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An improved series-parallel optimization approach for cooling water system
Applied Thermal Engineering 154 (2019) 368–379
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
An improved series-parallel optimization approach for cooling water system a
a,⁎
a
Shiqi Liu , Jinchun Song , Jia Shi , Bin Yang a b
T
a,b
School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China Shanghai Baosteel Company, Shanghai 201900, China
H I GH L IG H T S
configuration is proposed to reduce the water flow rate. • ATwoseries-parallel kinds of conditions with or without static pressure drop are considered. • A main-auxiliary pump structure is applied to reduce the operating consumption. • Heat exchanger and pump networks are optimized simultaneously. • The number of reusemain-auxiliary branch for the economic performance is analyzed. •
A R T I C LE I N FO
A B S T R A C T
Keywords: Cooling water system Series-parallel configuration Static pressure drop Reuse branch Mixed-integer nonlinear programming
Conventional cooling networks are usually operated in a parallel configuration. Although this configuration is easy operated, it results in inefficient use of water. A series configuration can effectively reduce water flow rate, while its arrangement is relatively complex, incurring more operating and capital cost. This paper proposes a water and energy conservation strategy to save energy cost and to simplify the network structure. Heat exchangers are arranged in series-parallel configuration to reduce the water flow rate for water conservation, and two kinds of installation conditions with or without static pressure drop are taken into account for energy conservation. A main-auxiliary pump structure is applied to reduce the operating consumption. The number of reuse branch for the economic performance is analyzed. Mixed-integer nonlinear programming based on a superstructure description is formulated by considering the configuration of pumps and heat exchanger networks simultaneously. Two case studies are employed to demonstrate the effectiveness of the proposed approach. The results show that the improved series-parallel configuration approach can save 10.9% of the total cost compared with the two-step sequential optimization and 4.7% compared with the simultaneous optimization method.
1. Introduction Cooling water system is one of the major water-using operations in industries to dissipate heat of process industries, power stations and chemical plants. Generally, the cooling water system is consisted by three major components: a heat exchanger network, a pump network and cooling towers. The irrational design will greatly increase the capital cost and the non-economic operation will lead to enormous consumption of energy and water. Therefore, the optimization of cooling water system is of great significance to improve the energy efficiency and economic performance of the industrial system. Research in this field has primitively focused on individual components but not the entire cooling water system. Traditional heat exchanger networks were invariably designed with parallel configuration, which might provoke a tremendous amount of fresh water consumption ⁎
and an inefficient use of cooling tower capacity. Therefore, reducing water consumption is the foremost concern of researchers. Approaches by using pinch technology [1,2] in a series configuration is proposed to reduce water consumption. Then more factors and approaches are considered such as pinch migration [3], heat exchanger and pipe constraints [4], waste water minimization [5], cooling tower performance [6,7], chilled water network [8], unified targeting algorithm [9]. Changing parallel configuration into series can reduce water flow rate and raise the outlet temperature significantly. However, if the network is complex or new heat exchangers are added into the network, the configuration might be retrofitted again. So pinch design approach might be a lack of flexibility. Then, mathematical models were established by using intermediate main approach [10], supplier and receiver arrangement [11], system mechanism analysis [12], steady state response analysis [13], or several stage arrangements [14–17]. Even
Corresponding author. E-mail address: [email protected] (J. Song).
https://doi.org/10.1016/j.applthermaleng.2019.03.048 Received 31 October 2018; Received in revised form 25 February 2019; Accepted 11 March 2019 Available online 18 March 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.
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temperature and cold stream outlet temperature, ℃ mass flow rate, kg/s mass flow rate of cold stream i, kg/s inlet mass flow rate from heat exchanger j to heat exchanger i, kg/s fout (i, j ) outlet mass flow rate from heat exchanger i to heat exchanger j, kg/s ftin (i) mass flow rate of fresh water from cooling tower to heat exchanger i, kg/s ftout (i) mass flow rate of useless water from heat exchanger i to cooling tower, kg/s ftotal total inlet/outlet mass flow rate, kg/s Hmain main pump pressure head, m Hauxi (i) auxiliary pump pressure head, m H (i) pressure head of HE i, m Kt (i) parameter used in relating physical properties to pressure drop for heat exchanger i Pressure drop coefficient of pipeline kpipe operating cost of pump, $ OCpump pressure head at point i Pi P i' pressure head at point i′ PA pressure head at point A PB pressure head at point B pressure head at point C of heat exchanger i PC(i) pressure head at point D of heat exchanger i PD(i) PE pressure head at point E PF pressure head at point F Tcout (i) outlet temperature of cold stream i, ℃ Tcin (i) inlet temperature of cold stream i, ℃ outlet temperature of heat exchanger network, ℃ tout total weight of the network, kg W binary variables between point i and i′ Y i', i Δpe (i) pressure drop of heat exchanger i, Pa pressure drop between point i and i’ ΔP i', i ΔpAB pressure drop between point A and B ΔpBC(i) pressure drop between point B and C for heat exchanger i ΔpC(i)D(i) pressure drop between point C and D for heat exchanger i ΔpD(i)C(j) pressure drop between point D for heat exchanger i and point C for heat exchanger j ΔpD(i)E pressure drop between point D for heat exchanger i and point E ΔpEF pressure drop between point E and F Δppipe (AB) pipe pressure drop between point A and B Δppipe (BC(i)) pipe pressure drop between point B and C for heat exchanger i Δppipe (D(i)C(j )) pipe pressure drop between point D for heat exchanger i and point C for heat exchanger j Δppipe (D(i)E) pipe pressure drop between point D for heat exchanger i and point E Δppipe (EF) pipe pressure drop between point E and F Δpmainpump pressure drop of main pump Δpauxipump pressure drop of auxiliary pump
Nomenclature Sets
f fc (i) fin (i, j )
I={i , j | i and j are heat exchangers}
Parameters
Af annualized factor Ap unit price per kilogram for pipe, $/kg cpc specific heat capacity of cooling water, kJ/(kg ℃) internal tube diameter, m di external tube diameter, m do ew unit cost of water, $/m3 ee unit cost of electricity, $/kW·h heat capacity of hot stream i, kW/℃ Fcp (i) g gravity constant, m/s2 annual operating time, h/year ht Htower installation height of cooling tower h (i) film transfer coefficient of hot stream i, W/(m2 ℃) film transfer coefficient of cooling water, W/(m2 ℃) hw Hins (i) installation height of heat exchanger i, m Height difference between HE i and HE j Hdi, j Hdi,positive Positive Height difference between HE i and HE j j conductivity of cooling water, W/m ℃ kt LV a large value L (i) lower bound mass flow rate of heat exchanger i, kg/s pipe length, m l Lpmain lower bound pressure head of main pump, m Lpauxi lower bound pressure head of auxiliary pump, m maximum number of reuse branch Nmax q (i) heat load of hot stream i, kW Reynolds number Re inlet temperature of heat exchanger network, ℃ tin Thout (i) outlet temperature of hot stream i, ℃ Thin (i) inlet temperature of hot stream i, ℃ U (i) upper bound mass flow rate of heat exchanger i, kg/s Upmain upper bound pressure head of main pump, m Upauxi upper bound pressure head of auxiliary pump, m v pipe velocity, m/s Cfpump, Cpump, γ capital parameter of pump capital cost parameter of heat exchanger x , y, z ΔTmin temperature approach between two streams, ℃ Variables exchange area for heat exchanger i, m2 binary variable for mass flow rate from heat exchanger j to heat exchanger i Bout (i, j ) binary variable for mass flow rate from heat exchanger i to heat exchanger j Bpump (i) binary variables of auxiliary pumps Btin (i) binary variable for mass flow rate from cooling tower to heat exchanger i Btout (i) binary variable for mass flow rate from heat exchanger i to cooling tower capital cost of pump, $ CCpump CCwater fresh water cost, $ pipe cost, $ CCpipe CCexch capital cost of heat exchanger, $ din (i) temperature approach between hot stream outlet temperature and cold stream inlet temperature, ℃ dout (i) temperature approach between hot stream inlet
Ar (i) Bin (i, j )
Greek letters
φt μ ρ ω η ρp
369
viscosity correction cooling water viscosity, Ns/m2 density of cooling water, kg/m3 pipe parameter efficiency of pumps density of pipe, kg/m3
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pressure head of the cooler were taken into account. Then, Ma et al. [29] analyzed energy recovery strategies for auxiliary pumps and hydro turbines. Bonvin et al. [30] proposed pump scheduling in a class of branched water networks, which achieved significant savings in terms of operating costs and energy. Even though the consideration of flow rate and pressure drop had been reported in most of the published work, the design issues associated with flow rate, pressure drop, pumping and installation had not been thoroughly addressed. This paper comprehensively studies the water and energy conservation strategy to minimize the capital cost and operating cost for the cooling water system. In term of water conservation, heat exchangers are arranged in series-parallel configuration so that water can be reused greatly and the inlet temperature of the cooling tower can be increased, thereby improving the cooling tower's efficiency. For energy conservation, the configuration is carried out by combining a main pump with auxiliary pumps. The pressure drop of pipes and heat exchangers is optimized to reduce pump operating cost. Two kinds of installation conditions (with or without the static pressure drop) are taken into account. Reducing the number of branches can effectively reduce the capital cost and simplify pump and heat exchanger network. Changing the structure of the pump network can effectively reduce the operating cost. Adding auxiliary pumps [24] might reduce operating cost but increase the capital cost. The selection of main and auxiliary pumps has an important impact on reducing the economic cost. The economic cost including pump operating and capital cost, pipe and heat exchanger capital cost, water cost must be taken into consideration. The optimization network is a complicated structure which includes intricate connections between its components. A simplified solution strategy has been taken to effectively solve computational problems associated with a mixed-integer nonlinear programming (MINLP) model. In this automated optimization approach, the water consumption and system operating and capital cost can reach the optimal value simultaneously. It is efficient to improve the economic performance and reduce the operating and capital cost of the cooling water system.
though these approaches are novel, they are less functional than the series configuration because of the intricate arrangements. The above work can significantly reduce the water flow rate, but they ignore the impact of pressure drop along pipes and heat exchangers, leading to high pump operating costs. Kim and Smith [2] considered an approach called “critical path algorithm” to search for the maximum pressure path and developed an automated design approach for cooling water networks. Gololo et al. [18–20] set a multiple cooling tower and heat exchanger network model, in which four cases with or without a specific source or sink and the pressure drop constraints were taken into consideration. These works mainly concentrate on the frictional pressure drop but ignore the static pressure drop. Their configurations require a higher pressure head for the main pump, which may provoke a higher operating cost for the pump network. Then location [21] and installation [22] were added to the optimization of cooling water system with multiple towers which discussed the interaction of various aspects in the cooling water system. But these approaches will increase the number of branches between towers and heat exchangers, resulting in additional pipe capital cost, low efficiency of cooling towers and complex pipe network configuration. In order to optimize pump operating cost through considering both the pressure head and flow rate, adding auxiliary pumps in the pump network can avoid the energy penalty by reducing the opening of valves for some heat exchangers. Therefore the pressure head of the main pump can be reduced. Hung et al. [23] optimized the water system considering pressure drop in pipes and added both a main pump and auxiliary pumps into the water network. The results showed that the cost could be much reduced by adding auxiliary pumps to the network. Sun et al. [24] optimized the model by using simulated annealing algorithm considering the static pressure drop in heat exchangers. The results indicated that adding auxiliary pumps might reduce the pump operating cost when the heat exchangers were installed at high positions. Then, Sun J et al. [25] optimized the cooling water system model by changing the model from parallel to series-parallel. Castro et al. [26] established an optimization model to minimize the operating cost of the cooling water system. Ma et al. [27,28] optimized the pump work and cooler networks simultaneously in which both the installation and the
Fig. 1. Superstructure of the cooling water system in the same installation height. 370
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Fig. 2. Superstructure of the cooling water system in different installation height.
2. Problem statement
is the specific heat capacity of hot stream i.
q (i) = (Thin (i) − Thout (i))·Fcp (i)
The superstructure of the cooling water system is illustrated in Fig. 1 and Fig. 2. In Fig. 1, pumps and heat exchangers are located at the same ground altitude. The pump head is composed of the heat exchanger and pipe pressure drop and the static pressure drop in the cooling tower. Energy conservation with auxiliary pumps is not necessary due to the fact that the heat exchanger is at ground altitude without static pressure drop. In Fig. 2, heat exchangers in various units have different installation altitude, leading to different static pressure head. A pump network with a main pump and auxiliary pumps is proposed to minimize the operating cost. In the cooling water system, each stream has different heat and distance parameters. The flow rates, inlet and outlet temperatures of hot streams and film transfer coefficients of both hot and cold streams are known. The flow rate of heat exchangers, pressure head loss of each branches and additional pressure head of installation height are optimized simultaneously to reduce economic consumption. Variables in this paper include mass flow rate, outlet temperature, area of the heat exchangers, pressure drop of exchangers and pipes, the pressure head of the main pump and auxiliary pumps, and reuse branches. The aim of this paper is to identify a superstructure of cooling water system to minimize the annual cost. The mathematical model developed in this paper employs several assumptions as follows:
(1)
Generally, to satisfy the requirements of heat transfer, the temperature approaches between cold and hot streams are required to be larger than the minimum temperature difference. And the inlet temperature of cooling water should be lower than the outlet temperature, as is shown as Eqs. (2)–(4).
ΔTmin ≤ Thout (i) − Tcin (i) = din (i)
(2)
ΔTmin ≤ Thin (i) − Tcout (i) = dout (i)
(3)
Tcout (i) ≥ Tcin (i)
(4)
In HE i, the heat load between cold and hot stream i are the same in Eq. (5), where cpc is the specific heat capacity of cooling water.
q (i) = (Tcout (i) − Tcin (i))·cpc·fc (i)
(5)
Point C(i) is to blend fresh water and other cold streams. The water flow rate of HE i is made up of fresh water and the reuse water from other heat exchangers. The mass flow rate constraint is shown as Eq. (6). Point D(i) facilitates the distribution of the cold stream to other heat exchangers as well as to the cooling tower. The mass flow rate constraint is shown as Eq. (7). n
fc (i) =
∑
fin (i, j ) + ftin (i)
j = 1, j ≠ i
1. Each hot stream corresponds to only one cold stream and the cold stream only includes water. 2. The exchangers are all 1-1 counter current shell and tube exchangers. The cold streams flow along the tube side and hot streams flow along the shell side. 3. The specific heat capacities of both hot and cold streams are consistent throughout the temperature range. 4. One cooling tower can satisfy the cooling requirements of all heat exchangers and water loss and make-up water can be ignored.
(6)
n
fc (i) =
∑
fout (i, j ) + ftout (i)
j = 1, j ≠ i
(7)
In the above equations, fin (i, j ) is inlet mass water flow rate from HE j to HE i and fout (i, j ) is outlet mass water flow rate from HE i to HE j. Eq. (8) is the constraint between inlet and outlet mass flow rate. ftin (i) is the fresh water from the cooling water. ftout (i) is the hot water to the cooling tower. The total mass flow rate is indicated by Eq. (9).
fin (i, j ) = fout (j, i) n
3. Mathematical model
ftotal =
∑ ftin (i) = ∑ ftout (i) i=1
3.1. Temperature and flow rate constraints of the HE network
(8) n i=1
(9)
Due to the different inlet temperature, some heat exchangers might not need fresh water or reuse water.Bin (i, j ) and Bout (i, j ) are composed by binary variables to limit reuse water from HE i to HE j or from HE j to HE i. AndBtin (i) , Btout (i) are binary variables to limit the inlet and outlet mass flow rate from/to the cooling tower. For HE i and j, The upper and lower bounds of the mass flow constraints are shown as follows:
In Figs. 1 and 2, Point A-F are placed to guarantee the logical location and to calculate expediently. The heat exchanger (HE) and pump network start at point A and end at point F. Depending on the known parameters of hot streams, the constraints condition of cold streams is determined. The transfer heat in hot stream i is as follows, where Fcp (i) 371
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Table 2 Physical property data.
Fig. 3. Pressure drop constraints of heat exchangers and pipes.
L (i)·Bin (i, j ) ≤ fin (i, j ) ≤ U (i)·Bin (i, j )
(10)
L (i)·Bout (i, j ) ≤ fout (i, j ) ≤ U (i)·Bout (i, j )
(11)
L (i)·Btin (i) ≤ ftin (i) ≤ U (i)·Btin (i)
(12)
L (i)·Btout (i) ≤ ftout (i) ≤ U (i)·Btout (i)
(13)
Bin (i, j ) = Bout (i, j ) = 0
(14)
i=j
L (i) and U (i) are the lower and upper bounds mass flow rates of HE i. The number of reuse branches should be limited to a specific value, which is as follows: n
n
∑i =1 ∑ j=1 Bin (i, j) ≤ Nmax n
(15)
n
∑i =1 ∑ j=1 Bout (i, j) ≤ Nmax
(16)
At the mixing point, the mixing temperature should satisfy the following equation.tin and tout represent for the inlet and outlet temperature of HE network, respectively. n
∑
fc (i)·Tcin (i) =
(fin (i, j )·Tcout (j )) + tin·ftin (i)
(17)
j = 1, j ≠ i
n
tout · ∑ ftout (i) = i=1
Items
Data
Annualized factor Af Unit price per kilogram for pipe Ap
0.298 0.78 $/kg
Specific heat capacity of cooling water cpc Unit cost of water ew Unit cost of electricity ee Gravity constant g Annual operating time ht Film transfer coefficient of cooling water hw conductivity of cooling water kt A large value LV Lower bound mass flow rate of heat exchanger iL (i) Inlet temperature of heat exchanger network tin Lower bound pressure head of main pump, Lpmain Upper bound pressure head of main pump, Upmain Lower bound pressure head of auxiliary pump, Lpauxi Upper bound pressure head of auxiliary pump, Upauxi Maximum number of reuse branch Temperature approach between two streams, ΔTmin Viscosity correction, φt Cooling water viscosity, μ Density of cooling water, ρ Density of pipe, ρp
4.18 kJ/(kg°C) 2 × 10-6$/m3 0.1 $/kW·h 9.8 m/s2 8000 h/year 2500 W/m2°C 0.6 W/m°C 10,000 2 kg/s
Efficiency of pumps Pipe length (m), internal diameter (m), water velocity (m/s) for pipe AB Pipe length (m), internal diameter (m),water velocity (m/s) for pipe EF Internal diameter (m),water velocity (m/s) for pipe BC(i) and D(i)E Installation height of cooling tower, Htower Heat exchanger capital cost, CCexch
0.7 100, 0.9, 3
∑i = 1 (30, 000 + 1000Ar (i)0.5 )
Pump capital cost, CCpump
2000 + 5(f Δp/ ρ)0.68
20 °C 10 m 35 m 0m 30 m 4 10 °C 1.05 10-3Ns/m2 995.7 kg/m3 1340 kg/m3
120, 0.9, 3 0.6, 1.5 8m n
n
∑ (ftout (i)·Tcout (i))
For HE i, the area can be calculated according to the film transfer coefficient of the cold stream and the hot stream.
(18)
i=1
The improved series-parallel configuration based on the above equations can reduce the water consumption of the system obviously due to the water reuse. But in a series configuration, the pressure drop will increase significantly, which will increase the operating cost of pumps [2]. So the pressure drop is an essential aspect for system optimization.
[din (i )·dout (i )·
din (i) + dout (i) 1/3 ] 2
·(
1 1 ) + h (i) hw
(20)
Constant Kt (i) can be calculated as follows:
Kt (i) = 3.2. Pressure drop constraints of the HE network
φt4.5 ·di0.5·μ11/6 d · i 2.5 0.023 ·fc (i)·ρ ·kt7/3·cpc7/6 do
(21)
The pressure drop of heat exchangers can be calculated in the above Eqs. (19)–(21). And the pipe pressure drop can be calculated as follows [24].
On the tube side of cold streams, the pressure drop of the HE i is related to constant Kt (i) , contact area Ar (i) , as well as film transfer coefficient[27], which can be calculated as follows:
Δpe (i ) = Kt (i )·Ar (i )·hw 3.5
q (i)
Ar (i) =
Δppipe = kpipe·
(19)
f2 ρ2
(22)
Table 1 Hot stream data. Stream i
Thin(i)/°C
Thout(i)/°C
Fcp (i) / (kW/℃)
q(i) /kW
h(i)/ (W/m2℃)
Pipe length for BC(i)/m
Pipe length for D(i)E/m
Case Study 1
1 2 3 4
50 55 75 75
30 45 45 60
75 400 100 200
1500 4000 3000 3000
785 660 865 966
100 100 125 150
175 200 175 150
Case Study 2
1 2 3 4 5
50 50 105 85 90
30 40 40 65 55
200 150 60 100 80
4000 1500 3900 2000 2800
854 743 720 1352 450
100 100 125 175 100
100 125 175 150 100
372
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kpipe =
8ρ ·l·ω π 2·di5
ω = 0.0032 +
follows. Δpmainpump and Δpauxipump (i) are the pressure head of the main pump and the auxiliary pump i respectively.
(23)
0.221 Re 0.237
(24)
ρ · di · v Re = μ
(25)
The pressure drop can be calculated according to the above equations. But if the network is unknown, the total pressure drop between point B and point E may not be calculated because whether the pipes exist is not certain. The pressure drop of the system is determined not only by the individual pipes and heat exchangers, but also by the configuration of the network. So it is not sufficient to calculate the pressure drop automatically by the above equation. According to the “critical path algorithm” [19–22], the overall pressure drop can be obtained by transferring the unknown constraints into inequality constraints. The critical path algorithm is used to find a maximum pressure drop path between two points. Suppose i and i′ are a starting point and an ending point respectively. Eq. (26) should be satisfied if the pipe exists between two points. Otherwise, logic constraints are formulated by combining the binary variables with a sufficiently large value to consider whether a connection between a node is activated (Y i', i = 1) or not (Y i', i = 0). In Eq. (27), after introducing a large value (LV), if a node connected with other node (Y i', i = 1), inequality constraints are strictly activated. Otherwise (Y i', i = 0), the inequality constraints are relaxed by a sufficiently value, so the inequality constraints are strictly activated as well. In this solving strategy, the maximum pressure drop path can be determined in the optimization progress.
P i' − Pi ≥ ΔP i', i
(
)
(28)
PB − PC(i) + Δpauxipump (i) + LV ·(1 − Btin (i)) ≥ ΔpBC(i)
(29)
PC(i) − PD(i) = ΔpC(i)D(i)
(30)
PD(i) − PC(j) + LV ·(1 − Bout (i, j )) ≥ ΔpD(i)C(j)
(31)
PD(j) − PE + LV ·(1 − Btout (j )) ≥ ΔpD(j)E
(32)
PE − PF = ΔpEF
(33)
3.3. Installation height and pump network constraints The influence of static pressure drop is discussed in the following two scenarios. In the first scenario, due to the low installation position of the heat exchanger, the static pressure drop can be ignored. The pump head must satisfy the static pressure head of the cooling tower. For the right side of Eqs. (28)–(33), the pressure drop between two points can be calculated as Eqs. (34)–(39).
(26)
P i' − Pi + LV · 1 − Y i', i ≥ ΔP i', i
PA − PB + Δpmainpump = ΔpAB
(27)
ΔpAB = Δppipe (AB)
(34)
ΔpBC(i) = Δppipe (BC(i))
(35)
ΔpC(i)D(i) = Δpe (i)
(36)
ΔpD(i)C(j) = Δppipe (D(i )C(j ))
(37)
ΔpD(i)E = Δppipe (D(i)E)
(38)
ΔpEF = Δppipe (EF) + Htower ·ρg
(39)
In the second scenario, the static pressure drop of heat exchangers cannot be ignored. Adding auxiliary pumps can reduce the operating cost of the main pump if the static pressure drop is high. Ma [27,28] discussed when the heat exchangers were arranged at different
Fig. 3 shows the pressure drop constraints in one of the possible branches according to the “critical path algorithm”. The pressure drop inequality constraints between point A and point F can be calculated as
Fig. 4. Network configuration for scenario 1, 2 and 3. 373
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Table 3 Comparison of three different configurations. Parameters
Parallel configuration
Series configuration
Series-parallel configuration
Total flowrate (kg/s) Number of reuse branch Total outlet temperature (℃) Total area of heat exchangers (m2) Main pump pressure head (m) Cost of cooling water ($) Operational cost of main pump ($) Annual capital cost of main pump ($) Annual capital cost of heat exchangers ($) Annual pipe cost ($) Total cost ($)
185.3 0 40.7 1109 9.3 10673.3 19252.6 1712.7 54730.3 30822.1 117191.0
67.8 3 61.4 1618 11.3 3905.3 8559.3 1239.5 59561.3 26203.1 99468.5
74.9 3 56.7 1050 9.3 4315.5 7756.3 1197.8 55659.5 21122.2 90051.3
Fig. 5. Two-step sequential optimization configuration.
Δpmainpump + Δpauxipump (i) ≥ H (i)·ρg
installation height, the pump head depended on the location of the highest heat exchanger in the branch. Without knowing the configuration of the branches, the total static pressure head can be calculated automatically according to the following equations.
Hdi, j = Hins (i) − Hins (j )
In scenario 2, for the right side of Eqs. (28)–(33), the pressure drop between two points can be calculated as Eqs. (44)–(49).
(40)
Hdi, j is the height difference between HE i and HE j. When the height of HE i is higher than HE j, Hdi, j is positive. Otherwise, Hdi, j is negative. Hdi,positive can be defined by the following equation. j
Hdi,positive = max(0, Hdi, j ) j
+
(44)
ΔpBC(i) = Δppipe (BC(i)) + Hins (i)
(45)
ΔpD(i)C(j) = Δppipe (D(i )C(j )) + Bout (i,
(41)
(46)
j )·Hdi,positive j
(47)
ΔpD(i)E = Δppipe (D(i)E)
(48)
ΔpEF = Δppipe (EF) + Htower ·ρg
(49)
Binary variable Bpump (i) is used for confirming the existence of the auxiliary pumps. Bpump (i) = 1 means there is an auxiliary pump between point B and point C(i), otherwise not. The auxiliary pump should be arranged at the beginning of the heat exchanger network, in other words, if the fresh water pipeline BC(i) does not exist, the auxiliary pump for the pipeline of HE i will not exist correspondingly.
n
∑ j=1,j≠i Bout (i, j)·Hdi,positive j
n
+ (PC(i) − PF )/(ρg) ∑ j=1,j≠i Bin (i, j)·Hdi,positive j
ΔpAB = Δppipe (AB)
ΔpC(i)D(i) = Δpe (i)
For HE i, the total head is the sum of height head and pressure head. The height head of HE i is produced by the highest position head in the branch of the HE i, as the following equation.
H (i) = Hins (i) +
(43)
(42)
The pressure head provided by the main pumpΔpmainpump and the auxiliary pump Δpauxipump (i) must satisfy the head requirement of the HE i.
Lpmain ≤ Δpmainpump ≤ Upmain 374
(50)
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Fig. 6. Simultaneous optimization configuration.
Table 4 Physical and economic data between different approaches. Parameters
Two-step sequential optimization
Simultaneous optimization
Improved series-parallel optimization
Total flow rate (kg/s) Total outlet temperature (°C) Total area of heat exchangers (m2) Main pump pressure head (m) Cost of cooling water ($) Operational cost of main pump ($) Annual capital cost of main pump ($) Operational cost of auxiliary pump ($) Annual capital cost of auxiliary pump ($) Annual capital cost of heat exchangers ($) Total cost ($)
115.8 49.3 1214.7 19.8 6671.8 26803.9 1992.1 8423.7 1231.5 64859.9 109982.8
98.4 54.5 1406.3 20.6 5669.5 24357.1 1904.2 1990.4 834.3 68108.7 102864.2
84.5 60.2 1612.2 19.5 4865.8 18444.2 1680.6 1301.0 1447.0 70299.5 98038.1
Lpauxi ·Bpump (i)·Btin (i) ≤ Δpauxipump (i) ≤ Upauxi ·Bpump (i)·Btin (i)
(51)
Hauxi (i) =
Δpauxipump (i) ρg
(55)
CCpump represents for the pump capital cost, Af is an annualized factor considering life expectancy. The total capital cost is as follows.
3.4. Objective function The objective of this optimization model is to minimize total annualized cost. The objective function is the sum of freshwater cost, the cost of pipelines and heat exchangers and the capital and operating cost of the pumps.
CCpump = Af ·[Cfpump + Cpump·(Hmain·ftotal ·g)γ ] + Af ·Bpump (i)·
obj function = OCpump + CCpump + CCpipe + CCexch + CCwater
CCexch = Af · ∑
OCpump = OCmain + OCauxi = ( Bpump (i))·ht ·ee
Hmain·ftotal ·g 1000η
+
n
∑i =1
Hauxi (i)·ftin (i)·g 1000η
n
∑i =1 [Cfpump + Cpump·(Hauxi (i)·ftin (i)·g)γ ] The capital cost of heat exchangers is as follows. n
(52)
i=1
(53)
CCpipe = Af ·W ·Ap W=
Δpmainpump ρg
(x + y·Ar z (i))
(57)
The capital cost of a pipeline is equal to the product of weight and price.W represents for the network total weight, Ap represents for the unit price per kilogram.
·
In the above equation, OCpump represents for the pump operating cost, the pump head Hmain and Hauxi (i) can be calculated as follows.
Hmain =
(56)
di =
(54) 375
π 2 (do − di2)·l·ρp 4 4f + 0.3 πρv
(58) (59)
(60)
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Fig. 7. Improved series-parallel configuration with one reuse branch.
Fig. 8. Improved series-parallel configuration with two reuse branches.
do = 1.101di + 0.006
series–parallel superstructure, temperature and flow rate constraints (Eqs. (1)–(18)), pressure drop constraints (Eqs. (19)–(49)), as well as main and auxiliary pump constraints (Eqs. (28), (29), (43) , (50), (51), (54), (55)) are taken into consideration. In this study, a mixed-integer nonlinear programming (MINLP) model is employed. Highly nonlinear terms included in the model present a major difficulty to solve the optimization problem directly, and a solution strategy to cope with the nonlinear equations is necessary. The model is deployed in software GAMS, and the solver DICOPT is used to solve the MINLP problem. Since the solution by using DICOPT is in the vicinity of the global optimum, it is hard to get a global optimal. It is critical to simplify the constraints. Generally, the velocity of
(61)
The fresh water cost is calculated as below.
CCwater = ftotal ·e w ·ht
(62)
3.5. Solution strategy This paper presents a series-parallel optimization model for circulating water system. The objective function (Eq. (52)) is to minimize total annual cost, including pump operating and capital cost (Eqs. (53)–(56)), heat exchanger capital cost (Eq. (57)), pipe capital cost (Eqs. (58)–(61)) and fresh water cost (Eq. (62)). In this improved 376
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Fig. 9. Improved series-parallel configuration with three reuse branches.
Table 5 Economic data of different number of reuse branches. Parameters
One reuse branch
Two reuse branches
Three reuse branches
Cost of cooling water ($) Operational cost of main pump ($) Annual capital cost of main pump ($) Operational cost of auxiliary pump ($) Annual capital cost of auxiliary pump ($) Annual capital cost of heat exchangers ($) Pipe cost ($) Total cost ($)
5133.1 19457.6 1720.8 1301.0 1447.0 69843.4 30360.2 129263.1
4865.8 18444.2 1680.6 1301.0 1447.0 70299.5 32207.8 130245.9
4363.0 16760.3 1612.3 6669.6 3124.3 70967.9 28974.5 132471.9
designed considering minimum water mass flow rate. This methodology maximizes the amount of reuse water and improves the outlet temperature of the heat exchanger network, which is beneficial to improve the performance of the cooling tower. But the pressure drop will increase, which will be unfavorable to the economy of pumps. The optimal series configuration is illustrated in Fig. 4.
water in the pipe is constant in the range of 1.5–3 m/s. In Eq. (23), pipe diameters are given to simplify the calculation by reducing the nonlinearity of the equation. 4. Case study 1 In this case study, the discussion is mainly about the comparison of series configuration, parallel configuration and series-parallel configuration. As the assumption above, the hot streams flow along the shell side and the cooling water flows along the tube side. Each hot process stream corresponds to only one heat exchanger. Hot stream data, physical properties data and pipe installation data is shown in Table 1, Table 2 respectively.
4.3. Scenario 3: series–parallel configuration According to the automatic optimization model in this paper, the improved series-parallel configuration is presented in Fig. 4. Table 3 compares the network data and economic costs of three different configurations. In scenario 1, there is no recycling of water. The total water mass flow rate is 185.3 kg/s. The cooling water cost and pipe cost are extremely high opposed to the other two configurations. In scenario 2, the main pump pressure head is high (11.3 m) compared with the other two configurations (9.3 m). Excessive pressure drops lead to relatively high annual capital cost of heat exchangers and pipes. In scenario 3, the improved series-parallel optimization approach can obviously improve the economic performance. The series-parallel optimization method saves 9.5% of the total annual cost compared with the series configuration and 23.2% compared with the parallel configuration. The above discussion does not consider the static pressure drop. Case study 2 will further discuss the arrangement of auxiliary pumps
4.1. Scenario 1: original parallel configuration The traditional parallel configuration is widely used in the industrial system. In this paper, parallel configuration means that both binary variablesBin (i, j ) andBout (i, j ) are zero. In other words, water reuses between heat exchangers are ignored. The optimal parallel configuration is presented in Fig. 4. 4.2. Scenario 2: series configuration According to the pinch technology, the heat exchanger network is 377
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the series configuration and the parallel configuration respectively. In case study 2, auxiliary pumps can effectively reduce the economic cost of the total system. The proposed series-parallel configuration saves 10.9% of the total economic cost compared with the two-step sequential optimization and 4.7% compared with the simultaneous optimization. The number of branches of reuse water will affect the economic performance, and its optimal value will be determined by the optimization results. One reuse branch is the optimal configuration, which saves 0.7% of the total cost compared with two reuse branches and 2.4% compared with three reuse branches. Adding reuse branches appropriately can reduce the pipe cost of the cooling water system. The improved series-parallel approach proposed in this paper is of great significance to improve the economic performance of the cooling system.
with different static pressure drop of heat exchangers. The influence of the number of reuse branches on the economic cost of the system is also analyzed. 5. Case study 2 Case studies performed by Sun et al. [25] and Ma et al. [27] were employed to verify the effectiveness of this approach. Hot stream data, physical properties data and pipe installation data are shown in Tables 1 and 2. In the two-step sequential optimization method [25], network configuration is changed. The first step is to use a thermodynamic approach to obtain the optimal cooler network. In the second step, the hydraulic model is established to obtain the optimal pump configuration with auxiliary pumps installed in parallel branch pipes. This approach is novel but circumscribed because the heat exchanger network and the pump network may not be optimized simultaneously and the water and electricity consumption is relatively high. The two-step sequential configuration is shown in Fig. 5. In the simultaneous optimization method [27], the intrinsic connection between the pump cost, cooler cost, and cooling water flow rate, pressure drop can be optimized simultaneously. But due to the lack of specific splitters and mixers, the arrangement of heat exchangers has some limitations, making it difficult to achieve the optimal water flow rate. The simultaneous configuration is shown as Fig. 6. Table 4 compares physical and economic data between different approaches. In the two-step sequential optimization approach, the water flow rate is relatively high compared with the other two approaches, leading to a higher cooling water cost and pump cost correspondingly. In simultaneous optimization approach, water flow rate and pump cost reduce at a certain extent. But the total cost does not change considerably due to high capital cost of heat exchangers. In the improved series-parallel model optimization approach, economic performance improves a lot because of water reuse. The number of reuse branch is two and the configuration is shown as Fig. 8. The water flow rate is 84.5 kg/s, which is 31.3 kg/s lower than two-step sequential optimization and 13.9 kg/s lower than simultaneous optimization. The operational cost of main pump decreases obviously ($18444.2) because of the reduction of the total water flow rate and the addition of auxiliary pumps. The improved series-parallel configuration saves 10.9% of the total economic cost compared with the two-step sequential optimization and 4.7% compared with the simultaneous optimization. Next, the influence of numbers of reuse branch on the economic performance is proposed. Figs. 7, 8 and 9 show the series-parallel configuration with one, two and three reuse branches, respectively. When the pipe cost is ignored, the economic performance is the optimal when the number of reuse branch is two ($98038.1), compared with one reuse branch ($98902.9) and three reuse branches ($103497.4). But when the pipe cost is considered, one reuse branch is the optimal configuration, which saves 0.7% of the total cost compared with two reuse branches and 2.4% compared with three reuse branches. Increasing the number of reuse branch may cause a higher total cost. The pipe cost is the optimal with three reuse branches, which means that there is no significant relationship between pipe cost and the number of reuse branches. The reuse branch pipeline increases, but the inlet and outlet pipeline reduces simultaneously (see Table 5).
Acknowledgement This work was financially supported by Science and Technology Commission Shanghai Municipality (13dz1201700) fund. References [1] W.C.J. Kuo, R. Smith, Design of water-using systems involving regeneration, Process Saf. Environ. Prot. 76 (2) (1998) 94–114. [2] W.C.J. Kuo, R. Smith, Designing for the interactions between water-use and effluent treatment, Chem. Eng. Res. Des. 76 (3) (1998) 287–301. [3] J.K. Kim, R. Smith, Cooling water system design, Chem. Eng. Sci. 56 (12) (2001) 3641–3658. [4] J.K. Kim, R. Smith, Automated retrofit design of cooling-water systems, Aiche J. 49 (7) (2003) 1712–1730. [5] J.K. Kim, R. Smith, Cooling system design for water and waste water minimization, Ind. Eng. Chem. Res. 43 (2) (2004) 608–613. [6] A. Ataei, M.H. Panjeshahi, R. Parand, et al., Application of an optimum design of cooling water system by regeneration concept and pinch technology for water and energy conservation, J. Appl. Sci. 9 (10) (2009) 1847–1858. [7] M.H. Panjeshahi, A. Ataei, M. Gharaie, et al., Optimum design of cooling water systems for energy and water conservation, Chem. Eng. Res. Des. 87 (2) (2009) 200–209. [8] D.C.Y. Foo, D.K.S. Ng, M.K.Y. Leong, et al., Targeting and design of chilled water network, Appl. Energy 134 (2014) 589–599. [9] A.U. Shenoy, U.V. Shenoy, Targeting and design of CWNs (cooling water networks), Energy 55 (1) (2013) 1033–1043. [10] X. Feng, R. Shen, B. Wang, Recirculating cooling-water network with an intermediate cooling-water main, Energy Fuels 19 (4) (2005) 1723–1728. [11] Y. Wang, K.H. Chu, Z. Wang, Two-step methodology for retrofit design of cooling water networks, Ind. Eng. Chem. Res. 53 (1) (2014) 274–286. [12] X. Zhu, F. Wang, D. Niu, et al., Integrated modeling and operation optimization of circulating cooling water system based on superstructure, Appl. Therm. Eng. 127 (2017) 1382–1390. [13] M. Picon-Nunez, C. Nila-Gasca, A. Morales-Fuentes, Simplified model for the determination of the steady state response of cooling systems, Appl. Therm. Eng. 27 (7) (2007) 1173–1181. [14] J.M. Ponce-Ortega, M. Serna-González, A. Jiménez-Gutiérrez, Optimization model for re-circulating cooling water systems, Comput. Chem. Eng. 34 (2) (2010) 177–195. [15] J.M. Ponce-Ortega, M. Serna-Gonzalez, A. Jiménez-Gutiérrez, A disjunctive programming model for simultaneous synthesis and detailed design of cooling networks, Ind. Eng. Chem. Res. 48 (6) (2009) 2991–3003. [16] J.M. Ponce-Ortega, M. Serna-Gonzalez, A. Jiménez-Gutiérrez, MINLP synthesis of optimal cooling networks, Chem. Eng. Sci. 62 (21) (2007) 5728–5735. [17] E. Rubiocastro, Medardo SernaGonzález, José María PonceOrtega, et al., Synthesis of cooling water systems with multiple cooling towers, Appl. Therm. Eng. 50 (1) (2013) 957–974. [18] K.V. Gololo, T. Majozi, On synthesis and optimization of cooling water systems with multiple cooling towers, Ind. Eng. Chem. Res. 50 (7) (2011) 3775–3787. [19] K.V. Gololo, T. Majozi, Complex cooling water systems optimization with pressure drop consideration, Ind. Eng. Chem. Res. 52 (22) (2013) 7056–7065. [20] K.V. Gololo, Analysis synthesis and optimization of complex cooling water systems, Modern Electron. Technique (2013). [21] C.C.S. Reddy, Naidut, et al., Holistic approach for retrofit design of cooling water networks, Ind. Eng. Chem. Res. 52 (36) (2013) 13059–13078. [22] C. Zheng, X. Chen, L. Zhu, et al., Simultaneous design of pump network and cooling tower allocations for cooling water system synthesis, Energy (2018) S0360544218303840. [23] S.W. Hung, J.K. Kim, Optimization of water systems with the consideration of pressure drop and pumping, Ind. Eng. Chem. Res. 51 (2) (2012) 848–859. [24] J. Sun, X. Feng, Y. Wang, et al., Pump network optimization for a cooling water system, Energy 67 (2014) 506–512. [25] J. Sun, X. Feng, Y. Wang, Cooling-water system optimisation with a novel two-step
6. Conclusion A superstructure model combining heat exchanger network and pump network is established in this paper. The heat exchanger network adopts a series-parallel configuration with the consideration of the pressure drop, flow rate, temperature and installation position. The main pump and auxiliary pumps are combined in the water pumping network. In case study 1, using the series-parallel optimization approach can save 9.5% and 23.2% of the total annual cost compared with 378
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(2018) S0360544218304274. [29] J. Ma, Y. Wang, X. Feng, Energy recovery in cooling water system by hydro turbines, Energy 139 (2017) 329–340. [30] G. Bonvin, S. Demassey, C.L. Pape, et al., A convex mathematical program for pump scheduling in a class of branched water networks, Appl. Energy 185 (2017) 1702–1711.
sequential method, Appl. Therm. Eng. 89 (2015) S1359431115000174. [26] M.M. Castro, T.W. Song, J.M. Pinto, Minimization of operational costs in cooling water systems, Chem. Eng. Res. Des. 78 (2) (2000) 192–201. [27] J. Ma, Y. Wang, X. Feng, Simultaneous optimization of pump and cooler networks in a cooling water system, Appl. Therm. Eng. 125 (2017) 377–385. [28] J. Ma, Y. Wang, X. Feng, Optimization of multi-plants cooling water system, Energy
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
An improved series-parallel optimization approach for cooling water system a
a,⁎
a
Shiqi Liu , Jinchun Song , Jia Shi , Bin Yang a b
T
a,b
School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China Shanghai Baosteel Company, Shanghai 201900, China
H I GH L IG H T S
configuration is proposed to reduce the water flow rate. • ATwoseries-parallel kinds of conditions with or without static pressure drop are considered. • A main-auxiliary pump structure is applied to reduce the operating consumption. • Heat exchanger and pump networks are optimized simultaneously. • The number of reusemain-auxiliary branch for the economic performance is analyzed. •
A R T I C LE I N FO
A B S T R A C T
Keywords: Cooling water system Series-parallel configuration Static pressure drop Reuse branch Mixed-integer nonlinear programming
Conventional cooling networks are usually operated in a parallel configuration. Although this configuration is easy operated, it results in inefficient use of water. A series configuration can effectively reduce water flow rate, while its arrangement is relatively complex, incurring more operating and capital cost. This paper proposes a water and energy conservation strategy to save energy cost and to simplify the network structure. Heat exchangers are arranged in series-parallel configuration to reduce the water flow rate for water conservation, and two kinds of installation conditions with or without static pressure drop are taken into account for energy conservation. A main-auxiliary pump structure is applied to reduce the operating consumption. The number of reuse branch for the economic performance is analyzed. Mixed-integer nonlinear programming based on a superstructure description is formulated by considering the configuration of pumps and heat exchanger networks simultaneously. Two case studies are employed to demonstrate the effectiveness of the proposed approach. The results show that the improved series-parallel configuration approach can save 10.9% of the total cost compared with the two-step sequential optimization and 4.7% compared with the simultaneous optimization method.
1. Introduction Cooling water system is one of the major water-using operations in industries to dissipate heat of process industries, power stations and chemical plants. Generally, the cooling water system is consisted by three major components: a heat exchanger network, a pump network and cooling towers. The irrational design will greatly increase the capital cost and the non-economic operation will lead to enormous consumption of energy and water. Therefore, the optimization of cooling water system is of great significance to improve the energy efficiency and economic performance of the industrial system. Research in this field has primitively focused on individual components but not the entire cooling water system. Traditional heat exchanger networks were invariably designed with parallel configuration, which might provoke a tremendous amount of fresh water consumption ⁎
and an inefficient use of cooling tower capacity. Therefore, reducing water consumption is the foremost concern of researchers. Approaches by using pinch technology [1,2] in a series configuration is proposed to reduce water consumption. Then more factors and approaches are considered such as pinch migration [3], heat exchanger and pipe constraints [4], waste water minimization [5], cooling tower performance [6,7], chilled water network [8], unified targeting algorithm [9]. Changing parallel configuration into series can reduce water flow rate and raise the outlet temperature significantly. However, if the network is complex or new heat exchangers are added into the network, the configuration might be retrofitted again. So pinch design approach might be a lack of flexibility. Then, mathematical models were established by using intermediate main approach [10], supplier and receiver arrangement [11], system mechanism analysis [12], steady state response analysis [13], or several stage arrangements [14–17]. Even
Corresponding author. E-mail address: [email protected] (J. Song).
https://doi.org/10.1016/j.applthermaleng.2019.03.048 Received 31 October 2018; Received in revised form 25 February 2019; Accepted 11 March 2019 Available online 18 March 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.
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temperature and cold stream outlet temperature, ℃ mass flow rate, kg/s mass flow rate of cold stream i, kg/s inlet mass flow rate from heat exchanger j to heat exchanger i, kg/s fout (i, j ) outlet mass flow rate from heat exchanger i to heat exchanger j, kg/s ftin (i) mass flow rate of fresh water from cooling tower to heat exchanger i, kg/s ftout (i) mass flow rate of useless water from heat exchanger i to cooling tower, kg/s ftotal total inlet/outlet mass flow rate, kg/s Hmain main pump pressure head, m Hauxi (i) auxiliary pump pressure head, m H (i) pressure head of HE i, m Kt (i) parameter used in relating physical properties to pressure drop for heat exchanger i Pressure drop coefficient of pipeline kpipe operating cost of pump, $ OCpump pressure head at point i Pi P i' pressure head at point i′ PA pressure head at point A PB pressure head at point B pressure head at point C of heat exchanger i PC(i) pressure head at point D of heat exchanger i PD(i) PE pressure head at point E PF pressure head at point F Tcout (i) outlet temperature of cold stream i, ℃ Tcin (i) inlet temperature of cold stream i, ℃ outlet temperature of heat exchanger network, ℃ tout total weight of the network, kg W binary variables between point i and i′ Y i', i Δpe (i) pressure drop of heat exchanger i, Pa pressure drop between point i and i’ ΔP i', i ΔpAB pressure drop between point A and B ΔpBC(i) pressure drop between point B and C for heat exchanger i ΔpC(i)D(i) pressure drop between point C and D for heat exchanger i ΔpD(i)C(j) pressure drop between point D for heat exchanger i and point C for heat exchanger j ΔpD(i)E pressure drop between point D for heat exchanger i and point E ΔpEF pressure drop between point E and F Δppipe (AB) pipe pressure drop between point A and B Δppipe (BC(i)) pipe pressure drop between point B and C for heat exchanger i Δppipe (D(i)C(j )) pipe pressure drop between point D for heat exchanger i and point C for heat exchanger j Δppipe (D(i)E) pipe pressure drop between point D for heat exchanger i and point E Δppipe (EF) pipe pressure drop between point E and F Δpmainpump pressure drop of main pump Δpauxipump pressure drop of auxiliary pump
Nomenclature Sets
f fc (i) fin (i, j )
I={i , j | i and j are heat exchangers}
Parameters
Af annualized factor Ap unit price per kilogram for pipe, $/kg cpc specific heat capacity of cooling water, kJ/(kg ℃) internal tube diameter, m di external tube diameter, m do ew unit cost of water, $/m3 ee unit cost of electricity, $/kW·h heat capacity of hot stream i, kW/℃ Fcp (i) g gravity constant, m/s2 annual operating time, h/year ht Htower installation height of cooling tower h (i) film transfer coefficient of hot stream i, W/(m2 ℃) film transfer coefficient of cooling water, W/(m2 ℃) hw Hins (i) installation height of heat exchanger i, m Height difference between HE i and HE j Hdi, j Hdi,positive Positive Height difference between HE i and HE j j conductivity of cooling water, W/m ℃ kt LV a large value L (i) lower bound mass flow rate of heat exchanger i, kg/s pipe length, m l Lpmain lower bound pressure head of main pump, m Lpauxi lower bound pressure head of auxiliary pump, m maximum number of reuse branch Nmax q (i) heat load of hot stream i, kW Reynolds number Re inlet temperature of heat exchanger network, ℃ tin Thout (i) outlet temperature of hot stream i, ℃ Thin (i) inlet temperature of hot stream i, ℃ U (i) upper bound mass flow rate of heat exchanger i, kg/s Upmain upper bound pressure head of main pump, m Upauxi upper bound pressure head of auxiliary pump, m v pipe velocity, m/s Cfpump, Cpump, γ capital parameter of pump capital cost parameter of heat exchanger x , y, z ΔTmin temperature approach between two streams, ℃ Variables exchange area for heat exchanger i, m2 binary variable for mass flow rate from heat exchanger j to heat exchanger i Bout (i, j ) binary variable for mass flow rate from heat exchanger i to heat exchanger j Bpump (i) binary variables of auxiliary pumps Btin (i) binary variable for mass flow rate from cooling tower to heat exchanger i Btout (i) binary variable for mass flow rate from heat exchanger i to cooling tower capital cost of pump, $ CCpump CCwater fresh water cost, $ pipe cost, $ CCpipe CCexch capital cost of heat exchanger, $ din (i) temperature approach between hot stream outlet temperature and cold stream inlet temperature, ℃ dout (i) temperature approach between hot stream inlet
Ar (i) Bin (i, j )
Greek letters
φt μ ρ ω η ρp
369
viscosity correction cooling water viscosity, Ns/m2 density of cooling water, kg/m3 pipe parameter efficiency of pumps density of pipe, kg/m3
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pressure head of the cooler were taken into account. Then, Ma et al. [29] analyzed energy recovery strategies for auxiliary pumps and hydro turbines. Bonvin et al. [30] proposed pump scheduling in a class of branched water networks, which achieved significant savings in terms of operating costs and energy. Even though the consideration of flow rate and pressure drop had been reported in most of the published work, the design issues associated with flow rate, pressure drop, pumping and installation had not been thoroughly addressed. This paper comprehensively studies the water and energy conservation strategy to minimize the capital cost and operating cost for the cooling water system. In term of water conservation, heat exchangers are arranged in series-parallel configuration so that water can be reused greatly and the inlet temperature of the cooling tower can be increased, thereby improving the cooling tower's efficiency. For energy conservation, the configuration is carried out by combining a main pump with auxiliary pumps. The pressure drop of pipes and heat exchangers is optimized to reduce pump operating cost. Two kinds of installation conditions (with or without the static pressure drop) are taken into account. Reducing the number of branches can effectively reduce the capital cost and simplify pump and heat exchanger network. Changing the structure of the pump network can effectively reduce the operating cost. Adding auxiliary pumps [24] might reduce operating cost but increase the capital cost. The selection of main and auxiliary pumps has an important impact on reducing the economic cost. The economic cost including pump operating and capital cost, pipe and heat exchanger capital cost, water cost must be taken into consideration. The optimization network is a complicated structure which includes intricate connections between its components. A simplified solution strategy has been taken to effectively solve computational problems associated with a mixed-integer nonlinear programming (MINLP) model. In this automated optimization approach, the water consumption and system operating and capital cost can reach the optimal value simultaneously. It is efficient to improve the economic performance and reduce the operating and capital cost of the cooling water system.
though these approaches are novel, they are less functional than the series configuration because of the intricate arrangements. The above work can significantly reduce the water flow rate, but they ignore the impact of pressure drop along pipes and heat exchangers, leading to high pump operating costs. Kim and Smith [2] considered an approach called “critical path algorithm” to search for the maximum pressure path and developed an automated design approach for cooling water networks. Gololo et al. [18–20] set a multiple cooling tower and heat exchanger network model, in which four cases with or without a specific source or sink and the pressure drop constraints were taken into consideration. These works mainly concentrate on the frictional pressure drop but ignore the static pressure drop. Their configurations require a higher pressure head for the main pump, which may provoke a higher operating cost for the pump network. Then location [21] and installation [22] were added to the optimization of cooling water system with multiple towers which discussed the interaction of various aspects in the cooling water system. But these approaches will increase the number of branches between towers and heat exchangers, resulting in additional pipe capital cost, low efficiency of cooling towers and complex pipe network configuration. In order to optimize pump operating cost through considering both the pressure head and flow rate, adding auxiliary pumps in the pump network can avoid the energy penalty by reducing the opening of valves for some heat exchangers. Therefore the pressure head of the main pump can be reduced. Hung et al. [23] optimized the water system considering pressure drop in pipes and added both a main pump and auxiliary pumps into the water network. The results showed that the cost could be much reduced by adding auxiliary pumps to the network. Sun et al. [24] optimized the model by using simulated annealing algorithm considering the static pressure drop in heat exchangers. The results indicated that adding auxiliary pumps might reduce the pump operating cost when the heat exchangers were installed at high positions. Then, Sun J et al. [25] optimized the cooling water system model by changing the model from parallel to series-parallel. Castro et al. [26] established an optimization model to minimize the operating cost of the cooling water system. Ma et al. [27,28] optimized the pump work and cooler networks simultaneously in which both the installation and the
Fig. 1. Superstructure of the cooling water system in the same installation height. 370
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Fig. 2. Superstructure of the cooling water system in different installation height.
2. Problem statement
is the specific heat capacity of hot stream i.
q (i) = (Thin (i) − Thout (i))·Fcp (i)
The superstructure of the cooling water system is illustrated in Fig. 1 and Fig. 2. In Fig. 1, pumps and heat exchangers are located at the same ground altitude. The pump head is composed of the heat exchanger and pipe pressure drop and the static pressure drop in the cooling tower. Energy conservation with auxiliary pumps is not necessary due to the fact that the heat exchanger is at ground altitude without static pressure drop. In Fig. 2, heat exchangers in various units have different installation altitude, leading to different static pressure head. A pump network with a main pump and auxiliary pumps is proposed to minimize the operating cost. In the cooling water system, each stream has different heat and distance parameters. The flow rates, inlet and outlet temperatures of hot streams and film transfer coefficients of both hot and cold streams are known. The flow rate of heat exchangers, pressure head loss of each branches and additional pressure head of installation height are optimized simultaneously to reduce economic consumption. Variables in this paper include mass flow rate, outlet temperature, area of the heat exchangers, pressure drop of exchangers and pipes, the pressure head of the main pump and auxiliary pumps, and reuse branches. The aim of this paper is to identify a superstructure of cooling water system to minimize the annual cost. The mathematical model developed in this paper employs several assumptions as follows:
(1)
Generally, to satisfy the requirements of heat transfer, the temperature approaches between cold and hot streams are required to be larger than the minimum temperature difference. And the inlet temperature of cooling water should be lower than the outlet temperature, as is shown as Eqs. (2)–(4).
ΔTmin ≤ Thout (i) − Tcin (i) = din (i)
(2)
ΔTmin ≤ Thin (i) − Tcout (i) = dout (i)
(3)
Tcout (i) ≥ Tcin (i)
(4)
In HE i, the heat load between cold and hot stream i are the same in Eq. (5), where cpc is the specific heat capacity of cooling water.
q (i) = (Tcout (i) − Tcin (i))·cpc·fc (i)
(5)
Point C(i) is to blend fresh water and other cold streams. The water flow rate of HE i is made up of fresh water and the reuse water from other heat exchangers. The mass flow rate constraint is shown as Eq. (6). Point D(i) facilitates the distribution of the cold stream to other heat exchangers as well as to the cooling tower. The mass flow rate constraint is shown as Eq. (7). n
fc (i) =
∑
fin (i, j ) + ftin (i)
j = 1, j ≠ i
1. Each hot stream corresponds to only one cold stream and the cold stream only includes water. 2. The exchangers are all 1-1 counter current shell and tube exchangers. The cold streams flow along the tube side and hot streams flow along the shell side. 3. The specific heat capacities of both hot and cold streams are consistent throughout the temperature range. 4. One cooling tower can satisfy the cooling requirements of all heat exchangers and water loss and make-up water can be ignored.
(6)
n
fc (i) =
∑
fout (i, j ) + ftout (i)
j = 1, j ≠ i
(7)
In the above equations, fin (i, j ) is inlet mass water flow rate from HE j to HE i and fout (i, j ) is outlet mass water flow rate from HE i to HE j. Eq. (8) is the constraint between inlet and outlet mass flow rate. ftin (i) is the fresh water from the cooling water. ftout (i) is the hot water to the cooling tower. The total mass flow rate is indicated by Eq. (9).
fin (i, j ) = fout (j, i) n
3. Mathematical model
ftotal =
∑ ftin (i) = ∑ ftout (i) i=1
3.1. Temperature and flow rate constraints of the HE network
(8) n i=1
(9)
Due to the different inlet temperature, some heat exchangers might not need fresh water or reuse water.Bin (i, j ) and Bout (i, j ) are composed by binary variables to limit reuse water from HE i to HE j or from HE j to HE i. AndBtin (i) , Btout (i) are binary variables to limit the inlet and outlet mass flow rate from/to the cooling tower. For HE i and j, The upper and lower bounds of the mass flow constraints are shown as follows:
In Figs. 1 and 2, Point A-F are placed to guarantee the logical location and to calculate expediently. The heat exchanger (HE) and pump network start at point A and end at point F. Depending on the known parameters of hot streams, the constraints condition of cold streams is determined. The transfer heat in hot stream i is as follows, where Fcp (i) 371
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Table 2 Physical property data.
Fig. 3. Pressure drop constraints of heat exchangers and pipes.
L (i)·Bin (i, j ) ≤ fin (i, j ) ≤ U (i)·Bin (i, j )
(10)
L (i)·Bout (i, j ) ≤ fout (i, j ) ≤ U (i)·Bout (i, j )
(11)
L (i)·Btin (i) ≤ ftin (i) ≤ U (i)·Btin (i)
(12)
L (i)·Btout (i) ≤ ftout (i) ≤ U (i)·Btout (i)
(13)
Bin (i, j ) = Bout (i, j ) = 0
(14)
i=j
L (i) and U (i) are the lower and upper bounds mass flow rates of HE i. The number of reuse branches should be limited to a specific value, which is as follows: n
n
∑i =1 ∑ j=1 Bin (i, j) ≤ Nmax n
(15)
n
∑i =1 ∑ j=1 Bout (i, j) ≤ Nmax
(16)
At the mixing point, the mixing temperature should satisfy the following equation.tin and tout represent for the inlet and outlet temperature of HE network, respectively. n
∑
fc (i)·Tcin (i) =
(fin (i, j )·Tcout (j )) + tin·ftin (i)
(17)
j = 1, j ≠ i
n
tout · ∑ ftout (i) = i=1
Items
Data
Annualized factor Af Unit price per kilogram for pipe Ap
0.298 0.78 $/kg
Specific heat capacity of cooling water cpc Unit cost of water ew Unit cost of electricity ee Gravity constant g Annual operating time ht Film transfer coefficient of cooling water hw conductivity of cooling water kt A large value LV Lower bound mass flow rate of heat exchanger iL (i) Inlet temperature of heat exchanger network tin Lower bound pressure head of main pump, Lpmain Upper bound pressure head of main pump, Upmain Lower bound pressure head of auxiliary pump, Lpauxi Upper bound pressure head of auxiliary pump, Upauxi Maximum number of reuse branch Temperature approach between two streams, ΔTmin Viscosity correction, φt Cooling water viscosity, μ Density of cooling water, ρ Density of pipe, ρp
4.18 kJ/(kg°C) 2 × 10-6$/m3 0.1 $/kW·h 9.8 m/s2 8000 h/year 2500 W/m2°C 0.6 W/m°C 10,000 2 kg/s
Efficiency of pumps Pipe length (m), internal diameter (m), water velocity (m/s) for pipe AB Pipe length (m), internal diameter (m),water velocity (m/s) for pipe EF Internal diameter (m),water velocity (m/s) for pipe BC(i) and D(i)E Installation height of cooling tower, Htower Heat exchanger capital cost, CCexch
0.7 100, 0.9, 3
∑i = 1 (30, 000 + 1000Ar (i)0.5 )
Pump capital cost, CCpump
2000 + 5(f Δp/ ρ)0.68
20 °C 10 m 35 m 0m 30 m 4 10 °C 1.05 10-3Ns/m2 995.7 kg/m3 1340 kg/m3
120, 0.9, 3 0.6, 1.5 8m n
n
∑ (ftout (i)·Tcout (i))
For HE i, the area can be calculated according to the film transfer coefficient of the cold stream and the hot stream.
(18)
i=1
The improved series-parallel configuration based on the above equations can reduce the water consumption of the system obviously due to the water reuse. But in a series configuration, the pressure drop will increase significantly, which will increase the operating cost of pumps [2]. So the pressure drop is an essential aspect for system optimization.
[din (i )·dout (i )·
din (i) + dout (i) 1/3 ] 2
·(
1 1 ) + h (i) hw
(20)
Constant Kt (i) can be calculated as follows:
Kt (i) = 3.2. Pressure drop constraints of the HE network
φt4.5 ·di0.5·μ11/6 d · i 2.5 0.023 ·fc (i)·ρ ·kt7/3·cpc7/6 do
(21)
The pressure drop of heat exchangers can be calculated in the above Eqs. (19)–(21). And the pipe pressure drop can be calculated as follows [24].
On the tube side of cold streams, the pressure drop of the HE i is related to constant Kt (i) , contact area Ar (i) , as well as film transfer coefficient[27], which can be calculated as follows:
Δpe (i ) = Kt (i )·Ar (i )·hw 3.5
q (i)
Ar (i) =
Δppipe = kpipe·
(19)
f2 ρ2
(22)
Table 1 Hot stream data. Stream i
Thin(i)/°C
Thout(i)/°C
Fcp (i) / (kW/℃)
q(i) /kW
h(i)/ (W/m2℃)
Pipe length for BC(i)/m
Pipe length for D(i)E/m
Case Study 1
1 2 3 4
50 55 75 75
30 45 45 60
75 400 100 200
1500 4000 3000 3000
785 660 865 966
100 100 125 150
175 200 175 150
Case Study 2
1 2 3 4 5
50 50 105 85 90
30 40 40 65 55
200 150 60 100 80
4000 1500 3900 2000 2800
854 743 720 1352 450
100 100 125 175 100
100 125 175 150 100
372
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kpipe =
8ρ ·l·ω π 2·di5
ω = 0.0032 +
follows. Δpmainpump and Δpauxipump (i) are the pressure head of the main pump and the auxiliary pump i respectively.
(23)
0.221 Re 0.237
(24)
ρ · di · v Re = μ
(25)
The pressure drop can be calculated according to the above equations. But if the network is unknown, the total pressure drop between point B and point E may not be calculated because whether the pipes exist is not certain. The pressure drop of the system is determined not only by the individual pipes and heat exchangers, but also by the configuration of the network. So it is not sufficient to calculate the pressure drop automatically by the above equation. According to the “critical path algorithm” [19–22], the overall pressure drop can be obtained by transferring the unknown constraints into inequality constraints. The critical path algorithm is used to find a maximum pressure drop path between two points. Suppose i and i′ are a starting point and an ending point respectively. Eq. (26) should be satisfied if the pipe exists between two points. Otherwise, logic constraints are formulated by combining the binary variables with a sufficiently large value to consider whether a connection between a node is activated (Y i', i = 1) or not (Y i', i = 0). In Eq. (27), after introducing a large value (LV), if a node connected with other node (Y i', i = 1), inequality constraints are strictly activated. Otherwise (Y i', i = 0), the inequality constraints are relaxed by a sufficiently value, so the inequality constraints are strictly activated as well. In this solving strategy, the maximum pressure drop path can be determined in the optimization progress.
P i' − Pi ≥ ΔP i', i
(
)
(28)
PB − PC(i) + Δpauxipump (i) + LV ·(1 − Btin (i)) ≥ ΔpBC(i)
(29)
PC(i) − PD(i) = ΔpC(i)D(i)
(30)
PD(i) − PC(j) + LV ·(1 − Bout (i, j )) ≥ ΔpD(i)C(j)
(31)
PD(j) − PE + LV ·(1 − Btout (j )) ≥ ΔpD(j)E
(32)
PE − PF = ΔpEF
(33)
3.3. Installation height and pump network constraints The influence of static pressure drop is discussed in the following two scenarios. In the first scenario, due to the low installation position of the heat exchanger, the static pressure drop can be ignored. The pump head must satisfy the static pressure head of the cooling tower. For the right side of Eqs. (28)–(33), the pressure drop between two points can be calculated as Eqs. (34)–(39).
(26)
P i' − Pi + LV · 1 − Y i', i ≥ ΔP i', i
PA − PB + Δpmainpump = ΔpAB
(27)
ΔpAB = Δppipe (AB)
(34)
ΔpBC(i) = Δppipe (BC(i))
(35)
ΔpC(i)D(i) = Δpe (i)
(36)
ΔpD(i)C(j) = Δppipe (D(i )C(j ))
(37)
ΔpD(i)E = Δppipe (D(i)E)
(38)
ΔpEF = Δppipe (EF) + Htower ·ρg
(39)
In the second scenario, the static pressure drop of heat exchangers cannot be ignored. Adding auxiliary pumps can reduce the operating cost of the main pump if the static pressure drop is high. Ma [27,28] discussed when the heat exchangers were arranged at different
Fig. 3 shows the pressure drop constraints in one of the possible branches according to the “critical path algorithm”. The pressure drop inequality constraints between point A and point F can be calculated as
Fig. 4. Network configuration for scenario 1, 2 and 3. 373
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Table 3 Comparison of three different configurations. Parameters
Parallel configuration
Series configuration
Series-parallel configuration
Total flowrate (kg/s) Number of reuse branch Total outlet temperature (℃) Total area of heat exchangers (m2) Main pump pressure head (m) Cost of cooling water ($) Operational cost of main pump ($) Annual capital cost of main pump ($) Annual capital cost of heat exchangers ($) Annual pipe cost ($) Total cost ($)
185.3 0 40.7 1109 9.3 10673.3 19252.6 1712.7 54730.3 30822.1 117191.0
67.8 3 61.4 1618 11.3 3905.3 8559.3 1239.5 59561.3 26203.1 99468.5
74.9 3 56.7 1050 9.3 4315.5 7756.3 1197.8 55659.5 21122.2 90051.3
Fig. 5. Two-step sequential optimization configuration.
Δpmainpump + Δpauxipump (i) ≥ H (i)·ρg
installation height, the pump head depended on the location of the highest heat exchanger in the branch. Without knowing the configuration of the branches, the total static pressure head can be calculated automatically according to the following equations.
Hdi, j = Hins (i) − Hins (j )
In scenario 2, for the right side of Eqs. (28)–(33), the pressure drop between two points can be calculated as Eqs. (44)–(49).
(40)
Hdi, j is the height difference between HE i and HE j. When the height of HE i is higher than HE j, Hdi, j is positive. Otherwise, Hdi, j is negative. Hdi,positive can be defined by the following equation. j
Hdi,positive = max(0, Hdi, j ) j
+
(44)
ΔpBC(i) = Δppipe (BC(i)) + Hins (i)
(45)
ΔpD(i)C(j) = Δppipe (D(i )C(j )) + Bout (i,
(41)
(46)
j )·Hdi,positive j
(47)
ΔpD(i)E = Δppipe (D(i)E)
(48)
ΔpEF = Δppipe (EF) + Htower ·ρg
(49)
Binary variable Bpump (i) is used for confirming the existence of the auxiliary pumps. Bpump (i) = 1 means there is an auxiliary pump between point B and point C(i), otherwise not. The auxiliary pump should be arranged at the beginning of the heat exchanger network, in other words, if the fresh water pipeline BC(i) does not exist, the auxiliary pump for the pipeline of HE i will not exist correspondingly.
n
∑ j=1,j≠i Bout (i, j)·Hdi,positive j
n
+ (PC(i) − PF )/(ρg) ∑ j=1,j≠i Bin (i, j)·Hdi,positive j
ΔpAB = Δppipe (AB)
ΔpC(i)D(i) = Δpe (i)
For HE i, the total head is the sum of height head and pressure head. The height head of HE i is produced by the highest position head in the branch of the HE i, as the following equation.
H (i) = Hins (i) +
(43)
(42)
The pressure head provided by the main pumpΔpmainpump and the auxiliary pump Δpauxipump (i) must satisfy the head requirement of the HE i.
Lpmain ≤ Δpmainpump ≤ Upmain 374
(50)
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Fig. 6. Simultaneous optimization configuration.
Table 4 Physical and economic data between different approaches. Parameters
Two-step sequential optimization
Simultaneous optimization
Improved series-parallel optimization
Total flow rate (kg/s) Total outlet temperature (°C) Total area of heat exchangers (m2) Main pump pressure head (m) Cost of cooling water ($) Operational cost of main pump ($) Annual capital cost of main pump ($) Operational cost of auxiliary pump ($) Annual capital cost of auxiliary pump ($) Annual capital cost of heat exchangers ($) Total cost ($)
115.8 49.3 1214.7 19.8 6671.8 26803.9 1992.1 8423.7 1231.5 64859.9 109982.8
98.4 54.5 1406.3 20.6 5669.5 24357.1 1904.2 1990.4 834.3 68108.7 102864.2
84.5 60.2 1612.2 19.5 4865.8 18444.2 1680.6 1301.0 1447.0 70299.5 98038.1
Lpauxi ·Bpump (i)·Btin (i) ≤ Δpauxipump (i) ≤ Upauxi ·Bpump (i)·Btin (i)
(51)
Hauxi (i) =
Δpauxipump (i) ρg
(55)
CCpump represents for the pump capital cost, Af is an annualized factor considering life expectancy. The total capital cost is as follows.
3.4. Objective function The objective of this optimization model is to minimize total annualized cost. The objective function is the sum of freshwater cost, the cost of pipelines and heat exchangers and the capital and operating cost of the pumps.
CCpump = Af ·[Cfpump + Cpump·(Hmain·ftotal ·g)γ ] + Af ·Bpump (i)·
obj function = OCpump + CCpump + CCpipe + CCexch + CCwater
CCexch = Af · ∑
OCpump = OCmain + OCauxi = ( Bpump (i))·ht ·ee
Hmain·ftotal ·g 1000η
+
n
∑i =1
Hauxi (i)·ftin (i)·g 1000η
n
∑i =1 [Cfpump + Cpump·(Hauxi (i)·ftin (i)·g)γ ] The capital cost of heat exchangers is as follows. n
(52)
i=1
(53)
CCpipe = Af ·W ·Ap W=
Δpmainpump ρg
(x + y·Ar z (i))
(57)
The capital cost of a pipeline is equal to the product of weight and price.W represents for the network total weight, Ap represents for the unit price per kilogram.
·
In the above equation, OCpump represents for the pump operating cost, the pump head Hmain and Hauxi (i) can be calculated as follows.
Hmain =
(56)
di =
(54) 375
π 2 (do − di2)·l·ρp 4 4f + 0.3 πρv
(58) (59)
(60)
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Fig. 7. Improved series-parallel configuration with one reuse branch.
Fig. 8. Improved series-parallel configuration with two reuse branches.
do = 1.101di + 0.006
series–parallel superstructure, temperature and flow rate constraints (Eqs. (1)–(18)), pressure drop constraints (Eqs. (19)–(49)), as well as main and auxiliary pump constraints (Eqs. (28), (29), (43) , (50), (51), (54), (55)) are taken into consideration. In this study, a mixed-integer nonlinear programming (MINLP) model is employed. Highly nonlinear terms included in the model present a major difficulty to solve the optimization problem directly, and a solution strategy to cope with the nonlinear equations is necessary. The model is deployed in software GAMS, and the solver DICOPT is used to solve the MINLP problem. Since the solution by using DICOPT is in the vicinity of the global optimum, it is hard to get a global optimal. It is critical to simplify the constraints. Generally, the velocity of
(61)
The fresh water cost is calculated as below.
CCwater = ftotal ·e w ·ht
(62)
3.5. Solution strategy This paper presents a series-parallel optimization model for circulating water system. The objective function (Eq. (52)) is to minimize total annual cost, including pump operating and capital cost (Eqs. (53)–(56)), heat exchanger capital cost (Eq. (57)), pipe capital cost (Eqs. (58)–(61)) and fresh water cost (Eq. (62)). In this improved 376
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Fig. 9. Improved series-parallel configuration with three reuse branches.
Table 5 Economic data of different number of reuse branches. Parameters
One reuse branch
Two reuse branches
Three reuse branches
Cost of cooling water ($) Operational cost of main pump ($) Annual capital cost of main pump ($) Operational cost of auxiliary pump ($) Annual capital cost of auxiliary pump ($) Annual capital cost of heat exchangers ($) Pipe cost ($) Total cost ($)
5133.1 19457.6 1720.8 1301.0 1447.0 69843.4 30360.2 129263.1
4865.8 18444.2 1680.6 1301.0 1447.0 70299.5 32207.8 130245.9
4363.0 16760.3 1612.3 6669.6 3124.3 70967.9 28974.5 132471.9
designed considering minimum water mass flow rate. This methodology maximizes the amount of reuse water and improves the outlet temperature of the heat exchanger network, which is beneficial to improve the performance of the cooling tower. But the pressure drop will increase, which will be unfavorable to the economy of pumps. The optimal series configuration is illustrated in Fig. 4.
water in the pipe is constant in the range of 1.5–3 m/s. In Eq. (23), pipe diameters are given to simplify the calculation by reducing the nonlinearity of the equation. 4. Case study 1 In this case study, the discussion is mainly about the comparison of series configuration, parallel configuration and series-parallel configuration. As the assumption above, the hot streams flow along the shell side and the cooling water flows along the tube side. Each hot process stream corresponds to only one heat exchanger. Hot stream data, physical properties data and pipe installation data is shown in Table 1, Table 2 respectively.
4.3. Scenario 3: series–parallel configuration According to the automatic optimization model in this paper, the improved series-parallel configuration is presented in Fig. 4. Table 3 compares the network data and economic costs of three different configurations. In scenario 1, there is no recycling of water. The total water mass flow rate is 185.3 kg/s. The cooling water cost and pipe cost are extremely high opposed to the other two configurations. In scenario 2, the main pump pressure head is high (11.3 m) compared with the other two configurations (9.3 m). Excessive pressure drops lead to relatively high annual capital cost of heat exchangers and pipes. In scenario 3, the improved series-parallel optimization approach can obviously improve the economic performance. The series-parallel optimization method saves 9.5% of the total annual cost compared with the series configuration and 23.2% compared with the parallel configuration. The above discussion does not consider the static pressure drop. Case study 2 will further discuss the arrangement of auxiliary pumps
4.1. Scenario 1: original parallel configuration The traditional parallel configuration is widely used in the industrial system. In this paper, parallel configuration means that both binary variablesBin (i, j ) andBout (i, j ) are zero. In other words, water reuses between heat exchangers are ignored. The optimal parallel configuration is presented in Fig. 4. 4.2. Scenario 2: series configuration According to the pinch technology, the heat exchanger network is 377
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the series configuration and the parallel configuration respectively. In case study 2, auxiliary pumps can effectively reduce the economic cost of the total system. The proposed series-parallel configuration saves 10.9% of the total economic cost compared with the two-step sequential optimization and 4.7% compared with the simultaneous optimization. The number of branches of reuse water will affect the economic performance, and its optimal value will be determined by the optimization results. One reuse branch is the optimal configuration, which saves 0.7% of the total cost compared with two reuse branches and 2.4% compared with three reuse branches. Adding reuse branches appropriately can reduce the pipe cost of the cooling water system. The improved series-parallel approach proposed in this paper is of great significance to improve the economic performance of the cooling system.
with different static pressure drop of heat exchangers. The influence of the number of reuse branches on the economic cost of the system is also analyzed. 5. Case study 2 Case studies performed by Sun et al. [25] and Ma et al. [27] were employed to verify the effectiveness of this approach. Hot stream data, physical properties data and pipe installation data are shown in Tables 1 and 2. In the two-step sequential optimization method [25], network configuration is changed. The first step is to use a thermodynamic approach to obtain the optimal cooler network. In the second step, the hydraulic model is established to obtain the optimal pump configuration with auxiliary pumps installed in parallel branch pipes. This approach is novel but circumscribed because the heat exchanger network and the pump network may not be optimized simultaneously and the water and electricity consumption is relatively high. The two-step sequential configuration is shown in Fig. 5. In the simultaneous optimization method [27], the intrinsic connection between the pump cost, cooler cost, and cooling water flow rate, pressure drop can be optimized simultaneously. But due to the lack of specific splitters and mixers, the arrangement of heat exchangers has some limitations, making it difficult to achieve the optimal water flow rate. The simultaneous configuration is shown as Fig. 6. Table 4 compares physical and economic data between different approaches. In the two-step sequential optimization approach, the water flow rate is relatively high compared with the other two approaches, leading to a higher cooling water cost and pump cost correspondingly. In simultaneous optimization approach, water flow rate and pump cost reduce at a certain extent. But the total cost does not change considerably due to high capital cost of heat exchangers. In the improved series-parallel model optimization approach, economic performance improves a lot because of water reuse. The number of reuse branch is two and the configuration is shown as Fig. 8. The water flow rate is 84.5 kg/s, which is 31.3 kg/s lower than two-step sequential optimization and 13.9 kg/s lower than simultaneous optimization. The operational cost of main pump decreases obviously ($18444.2) because of the reduction of the total water flow rate and the addition of auxiliary pumps. The improved series-parallel configuration saves 10.9% of the total economic cost compared with the two-step sequential optimization and 4.7% compared with the simultaneous optimization. Next, the influence of numbers of reuse branch on the economic performance is proposed. Figs. 7, 8 and 9 show the series-parallel configuration with one, two and three reuse branches, respectively. When the pipe cost is ignored, the economic performance is the optimal when the number of reuse branch is two ($98038.1), compared with one reuse branch ($98902.9) and three reuse branches ($103497.4). But when the pipe cost is considered, one reuse branch is the optimal configuration, which saves 0.7% of the total cost compared with two reuse branches and 2.4% compared with three reuse branches. Increasing the number of reuse branch may cause a higher total cost. The pipe cost is the optimal with three reuse branches, which means that there is no significant relationship between pipe cost and the number of reuse branches. The reuse branch pipeline increases, but the inlet and outlet pipeline reduces simultaneously (see Table 5).
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6. Conclusion A superstructure model combining heat exchanger network and pump network is established in this paper. The heat exchanger network adopts a series-parallel configuration with the consideration of the pressure drop, flow rate, temperature and installation position. The main pump and auxiliary pumps are combined in the water pumping network. In case study 1, using the series-parallel optimization approach can save 9.5% and 23.2% of the total annual cost compared with 378
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