cooling water system with single source. Ponce-Ortega et al. (2010) also presented a mathematical model for synthesis of cooling water networks that was based on a stage wise superstructural approach. This work included the cooling tower model and the pressure drop for each cooler was considered. Panjeshahi and Ataei (2008) extended the work of Kim and Smith (2001) on cooling water system design by incorporating a comprehensive cooling tower model. Different approach was taken by Majozi and Moodley (2008) who developed a mathematical model for optimization of cooling water systems with multiple cooling towers. This work was later improved by Gololo and Majozi (2011) by incorporating the cooling tower model. In all the abovementioned work the topology of cooling water network was more complex thus prone to higher pressure drop than the conventional parallel design. Kim and Smith (2003) presented a paper on retrofit design of cooling water systems in which pressure drop was taken into consideration. The authors used graphical technique to
Pressure Drop Consideration in Cooling Water Systems with Multiple Cooling Towers
691
target the minimum circulating water flowrate and mathematical technique to design a cooling water network. This work was limited to one cooling source. This paper presents a mathematical technique for pressure drop optimization in cooling water systems consisting of multiple cooling towers. The proposed technique is based on the Critical Path Algorithm (CPA) and the superstructural approach. The CPA is used to select the cooling water network with minimum pressure drop whilst the superstructure allows for cooling water reuse. This technique was previously used by Kim and Smith (2003) to synthesize cooling water network with single source. However in this paper CPA is adapted for a cooling water network with multiple sources. Furthermore, the detailed cooling tower model is also incorporated.
2. Model development A two-step approach is employed to synthesize and optimize the cooling water system with multiple cooling towers considering pressure drop. The first step involves targeting of the minimum circulating water flowrate and in the second step the CPA is incorporated to synthesize the cooling water network with multiple cooling sources. The cooling tower model developed by Kröger (2004) is used to predict the outlet conditions of the cooling towers and the overall cooling towers effectiveness. The cooling water network model by Gololo and Majozi (2011) is improved by incorporating the modified heat exchangers and pipes pressure drop correlations of Nie and Zhu (1999) shown in Eq. (1) and Eq. (4) respectively. In this paper the correlation of Nie and Zhu (1999) is expressed in terms of mass flowrate. ΔP = N t1 m 1.8 + N t 2 m 2
(1)
2.8 0.2 where Nt1 = 1.115567μ ntp A 2.8 2.8 4.8
(2)
π
Nt2 =
ρN t d o di
20ntp 3 ρ
(3)
π 2 N t2 d i4
The line pressure drop is calculated from Eq. 4 (Kim and Smith, 2003). ΔP = N p
1
(4)
F p0.36
0.176 0.2 where N p = 188.318ρ μ L 1.8
π
(5)
The CPA is used to select the cooling water network with minimum pressure drop. Kim and Smith (2003) used the superstructure shown in Fig. 1(a). The superstructure is based on single source cooling water network. By modifying the superstructure for single source cooling water systems, a multiple sources superstructure is shown in Fig. 1(b). The CPA used by Kim and Smith (2003) is based on finding a path from source to sink with maximum pressure drop. The maximum pressure drop path is then minimized during optimization to obtain the network with minimum pressure drop. Eq. (6) is used to identify the maximum pressure drop path between the source and sink.
692
K.V. Gololo & T. Majozi.
Pm − Pn ≥ ΔPmn
(6)
Figure 1: Cooling water system superstructure; (a) Single source (b) Multiple sources
To cater for multiple sources and sinks, the superstructure in Fig. 1(b) is modified by using single imaginary source and sink as shown in Fig. 2. Eq. (7) is then used to define the pressure of source node n from the imaginary source node.
Figure 2: Multiple sources cooling water system superstructure
Eq. (8) to (11) represent the CPA adapted from Kim and Smith (2003). Eq. (12) defines the pressure at the imaginary sink node. From this equation the imaginary sink node will assume a value from all sink nodes with minimum pressure thus identifying a path with maximum pressure drop. The pressure drop of this critical path is then minimized to synthesize a cooling water network with minimum pressure drop. Sets i = { i |i is a cooling water using operation} n = { n |n is a cooling tower} PS ,img − PS ,n = ΔPimg,n
(7)
PS ,n − Pin,i + LV (1 − xn,i ) ≥ ΔPn,i
(8)
Pout,i ' − Pin,i + LV (1 − yi ',i ) ≥ ΔPi ',i
(9)
Pin,i − Pout,i = ΔPi
(10)
Pout,i − PE ,n + LV (1 − z i ,n ) ≥ ΔPi ,n
(11)
Pressure Drop Consideration in Cooling Water Systems with Multiple Cooling Towers
693
x , y and z are a binary variables indicating the existence of stream from any source n/ operation i' to operation i /sink n. LV is a large value. PE ,n − PE ,img ≥ ΔPn,img
(12)
The network topology with minimum pressure drop is then synthesized by minimizing the pressure drop between the imaginary source and sink shown in Eq. (13).
PS ,img − PE ,img = ΔP
(13)
A case study is considered which involves a cooling water system with dedicated cooling water sources and sinks. This implies that a set of heat exchanger can only be supplied by one cooling tower. No pre-mixing or post-splitting of cooling water return is allowed. However, reuse of water within the network is still allowed (Majozi and Moodley, 2008). The developed mathematical model consists of bilinear terms and binary variable thus rendering the models MINLP.
3. Solution Procedure The solution procedure involves linearization of nonlinear terms and using results from linearized model as a starting point for the exact MINLP. The bilinear terms in the cooling water network model are linearized using the Reformulation Linearization technique by Sherali and Alameddine (1992) as shown in the paper of Gololo and Majozi (2011). However, a different technique is used to linearize the pressure drop equations. Functions are first plotted within the operating range of the heat exchangers and the piecewise linearization is then used to approximate the nonlinear function (Kim and Smith, 2003).
4. Case Study Fig. 3 shows a cooling water system consisting of three cooling towers each supplying a set cooling water using operations. The total circulating water flowrate is 31.94 kg/s and the overall cooling towers effectiveness is 90%.
Figure 3: Cooling water system with multiple cooling towers (Majozi and Moodley, 2008) Fig. 4 shows synthesized cooling water system after the application of the proposed technique. The total circulating cooling water decreased by 22% due to the exploitation of reuse opportunities. The overall increase in cooling tower return temperature associated with decrease in overall circulating water flowrate resulted in a 4%
694
K.V. Gololo & T. Majozi.
improvement in effectiveness. The proposed methodology does not only debottleneck the cooling water system but also generate the network topology with the least pressure drop. This has a potential to minimize pumping cost associated with additional reuse/recycle streams. The pressure drop between sources and sinks ΔPS1, E1 , ΔPS 2, E 2 and ΔPS 3, E 3 is 19 kPa, 31 kPa and 43 kPa respectively.
Figure 4: Debottlenecked cooling water system with the minimum pressure drop
5. Conclusion The mathematical model for synthesis and optimization of cooling water systems with multiple cooling sources which takes into account the pressure drop is presented. The proposed technique is based on the CPA and the superstructural approach. The mathematical formulation developed yields MINLP structure. The case study showed a 22% decrease in circulating water flowrate due to the exploration of reuse opportunities. The return cooling tower temperature was increased which resulted in 4% improvement in the overall effectiveness. The proposed technique offers the opportunity to debottleneck the cooling water system with multiple cooling towers while maintaining minimum pressure drop and maximizing the overall cooling tower effectiveness.
References J.K. Kim, & R. Smith, (2001), Cooling water system design, Chem. Eng. Sci., 56, 3641-3658 J.M. Ponce-Ortega, M. Serna-González, & A. Jiménez-Gutiérrez, (2010), Optimization model for re-circulating cooling water systems, Comput. Chem. Eng., 34, 177-195 M.H. Panjeshahi, & A. Ataei, (2008), Application of an environmentally optimum cooling water system design in water and energy conservation, Int. J. Environ. Sci. Technol., 5(2), 251-262 T. Majozi, & A. Moodley, (2008), Simultaneous targeting and design for cooling water systems with multiple cooling water supplies, Comput. Chem. Eng., 32, 540-551 K.V. Gololo, & T. Majozi, (2011), On synthesis and optimization of cooling water systems with multiple cooling towers, Ind. Eng. Chem. Res., 50,(7), 3775-3787 J.K. Kim, & R. Smith, (2003), Automated retrofit design of cooling water system, AICHE J. ,Vol 49, No 7, 56, 1712-1730 D.G. Kröger, (2004), Air-cooled heat exchangers and cooling towers: mass transfer and evaporative cooling, Penn Well Corporation, USA X. Nie, & X.X. Xhu, (1999), Heat exchanger retrofit considering pressure drop and heat transfer enhancement, AICHE J., 45, 1239-1254 H.D. Sherali, & A. Alameddine, (1992), A new reformulation-linearization technique for bilinear programming problems, J. Global Optim., 2(4), 1992, 379-410
Pressure Drop Consideration in Cooling Water Systems with Multiple Cooling Towers Khunedi V. Gololoa,b, Thokozani Majozia a
Department of Chemical Engineering, University of Pretoria, Pretoria, South Africa Council for Scientific and Industrial Research, Advanced Modelling and Digital Science, Pretoria, South Africa b
Abstract Pressure drop consideration has shown to be an essential requirement for synthesis of cooling water network where reuse/recycle philosophy is employed. This is due to an increased network pressure drop associated with additional reuse/recycle streams. This paper presents a mathematical technique for pressure drop optimization in cooling water systems consisting of multiple cooling towers. The proposed technique is based on the Critical Path Algorithm (CPA) and the superstructural approach. The CPA is used to select the cooling water network with minimum pressure drop whilst the superstructure allows for cooling water reuse. This technique which was previously used in a cooling water network with single source is modified and applied in a cooling water network with multiple sources. The mathematical formulation exhibits a mixed integer nonlinear programming (MINLP) structure. The cooling tower model is used to predict the exit conditions of the cooling towers, given the inlet conditions from the cooling water network model. Keywords: Cooling water system, Pressure drop, Critical Path Algorithm, Optimization
1. Introduction Cooling water systems are used in many industries to remove waste heat from the process to the environment. Research in this area has focused mostly on optimization and synthesis of cooling water systems in which the technique of recycle and reuse is explored. In most cases the synthesized cooling water network is more complex thus resulting in a higher pressure drop. Kim and Smith (2001) used the graphical technique to debottleneck a cooling water system with single source. Ponce-Ortega et al. (2010) also presented a mathematical model for synthesis of cooling water networks that was based on a stage wise superstructural approach. This work included the cooling tower model and the pressure drop for each cooler was considered. Panjeshahi and Ataei (2008) extended the work of Kim and Smith (2001) on cooling water system design by incorporating a comprehensive cooling tower model. Different approach was taken by Majozi and Moodley (2008) who developed a mathematical model for optimization of cooling water systems with multiple cooling towers. This work was later improved by Gololo and Majozi (2011) by incorporating the cooling tower model. In all the abovementioned work the topology of cooling water network was more complex thus prone to higher pressure drop than the conventional parallel design. Kim and Smith (2003) presented a paper on retrofit design of cooling water systems in which pressure drop was taken into consideration. The authors used graphical technique to
Pressure Drop Consideration in Cooling Water Systems with Multiple Cooling Towers
691
target the minimum circulating water flowrate and mathematical technique to design a cooling water network. This work was limited to one cooling source. This paper presents a mathematical technique for pressure drop optimization in cooling water systems consisting of multiple cooling towers. The proposed technique is based on the Critical Path Algorithm (CPA) and the superstructural approach. The CPA is used to select the cooling water network with minimum pressure drop whilst the superstructure allows for cooling water reuse. This technique was previously used by Kim and Smith (2003) to synthesize cooling water network with single source. However in this paper CPA is adapted for a cooling water network with multiple sources. Furthermore, the detailed cooling tower model is also incorporated.
2. Model development A two-step approach is employed to synthesize and optimize the cooling water system with multiple cooling towers considering pressure drop. The first step involves targeting of the minimum circulating water flowrate and in the second step the CPA is incorporated to synthesize the cooling water network with multiple cooling sources. The cooling tower model developed by Kröger (2004) is used to predict the outlet conditions of the cooling towers and the overall cooling towers effectiveness. The cooling water network model by Gololo and Majozi (2011) is improved by incorporating the modified heat exchangers and pipes pressure drop correlations of Nie and Zhu (1999) shown in Eq. (1) and Eq. (4) respectively. In this paper the correlation of Nie and Zhu (1999) is expressed in terms of mass flowrate. ΔP = N t1 m 1.8 + N t 2 m 2
(1)
2.8 0.2 where Nt1 = 1.115567μ ntp A 2.8 2.8 4.8
(2)
π
Nt2 =
ρN t d o di
20ntp 3 ρ
(3)
π 2 N t2 d i4
The line pressure drop is calculated from Eq. 4 (Kim and Smith, 2003). ΔP = N p
1
(4)
F p0.36
0.176 0.2 where N p = 188.318ρ μ L 1.8
π
(5)
The CPA is used to select the cooling water network with minimum pressure drop. Kim and Smith (2003) used the superstructure shown in Fig. 1(a). The superstructure is based on single source cooling water network. By modifying the superstructure for single source cooling water systems, a multiple sources superstructure is shown in Fig. 1(b). The CPA used by Kim and Smith (2003) is based on finding a path from source to sink with maximum pressure drop. The maximum pressure drop path is then minimized during optimization to obtain the network with minimum pressure drop. Eq. (6) is used to identify the maximum pressure drop path between the source and sink.
692
K.V. Gololo & T. Majozi.
Pm − Pn ≥ ΔPmn
(6)
Figure 1: Cooling water system superstructure; (a) Single source (b) Multiple sources
To cater for multiple sources and sinks, the superstructure in Fig. 1(b) is modified by using single imaginary source and sink as shown in Fig. 2. Eq. (7) is then used to define the pressure of source node n from the imaginary source node.
Figure 2: Multiple sources cooling water system superstructure
Eq. (8) to (11) represent the CPA adapted from Kim and Smith (2003). Eq. (12) defines the pressure at the imaginary sink node. From this equation the imaginary sink node will assume a value from all sink nodes with minimum pressure thus identifying a path with maximum pressure drop. The pressure drop of this critical path is then minimized to synthesize a cooling water network with minimum pressure drop. Sets i = { i |i is a cooling water using operation} n = { n |n is a cooling tower} PS ,img − PS ,n = ΔPimg,n
(7)
PS ,n − Pin,i + LV (1 − xn,i ) ≥ ΔPn,i
(8)
Pout,i ' − Pin,i + LV (1 − yi ',i ) ≥ ΔPi ',i
(9)
Pin,i − Pout,i = ΔPi
(10)
Pout,i − PE ,n + LV (1 − z i ,n ) ≥ ΔPi ,n
(11)
Pressure Drop Consideration in Cooling Water Systems with Multiple Cooling Towers
693
x , y and z are a binary variables indicating the existence of stream from any source n/ operation i' to operation i /sink n. LV is a large value. PE ,n − PE ,img ≥ ΔPn,img
(12)
The network topology with minimum pressure drop is then synthesized by minimizing the pressure drop between the imaginary source and sink shown in Eq. (13).
PS ,img − PE ,img = ΔP
(13)
A case study is considered which involves a cooling water system with dedicated cooling water sources and sinks. This implies that a set of heat exchanger can only be supplied by one cooling tower. No pre-mixing or post-splitting of cooling water return is allowed. However, reuse of water within the network is still allowed (Majozi and Moodley, 2008). The developed mathematical model consists of bilinear terms and binary variable thus rendering the models MINLP.
3. Solution Procedure The solution procedure involves linearization of nonlinear terms and using results from linearized model as a starting point for the exact MINLP. The bilinear terms in the cooling water network model are linearized using the Reformulation Linearization technique by Sherali and Alameddine (1992) as shown in the paper of Gololo and Majozi (2011). However, a different technique is used to linearize the pressure drop equations. Functions are first plotted within the operating range of the heat exchangers and the piecewise linearization is then used to approximate the nonlinear function (Kim and Smith, 2003).
4. Case Study Fig. 3 shows a cooling water system consisting of three cooling towers each supplying a set cooling water using operations. The total circulating water flowrate is 31.94 kg/s and the overall cooling towers effectiveness is 90%.
Figure 3: Cooling water system with multiple cooling towers (Majozi and Moodley, 2008) Fig. 4 shows synthesized cooling water system after the application of the proposed technique. The total circulating cooling water decreased by 22% due to the exploitation of reuse opportunities. The overall increase in cooling tower return temperature associated with decrease in overall circulating water flowrate resulted in a 4%
694
K.V. Gololo & T. Majozi.
improvement in effectiveness. The proposed methodology does not only debottleneck the cooling water system but also generate the network topology with the least pressure drop. This has a potential to minimize pumping cost associated with additional reuse/recycle streams. The pressure drop between sources and sinks ΔPS1, E1 , ΔPS 2, E 2 and ΔPS 3, E 3 is 19 kPa, 31 kPa and 43 kPa respectively.
Figure 4: Debottlenecked cooling water system with the minimum pressure drop
5. Conclusion The mathematical model for synthesis and optimization of cooling water systems with multiple cooling sources which takes into account the pressure drop is presented. The proposed technique is based on the CPA and the superstructural approach. The mathematical formulation developed yields MINLP structure. The case study showed a 22% decrease in circulating water flowrate due to the exploration of reuse opportunities. The return cooling tower temperature was increased which resulted in 4% improvement in the overall effectiveness. The proposed technique offers the opportunity to debottleneck the cooling water system with multiple cooling towers while maintaining minimum pressure drop and maximizing the overall cooling tower effectiveness.
References J.K. Kim, & R. Smith, (2001), Cooling water system design, Chem. Eng. Sci., 56, 3641-3658 J.M. Ponce-Ortega, M. Serna-González, & A. Jiménez-Gutiérrez, (2010), Optimization model for re-circulating cooling water systems, Comput. Chem. Eng., 34, 177-195 M.H. Panjeshahi, & A. Ataei, (2008), Application of an environmentally optimum cooling water system design in water and energy conservation, Int. J. Environ. Sci. Technol., 5(2), 251-262 T. Majozi, & A. Moodley, (2008), Simultaneous targeting and design for cooling water systems with multiple cooling water supplies, Comput. Chem. Eng., 32, 540-551 K.V. Gololo, & T. Majozi, (2011), On synthesis and optimization of cooling water systems with multiple cooling towers, Ind. Eng. Chem. Res., 50,(7), 3775-3787 J.K. Kim, & R. Smith, (2003), Automated retrofit design of cooling water system, AICHE J. ,Vol 49, No 7, 56, 1712-1730 D.G. Kröger, (2004), Air-cooled heat exchangers and cooling towers: mass transfer and evaporative cooling, Penn Well Corporation, USA X. Nie, & X.X. Xhu, (1999), Heat exchanger retrofit considering pressure drop and heat transfer enhancement, AICHE J., 45, 1239-1254 H.D. Sherali, & A. Alameddine, (1992), A new reformulation-linearization technique for bilinear programming problems, J. Global Optim., 2(4), 1992, 379-410